Quantum Mechanical Continuum Solvation...

Quantum Mechanical Continuum Solvation Models

Jacopo Tomasi,*,† Benedetta Mennucci,† and Roberto Cammi‡

Dipartimento di Chimica e Chimica Industriale, Universita di Pisa, Via Risorgimento 35, 56126 Pisa, Italy, and Dipartimento di Chimica, Universita diParma, Viale delle Scienze 17/A, 43100 Parma, Italy

Received January 6, 2005

Contents1. Introduction 3000

1.1. Generalities about This Review 30001.2. Generalities about Continuum, Focused, and

Layered Models3001

2. Methodological Outlines of the Basic QMContinuum Model

3003

2.1. Definition of the Basic Model 30032.2. Cavity 30032.3. Solution of the Electrostatic Problem 3005

2.3.1. ASC Methods 30062.3.2. MPE Methods 30152.3.3. Generalized Born (GB) Approaches 30162.3.4. Finite Element (FE) and Finite Difference

(FD) Methods3018

2.4. Solution of the Quantum Mechanical Problem 30192.4.1. Intuitive Formulation of the Problem 30192.4.2. Electrostatic Operators 30202.4.3. Outlying Charge 30212.4.4. Definition of the Basic Energetic Quantity 30232.4.5. QM Descriptions beyond the HF

Approximation3024

3. Some Steps beyond the Basic Model 30253.1. Different Contributions to the Solvation

Potential3025

3.2. Use of Interactions in Continuum SolvationApproaches

3027

3.2.1. Cavity Formation Energy 30283.2.2. Repulsion Energy 30333.2.3. Dispersion Energy 30353.2.4. Charge Transfer Term 30363.2.5. Definition of the Cavities in the

Calculation of Solvation Energy3037

3.2.6. Contributions to the Solvation FreeEnergy Due to Thermal Motions of theSolute

3038

4. Nonuniformities in the Continuum Medium 30394.1. Dielectric Theory Including Nonlinear Effects 30394.2. Nonlocal Electrostatic Theories 30404.3. Nonuniformities around Small Ions 3041

4.3.1. ε(r) Models 30414.3.2. Layered Models 30414.3.3. Molecular Cluster Models 3042

4.4. Nonuniformities around Neutral Molecules 30434.5. Nonuniformities around Systems of Larger

Size3043

4.6. Systems with Phase Separation 30445. Nonequilibrium in Time-Dependent Solvation 3046

5.1. Dynamic Polarization Response 30475.2. Vertical Electronic Transitions 30475.3. Solvation Dynamics 30495.4. Spectral Line Broadening and Solvent

Fluctuations3053

5.5. Excitation Energy Transfers 30545.6. Time-Dependent QM Problem for Continuum

Solvation Models3056

6. Molecular Properties of Solvated Systems 30586.1. Energy Properties 3059

6.1.1. Geometrical Derivatives 30596.1.2. IR and Raman Intensities 30616.1.3. Surface-Enhanced IR and Raman 3062

6.2. Response Properties to Electric Fields 30636.2.1. QM Calculation of Polarizabilities of

Solvated Molecules3064

6.2.2. Definition of Effective Properties 30646.3. Response Properties to Magnetic Fields 3066

6.3.1. Nuclear Shielding 30666.3.2. Indirect Spin−Spin Coupling 30676.3.3. EPR Parameters 3068

6.4. Properties of Chiral Systems 30696.4.1. Electronic Circular Dichroism (ECD) 30696.4.2. Optical Rotation (OR) 30696.4.3. VCD and VROA 3070

7. Continuum and Discrete Models 30717.1. Continuum Methods within MD and MC

Simulations3072

7.2. Continuum Methods within ab Initio MolecularDynamics

3074

7.3. Mixed Continuum/Discrete Descriptions 30757.3.1. Solvated Supermolecule 30767.3.2. QM/MM/Continuum: ASC Version 30767.3.3. ONIOM/Continuum 30777.3.4. (Direct) Reaction Field Model 30787.3.5. Langevin Dipole 3078

7.4. Other Methods 30797.4.1. ASEP-MD 30797.4.2. RISM-SCF 3080

8. Concluding Remarks 3081

* Author to whom correspondence should be addressed ([email protected]).† Universita di Pisa.‡ Universita di Parma.

2999Chem. Rev. 2005, 105, 2999−3093

10.1021/cr9904009 CCC: $53.50 © 2005 American Chemical SocietyPublished on Web 07/26/2005

9. Acknowledgment 308410. References 3084

1. Introduction

1.1. Generalities about This ReviewThis review on continuum solvation models has

been preceded in Chemical Reviews by others ad-dressing the same subject. They are due to Tomasiand Persico1 (published in 1994), Cramer and Tru-hlar2 (published in 1999), and Luque and Orozco3

(published in 2000). These three reviews on the sametopic in a journal covering all of the aspects ofchemical research indicate the interest this topic hasfor a sizable portion of the chemical community.

Liquid solutions play in fact a fundamental role inchemistry, and this role has been amply acknowl-edged by Chemical Reviews since its very beginning.In the abundant number of reviews addressing dif-ferent aspects of chemistry in the liquid phase, thenumber of those centered on the theoretical andcomputational aspects of the study of liquid systemshas considerably increased in the past two decades.This reflects the increasing importance of computa-tional approaches in chemistry, an aspect of theevolution of scientific research chemistry shares withphysics, biology, engineering, geology, and all of theother branches of sciences, an evolution that isultimately due to the widespread availability ofefficient computers.

Computers have permitted the activation of manyapproaches in sciences that were dormant, or limitedin their applications to the level of simple model, forthe lack of appropriate computational tools; anexample in chemistry is given by the activation ofmethods based on quantum mechanics for the de-scription of isolated molecules, which have now

reached the degree of accuracy in the description ofmolecular structures that chemists require.

Computers have also completely modified the waysof doing theoretical and computational studies ofliquid systems, permitting the introduction of newapproaches, new concepts, and new ideas. The mostimportant innovations in this field are related to theuse of computer simulations that directly, or indi-rectly, are the basis of our present understanding ofcondensed systems. Simulations, initiated about 50years ago, have greatly evolved in the past 20 yearsand proceed now in covering all of the fields in whichcondensed matter occurs. One aspect of this evolution

Jacopo Tomasi received his “Laurea” degree in chemistry in 1958(University of Pisa) discussing a thesis on the theoretical determinationof the intensities of vibrational overtones. Since 1980 he has been a fullprofessor of physical chemistry at the University of Pisa. His researchinterests cover several aspects of theoretical chemistry with propensityto the formulation and elaboration of models based on ab initio quantumchemistry with a special emphasis on the exploitation of the interpretationof the phenomenon to obtain computational codes of easy use. Thisapproach has been applied to molecular interactions, chemical reactionmechanisms, photochemical processes, solvent effects on molecularresponse properties, and other related subjects. He has authored, orcoauthored, more than 300 scientific papers.

Benedetta Mennucci was born in Lucca, Italy, in 1969. She received her“Laurea” degree in chemistry in 1994 and her Ph.D. degree in chemistryfrom the University of Pisa in 1998 discussing a thesis on theoreticalmodels and computational applications of molecular phenomena involvingthe environment effect. In the same year she became assistant professorin the Department of Chemistry of the University of Pisa. Since 2002 shehas been associate professor of physical chemistry at the same institution.Her research interests focus on the elaboration of theoretical models andcomputational algorithms to describe molecular systems in condensedphase with particular attention to molecular properties and time-dependentphenomena. She has authored, or coauthored, more than 80 publications.

Roberto Cammi was born in Busseto, Italy, in 1954. In 1979, he wasawarded the degree of “Dottore in Chimica” at the University of Parma,discussing a thesis in theoretical chemistry. Since 1983 he has been aresearcher with the Institute of Physical Chemistry of the University fParma. In 2000 he became an associate professor in the Department ofGeneral and Inorganic Chemistry, Analytical Chemistry, Physical Chemistryof the University of Parma, and since 2002 he has been a full professorof theoretical chemistry of the University of Parma. He teaches physicalchemistry and theoretical chemistry. His research field is theoretical andcomputational chemistry, mainly the developments and applications ofquantum mechanical continuum methods to the study of solvent effectson molecular processes and properties. He has published more than 80research papers and 8 chapters of collective books.

3000 Chemical Reviews, 2005, Vol. 105, No. 8 Tomasi et al.

is of direct interest here: the merging of simulationswith quantum mechanical (QM) descriptions of mo-lecular structures. This merging, in progress forseveral years, has to overcome computational dif-ficulties, due to the computationally quite intensivenumerical procedures.

At this point of our rapid exposition we mayintroduce what we consider the second importantinnovation in the field of condensed systems madepossible by the use of computers: the continuummodels.

Continuum models were introduced more than acentury ago, in very simplified versions, givingresults of remarkable importance using computa-tional instruments no more complex than a slide rule.These models have been used for more than 50 years,and in more detailed versions they are still in use,but the essential step, opening new perspectives forthe study of solvent effects, has been its merging withthe QM descriptions of molecules. Continuum modelsare in fact the ideal conceptual framework to describesolvent effects within the QM approach, as will beshown in section 2.

This merging was initiated more than 30 years agoby a small group of young researchers (Claverie,Rivail, Tapia, and Tomasi), working in French andItalian laboratories gravitating around Paris andPisa. The initial stimulus was provided by therecognition that the QM description of the electro-static potential generated by the charge distributionof a molecule could represent a valid analytic andinterpretative tool to study intermolecular interac-tions.4

The perspectives opened by these findings wereelaborated independently by the various laboratoriesand led to the alternative theoretical methods5-8 thathave been examined in detail in the precedingreviews. In the present one we abandon the historicalperspective used in the first review1 to present andanalyze the methodological issues of the “modern”continuum solvation theory (modern in the sense thatis essentially based on the QM description of thesolute). The second2 and third3 reviews give consider-able space to methodological issues, but also payattention to applications: detailed surveys on theresults on chemical equilibria, spectra, and dynamicsof reactions are reported in the former, whereas, inthe latter, attention is centered on biomolecularsystems, also including in the survey simulationmethods with QM description of the solute.

The present review is again centered on method-ological issues, with a perspective focused on the mostrecent developments. The body of the theory devel-oped in the first 30 years of the “modern” solvationmethods is summarized, giving emphasis to thoseaspects that are at the basis of all recent extensionsand reformulations of the model. Some among themwere already considered in the previous reviews,reflecting, however, the provisional state of themethodological elaboration, now accomplished; otherscorrespond to new entries in the theory.

With respect to all of the previous reviews, here amuch larger space is devoted to three aspects thatwe consider to be important in future applications of

continuum models, namely, their use in studyingphenomena involving time-dependent solvation, onthe one hand, and molecular properties, on the otherhand, and their coupling to discrete models. None ofthese three aspects is completely new, but only in thepast few years have they acquired the necessaryreliability and accuracy. In parallel, the extension ofthe continuum models to treat complex condensedphases (ionic solutions, anisotropic dielectrics, het-erogeneous systems, liquid-gas and liquid-liquidinterfaces, crystals, etc.) has allowed us to enlargethe range of possible applications and to considerphenomena and processes that have been until nowthe exclusive property of computer simulations. Thetable of contents should give a schematic but clearproof of this new trend in continuum solvationmethods.

Throughout the text, we have also added remarksindicating other extensions of the continuum methodsthat are still in their infancy, or even in an earlierstage, but which seem to us to be possible and topromise a satisfactory reward. A further analysis offuture prospects will be done in the last sectiondedicated to comments and conclusions.

We conclude this introductory section with a shortexposition of the main features of the computationalstrategies based on continuum distributions. Theconsiderations reported here do not claim originality,but we find it convenient to report them to putcontinuum approaches in the right perspective.

1.2. Generalities about Continuum, Focused, andLayered Models

A continuum model in computational molecularsciences can be defined as a model in which a numberof the degrees of freedom of the constituent particles(a large number, indeed) are described in a continu-ous way, usually by means of a distribution function.

Continuum distributions are a very general con-cept. In the standard quantum mechanical descrip-tion of a single molecule M based on the usual Born-Oppenheimer (BO) approximation (the cornerstoneof the theory for molecular sciences), the electronicwave function, Ψ, is expressed in terms of one-electron wave functions, each depending on thecoordinates of a single electron. From this single-particle description a one-particle distribution func-tion is easily derived with an averaging operation,the one-electron density function FM

e (r), which con-tains a good deal of the information conveyed by theoriginal wave function. According to the formaltheory, one-electron FM

e (r) and two-electron FMe (r,r′)

density functions collect all of the elements necessaryfor a full exploitation of the QM basic calculation.Electron density functions are endowed with manyimportant formal properties, as they represent thekernels of integral equations from which propertiescan be derived. Actually, the formalism of the densitymatrices is completely equivalent to (and even morepowerful than) the usual wave function formalism.

By tradition, in basic quantum mechanics, theemphasis is not placed on the continuity of theelectronic distribution but on the discreteness of themolecular assembly. We are here interested in the

Quantum Mechanical Continuum Solvation Models Chemical Reviews, 2005, Vol. 105, No. 8 3001

application of the concept of continuum distributionfunctions to particles of different physical types,including electrons and nuclei as well. Continuumdistributions of this kind, often supplemented byconstraints acknowledging the existence of molecules,are of current use in statistical mechanics and findapplication, among others, in computer simulationsof pure liquids and solutions.

The continuum models we shall consider are in-termediate between the two extremes we have men-tioned, a continuum model for the electrons of asingle molecule, and a continuum model for a verylarge assembly of molecules. Our aim is to preservethe accuracy of the former in describing details of themolecule and the capability of the latter in stronglyreducing the degrees of freedom of large molecularassemblies. To do it in a proper way it is convenientto introduce another concept, that of focused models.

In focused models the interest of the enquirer fallson a limited portion of the whole system. There is alarge variety of systems and properties for which thefocusing approach can be profitably exploited. Asingle solute molecule in a dilute solution is just anexample, but many other examples can be cited: adefect inside a crystal, the superficial layer of a solid,the active part of an enzyme, the proton or energytransfer unit in a larger molecular assembly, andeven a single component of a homogeneous system,as a single molecule in a pure liquid or in a gas.

The definition of a focused model presents specificproblems for each different case; here, we shallhighlight some general aspects referring to solvationmodels.

The concept of a focused model can be translatedinto a simple formal expression. The whole systemis partitioned into two parts, which we define as thefocused part F, and the remainder R. The Hamilto-nian of the whole system may be written as

where f and r indicate the degrees of freedom ofthe F and R parts, respectively. To focus the modelmeans to treat the F part at a more detailed levelthan the R part. An important parameter in focusedmodels is the number of the degrees of freedom of R,which are not explicitly taken into account. Incontinuum solvation models the whole HR(r) term iseliminated and the total Hamiltonian is reduced toan effective Hamiltonian (EH) for the solute in theform

In this approach in fact, there is no need to get adetailed description of the solvent, it being sufficientto have a good description of the interaction. This issurely a considerable simplification. In other solva-tion models HR(r) is maintained and the focusingsimplifications involve the number of freedom de-grees within R. This is the case of the QM/MMmethods, in which a limited number of the degreesof freedom for each molecule within R are explicitlyconsidered, at least those involving position andorientation.

The elimination of the solvent Hamiltonian is notsufficient to eliminate the r degrees of freedom,because they appear in the interaction Hamiltonian.An almost complete elimination can be obtained byintroducing an appropriate solvent response functionthat we indicate here with the symbol Q(rb,rb′), where(rb) is no more the whole set of solvent coordinates,but just a position vector.

The solvent response function is similar in natureto the response functions that can be derived fromthe electron density function in the case of a mol-ecule. The density here is that of the liquid systemR, and Q(rb,rb′) is, in the more complete formulations,expressed as a sum of separate terms each relatedto a different component of the solute-solvent inter-action.

We shall be more specific in section 3, but for thisgeneral discussion we limit ourselves to the electro-static contribution. The response to be considered inthis term is that with respect to an external electricfield. As in classical electrostatics, the polarizationfunction PB of the medium is proportional to theexternal field

and the kernel response function we have to use isrelated to the function we have here introduced,namely, the permittivity, ε. The expression of ε maybe quite simple, just a numerical constant, or morecomplex, according to the model one uses. This pointwill be fully developed in section 4, in which also thebasic assumption of linearity introduced with eq 4will be reconsidered, but here it is sufficient toremark that the whole set of solvent coordinates ris replaced by a function depending only on oneparameter, the position vector rb, or by a couple ofposition vectors (rb,rb′).

The quest for the accuracy requested in chemicalapplications of such focused models suggests theintroduction of additional features, different accord-ing to the various cases. Within this very largevariety of proposals a third concept emerges for itsgenerality, the concept of layering.

Layering can be considered a generalization offocusing. The material components of the models arepartitioned into several parts, or layers, because oftenthese parts are defined in a concentric way, encirclingthe part of main interest. Each layer is defined at agiven level of accuracy in the description of thematerial system and with the appropriate reductionof the degrees of freedom. There is a large variety oflayering; for simplicity, they can be denoted withabbreviations, as, for example, QM/QM/Cont or QM/MM/Cont for a couple of three-layer models in whichthe inner layer is treated at a given QM level, thesecond at a lower QM or at a molecular mechanicslevel, respectively, and the third using a continuumapproach. Some type of layering involving as chainend the continuum description will be examined insection 7.

HFR(f,r) ) HF(f) + HR(r) + Hint(f,r) (1)

HeffFR(f,r) ) HF(f) + Hint(f,r) (2)

HeffFR(f,r) ) HF(f) + Vint[f, Q(rb,rb′)] (3)

PB ) ε - 14π

EB (4)


2. Methodological Outlines of the Basic QMContinuum Model

This topic has been amply presented in the firstreview on the argument,1 complemented in a secondreview,2 and summarized in many other places,including textbooks.9-12 For this reason we shallavoid an exhaustive presentation of the whole litera-ture but shall only present the essential elements.By contrast, we shall pay attention to all topics thathave been the subject of discussion in recent yearsand/or new methodological proposals.

In the next subsection we shall identify the char-acteristics that define the “basic QM continuummodel”: these in principle completely determine themodel we are interested in but, in practice, they areseldom completely fulfilled. We thus suggest that thereader consider this definition more as a “literal”convenience and a research of formal clarity ratherthan as a description of a real methodology. As amatter of fact, in this section, there will be manyoccasions to present models that do not completelyconform to this basic definition.

2.1. Definition of the Basic ModelFor the basic QM model we intend models with the

following characteristics:(1) The solute is described at a homogeneous QM

level. Computational procedures based on semiclas-sical or classical descriptions of the solute will beconsidered as derivation of the basic QM model andbriefly examined. Other models based on layered QMdescriptions of the solute (including layers treatedall at a QM level or mixed QM and semiclassicaldescriptions) will be considered in section 7.

(2) The solute-solvent interactions are limited tothose of electrostatic origin. Other interaction termsexist, and they must be taken into account to have awell-balanced description of solvent effects (see sec-tion 3). This point has to be emphasized; our choiceof paying attention first to the electrostatic model,dictated by convenience of exposition, should not leadthe reader to the false conclusion that only electro-statics is important in solvation. Often the oppositehappens, and in addition some problems arising intreating the electrostatic term are greatly alleviatedby the consideration of other solute-solvent interac-tions.

(3) The model system is a very dilute solution. Inother words, it is composed of a single solute molecule(including, when convenient, some solvent molecules,the whole being treated as a supermolecule at ahomogeneous QM level) immersed in an infinitesolvent reservoir.

(4) The solvent is isotropic, at equilibrium at agiven temperature (and pressure). Possible exten-sions beyond the isotropic approximation will beconsidered mainly to show new potentialities of themost recent solvation models.

(5) Only the electronic ground state of the solutewill be considered. Extensions to other electronicstates will be considered in section 5.

(6) No dynamic effects will be considered in thebasic model. Under the heading of “dynamic effects”

there is so large a variety of important phenomenathat it would require a separate review. The mainaspects of these phenomena will be considered insections 5 and 6.

Once we have better defined what we intend withthe expression “basic QM continuum model”, we canconsider some of its essential elements.

2.2. CavityThe cavity is a basic concept in all continuum

models. The model in fact is composed of a molecule(or a few molecules), the solute, put into a void cavitywithin a continuous dielectric medium mimicking thesolvent. The shape and size of the cavity are differ-ently defined in the various versions of the continuummodels. As a general rule, a cavity should have aphysical meaning, such as that introduced by On-sager,13 and not be only a mathematical artifice asoften happens in other descriptions of solvent effects.On the physical meaning of Onsager’s cavity, see alsothe comments in ref 14. In particular, the cavityshould exclude the solvent and contain within itsboundaries the largest possible part of the solutecharge distribution. Here, for convenience, we divideit into its electronic and nuclear components:

Obviously these requirements are in contrast withthe description of the whole system given by any QMlevel. The electronic charge distribution of an isolatedmolecule, in fact, persists to infinity. In a condensedmedium the conditions on FM

e at large distances areless well-defined, but in any case there will be anoverlap with the charge distribution of the medium,not explicitly described in continuum models butexisting in real systems.

In continuum models, much attention has beenpaid to the portion of solute electronic charge outsidethe boundaries of the cavity; the terms “escapedcharge” and “outlying charge” are often used toindicate this portion of charge. This subject will betreated in due detail in section 2.4.3. Here we willassume that all of the solute charge distribution liesinside the cavity, which in turn has a size not so largeas to be in contrast with the solvent exclusionpostulate.

The optimal size of the cavity has thus been asubject of debate, and several definitions have beenproposed. The adopted definitions are the result of atradeoff between conflicting physical requirements.

The shape of the cavity has also been the object ofmany proposals. It is universally accepted that thecavity shape should reproduce as well as possible themolecular shape. Cavities not respecting this condi-tion may lead to deformations in the charge distribu-tion after solvent polarization, with large unrealisticeffects on the results, especially for properties. Here,once again, there is a tradeoff between computationalexigencies and the desire for better accuracy.

Computations are far simpler and faster whensimple shapes are used, such as spheres and el-lipsoids, but molecules are often far from having aspherical or ellipsoidal shape.

FM ) FMe + FM

n (5)


Quantum mechanical calculations of the molecularsurface can give a direct ab initio definition of thecavity.

An accurate description is based on the use of asurface of constant electronic density (isodensitysurface).15,16 Within this framework, one only needsto specify the isodensity level (typically in the rangeof 0.0004-0.001 au) and, thus, the cavity will bederived uniquely from the real electronic environ-ment. Such a cavity has been inserted into theGaussian computational package.17 Even if not largelyused at the moment, in our opinion the isodensitysurface represents an important definition of thecavity for continuum solvation models, and it willsurely receive a renewed interest in the coming years.

A different technique pioneered by Amovilli andMcWeeny18 and employed by Bentley and others19-24

is based on the calculation of the interaction energybetween a given molecule and an atomic probe(typically a rare gas atom, from He to Ar) placed atopportune positions in the outer molecular space.From these calculations, a set of three-dimensionalisoenergy surfaces is determined. As described below,two kinds of surfaces are of interest, the solvent-accessible surface (SAS) and the solvent-excludedsurface (SES). The first can be directly obtained fromthese calculations, whereas the second requires anadditional assumption. According to Bentley the SEScan be determined from the electronic density func-tion of the system constituted by the molecule andthe probe. Following the AIM topological analysis,25

the points of the surfaces can be identified in thesaddle points [also indicated as (3,-1) bond criticalpoints] of such a function.

A connection of these surfaces with the thermalenergy kT allows one to define T-dependent molec-ular surfaces and cavities. This technique is ofpotential interest as a benchmark for solvent calcula-tions far from the ambient temperature; otherwise,the approach is too costly to be used in standardapplications.

The generally adopted compromise between ana-lytical but too simple and realistic but computation-ally expensive cavities is based on the definition ofthe cavity as an interlocked superposition of atomicspheres with radii near the van der Waals (vdW)values (the precise determination of such radii isrelated to the problem of the cavity size).

The most used set of vdW radii in the chemicalliterature is that defined by Bondi26 (∼5000 citationsin the past 15 years). This set was confirmed as therecommended one some years ago,27 after the exami-nation of a quite large number of intermolecularcontact distances drawn from the Cambridge Struc-tural Database. For his tabulation, Bondi used dataof other origin, mainly addressing the hard volumeof the molecule and not the non-covalent contactdistances. The data drawn from 28403 crystal struc-tures confirm the Bondi values, with the only excep-tion the hydrogen radius, set by Bondi at 1.2 Å, whichis probably too high by 0.1 Å.

Another tabulation of vdW radii frequently usedis that of Pauling,28 available in many tabulations of

physicochemical data, for example, in the CRCHandbook.29

A third set of values has been inserted in the latestversions of the Gaussian package.17 It is drawn fromthe compilation of data for the universal force field(UFF),30 and it covers the whole periodic tableincluding groups not present in Bondi’s or Pauling’stabulations.

The presence of small differences in the radiusvalues currently used in continuum models and theconsequent effect on the solvation energies deservea few words of comment. Chemical applications ofmolecular calculations generally involve trends, inparticular, comparisons of selected properties com-puted for different systems. In the particular case ofsolvation calculations, the comparisons may involveproperties of different solutes in the same solvent orof the same solute in different solvents. Absolutevalues at a precision comparable with that of veryaccurate experiments are rarely requested. For thisreason, the selection of the hard radii to use amongthe recommended tabulations seems to us to be nota critical issue. Attention has to be paid, however, tomaintain a coherent choice for all of the calculationsand to avoid comparisons among results obtainedwith different definitions of the radii.

Molecules often have an irregular shape, and theoccurrence of small portions of space on their periph-ery where solvent molecules cannot penetrate is nota rare event. This intuitive consideration is at thebasis of two definitions, those of solvent-excludingand solvent-accessible surface (SES and SAS, respect-ively).31-35

Both introduce in the surface (and in the volume)changes to the vdW description related, in a differentway, to the finite size of the solvent molecules. Inboth cases, the solvent molecule is reduced to asphere, with a volume equal to the vdW volume(other definitions of this radius have been used, butthis seems to be the most consistent definition). Thepositions assumed by the center of a solvent sphererolling on the vdW surface of the solute define theSA surface, that is, the surface enclosing the volumein which the solvent center cannot enter. The samesphere used as a contact probe on the solute surfacedefines the SE surface, that is, the surface enclosingthe volume in which the whole solvent moleculecannot penetrate (see Figure 1 for a schematicdrawing of the different surfaces for the same mol-ecule).

In the literature, the SES is also called “smoothmolecular surface” or “Connolly surface”, due toConnolly’s fundamental work in this field. Indeed, theSE surface developed by Connolly32,36 can be consid-ered to be the prototype for the computational studyof molecular surfaces. Visualization and handling ofsurfaces have given origin to a very large literaturethat cannot be reviewed here. The reader can bereferred to Connolly’s website37 for a clear and concisereview accompanied by a sizable selection of refer-ences, adjourned at 1996 (430 entries). Connolly’ssurfaces have been applied to a very large variety ofproblems, and they have been also used to compute


solvation energies with continuum models (generallyof classical type). The probe sphere divides the wholeSE surface into pieces of three types: the convexpatches in which the probe touches just one sphereof the hard vdW shape function, the toroidal patchesin which the probe touches two spheres of the hardbody, and the concave (reentrant) patches in whichthe probe touches three spheres. An analytical ex-pression for this shape, easy to visualize on acomputer screen with a probe provided with markersthat put dots on the SES, has been given by Connollywithin a short time from the first computer imple-mentation of the procedure (1979-1981), and it isstill in use, with some modifications. The analyticaldescription presents some problems, among which wemention a few: the intersection between a torus anda sphere is described by a fourth-degree equation, forwhich the available solvers are not sufficiently ro-bust; the SES may present singularities and cusps;these last problems are better treated with methodsdeveloped by Gogonea38,39 and by Sanner et al.40 Thelatter have developed a computational code calledMSMS,41 standing for Michael Sanner’s MolecularSurface, which has received much attention, espe-cially among biochemistry-oriented computationalresearchers. MSMS computes, for a given set ofspheres and a probe radius, the reduced surface andthe analytical model of the SES. The MSMS algo-rithm can also compute a triangulation of the SESwith a user-specified density of vertices.

Besides Connolly’s SES, another SES-like surfacewill be reviewed here as of current use in QMcontinuum solvation methods. This alternative sur-face is defined following a different strategy, origi-nally conceived in Pisa around 1984 and finalized in1986 by Pascual-Ahuir in his Ph.D. research.42 Thissurface-building method, known as GEPOL,43-45 isbased on a sequential examination of the distanceamong the centers of each couple of hard vdWspheres and comparison of it with the solvent probediameter. If the distance is such that the probecannot pass between the two hard spheres, additionalspheres are added. Only three cases are possible,each corresponding to a different positioning of theadditional spheres, each with the opportune radius(position and radius are determined with very simple

and unambiguous algorithms). The whole set ofspheres, the original vdW spheres and those added,is subjected again to the same sequential examina-tion to add new spheres (second-generation spheres)to smooth the surface. The program originally writtenby Pascual-Ahuir introduced thresholds and optionsto keep the number of additional spheres withinreasonable limits.

In GEPOL, the final surface is thus always theresult of the intersections of spheres and, in thissense, it can be seen as an alternative version of theSES made only by convex elements.

To complete the section on the definition of thecavity, we recall an alternative strategy to define vander Waals, solvent-accessible, and solvent-excludingmolecular surfaces originally formulated by Pomelliin 1994-1995 for his master’s thesis. This strategy,known as DEFPOL,46,47 has never been implementedin publicly released computational packages, andthus its use is limited to a few examples; however, itstill presents some aspects that are worth recallinghere. The basic strategy consists of progressivedeformations of a regular polyhedron with the desirednumber of faces (triangular faces are preferred)inscribed into a sphere centered on the mass centerof the molecule. The sphere is deformed into theinertial ellipsoid and enlarged so as to have all shapefunctions of the molecule within it. Each vertex ofthe deformed polyhedron is then shifted along theline connecting the initial position with the origin ofthe coordinates until it lies on the surface of theshape function. The polyhedron is so transformed intoa corrugated polyhedron topologically equivalent tothe initial one, with faces still defined as triangles.The center of each triangle is then shifted along theaxis orthogonal to the triangle until it touches thesurface of the shape function. In the cases in whichthe volume of the tetrahedron defined by the threedisplaced vertices and the displaced center is largerthan a given threshold, the procedure is repeated ona finer scale on the three triangles having as verticesthe original ones and the triangle center. The finalstep consists of transforming the flat triangles intospherical triangles, each with the appropriate cur-vature.

2.3. Solution of the Electrostatic Problem

The physics of the electrostatic solute-solventinteraction is simple. The charge distribution FM ofthe solute, inside the cavity, polarizes the dielectriccontinuum, which in turn polarizes the solute chargedistribution. This definition of the interaction corre-sponds to a self-consistent process, which is numeri-cally solved following an iterative procedure. It isimportant to remark that the corresponding interac-tion potential is the one we shall put in the Hamil-tonian of the model. As this potential depends on thefinal value of FM reached at the end of this iterativeprocedure, the Hamiltonian (previously introducedas an “effective Hamiltonian”, see eq 2) thus turnsout to be nonlinear. This formal aspect has importantconsequences in the elaboration and use of thecomputational results (see section 2.4.4).

Figure 1. Solvent accessible surface (SAS) traced out bythe center of the probe representing a solvent molecule.The solvent excluded surface (SES) is the topologicalboundary of the union of all possible probes that do notoverlap with the molecule.


Reference is often made to the solvent reaction fieldfor the interaction potential obtained with continuummodels (and also with models using explicit solventmolecules). This label has a historical reason, beingrelated to Onsager’s seminal paper13 in which thesolute was reduced to a polarizable point dipole andthe electrostatic interaction between a polarizablemedium and a dipole was expressed in terms of anelectrostatic field, having its origin in the polarizationof the dielectric. Actually, it is now convenient tospeak in terms of the solvent reaction potential,because a potential is the term we have to introduceinto the Hamiltonian.

The basic model requires the solution of a classicalelectrostatic problem (Poisson problem) nested withina QM framework. Let us consider the electrostaticproblem first.

In our simplified model the general Poisson equa-tion

can be noticeably simplified to

where C is the portion of space occupied by the cavity,ε is the dielectric function (actually a constant) withinthe medium, and V is the sum of the electrostaticpotential VM generated by the charge distribution FMand the reaction potential VR generated by thepolarization of the dielectric medium:

Observe that we have assumed that all of the realcharges of the system (i.e., those described by FM) areinside the cavity. Therefore, this basic model is notformally valid for liquid systems having real chargesin the bulk of the medium (as is the case, e.g., forionic solutions) and also, strictly speaking, for thetiny portions of the electronic component of FM lyingout of the cavity (see section 2.4.4 for a more detailedanalysis). It is valid, however, for systems having amultiplicity of cavities C1, C2, ..., Cn, each containinga different FM1, FM2, ..., FMn charge distribution.

Equations 7 and 8 are accompanied by two sets ofboundary conditions, the first at infinity and thesecond on the cavity surface. At infinity we have

with finite values for R and â. These conditionsensure the harmonic behavior of the solution, butthey have to be specifically invoked in just one of themethods we shall examine; in the other cases, theconditions are automatically satisfied in our basicQM model.

More important, from a practical point of view, arethe conditions at the cavity surface Γ. They may beconcisely expressed as jump conditions:

The jump condition (eq 12) expresses the continuityof the potential across the surface, a condition validalso for other dielectric systems we shall considerlater:

The second jump condition (eq 13) involves thediscontinuity of the component of the field (expressedas a gradient of V) that is perpendicular to the cavitysurface. In the model we are considering here, acavity with a dielectric constant equal to 1 and anexternal medium with ε (a finite value >1), thiscondition leads to

where nb is the outward-pointing vector perpendicularto the cavity surface.

Equations 7-15 are the basic elements to use inthe elaboration of solvation methods according tostandard electrostatics. In the first review1 the ap-proaches in use were classified into six categories,namely, (1) the apparent surface charge (ASC) meth-ods, (2) the multipole expansion (MPE) methods, (3)the generalized Born approximation (GBA), (4) theimage charge (IMC) methods, (5) the finite elementmethods (FEM), and (6) the finite difference methods(FDM).

We maintain here this classification, but with therequired modifications due to the important develop-ments achieved in the past years in almost all of thecategories. Only in the category of the IMC methodsare no new important developments to be found, atleast within the framework of molecular calculations,and thus this category of methods will be not con-sidered in the present review: the interested readeris referred to our previous review.1

2.3.1. ASC MethodsFrom the jump condition (eq 15) one may derive

an auxiliary quantity that defines all of the ASCmethods: an apparent surface charge σ(s) spread onthe cavity surface. We are using the symbol s for theposition variable, to emphasize that this chargedistribution is limited to the surface Γ.

The definition of the ASC is not unequivocal butchanges in alternative versions of the model. In allcases, however, the ASC defines a potential over thewhole space:

This potential is exactly the reaction potential VR ofeq 9. There are no approximations in this elabora-tion: the definition of VR given with eq 16 is exact

- ∇B[ε(rb)∇VB(rb)] ) 4πFM(rb) (6)

-∇2V(rb) ) 4πFM(rb) within C (7)

-ε∇2V(rb) ) 0 outside C (8)

V(rb) ) VM(rb) + VR(rb) (9)

limrf∞

rV(r) ) R (10)

limrf∞

r2V(r) ) â (11)

[V] ) 0 on Γ (12)

[∂V] ) 0 on Γ (13)

[V] ) Vin - Vout ) 0 (14)

[∂V] ) (∂V∂n)in - ε (∂V

∂n)out) 0 (15)

Vσ(rb) ) ∫Γ

σ(sb)|rb - sb|d

2s (16)


when σ(s) is defined according to the proper electro-static equations.

The reduction of the source of the reaction potentialto a charge distribution limited to a closed surfacegreatly simplifies the electrostatic problem withrespect to other formulations in which the wholedielectric medium is considered as the source of thereaction potential. Despite this remarkable simpli-fication, the integration of eq 16 over a surface ofcomplex shape is computationally challenging. Thesolutions are generally based on a discretization ofthe integral into a finite number of elements. Thistechnique may be profitably linked to the boundaryelement method (BEM), a numerical technique widelyused in physics and engineering to solve complexdifferential equations via numerical integrationof integral equations (see the Website http://www.boundary-element-method.com/ for a global viewon literature and applications of this method).

The cavity surface Γ is approximated in terms of aset of finite elements (called tesserae) small enoughto consider σ(s) almost constant within each tessera.With σ(s) completely defined point-by-point, it ispossible to define a set of point charges, qk, in termsof the local value of σ(s) on each of these tesseraetimes the corresponding area Ak. The integral of eq16 is thus transformed in the following finite sum:

Actually the local value of the potential necessary todefine qk also depends on the whole set of the surfacecharges, and so the correct values of the surfacecharges, and the correct expression of the reactionpotential, are to be obtained through an iterativeprocedure. These aspects will be examined in section2.3.1.5; now we present the most important ASCmodels. In this presentation we shall not make useof the BEM version of the ASC equations involvingthe point charges qk but instead the original ones interms of a continuous surface charge σ(s). A descrip-tion of the formal and practical aspects related to theuse of the BEM approach for ASC methods will begiven in section 2.3.1.5.

2.3.1.1. Polarizable Continuum Model (PCM):Original Formulation. PCM, the oldest ASCmethod, at present is no more a single code, butrather a set of codes, all based on the same philoso-phy and sharing many features, some specialized forsome specific purposes, others of general use, butwith differences deserving mention.

The original PCM version was published in 1981,after some years of elaboration,8 and subsequentlyimplemented in local and official versions of variousQM computational packages.48 More recently, PCMwas renamed D-PCM (D stands for dielectric)49 todistinguish it from the two successive reformulations(CPCM and IEFPCM) that we shall present in thenext sections. This acronym is not completely correctas also the other reformulations refer to dielectricmedia (directly or indirectly); however, we cannotforget that it has become of common use in theseyears, and, thus, in the present review, we will adopt

DPCM to refer to the first version of the model,whereas PCM will be used to refer to the entirefamily of models.

DPCM, like all members of the PCM family, is ableto describe an unlimited number of solutes, eachequipped with its own cavity and ASC, interactingamong them through the dielectric. In this way,DPCM permits an extension of the basic model toassociation-dissociation phenomena, molecular clus-tering, etc., and it can account for a continuous shiftfrom a single cavity to two cavities during a dissocia-tion and the merging of two or more cavities duringassociation. In parallel, it permits an extension tomodels in which the medium is composed by a set ofnonoverlapping dielectric regions at different permit-tivity, constant within each region.

The reason for this versatility is in the use of theASC approach in an unsophisticated version. Tobetter appreciate this point, let us look again at thebasic electrostatics from a different viewpoint.

For systems composed by regions at constantisotropic permittivity (including systems composedof a single isotropic solvent with multiple cavities),the polarization vector is given by the gradient of thetotal potential V(r) (including also that deriving fromapparent charges)

where εi is the dielectric constant of the region i.At the boundary of two regions i and j, there is an

ASC distribution given by

where nbij is the unit vector at the boundary surfacepointing from medium i to medium j. The basic ASCmodel is so transformed into a similar system, withseveral ASCs that must be treated all on the samefooting.

The basic PCM definition may be derived from thegeneral expression 18 by taking into account the factsthat in the basic case εi ) 1 and εj ) 1 and that wehave computed the gradient on the internal (in) partof the surface, namely

where nb indicates the unit vector perpendicular tothe cavity surface and pointing outward.

After its first formulation, the DPCM was revisedmany times both in its theoretical aspects and in itsnumerical implementation; among all of these revi-sions, a fundamental one was proposed in 1995 byCammi and Tomasi50 when a new, and more efficient,computational strategy was defined to solve the BEMequivalent of eq 20 (see section 2.3.1.5 for moredetails and comments on this strategy).

Since its first presentation in 1981, DPCM hasbeen “adopted” by many groups, which have thuslargely contributed to its diffusion and its develop-ment. Some of these extensions will be examined infollowing sections of this review, but here we cannot

Vσ(rb) = ∑k

σ(sbk)Ak

|rb - sbk|) ∑

k

qk

|rb - sbk|(17)

PBi(rb) ) -εi - 1

4π∇VB(rb) (18)

σij ) - (PBj - PBi)‚nbij (19)

σ(s) ) ε - 14πε

∂

∂nb(VM + Vσ)in (20)


forget to cite the work of three groups that havegiven, and continue to give, important contributionsto show potentialities of DPCM.

First we cite the extensive and systematic workdone in Barcelona by Luque, Orozco, and co-workers.The work of this group has to be here mentioned forseveral reasons. Their reformulation of DPCM, knownby the acronym MST from the names of the threeauthors of the first PCM paper, has been applied toalmost all aspects of solvation problems, with specialattention to organic and biological systems; theamount of results of remarkable quality is the largestgiven by a single group. The papers of this vastliterature devoted to methodological innovations (themain theme of this review) are abundant, all deserv-ing attention. For obvious reasons, only a few51-54 arecited here, and the reader is referred to a recentreview3 for many others. Barcelona’s group has spentefforts during the past 10 years to analyze theperformances of the method in many ways and tovalidate it by comparisons with other procedures:simulations of different type, other continuum mod-els. Among them we cite a comparison with twomethods we shall examine in the following pages, theMPE method of Nancy (section 2.3.2) and the GBmethod of Minneapolis (section 2.3.3), performed incollaboration with the authors of such methods.55 Inthe following, we shall cite other papers by Luqueand Orozco, but what stimulated us here is tomanifest the appreciation for the considerable workin developing solvation methods, through the ex-ample of numerous impeccable applications, the workof analysis and validation, the suggestion of meth-odological innovations improving or extending thepotentialities of the approach, and the activity ofpopularizing it by means of several short (or long)review papers.

Another important contribution to the developmentof the DPCM has been done by Basilevsky and co-workers; in their numerous works on solvationphenomena they started from the original definitionof DPCM to develop a parallel model known by theacronym BKO,56 standing for Born-Kirkwood-On-sager. During the years, this model has been appliedto many different problems. Here it is not possibleto report the vast literature, but in the followingsections we shall analyze in detail the main develop-ments introduced by Basilevsky and co-workers,namely, the reformulation of continuum models tononlocal dielectrics, their application to studies ofsolvation dynamics, and their extension to mixedcontinuum/discrete approaches. In these sections, weshall also report all correspondent references.

A last contribution we quote here is that of theresearch team of the Gaussian computational pack-age.17 This team soon realized the potentialities andcomputational interest of the ASC approach to sol-vation, in the form given by DPCM. A version of theoriginal model was rewritten by Gaussian and in-serted in some releases around 1990, but the officialcollaboration between the PCM group and Gaussianstarted only at the end of 1996, and it has continuedsince then: more details on the results of thiscollaboration will be given in the following sections.

The Gaussian team has also independently elabo-rated a different PCM-like model based upon thepreviously mentioned isodensity surfaces (see section2.2). This version of PCM is generally known as theisodensity-PCM (IPCM) model.16 The same model hasbeen further extended to allow the isodensity surfaceat each SCF iteration to be varied; that is, the cavityis not fixed, even once the geometry is fixed (as isthe case in the standard PCM model), but it isrelaxed to the isodensity of the solvated molecule ateach SCF iteration. This further development isgenerally known as self-consistent isodensity (SCI-PCM).57 Both IPCM and SCIPM models are availablein the Gaussian computational package.17

2.3.1.2. Conductor-like Screening Model (COS-MO). In this method, originally devised by Klamt andSchuurmann,58 the dielectric constant of the mediumis changed from the specific finite value ε, charac-teristic of each solvent, to ε ) ∞. This value corre-sponds to that of a conductor, and this changestrongly modifies the boundary conditions of theelectrostatic problem. The most important effect isthat the total potential V(r) of eq 9 cancels out onthe cavity surface. From this condition it follows thatthe ASC is determined by the local value of theelectrostatic potential instead of the normal compo-nent of its gradient (as in eq 20). To recover theeffects of the finite value of the dielectric constant ofthe medium, the ideal unscreened charge density, σ*,corresponding to ε ) ∞, is finally scaled by a properfunction of ε, namely

The scaling function, f(ε), has been empirically de-termined by comparing COSMO (unscaled) and cor-rect electrostatic solute-solvent energies. The sug-gested formula is of the type

with k small.In the original paper about COSMO, Klamt sug-

gested k ) 0.5,58 with the remark that k seems todepend on the cavity shape and on the distributionof charges in the solute. Other versions of COSMOproposed in the following years adopted other valuesfor k. Truong and Stefanovic, in their GCOSMO,59,60

and Cossi and Barone, in their C-PCM61 (this pro-gram is a part of the PCM suite of programs),proposed k ) 0 on the basis of an analogy with theGauss law. Subsequently, Cossi et al.62 compared thesolvation free energies for neutral and chargedsolutes in water and in CCl4, and they found that thechoice of k is irrelevant for water but important inCCl4: for neutral solutes the best agreement isobtained for k ) 0.5, whereas for charged moleculesit is preferable to use k ) 0. Pye and Ziegler63 in theirimplementation of COSMO in the ADF computa-tional package64,65 use k ) 0 as default, but leave theusers free to select the value they prefer. Chipman,in a recent comparison of several continuum mod-els,66 gives results for both k ) 0 and 0.5, remarkingthat k ) 0 seems to be somewhat better.

σ(s) ) f(ε)σ*(s) (21)

f(ε) ) ε - 1ε + k

(22)


We note that the corrections introduced by thisfactor are quite small for solvents with high dielectricconstant, and the final output of the energy is almostinsensitive to changes in the k parameter. Things area bit more critical for solvents of low ε and, inparticular, for the cases in which one has to sepa-rately compute the inertial and the electronic com-ponent of the polarization vector PB (see sections 5.1and 5.2); in these cases attention has to be paid inthe use of solvation codes of the COSMO family.

An extension of the COSMO model, called COSMO-RS (conductor-like screening model for real solvents),has been proposed by Klamt,67 starting from theanalysis that solvents do not behave linearly whenwe consider strong electric fields on the molecularsurfaces of fairly polar solutes. COSMO-RS is atheory that describes the interactions in a fluid aslocal contact interactions of molecular surfaces, theinteraction energies being quantified by the valuesof the two screening charge densities that form amolecular contact. Having reduced all interactionsto local interactions of pairs of molecular surfacepieces, one can consider the ensemble of interactingmolecules as an ensemble of independently interact-ing surface segments. COSMO-RS has become apredictive method for the thermodynamic propertiesof pure and mixed fluids. In contrast to groupcontribution methods, which depend on an extremelylarge number of experimental data, COSMO-RScalculates the thermodynamic data from molecularsurface polarity distributions, which result fromquantum chemical calculations of the individualcompounds in the mixture. The different interactionsof molecules in a liquid, that is, electrostatic interac-tions, hydrogen bonding, and dispersion, are repre-sented as functions of surface polarities of thepartners. Using an efficient thermodynamic solutionfor such pairwise surface interactions, COSMO-RSfinally converts the molecular polarity informationinto standard thermodynamic data of fluids, that is,vapor pressures, activity coefficients, excess proper-ties, etc.;68-72 the corresponding computational codeis called COSMOtherm.73

2.3.1.3. Integral Equation Formalism (IEF).With the IEF method74-76 originally formulated byCances and Mennucci in 1997, we come back againto methods involving solutes in a liquid phase andusing the characterization of the solvent given bymacroscopic properties of the specific liquid (such asthe permittivity, ε). Actually, IEF is a method moregeneral than DPCM, or COSMO, and the case of anisotropic dielectric is nothing more than one of itspossible applications, as we shall show below. To-gether with the previously mentioned DPCM andCPCM, IEF is a member of the PCM suite ofsolvation codes implemented in Gaussian since its1998 version and, in the latest G03 version,17 it hasbecome the default PCM formulation.

As in the two other ASC methods we have de-scribed in the previous sections, also for IEF we startfrom the electrostatic system (eq 6) and we introducethe decomposition (eq 9) of the potential V in the sumof the electrostatic potential generated by the charge

distribution FM in vacuo, VM, and the “reactionpotential”, VR.

In the IEF, however, the potentials are redefinedin terms of the proper Green functions; we recall thatthe Green function of an electrostatic problem G(x,y)is the potential produced in x by a unit point chargelocated in y (here with x and y we indicate twogeneral positions in the space, but for simplicity’ssake we drop the vector symbol).

If we indicate by G(x,y) ) 1/|x - y| the Greenfunction corresponding to the operator -∇2 (namely,its kernel), by Gs(x,y) the Green kernel of the operator-∇(ε∇), and GR(x,y) ) Gs(x,y) - G(x,y), the followingrelationships arise:

Exploiting some basic results of the theory ofintegral equations (see ref 76 for details), it can beproved that the reaction potential, VR, which satisfies

can be represented as a single layer potential

where the surface charge σ is the unique solution tothe equation

with A and g being two integral operators defined as

In the two last relationships we have introduced theoperators Sa, Da, and Da

/ with a ) e or i standing forexternal (i.e., outside the cavity) and internal (i.e.,inside the cavity), respectively. These operators areformally defined for σ by

where εa ) 1 for a ) i and εa ) ε (the solventpermittivity) for a ) e.

We note that eq 27 is different with respect to thatreported in the original papers74,76 because here we

V(x) ) ∫R3 Gs(x,y)FM(y) dy

VM(x) ) ∫R3 G(x,y)FM(y) dy (23)

VR(x) ) ∫R3 GR(x,y)FM(y) dy

- ∇2VR ) 0 in C and outside C

[VR] ) 0 on Γ (24)

VR f 0 at infinity

VR(x) ) ∫Γ

σ(y)|x - y| dy ∀ x ∈ R3 (25)

Aσ ) - g (26)

A ) (2π - De)Si + Se(2π + Di/)

g ) (2π - De)VM + Se(∂VM/∂n)(27)

(Saσ)(x) ) ∫Γ Ga(x,y)σ(y) dy

(Daσ)(x) ) ∫Γ [(εa∇ByGa(x,y))‚nb(y)]σ(y) dy (28)

(Da/σ)(x) ) ∫Γ [(εa∇BxGa(x,y))‚nb(x)]σ(y) dy


have used a different form of the Poisson eq 6including a 4π factor on the right-hand side. This isreflected in the Green functions, Ga, and thus in thecorresponding operators Sa and Da, which now do notinclude the 1/4π factor.

The operators defined in eq 28 are well-known inthe theory of integral equations: they are three ofthe four components of the Calderon projector.77 Werecall some of their properties: the operator Si is self-adjoint, and Di

/ is the adjoint of Di for the scalarproduct. Besides, SiDi

/ ) DiSi.Equation 26 may be further simplified using the

equality (2π - Di)VM + Si(∂VM/∂n) ) 0; in this waythe expression for g in eq 27 can be rewritten as78

and thus the surface charge σ depends only on thepotential VM (and no longer on a normal componentof the field) exactly as in the COSMO approach. Thissimplification is important from both numerical andformal points of view.

Numerically, it is advantageous not only becausethe calculation of a single scalar function (the poten-tial) is computationally less demanding than theparallel calculation of both the potential and thevectorial electric field, but also because the potentialis less sensitive than the field to numerical instabili-ties that can appear when we introduce a BEMapproach to solve the electrostatic equations. Froma formal point of view, the reformulation of the IEFASC in terms of only the potential is importantbecause it represents an implicit correction of theerror due to the fraction of solute electric chargediffusing outside the cavity (namely, the previouslydefined outlying charge). The details on this issue willbe given in section 2.4.3. Here it is worth reportingthe final equations for the IEF ASC in the case ofisotropic solvents; these, in fact, will be useful in thefollowing when we shall compare IEF to other ASCmethods. Namely, using the relations, Se ) Si/ε, Di) De, we can transform the IEF operators A of eq 27and g of eq 29 in

and thus eq 26, defining the ASC, reduces to78

where, once again, we have used the relationship Si

Di/ ) DiSi. In the following, this version of IEF will

be denoted IEF(V) to indicate that only the soluteelectrostatic potential (VM) is required to determinethe ASCs.

From eqs 24-29 it can be seen that the IEFapproach is completely general, in the sense that itcan be applied without the need of modifying eitherthe basic aspects of the model or the basic equations(26-29) to all of those systems for which the Greenfunctions inside and outside the cavity are known.As a matter of fact, as the interior form of G is always

known, Gi(x,y) ) 1/4π|x - y|, the problem is shiftedto the evaluation of the exterior, Ge(x,y). Analyticalexpressions of this function are available for standardisotropic solvents characterized by a constant andscalar permittivity ε [namely Ge(x,y) ) 1/(ε|x - y|)],but also for anisotropic environments characterizedby a tensorial but constant permittivity E for ionicsolutions described in terms of the linearized Pois-son-Boltzmann equation (see section 2.3.1.6 forfurther details) and for sharp planar liquid-metalinterfaces.79 Obviously, it is not always possible tohave analytical Green functions. However, in severalcases the Green function can be effectively builtnumerically, and thus the IEF approach can begeneralized to many other environments as, forexample, a diffuse interface with an electric permit-tivity depending on the position.80

As a final comment it is worth noting that IEFcontains as subcases both the DPCM and the COS-MO models. In the first case we have to consider anisotropic solvent with a scalar permittivity and usethe electrostatic equation already introduced aboveto rewrite the IEF ASC eq 29 in terms of thepotential, in the opposite way so as to keep (∂VM/∂n)instead. To recover COSMO ASC, on the other hand,we have to consider that ε f ∞, and thus all of theterms in eqs 27 and 29 involving the Se operator canbe neglected (Se is in fact proportional to 1/ε).

2.3.1.4. Surface and Volume Polarization forElectrostatic [SVPE and SS(V)PE]. From 1997 todate, Chipman and co-workers have developed aseries of continuum models81-85 that have, as astarting point, the consideration that unconstrainedquantum mechanical calculation of solute chargedensity generally produces a tail that penetratesoutside the cavity into the solvent region (the “outly-ing charge” mentioned in the previous sections).Exact solution of Poisson’s equation in this situationwould require invocation of an apparent volumepolarization charge density lying outside the cavityin addition to the apparent surface polarizationcharge density lying on the cavity.86 As a conse-quence, also the reaction potential heretofore writtenin terms of an apparent surface charge has to besupplemented with a term due to the apparentvolume charge, namely, following the notation usedby Chipman

where the integration is over the whole volumeexcluding the molecular cavity.

The acronym used to indicate this formulation wasSVPE, meaning that both surface and volume polar-ization for electrostatic interactions were included.Contrary to the previous ASC models, this methodrequires a discretization of both the cavity surfaceand the exterior volume; in particular, the volumepolarization charge density â was approximated bya collection of point charges located at various nodes

VR(x) ) Vσ(x) + Vâ(x)

Vâ(x) ) ∫ext

â(y)|x - y| dy ) - (ε - 1

ε )∫ext

FM(y)|x - y| dy

(32)

g ) [(2π - De) - SeSi-1(2π - Di)]VM (29)

A ) (1 - 1/ε)[2π(ε + 1)/(ε - 1) - Di]Si

g ) (1 - 1/ε)(2π - Di)VM(30)

[2π(ε + 1ε - 1) - Di]Siσ ) - (2π - Di)VM (31)


on a series of layers covering the exterior region, asdescribed in detail in ref 82.

The exact SVPE method is laborious to implementand time-consuming, because it utilizes a volumepolarization potential arising from a discontinuousvolume charge density. With a cavity surface that isadapted to the detailed nonspherical shape of ageneral molecular solute, this leads to difficult inte-grations over just part of the full three-dimensionalspace.

To avoid this large complexity a simpler ap-proximate solution that involves only apparent sur-face charge distributions was subsequently intro-duced. Here, we limit ourselves to summarize themain aspects of this method denoted “surface andsimulation of volume polarization for electrostatics”,SS(V)PE.

The SS(V)PE method originated with the demon-stration81 that all direct and indirect effects of thevolume polarization charge density on the exactSVPE reaction potential can be exactly representedat all points inside the cavity (which is the mostimportant region) by simulating the explicit volumecharge density in terms of an additional surfacecharge density so that the total apparent surfacecharge satisfies the equation84

where we have used the same notation introducedfor IEF in the previous section.

The correspondence between SS(V)PE and IEF isnot only of notation: the two methods in fact coincidewhen the isotropic IEF(V) eq 31 is considered; toshow this, it is sufficient to multiply both sides of eq33 by 2π(ε + 1)/(ε - 1) and to use the relationship Si

Di/ ) DiSi. The formal equivalence of the two meth-

ods was published87 only one year after the publica-tion of the SS(V)PE original paper,84 and thus thetwo methods are often considered as two alternativeASC methods. As a matter of fact, when the wholesolute charge is contained within the cavity, SS(V)-PE is equivalent also to DPCM (besides IEF), butwhen a part of this charge lies outside this, equiva-lence with DPCM is no longer valid; on the contrary,the equivalence with IEF(V) (see the end of section2.3.1.3) remains in all cases. The late recognition ofthe equivalence of the two approaches is also reflectedin a further PCM approach proposed by Cossi et al.88

In fact, this model, indicated as implicit volumecharges PCM (or IVCPCM), was presented by theauthors as the PCM reformulation of the SS(V)PEby Chipman but, in practice, it exactly coincides withthe IEF version of PCM. Unfortunately, also inpapers that appeared later, the authors never clari-fied this point, and thus the two methods continuedto be described as different approaches.89

2.3.1.5. Discretization of the Cavity Surfaceand the ASC-BEM Equations. In the presentationof the ASC models we have adopted the original

formalism using a continuous surface charge σ. Here,we go a step farther and present a description of theformal and practical aspects related to the use of theBEM version of the ASC equations involving thepoint charges qk. It is interesting to note that BEMis a relatively young technique (the acronym was firstused in 197790), but now a very large literature,including books and dedicated international confer-ences (the principal conference on BEM conducts its27th meeting in 200591), can be found on this verypowerful mathematical approach.

As introduced at the beginning of section 2.3.1, thedefinition of ASC elements uses the discretization ofthe surface into a finite number of elements (usuallycalled tesserae), which is the cornerstone of BEM,and for this reason it may be considered an applica-tion of a BEM procedure (or, better, of the indirectBEM procedure; see also section 2.3.1.6). Actually itis not a trivial application, because the original BEMwas limited to cases without internal sources, thatis, based on the Laplace equation ∇2V(r) ) 0 insteadof the Poisson equation (7). Other features of BEMhave been considered in more recent versions of ASCmethods, but it must be said that the large wealthof suggestions available in the BEM literature hasnot yet been fully exploited in the continuum solva-tion methods.

The necessary preliminary step in BEM-ASC strat-egy is the generation of the surface finite elements.This step can be achieved in many different ways;here, in particular, we shall focus on the algorithmdeveloped for the GEPOL cavity (see section 2.2).

In the original formulation, the GEPOL tesseraewere generated by inscribing a polyhedron with 60faces within each sphere, projecting the faces ontothe spherical surface, and discarding those that werefully inside an intersecting sphere. If a polyhedronface happened to be partially inside a neighboringsphere, the tessera was cut into smaller triangles,and only those triangles that were completely ex-posed were kept; in this way the final tessera wasrepresented by a complex polygon.

GEPOL has been considerably improved during theyears; we mention here an improved scheme to fillthe solvent-excluded space92 and the algorithmthrough which the exposed portion of a tesserapartially inside other spheres is analytically cut byadding edges to the tessera being generated.93 In thisway, all of the tesserae are defined in terms ofconnected arc segments, and their surface areas canbe analytically computed using the Gauss-Bonnetformula so that the analytical derivatives of all thetesserae can be computed. Another important im-provement is the introduction of flexible tessellation94

and symmetry-adapted tessellations.95,96 This ad-vanced version of GEPOL is available in GAMESS97,98

and Gaussian17 computational codes. The last revi-sion of Gaussian (Gaussian 03) includes an even moreadvanced version of GEPOL; this recent re-imple-mentation has been realized by Scalmani et al.99 and,among different important numerical improvements,it presents also a linear scaling computational cost,thus extending the range of applicability to very largemolecular systems.

TσSS(V)PE ) (ε - 1ε + 1)[Di

2π- I]VM

T ) Si - (ε - 1ε + 1) 1

4π(DiSi + SiDi

/)

(33)


Recently, two independent novel schemes havebeen proposed100,101 to sample the GEPOL molecularsurface without an explicit discretization of the cavitysurface, which may lead to unphysical discontinui-ties. In both of these schemes the sampling of thesurface is done on each original sphere, and it is notchanged after all of the spheres have been combinedto construct the final cavity. Each sampling point isthen associated with a “weight” (which in the simplescheme coincides with the area of the tessera) andto a switching function, the definitions of which aredifferent in the two methods. In both cases, however,the main result is that no sampling points andweights will be discarded; their number remainsconstant also during a geometry optimization whenchanges in the solute geometry induce changes in therelative position of the atom-centered spheres.

Once the cavity surface has been partitioned intesserae small enough to consider σ(s) almost con-stant within each tessera, it is possible to define aset of point charges qk in terms of the local value ofσ(s) on each of these tesserae times the correspondingarea. In doing this the electrostatic equation pre-sented in sections 2.3.1.1-2.3.1.4 for σ(s) within thevarious ASC methods can be rewritten as a set of T(with T equal to the number of tesserae) coupledequations, which can be recast in a matrix form ofthe type

where K is a square matrix T × T collecting thecavity geometrical factors (the tesserae representa-tive points sk and the corresponding areas) and thedielectric constant of the medium and q and f arecolumn matrices, the first containing the unknown

charges and the second the values of the properelectrostatic quantity, namely, the normal componentof the electric field En or the electrostatic potentialV, calculated at the tesserae. Equation 34 representsthe electrostatic BEM problem we have to solve.

The elaboration of the problem giving origin to anequation of type 34 has been done in a large numberof ways, in practice one for each ASC method, andalso in several different ways for the same method.There is no need to repeat here the various elabora-tions; instead, we provide a table with the expres-sions of the elements of the K and f matrices for themost important variants of ASC methods, namely,DPCM,CPCM, IEFPCM, and SS(V)PE. As far asconcerns IEFPCM, two different sets of expressionsare given, the first referring to standard isotropicsolvents (characterized by a constant scalar permit-tivity ε) and the second for anisotropic solvents (i.e.,characterized by a constant but tensorial permittivityε) and for ionic solutions (in the limit of a linearizedPB scheme, see eq 43). We also note that in all cases,IEFPCM equations have been rewritten in the IEF-(V) form, that is, using only f ) V and not thecombination of V and En as in the first formulationof the method; this reformulation was originallyproposed for the isotropic case only, but recently ithas been generalized to anisotropic dielectrics;102

here, we present for the first time the parallel versionfor the linearized Poisson-Boltzmann scheme (seealso section 2.3.1.6).

In Table 1 we have reported only the off-diagonalelements of the D and S matrices involved in thedifferent versions, because different numerical solu-tions have been proposed for the diagonal elements.In particular, for S the following approximation is

Table 1. Matrices To Be Used in Equation 34 To Get the Apparent Charges in Various ASC Methods

K f

DPCM (2πε + 1ε - 1

A-1 - D*)-1

with Dij )(sbi - sbj)‚nbj

|sbi - sbj|398ifj

Dij/ En

COSMO [CPCM] S-1 with Sij ) 1|sbi - sbj|

V

IEFPCM(iso) [IVPCM] [2π(ε + 1ε - 1)A-1 - D]S-1

[2πA-1 - D] V

SS(V)PE [2π(ε + 1ε - 1)A-1 S - A-1 (DAS + SAD*)

2 ]-1[2πA-1 - D]

IEFPCM [2πA-1 - De]S + Se[2πA-1 + D*]-1[2πA-1 - De] -

SeS-1[2πA-1 - D]

V

aniso[De]ij )

(sbi - sbj)‚nbj

xdet E[(ε-1‚(sbi - sbj))‚(sbi - sbj)]3/2

[Se]ij ) 1xdet E[(ε-1‚(sbi - sbj))‚(sbi - sbj)]

1/2

ionic[De]ij )

exp[-k|sbi - sbj|][1 + k|sbi - sbj|](sbi - sbj)‚nbj

|sbi - sbj|3

[Se]ij )exp[-k|sbi - sbj|]

ε|sbi - sbj|

q ) -Kf (34)


generally used

where ai indicates the area of tessera i and k is anumerical parameter. This equation derives from theexact formula of a small circular cap section of asphere or a flat circular element of any size for whichk ) 1. To improve the approximation in the case of aspherical element such as those used in ASC meth-ods, a different value of k is used. The constant k wasoriginally determined58 to have the value 1.07 on thebasis of the numerical evaluation for a sphere dividedinto various numbers of equivalent finite segments,but then, after a more accurate fitting, this value wasrevised to 1.0694, and this is the value implemented,for example, in the latest version of CPCM andIEFPCM in Gaussian code.103 For the D matrix, itcan be shown that, if the tessera is placed on asphere, then75

where RI is the radius of such a sphere.An alternative expression for Dii has been adopted

by Foresman et al.,16 by Chipman,83,104,105 and byCossi et al.88 following the original proposal byPurisima;106,107 this expression uses the sum of theoff-diagonal elements, namely

In the case of IEFPCM for anisotropic dielectrics andionic solutions, no analytical expressions can be givenfor the diagonal elements of the De and Se matrices,and thus the procedure that has been implementedis based on a Gaussian integration scheme.74

The linear system (eq 34) can be solved either bymatrix inversion or iteratively.50 The cost of comput-ing the ASC terms used to be usually negligible ifcompared to other steps in an ab initio calculation(e.g., Fock matrix building and diagonalization).Nowadays, this is no longer true as, in the field ofab initio methods, fast algorithms for which the costgrows linearly with the system size [e.g., the fastmultipole method (FMM)] are beginning to be widelyavailable and applied.108-111 On the other hand, thesignificant research effort being devoted to hybridquantum/classical approaches (see section 7) is driv-ing toward the handling of large (partly) classicalsolutes, which could easily involve thousands ofatoms.

Given these methodological and algorithmic ad-vances, the cost of calculating the ASCs could easilybecome the computational bottleneck. Indeed, when-ever fast and linear scaling algorithms, such as theFMM, are used to compute the electrostatic potential/field on the tesserae, the cost of evaluating theinteraction of the polarization charges among them-selves will grow more steeply and will rapidly becomedominant as the number of tesserae increases.

Recently, an efficient linearly scaling procedure hasbeen formulated112 (and implemented in the latestrevision of Gaussian) by exploiting the FMM forcomputing electrostatic interactions. Such a proce-dure has been formulated for DPCM, CPCM, andIEFPCM models, and it is based on an iterativesolution of the linear system (eq 34). Here, we presentthe basic equations just for the DPCM case as moreintuitive, but we refer the interested reader to thereference paper for the corresponding equations forCPCM and IEFPCM. By properly rewriting thesystem (eq 34), the iterative equation defining theapparent charges becomes

where

and q(n-1) and q(n) are the charges at the (n - 1)thand nth iteration, respectively. The definitions ofzi/[xj] and z[x] (eqs 39) state that these ‘‘kernels” cost

formally as the square of the number of tesserae T.However, taking a closer look at those expressions,it is easy to realize that both quantities can becomputed in the framework of the FMM algorithm,thus achieving linear-scaling computational cost. Inparticular, zi

/[xj] is the normal component of theelectric field at the ith tessera generated by the(pseudo)charges xj located at the other T - 1 tesserae,and z[x] is the sum of the ‘‘normal fields” generatedby the (pseudo)charges xj at the other T - 1 tesserae.The performances of different iterative solvers andpreconditioning strategies have been assessed for thedifferent PCM models.

We conclude this section by mentioning an alterna-tive approach to define the apparent charges formu-lated by York and Karplus113 within the COSMOformalism. The method uses smooth Gaussian basisfunctions of the form

to represent the electrostatic potential and surfaceelement interaction matrices and circumvents theCoulomb singularity problem due to overlappingsurface elements represented by point charges. Here,rk is the coordinate of the surface element k, and úkis the Gaussian exponent that is adjusted to obtainthe exact Born ion solvation energy.

For the appearance and disappearance of surfaceelements to occur smoothly as a function of geo-metrical changes, a switching layer around each atomis introduced. The switching layer serves to “turn off”or “turn on” the surface elements associated withother atoms as they pass into or out of the layer.

Very recently, this approach has been extended tothe PCM suite of models by Scalmani and Frisch114

Sii ) k x4πai

(35)

Dii ) kx4πai

2RI(36)

Dii ) - (2π + ∑j*i

Dijaj)1

ai

(37)

qi(n) )[1

ai( 4πε - 1

+ zi[aj])]-1[-(En)i + zi

/[qj(n-1)]] (38)

zi/[xj] ) ∑

j*iDij

/xj

zi[xj] ) ∑j*i

Dijxj

(39)

φk(|rb - rbk|) ) (úk2

π )3/2

e-úk/2|rb-rbk|2 (40)


within the Gaussian code. In this new formalism theS and D matrices reported in Table 1 can beexpressed in terms of two-center, two-electron inte-grals and their derivatives, for example

where

2.3.1.6. More on the BEM. When one wants toextend the basic model to solutions with nonzero ionicstrength (i.e., to salt solutions), additional problemsappear. One way to treat these problems consists ofadopting other formulations of the BEM approachsuch as those reported here below. These methodscan be applied also to zero ionic strength, and for thisreason they are considered here, in a section of thereview dedicated to isotropic liquids without dis-persed ions. The basic electrostatic equations modifyas

where k accounts for the ion screening: its inverseis known as the Debye screening length (1/k), namely,k2 ) 8πe2I/εkT (I is the ionic strength ∑i zi

2ci/2 for adissolved salt containing two or more types of ion i,each having charge zie and an average bulk concen-tration ci).

Equation 43, generally known as the linearizedPoisson-Boltzmann (LPB) equation, is the result ofthe Debye-Huckel approximation applicable in thecase of low potentials, a condition approached at lowconcentrations. The LPB equation is obtained bytaking the linear term of the Taylor expansion of thehyperbolic sine function which, according to a treat-ment of statistical physics in the thermodynamicequilibrium approximation, represents the distribu-tion of the mobile ions in the field of the electrostaticpotential V.

The LPB equation can be solved analytically forsimple cavity shapes such as a sphere.115 However,for a general cavity of more complicated shape (suchas that adapted to the molecular solute), it must besolved numerically. Many different models have beendeveloped so far. In this section, we are interestedin the specific class of approaches, which determinesthe total electrostatic potential indirectly by usingBEM techniques. To better analyze this class ofapproaches it is useful to introduce some furtherinformation about BEM that has not been presentedyet.

To derive the BEM, one must replace the partialdifferential equation that governs the solution in adomain by an equation that governs the solution onthe boundary alone. There are two fundamentalapproaches to the derivation of an integral equationformulation of a partial differential equation. Thefirst is often termed the direct method, and theintegral equations are derived through the applica-tion of Green’s second theorem. The other method iscalled the indirect method and is based on theassumption that the solution can be expressed interms of a source density function defined on theboundary.

Among the indirect methods, we can, for example,mention the IEF and the parallel SS(V)PE ap-proaches described in the previous sections. Both ofthese two methods have been generalized to treat theLPB problem,75,116,117 and, as for the basic isotropicsolvent, only a single-layer (charge) distribution σ(s)is exploited. We note here that, for problems involv-ing ionic effects, the use of indirect approaches is newand thus not largely used until now (to the best ofour knowledge the original implementation of IEFin 1997 is the first example). A far more used BEMmethod is the direct one. To explain this method letus consider the electrostatic equations (42 and 43)governing the domain C bounded by the surface Γ.By applying Green’s second theorem, these equationscan be replaced by integral equations; for example,eq 43 becomes

where the function G is the Green function exp(-k|x- y|)/|x - y|.

The surface integral on the right-hand side of eq44 is called the single-layer potential, whereas theone on the left is called the double-layer potential.For this reason, the direct BEM approaches are alsoindicated as models employing both single-layer anddouble-layer distributions over the surface cavity.

The power of this formulation lies in the fact thatit relates the potential V and its derivative on theboundary alone; no reference is made to V at pointsin the domain. To numerically solve the integralequations of the type of eq 44, the first step is topartition the surface into surface elements (i.e., todefine an N-point grid on the surface).118-124 Trian-gular and quadrilateral elements are commonly usedfor this purpose. As a second step, the unknownfunctions V(r) and ∂V(r)/∂n are approximated bycontinuous trial functions over each boundary ele-ment. Generally, polynomials determined by theirvalues at the N grid points (the nodes) are used. Inthis way, the integrals become sums of integrals overelements, which results in a set of linear equationsfor the polynomial coefficients. This set of equationscan then be written as a single matrix equation ofdimension 2N.125,126 A fast multipole algorithm canbe used to reduce computational costs,107,127,128 or onecan apply an iterative procedure on a pair of coupledN × N finite matrix linear equations in everyiteration.129

Sij ) ⟨φi|φj⟩ )erf(úij|sbi - sbj|)

|sbi - sbj|

Dij )∂Sij

∂sbjnj )

(sbi - sbj)nj

|sbi - sbj|3 [erf(úij|sbi - sbj|) -

2xπ

úij|sbi - sbj| e-(úij|sbi-sbj|)2]

(41)

úij ) úiúj/xúi2 + új

2

-∇2V(rb) ) 4πFM(rb) within C (42)

- ε(∇2 - k2)V(rb) ) 0 outside C (43)

∫Γ

∂G(x,y)∂ny

V(y) dy + 12

V(y) ) ∫ΓG(x,y)∂V(x,y)

∂nydy

(44)


2.3.2. MPE MethodsThe seminal papers of Kirkwood115,130 and On-

sager131 have provided inspiration for various con-tinuum solvation models based on a multipole ex-pansion (MPE) of the solute charge distribution.Here, we present these models starting from thesimplest one (a dipole inside a sphere) and passingto those using multipole expansions and/or cavitiesof more complex form.

Appreciable popularity was gained in past yearsby the Onsager-SCRF code, elaborated by Wibergand co-workers132,133 for the Gaussian computationalcode.103 The Onsager model is the simplest versionof the MPE approach. Solvation is described in termsof a dipole moment, drawn in an iterative way fromQM calculations on the molecule. The appealingfeature of Onsager-SCRF was that it permitted oneto directly exploit almost all of the computationalfacilities of the Gaussian packages. For this reason,and for its very limited computational cost, it is stillin use by people not requiring an accurate descriptionof solvation effects but just a guess or a qualitativecorrection to the values obtained for the isolatedmolecule. Users must be aware of the limitations ofthe approach, of the unphysical deformation of thesolute charge distribution it may induce, and of othershortcomings specific of the approach, such as thelack of solvation for solutes with zero permanentdipole.

A more general model going beyond the dipoleapproximation is that developed by Mikkelsen andco-workers.134 This model starts from the idea pro-posed by Kirkwood to describe the interaction be-tween a set of classical charges (described in termsof a multipole expansion) enclosed in a sphericalcavity embedded in a structureless polarizable di-electric medium described by the macroscopic dielec-tric constant ε. Mikkelsen applies the model to QMcharge distributions instead of classical point charges,as in the original Kirkwood paper, and the effects ofthe solvent polarization are described in terms ofproper QM operators to be added to the Hamiltonianof the isolated system (i.e., adopting the same strat-egy introduced in section 2.3 to define the “effectiveHamiltonians”).

Even if the model is limited to spherical cavities,it presents very advanced features especially as faras concerns the QM aspects. The model has beenimplemented in the Dalton quantum chemistry pro-gram,135 and it has been used to study solvent effectson a large number of molecular response properties;it has also been generalized to several QM ap-proaches including multiconfigurational self-consis-tent field and coupled cluster methods136-143 (see alsosection 2.4.6 and section 6).

The most complete, and also the most largely usedMPE method, is that developed in Nancy by Rivailand co-workers.6,144 Historically, this method repre-sents the first example of QM continuum solvationmethods, but over the years it has been continuouslydeveloped, and it continues its evolution.145 In somecases the acronym SCRF (self-consistent reactionfield) is also used to indicate this specific method,even if it also identifies the larger class of solvation

methods implying the use of a reaction potential tobe self-consistently solved together with the solutecharge.

Here, we shall consider three steps in the evolutionof this MPE method: (i) a one-center expansion fora spherical cavity (in this case it coincides with thatdeveloped by Mikkelsen); (ii) its extension to ellipsoidcavities; and, finally, (iii) its extension to multicenterexpansion and to cavities of general shape.

(i) The expression of the multipole expansion forthe potential of a charge distribution placed into asphere of radius a, immersed into a continuum withdielectric constant ε, has been known for a long timeand can be found in numerous textbooks. We reportthe expression given by Bottcher146 for the potentialinside the cavity, namely

where the coefficients fl, called reaction field coef-ficients, have the following form:

Ylm(θ,æ) are the spherical harmonic functions,

whereas Mlm is a multipole moment of the solute

charge distribution. Note that an explicit use of theboundary conditions at r f ∞ (eqs 10 and 11) hasbeen made here. The second term of eq 45 representsthe solvent reaction potential acting on the portionof space occupied by the solute.

The QM reformulation of the problem given byRivail consists of using quantum mechanical ap-proaches to compute the elements of the multipoleexpansion of FM and in introducing the correspondingVr terms in the Hamiltonian to get a new solvent-modified FM, until convergence. This process, as inthe ASC methods, is of iterative nature.

(ii) Rivail and co-workers rapidly abandoned thespherical cavity of the first formulation of the modelto pass to ellipsoids. The shape of the cavity is acritical factor in continuum methods, and surely anellipsoidal cavity, with its variable parameters, ismore suited than a sphere to describe solvent effectson molecules. To do this extension, Rivail’s group wascompelled to develop a part of the mathematicaltheory on multipole expansions (in terms of el-lipsoidal harmonics), which was almost completelyneglected before. We shall use now the simplifiednotation introduced by Rivail and valid for bothellipsoids and spheres.

The solute-solvent interaction energy can be writ-ten in the form

where Rlm is the component of the reaction field

corresponding to multipole moment Mlm. There is a

V(r) ) VM + Vr ) ∑l)0

∞ 1

rl+1∑

m)-l

l

Mlm Yl

m(θ,æ) +

∑l)0

∞

flrl ∑m)-l

l

Mlm Yl

m(θ,æ) (45)

fl ) -(l + 1)(ε - 1)l + (l + 1)ε

1a2l+1

(46)

WMS ) ∑l

∑m

Mlm Rl

m (47)


linear relationship between each (lm) element of thecharge distribution expansion and the correspondingelement of the reaction potential, namely

where we have introduced the Einstein conventionof summation over the repeated indices. The coef-ficients fll′

mm′ are the generalization of the reactionfield coefficients fl defined in eq 46; we note, in fact,that for a sphere values of fll′

mm′ are no longer de-pendent on m and are nonzero only when l′ ) l. Also,in the more general case of an ellipsoid the coef-ficients fll′

mm′ have an analytical expression depend-ing only on the geometrical parameters of the cavityand on the dielectric constant. The Nancy MPE codesnow in use truncate the multipole expansion to lmax) 6. Only recently have results with a larger lmax beenpublished.55

(iii) The reaction field components, expanded overa set of centers I, J, ... have the following expression

whereas the analogue of eq 47 is

In this more general case of a multicenter expansion,the reaction factors fll′

mm′ do not have an analyticalform and have to be computed numerically using theclassical electrostatic conditions at the boundary ofthe cavity.

In the latest version of the Nancy MPE method,145

the ellipsoidal cavity has been abandoned, shiftingto a molecular-shaped cavity such as that used inPCM. For this cavity, the number of points selectedto fix the reaction factors is relatively large (usuallyof the order of 1500 or more), but this large amountof numerical values (quite simply individually com-puted, however) is efficiently compacted into linearequations, put into a matrix form, and solved withstandard methods. The large number of points en-sures rotational stability with respect to the selectionof sampling points, a problem that disturbs othersimilar fitting procedures, such as the definition ofthe ESP atomic charges, that is, the charges fittedon the electrostatic potential, and some early versionsof PCM.

Another point of general methodological interestin this latest version of the Nancy MPE code involvesthe selection of the multipole multicenter distribu-tion. Because the partition is arbitrary, there is aninfinite number of ways of defining the distribution,but they are not equivalent from a computationalviewpoint. The literature on intermolecular interac-tion is rich with proposals and debates about thechoice of the multipole distribution. Rivail and co-workers have selected and tested three differentdistributions. In the first, called GAD (Gaussianaveraged distribution), each multipole is distributed

over the N centers (the number of centers corre-sponds to the number of nuclei) with a Gaussianweight, with the translation expressed in terms ofsolid harmonics. In the second, called MSP (Mul-liken-Sokalski-Poirier), use is made, for the elec-tronic component, of the one-electron integrals of thepertinent øµ

/øv elementary charge distribution forthe multipole moment operator, namely

This expression is similar to that used in precedingone-center versions of the method. The zeroth ordermoment is taken equal to the Mulliken charge of theatom.

In the last expansion, called LSP, use is made ofthe Lowdin orthogonalization procedure applied toexpression 51.

All three expansions are convergent, but GAD andMSP have a better convergence, MSP being the lesstime-consuming. The solvation code is accompaniedby analytical derivatives with respect to the nuclearcoordinates,145,147 a feature essential for the perfor-mance of geometry optimizations in solution (seesection 6.1.1).

We conclude this section by noting that this recentre-elaboration of the Nancy MPE code shows thatprevious criticisms about the unreliability of MPEcodes for large molecules are not justified. Indeed,results are obtained that are of comparable qualityto those of ASC codes and still with comparablecomputational times.

2.3.3. Generalized Born (GB) ApproachesThe GB methods can be considered as an extreme

case of multicenter MPE, with multipoles truncatedat the first term, the charge, and the centers ofexpansion placed on all of the nuclei.

The name comes from an affiliation with the oldBorn model, which gives the interaction free energyof a spherical ion inside a dielectric148

where a is the sphere radius.The passage from a single sphere to a multiplicity

of interacting spheres, each with a charge at itscenter, required a considerable amount of ingenuity.Earlier versions of the GB approach were describedin the 1994 review, and we start here from thedefinition given by Still and co-workers149 of anempirical Coulomb operator 1/fGB that has to beapplied to all pairs qi and qj of atomic solute charges,namely, by generalizing the free energy (eq 52) to

The expression of fGB was not defined uniquely in this

Rlm) ∑

l′∑m′

fll′mm′ Ml′

m′ ) fll′mm′ Ml′

m′ (48)

Rlm(I) ) ∑

Jfll′mm′(I,J)Ml′

m′(J) (49)

WMS ) ∑I,J

Mlm(I)fll′

mm′(I,J)Ml′m′(J) (50)

Mlm(I) ) ∑

µ∈I∑ν∈I

Pµν⟨µ|Mlm|ν⟩ (51)

∆G ) - q2

2a(ε - 1ε ) (52)

∆G ) -(ε - 1

ε) ∑

i,j)1

qiqj

2fGB

(53)


seminal paper, but a simple effective expression wasgiven

where Rij ) (RjRj)0.5 is the geometrical mean of thepertinent pair of the so-called generalized Born radiiRi (which may be interpreted roughly as the distancefrom each atom to the dielectric boundary). Theexponent rij

2/2Rij)2 is a damping factor between thecharges placed on atoms i and j, leading smoothly tothe Born formula, when the spheres are separatedand placed at very large distances among them, andto the normal Coulomb operator, when two spheresmerge.150

We note that when eq 53 is applied to a two-dielectric system in which the set of charges occupiesa molecular cavity with a dielectric constant εpsurrounded by a uniform high-dielectric continuumenvironment with a dielectric constant εw, the pre-factor becomes (1/εp-1/εw). An internal dielectric ofεp ) 1 is appropriate for simulations when dipolefluctuations occur explicitly as part of the model. Alarger internal dielectric constant is more appropriatewhen the energies of minimized or averaged struc-tures are evaluated.

A fundamental problem arising in the GB ap-proach, and present in all of its successive formula-tions, is the definition of the appropriate Born radiiRi.151-155

For every charged atom i, the effective Born radiusRi is related to the effective Born free energy ofsolvation, ∆Gi, of a reference system through theBorn formula (eq 52), where the reference system is,by definition, the same as the original solute/continuum solvent system, except that in the refer-ence system atom i bears unit charge and all otheratoms are neutral. Exact calculation of Ri for everycharged atom in a macromolecule through numericalsolution of the Poisson-Boltzmann equation is time-consuming and of little practical use. The potentialof GB models has come from the progress in theformulation of approximate methods to compute Riwith sufficient accuracy and efficiency. An analyticalformula proposed by Qiu et al.156 forms the basis ofseveral recent parametrizations and applications ofthe GB model for protein systems.150,157-160 In Qiu’swork, the parameters contained in the formula havebeen optimized to minimize the differences betweenthe effective Born radii calculated by the finitedifference Poisson-Boltzmann (FDPB) method (seesection 2.3.5) and by formula 53. Besides the ap-proach of Qiu et al., Hawkins et al. have proposedanother approximate pairwise method to compute theeffective Born radius analytically.161,162

There are many different GB models, and still theyare the subject of active research, especially in thesimulation of macromolecules150,163-165 such as nucleicacids and proteins. Here, we obviously cannot men-tion all of them and analyze their specificities.Moreover, most of these models are formulatedwithin classical descriptions and thus are beyond thescope of the present section (see section 7.1 instead).

GB methods have been also successfully applied toQM descriptions of the solute.

The best known and more widely used QM GBmethod is that developed by Cramer and Truhlar inMinneapolis.166 This method has been elaborated anddistributed in a large variety of versions, with dif-ferent names. At present, these models are collectedunder the collective name SMx, where x stands foran alphanumeric code indicating the version, withits specific features (the most used recent codes areSM5.42R and SM5.43R), but in the past other nameswere used such as AMSOL. In one of the most recentpapers167 19 versions of the method are reported, butother versions are not present in this list.

The first versions known under the name AMSOLwere all developed for various semiempirical methods(AMSOL still refers to a semiempirical code168),whereas the SMx suite of programs permits ab initioQM calculations at several levels of the theory.169-172

Some SMx programs are available in the GAMESS-PLUS,173 which is an add-on module to theGAMESS,97,98 and in HONDOPLUS174,175 program.The latest version of the method is called SMx-GAUSS,176 and it may use Gaussian output files orbe connected with Gaussian 0317 to exploit the manyfeatures this last program contains. The objective ofmaking the SMx method more accessible to users isnot the only, nor the main, reason for so large anumber of versions. The real reasons will be shownlater in this subsection.

We have not yet examined how the solvationelectrostatic problem is linked to the QM one incontinuum methods. This will be done in detail insection 2.4, but it is convenient to anticipate heresome aspects to better present the SMx models.

The Fock operator of the solute in vacuo (forsimplicity we limit ourselves to the Hartree-Fockcase, even if SMx codes can treat higher levels of thequantum molecular theory) is modified by a solute-solvent interaction potential describing the electro-static interaction according to the GB model, that is,in terms of atomic charges qi and of the modifiedCoulomb operator (see eq 53). The qi charges arederived from the solute electronic wave function viaa population analysis performed in different ways inthe various versions of the method. First, Mullikenpopulations were used, then Lowdin charges, andfinally charges that according to the Cramer andTruhlar definitions are of classes III and IV. The classIV charges177 are derived from Mulliken NDDOcharges with a mapping to better describe physicalobservables. Energy minimization cycles in the so-defined Fock equation lead to self-consistent modi-fication of the set of qi charges and to the introductionof solvent polarization in the solute (see also section2.4.5 for further comments). The free energy of theelectronic polarization, called GP, can be so computed.

Another term of the free energy difference withrespect to the isolated molecule is also computed.This term, called ∆EEN, corresponds to the changein the electronic and nuclear energy of the soluteupon relaxation from the gas-phase geometry. Thesetwo terms are combined to give the electrostatic

fGB ) xrij2 + Rij

2 e-Dij (54)


contribution to the solvation free energy:

We recall that, to compute the complete solvation freeenergy Gsol, other nonelectrostatic contributions haveto be included in the model (see section 3). Thisextension has been done in the SMx set of codes byadding to the free energy of eq 55 further terms(usually indicated with the abbreviation CDS, forcavitation, dispersion, and solvent structural) pro-portional to the solvent-accessible surface area of thesolute.

In all stages of the evolution of the Cramer-Truhlar solvation code a complete parametrization,or reparametrization, has been performed on verylarge data sets, generally of some thousands ofsolute-solvent couples, with a coupled fitting forGCDS and ∆GENP. These parametrizations are quitespecific in the sense that for each solvent they arespecific for the type of QM calculation (AM1, PM3,HF, DFT, according to the functional, etc.), for thebasis set in ab initio methods, for the definition ofthe qi model (Mulliken, Lowdin, class III, class IV),for the parameters inside the coulomb operator 1/fGB) gkk′ for the Born radius and for the definition ofthe solvent accessible surface (an alternative defini-tion of SAS has been also given).178 In addition,reparametrization has been also performed for spe-cific classes of phenomena (solvent partition equilib-ria, solvatochromic effects, etc.). This remarkableattention to the fitting procedures explains the largenumber of different versions, each about a specificcombination of the above-mentioned parameters ofthe computation. We note that the accurate docu-mentation available for the users is a fundamentalhelp for selecting the version of the programs bestsuited for a specific study.

2.3.4. Finite Element (FE) and Finite Difference (FD)Methods

The integral equation we encounter in solvationelectrostatics can be solved with the FE method,which makes use of numerical solutions of theintegral over the whole space, divided into a numberof finite volumes, or domains, here called elements.The philosophy is similar to that of BEM, with thedifference being that the reduction to surface inte-grals is not used here. FEM is older than BEM, andfor many years it has been the approach more usedin computational engineering and physics. It contin-ues to be amply used, and for FEM (as we remarkedfor BEM) there is available a large wealth of elabora-tions that could be profitably exploited in computa-tional chemistry.

Nowadays, FEM is widely used in many fields,such as physics and engineering, but there are alsoapplications in computational chemistry (and inparticular in the study of solvation); the best knownexamples are the method elaborated by Friesner’sgroup179-181 and implemented in the integrated suiteof programs Maestro,182 with the acronym PBF, andthe adaptive Poisson-Boltzmann solver (APBS) byMcCammon and co-workers.183

In particular, in the PBF method, the discretizationof the linearized Poisson-Boltzmann (PB) problem(eq 43) is performed by expanding the electrostaticpotential in a linear combination of basis functionsand sampling the three-dimensional space over whichthe calculation is to be carried out with a set of gridpoints such that there is a one-to-one correspondencebetween the set of basis functions and the set of gridpoints. The coefficients of the expansion are thevalues of the potential at the grid vertices. Thismethod is equivalent to that described in section2.3.1.6, but this time the grid is over the three-dimensional space and not limited to a surface.

The finite elements are defined using a Lebedevquadrature grid combined with an accurate definitionof the cavity including the re-entrant portions of thesolvent accessible surface, or SAS (see section 2.2).The finite elements are defined in an adaptive wayusing a triangulation front-marching algorithm. Theuse of this three-dimensional mesh leads to a sparsesystem of equations that are solved iteratively. Wenote that the density of the corresponding matrixdepends on the connectivity of the mesh, and in thisthree-dimensional triangulation this connectivity islocal; that is, a grid vertex is coupled only to itsneighbors. As a result, the system of equations tosolve is sparse. This contrasts with boundary-ele-ment-based approaches in which the reduction indimensionality achieved results in every surfaceelement interacting with every other surface element.The resulting system of equations is therefore alwaysdense.

The method used for calculating the polarizationcontribution to the solvation energy starts from theequation

where Γ is the SAS, Rion3 the ion accessible region,

Fion is the density of free ions in the solvent, and V0

is the electrostatic potential produced by the fixedcharge distribution of the solute. The polarizationcharge σ is calculated using the discontinuity of theelectric field at the dielectric interface and thepolarization vector, namely

We note that when the contribution of the free ionterm to the polarization energy is extremely smallcompared to the reaction field term, the methodcoincides with the DPCM method we have describedin section 2.3.1.1. In the more general case, however,the volume term remains, and the finite elementdiscretization provides a simple scheme for evaluat-ing this term by Gaussian quadrature.

In the numerical solution of integral equations bothBEM and FEM approaches have advantages anddisadvantages, but for the specific case we are nowconsidering there are no doubts: BEM is superior forsystems characterized by three-dimensional homo-geneous domains (especially if infinite or semi-infinite), and this is exactly the case of dilute

∆GENP ) ∆EEN + GP (55)

Gpol ) 12 ∫Γ σV0 d3rb + 1

2 ∫Rion3 V0 d3rb (56)

σ(rb) ) 14π

(∇BVi0 - ∇BVe

0)‚nb (57)


solutions. We give here some reasons supporting thisstatement.

BEM is capable of giving values of the unknown(in our case the sources of the reaction electrostaticpotential) in the interior of the domain in a pointlikeform, thus avoiding the interelement continuityproblems that disturb FEM applications to solvation.The reduction of the spatial problem from three totwo dimensions greatly facilitates mesh refinementstudies, and it leads to a system of algebraic equa-tions much smaller than in an equivalent FEMformulation. This advantage is partly compensatedby the sparcity of FEM linear equations, which isexploited in the PBF method described above.

Comparison between the two approaches could leadto different conclusions in some specific solvationcases. BEM is less efficient than FEM for systemswith one or two spatial dimensions extremely smallwith respect to the others but dimensionally effective,as is the case of bicontinuous liquid systems. BEMhas difficulties in treating problems exhibiting rapidchanges in the physical properties of a domain. Thislast problem has been excellently solved in solvationchemistry for the inhomogeneities and anisotropiesof the molecules inside the cavity, but, for example,it remains a problem for the solvent anisotropies dueto nonlinear dielectric saturation.

Although BEM and FEM aim at giving a numericalsolution of the integral equations, the FD methodsaim at solving the differential equations. The dif-ferential equation to solve is, once again, the basicelectrostatic PB problem (eq 43); we note that, withFD methods, also the nonlinear version of such anequation can be solved.

The FD methods have a long history in continuumsolvation, and they are very popular, especially inclassical version making no explicit use of QMmethods.184-186

In these approaches, a discrete approximation tothe governing partial PB differential equations isobtained through a volume-filling grid. Ideally, aboundary-conforming grid (i.e., one that does notintersect the molecular surface) is preferred, but sucha grid is difficult to generate for a complex molecularshape. Therefore, in most implementations, such asthe widely used UHBD187 (University of HoustonBrownian Dynamic)188-190 and DelPhi191 code (nowincluded in Insight II package),192,193 a regular latticeis laid over the molecule and cells are allowed to goover the molecular surface. A regular lattice arrange-ment is also useful for an efficient multigrid solutiontechnique for solving the algebraic equations result-ing from the discretization of the partial differentialequations.194

A further FDM approach has been developed byBashford, and it has been implemented in a compu-tational code known as MEAD (macroscopic electro-statics with atomic detail).195 The MEAD electrostaticmodel has been coupled with QM techniques: aninner region containing the active-site atoms istreated quantum mechanically, whereas the sur-rounding region is treated as a classical electrostaticsystem that generates a reaction field and, possibly,a field due to permanent atomic charges of the

protein. This technique has been used to calculateredox properties196 and pKa values.197

There are many similarities between FE and FDmethods. In both cases, a grid of points covering thespace is defined. In FE procedures, these points haveto be considered as representative points of a finitethree-dimensional domain, and in the FD procedures,nodes in which the differential equation has to besolved. In addition, the finite element method differsfrom the finite difference approach in that variationalprinciples rather than finite difference approxima-tions are used to derive the discrete equations.198,199

A major reason for adopting a finite element ap-proach is that it accommodates unstructured grids,which offer improved geometric flexibility and vari-able mesh spacing, albeit at considerably higher per-node storage and CPU costs than a regular latticearrangement.

2.4. Solution of the Quantum Mechanical ProblemAs described in section 1, the Hamiltonian appear-

ing in the Schrodinger equation of the basic modelis the effective Hamiltonian of eq 3, namely

This effective Hamiltonian is composed by two terms,the Hamiltonian of the solute HM

0 (i.e., the focusedpart M of the model) and the solute-solvent interac-tion term Vint (i.e., the solvent reaction potential);

The definition of Vint depends on the method em-ployed to set the electrostatic problem, which has tobe solved within the framework of the QM eq 58.

To describe this procedure in more detail let us firstconsider an intuitive formulation of the problem.

2.4.1. Intuitive Formulation of the ProblemBeing interested in a QM problem, let us introduce

the currently used Born-Oppenheimer (BO) ap-proximation, which relies on a partition of the solutevariables into electronic and nuclear coordinates.

The solute charge distribution is convenientlydivided into electronic and nuclear components FM-(r) ) FM

e (r) + FMn (r). We recall that the QM procedure

does not affect the nuclear component of FM, althoughit will modify the electronic component with respectto a starting provisional definition. This modificationis done in an iterative way under the action of Vint,which is in turn modified in the iterative cycle.

Let us look now at the structure of Vint in eq 59.This operator can be divided into four terms havinga similarity with the two-, one-, and zero-electronterms present in the solute Hamiltonian.

To show it in an intuitive way, we consider thesolute-solvent interaction energy Uint given as theintegral of the reaction potential times the wholecharge distribution FM.

The interaction potential has, as sources, the twocomponents of FM, and thus it is composed of twoterms, one stemming from the electronic distributionof the solute M and one from its nuclear distribution.

Heff|Ψ⟩ ) E|Ψ⟩ (58)

Heff ) HM0 + Vint (59)


Uint can thus be partitioned into four terms

where Uxy corresponds to the interaction energybetween the component of the interaction potentialhaving as source FM

x (r), namely, Vintx , and the charge

distribution FMy (r).

As in all QM applications to molecular systems, thesolution of the Schrodinger eq 58 is based on anexpression of the unknown wave function Ψ in termsof molecular orbitals expressed over a finite basis setø. Within this framework, it is convenient toconsider the FM

e charge distribution expressed assums of contributions due to the elementary chargedistributions ø/µøv, whereas FM

n (r) is

Following this formalism, three different QM op-erators appear, namely, Vnn, Vne (it may be shownthat Une and Uen are formally identical), and Vee.These correspond to zero-, one-, and two-electronoperators appearing in HM

0 , respectively. We notethat the zero-order term gives rise to an energeticcontribution Unn, which is analogous to the nuclei-nuclei repulsion energy Vnn and, thus, it is generallyadded as a constant energy shift term in HM

0 . Theconclusion of this analysis is that we have intuitivelydefined four operators (reduced in practice to two,plus a constant term) which constitute the operatorVint of eq 59.

Before proceeding further in this presentation, wenote that this analysis has a direct link with the ASCapproaches, but it is valid also for the MPE methods,with minor changes, and for the FE and FD methods,as well. Note that the QM versions of the last twoapproaches describe the interaction potential interms of cavity apparent surface charges, and thusthey exploit the same features of ASC methods. Bycontrast, for GB methods the formal setting is dif-ferent. These methods reduce, in fact, the solutedensity function to atomic charges that are treatedin terms of the generalized Coulomb operators wehave already introduced (see section 2.2.3).

2.4.2. Electrostatic OperatorsThe next step consists of solving the QM problem

with the determination of the electrostatic interactionoperators nested within it. It is convenient now tomake explicit use of the apparent surface charges,because they simplify the exposition; as stated above,this does not represent a real restriction as almostall of the QM solvation methods follow the same basicpattern.

The most naive (and transparent) formulation ofthe process of mutual interaction between real andapparent charges is that used in the first version48

of DPCM. We recall it here, even though it wasalready presented in the 1994 review, as helpful inthe understanding of the basic aspects of the mutualpolarization process.

One starts from a given approximation of FMe (let

us call it FM0 ) that could be a guess, or the correct

description of FMe without the solvent, and obtains a

provisional description of the apparent surface chargedensity, or better, of a set of apparent point chargesthat we denote here qk

o,o. These charges are notcorrect, even for a fixed unpolarized description ofthe solute charge density, because their mutualinteraction has not been considered in this zero-orderdescription. To get this contribution, called mutualpolarization of the apparent charges, an iterativecycle of the PCM eq 20 (including the self-polarizationof each qk) must be performed at fixed FM

0 . The resultis a new set of charges qk

o,f, where f stands forfinal. The qk

o,f charges are used to define the firstapproximation to Vint, and a first QM cycle is per-formed to solve eq 58. With the new FM

1 the innerloop of mutual ASC polarization is performed, againgiving origin to a qk

1,f set of charges. The procedureis continued until self-consistency.

We remark that, in this formulation, we havecollected into a single set of one-electron operatorsall of the interaction operators we have defined inthe preceding section, and, in parallel, we have putin the qk set both the apparent charges related tothe electrons and nuclei of M. This is an apparentsimplification as all of the operators are indeedpresent.

It is interesting here to note that this nesting ofthe electrostatic problem in the QM framework isperformed in a similar way in all continuum QMsolvation codes, including GB methods, in whichthere is an iterative updating of the atomic chargesduring the QM cycle.

We report here the Schrodinger equation of theASC version of the basic model with the introductionof a new formalism to make the exposition moregeneral:

With the superscript R we indicate that the corre-sponding operator is related to the solvent reactionpotential and with the subscripts r and rr′ the one-or two-body nature of the operator. The conventionof summation over repeated indexes followed byintegration has been adopted. The emphasis givenin this formula to the density operators is related toa formal problem arising in focused models: the maincomponent of the system, M, interacting with theremainder does not have a wave function satisfyingall of the rules of canonical QM. Only the densityfunction has a more definite status (see ref 200 forcomments on this point). For this reason we use onlydensity operators and expectation values in expres-sion 62.

The Fre Vr

R operator describes the two componentsof the interaction energy we have previously calledUen and Une. In more advanced formulations ofcontinuum models going beyond the electrostaticdescription, other components are collected in this

Uint ) Uee + Uen + Une + Unn (60)

FMn ) ∑

R

nucl

ZRδ(r - RR) (61)

Heff|Ψ⟩ ) [HM0 + Fr

e VrR + Fr

e Vrr′R ⟨Ψ|Fr′

e |Ψ⟩]|Ψ⟩ )E|Ψ⟩ (62)


term as we shall show in section 3. VrR is sometimes

called the solvent permanent potential, to emphasizethe fact that in performing an iterative calculationof |Ψ⟩ in the BO approximation this potential remainsunchanged.

The FreVn′

R⟨Ψ|Fre

′|Ψ⟩ operator corresponds to theenergy contribution, which we previously called Uee.This operator changes during the iterative solutionof the equation. Vn′

R is said to be the response func-tion of the reaction potential. It is important to notethat this term induces a nonlinear character to eq62. Once again, in passing from the basic electrostaticmodel to more advanced formulations, other contri-butions are collected in this term as we shall showin section 3.

The constant energy terms corresponding to Unn

and to nuclear repulsion are not reported in eq 62.Following a historical hierarchy to get molecular

wave functions, we introduce here the Hartree-Fock(HF) approach. The generalization to the densityfunctional (DF) approach is straightforward; the onlything to add is the proper exchange correlation terminsteadsor together, if specific hybrid DFs are usedsof the HF one. Extensions to post-HF methods willbe described in section 2.4.5.

In this framework we have to define the Fockoperator for our model. As before, we adopt here anexpansion of this operator over a finite basis set ø,and thus all of the operators are given in terms oftheir matrices in such a basis.

The Fock matrix reads

The first two terms correspond to HM0 , the third to Fr

e

Vn′R, and the last to Fr

eVn′R⟨Ψ|Fr

e′|Ψ⟩.

We take for granted the reader’s familiarity withthe standard HF procedure and formalism. We recallthat all of the square matrices of eq 63 have thedimensions of the expansion basis set and that P isthe matrix formulation of the one-electron densityfunction over the same basis set. According to thestandard conventions P has been placed as a sort ofargument to G0 to recall that each element of G0

depends on P. For analogy, we have made explicit asimilar dependence for the elements of XR.

We remark that also the standard HF equation isnonlinear in character and that in the elaboration ofthis method its nonlinearity is properly treated. Thenew term XR(P) adds an additional nonlinearity ofdifferent origin but of similar formal nature, whichhas to be treated in an appropriate way. This factwas not immediately recognized in the old versionsof continuum QM methods, giving origin to debatesabout the correct use of the solute solvent interactionenergy. This point will be treated later, in section2.4.4.

Notice that, as in the previous analysis of theSchrodinger eq 58, in the Fock matrix expression (eq63) we have used a single term to describe the one-electron solvent term. We remark, however, that inthe original formulation two matrices, jR and yR, were

used, namely

In both expressions the summation runs over all ofthe tesserae (each tessera is a single site whereapparent charges are located), Vµν(sk) is the potentialof the øµ

/øν elementary charge distribution computedat the tessera’s representative point, Vn(sk) is thepotential given by the nuclear charges, computedagain at the same point, qn(sk) is the apparent chargeat position sk deriving from the solute nuclear chargedistribution, and qµν

e (sk) is the apparent charge atthe same position, deriving from øµ

/øν. The two ma-trices (eqs 64 and 65) are formally identical, as saidbefore, and thus in eq 63 we have replaced them withthe single matrix

We note that in computational practice, the morecomputationally effective expression (eq 64) is gener-ally used.

The elements of the second solvent term in theFock matix (eq 63) can be put in the form

with

In this way, we have rewritten all of the solventinteraction elements of the Fock matrix in terms ofthe unknown qe and qn apparent charges (the last,not being modified in the SCF cycle, can immediatelybe separately computed).

2.4.3. Outlying ChargeIn the previous sections we have introduced a

phenomenon always present when we use a QMdescription of the solute charge density, namely, thephenomenon we have previously designated “es-caped” or “outlying” charge. This is due to the factthat a portion of the electronic charge of the soluteis always lying outside any cavity of reasonable size.

In all methods that exploit cavities and boundariesto separate solute and solvent domains and QMcharge distributions to describe the solute, this effectshould be taken into account, but it becomes particu-larly significant in ASC methods, or better in theolder formulations of the ASC methods. Let us tryto better understand this statement.

ASC methods based on the standard definition ofthe apparent surface charge σ in terms of thepolarization vector and thus of the normal componentof the electric field at the boundary Γ (see eqs 19 and

jµνR ) ∑

kVµν(sbk)q

n(sbk) (64)

yµνR ) ∑

kVn(sbk)qµν

e (sbk) (65)

hR ) 1/2(jR + yR) (66)

XµνR ) ∑

kVµν(sbk)q

e(sbk) (67)

qe(sbk) ) ∑µν

Pµνqµνe (sbk) (68)

F ) h0 + G°(P) + hR + XR(P) (63)


20) should, in fact, satisfy the Gauss theorem, namely

where QM is the total charge of the solute (i.e., zerofor neutral solutes). This equation is indeed a verysimple and direct numerical test to check the amountof the outlying charge; in fact, if the sum of theapparent point charges, that is, the discretized formof the left-hand integral above, does not fulfill eq 69,then we can assume that the discrepancy is due tothe portion of the electronic charge of the solute lyingoutside the cavity.

Since the very first formulation of PCM (i.e., theD version of this model described in section 2.3.1.1),this possible discrepancy was recognized and strate-gies were introduced to correct the apparent chargesso that eq 69 was satisfied. These strategies werecalled “renormalization of the apparent charge”, andthey simply consisted in spreading an additional ∆q′charge over the tesserae, with a simple recipe tryingto respect the correct spatial distribution of thesurface charges (in a dipolar molecule, for example,there are positive and negative regions of the appar-ent charge). The various formulas corresponding tothe different strategies can be found in the 1994review, in which some typographical errors of theoriginal PCM paper48 were corrected.

For several years this renormalization was consid-ered to be sufficient, taking into account the moderatequality of the ab initio calculations possible at thattime. Variants were elaborated by others, but pre-sented no advantages (the interested readers mayfind a synopsis in the 1994 review). Things changedin the 1990s: the availability of more powerfulcomputers and progress in ab initio methods madeit possible to extend the QM solvation codes to moreambitious goals, in particular, geometry optimiza-tions and calculation of molecular response properties(see section 6). To reach these goals, there is the needof accurate codes for the analytic evaluation ofderivatives with respect to the nuclear coordinatesas well as to external parameters: both types ofderivatives are sensitive to the outlying charge.Therefore, the research on this subject acquired amore oriented and practical objective.

To have an appropriate perspective of the outlyingcharge problem, it is convenient to compare it withthat of ionic solutions. In both cases there is a realcharge distribution out of the cavity, but the ioncharge distribution is distributed in the whole me-dium, and it has a zone of zero density just aroundthe cavity surface, whereas the solute electronic“tails” rapidly decrease from the cavity surface goingto larger distances and they give origin to a volumepolarization effect limited to a small region encirclingthe solute. In general, this density is anisotropic, inthe sense that its intensity (and its decaying withthe distance) changes in different portions of thecavity surface, being strictly related to the chargedistribution of the nearby solute’s groups. The prob-lem of the outlying charge, and of the connected

volume polarization, appears to be more delicate totreat than that of ionic solutions.

In addition to this electronic outlying charge, thereare other possible sources for the discrepancy of theapparent charges with respect to the Gauss law (eq69). These are related to the implementation of thecomputational procedure, in particular, the numberand relative positions of the sampling points, and tothe expressions used to define the mutual polariza-tion of the apparent charges. For this reason, in 1995the two components of the solute charge distribution,electrons and nuclei, started to be considered as twodifferent sources of apparent charges, which in turnwere separated into two sets, each one checked withthe proper Gauss law.50 Nuclei, in fact, are surelywithin the cavity, and they do not exhibit outlyingcharges, so in this case the discrepancy has to beattributed to the numerical aspects. Actually, wefound that such discrepancy for the nuclei-inducedapparent charges depends on the accuracy in thetessellation (density of points) and on the localgeometry of the surface. The main source of theseerrors could be eliminated by using denser meshes,and in fact in the most recent versions of the ASCmethods (such as those implemented in the Gauss-ian103 suite of programs) the number of tesserae hassignificantly increased with respect to the olderversions with a parallel increase in the accuracy ofthe results: now the nuclear contribution to the erroron the apparent charges is much smaller than theelectronic one. Obviously, the same numerical errorsoccur also in the electronic apparent charges, and soa new renormalization scheme of the apparent chargewas introduced, in which two distinct correctionswere used for nuclei-induced and electron charges.50

This improvement in renormalization, however,was not sufficient to completely eliminate the un-wanted effects due to outlying charge. A direct attackto the problem, without resorting to empirical cor-rections and based on the integration of the volumepotential, was undertaken by the PCM group, byKlamt, and by Chipman.

Within the DPCM approach, an application of theBEM procedures to build up a double-charge modeltaking into account also the outlying charge wasexplored by two of the present authors.86 Applyingelectrostatic laws, the presence of the outlying chargeshould give origin to an apparent (or bound) chargedistribution Fb in the volume of the dielectric

where FM is the real charge distribution out of thecavity. For what was said above about the fast decayof FM, to get a good representation of Fb one shoulduse a very refined partition of the volume with anextremely fine grid of points. To overcome thisdifficulty, we resorted to a mathematical treatment.As here, it is important to describe the effects thatthe apparent volume charge exerts on the solute, thatis, the values of the corresponding reaction potentialinside the cavity; then, such reaction potential canbe mimicked by that due to an additional apparentsurface charge. By using the discretization procedure

∫Γ σ(sb) dsb ) - ε - 14πε

∫Γ EB(sb)‚nb(sb) dsb ) - ε - 1ε

QM

(69)

Fb(rb) ) - ε - 1ε

FM(rb) for r outside C (70)


used in standard ASC methods, this new procedureleads to a second set of charges, placed at the samepoints as the original ones. The dimension of theproblem is left unchanged, because the two sets ofsurface charges are put on the same locations, andtheir sum can be used in defining the reactionpotential.

This method has been considered by some as anovel renormalization procedure, but actually it isbased on a development of the electrostatic modelreleasing one of the constraints we set at the begin-ning of this section and giving for the added feature,the outlying charge, a treatment in line with theBEM.

A parallel approach was proposed by Klamt andJonas201 within the COSMO model: after a normalCOSMO calculation an additional cavity is con-structed around the solute molecule lying at anappropriate distance further from the original one.By calculating the potential arising on the segmentsof this outer cavity from the solute and the COSMOscreening charges on the inner cavity, a further setof charges is obtained and added to the primary onesto get the correct solute-solvent interaction energy.

All of these approaches were subsequently over-come by the elaboration of the IEF method (seesection 2.3.1.3), which opened new perspectives forthe description of outlying charge effects. This aspectof IEF was clearly expressed only in 2001202 spurredby the important analysis performed by Chipmanstarting from 1997.81-84 In particular, in an articleappearing in 2000 in the Journal of Chemical Phys-ics, Chipman203 proposed and examined a new trac-table formulation to get an accurate approximationof volume polarization effects arising from the es-caped charge without making a calculation in theouter volume. Following a strategy similar to thatin the method described above,86 his proposal was toadd to the standard ASC density an additionalsurface charge density giving rise, inside the cavity,to an electrostatic potential approximately equivalentto that arising from the apparent (or bound) chargeFb(r). In this way, as most of the solute electrons areactually confined inside the cavity, the most impor-tant effects of the volume polarization are kept, andthe computational effort is limited because onlysurface charges are used. This proposal was trans-lated into a model, namely, the SS(V)PE model wehave described in section 2.3.1.4.

As said before, the SS(V)PE and IEF models (whenrewritten in terms of the potential only) are formallyequivalent; the analysis started by Chipman, andcontinued by Cances and Mennucci, has clearlyshown that both of these models, contrary to DPCMor other ASC methods, take into account the effectsarising from the outlying charge. Besides the obviousgreater accuracy in the representation of the reactionfield, this specificity has also an immediate practicalconsequence, namely, that IEF and SS(V)PE do notneed any renormalization of the surface charge.

2.4.4. Definition of the Basic Energetic Quantity

To use the solution of the QM equations in con-tinuum solvation methods, some important formal

aspects have to be reconsidered. To better understandthese specificities, it is convenient to start from abrief summary of what is done in QM calculations ofan isolated molecule.

The Fock operator

is used to determine the function |ΨHF0 ⟩ correspond-

ing to the variational approximation to the exactwave function for the system specified by H0. Thiscorrespondence is determined by minimizing theappropriate energy functional E[Ψ], namely

or, in a matrix form

where we have used the same formalism used insection 2.4.2 and we have introduced the traceoperator (“tr”). Obviously, the nuclear repulsionenergy, Vnn, in the BO approximation is a constantfactor. We remark that in eq 73 there is a factor of1/2 in front of the two-electron contribution. Thisfactor is justified in textbooks by the need to avoid adouble counting of the interactions, but this doublecounting has its origin in the nonlinearity of the HFequation.

Let us now pass to continuum models. As before,also here the new Fock operator defined in eq 63determining the new solution |ΨHF

S ⟩ is obtained byminimizing the appropriate functional. However,now, the kernel of this functional is not the Hamil-tonian Heff given in eq 59 but rather Heff - Vint/2, andthus the energy of the system is given by

that, expressed in a matrix form similar to that usedfor EHF

0 , reads

where the solvent matrices, j, y, and X, are thosedefined in section 2.4.2 (here we have only droppedthe “R” superscript).

We have added half of the solute-solvent interac-tion term related exclusively to nuclei, which in theBO approximation is constant.

Similar expressions hold for other levels of the QMmolecular theory: the factor 1/2 is present in all casesof linear dielectric responses; for this reason, in thefollowing, we shall drop the HF index.

The reason for the change of the functional we haveintroduced lies in the nonlinear nature of the effectiveHamiltonian. This Hamiltonian has in fact an explicitdependence on the charge distribution of the solute,expressed in terms of FM

e , which is the one-body

F0 ) h0 + G0(P) (71)

EHF0 ) ⟨ΨHF|H0|ΨHF⟩ (72)

EHF0 ) tr Ph0 + 1

2tr PG0(P) + Vnn (73)

GHFS ) ⟨ΨHF|Heff - 1

2Vint|ΨHF⟩ (74)

GHFS ) tr Ph0 + 1

2tr PG0(P) + 1

2tr P(j + y) +

12

tr PX(P) + Vnn + 12

Unn (75)


contraction of |ΨHFS ⟩⟨ΨHF

S |, and thus it is nonlinear.It must be added that this nonlinearity is of the firstorder, in the sense that the interaction operatordepends only on the first power of FM

e . The model isin fact based on an approximation motivated byphysical considerations. The solvent reaction fieldhas, in general, a relatively low strength, and thus alinear response regime is sufficient to describe itseffects. For this reason, the expansion of the interac-tion potential in powers of the reaction field is limitedto the first order in almost all of the continuummethods. In other words, it may be said that thesemethods treat the linear aspect of a nonlinearproblem. True nonlinear solvation effects are, how-ever, present. Generally, they are treated in otherways, for example, making reference to phenomeno-logical concepts, such as dielectric saturation orelectrostriction, and directly using higher terms inexpansion of the interaction potentials (more detailson these aspects will be treated in section 4).

Some comments about nonlinearities in the Hamil-tonian are here added. This subject is rarely treatedin an explicit way in quantum chemistry, despite thelarge use of the Fock operator, which is nonlinear,and, in the case of ionic solutions, of the use of thePoisson-Boltzmann approach, which also leads to anonlinear problem. The case we are considering hereis called scalar nonlinearity (in the mathematicalliterature it is also called “nonlocal nonlinearity”):204 this means that the operators are of the form P(u)) ⟨Au,u⟩Bu, where A and B are linear operators and⟨.,.⟩ is the inner product in a Hilbert space. Theliterature on scalar nonlinearities applied to chemicalproblems is quite scarce (we cite here a fewpapers7,205-210), but the results justified by this ap-proach are of universal use in solvation methods.

Let us now come back to eq 74. Attentive readerswill have noticed that we have changed the symbolfor the energy. The symbol G emphasizes the factthat this energy has the status of a free energy. Theexplicit identification of the functional (eq 74) withthe free energy was first done by Yomosa,205 to thebest of our knowledge. In the 1994 review alternativejustifications for the factor 1/2 in the expression of theenergy were given, deduced from the perturbationtheory, by statistical thermodynamics and by clas-sical electrostatics, all valid for a linear response ofthe dielectric. We report here only a considerationbased on classical electrostatics. Half of the workspent to insert a charge distribution (i.e., a molecule)into a cavity within a dielectric corresponds to thepolarization of the dielectric itself, and it cannot berecovered by taking the molecule away and restoringit in its initial position. This half of the spent workis irreversible, and it has to be subtracted from theenergy of the insertion process to get the free energy(or the chemical potential).

2.4.5. QM Descriptions beyond the HF Approximation

In the past few years, a large effort has beendevoted to extending solvation models to QM tech-niques of increasing accuracy. In this way, modelsoriginally limited to the Hartree-Fock level can benow used in many post-SCF calculations. This com-

putational extension has been accompanied by areformulation of the various QM theories so as toinclude the specifics of the solvation model. Such arequirement is particularly pressing for those modelsin which an effective Hamiltonian is introduced.Electron correlation is more commonly introducedinto these techniques using CI or MCSCF and DFTmethods.

DFT does not require any specific modification withrespect to the formalism presented in the previoussection, as far as concerns the solvation terms, andthus there are numerous examples of DFT applica-tions of continuum solvation models. As a matter offact, continuum solvation methods coupled to DFTare becoming the routine approach for studies ofsolvated systems.

Applications of continuum solvation approaches toCI or MCSCF wave functions have quite a longhistory. In 1988 Mikkelsen et al.134 presented aMCSCF version of their multipole approach (seesection 2.3.2) first limited to equilibrium solvationand then extended to nonequilibrium descriptions137

(see section 5) and to response methods136 (see section6). A similar history is presented by the family ofPCM approaches, which have been first applied toequilibrium211 (including gradients212) and nonequi-librium78,213,214 and subsequently to the responsemethod.215

Also, configuration interaction (CI) approachessoon attracted the interests of researchers workingin the field of solvation methods.205,216-218 In this case,however, some delicate issues not present for isolatedsystems appear.

As remarked by Houjou et al.,219 an important pointto analyze is the physical meaning given to theexcitation energy of solvated systems when obtaineddirectly from the CI calculation. Karelson and Zern-er220 have indicated that the solvent effect is incor-porated ‘‘automatically” into the CI only by addingsolute-solvent interaction term(s) to the Fock matrixfor the in vacuo state, whereas the resulting excita-tion energy must be corrected for the electronicrelaxation and for the costs required for the polariza-tion of solvent. Similarly, Klamt derived the correc-tion term corresponding to the electronic relaxation,denoted ‘‘electronic screening correction” in ref 221,and the diagonal correction in the CI matrix.

In addition, the nonlinear dependence of the sol-vent reaction field operators on the solute wavefunction introduces a new element of complexity. Ina CI scheme, this nonlinearity is solved through aniteration procedure; at each step of the iteration, thesolvent-induced component of the effective Hamilto-nian is computed by exploiting the first-order densitymatrix of the preceding step. The main consequenceof a correct application of this scheme is that eachelectronic state, ground and excited, requires aseparate calculation involving an iteration optimizedon the specific state of interest. This procedure hasbeen adopted by Mennucci et al.78 in their IEFPCM-CI approach as well in the successive extension tomultireference CI wave functions (including pertur-bative corrections)222 within the framework of theCIPSI method.223-225


Other examples of correlated ab initio solventmethods have employed Møller-Plesset (MP2) orcoupled cluster (CC) theory for the description of theelectronic structure of the solute.

Perturbation theory within solvation schemes hasbeen originally considered by Tapia and Goscinski7-at the CNDO level. An ab initio version of theMøller-Plesset perturbation theory within the DPCMsolvation approach was introduced years ago byOlivares et al.226-231 following some intuitive consid-erations based on the fact that the electron correla-tion which modifies both the HF solute chargedistribution and the solvent reaction potential de-pending on it can be back-modified by the solvent.To decouple these combined effects, the authorsintroduced three alternative schemes: (1) the non-iterative “energy-only” scheme (PTE), in which thesolvated HF orbitals are used to calculate MPncorrelation correction; (2) the density-only scheme(PTD), in which the vacuum MPn density matrix isused to evaluate the reaction field; and (3) theiterative (PTED) scheme, in which the correlatedelectronic density is used to make the reaction fieldself-consistent. With regard to the last scheme, somefurther comments are required. The PTED schemeleading to a comprehensive description of the effectsseparately considered by PTD and PTE is rathercumbersome and not suited for the calculation ofanalytical derivatives, even at the lowest order of theMP perturbation theory. In addition, it has beenshown that a rigorous application of the perturbationtheory, in which the nth-order correction to theenergy is based on the (n - 1)th-order density, thecorrect MP2 solute-solvent energy has to be calcu-lated with the solvent reaction field due to theHartree-Fock electron density.232,233 This means thatthe PTED scheme at the MPn level is not in analogywith standard vacuum Møller-Plesset perturbationtheory as terms of higher than nth order are in-cluded.

Other MP2-based solvent methods consist of theMP2-PCM,234 the Onsager MP2-SCRF,132 and a mul-tipole MP2-SCRF model,235 Subsequently, still in thePCM framework, MP2 gradients and Hessians havebeen developed.234,236

In recent years CC methods have proven to beamong the most successful electronic structure meth-ods for molecules in a vacuum. Despite this largespreading of CC methods, there has not yet been aparallel effort to extend them to introduce theinteractions between a molecule and a surroundingsolvent. The only example appearing so far in theliterature is the CCSCRF developed by Christiansenand Mikkelsen,141 using the multipole solvation ap-proach described in section 2.3.2; the same schemehas been also extended to CC response methodincluding both equilibrium and nonequilibrium sol-vation.140

To conclude this brief overview, we cite some recentand less recent extensions of continuum models tothe valence bond (VB) method.

One of the first successes in incorporating solvationeffects into a VB method was the empirical valencebond (EVB) method formulated by Warshel and

Weiss in 1980.237 However, at the beginning of the1990s Kim and Hynes238-240 developed a method forcalculating the electronic structure of a solute andits reaction pathways, in a manner that incorporatesnonequilibrium and equilibrium solvation effectswithin a continuum approach. Later, Amovilli et al.211

presented a method to carry out VB analysis ofcomplete active space-self-consistent field wave func-tions in aqueous solution by using the DPCM ap-proach.14 Another method of the same family ofsolvation methods, namely, IEF of section 2.3.1.3, hasrecently been used to develop an ab initio VB solva-tion method.241 According to this approach, to incor-porate the solvent effect into the VB scheme, thestate wave function is expressed in the usual termsas a linear combination of VB structures, but, now,these VB structures are optimized and interact withone another in the presence of a polarizing field ofthe solvent. The Schrodinger equation for the VBstructures is solved directly by a self-consistentprocedure. This iterative scheme is exactly equivalentto that described above for CI wave functions.

3. Some Steps beyond the Basic ModelThe presence of nonelectrostatic terms in the

interaction between a solute and the surroundingsolvent was well-known to the pioneers of the firstQM solvation models (see section 1). In fact, the firstmodel proposed by Claverie in 1972 addressed dis-persion terms only,5 and Rivail in his first attemptin 19736 included both electrostatic and dispersionterms.

In our exposition we have privileged the electro-static term. Historically, continuum models mainlyderive from the seminal models of Kirkwood andOnsager, which paid attention mostly to the electro-static term, and this particular attention to electro-statics has been maintained in the successive elabo-ration of methods, of both classical and QM types.The practical reason for this is that water surely isthe most important liquid medium, and for dipolarsolvents the dominant term in the solute-solventinteraction is the electrostatic one. In many codesnonelectrostatic terms are considered as correctionsto the dominant term for which a cursory descriptionmay be considered to be sufficient. In many publishedstudies, it must be added, nonelectrostatic terms arecompletely ignored.

Actually, the nonelectrostatic terms cannot becursorily treated, or ignored, in a good solvationmethod. There is a delicate balance between electro-static and nonelectrostatic energy components, ofopposite sign, that must be accurately evaluated bothin the polar and in the nonpolar solvents in order tohave descriptions which could lead to severely biasedpredictions of properties.

We shall examine now the nature and basic char-acteristics of these nonelectrostatic interaction terms.

3.1. Different Contributions to the SolvationPotential

The analysis we shall sketch here is not limited tocontinuum models but also involves discrete solvation


models. Differences between the two approaches willbe commented on at the appropriate place.

The basic information on the nature of molecularinteractions giving origin to the solvent effect derivesfrom the theoretical analysis and interpretation ofthe interactions occurring in molecular clusters.

There is an extremely large literature on thissubject, covering a span of more that 50 years, thatcannot be summarized here. Studies have progressedfrom simple systems, dimers composed of two verysimple molecules, increasing step by step the numberof components of the aggregate and the internalcomplexity of the molecules. Several aspects of thesestudies are pertinent to our subject, but we shall limitourselves to the basic information derived frombimolecular associations, complemented by someaspects derived from the analyses of trimers andlarger aggregates.

The interaction energy between two molecules Aand B can be expressed as the difference between theenergy of the supermolecule and that of the twoseparate partners:

This energy has been analyzed in a significantnumber of cases using a variety of techniques basedon perturbation theory approaches as well as varia-tional techniques. The details and the variantsamong the different approaches are not of interesthere. What is of interest is to examine a partition of∆E(AB) into terms each having a physical interpre-tation and the possibility of being individually com-puted. This partition reads

The Coulomb term, Ecoul, is defined as the electro-static energy among the two unperturbed (i.e., notpolarized) partners. This term is very sensitive to themutual orientation of the two partners and stronglyanisotropic at short distances. The interaction maybe positive or negative and, for a given A-B system,it may change sign according to the orientation whenat least one of the two molecules has a (local) chargedistribution with a dipole. The Coulomb energycontinues to be numerically important also at largemonomer separations; actually, it is the term havingthe highest long-range character. The typical distancedependency is R-1.

The polarization term, Epol, has its origin in themutual electrostatic polarization of the two chargedistributions, and it is everywhere negative. At shortmonomer separations, it is moderately anisotropic,being sensitive to the mutual orientation of the twopartners. Epol has a relatively long-range character:the typical distance dependency is R-4. Epol(AB) canbe easily decomposed into two terms, the polarizationof A under the influence of B, Epol(A r B), and thepolarization of B under the influence of A, Epol(B rA).

The exchange term, Eexc, is related to the presencein the QM formulation of intermonomer antisymme-trizer operators, which ensure the complete antisym-metry of the electrons in the AB dimer. Complete

antisymmetry means to introduce repulsion forcesbetween the electron distribution of the monomersrelated to the Pauli exclusion principle. Eexc is posi-tive for every orientation of the partners, roughlyproportional to the mutual overlap of the two chargedistributions, and rapidly decaying with the distance.The typical distance dependency is dictated by a R-12

term.The dispersion term, Edis, can be related to dy-

namical effects, as the formation of an instantaneousdipole moment in a monomer due to the motion ofthe electrons. This dipole polarizes the charge dis-tribution of the other partner and gives origin to aninteraction energy similar in some sense to Epol. It isa distinct phenomenon, however, and must be sepa-rately treated. Dispersion energies are negative,relatively weak, and decaying as R-6.

The charge transfer term, ECT, is not present in thestandard analyses of ∆E based on perturbationtheory, whereas it appears in the decompositionsbased on the variational approach [i.e., starting fromthe variational calculation of the E(AB) energy] in aform that seems to be a bit artificial, being presentalso in symmetric AA dimers under the form ofmonomer contributions which cancel each other. Inchemistry, charge transfer effects are quite importantin large classes of chemical reactions; these, whenconsidered under the viewpoint of a theory address-ing the general characteristics of solutions, belong tothe category of “rare events”, and they will not betreated here. However, we also note that smallamounts of charge transfer, not related to chemicalreactions, can play a role in assessing solvent effectson some molecular properties, and in the elaborationof solvation models we have surely to pay attentionto these interactions even if, at present, they haveattracted very limited attention (see section 3.2.4).

This very schematic presentation of the decomposi-tion of dimeric interactions is far from being com-plete, and our presentation of decomposition resultsshould be extended to the more complex analysesperformed on trimers and larger aggregates. As amatter of fact, here we shall be even more concise,reporting only the conclusions that seem more inter-esting for liquids.

(1) There are no new types of interactions toinclude in the model. Specific interactions, ofteninvoked or described with empirical ad hoc formulas,as, for example, hydrogen bonds, are in this approachdescribed in terms of the contributions we havealready presented.

(2) All of the contributions, with the exception ofEcoul, are nonadditive, and the weights of such non-additivity ares very different for the different terms.The larger nonadditivity, both in weight and inrange, is that due to the polarization energy. Mol-ecules in an assembly are strongly influenced bymutual many-body polarization effects. The nonad-ditivity of exchange is in se very short-ranged. Thenonadditivity of dispersion is moderate and plays animportant role in some special cases only, as, forexample, in clusters composed of rare gas atoms inwhich classical electrostatic terms are zero. Thenonadditivity of charge transfer has been less stud-

∆E(AB) ) E(AB) - E(A) - E(B) (76)

∆E(AB) ) Ecoul + Epol + Eexc + Edis + ECT (77)


ied. It should be a relatively short-range effect,especially for the energy.

3.2. Use of Interactions in Continuum SolvationApproaches

We have now the elements to give a more detailedappraisal of the use of these interactions in thecontinuum description of solute-solvent interactions.Among the various solvation methods, reference willbe made mostly to the PCM, which is the mostcomplete in this field, and to GB methods (see section2.3.3) that have paid much attention to the consid-eration of nonelectrostatic terms.

Continuum methods start from the definition of aliquid, the pure solvent, equilibrated at the givenmacroscopic conditions (temperature and pressure,for instance). The solute molecule must be insertedwithin this liquid. This specific aspect of continuummodels gives rise to a problem that can be solved byintroducing in the definition of the overall intermo-lecular interaction energy (see eq 77) the concept ofthe energy of formation of a suitable cavity withinthe liquid. We shall call it Gcav.

The other contributions to the interaction energycan be collected into three terms, named electrostatic,repulsion, and dispersion. There are some changeswith respect to the partition given in eq 77. Coulomband polarization terms are grouped together; theseparate calculations of the two contributions can bedone (and it has been done in the first DPCMversion), but this separation is of no practical helpin the QM calculations. In any case, it may be usedfor more detailed analyses about the origin of theinteractions. The repulsion term takes into accountthe presence of the cavity; the physical origin of theterm is unchanged, but all solvent molecules are keptout of the cavity. The dispersion term, too, is leftunchanged, and it involves interactions with solventmolecules out of the cavity.

We have now the elements to write the interactionenergy decomposed into four terms, the three wehave just mentioned and the cavity formation term.Before doing this, we have, however, to consider twoadditional points.

The first is related to the use of ab initio QMmethods. These methods give the molecular energymeasured with respect to a quite peculiar reference,that is, noninteracting electrons and nuclei. Thesecond point to consider is related to the existenceof free energy contributions due to other degrees offreedom which are present in the focused model butare neglected in the BO approximation we have usedso far. These degrees of freedom are related to thethermal motions of nuclei. We shall call Gtm thecorresponding free energy contribution. More com-ments will be made below, but we may now give anexplicit definition of the energy in continuum ab initiomodels and of the solvation free energy, two relatedbut not identical concepts.

Definition 1: Free Energy of the Solution in abInitio Continuum Models. The energy of a systemcomposed by a liquid and a solute is defined makingreference to an ideal system composed of the pureliquid at equilibrium under given values of the

macroscopic parameters (pressure, temperature) andby the appropriate number of noninteracting elec-trons and nuclei necessary to describe the solute andat zero kinetic energy. Putting the energy of thereference system equal to zero, the free energy of thesolution at the equilibrium is given by the followingexpression:

The definition of the solvation free energy givenbelow can be considered as a supplementary defini-tion drawn from eq 78, but it is the primary definitionfor computational methods, which do not allow anevaluation of the solute molecular energy. This is thecase for computational procedures based on semiem-pirical QM methods or the use of classical models.

Definition 2. Solvation Free Energy. The solvationfree energy, measured with respect to a systemcomposed of the pure unperturbed liquid at equilib-rium and by the solute molecule(s) in a separatephase considered as an ideal gas, is given by thefollowing expression:

If we define ∆Gel ) Gel - G0, where G0 ≡ E0 is the abinitio energy of the isolated molecule in the BOapproximation and Gel is the analogous quantitycomputed in solution, we may obtain the first termof eq 79 from ab initio calculations. ∆Gel can becomputed directly in all non ab initio calculations.The term ∆Gtm of eq 79, being a difference betweentwo analogous terms related to the molecule insolution and in the gas phase, may be simpler tocompute than the Gtm term in eq 78 because anypartial contributions cancel each other at a goodextent. This fact is often exploited when calculationsaddress solvation energy problems.

In definition 2 we have paid a bit more attentionto the thermodynamic definition of the referencestate. The reason the P∆V term was not inserted indefinition 1 will be mentioned later (section 3.2.6).

The second comment involves the internal geom-etry of the solute. In passing from the gas phase tothe solution there is always some change in theinternal geometry, which often cannot be neglectedwhen the definition ∆Gel ) Gel - G0 is used. Thisproblem does not exist for definition 1 because noreference is made to molecules in the gas phase ineq 78. Semiempirical and classical models shouldconsider the energy term due to changes in theequilibrium geometry, but this sometimes is not done.

The first three terms in the right-hand side of eq78 can be computed all at the same time with aunique QM calculation based on the appropriateSchrodinger equation, or they can be computedseparately with semiclassical models. Gcav must becomputed separately; the same holds for Gtm. It isworth remarking that contributions to the free energydue to thermal motions of nuclei are separatelytreated also in the QM descriptions of isolatedmolecules; the same standard techniques (largelysimplified with respect to a full QM treatment, rarely

G ) Gel + Grep + Gdis + Gcav + Gtm (78)

∆Gsol + ∆Gel + Grep + Gdis + Gcav + ∆Gtm + P∆V(79)


used) are employed for molecules both in gas phaseand in solution.

The inclusion of nonelectrostatic terms (namely,the cavity formation term, Gcav, and the attractivevan der Waals solvent-solute interaction terms,GvdW) in GB methods described in section 2.3.3 isgenerally done in terms of an empirical functionbased on the surface area of the solute. For thisreason, the GB methods are often denoted with theGB/SA abbreviation.156 In particular, the GB/SAmodel computes Gcav + GvdW together by evaluatingsolvent accessible surface areas

where SAk is the total solvent accessible surface areaof all atoms of type k, and σk is an empiricallydetermined atomic solvation parameter.

The effort of defining parameters, and of fixingtheir appropriate numerical values, is an exactingtask, amply repaid, however, by the computationalefficiency of the method.

In a specific family of GB methods, namely, thatof the SMx models developed by Cramer and Truhlardescribed in section 2.3.3, the nonelectrostatic con-tributions are described in terms of an empiricalexpression, which has been differently parametrizedin the different versions of the model. We refer hereto the universal solvation model, which is “universal”in the sense that a unique code offers options for alarge number of solvents of different type, each withits own set of parameters. The SM5.42R and SM5.43Rcodes we have mentioned in section 2.3.3 are ofuniversal type. In this framework, the nonelectro-static part of the solvation free energy, usuallyindicated as GCDS, depends on a fitting over sixsolvent descriptors: the refractive index (at thewavelength of the Na D line), Abraham’s hydrogenbond acidity and bond basicity parameter,242 thereduced surface tension, the square of the fractionof non-hydrogenic solvent atoms that are aromaticcarbon atoms (carbon aromaticity), and the squareof the fraction of non-hydrogenic solvent atoms thatare F, Cl, or Br (electronegative halogenicity). Moredetails can be found in the already cited paper.167

We now examine the separate components first andthen give the unified QM definition of the procedurebased on a Hamiltonian collecting all of the interac-tion terms.

3.2.1. Cavity Formation Energy

The idea of formally breaking the process oftransfer of a molecule from the gas phase to a liquidinto two steps, first the formation of a cavity in thebulk of the liquid and then the insertion of themolecule in the cavity, dates back to 1937.243 Thebalance between the reversible work of these twosteps gives, according to this model, the excess freeenergy of the process or, in other words, the solvationfree energy. This device to evaluate solvation energyhas been used for many decades, without resortingto detailed molecular descriptions of the system.

The different formulations of methods to computethe energy of cavity formation have been amplydescribed in the 1994 review,1 to which readers arereferred. There are reasons, however, to reconsiderthis subject in the present review.

There is no physical counterpart to this idealformation of an empty cavity, and as a consequence,there are no experimental values to compare with theresults of the models. Actually, no direct checks arepossible for any of the contributions into which thesolvation energy can be divided, and this fact makesharder the task of getting an accurate modeling ofthe property in which all of the components have awell-defined meaning. This accuracy in the elabora-tion of the model is a necessary step to go beyondmodels limited to the appraisal of solvation energy.There are many other properties to study and manyother applications of continuum solvation models, forwhich accurate descriptions of the separate contribu-tions to the solvation effects are necessary. There arealso methods that combine cavity formation energywith other short-range interaction terms: this is adifferent philosophy to formulate continuum solventmodels, reasonable, efficient, and giving good resultsfor properties more directly related to the solvationenergy. If extensions of the models belonging to thiscategory to other properties are to be sought, theempirical expressions in use will require modifica-tions.

With regard to the cavity formation energy, theattempts to arrive at reliable values for Gcav have aprobability of success relatively higher than for theother terms, because the interaction potentials aresimpler in this case. The way to get these reliablevalues is based on the use of computer simulationsof liquids, a subject we examine here below.

3.2.1.1. Computer Simulations and Gcav Val-ues. A strategy to get Gcav values via computersimulations is far simpler than similar strategies forthe other components of the solvation energy. Gcav

in fact depends on the solvent only, and the numberof solvents, even if large, is far smaller than thenumber of possible combinations of solute-solventsystems. For each solvent, the number of degrees offreedom to scan in a systematic way is limited: themacroscopic variables (T and P, for example) and thesize and shape of the cavity.

There are several fields in chemistry in which theevaluation of the energy of formation of a microscopiccavity is of interest. Some among them involve casesin which a cavity is spontaneously formed under theaction of an external agent; we cite as examples theformation of cavities (or microbubbles) under theaction of ultrasound244 and bubble formation atliquid-solid interfaces,245 topics of remarkable con-ceptual and practical interest, but to which ourcontinuum models have not yet been applied. Itseems to us appropriate to reserve the expression“cavitation energy” (used in the older literature onsolvation models) to phenomena in which real cavi-ties are formed and to use for our cavities, whichrepresent a formal ideal step in solvation, the expres-sion “cavity formation energy”.

Gcav + GvdW ) ∑k)1

N

σk(SAk) (80)


The cavity formation energy, in the meaning usedby Uhlig in 1937,243 plays an important role in thetheory of hydrophobicity. We shall not review hydro-phobicity theories here, because they encompasscontinuum models, and we shall consider hydropho-bic solvation only as a specific case of the moregeneral field of solvation phenomena. This remarkis not out of line here, because the majority ofavailable calculations of Gcav via computer simula-tions246-270 have been motivated by hydrophobicitystudies, and this introduces a bias to the definitionof the simulation, inasmuch as the interests ofhydrophobicity are not coincident with those ofgeneral solvation.

In fact, from simulations we would have informa-tion about a large number of topics, necessary tomodel methods for the calculation of Gcav for thewhole domain of solvation problems. The topics ofinterest range from the chemical nature of the solventand of its physicochemical properties to the physicalmacroscopic characteristics of the liquid (tempera-ture, pressure, and also externally applied factors,as electric fields and hydrodynamic flows), to the formof the cavity that must faithfully reproduce theextreme variety of sizes and shapes solutes have.

It would be extremely convenient to have a satis-factory analysis of these factors before we begin todevelop Gcav computational methods. Parenthetically,we remark that we need analyses and not tables ofspecific values, because in the chemical applicationsaddressed by continuum models there is often theneed of examining the whole solvation free energy,and hence Gcav, for a sizable number of solutegeometries. It would be impractical and in contradic-tion with the basic philosophy of continuum modelsto use computer intensive methods, as simulationsare, to compute a single element of the solvationenergy at each geometry of the solute. Simulationsare to be mainly used as validation tests of simplerexpressions describing Gcav and are better if they areof analytical type.

Of course, this complete analysis has not beenperformed, but the available simulations have beenof great help in assessing the relative merits of thevarious methods we shall examine in the remainderof this section.

There are at present a sizable number of Gcavvalues from simulations for small cavities, in generalspherical but with a few examples of nonsphericalcavities,258,262 in various solvents. These latter simu-lations are mostly due to Hofinger and Zerbetto,269-271

who have recently added to the half score of solventsconsidered in the past 12 additional solvents, all ofchemical relevance. Some of these simulations alsogive information about temperature and densitydependence. These results validate the guesses madeyears ago (and discussed in the following pages)about the general validity of some simple models,dispelling doubts and criticisms and ending incon-clusive discussions.

One may wonder why after more than 30 years ofcomputer simulations only now is there a satisfactorysituation (at least for small cavities). One reason isthat only in recent years has progress in computa-

tional machineries permitted a more efficient evalu-ation of Gcav values. In 1998 Levy and Gallicchio272

remarked that the intensive computational demandof simulations raised obstacles to the systematicstudy of solvation in the preceding years. Now thesituation is greatly improved, and we are expectingfrom simulations additional help, especially for largercavities and for liquids far from the standard ther-modynamic state.

We cannot wait, however, and to proceed we haveto base our decisions on other criteria such as indirectanalyses of the large wealth of Gcav values computedwith simpler models.

The most extensive analyses performed thus farare, to the best of our knowledge, those performedby the Luque and Orozco group in Barcelona. Werefer interested readers to one273 of their papers asit reflects the evolution of the cavity formationmethods over the past 10 years.

In our 1994 review1 we analyzed a variety ofmethods; many have been abandoned and, in fact,only two approaches have been considered in succes-sive papers: those based on statistical mechanics andthose based on the area of the solute exposed to thesolvent. Here we shall consider both approaches.

3.2.1.2. Statistical Mechanics and Gcav Values.The elaboration of PCM has strongly relied onstatistical methods to supplement the basic electro-static code.274 For the cavity formation two ap-proaches have been presented, one based on thecavity surface and the other, which is the subject ofthis section, based on a statistical mechanics ap-proach, namely, the scaled particle theory (SPT),introduced and elaborated over the years by Re-iss.275,276 We may anticipate that the analyticalexpressions of Gcav obtained from the SPT versiongive values in good agreement with numerical simu-lations.

This is not the appropriate place to fully explorethe great merits of the statistical mechanics methodsfor our understanding of the properties of liquids. Ashort comment is, however, necessary to put SPT,and the use we are making of this theory, in theappropriate context.

Statistical mechanics methods encountered dif-ficulties in passing from gases to liquids, and toovercome these difficulties very simplified modelswere adopted. What is of interest here involves thereduction of the fluid particles (atoms or molecules)to hard spheres, neglecting the portion of the interac-tion called the soft portion (i.e., the electrostatic anddispersion terms which, in this review, we are moreinterested in). Hard sphere models were extensivelyused to study the various aspects of the theory forfluid systems. Actually, the detailed and satisfactorydescription of the properties of the hard sphere fluidswas found to be unable to provide numericallyacceptable estimates of the properties of real fluids.The connection between the worlds of hard spheresand of real liquids was given by the SP theory, amodel based on hard spheres again, but with radiisuitably modified with a scaling procedure to satisfysome macroscopic experimental quantity. SPT is arigorous theory (in the sense that there are no


additional adjustable parameters), trading its ef-ficiency in giving numerical results comparable withexperiments with a loss of generality with respect tothe formally exact hard sphere model.

Here, we are not interested in studying the generalproperties of liquids, nor the applications of SPT asa general albeit simplified theory for liquids, but inthe specific application of SPT to define the freeenergy of formation of a cavity. Cavity formationindeed is the starting point of SPT. Work is requiredto exclude the centers of molecules from any specifiedregion. A cavity is defined as a region in which thecenters of the solvent molecules, described as hardspheres of radius RS, are excluded. The radius of thecavity is RMS ) RS + RM, where RM is the radius ofthe hard sphere representing in our case the solute.The reversible work necessary to form this cavity iscomputed with an application of the so-called charg-ing process, in which an integration is performed onthe internal energy of the system in terms of acontinuous parameter λ ranging from 0 to 1. Herethe parameter λ is related to the radius of theinserted particle, put equal to zero at λ ) 0 andarriving at RM at the upper limit of the integration.Two parameters are necessary to determine thereversible work required to make this cavity, theratio of the two hard sphere radii RM/RS, and thereduced number density of the solvent F ) N/Vs(where N and Vs are, respectively, Avogadro’s numberand the molar volume of the liquid) entering in theauxiliary function y:

The analytical form of the free energy of cavityformation was given by Pierotti in terms of anexpression in terms of RMS truncated at the thirdorder.277-279 The complete Pierotti expression can befound in his original papers as well as in our 1994review.1 We report here the somewhat simplifiedform used in PCM:

The simplifications we have introduced are due tothe negligible values of the contributions related tothe pressure P for liquids at ambient pressure (seecomments, e.g., in ref 255).

The enthalpy of cavity formation can also becomputed within this approach. The simplest way isto use the Helmholtz equation

with an appropriate evaluation of the derivative.280

We note, however, that careful attention has to bepaid to the enthalpy and entropy values obtained insuch a way. There is in fact a sizable literature aboutlimitations and defects in the Pierotti-derived for-

mulas for entropy and enthalpy. Free energy is amore robust quantity, and an indiscriminate use ofentropies and enthalpies obtained by stressing theinitial conditions set in Pierotti’s papers is to beavoided or performed after an accurate scrutiny ofthese remarks.

Actually, little is left of the original SPT in thecurrent use of Pierotti’s formulas for the calculationof Gcav made in continuum solvation methods. Alter-native formulations, closer to the SPT model, en-counter some problems. For example, SPT defines ahard sphere pressure for the liquid, which may bevery different from the actual pressure experiencedby the real liquid one is simulating. This fact intro-duces a dilemma in the definition of Gcav (see Tangand Bloomfield281 for a detailed and critical analysisof these problems).

The main reason that the formulas in use are onlyloosely related to the original SPT derivation lies inthe definition of the hard sphere radii. As mentionedabove, SPT radii are not exactly the hard sphere radiiused in the early statistical treatments of monatomicfluids (which were addressing real atoms deprivedof all “soft” interactions) but rather formal sphereradii calibrated on a property of the real fluid. Inprinciple, SPT hard spheres are not limited tospherical (or almost spherical) solvent molecules, andin fact the theory has been applied also to liquidscomposed of molecules far from sphericity, such asbenzene or n-hexane. The essential point is that SPTreduces the interparticle interaction to an isotropicform in which directional properties are mediated(and soft interactions are neglected in the initial stepof application of the method).

Hard spheres describing the properties of liquidshave been obtained by making reference to a varietyof experimental data, such as those measuring vaporpressure,276 compressibility,282 heats of vaporiza-tion,279,283 surface tension,284 and solubility,285 andfrom the cell theory of liquids.286 To be rigorous, notall of the cited papers report hard sphere radiiobtained with SPT; some are related to other elabo-rations of experimental data bearing some similaritywith the SPT. In particular, use if often made of twodifferent equations of state, the so-called Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL)287,288

or the simpler Carnahan-Starling (CS)289 valid fora single-component fluid, which are considered to besuperior to the SPT in predicting the free energy ofsolvation in a hard sphere fluid.290

There are available several collections of hardsphere radii. Noteworthy are those given by Ben-Amotz and Willis291 (supplemented by group volumeincrements) and the two given by Y. Marcus.292,293

Also note that different sources give different valuesfor the hard sphere radii. For water, for example, theradii range from 1.25 to 1.58 Å, and Tang andBloomfield281 cite eight different papers using differ-ent radii.

To these hard sphere radii we may add othersderived from models elaborating experimental datawith spherical potentials containing also a separatesoft interaction component The most famous modelis the (6-12) Lennard-Jones (LJ) model. There is a

y ) π6

(2RS)3F (81)

Gcav ) RT- ln(1 - y) + 3y1 - y(RM

RS) +

[ 3y1 - y

+ 92( y

1 - y)2](RM

RS)2 (82)

Hcav ) Gcav - T(∂Gcav

∂T )P

(83)


wealth of LJ values in the literature. This variety ofnumerical values could raise doubts in a potentialuser, but the situation is not as bad as it mightappear. Two remarks are in order. The first is thatit has been argued285,294-296 that the most consistentradii to use in SPT studies are those obtained withPierotti’s technique, and in fact the radii for waterobtained by fitting results of independent computersimulations are in a restricted range around thevalue given by Pierotti. The second remark is thatwe are here not interested in a full use of the SPtheory nor in other possible applications such as thedefinition of the solvent accessible surface (see section2.2) for which the hard solvent radius is used, butsolely in the solvent radius to use for the calculationof Gcav. We shall come back to this last point afterexamining the other face of the problem, that is, thedefinition of the radius for the solute.

The variety of solutes a continuum solvation modelshould be able to describe is so large that compila-tions of hard sphere radii, even if supplemented bygroup increments, will never be sufficient. The solu-tion to this problem has been found by abandoningSPT and making reference to another molecularmodel, that of van der Waals radii. Chemists are wellaccustomed to using van der Waals radii, which havea longer history than SPT hard spheres and havebeen empirically defined in a large variety of ways.There is no unique set of van der Waals radii, butsuccessive refinements confirm the substantial valid-ity of the older compilations (see section 2.2). Inaddition, there are reliable computational techniquesable to give independent descriptions of the molecularvan der Waals surfaces and also indirect hints thatmay be used to compute Gcav values.

We have now arrived at defining solute shapes, andthe corresponding cavities, in terms of atomic orgroup interlocking spheres, related to van der Waalsvalues. To do this Gcav calculation, a further step isstill necessary. The geometrical quantity given by vander Waals-like models of a generic solute does notcorrespond to the spherical shape considered inPierotti’s formula.

The direct use of a cavity modeled in terms ofinterlocking spheres for the calculation of Gcav ac-cording to Pierotti’s scheme has been performed byIrisa.297 Limiting ourselves for the moment to solutesof small size, there are two reasonable ways ofreducing an irregular van der Waals envelope to asphere; they consist of defining the radius RM of thissphere in terms of the surface SM of the envelope orin terms of its volume VM:

In an already cited paper,273 Colominas et al.examined this problem in a broader context involvingthe different definitions of Gcav. The discussion isbased on the statistical analysis of actual valuesfound with different methods in a sizable number of

solutes in several solvents. For the specific questionwe are considering, these authors find no appreciabledifferences in the performances of the two definitions(called SPT-S and SPT-V, respectively) that are, onthe whole, slightly superior to the other examineddefinitions.

Another point considered in this analysis involvesthe size consistency of the various procedures. Sizeconsistency is not really relevant when the study islimited to a single molecule, but it becomes importantwhen one considers dissociation or association pro-cesses. In both cases, the Gcav value should changesmoothly along the coordinate describing the process.Both SPT-S and SPT-V lack this property. This pointwas intensely discussed in Pisa in the early 1980suntil Claverie suggested a modification of Pierotti’sformula, which he separately published later;298 thismodification was immediately inserted in the Pisasolvation model PCM (see section 2.3.1.1). Accordingto this modification, the cavity formation energy isgiven by a sum of contributions each related to asingle-sphere SPT term weighted by the area of thesingle sphere exposed to the solvent:

This formula has been called the Pierotti-Claverieexpression for the cavity free energy and is indicatedwith the acronym C-SPT in the analysis of Gcavreported in the cited paper.273 In the first implemen-tation of formula 86, the number N of spherescorresponded to the number of atoms in the solute,and their radii were fixed in agreement with theselected definition of van der Waals parameters.26 Wenote that, in successive implementations, N is notnecessarily equal to the total number of solute atoms,but it can be smaller as now united-atom cavities aregenerally used (see section 3.2.5). Indeed, the choiceof the cavity used to compute Gcav in continuummethods has never been uniquely determined andthus different choices have been used. To try to clarifythe large number of available options, it is useful tointroduce the expressions one cavity and two cavity.The first refers to all calculations in which just asingle cavity is used for Gcav and ∆Gel, whereas thesecond refers to calculations in which two differentcavities are used to compute Gcav and ∆Gel (seesection 3.2.5 for more details). In general, the two-cavity calculations are to be preferred.

To conclude, we may say that statistical analyses273

and computer simulations, in particular those per-formed with a molecularly shaped cavity,258 agree inconsidering SPT-V and SPT-S to be the best proce-dures, almost equivalent and marginally better thanC-SPT. The size consistency exhibited by this latterseems to us an important factor to keep, even if theprice is a minor difference in the value of the cavityformation energy.

We have here sketched the present situation forsmall solutes for which the algorithms in use workrelatively well for solvents with molecules of smallsize, such as water, irrespective of whether thesemolecules form hydrogen bonds in the bulk. For other

RM ) xSM

4π(84)

RM ) x3 3VM

4π(85)

Gcav ) ∑k)1

N Ak

4πR2k

Gcav(Rk) (86)


solvents, in particular those composed of moleculesof larger size, rarely considered in theoretical model-ing but often used in chemistry, little is known.Recent simulations performed by Hofinger and Zer-betto270 on organic solvents seem to confirm thevalidity of the SPT procedure.

For solutes of larger size, no accurate simulationsof Gcav are available. Studies on hydrophobicity rangefrom very small solutes, such as rare gases andmethane, to large, mesoscopic and macroscopic, bod-ies. The hypothesis of a crossover of two regimes forthe cavity formation energy, the first scaling as thevolume of the cavity and acting for small cavity radiiand the second scaling as the surface and acting forlarger cavities, has been strongly advocated.266,299 Thecrossover from the first to the second regime shouldoccur at a molecular length scale. Here, we suggestthat for small cavities the two regimes should leadto similar cavity formation energies, but for largercavities the scaling of Gcav with the surface of thesolute should be more probable.

As a final note about SPT, we recall that it is notlimited to spheres. Extensions of the theory to otherconvex shapes have been known for a long time. Thereaders are referred to papers by Gibbons300 for apreliminary examination of a variety of shapes, byCotter and Martire301,302 and by Lasher303 for exten-sion to liquid crystals using spherocylinders, and byBoublik304,305 for more analyses of the equation ofstate based on hard dumbbells and spherocylinders.The examples we have reported involve applicationsof SPT to homogeneous fluids, in which all particleshave the same shape. A limited use of these modelshas been done thus far for continuum calculations,the only exceptions we know being the PCM studieson the determination of the cavity formation energyin nematic liquid crystals.306

3.2.1.3. Information Theory Method. The sta-tistical concepts underlying SPT have been used byPratt and co-workers307,308 to formulate an interestingapproximation to the calculation of Gcav.

The key point in this method is the relationshipconnecting the free energy of formation of a cavitywith the probability p0 of finding in the liquid anempty cavity of given size and shape:

In this method p0 is considered as the first memberof a probability distribution where the term pn is theprobability of finding n solvent centers in the volumev corresponding to the selected cavity. To get esti-mates of the pn values, use is made of simulation data(performed over a model of the bulk liquid) to get themoments of the fluctuations in the number of solventcenters. The moments of the fluctuation in theparticle numbers are given by the following thermalaverages:

The first three moments, namely, ⟨1⟩, ⟨n⟩, and ⟨n2⟩,can be obtained from experimental quantities [the

first is a constant, the second derives from the densityF, and the third derives from the radial distributionfunction gCC(r) of the molecular centers, upon inte-gration]; the other moments have to be derived fromsimulations under the normalization constraintsgiven by the three first moments.

An approximation introduced at this point gives thename to the method: ITM, which means informationtheory method. Following the IT rules, it is possibleto define the “best-possible” estimate of the prob-abilities, in the form of a set pn, which maximizesthe information entropy, under the constraint givenby the first three moments.

This definition leads to

where pn(0) collects the set of values of an empiri-

cally chosen “default” initial model and the λi valuesare Lagrangian multipliers satisfying the momentconditions. Further details can be found in the above-cited references. It is sufficient to add that a flatdefault distribution (all pn

(0) ) 1, for n e nmax), givingrise to a Gaussian distribution, has been shown tobe sufficient to give good results for the cavityformation energy, expressed in the form

where σ is the standard deviation of the data usedin the elaboration.

The method is not limited to the calculation of Gcavor to water as the liquid system. It may be consideredas a new approach to study the physicochemicalproperties of liquids. Potentialities and limitationsof the method are not yet fully explored, despite thesizable work done by Pratt and co-workers in recentyears, in particular for the interpretation of thehydrophobic effect.309 Of remarkable interest is alsothe formulation recently proposed by Basilevsky andco-workers,310,311 which introduces a binomial distri-bution law for a number of solvent particles occupy-ing a given volume (the cavity). The resulting distri-bution consists of two binomial peaks proportionalto the volume and to the surface area of the cavity,respectively. This formulation is, in our opinion, aninteresting way to combine cavity formation pro-cesses for small and large solutes.

3.2.1.4. Surface Tension and Gcav Values. As analternative to the statistical method described in theprevious section, we describe here one based on thesurface of the solute exposed to the solvent.

In his seminal paper Uhlig243 used the surfacetension γ of the pure solvent to evaluate Gcav:

This expression has been used for years withoutchange, despite the progress in the understanding ofcurvature and microscopic effects on γ initiatedaround 1950312,313 and considerably improved in morerecent years in the specular problem of the surfacetension for microscopic droplets. A more detailed

Gcav ) -kT ln p0 (87)

⟨nk⟩ ) ∑n)0

∞

pnnk (88)

pn ) pn(0) exp(λ0 + λ1n + λ2n

2) (89)

Gcav ) kTF2ν2

2σ2+ ... (90)

Gcav ) 4πγR2M (91)


version of the Uhlig formula, including microscopiccurvature corrections to γ, was elaborated by Sinano-glu in a series of papers spanning the period 1967-1982.314-317 In 1981-1982 a version of the Sinanoglumodels was introduced into PCM, and this code wasretained in more recent versions of our program. Theuse of Gcav values obtained with this formula was,however, abandoned because the examination of thetrends of this quantity with respect to some param-eters (radius of the spherical cavity, temperature,pressure) was not numerically convincing.

Years later, a Monte Carlo simulation258 showedthat the corrections introduced by Sinanoglu for themicroscopic curvature were actually worse than thedescription of Gcav given by surface tension withoutcorrections. This study, in addition, indicated that themost appropriate surface to use in combination withγ to get Gcav lies between the van der Waals surfaceand the solvent accessible surface (SAS). This re-mark, we must stress, is based only on a single setof simulations performed on spheres of relativelysmall size in water.

Another approach to improve the Uhlig formulawas proposed by Tunon et al.318 It consists of theintroduction of a correction term to eq 91 based onother properties of the solvent, namely, the numberdensity of the solvent, ns, and the volume of a solventmolecule, Vs:

This formula is still in use and has been applied to asizable number of studies, performed both withRivail’s MPE method319-321 and with PCM.53,273,322

3.2.2. Repulsion EnergyIn our interpretation of the solvation energy, Grep

is the portion of the steric contribution to Gsol afterthe ideal process of making the cavity has beenperformed. We have remarked in passing that eachterm of the interaction free energy, isolated or incombination with others, can be described in termsof a charging process with respect to an appropriateparameter. The parameter for Gcav we have selectedcorresponds to a length, the radius, or the combina-tion of radii of the hard sphere cavity. This chargingprocess produces a considerable change in the solventdistribution function; a cavity, not existing before, isnow present. Ideally, according to the Uhlig model,the presence of this new element should not perturbthe distribution function in the other parts of thesolvent specimen. Actually, it is not so: hard spherecavity simulations introduce a perturbation on thewhole solvent distribution, because the macroscopicparameters are kept fixed in the simulation. Thiseffect is of little entity on the whole macroscopicliquid system, and we may discard it in our consid-erations, as we have implicitly discarded anothersimilar effect, that of using, for a very dilute solution,properties of the pure solvent.

The next step in our calculation of the solvationenergy consists of using the solvent distributionmodified by the cavity to perform the chargingprocesses for the other components of the solvation

energy. With PCM it is possible to collect all of theremaining charging processes into a single process,performed at the QM level with the aid of anappropriate Schrodinger equation. This unique fea-ture of PCM, that is, the unification of differentprocesses into a single step (or, in other words, totreat repulsion, dispersion, and electrostatic contri-butions on the same footing), considerably reducesthe problem of couplings. In fact, in a series ofdifferent charging processes performed in sequence,one has to use, for the second process, the solventdistribution function modified by the first. There arecouplings among charge processes that are neglectedin this sequence, and the final result depends to someextent on the chosen sequence. The effect of thesecouplings has been little studied; an example can befound in ref 323. Almost all continuum solvationmodels neglect these couplings, with the exceptionof those, such as Cramer and Truhlar’s generalizedBorn model (see section 2.3.3), which replace explicitcalculations with numerical parameters drawn fromexperiment. In the other methods, an approximationis tacitly assumed: we have called it the “uniformapproximation”,323 and it consists of taking thesolvent distribution constant equal to that describingthe bulk unperturbed solvent in all charging pro-cesses.

Having so established a common framework for allof the contributions to Gsol, we now examine in moredetail how Grep has been described. We start from theQM description, which is not used independently butcombined with dispersion and electrostatic terms, aswill be shown later, and we move on to nonquantummethods that permit a direct evaluation of this term.

The theory for the repulsion term derives from thetheory of the exchange term as expressed in theperturbation theory treatment of noncovalent inter-actions. The application of perturbation theories tothe description of molecular interactions is a quitecomplex problem, and the main source of this com-plexity derives from additional elements in the an-tisymmetry operator for the electronic wave function,which gives rise to the exchange terms. The difficul-ties found in describing the electron exchange arerelated to the difficulties quantum mechanics findsin describing a system M interacting with others. Anexact expression for exchange contributions to theinteraction energy is not available. There are excel-lent approximations that have given good results formolecular systems of small dimensions, and there isa hierarchy of descriptions introducing more andmore simplifications and ending with very simpleexpressions. An example of this last level of accuracyis the r-12 term used in the Lennard-Jones site-sitepotential.

The expression used in PCM due to Amovilli andMennucci324 is based on a dissection of the exchangeenergy for a dimer into terms depending on theelectronic densities of the partners.325 The elaborationof this expression, derived for the interaction energyof two molecules, to get Grep uses the replacement ofthe electron density of the second partner (the firstis the solute) with a one-electron charge distributionaveraged over the whole body of the solvent (the

Gcav ) 4πγR2M - ln(1 - Vsns) (92)


details can be found in the reference paper). Whenfollowed by proper mathematical manipulations, thiselaboration reduces the double integral over thewhole space to a simple integral of the solute chargedistribution lying outside the cavity C multiplied bya constant factor, namely

where the subscript M refers to the solute (and FMindicates its electronic density), S refers to thesolvent, FS is the density of the solvent relative todensity of water at 298 K, and nval

S and MS are thenumber of valence electrons and the molecularweight of the solvent, respectively.

The integral in eq 93, which represents the portionof solute charge lying outside the cavity, is rewrittenas the difference of total solute electronic charge (i.e.,the number of electrons) and the electronic chargeinside the cavity. By applying the Gauss theorem,we obtain that the internal charge is 1/4π ∫Γ EBM(sb)‚nb(sb) d2s, where nb is the outward unit vector perpen-dicular to the cavity surface at point sb and EM thesolute electric field due to the electrons only. The finalstep of the procedure is analogous to that used forother contributions to the solvation energy (includedthe classical expression of Grep we shall examinebelow): the surface integral is reduced to a summa-tion expressed in terms of contributions of singletesserae, defined in terms of the local value of thecomponent of the solute electric field normal to thecavity surface. To use the same tesserae used in thecalculation of the electrostatic solvation energy,which uses a cavity larger than that defined by thevan der Waals radii, the single terms are scaled by aproper numerical factor.

This method permits us to introduce the repulsioncontribution in the effective Hamiltonian, in the formof a new one-electron operator that we shall call hrep,which will be inserted in the Fe

r VrR term of the

definition (eq 62) of our Schrodinger equation. Thepresence of this operator acts to modify the solutewave function. Within the same approach used insection 2.4.2, this operator becomes a matrix definedon the chosen basis set ø, namely

where S is the overlap matrix in the same basis and

Eru being the electric field integrals on ø.Using the operator (eq 94), the repulsion contribu-

tion to the solvation free energy defined in eq 93becomes

where P is the density matrix obtained by solving

the effective Schrodinger eq 62 in which we haveadded the repulsion term.

We have thus introduced a formulation in whichthe repulsion term is treated on the same footing asthe electrostatic one, with a completely negligibleadditional computational cost, because all of theelements necessary for this new term are alreadypresent in the electrostatic calculation. This newterm does not present a problem for the analyticalderivatives (even if the formula has never beenimplemented), and it noticeably reduces the amountof the outlying solute charge.

PCM codes continue to present an alternativeversion for the calculation of Grep,326 which we do notpresent here, because it was throroughly analyzedin the 1994 review,1 to which we refer interestedreaders. Here we recall only that this formulation iscompletely based on empirical classical potentials forthe repulsion interaction and thus, in this case,repulsion effects cannot modify the solute wavefunction. Once again, the repulsion free energy canbe written as a sum of volume integrals, which maybe reduced to surface integrals, making use ofauxiliary vector functions. The surface integrals arethen expressed as discrete sums over the k tesseraeof the cavity surface Γ, each with an area ak:

with sites m and s corresponding to solvent nucleiand numerical parameters dms empirically fixed withsome criteria.

This classical version of the repulsion energy isfaster than the QM version. As a matter of fact, itscomputational cost is practically zero if the tessella-tion of the cavity is available; otherwise, it corre-sponds to the cost of making this tessellation.

The molecular surface is of SAS type, that is,defined as the surface described by a rolling sphereon the molecular surface (see Figure 1). The radiusof this rolling sphere should correspond to theminimal distance each site m can reach from thesolute, so there is the need to define a surface for eachm (for example, with methane as solvent, the sphereto the carbon atom contribution had a radius largerthan that for the H contribution by an amount equalto the RCH distance).

We consider it convenient to maintain this classicalversion in PCM codes, because the method can alsobe used for semiclassical descriptions of the solute,for which an explicit solution of a Schrodinger equa-tion is not necessary, and it would be senseless torequire the use of QM algorithms for a singlecontribution. We remind the reader, however, thatthis classical version does not cooperate with theothers to define the solvent reaction potential, andit cannot change the effects that this potential hason the electronic charge distribution of the solute.

Grep ) Fs ∑k∈Γ

∑s∈S

Ns ∑m∈M

akABms(rbmk)‚nbk (96)

ABms(rbms) ) - 19

dms(12)

rms12

rbms (97)

Grep ) R ∫r∉C FM(rb) d3r

R ) kFS

nvalS

MS

(93)

(hrep)ru ) R[S - S(in)]ru (94)

Sru(in) ) 1/4π ∫Γ EBru(sb)‚nb(sb) d2s

Grep ) tr Phrep (95)


3.2.3. Dispersion EnergyThe dispersion energy term is often collected with

the repulsion term into a unique term defining theso-called van der Waals contribution to the interac-tion energy. Actually, dispersion and repulsion havedifferent physical origins, different ranges of action,different sensitivities to fine details of the electroniccharge distribution, and opposite signs. It seems tous better to consider the two terms separately, bothfor the analysis and for the calculation.

The literature on dispersion is quite large, evenlarger than the abundant literature on exchange-repulsion interactions. There is no need of repeatinghere the detailed historical analysis presented in our1994 review.1 However, we recall the strong influencethat a seminal set of papers by Linder327 has had inthe elaboration of QM continuum solvation models.These papers have subsequently influenced anotherapproach formulated by Hunt328-331 starting from therevival of a remark about the electrostatic nature ofdispersion forces expressed by Feynman years ago332

and after called “Feynman’s conjecture”.333 Hunt’spapers have, in our opinion, a strong connection withcontinuum solvation models, and thus they might beexploited soon to develop new methodologies.

We shall begin the exposition starting from the abinitio codes inserted in PCM, continuing with otherQM formulations, and then finishing with simplersemiempirical recipes.

To insert dispersion terms in the Hamiltonian andthus to account for dispersion effects on the solutecharge distribution and for possible coupling withother solute-solvent interactions, we need an ana-lytical expression with explicit dependence on thesolute charge density. This goal can be reached indifferent ways, with the introduction of some ap-proximations.

The method implemented in PCM is based on theformulation of the theory given by McWeeny200 andon the use of dynamic polarizabilities. The steps ofthe procedure adopted for PCM can be found in thesource papers.324,334 The final formula of the solute-solvent dispersion free energy in the usual frameworkof a molecular orbital description of the wave functionreads

where the subscripts refer to elements of the selectedbasis set ø.

The term [rs|tu] has the aspect of a two-electronintegral, but simply involves combinations of one-electron integrals in the following way

where the integration is done on the cavity surfaceand Vab and Eab represent the potential and thenormal component of the electric field computed atthe same point of the surface when expressed on thechosen basis set.

The Pru coefficients in eq 98 are elements of thefirst-order solute density matrix P expressed in thesame basis set, S is the overlap matrix in the samebasis, and â has the expression

where nS is the refractive index of the solvent, IS itsfirst ionization potential, and ωM the solute averagetransition energy.

Expression 98 may be rewritten in a matrix form,making use, once again, of the trace operator andintroducing two new matrices with integrals per-formed over the basis set

where the elements of the two matrices we have hereintroduced have the following form:

The integrals defining these matrix elements (see eq99) can be computed in a discrete form, making useof the tessellation of the cavity surface used tocompute the apparent charges in PCM continuummethods.

We have to remark that the nature of â indicatesthe approximations introduced in this elaboration ofthe dispersion energy operators: among them one isspecific for the solute, the average transition energy.Because the most important transitions in polariza-tion of molecules occur between “frontier” orbitalssthe highest occupied and the lowest virtual orbitalssthe average transition energy can be approximatedby defining a proper window around the energy ofthe highest occupied orbital.

The computational demand is higher for dispersionthan for repulsion as the basic integrals are morecomplex in the dispersion term, which also includesa second term (Xdis) not present in repulsion. Inaddition, Gdis turns out to be quite sensitive to thequality of the basis set; to have good values of thisquantity we need to use basis sets decidedly largerthat those necessary for geometry optimization andfor the calculations of the other components of thesolvation energy.

There are other methods, developed at varioustimes, to get dispersion energy contributions with abinitio calculations (see the 1994 review for thecomplete list).

Perturbation theory, applied first to atoms andthen to small molecules, led to the definition of asimple formula for the expansion of the dispersionenergy between two atoms in terms of the inversepowers of the interatomic distance, with coefficients

Gdis ) -â

2∑rstu

[rs|tu]Pru(S-1)st +â

4∑rstu

[rs|tu]PruPst

(98)

[rs|tu] ) 12 ∫Γ dsb[Vrs(sb)Etu(sb) + Vtu(sb)Ers(sb)]

(99)

â )(nS

2 - 1)

4πnS(nS +ωM

IS)

(100)

Gdis ) tr P[hdis + 1/2Xdis(P)] (101)

(hdis)ru ) -â

2∑st

[rs|tu](S-1)st (102)

(Xdis)ru(P) )â

2∑st

[rs|tu]Pst (103)


drawn from experiments as well as from calculations:

The assumption that dispersion energy in morecomplex systems can be described as a sum of atom-atom terms has been expressed, and accepted as areasonable approximation, for many years (see ref335 and references cited therein). Accepting thisapproximation and indicating by m the atoms of thesolute and by s the atoms of the solvent molecules,the dispersion energy can be written as a sum ofvolume integrals, which in turn may be reduced tosurface integrals with the aid of appropriate vectorfunctions ABdis,ms

(n) (rbms) and reduced to a finite summa-tion of elements defined for each tessera of the cavitysurface. The procedure exactly parallels that in usefor the repulsion term, and there is no need to repeatwhat has been summarized in the preceding sectionand in the 1994 review.1 Here, it is sufficient to saythat the form of the dispersion contribution to thesolvation free energy is exactly equivalent to thatreported in eq 96 for the repulsion interaction, butthis time the vector functions ABdis,ms

(n) (rbms) become

The serious limits of the use of atom-atom terms forthe description of interactions within small clusterscomposed of polyatomic molecules are well-known.In passing to solutions the constraints imposed bythe model (a fixed cavity, a continuous distributionof the solvent molecules) a part of the artifactsspecific to the atom-atom model are eliminated.However, the final values of dispersion and repulsionenergies depend on the values of the coefficients usedin the potential. The availability of QM versions forGdis (as well as for Grep) makes it possible to define acomputational strategy to get independent values forthese coefficients.336

3.2.4. Charge Transfer TermAmong the various components of the intermolecu-

lar interaction energy in small clusters we havementioned in section 3.1, there was a charge transferterm that was not included in the decomposition ofthe solvation energy we presented in section 3.2.Indeed, charge transfer (CT) is an important termin the decomposition of the variational definition ofthe interaction energy ∆E, even if it is not includedin the standard definition of this quantity performedwith the aid of perturbation theory.

We have not put a CT term in the decompositionof ∆Gsol because CT effects are quite specific anddirectional and, thus, not appropriate for descriptionby continuum models that are essentially based onan averaged distribution of solvent molecules. How-ever, ionic and some strongly polar solutes tend toaccumulate electronic charge from the media ordistribute it to the media, irrespective of the local

orientation of the nearby solvent molecules. Envisag-ing processes of just this type, Gogonea and Merz337,338

have proposed an addition to QM ASC methodsincluding CT effects. The procedure starts from astandard continuum BEM calculation giving theapparent charge distribution, called by the authorsσRF to signify that it depends on the reaction field(RF). The CT process is mimicked by introducing anadditional CT charge, σCT, spread over the samecavity surface. This charge is related to a new conceptintroduced in the QM formulation of the theory, thefractional electron pair (FEP). The FEP is added tothe LUMO, or deleted from the HOMO of the solute,with the continuum medium acting as a generatoror a sink of electronic charge, respectively.

The whole calculation consists of an iterativescheme which can be schematized in three steps.

First, the dielectric is charged by distributing σCTas a charge density onto the BEM grid. Then RFcharge densities are obtained from the BEM equa-tions

where σRF and σCT are vector quantities containingthe set of RF and CT charge densities, respectively,and ε0 and εi are the solvent and solute dielectricconstants. In eq 106 T is a vector having elementsthat are normal components of the electric fieldgenerated by the solute charge density on the bound-ary elements, and the Kij terms depend only on theshape of the dielectric interface and have the form

where i and j are boundary elements and ai is thecorresponding area.

Finally, the RF and CT charge densities are fedinto the Hamiltonian, and the solute is (dis)chargedby adjusting the density matrix during the SCFprocedure (FEP creation) such that the total chargeon the solute is less qCT

where k is either the HOMO or the LUMO orbital,which is determined by the direction of charge flow,and qCT is the pointlike equivalent of the σCT at eachboundary element, with

The whole procedure has to be iteratively repeatedto get convergence on the SCF energy. The finalexpression of ∆Gsol has an additional term, called GCT,whereas the other nonelectrostatic terms are de-scribed with the standard procedures we have docu-mented in this section.

Vdis(rms) ) ∑n)6,8,10

-dms

(n)

rmsn

(104)

ABdis,ms ) -dms

(6)

3rms6

rbms (105)

(2πI -εi - ε0

εi + ε0K)σRF )

εi - ε0

εi + ε0[T - 1

ε0(2πI - K)σCT]

(106)

Kij ) ∫ai

(rbi - rbj)‚n

|rbi - rbj|3dai (107)

Pµν ) 2 ∑i

occ

cµicνi + qCTcµkcνk (108)

qCT )

<0 charge flow: solute f dielectric (k)HOMO)>0 charge flow: dielectric f solute (k)LUMO)


We have reported some details of this method,surely not sufficient to provide readers with all of theelements for a full understanding of the procedurebut sufficient to show that, in the case of need,extensions of continuum solvation methods to phe-nomena in which the medium acts as a sink or asource of electronic charge may be implemented.Phenomena that could be studied with models of thistype can be easily identified. Gogonea and Merz givesome examples. Here we add two other examples.

In the study of molecular anions of relativelycomplex structure there is a growing interest inexamining the relationships between standard va-lence-type anions and dipole-bound anions.339,340

Computational models in which the medium effectis expressed only in terms of a few solvent moleculesplaced at the positions in which they more stronglystabilize the systems could unduly alter the relativestabilities of the two types of anions. Artifacts ofsimilar type, due to the use of small optimizedclusters, are well-known in systems ruled by complexequilibria of hydrogen bonds. The field of molecularanions is far less explored than that of hydrogen-bonded systems, and so artifacts in the descriptionof these relative labile systems are less easy to detect.It is possible that an appropriate continuum-discretemodel including CT with the continuum could leadto a more balanced appreciation of the relativestabilities of the various species. To close this remarkon molecular anions, we note that the scant experi-mental evidence indicates that other physical effects,such as electronic polarization and dispersion, couldplay an important role.340 Both effects can be de-scribed with fair accuracy and with limited compu-tational efforts by using continuum models.

Rydberg states, so common in isolated molecules,are rarely detected for molecules in condensed media.The large size of the Rydberg orbitals would lead toa considerable spread of the corresponding electroniccharge on the solvent molecules. This hypotheticalphenomenon is surely counterbalanced by repulsionforces related to the exchange, or Pauli, effects, butthe possibility of a partial CT to the medium cannotbe a priori excluded, and this possibility has not beenexplored, at the best of our knowledge.

The extensions of the continuum model we havehere considered violate the basic assumption ofadopting a Hartree partition between solute andsolvent. The use of this partition, shared by allcontinuum and discrete solvation methods (with theexception of solvent molecules included in a QMsolvation cluster), means that the solute chargedistribution FM may be polarized, or distorted, by theinteraction with the medium, but with its integralvalue left unchanged. This assumption is not adogma, however, and can be relaxed, opening newperspectives to continuum models. The Gogonea andMerz model is too crude to be directly used to exploresuch perspectives, but theoretical chemists are ac-customed to improving and refining crude models.

3.2.5. Definition of the Cavities in the Calculation ofSolvation Energy

In the preceding sections we have encountereddifferent types of cavities and given some remarks

about them. We have now more elements to completethe presentation reported in section 2.2 with acomprehensive display of the cavities in use for thedifferent terms of the solvation energy.

Cavity Formation Energy. The surface is of vdWtype when the SPT approach is used, whereas acavity with a size between vdW and SAS is recom-mended when a surface tension approach is used (seesection 3.2.1.2).

Repulsion Energy. In semiclassical methods thecavity surface is of SA type. Actually some versionsof PCM use a hierarchy of SA surfaces, for thereasons explained in section 3.2.2. When the QMrepulsion method is used, the cavity is brought tocoincide with the cavity used for the electrostatic withthe introduction of a simple scaling in the numericalfactor R defined in eq 93.

Dispersion Energy. The same remarks reported forthe repulsion term hold also for the dispersion term,both for the semiclassical model and for the QMmodel.

We note that in the parametrized classical models,which calculate Gdis, Grep, and Gcav as unique contri-butions (see, for example, section 2.3.3), the SAsurface is generally used.

Electrostatic Energy. There are codes using cavitiesof regular shape. For cavities of molecular shape thesemiclassical methods use both SA and SE cavities.In QM calculations use is made of SESs. However,in most cases (e.g., by adopting the GEPOL ap-proach), such surface is not obtained as suggestedby Connolly (i.e., considering the surface contactdetermined by a rolling probe sphere) but as thesurface obtained by interlocking spheres of properdimension and position (see section 2.2). In this case,a scaling factor (f > 1) is generally introduced for thevdW radii. The standard value for f is 1.2: this factorwas originally introduced in the first PCM paper48

on the basis of considerations and tests that havebeen summarized on several occasions (see, e.g., ref341). Here it is sufficient to recall that these tests,although of different types, were of relatively lownumerical accuracy and the value f ) 1.2 waspresented as a provisional value. Tests performed inthe following years confirmed that f ) 1.2 was a goodguess, and this value is still in use. There are,however, other suggested values for f. Some amongthem are based on the measure of some propertiesof the electron distribution of the solute.342 Thesedefinitions have the defect of asking additionalcalculations, a strategy in contrast with the basicphilosophy of continuum methods, as we have com-mented in the preceding sections.

Other suggestions are about the modification of thevalue of f for some specific group.343 A considerablenumber of studies have been performed by Luque andOrozco and co-workers for different solvents. Accord-ing to their analysis the electrostatic term has to bedetermined by using a solvent-exposed surface cre-ated by scaling the atomic radii (taken from thePauling set) by a solvent-dependent factor, whichadopts values of 1.25, 1.50, 1.60, and 1.80 for neutralcompounds in water,344 octanol,345 chloroform,346 andcarbon tetrachloride, respectively. For small monova-


lent ionic compounds, the same authors347 suggestedthat the solute cavity be reduced with respect toneutral species: the optimum cavity for the hydrationof ions was defined by scaling the atomic radii by afactor of 1.10-1.15. This reduction was justified bythe fact that due to strong electrostatic interactionssolvent molecules are in general closest to the atomsof charged molecules rather than those of the neutralones. Very recently, a further refinement has beenintroduced;348 instead of reducing the whole cavityof the solute, which might be appropriate for smallsize ions, the scaling down is now limited to the heavyatom (and the hydrogen atoms attached to it) thatbears the formal charge.

In 1997 Barone et al.349 presented a set of rulesfor determining the atomic radii of spheres used byGEPOL to build the molecular cavities within theDPCM solvation model. The procedure was appliedto compute the hydration free energy for moleculescontaining H, C, N, O, F, P, S, Cl, Br, and I at theHartree-Fock computational level with a mediumsize basis set [6-31G(d) with eventual diffuse func-tions for anionic species]. This new definition of thecavity, which was referred to as the united atommodel for Hartree-Fock (UAHF), was based onchemical considerations, basically hybridization, for-mal charge, and first-neighbor inductive effect. TheUAHF parameters were obtained by imposing twoinitial conditions; the hydrogens do not have indi-vidual spheres, but they are included in the spheresof the heavy atoms to which they are bonded (unitedatom model), and the elements of each periodic tablerow have the same ‘‘basic” radius, modified by themolecular environment. As previously suggested byOrozco and Luque, ions present atomic radii differentfrom (and smaller than) those of the correspondingneutral molecules.

Due to its implementation in the Gaussian code,17

the UAHF method has been used in many studies;it is, however, important to note here that thismethod works well when all of the conditions usedin the original fitting procedure are maintained,namely, the use of DPCM solvation model and theHF/6-31G(d) level of calculation. Extensions to othermore recent versions of PCM (CPCM or IEFPCM) orto other QM levels (for example, DFT), or only toother, larger, basis sets, in fact, cannot reproduce thegood results of the original version. It is also impor-tant to note here that, due to this limitation, the newversion of Gaussian code (G03) no longer considersUAHF cavities as the recommended (or default)cavities for PCM (which, in turn, is no longer DPCMas default but IEFPCM) and, thus, careful attentionhas to be paid by all users.

We terminate this brief excursion on differentproposals to define cavity radii by quoting a paperby Dupuis and co-workers350 in which they describea new protocol to define the radii of the spheres tobe used for oxoanions, XOm

n-. These radii are deter-mined by simple empirically based expressions in-volving effective atomic charges of the solute atomsthat fit the solute molecular electrostatic potentialand a bond-length-dependent factor to account foratomic size and hybridization.

3.2.6. Contributions to the Solvation Free Energy Due toThermal Motions of the Solute

We have so far considered the contributions to thesolvation energy that may be computed at fixedgeometry of the solute (BO approximation). Thesecalculations can be extended to the whole space ofthe nuclear conformations, a 3N-6 space, that willbe indicated by R. The energy G(R) (definition 1,see eq 78) defines the PES surface of the focusedsystem in solution. We remark that to act as PES,the ∆Gsol(R) function (definition 2, eq 79) must besupplemented by an E0(R) function, which has to beindependently computed.

The G(R) function replaces the corresponding E(R)function defined for the isolated system. The differ-ence is in the physical nature of the two surfaces (acomponent of the free energy versus an internalenergy) and not in their use. We remark, however,that for the study of some phenomena, it is conve-nient to expand the R space to an RxS spaceincluding dynamical solvent coordinates.351-357 Thedefinition of “solvent coordinates” is indeed a veryimportant subject to which continuum solvationmethods have been successfully applied. In thisreview, we have decided not to consider this topic,not because of less interest but just because thesubjects to treat are so many that we had necessarilyto make a selection. Therefore, here, we shall notconsider dynamical couplings of solute and solventcoordinates and we shall limit ourselves to analyzeG(R).

In analogy with what is done for the E(R), eachvalue of G(R) has to be supplemented with additionalcontributions due to the internal and external mo-tions of the nuclei. We are here only interested inthe contributions permitting the calculation of thefree energy. For E(R) these contributions depend onthe temperature; for G(R) the temperature has beenalready fixed in the definition of the PES (thedielectric constant ε as well other solvent propertiesused in the calculations depend on T). The calculationof the free energy is often limited to the critical pointsof the PES corresponding to local minima and totransition states (saddle points of the first type: SP1)for which the calculation of the internal motionscontributions (i.e., vibrations) is computationallysimpler and similar in vacuo and in solution.

For the contributions due to the external motions(translation and rotation of the whole molecule), theprocedure in solution is not so simple as for moleculesin the gas phase. It is convenient to make explicitreference to Ben-Naim’s definition of the solvationprocess.358 According to Ben-Naim, the basic quantityis the free energy change ∆Gsol

/ for transferring thesolute from a fixed position in the ideal gas phase toa fixed position in the liquid phase. The constraintof fixed position in the liquid is released later, withthe addition of a further contribution to the energy,called “liberation free energy”, related to the momen-tum partition function ΛM

3 and to the number den-sity FM. The explicit formula is FMΛM

3 . An analogousrelease of the fixed position is introduced for M in


the gas phase, and the resulting solvation free energyis

where the indices g and s refer to M in gas phaseand in solution, respectively.

The freedom one has in defining standard statesmay be exploited to adopt, for the transfer process,those that give the simplest relationship between ∆Gsol

/ and ∆Gsol. This process is defined by Ben-Naimas the “F process”, and, in practice, it corresponds tothe use of unitary molar concentrations for bothstates. In this case we have ∆G(F process) ) ∆Gsol

/ .Note that this noticeable simplification is only validfor the Gibbs free energy; for other thermodynamicfunctions the equality between starred and nons-tarred quantities does not exist. Within this process,we may write the explicit and complete expressionof the solvation energy as

The last two terms have been grouped in eq 79 intothe ∆Gtm term. The contributions related to themolecular motions are expressed in terms of themicroscopic rotational and vibrational partition func-tion of M (the last term also includes zero vibrationalenergy contributions). The usual recipes and simpli-fications used for gas phase calculations can be usedhere; the only term requiring some specific modelingfor solutions is the rotational contribution.

The last term we have to consider is the P∆Vcontribution we have included in eq 79. With ourchoice of standard states, the value of this contribu-tion is zero. It assumes other values when otherreference states are adopted: the contribution to theenergy due to a change of reference is called the craticterm.

There is an additional formal point to make con-cerning the free energy and the thermodynamicensemble used.

The expressions we have considered refer to the(TPN) ensemble, in which G (Gibbs free energy) isthe leading free energy term. In this case, theHelmholtz free energy A is given by the followingrelationship:

Ben-Naim gives359 some numerical examples toshow that in normal cases the last term of thisequation is exceedingly small; the ratio P∆V*/∆G*ranges in the interval of (1-0.1) × 10-3.

In the canonical ensemble (TVN) the leading freeenergy is that of Helmholtz, but when the averagevolume ⟨V⟩ in the (TP) ensemble is taken to be equal

to the volume V in the canonical ensemble, thefollowing relationship applies

even if the two quantities are defined with twodifferent processes.

What we have here reported should be sufficientto indicate that people performing calculations onsolvent effects with computer simulations may useeither ensemble and either definition of the freeenergy to arrive at practically the same results.Continuum methods, instead, are characterized bythe use of dielectric properties measured at givenvalues of T and P. Thus, the thermodynamic en-semble they formally refer to is the TPN, but fromthe numerical point of view there is an almostcomplete agreement with the alternative TVN asliquids are practically incompressible. We note thatan exception exists, for example, the supercriticalfluids, as for them the incompressibility approxima-tion is not valid.

Thermodynamics, however, is a rigorous disciplinehiding many subtleties, and also in the apparentlysimple definition of the basic solvation phenomenathere have been fierce debates. We recommendfamiliarity with Ben-Naim’s books;358,359 his approachhas in fact important merits: it is applicable to liquidsystems of any type, without limitations in theconcentrations; it allows the use of microscopic parti-tion functions; it avoids assumptions about thestructure of the liquid and the introduction of arti-ficial concepts such as the free volume available tothe solute; it can be applied to heterogeneous sys-tems; and it gives, as we have seen, a transparentconnection between the computational continuumprocedures and a sound thermodynamic formulationof the solvation process.

4. Nonuniformities in the Continuum MediumWe reconsider the electrostatic aspects but with

extensions toward a more detailed description. Weshall consider, once again, liquid media at equilibri-um, but we shall release the constraint introducedin section 2 of a uniform and homogeneous medium.

The reasons suggesting, or compelling, the releaseof this constraint are numerous and of very differenttypes. We shall not present a unified description ofthese reasons, preferring to give first a synopsis ofsome aspects of the dielectric theory for molecularliquids we have discarded in section 2 and then toanalyze three specific problems, namely, nonlocaleffects, nonuniformities around ionic and neutralsolutes, and systems with phase separation. Beforeentering into the details of these phenomena, webriefly review the main aspects (and the relatedliterature) of the dielectric theory going beyond thelinear approximation.

4.1. Dielectric Theory Including Nonlinear EffectsThe electrostatic part of the solvation model we

have thus far considered is based on the assumptionof a linear relationship between the medium polar-ization function PB and the strength of the electric

∆Gsol ) ∆Gsol/ - RT ln

nM,gΛM,g3

nM.sΛM,s3

(109)

∆Gsol ) ∆Gel + Gcav + Grep + Gdis + RT lnqrot,g

qrot,s+

RTqvib,g

qvib,s(110)

∆G*(TPN) ) ∆A*(TPN) + P∆V*(TPN) (111)

∆G*(TPN) ) ∆A*(TVN) (112)


field EB acting on the medium and giving rise to thispolarization (see eq 4). In other words, the model isbased on the linear regime of the dielectric responseof the medium. To consider nonlinear responseregimes, one has to develop PB in powers of EB,according to the standard practice of response func-tions. This analysis has been done by physicists, withmeasurements on macroscopic dielectrics, since the19th century. The effects measurable at the macro-scopic level can be described with only the firstnonlinear term. The experimental determination ofthe nonlinearity coefficient is usually performed bysuperimposing a weak alternating field to a strongstatic field and measuring the slope of the dielectricdisplacement function DB, which has the same non-linear behavior as PB

where ø(3) is the third-order susceptibility of themedium. We note that the first nonlinear term isrelated to the third-order susceptibility as PB issymmetric with respect to a change of sign of the fieldEB and thus only odd powers appear.

The quantity that can be measured is an incre-mental dielectric constant, which depends on thefield, namely

From eq 114 it follows that nonlinear effects can bedescribed in terms of the quantity ∆ε/E2 ) 12πø(3).Usually this term is negative; that is, the realpermittivity is smaller than expected on the basis ofa linear response (the “normal saturation” effect), butit may also be positive; several effects may in factcontribute to determine the experimental value ofthis term.

The main problem in the formulation of electro-static theories based on nonlinear effects lies in thefact that the Poisson equation found to be valid forlinear dielectrics in all regions without charge doesnot hold in the nonlinear regime. The solution of thiselectrostatic problem is quite complex, and a fulltheory has never been completely settled.360

Here, we shall limit our attention to the staticaspects of this problem for liquids, in particular, forionic solutions, for which there is an abundantmethodological literature covering almost a century.

The formal basis was the elaboration of the dielec-tric problem with a molecular description in a con-tinuum framework given by Debye,361,362 who re-placed the external field EB acting on a dipole of themedium with the internal field of Lorentz363 andintroduced an expansion of the Langevin function(conceived for magnetic moments). Similar but notidentical versions of the theory, making use of anexpansion of the Langevin function, were publishedby Webb364 (who also introduced electrostrictioneffects), Sack,365,366 and Ingold.367 The main limita-tions of these approaches were mostly due to the useof the Lorentz field, which is not adequate for thisscope.

Substantial advances were given by the Onsagertheory,131 with the separation of the reaction field

from the internal field and the definition of a cavityfield, by Kirwood130 and later by Frolich,368whoimproved the theory of the dielectric constant withstatistical mechanics arguments. This collective workpaved the way to the elaboration of better theoriesof liquid dielectrics with saturation effects, done byBooth,369 which has been amply used but also criti-cized,370 and then by Harris and Alder.371

This brings us to the beginning of the second halfof the past century, immediately before the adventof the era of computers. Computer simulations andadvances in the integral equation formulations ofstatistical mechanics have opened new perspectivesto the study of liquid systems. In addition, newaspects have been put in evidence by the impressiveadvances in the experimental techniques, especiallyof spectroscopic type. This evolution in researchthemes and techniques could open another perspec-tive for the use of continuum solvation models towarda more detailed description than in the past, and thisis the reason we are here considering fine details ofthe picture, such as nonlinearities and nonlocaleffects.

4.2. Nonlocal Electrostatic TheoriesThe nonlocal electrostatic theory has been intro-

duced in the studies of solvation by Dogonadze andKornyshev,372,373 borrowing it from solid state physics.The basic point in this approach is the introductionof nonlocal structure in the susceptibility connectingthe polarization vector PB to the electric field EB. Inthis formulation the celebrated relation (eq 4) be-comes

The susceptibility is no longer a constant factor,namely, (ε - 1)/4π, as in the standard description ofhomogeneous isotropic solutions, but the kernel of anintegral equation depending on the field felt at allpositions of the medium.

The nonlocal electrostatic theory gives an alterna-tive approach to the description of the local anisot-ropy of the medium traditionally interpreted asnonlinear dielectric saturation effects. The nonlocaltheory in use is a linear theory that can be appliedby retaining almost all of the elements of the stan-dard continuum solvation energy, such as the mo-lecular cavity and the nonelectrostatic terms.

During the years, the theory has noticeably pro-gressed since its first formulation given in the alreadycited papers. Kornyshev himself has further devel-oped its original theory,374-376 but the formal andcomputational advances most directly related tocontinuum methods are due to Basilevsky andParsons,377-379 who developed a model to account fornonlocal effects within an ASC framework. Accordingto this model, the solute charge density is embeddedin a cavity, whereas the external volume is describedin terms of the equations of nonlocal electrostatics.The susceptibility kernel is then described as a stepfunction on the boundary surface of the cavity,namely, ø(|rb - rb′|), if rb and rb′ are outside the boundaryand 0 if inside. This model, accounting for matching

DB ) EB + 4πPB ) (ε + 4πø(3)E2)EB (113)

εE ) ∇EB‚DB ) ε + 12πø(3)E2 (114)

PB(rb) ) ∫ d3rb′ø(|rb - rb′|)EB(rb′) (115)


conditions on the boundary surface, can be solvedanalytically for smooth Lorentzian models of thedielectric function ø(k), which is the Fourier trans-form of the kernel ø(|rb - rb′|). However, studies ofmicroscopic molecular solvent models have broughtinto doubt the validity of the Lorentzian solventmodel, suggesting that ø(k) has a pole structureleading to an oscillatory behavior in the correspond-ing polarizability kernel ø(|rb - rb′|). The most strikingresult of the pole model, as applied to solvation effectsfor spherical ions, is an irregular behavior in thesolvation energy as a function of ion radius a.Following the analysis made by Basilevsky andParsons, it appears that the addition of pole termsto a Lorentzian function for ø(k) violates the positivedefiniteness of the integral operator governing thebasic equation of the theory: this is the fundamentalorigin of the instability. According to the two authors,oscillations introduced by poles in ø(k), related tospatial correlations inside a solvent, are incompatiblewith the frozen solute cavity model, in which thecavity radius is considered as an intrinsic propertyof a given solute ion. They thus introduce modifica-tions of the cavity model to get more flexibility byallowing for the mutual adjustment of the cavity sizeand solvent structure in the close vicinity of the ion.Doing this, they almost completely return to a theorywith a positive-definite kernel in which no instabili-ties arise.

4.3. Nonuniformities around Small IonsThe phenomenon most thoroughly studied in con-

nection with the problems we are considering in thissection is that of solvation of small ions. The firstreason is the importance ionic solutions have inchemistry; the second is the simplicity of the modelsfor isolated ions, an aspect making easier the formu-lation of theories to extend later to more complexsolutes.

4.3.1. ε(r) ModelsThe systematic calculations of the solvation free

energy of ions date back to Born’s seminal paper.148

The use of this model was appealing because of itsmathematical elegance, and in fact it was immedi-ately applied to a variety of solvents. However, a fewyears later, it was realized that the Born equationonly qualitatively accounts for much of the availableexperimental data. The most evident deviation withrespect to such data was a deviation toward toonegative free energy values. Attempts aiming ateliminating the “errors” of the model by increasingthe ionic radius beyond its crystallographic valuehave been performed.380-384 This correction consider-ably reduced the errors, but the increase of the radiusfor highly charged ions was excessively large, and thelack of a serious justification of the procedure (despiteseveral attempts) led the scientists to consider thisapproach purely empirical and of no help for anexplanation of the phenomenon.

In addition to the ionic radius, the Born modelinvolves a second parameter, namely, the dielectricconstant. An interpretation of the “failures” of theBorn prediction in terms of nonlinear effects in the

dielectric response was immediately advanced (actu-ally the ions were considered to be the only source ofnonlinearities in liquids not subjected to externalfields).385 We have already presented the evolutionin the elaboration of a dielectric theory for dipolarliquids. Following this evolution, a permittivity func-tion ε(r), replacing the dielectric constant (or permit-tivity) ε with a function of the distance from the ion,was introduced. We note that the spherical symmetryof the ionic systems permits the reduction of theposition argument of the function, rb, to a singleparameter only, the distance r.

The ε(r) function is an empirical function; thetheory of dielectrics is not able to give the detailsnecessary for an accurate definition of this function.It is in fact an approximate and not fully developedtheory, as we have seen in section 4.1, collecting intoa single term (∆ε/E2 ) 12πø(3)) contributions ofdifferent physical origins. On the other hand, empiri-cal determinations suffered from the scarcity ofexperimental data, which in practice are limited tothe integral values of the thermodynamic solvationfunctions and thus subject to uncertainties and alsoincluding nonelectrostatic contributions. As a conse-quence, the attempt to define appropriate analyticalforms of ε(r) was a delicate task requiring an accurateevaluation of the available data and some ingenuity.The papers we cite here383,386-398 have been selectedfrom a longer list because they present analyses andremarks that might be exploited in further elabora-tions of the model. Having in view these possibledevelopments, it is worth citing some papers byEhrenson, who has given illuminating analyses onthis subject.399-401

4.3.2. Layered Models

Another approach to describe the nonuniformities(or local anisotropy) of the medium around ions isbased on the introduction of a layering of the solvent,with each layer characterized by a different dielectricconstant.

Layers in continuum models were first introducedby Kirkwood130 and by Oster402 to treat ionic solutionsand mixed solvent around a dipolar solute, respec-tively. Here we consider layers with progressivelyincreasing permittivities to mimic the dielectricsaturation effects.

Analytical formulations of the layered continuumelectrostatic solvation model were first given byBeveridge and Schnuelle403 and then by Abraham etal.404 in the framework of the MPE formulation (seesection 2.3.2). The one-layer version of these modelshas been employed by Abraham in a set ofpapers405-410 on monatomic ions in a set of solventsto get appreciably accurate values of the thermody-namic solvation functions and a good deal of informa-tion about the properties of the solutions. The layeredmodel, called “dual solvent region” method, has beendescribed by Ehrenson in the above-cited criticalexamination of dielectric saturation models as sim-pler and more transparent and giving results com-parable to the those of the best ε(r) models. The“transparency” noted by Ehrenson is due to thevalues of the two additional parameters present in


this model, the value of the dielectric constant in thelayer, and the thickness of the layer, which have beennumerically optimized in the context of the citedstudies.399-401 In the dual solvent region model, thedielectric constant of the layer corresponds, in a goodapproximation, to the optical dielectric constant ofthe solvent and its thickness to the solvent moleculardiameter. The numerical optimization leads to adescription of the phenomenon acceptable on intui-tive grounds; the first solvent layer is immobilizedand dipole-oriented, contributing to the polarizationwith its electronic term only. Of course, there iscompensation of effects, but these findings reinforcedthe opinion, already expressed in the literature, thatpure continuum models could exploit the differencebetween the electronic and total permittivity tointroduce effects due to electric saturation.

Considerations of this type have led Basilevsky andco-workers to formulate a layered model withoutconstraints on the symmetry of the two surfaces (itis based on an ASC approach).411-413 The approach,called the frequency-resolved cavity model (FRCM),is based on the use of the electronic permittivity inthe first zone and on the static value in the outerspace; more details on this method will be given insection 5.3.

4.3.3. Molecular Cluster Models

A third way to describe dielectric saturation effectsaround ions consists of replacing the continuousdielectric in the vicinity of the ion with solventmolecules. We are here at the border between twoapproaches, the continuum approach, which addssome discrete molecules in critical positions, and thediscrete model, which replaces distant molecules witha continuous distribution. We shall return to thissubject, with a discussion of where to put the borderbetween the two approaches (see section 7). Here weare in the continuum domain with the addition ofdiscrete elements.

Attempts at inserting some discrete solvent mol-ecules in the Born-like description initiated very earlyto try to eliminate the “failures” of this model; wereport here a selection of some old references.390,414-419

By comparing these references with those reportedin section 4.3.1 (more recent), it appears that themain attention has shifted from solvation energiesto the structure of the liquid around the ion. Thesereferences are also testimony to the beginning of theuse of QM calculations in the study of condensedsystems. Actually, the discrete solvent molecules usedin these, and in other successive papers, were mod-eled at a level that was practically unable to describethe electronic solute polarization considered to beessential in the models of the preceding subsections.

We have already described several shortcomingsof the use of semiclassical discrete solvent models forsimulations. In the present case, the need for usingsimplified models is less justified, because the num-ber of molecules is small.

The models to be used here should combine thefeatures necessary to describe molecules in a liquid,which are different from those of molecules interact-ing in the gas phase, and the features specific for

molecules feeling the strong field of the ion. Anattempt to formulate a procedure able to introduceboth features in semiclassical models was elaboratedby Floris et al.420-424 According to this procedure, abinitio effective pair potentials can be obtained usinga continuum solvation model (the PCM in the citedpapers). In this way, many-body effects are includedin the ion-solvent potential at the same time keepingthe computational convenience and simplicity of two-body functions in the development process. Thesepotentials are then used in fully discrete computersimulations. Molecular dynamics in particular isappropriate for the study of the dynamical featuresof the system, such as the persistence time of asolvent molecule in the first or second solvation shell,the determination of the diffusion coefficients, thespectra of the solvation shell, and the dynamics ofexchange of solvent molecules of different nature inmixing solvents. All of the aspects we have herementioned have been studied for a variety of cationsin the above-cited papers, to which we add here amore recent one on Cd(II) in aqueous solution.425

Other semiclassical potentials including polariza-tion have been employed. We cite among them theeffective fragment potential (EFP) developed by Gor-don and collaborators,426 which has given good resultsin modeling solvent molecules, especially for neutralsolute (see also section 7.3.2). The EFP descriptionpresents shortcomings when there is remarkablecharge transfer such as in anions and in somecations.427,428

The use of a full QM description of the cluster,however, is advisable when the property underexamination is suspected to be sensitive to layeringeffects. It is not yet fully ascertained when anaccurate and well-balanced description of the solva-tion cluster is compulsory, but surely it is convenientto have available descriptions of this type. An ap-preciable number of studies have been performedon isolated metal-solvent clusters of the type[MSn]k+,429-433 but often, when the number n ofsolvent molecules reaches or exceeds the numbernecessary for the first solvation shell, the geometrydoes not correspond to that found in solution. Theconcept of a “hydrated ion” as a new species formedin solution by the ion and a limited number of watermolecules is now widely accepted.434 A simplifiedstrategy is often employed to study these systems.It consists of performing a geometry optimization ofthe hydrated ion in the gas phase followed by arefinement of the geometry with the hydrated ionwithin the continuum solvent.435,436 It is noteworthythat these geometry optimizations of the clusters aresensitive to details in the continuum solvation model.This fact was pointed out by Sanchez Marcos and co-workers,436 who noticed that there was ambiguity inthe results according to the procedure used. Inparticular, it was shown that the incorrect lengthen-ing of the metal-oxygen distance found in someoptimizations with continuum models was due to thespherical shape of the cavity used in those models;with a molecular shaped cavity the correct shorteningis in fact recovered.


4.4. Nonuniformities around Neutral MoleculesSome of the early attempts to model dielectric

saturation for ions in solution were also extended tosmall neutral molecules, reduced to a point dipole.However, the electric field produced by a dipole isdecidedly smaller than that produced by a localizedcharge, and thus the saturation effect on the solva-tion energy is far smaller; this induced the consid-eration of ions only. More recently, the progress incomputational methods and the availability of dataon time-dependent phenomena have renewed theinterest in larger and more complex molecular sys-tems. For example, the method of Basilevsky and co-workers411-413 considered in section 4.3.2 is notlimited to atomic ions, but it can be extended tomolecular systems, charged or uncharged.

A method addressing equilibrium solvation ener-gies of polyatomic molecules has been recently pro-posed by Edholm et al.437 The description of thedielectric nonlinearity is based on a modification ofthe Langevin-Debye model,438 with a finite differ-ence method applied to calculate the spatial depen-dence of the electrostatic potential using a quite finegrid borrowed from DFT methods and limited to alayer of 1.2 nm (the solvent outside this layer isdescribed with a linear continuum). The method alsocontains the calculation of the nonelectrostatic terms,van der Waals and cavity formation, according to theschemes discussed in section 3. The solute is de-scribed at fixed geometry, not influenced by thesolvent, and with atomic charges drawn from theOPLS-AA force field.439,440 In addition, solvent polar-ization effects are neglected. The hydration freeenergy values for around 200 neutral organic com-pounds with their decomposition into electrostatic,van der Waals, cavity formation, and nonlinearelectrostatic terms are reported. The nonlinear con-tribution is decidedly smaller that the cavity forma-tion and van der Waals terms, but comparable to oreven larger than the sum of these two terms, which,in water, compensate each other to a good extent. Theauthors remark that the extensive parametrizationused in the Cramer and Truhlar SMx models (seesection 2.3.3) should implicitly include nonlineareffects and that PCM (see section 2.3.1) uses adjustedcavity radii, which will quench the saturation effects.In our 1994 review, we explained the origin of thescaling factor, applied both to polar and nonpolarmolecules, and we confirm here that neither dielectricsaturation nor electrostriction (the opposite termrelated to nonlinearity) have been considered infixing this scale factor, but we accept the remarkabout the quenching.

Sandberg and Edholm have expressed two otherformulations of the nonlinear response solvation. Onealready cited paper398 involves spherical ions andgives an analytical formulation exploiting the sim-plifications made possible by the symmetry of theproblem. The other441 uses for the charge distributiona dipole, derived from the GROMACS force field442

and specialized for amino acids. These methods seemto be still in evolution, and it is convenient to awaitfuture developments to assess their validity to com-pute solvation energies.

4.5. Nonuniformities around Systems of LargerSize

In extending our survey to larger molecules weenter into a field dominated by computations onbiomolecules. The solvation methods we have de-scribed also apply to biomolecules, and we do notintend to open here a chapter dedicated to this veryimportant domain of computational chemistry, forwhich there is a wealth of specialized reports. Ourattention will be limited to the very specific aspectof the description of nonuniformities arising from thelarge size of biomolecules.

There is a long tradition of using molecular me-chanics (MM) methods to describe large molecules.Molecular mechanics, we remind, is based on empiri-cal force fields in which classical terms related tointermolecular energy deformations are accompaniedby site-site non-covalent interactions of van derWaals and electrostatic type. These last are ex-pressed in terms of point charge interactions, whichare not screened because electronic polarization ismissing in standard MM models. This defect of themodel has been empirically corrected by introducingin the calculation of Coulombic interactions a dielec-tric constant, with values ranging from 2 to 20 (thedefinition of a proper value of ε to be used withbiomolecules has given rise to considerable contro-versy).443-446 The fact that a “constant” value of ε wasnot sufficient, at least for groups bearing a netcharge, was soon noticed. In 1923 Bjerrum447 pro-posed the use of a distance-dependent dielectricfunction to describe the ionization constants of bi-functional organic acids. Bjerrum’s proposal was laterelaborated by Kirkwood and Westheimer448 andthereafter amply used, especially for the calculationof pKa of titrable groups in proteins. Numerouschecks have been done, and several modificationshave been proposed. There is in the literature asizable number of variants of a linear form of thedistance-dependent dielectric function: ε(r) ) kr, withk determined in different ways.

However, a linear relationship of this kind is notappropriate to describe the saturation effects deter-mining the nonlinearity (see section 4.1). Manyproposals have been suggested449-462 to take intoaccount the physical origin of the problem using oneamong the various models we have reviewed, ordefining, through other ways, ε(r) functions with asigmoidal shape.

Here we cannot discuss all of these proposals, butwe can point out some authors whose work has hada larger impact on the literature. Warshel et al.463

used experimental pKa values and redox potentialsto derive a distance-dependent dielectric function,and Mehler and Eichele464,465 introduced a sigmoidaldistance-dependent dielectric permittivity function tocalculate the electrostatic effects in water accessibleregions of proteins. For metal ion-nucleic acid in-teractions, Hingerty et al.466 proposed a modificationof Debye’s distance-dependent dielectric function,which is sigmoidal and rises rapidly with distance.An alternative mathematical expression for thisfunction was proposed by Ramstein and Lavery.467

Guarnieri et al.468 proposed an extension of Mehler


and Eichele’s work, in which the position dependenceof the screening is partly accounted for by usingdifferent values of a parameter for buried and forexposed atoms. More recently, in an attempt at amore generally applicable function, Sandberg andEdholm469 proposed a modification of the function ofWarshel et al., with parameters depending on themean distance of the atoms from the protein surface.Hassan et al.,470 following a formulation originallydeveloped by Bucher and Porter471 and subsequentlyby Ehrenson,472 obtained a reaction-field-correctedform of the radial-dependent permittivity profile.

In our rapid exposition we have simplified severalaspects of the methodological exposition of the useof “screened” Coulomb potentials (for example, nomention has been made of the use of models withoutand with solvent cavity). Interested readers arereferred to the above-cited papers to which we addhere an older but rich review by Mehler,473 butfurther comments as well as additional referenceswill be reported in section 7.1.

4.6. Systems with Phase SeparationWe consider another class of systems to which the

concepts related to nonuniformities in the localproperties of the liquid can be applied, namely,systems with phase separation boundaries. This is avery large class of systems, collecting elements ofquite disparate nature. We anticipate that in somecases the explicit consideration of local deviations ofthe liquid properties with respect to the bulk iscompelling, whereas in others the description of thesechanges is optional, in the sense that they need tobe considered only when one aims at a more accuratedescription of fine details not given by simplerdescriptions of the medium.

Local changes involve several aspects of the liquid,the density, the local concentration of dissolvedspecies (an aspect of particular relevance in ionicsolutions), the viscosity, and other parameters relatedto the motion of solvent molecules, and, of course, thedielectric function to which our attention will be herelimited. Before entering into details, we need tointroduce additional characterizing elements in ourset of systems, thus far defined only in terms of thepresence of phase boundaries. In this way, in fact,we can obtain subclasses that are more homogeneouswith respect to the methodological tools to use. Weshall limit ourselves to a rough level of classification,finer classification being necessary only for morespecific reviews.

An important feature, useful for our attempt ofclassification, is the length scale for the boundaries.This feature has been proposed to define ‘‘complexliquids” in which the appropriate scale length is inthe mesoscopic range.474 This definition of complexliquids collects, for example, colloidal suspensions,micellar solutions, microemulsions, closed bilayers,foams, gels, and other variants of the basic types wehave here mentioned as bicontinuous liquids. Weshall consider the complex liquids no further in thisreview, but we note that a description of complexliquids coherent with continuum models can be given,even if not in a straightforward way.

The local effects due to the surface play a far moreimportant role in systems with multiple mesoscopicphase boundaries than in systems with a single phaseboundary, and in general there will be the need ofextending the model to include mutual interactionsamong the mesoscopic domains, an extension thatwas not yet considered in the QM continuum models.We have here mentioned multiple mesoscopic sys-tems because in our opinion they represent one ofthe most important perspectives of development ofcontinuum models

Other subclasses with macro- or mesoscopic lengthscale for the boundary may be characterized by thenumber of phases, by their geometry, and by thephysical and chemical nature of the components.Some among them (liquid surfaces, pores, thin films,etc.) are the subject of intensive studies, deservingseparate reviews.

Let us here consider some examples of two-phasesystems, namely, systems with a liquid/gas, liquid/liquid, or liquid/solid surface. The continuum modelwe have examined in the preceding sections can beapplied to these systems with minor changes. Wethus return to models in which the focused compo-nent continues to be a single molecule or a smallcluster, whereas the remainder has a more complexdefinition, possibly including local modifications nearthe boundary (for the definition of focused models seesection 1).

The scientific and practical interest of two-phasesystems has led to a detailed experimental charac-terization of at least some among them, and it hasspurred a considerable number of attempts of theo-retical modeling also including continuum models.These models exhibit specific features as the char-acteristics of two-layer models may be quite different.For example, the characteristics to model in processesoccurring at the electrode surface are remarkablydifferent from those necessary in the study of theeffects of a metal on the spectroscopic properties of achromophore, and both have little to do with thoserequired to model processes and equilibria in aerosolsof environmental interest. For this reason we limitourselves to a few selected examples, covering thethree types of two-phase systems mentioned aboveand bearing a strict connection with the methodspresented in sections 2 and 3.

As remarked above, in some cases the explicit useof local corrections to the bulk properties is notcompulsory, and we start to examine some methodsof this simpler type.

An ASC method, based on PCM and aiming atstudying systems of this type, has been elaboratedyears ago by Sakurai and co-workers in Japan.475-477

The description of the medium is given in terms ofnonoverlapping portions of a homogeneous con-tinuum with different values of the dielectric constantwithin each portion. The code was later improved219,478

and used for a number of studies, mostly of organic-biologic interest, from which the versatility of theapproach may be appreciated.479-483

Among the successive variants of Sakurai’s proce-dures, one deserves explicit mention in this re-view.484,485 It involves a QM/MM method in which the


feature of separate dielectric domains is exploited toallow instantaneous differential polarization in theprotein fragments. In this model, called the polariz-able mosaic model (PMM), every covalent bond in theMM region is replaced by a cylindrical stick made ofdielectric, the permittivity of which depends on thenature of the original bond. As a consequence, aprotein molecule is represented by a mosaic consist-ing of cylinders with various polarizabilities. Thismodel offers a perspective of solution to two kinds ofproblems. First, other bodies can be described in afirst approximation as a whole region, introducingthen local anisotropies with a similar proceduresas,for example, a relatively homogeneous membranewith local elements involved in some local processes.Second, the introduction of mutual polarization of theclassical elements in QM/MM simulations is a com-putational challenge, and this approach is a possibleway of solving this problem.

The first version of PCM, namely DPCM, was alsoused to study problems characterized by the presenceof a boundary. The first paper, published in 1986,486

presented a model addressing energy changes in longDNA fragments subjected to large-scale deformationsand to local opening of the double helix. The problemsposed by this model are representative of thoseoccurring in “complex liquids” and in related systems.Around the double helix (a polyion) there is a portionof solvent showing large nonlinear dielectric effects,combined with partial counterion condensation, andboth effects change when the double helix is bent oropen. In this model, the local nonlinear dielectriceffects were described in terms of the layered modeldescribed in section 4.3.2. Another application wason the effects of partial solvation in modifying themolecular electrostatic potential of a substrate placedin the cleft of the active site of a model enzyme.487

Other studies involved polar solutes (with hydropho-bic tail) placed near the flat separation surface of twoimmiscible liquids (or at a liquid/vacuum separa-tion).488-491

In all of these DPCM studies, use was made of atessellation of the boundary (of planar infinite,spherical, or irregular shape, according to the case).More recently, the same systems were reformulatedwithin the IEF version of PCM.80 Within this formal-ism, by introducing novel Green-type operators ofanalytical form, tessellations of the infinite plane (orthe spherical boundary) separating the two phasesare no longer required, and thus the dimension of theproblem is identical to that of a homogeneous andisotropic solution. We note that full analytical Greenoperators are relatively easy to obtain for surfaceswith a regular shape but not for more complexshapes. In these cases it is thus useful to introduceGreen operators of semianalytical form (requiring apartial numerical integration) or to combine the IEFformalism with the original BEM formulation basedon a tessellation of the boundary surface.

The use of the IEFPCM formalism also allows usto introduce an ε(r) function for the liquid portion.The present version of the computational code candeal with ε(r) functions limited to a single spatialvariable (in practice, all of the models considered in

section 4.3.1); these kinds of functions, for example,are well suited for planar surfaces and spheres. Theextension to some other cases, such as cylindricalsurfaces or liquid with a simple surface combinedwith a tensorial permittivity, is immediate, but nottested yet. The extension to more anisotropic per-mittivities exhibiting a dependence on all threespatial coordinates would require a more intensivecomputational elaboration, but it is surely feasible.This model, which eliminates all of the divergencesoccurring in simpler image charge models with asharp surface, has thus far been applied to thecrossing of liquid/vapor and liquid/liquid surfaces byneutral and charged molecules, namely, a two-surfacemodel mimicking a membrane has been studied,80 aswell as the changes of the polarizability of the halideanions passing from water to air.492 In both cases thefunction, ε(r), has been modeled on the correspondingdensity profile, F(r), derived from simulations, as-suming proportionality between the two functions.

Another continuum electrostatic model to treatsolute at interfaces has been elaborated by Benjamin,especially to compute the spectral line shape at liquidinterfaces and surfaces.493 In this model based on theimage charge approach, the interface is described asa sharp boundary separating two bulk media, eachcharacterized by different optical and static dielectricconstants, and the chromophore is modeled as a pairof spherical cavities in which two equal and oppositepoint charges are embedded.

As a matter of fact, Benjamin’s contribution to theknowledge of interfaces and their modelization goeswell beyond this model.493-499 His main activity hasin fact involved molecular dynamics (MD) simula-tions and has been focused on the study of electronicproperties of chromophores as a function of theirlocation at the interface.

Let us now pass to another kind of system withboundaries, namely, that represented by lipid bilayermembranes. These systems have been treated withinthe framework of continuum electrostatics by con-sidering a slab of low dielectric material intended torepresent the hydrocarbon tails of the lipid molecules,surrounded by high dielectric material intended torepresent the headgroups and aqueous solution. Asa matter of fact, a solvated lipid bilayer consideredat the nanometer length scale is an example of asystem nonuniform in one dimension, that is, alongthe axis normal to the bilayer. In general, thedielectric permittivity for such a system will be afunction of the position along one axis. Furthermore,it will not be a scalar, as has been assumed invirtually all applications of continuum electrostaticsto molecular systems, but will be a second-ranktensor.

Several authors have presented methods for cal-culating the local dielectric constant for a nonuniformsystem. King, Lee, and Warshel500 and Simonson andco-workers450,501 have presented methods for deter-mining the dielectric constant of a spherically sym-metric but nonuniform system and applied them toproteins solvated in water. These methods are basedon a partition of the system into two or moreconcentric spherical shells. Once an estimate of the


dielectric constant in the outer shells is made, thedielectric constant in the inner shell can be calculatedeither from fluctuations of the total dipole momentfor that region or by direct observation of the polar-ization response to a uniform field. A dielectric profilefor a lipid bilayer in solution was presented earlierby Zhou and Schulten.502 Their approach was todivide the membrane into slices and compute thepermittivity for each slice with the usual expressionfor a uniform, isotropic system. The implicit assump-tion is that the slices do not interact, which seemshard to justify. More recently, Stern and Feller503

presented a method for calculating the static dielec-tric permittivity profile for a system nonlinear in onedimension under the periodic boundary conditionsused in computer simulations.

As a last example of systems with phase separa-tion, we consider liquid/metal systems with particularattention to the surface enhancing (SE) effects of ametal on the photophysical properties of a molecularchromophore placed in the vicinity of the surface.Some of these studies will be examined in more detailin sections 5 and 6, and here we limit ourselves topresenting some methodological aspects related tothe theme of this section.

The two models we consider describe the metal onthe same footing as the solvent in homogeneoussolutions, in the sense that the metal contributionsto the proprieties of the focused part (the chro-mophore) are treated in terms of a continuousresponse function without explicit consideration ofthe atomistic structure of the metal itself. Thedefinition of the dielectric function for a metal speci-men is more complex than for a liquid, and thusspecific modifications have to be introduced.

In the first model79,504-509 we cite (which belongsto the family of PCM methods), the metal is consid-ered to be a perfect conductor for time-independentexternal electric fields. To study photophysical prop-erties, however, one needs to introduce a frequency-dependent electrical perturbation and thus, in thiscase, the metal is considered as a dielectric with afrequency-dependent dielectric function. The startingpoint to model this basic quantity may be given bythe Drude theory presented in all textbooks of solidstate physics,510 which gives an expression of thedielectric function suitable for the bulk of the metal.The metal specimens used in SE measurements,however, often have a finite size, and are all subjectedto other sources of electric potential, static as wellas frequency dependent, related to the presence of achromophore of molecular size. The finite size ofmetal specimens first affects the relaxation time,which surely has not the value of the bulk used inthe Drude theory. To account for these effects, acorrection based on the Fermi velocity of the electronshas been introduced in the model. In addition, thedielectric response of the electrons in the metal hasa nonlocal character, in the sense that the polariza-tion vector at a given point depends on the value ofthe electric field over all other points. This nonlocal-ity, completely neglected in the original Drude for-mula, has been partially described in the model506

making use of a hydrodynamic model511 to introduce

a correction in the Drude expression, or, more ac-curately,508 using the Lindhard theory512 with Mer-min corrections for the finite relaxation time.513

Another correction introduced in the model involvesthe effect of surface roughness, found to be importantto realistically describe the numerical output of someexperiments. To this end a modified version of theGreen function proposed by Rahman and Maradu-din514 has been employed.508

The second model515-517 we cite assumes that themolecular system is enclosed in a half-sphericalcavity embedded in a linear, homogeneous, isotropicdielectric medium and adsorbed on the surface of aperfect conductor. The effects of the media on theproperties of the molecular system are describedusing the method of image charges. According to thismethod, a charge located in the cavity induces threecharges in the surrounding dielectric environment:one charge in the metal surface, due to the perfectconductor/vacuum interface, a second charge due tothe half-spherical environment between the vacuumand the dielectric medium, and a third charge thatis the image in the metal due to the interactionsbetween the perfect conductor and the inducedcharges in the solvent.

5. Nonequilibrium in Time-Dependent SolvationIn the sections 3 and 4 we have presented exten-

sions of the basic model but with the common featureof retaining the condition of equilibrium betweensolute and solvent. In this section we instead reviewthe physics we have to change and/or add to this basicmodel to properly describe time-dependent (TD), ornonequilibrium, processes.

TD processes embrace a large variety of phenom-ena, which span an impressive range of time scales(from 10-15 s for linear and nonlinear spectroscopyto 10-2 s for diffusion-controlled reactions in diluteaqueous solution). Obviously this broad range in-volves many physical aspects of solvation. Here, welimit ourselves to consider only those related to themost recent progress in the description of the dy-namic effects in QM solvation models.

Rigorously, the dynamic evolution of the molecularsolute should require that we postulate the properTD QM equations, which generalize the time-inde-pendent effective Schrodinger equation introduced insection 2.4 for the “static” basic model.

As a matter of fact, not all dynamic processesrequire an explicit TD description. For example, ifthe time dependence of the solute-solvent interac-tions is sufficiently slow (as in the case of diffusiverelaxation processes of the slow degrees of freedomof the solvent), it is possible to apply the adiabaticapproximation and the state of the solute at time tcan be obtained as the eigenstate of H(t) using time-independent formalism. In contrast, the dynamic QMequation for the solute must be explicitly introducedin describing its response properties to TD electro-magnetic perturbations. This aspect will be consid-ered in section 6 devoted to solvent effects on soluteresponse properties.

In the presence of a time evolution of the solutewe have also to extend the description of the solvent


response properties to the time (i.e., frequency)domain. In principle, we should consider all of thesolute-solvent interactions described in section 3,but here we shall limit ourselves to the electrostaticcomponent only. This is in fact the most importantwhen a dynamic evolution has to be studied.

5.1. Dynamic Polarization ResponseThe responses of the microscopic particles (mol-

ecules, atoms, electrons) of the solvent required toreach a certain equilibrium value of the polarizationhave specific characteristic times (CT). When thesolute charge distribution varies appreciably withina period of the same order as these CTs, the responseof the microscopic particles will not be sufficientlyrapid to build up a new equilibrium polarization, andthe actual value of the polarization will lag behindthe changing charge distribution.

The actual polarization P(t) is determined by theMaxwell field in the medium not only at time t butalso at a previous time t′, in a form expressed by theintegral518,519

where the kernel G(t - t′) is the solvent responsefunction and E(t′) the Maxwell field. Transformingthe convolution integral (eq 116) to the Fourierfrequency space we obtain520

where ε(ω) is the complex dielectric constant

The real part ε′(ω) is the frequency-dependentdielectric constant, describing the component of thepolarization density in phase with the oscillating fieldwhile the complex part, or loss factor, ε′′(ω), deter-mines the component of the polarization with a phasedifference of π/2 with respect to the Maxwell field,giving rise to the loss of energy of the electric fieldin the medium.

The frequency-dependent dielectric constant ε′(ω)and the loss factor ε′′(ω) can be determined byexperimental or theoretical methods.518 For mostpolar solvents, at low frequency, ε′(ω) is equal to thestatic dielectric constant ε, and ε′′(ω) is zero. As thefrequency increases, ε′(ω) initially decreases ratherslowly, then, in the IR, vis, and UV range of frequen-cies, there are sharp increases followed by a decrease.The loss factor ε′′(ω) has peaks in the neighborhoodof the frequencies where ε′(ω) changes.

The explanation of the correspondence in thebehavior of all polar compounds in an electromag-netic field is found in the fact that the electricpolarization is built up of three parts, the orienta-tional, the atomic, and the electronic polarization,each part corresponding to motions of a different kindof particle (molecules, atoms, electrons, respectively)with different CTs. As the frequency increases, theso-called nonequilibrium effects appear subsequently

in the different contributions of the polarization.Orientational contribution (CT below 10-12 s) willfirst start to lag behind the variations of the electro-magnetic field; next, atoms (CT around 10-14 s) willnot be able to follow the field, and finally, at veryhigh frequencies, electrons (CT around 10-16 s) alsowill lag behind the electromagnetic field.

Orientational polarization arises from differentrelaxation processes but, if we assume that it can bedescribed in terms of only one of these processes,characterized by a relaxation time τ, then the fre-quency dependence of the dielectric constant may beadequately described by the Debye equation

where ε∞ is the dielectric constant at a frequency atwhich the orientational polarization can no longerfollow the changes in the field; in other words, it isthe dielectric constant characteristic of the inducedpolarization.

Also, the behavior of the induced polarization,connected with atoms and electrons motions, in time-dependent fields can be described phenomenologicallyin terms of a number of resonance processes. Atfrequencies corresponding to the characteristic timesof the intramolecular motions by which the inducedpolarization occurs, there are sharp absorption lines,due to the discrete energy levels for these motions.By approximating these absorptions with delta func-tions, we obtain for the frequency dependence of theimaginary part

whereas the corresponding frequency dependence ofthe real part is obtained from one of the Kronig-Kramers relationships:

In the following sections we shall present twoclasses of phenomena for which the concepts hereintroduced about orientational and induced polariza-tion and nonequilibrium become of fundamentalimportance. These two classes can be seen as twofaces of the same, more general, problem of solvationdynamics, but at the same time they present verydifferent aspects. The first is in fact limited to a veryshort time scale, namely, that involved in a verticalelectronic transition, and thus only the inducedpolarization will be involved, whereas the secondcovers much longer times, thus allowing a completeanalysis of the solvent dynamic response due to bothresonance and relaxation processes.

5.2. Vertical Electronic TransitionsAll of the phenomena in which the solute undergoes

a sudden change in its charge distribution require a

PB(t) ) ∫-∞t

dt′G(t - t′)EB(t′) (116)

PB(ω) )ε(ω) - 1

4πEB(ω) (117)

ε(ω) ) ε′(ω) + iε′′(ω) (118)

ε′(ω) ) ε∞ +ε - ε∞

1 + ω2τ2(119)

ε′′(ω) )(ε - ε∞)ωτ

1 + ω2τ2(120)

ε′′(ω) ) ∑k

Akδ(ω - ωk) (121)

ε′(ω) ) 1 +2

π∑k

Akωk

ω2k - ω2

(122)


specific analysis when in the presence of a polarizablesolvent. In this case, in fact, it becomes fundamentalto properly account for the dynamic response of thesolvent.

This class of phenomenon is of very general occur-rence; here, in particular, we shall focus on thevertical electronic transitions, but electron and en-ergy transfers, or nuclear vibrations, are also ex-amples of the same class.

Historically, in continuum models large use hasbeen made of the approximation according to whichit is sufficient to decompose the response functioninto two terms. Within this approximation, alsoknown as the Pekar or Marcus partition, the polar-ization vector PB described in the previous sectionbecomes

where fast indicates the part of the solvent responsethat always follows the dynamics of the process andslow refers to the remaining inertial term. Suchsplitting in the medium response gives rise to the so-called “nonequilibrium” regime. Obviously, what isfast and what is slow depends on the specific dynamicphenomenon under study.

In a very fast process such as the vertical transitionleading to a change of the solute electronic state viaphoton absorption or emission, Pfast can be reducedto the term related to the response of the solventelectrons, whereas Pslow collects all of the other termsrelated to the various nuclear degrees of freedom ofthe solvent.

The operative partition of the total polarization canbe performed using two alternative, but equivalent,schemes, which we prefer to indicate without usingMarcus and Pekar names because in the literaturethere is an annoying confusion about these names.

One scheme, which we shall call partition I, wasproposed by Marcus in 1956 in his landmark paper521

on the nonequilibrium polarization in a classical

continuum solvation framework. The Marcus parti-tion has been extended to QM continuum models byseveral authors78,137,213,221,238,239,522,523 to properly de-scribe excited/de-excited electronic states after avertical (Franck-Condon) transition. In this scheme,eq 123 is rewritten as

where superscripts E and A refer to the “early” (orinitial) and the “actual” time in relation to thechronological order of the transition process. In thispartition, the previous slow and fast indices arereplaced by the subscripts or and el referring to“orientational” and “electronic” polarization responseof the solvent, respectively.

In the second scheme, which we shall indicate aspartition II, eq 123 has to be rewritten as

where the subscripts in and dyn refer now to an“inertial” and a “dynamic” polarization response ofthe solvent, respectively. The use of eq 125 withinthe continuum solvation methods has been appliedboth to classical and to QM approaches.56,214,216,219,524

The differences between the two schemes arerelated to the fact that, in partition I, the divisioninto slow and fast contributions is done in terms ofphysical degrees of freedom (namely, those of thesolvent nuclei and those of the solvent electrons),whereas in partition II, the concept of dynamic andinertial response is exploited. This formal differenceis reflected in the operative equations determiningthe two contributions to PB as, in II, the slow term(Pin) includes not only the contributions due to theslow degrees of freedom but also the part of the fastcomponent that is in equilibrium with the slowpolarization, whereas, in I, the latter component iscontained in the fast term Pel (see Table 2 in whichwe compare the electrostatic equations identifyingthe two schemes).

Table 2. Basic Equations within Partitions I and IIa,b

partition I partition II

PBA ) PBorE + PBel

A PBA ) PBinE + PBdyn

A

PBorE )

øor

øPBE(ε0) PBin

E ) PBE(ø) - PBdynE (øel)

PBelA ) øelEB[FA,PBor

E , PBelA] PBdyn

X (øel) ) øelEB[FX,PBdynX ] with X ) A, E

-∇2V ) 4πFA -∇2V ) 4πFA

-ε∞∇2V ) 0 -ε∞∇2V ) 0Vin - Vout ) 0 Vin - Vout ) 0

(∂V∂n)in

- ε∞ (∂V∂n)out

) 4πσorE (∂V

∂n)in- ε∞ (∂V

∂n)out) 0

Gneq ) G∞or-el + ∫FAVor

E dr3 - 12 ∫FEVor

E dr3 -

12 ∫Vor

E (σelE - σel

A) dr2

Gneq ) G∞in-dyn + ∫FAVin

E dr3 - 12 ∫FEVin

E dr3

G∞or-el ) E0

A + 12 ∫FAVel

A dr3 G∞in-dyn ) E0

A + 12 ∫FAVdyn

A dr3

a ø ) (ε0 - 1)/4π, øel ) (ε∞ - 1)/4π and øor ) ø - øel. b Even if not indicated, the potential V has the same sources of thecorresponding electric field EB.

PB ≈ PBfast + PBslow (123)

PBA ) PBorE + PBel

A (124)

PBA ) PBinE + PBdyn

A (125)


To better illustrate this aspect, let us consider twopolarizable dipoles having centers fixed at r and r′,both placed in a homogeneous external field. Thiselectric field determines the mean orientation of thedipole at r′. The induced dipole in r has two contribu-tions, the first resulting from the external field andthe second originating from the interaction with themean orientation of the dipole in r′. If the externalfield suddenly changes so that the mean orientationof the dipole in r′ has not enough time to relax, theinduced dipole in r will change only as far as concernsthe component related to the interaction with theexternal field, whereas the second component willmaintain the value corresponding to the initial field.As a consequence only the first component presentsa dynamic character in response to the changes ofthe external field.

It is important to stress that the two schemes I andII differ in the form of the slow and fast componentsof the polarization, but they are practically andphysically equivalent (in the limit of a linear responseregime) in the sense that they give the same valueof the total reaction potential and, thus, the sameeffects on the solute as well as the same interactionenergies. This equivalence has not been alwaysrecognized in the literature,221,525 and thus someconfusion exists on the use of these two partitions.However, as shown by Cammi and Tomasi522 and,more explicitly, by Aguilar,526 such confusion disap-pears when the proper expression for the nonequi-librium free energy functional is used in each ap-proach.

The use of one partition instead of the other isdictated only by the requirements of the specificcomputational code in which the model is imple-mented; for example, in Gaussian,17 the nonequilib-rium version of PCM has been implemented in theframework of partition I, exactly as the nonequilib-rium version of the MPE implemented in Dalton.135

In addition to the proper description of the solventresponse, another aspect must be considered whenwe simulate electronic excitations of solvated mol-ecules. This additional aspect, however, appears onlywhen a QM description is used.

Within the continuum solvation framework, as inthe case of isolated molecules, it is practice tocompute the excitation energies (or better excitationfree energies) with two different QM approaches: thestate-specific (SS) method and the linear-response(LR) method. The former has a long tradition and isrelated to the classical theory of the solvatochromiceffects (see references in our 1994 review1). The latterwas introduced later in connection with the develop-ment of the linear response theory for continuumsolvation models,136,138,140,215,527-529 which will be de-scribed in section 5.6. Here, however, we anticipatea relevant aspect of the theory when compared to theSS method for the calculation of transition energiesof solvated molecules.

The state-specific method solves the nonlinearSchrodinger equation for the state of interest (groundand excited state) usually within CI or MCSCFdescriptions, and it postulates that the transitionenergies are differences between the corresponding

values of the free energy functional, the basic ener-getic quantity of the QM continuum models. Asdiscussed in section 2.4.6, the nonlinear character ofthe reaction potential requires the introduction in theSS approaches of an iteration procedure not presentin parallel calculations on isolated systems. At eachstep of this iteration, the additional solvent-inducedcomponent of the effective Hamiltonian is computedby exploiting the first-order density matrix of thepreceding step, and the resulting energy, E )⟨Ψ|Heff|Ψ⟩, is corrected for the work required topolarize the dielectric G ) E - (1/2)⟨Ψ|VR|Ψ⟩ (we notethat the inclusion of the nonequilibrium needs somefurther refinements, as indicated by all of the mostdetailed papers on this subject mentioned above). Animportant consequence (and disadvantage) of thisiterative approach is that one has to solve a differentproblem for each different state instead of obtainingthe energies of all states in a single run as for isolatedsystems.

A different analysis applies to the LR approach (ineither Tamm-Dancoff, RPA, or TDDFT version) inwhich the excitation energies, defined as singularitiesof the frequency-dependent linear response functionsof the solvated molecule in the ground state, aredirectly determined by avoiding explicit calculationof the excited state wave function (see section 5.6).In this case, the iterative scheme of the SS ap-proaches is no longer necessary, and the wholespectrum of excitation energies can be obtained in asingle run as for isolated systems.

Although it has been demonstrated that for anisolated molecule the SS and LR methods are equiva-lent (in the limit of the exact solution of the corre-sponding equations), a formal comparison for mol-ecules described by QM continuum models showsthat this equivalence is no longer valid. By using anonequilibrium approach in both frameworks, it canbe established that these intrinsic differences arerelated only to the interaction of the solute with thefast response of the solvent medium, and they areconnected with the basic assumption of a Hartreepartition between solute and solvent.530 The analysisof this aspect is quite new, and thus further inves-tigations will surely appear in the future.

5.3. Solvation DynamicsSolvation dynamics is the time-dependent, non-

equilibrium, relaxation of the solvent due to aninstantaneous change in the interactions between thesolvent molecules and a different, newly createdelectronic state on the solute. Upon excitation of thesolute, the surrounding solvent molecules, whichwere initially equilibrated with the ground electronicstate of the solute, are in a nonequilibrium state andwill adjust their positions and momenta in order torelax to equilibrium with the new interactions as-sociated from the new electronic state. This relax-ation is driven by a decrease in the solvent systeminteraction energy, and as a result, any measurementthat probes this energy will mirror the underlyingsolvation dynamics of the system.

Among the techniques used to investigate solvationdynamics, the time-resolved fluorescence Stokes shift


(TRFSS) measurement is the most widely used. TheTRFSS technique measures the time-dependent redshifting (Stokes shift) of the emission spectrum of thesolute as the system relaxes. A time-dependentStokes shift (TDSS) function, S(t), can then beconstructed that describes the spectral diffusion ofthe excited molecules and contemporaneously moni-tors the response of the solvation to the new soluteexcited state

where ν(t) is the peak of the fluorescence spectrumat time t and ν(0) and ν(∞) are the values immediatelyafter the excitation and at sufficiently long times forthe solvent to re-equilibrate to the solute excitedstate.

To apply theoretical models to the study of solva-tion dynamics, it is fundamental to connect thenonequilibrium function S(t) to quantities related tothe solute-solvent interaction energy: this quantityis represented by the time correlation function⟨δE(0)δE(t)⟩ that represents the ensemble-averaged(“⟨ ⟩”) behavior of the solvent fluctuations. In fact, ifone observes the interaction energy between a solutemolecule and its solvent surroundings as a functionof time, E(t), one finds that it fluctuates rapidly aboutsome average value ⟨E⟩. The time dependence of thesefluctuations, δE(t) ) E(t) - ⟨E⟩, is intimately con-nected to the frictional forces that mediate a varietyof dynamical processes that take place in a solventenvironment.

Assuming that the perturbation caused by theelectronic transition is not too large, a simple linearresponse calculation provides the relationship

The physical content of this “fluctuation-dissipation”connection is that, if the perturbation is not too large(or if the interactions are harmonic), relaxation of the“forced” departure from equilibrium produced by theelectronic transition is not different from the relax-ation of a naturally occurring fluctuation in theunperturbed system.

The relationship (eq 127) between S(t) and thefunction C∆E(t) (generally known as the solvation timecorrelation function, STCF) is the basis of manytheoretical models that have provided importantphysical insights into the solvation dynamics.

The dynamical mean spherical approximation(MSA)531,532 as applied by Wolynes and others533-535

was the first and most influential of theories toaddress solvation dynamics. Other approaches havebeen devised.536-540 These treatments involve merg-ing liquid structure theory with linear models ofrelaxation, often incorporating empirical informationsuch as the frequency-dependent dielectric constantof the bulk solvent, ε(ω). In particular, much atten-

tion has been devoted to treating the spatial depen-dence of the dielectric response function, ε(k,ω),which includes the molecular nature of the solvent.

A different approach, employing the perspective ofa continuum model, was originally developed in 1997by Marcus and co-workers541 to calculate the STCFof eq 127.

The continuum model used by Marcus and co-workers is that of a dipole in a spherical cavity or anellipsoid filled with dipole density. In both cases theentire dielectric spectral response function ε(ω),without explicitly considering the spatial dependenceof dielectric relaxation, is exploited. The startingassumption is that the optical excitation of the solutemolecule occurs instantaneously and the dipole mo-ment of the molecule is changed from µg to µe at timet ) 0. Thus, the dipole moment of the solute at timet, µ(t), can be written as µ(t) ) µg + θ(t)(µe - µg),where θ(t) is the unit step function. Following theOnsager model, the energy of interaction of such adipole with the solvent is

where R(t) is the reaction field at time t due to thesurrounding solvent acting on the solute dipole. Thereaction field R(t) can be obtained from linear re-sponse theory as

where r(t - t′) is the Onsager reaction field expres-sion for a point dipole in a sphere.

The resulting solvation energy difference betweenthe excited state and the ground state molecule, attime t, can be thus rewritten as

where all of the constant terms have been eliminatedand Im indicates the imaginary part. To obtain eq130, one has to introduce first the Fourier transformof ∆E(t) by using the complex Fourier transform ofθ(t) and then the inverse Fourier transform of theresulting equation.

Basically, eq 130 is resolved by numerically inte-grating on different frequencies and by knowing thedielectric permittivity in the whole frequency range.In the low-frequency region Debye’s formula is used,whereas for the rest, a spline fit to experimental datais used for both the real and imaginary parts of ε(ω).

In 1998 Song and Chandler542 reformulated thismodel so as to treat molecules of arbitrary shapesand more realistic charge distributions. The algo-rithm they developed is based on the Gaussian fieldformulation of dielectric continuum theory formu-lated by Song et al. in 1996.543 In such a formulation,the solvent is described in terms of a linear respond-ing field, that is, a Gaussian field, that is expelledfrom the region of space occupied by the solute. Inthe 1998 extension, a new reformulation of the time-

S(t) )ν(t) - ν(∞)ν(0) - ν(∞)

(126)

S(t) ≡ C∆E(t) ≡ ⟨δ∆E(0)δ∆E(t)⟩⟨δ∆E2⟩

≡ ∆E(t) - ∆E(∞)∆E(0) - ∆E(∞)

(127)

Esolv(t) ) - m(t)R(t) (128)

R(t) ) ∫-∞′

dt′r(t - t′)µ(t′) (129)

∆E(t) ) -[∫-∞′

dt′r(t - t′)θ(t′)]∆µ2 )

-4(∆µ)2

a3π∫0

∞dω cos ωt

ωIm[ ε(ω) - 1

2ε(ω) + 1] (130)


dependent solvation energy of a generic distributionof point charges (representing the chromophore) dueto the dielectric is reduced to an integral over thechromophore molecular surface by introducing atime-dependent generalization of the BEM. As anapplication, a comparison with experimental mea-surements of solvation relaxation functions for cou-marin 343- (C343-) in water and C153 in methanoland acetonitrile was reported. The required data forthe calculations are the atomic charge distributionchange of the chromophore from the electronic groundstate to its first electronically excited state, thechromophore’s molecular surface, and the frequency-dependent dielectric constant of the solvent.

More recently, the approach developed by Marcusand co-workers has been reformulated by Ingrossoet al.544 within the IEFPCM continuum model and,for the first time, coupled to a QM description of thesolute electronic states. In this case the time-depend-ent solvation energy, Esolv(t), is the electronic com-ponent of the solute-solvent electrostatic interaction(the nuclear part is a constant independent of timeif we assume a negligible role of solute nuclearrelaxation)

where Vel and qel are the vectors collecting the solutepotential at the surface and the solvent apparentcharges. By comparing eq 131 with the correspondingone (eq128) in the Onsager-like model, it is evidentthat the solute property of interest is no longer thedipole moment but the electrostatic potential, and thesolvent response is here represented in terms of thetime-dependent apparent charges. Thus, assumingfor V the same evolution used in the Marcus modelto define µ(t) in terms of the step function θ(t) andapplying the same mathematical framework of Fou-rier and anti-Fourier transformations, one obtainsthe IEFPCM equivalent of the right-hand side of eq130. The model has been applied to the calculationof the STCF for coumarin 153 in various polarsolvents (water, acetonitrile, dimethyl sulfoxide, andmethanol) described in terms of a generalized disper-sion relation (in the form of a Cole-Davidson orCole-Cole equation) for ε(ω) in the low-frequencylimit and a fit of experimental data at high frequen-cies. Using a configuration interaction approachincluding all singly substituted determinants (CIS)for the calculation of the coumarin excited states, allof the major features observed in experimentalresults of S(t), the initial inertial decay and themultiple exponential long time relaxation, are repro-duced almost quantitatively by the calculations.

Subsequently, the same model has been extendedto describe a general time-dependent relaxation ofthe solvent. This extension has been initially pro-posed within the DPCM scheme by Caricato et al.545

and then applied to the IEFPCM.546 Once again thestarting assumption is that the change in the solutemolecule occurs instantaneously (at time t ) 0) andthe corresponding variation of the electrostatic po-tential V between the initial (0) and the final (fin)state is a step change. The time-dependent solvent

polarization charges at a generic time t can thus bewritten as

where the vector q0 collects the polarization chargesin the initial solute-solvent equilibrium.

Under the assumption of linear response of thesolvent the variation of the polarization charges δqat time t due to a change in the electrostatic potentialat time t ) 0, one obtains

where R now indicates the PCM solvent responsefunction. As done before in eq 130, this expression istransformed in a numerical procedure by passingfrom the time domain to the frequency domain byusing Laplace transformations. This change is re-quired as the dielectric response of the solvent isdescribed in terms of its complex dielectric permit-tivity as a function of a frequency ε(ω). The use ofLaplace-transformed equations to pass from the timeto the frequency domain has the effect of simplifyingthe formalism and of allowing the straightforwarduse of the function ε(ω). Starting from eq 133, in fact,the variation of the charges becomes

where the definition of the matrix K for the differentPCM schemes can be found in Table 1.

To find the proper form for the frequency-depend-ent response function R(ω), it is useful to resort tothe iterative form of the PCM equations (see section2.3.1.5), obtaining

where

The integral in eq 136 can be solved analytically ifwe use the Debye expression for ε(ω); however, theapproach can be applied to any other functional formfor the complex dielectric permittivity (like, forexample, a multiple Debye, a Davidson-Cole, or aCole-Cole form) as well as to a combined schemeincluding a fit of experimental data for the high-frequency portion of ε(ω). The only practical differ-ence is that in the latter case, the integral in eq 136is solved.

Another continuum model used to describe solva-tion dynamics is the frequency-resolved cavity model(FRCM) formulated by Basilevsky and co-workers.547

Esolv(t) ) -Vel† qel(t) (131)

q(t) ) q0 + δq(∆V,t) (132)

δq(∆V,t) ) ∫-∞t

dt′R(t - t′)θ(t′)∆V (133)

δq(∆V,t) = δq′(∆V,t) + K∆V

δq′(∆V,t) ) 2π∫0

∞ dωω

Im[R(ω)]cos(ωt)∆V(134)

δq′i(n) ) 2π

Siigi(t)[bi + 1

2πzi[aj(Sjjδq′j

(n-1) +

yj[δq′k(n-1)])] - 1

2πzi[aj](Siiδq′i

(n-1) + yi[δq′i(n-1)])] -

Sii-1yi[δq′j

(n-1)] (135)

gi(t) = 2π∫0

∞ dωω

Im[( 4πε(ω)ε(ω) - 1)-1] cos(ωt) (136)


The FRCM model coincides with the DPCM schemefor the electrostatic equations of the apparent charges(it is known by the acronym BKO as described insection 2.3.1.1), but differs from it in the use of twosurfaces instead of a single one. The two surfaces arehere introduced to separate the inertialess (high-frequency) response of the medium from the inertial(low-frequency) one. Between the two surfaces themedium is modeled by the inertialess high-frequencydielectric constant ε∞; outside the outer cavity themedium is modeled by the static dielectric constantε0. The layer between the two surfaces correspondsto the first solvation shell.

Calculations in this scheme amount to simulta-neously solving two equations describing the surfacecharge densities on the two cavities, namely, σ∞ (onthe inner surface) and σ (on the outer). In general, asimplification is applied to the FRCM model, namely,the charge density on the outer surface is assumedto be sufficiently small for its influence on the innercharge density to be neglected. Using this approxi-mation, the equation describing σ∞ may be separatedfrom that determining σ, and thus it is identical tothe ordinary BKO equation but with a dielectricconstant now given by ε∞. σ∞ then enters as aparameter into the equation describing σ, whicheffectively becomes a BKO equation with scaledparameters.

Dynamics calculations are performed by replacingε0 with the complex number ε(ω), thereby splittingthe equation in σ into two equations describing itsreal and imaginary parts. Solving these equationsproceeds in a similar way as for the standard BKOcase. Once again, the STCF C∆E(t) is expressed interms of ∆E(t), the inverse Fourier transform of thereorganization energy E(ω), which can be consideredas a solvation energy for the charge distributiondifference between ground and excited state.

The one-cavity BKO and the two-cavity FRCMapproaches have been also extended to dynamicalnonlocal models for the dielectric function by usingthe equation

where the spatial dispersion is determined in termsof a Lorentzian model [∆(k) ) 1/(1 + λ2k2)]. Thisnonlocal model when applied to the calculation of thesolvation time correlation function C∆E(t) gives a long-time asymptote which differs markedly from that ofthe local theories and of the experiments. In thisregion, corresponding to the low-frequency limit ω f0 in the frequency domain, the STCF behavior iscompletely determined by the size difference betweenthe solvent and solute particles. A detailed analysisof these findings led the authors to observe that thedynamics for the nonlocal spherical model reducesto the single-exponential dynamics of the simplestdynamical continuum theory, and thus more realisticmodels of the nonlocal dielectric functions should beintroduced.

The examples we have described above demon-strate that reasonable modifications of the conven-tional continuum solvent model enable one to over-come the common belief that continuum medium

models are incapable of adequately treating solvationdynamics. The improved versions of the continuumtheories are in fact able to qualitatively monitor thesame picture of polar solvent dynamics as do molec-ular theories. Then, by properly adjusting the modelparameters (a feature that is common for bothmolecular and continuum models), experimental de-pendencies can be well reproduced. The key step inall continuum treatments is to use the complex-valued solvent frequency-dependent dielectric func-tion ε(ω) as a phenomenological characteristic ofsolvent dynamics.

Indeed, continuum treatments present an impor-tant advantage with respect to standard molecularsimulations, and this could become decisive soon:when combined with modern quantum chemicalcalculations, they allow a precise description ofelectronic structure and charge distributions in chemi-cally nontrivial solute species, whereas current mo-lecular treatments still invoke crude solute modelsand concentrate on a description of the details ofsolvent structure.

We finish this section by noting that almost all ofthese detailed studies on time scales and mechanismsof solvation dynamics have been limited to polarsolvents. For nonpolar solvation, with the notableexceptions of the works from Berg and co-workers548-550 and from the Maroncelli group,551,552

few experimental studies exist. In addition, thefeedback between theory and experiment that hasbeen so productive for polar solvation has not yetbeen achieved for nonpolar or, more generally, non-dipolar systems, where nonpolar systems refer tosystems that interact via weak nonelectrostatic forces,such as dispersive and repulsive interactions, whereasnon-dipolar systems include higher electrostatic mul-tipole (e.g., quadrupole or octupole) moments inaddition to the nonpolar interactions. There are anumber of reasons why non-(di)polar solvation hasbeen less deeply analyzed. First, there is no obviousmethod to extract the dynamical time scales in non-dipolar liquids from existing frequency domain datasuch as dielectric relaxation measurements, becausenon-dipolar systems do not exhibit significant dielec-tric signals. Second, the assumptions that greatlysimplify the understanding of dipolar solvation,linear response and the relative insensitivity of thetime scales to the specific nature of the probemolecule, may not hold for nonpolar or non-dipolarsolvation.539 In other words, the short-range natureof the interactions implies that the molecular proper-ties of the solvent-solute combination play a moreimportant role in the solvation dynamics of non-dipolar systems than in the dipolar case.

Despite this increased difficulty, there are somemodels that can successfully predict the long timedynamics observed in nonpolar systems.

For instance, Berg553,554 formulated a novel theoryto describe nonpolar solvation in which the responseof the nonpolar solvent is not based on dipolarinteractions, but instead treats the solvation dynam-ics as a re-equilibration of the solvent to a changingvolume of a solute. This model is an analogue ofcontinuum models of polar solvation, which predict

ε(k,ω) ) ε∞ + (ε(ω) - ε∞)∆k(k) (137)


solvation dynamics from the frequency-dependentdielectric constant, but here the solvation reponsefunction is calculated from the time- or frequency-dependent mechanical moduli of the solvent. Thehypothesis is the assumption that a change in thesolute’s effective size or shape upon electronic excita-tion is the primary interaction driving nonpolarsolvation. Following excitation, the solvent mustmove to allow the solute cavity to expand. Thismovement is modeled by treating the solvent as aviscoelastic (VE) continuum. This VE model predictstwo distinct time scales that can be associated withthe inertial and diffusive mechanisms and timescales. Comparisons with experimental data showthis model fits long time solvation dynamics well.

5.4. Spectral Line Broadening and SolventFluctuations

The progress in solvation dynamics we have sum-marized in the previous section has also stimulatedthe application of continuum models to the study ofthe line shape of bands corresponding to absorption/emission processes and of the solvent fluctuations.

Such fluctuations in the structure of the solvationshell surrounding the chromophore555 and the con-sequent variation in the local electric field lead to astatistical distribution of the energies of the electronictransitions. This phenomenon is called inhomoge-neous broadening and, in most cases, its extent ismuch greater than that of homogeneous broadeningdue to the existence of a continuous set of vibrationalsublevels in each electronic state. For obvious reasonswe focus here only on the inhomogeneous broadeningby giving a short review on the main issues relatedto its modelization.

A widely used approach to the analysis of solventspectral broadening is based on the model formulatedby Mukamel and co-workers556,557 in which the sol-vent is represented as a bath of Brownian oscillators.This model provides convenient analytical expres-sions for the line broadening function g(t). Imple-mentation of the Brownian oscillator model requiresonly two adjustable parameters: the frequency andthe variance of solvent-induced frequency fluctua-tions, which in turn is related to the solvent reorga-nization energy.558,559 The application of the Brown-ian oscillator model to solvent-induced line broadeningtypically assumes that the solvent modes coupled tothe electronic transition are overdamped, and thusthe time correlation function for solvent-inducedfrequency fluctuations follows an exponential relax-ation. However, as we have described in the previoussection, theory and experiments on solvent relaxationsuggest that at sub-picosecond times this relaxationis a Gaussian function of time, resulting from inertialsolvent motion, followed by exponential decay char-acteristic of diffusional motion on the picosecond timescale.

To account for this inertial part of solvent relax-ation, one can start from a phenomenological cor-relation function, which is a Gaussian function oftime. In the limit of linear solvent response, in whichthe solvent dynamics are assumed to be independentof the electronic state of the chromophore, the

frequency-frequency correlation function (FFCF) isdefined by

where δωeg(t) is the solvent-induced fluctuation in thefrequency ωeg of the electronic transition of thechromophore and the angle brackets indicate anequilibrium average over the solvent coordinates. Wenote that in this linear solvent response limit theFFCF of eq 138 is related to the function S(t)determined from time-resolved Stokes shift measure-ments as shown in eq 127 (namely, the previouslydefined STCF is the FFCF normalized to unity at t) 0).

Several very general relationships between opticalobservables can be obtained for such linearly re-sponding (Gaussian) solvents interacting with a fixedcharge distribution of the solute. More precisely,these spectroscopic observables can be related to thesolvent reorganization energy, λs.560 This is in factthe only parameter needed to characterize the ener-getic intensity of the solvent nuclear fluctuationscoupled to the solute.

The relationship between λs and the solvent-induced absorption, and emission, spectral width(σabs/em), can be written as561,562

where â ) 1/kT. This relationship is exact for anyfixed charge distribution of the solute in a linearlyresponding solvent.

Here, we do not want to enter in the large field oftheoretical models to determine the solvent reorga-nization energy, which followed the original Marcusformulation of electron transfer521 (some reviews canbe found in refs 563 and 564). Here, it is importantto recall that this concept is strictly connected to thedifference in solvation free energy between the initialnonequilibrium and the final equilibrium relaxedsolvent configurations originated by any fast processof solute charge redistribution such as the verticalelectronic transition described in section 5.2.240,412,545,565

The effect of relaxing the assumption of a fixedcharge distribution of the solute allowing the elec-tronic density to flow between the initial and finalstates (electronic delocalization) is to alter the soluteelectric field, thus modifying the solute-solventcoupling and the line shift, and, even more, theinhomogeneous broadening and the observed opticalwidth. This is because, in an electronically delocalizedsolute, the charge distribution changes instanta-neously with solvent configuration, so that eachconfiguration “sees” a different charge distribu-tion.566-568 This self-consistent action results in es-sentially nonlinear features of the solvent effect.569-571

We conclude this section by mentioning a differentclass of approaches to the study of the line shape insolvated systems, which explicitly account for thepositional fluctuations of solvent molecules aroundthe solute. For example, this is the approach devel-oped by Kato and co-workers572 within the RISM-SCFmodel (see section 7.4.2). In this framework thespectral bandwidths are estimated by calculating the

C(t) ) ⟨δωeg(t)ωeg(0)⟩ (138)

λabsw ) âσabs

2 /2 and λemw ) âσem

2 /2 (139)


derivatives of the solute-solvent radial distributionfunction analytically.

Within the framework of continuum models, wherethe solvent loses its molecular nature, things aremore complex. An interesting proposal to account forsolvent fluctuations still maintaining a continuumdescription has been made by Kim.567,573 Startingfrom the quantum dielectric continuum solvent for-mulation of Kim and Hynes,240 Kim develops amethod to include cavity size variation with thesolute electronic charge distribution and its thermalfluctuations. By employing a coherent state descrip-tion for the solvent electronic polarization, he obtainselectronically adiabatic free energies as a function ofthe cavity radius variable that measures the fluctu-ating cavity size and the solvent coordinates thatgauge the nonequilibrium solvent orientational po-larization. These define multidimensional electronicfree energy surfaces, upon which nuclear dynamicsoccur. The effects of the cavity size variation on theelectronic spectra are illustrated by using a simpletwo-state model description for the solute. It is foundthat even in a nonpolar solvent, there can be asignificant Stokes shift arising from the cavity sizerelaxation after the Franck-Condon transition. Thecavity size fluctuations can also make a non-negli-gible contribution to the spectral line broadening.

5.5. Excitation Energy Transfers

It is well-known that de-excitation of the electroni-cally excited states of molecules depends on manydifferent processes (the sum of the rate constants forthese processes is equal to the reciprocal of theexcited state lifetime). Most of these processes involveinteraction (collisions, excimer/exciplex formations,proton and electron transfers, and energy transfers)of an excited molecule M with another molecule. Thefluorescence characteristics (decay time and/or fluo-rescence quantum yield) of the excited molecule areaffected by the presence of the other molecule as aresult of competition between the intrinsic de-excita-tion and these intermolecular processes.

Of course, intramolecular excited state processescan also affect the fluorescence characteristics of agiven molecule: intramolecular charge transfer, in-ternal rotation (e.g., formation of twisted chargetransfer states), intramolecular proton transfer, etc.,as well as photochemical de-excitation can apply.

All of these possible mechanisms determining theexcited state lifetime can be strongly affected by themedium in which the molecules are embedded. Dueto this importance, many theoretical models havebeen formulated to account for medium effects onexcited state lifetimes. We cannot give an exhaustivereview as this would require a specific volume byitself, but we restrict the analysis to intermolecularphotophysical processes only and, in particulr, tothose involving energy transfers.

Nonradiative transfer of excitation energy requiressome interaction between a donor molecule (D) andan acceptor molecule (A), and it can occur if theemission spectrum of the donor overlaps the absorp-tion spectrum of the acceptor, so that several vibronic

transitions in the donor have practically the sameenergy as the corresponding transitions in the ac-ceptor. Such transitions are coupled, that is, inresonance. The term resonance energy transfer (RET)is often used. In some papers, the acronym FRET isalso used, denoting fluorescence resonance energytransfer.

Energy transfer can result from different interac-tion mechanisms.574 The interactions may be Cou-lombic and/or due to intermolecular orbital overlap.The Coulombic interactions consist of long-rangedipole-dipole interactions (Forster’s mechanism575)and short-range multipolar interactions. The interac-tions due to intermolecular orbital overlap, whichinclude electron exchange (Dexter’s mechanism576)and charge resonance interactions, are of course onlyshort range. For allowed transitions of D and A, theCoulombic interaction is predominant, even at shortdistances. For forbidden transitions of D and A (e.g.,in the case of transfer to triplet states), the Coulombicinteraction is negligible and the exchange mechanismprevails, but it is operative only at short distances(<10 Å) because it requires overlap of the molecularorbitals. In contrast, the Coulombic mechanism canstill be effective at large distances (up to 80-100 Å).

It is evident that both Coulombic and overlapinteractions will be heavily affected by the presenceof a solvent around and possibly between the twointeracting molecules.

In his unified formulation of EET,575 Forster in-troduced the effects of a dielectric medium, that is, asolvent. His idea was in fact that the Coulombinteraction between donor and acceptor, approxi-mated as a dipole-dipole interaction between transi-tion dipole moments of the donor and acceptormolecules, is screened by the presence of the dielec-tric, namely

where R is the distance between the donor andacceptor and n is the solvent refractive index.

The screening factor (1/n2) in eq 140, however, isjust an approximation of a part of the global effectsinduced by a polarizable environment on the EETprocess. The presence of a solvent, in fact, not onlyscreens the Coulomb interactions as formulated byForster but also affects all of the electronic propertiesof the interacting donor and acceptor.577

An example of a more detailed description of thesolvent effects on EET was given by Juzeliunas andAndrews,578 who reported a many-body descriptionof EET based on quantum electrodynamics (QED).By considering the energy transfer to be mediatedby bath polaritons (medium-dressed photons), thistheory accounts for the modification of the barecoupling tensor both by screening effects of themedium, as in the Forster theory, and by local fieldeffects. Also in this theory, however, a dipole ap-proximation is used to introduce medium effects, andin fact the expression they derive is essentiallyForster’s dipole-dipole coupling scaled by a prefactor,

VCoul ≈ 1n2[µbD‚µbA

R3-3

(µbD‚RB)(µbA‚RB)

R5 ] (140)


[(ε + 2)/3]2, where ε is the optical dielectric constantof the medium. Agranovich and Galanin obtained thesame prefactor in 1982 from basically the sameconsiderations in a classical theory.579 More recently,two QM models for EET including solvent effectshave appeared in the literature. Both models arebased on a time-dependent variational approach eventhough they follow two completely different paths.In the first model by Tretiak et al.580,581 a Hartree-Fock approximation is combined with semiempiricalHamiltonians, whereas in the second, by Hsu et al.,582

a DFT description is used. To account for solventeffects, both methods introduce a simplified con-tinuum description based on two important ap-proximations, one on the form of the molecular cavity,which is always limited to a sphere, and the otheron the solute electronic densities, which are repre-sented in terms of a dipolar (or a multipolar) expan-sion.

More recently, a new model has been presented byIozzi et al.583 This is still based on a continuumdescription of the solvent, exactly as in the modelsdescribed above, but it differs from them in twofundamental aspects. On the one hand, it introducesthe solvent effects in all of the steps of the QMcalculation of the EET process without requiring anysimplification in the representation of the molecularcharge distributions or the introduction of fixedprefactors, but defining proper operators to be addedto the Hamiltonian and to the resulting responseequations. On the other hand, it describes the mo-lecular cavity on the real three-dimensional structureof the interacting solutes without imposing an arti-ficial spherical boundary. In this framework, the EETprocess is described through the time-dependentdensity response theory (or TDDFT, see section 5.6)originally proposed in the paper by Hsu et al.,582

whereas the introduction of the solvent effects isrealized in terms of the IEFPCM described in section2.3.1.3.

The model considers two solvated chromophores,A and D, with a common resonance frequency, ω0,when not interacting (this condition is triviallyfulfilled when A ≡ D, but it is still realistic in thecase of either large or condensed systems, i.e.,whenever high density of nuclear states can befound). When their interaction is turned on, theirrespective transitions are no longer degenerate. Incontrast, two distinct transition frequencies ω+ andω-, appear. The splitting between these defines theenergy transfer coupling, JDA ) [ω+ - ω-]/2, whichcan be evaluated by computing the excitation ener-gies of the D x A system, as, for example, through aproper TDDFT scheme (see eq 152 in the followingsection).584-589 The alternative strategy used in themodel is based on a TDDFT-perturbative approach,582

which considers the D/A interaction as a perturbationand includes solvent effects defining an effectivecoupling matrix, which couples transitions of D withthose of A in the presence of a third body representedby the IEF dielectric medium. Within this frame-work, the first-order coupling J becomes a sum of twoterms, one of which is always present (i.e., also inisolated systems) and another that has an explicit

dependence on the medium, namely

and

where FT(r) indicates transition densities of thesolvated systems D and A in the absence of theirinteraction. In particular, J0 describes a solute-solute Coulomb and exchange correlation interaction(through the exchange correlation functional of theproper DFT scheme), corrected by an overlap contri-bution. The effects of the solvent on J0 are implicitlyincluded in the values of the transition properties ofthe two chromophores before the interaction betweenthe two is switched on. These properties can in factbe significantly modified by the “reaction field”produced by the polarized solvent. In addition, thesolvent explicitly enters into the definition of thecoupling through the term JIEF of eq 142, whichdescribes a chromophore-solvent-chromophore three-body interaction (we note that this term is originatedfrom the explicit solvent term defined in the TDDFT/IEFPCM eq 155 of the following section).

The JIEF term can account for effects due todynamic solvation by introducing the dynamic (εω)instead of the static permittivity in the definition ofthe transition apparent surface charges q(sk; εω,FT).These latter generalize the concept of ASC describedin section 2.3.1 to transition densities as the sourceof polarization, but their definition is exactly equiva-lent to that given when the source is represented bya standard state density.

There are two the fundamental advantages of theperturbative TDDFT/IEFPCM description of thecoupling: one of a physical nature and the othernumerical. The conceptual aspect, unique to thisapproach, is that the coupling is computed within ascheme coherently accounting for different kinds ofinteractions (namely, direct and indirectsi.e., solvent-mediated Coulomb, exchange, possibly including cor-relation effects and overlap interactions). In this way,both the reaction and the screening effects due to thesolvent are introduced in a unified model, whichmerges Forster- and Dexter-type approaches. Thenumerical advantage of the perturbative approachwith respect to the standard one in which the TDscheme for the supermolecule (D x A) is solved is thatonly the properties of the solvated chromophores (Dand A) are required to get the coupling.

This feature of the TDDFT-PCM perturbativeapproach becomes particularly important when oneapplies the model to the calculation of excitonicsplittings590 in conjugated organic materials formedby equivalent polymeric chains in a three-dimen-sional array.591 In this case, in fact, one can introduceboth short-range and long-range interchain effectssimply by computing the properties of a single chainembedded in a continuum medium characterized by

J0 ) ∫ dr ∫ dr′ FDT*(r′)( 1

|r - r′| +

gxc(r′,r,ω0))FAT(r) - ω0 ∫ dr FD

T*(r)FAT(r) (141)

JIEF ) ∑k (∫ dr FD

T*(r)1

|r - sk|)q(sk; εω,FAT) (142)


an anisotropic dielectric tensor (the two differentcomponents of the tensor correspond to the dielectricpermittivity along and perpendicular to the main axisof the polymer chain). This extension to anisotropicdielectrics is made possible by the use of the IEFPCMformalism presented in section 2.3.1.3.

Processes of energy transfer are also responsiblefor another interesting phenomenon strongly de-pendent on the environment. It is well-known thatthe presence of a metal body can strongly affect theresponse properties of a molecule placed in its closeproximity. In section 6.1.3, we shall examine the well-known surface-enhanced Raman scattering (SERS)for which enhancement factors of >10 orders in themagnitude of the Raman scattering of molecules closeto metal particles have been reported. As far asconcerns the excited state properties, the presenceof the metal can have also the opposite effect and isresponsible, in many cases, for a decrease of themolecular responses. This result is due to the factthat the excitation energy of the molecule can beefficiently transferred to the metal body (for example,via the resonant energy transfer we have just de-scribed) and can undergo, inside this medium, severaldissipation processes.

The great technological interest in these phenom-enona has led to the formulation of many theoreticalmodels. Most of them consider the molecule as apolarizable pointlike dipole.592-595 For metal-mol-ecule distances of less than a few nanometers (asituation that occurs in many experimental situa-tions) this approximation can be too rough and a QMdescription of the whole molecule would instead benecessary. On the other hand, for many of thephysical systems of interest, the metal shape is oneof the most important factors, and a proper descrip-tion of this aspect is compulsory.

To account for both of these demands, Corni andTomasi79,504,506 have formulated a QM method for thecalculation of dynamic response properties of amolecule in close proximity to metal bodies andpossibly in the presence of a solvent. Such a model(see also sections 4.6 and 6.1.3) treats the moleculeat a QM level and can explicitly consider metalparticles of complex shape. The metal (and thesolvent, if present) is described as a continuous bodycharacterized by its (dielectric) response propertiesto electric fields, both those imposed on the systemfrom outside and those arising from the molecularcharge distribution. The metal-molecule and solvent-molecule interactions are treated within the PCMapproach in its D and IEF versions. Recently, thisprocedure has been generalized for the calculationof the contribution to the molecular deexcitation ratedue to the metal;508,509 this generalization is basedon the response theory (applied at a TDHF or TDDFTmethod, see the following section), in which excitationenergies and lifetimes can be obtained from the realand imaginary parts of the poles of the linearresponse function. Strictly speaking, the lifetimesobtained in this way are related to the width of theabsorption peaks of the molecule.

5.6. Time-Dependent QM Problem for ContinuumSolvation Models

We conclude section 5 devoted to time-dependent(TD) solvation by considering in a more detailed wayall of the QM issues implicitly or explicitly cited inthe previous subsections more focused on the physicaland modelistic aspects of the phenomenon.

Once again, the key point is the nonlinear charac-ter of the effective Hamiltonian defining the molec-ular system in the presence of the solvent (see section2.4.5).

Here, in particular, such an effective Hamiltonianbecomes time dependent. There are a large varietyof processes in solution that can give rise to this timedependency. Here we limit ourselves to two cases: (a)the solvent dynamics relaxation and (b) the presenceof an external time-dependent perturbing field. Ingeneral, we can write

where HM0 is the Hamiltonian of the unperturbed

solute, Vint the solute-solvent interaction term, andW(t) a general time-dependent perturbation termthat drives the system.

In case a discussed in section 5.3 we have that W(t)) 0 and the time dependence of the solute-solventinteraction Vint(t) originates from dynamical pro-cesses involving inertial degrees of freedom of thesolvent. The time scale of these processes is ordersof magnitude higher than the time scale of theelectron dynamics of the solute, and an adiabaticapproximation can be used to follow the electronicstate of the solute, which can be obtained as eigen-state of the time-dependent effective Hamiltonian (eq143).

This approach is a direct extension of the QMproblem for the basic model (see eq 58) to the effectiveHamiltonian Heff(t) and, as shown in section 5.3, hasbeen used by various authors377,544,545,596 to study thetime correlation function for the time-dependentStoke’s shift following a vertical excitation and tostudy electron-transfer processes of molecular sol-utes.

For case b, which will be considered in more detailin section 6, the external perturbing fields of interestare usually optical field, and the study of responseproperties of a molecular solute to the field requiresan extension of the basic model to describe the time-dependent nonlinear Schrodinger equation for thesolute, namely

This extension has been formulated by Mikkelsen etal.136 for the MPE solvation model (see section 2.3.2)and by Cammi and Tomasi597 within the frameworkof the PCM model (see section 2.3.1).

Mikkelsen and co-workers have considered theresponse of a reference variational (HF/MCSFC)136-138,598 state by using the Frenkel variation prin-ciple for the TD-QM problem in the form of theEhrenfest equation, which describes the time evolu-

Heff(t) ) HM0 + Vint(t) + W(t) (143)

i∂Ψ(t)

∂t) [HM

0 + Vint(t) + W(t)]Ψ(t) (144)


tion of the expectation values for any operator B. TheE-equation has the form

where the expectation values ⟨..⟩ are determined fromthe time-dependent wave function Ψ(t) at time t. Thetime evolution of Ψ(t) may be determined by requir-ing that the E-equation (i.e. “the equation of motion”)be satisfied by the excitation/de-excitation operatorsthrough each order in the interaction operator. Byinserting such an expansion into the evolution of theexpectation value of an operator A, assuming A tobe time-independent in the Schrodinger picture, weobtain the linear and nonlinear response functions,A;Wω1.ω1+iε, ,A;Wω1,Wω2.ω1(ω2+2iε of the solute.

Cammi and Tomasi using the PCM model give aslightly more general approach to the problem byusing the Frenkel variation principle in the form ofthe stationarity equation

where G is the time-dependent generalization of thefree energy functional G introduced in section 2.4.4(eq 74). From this stationary condition a variety ofspecific time-dependent equations at HF, MCSCF,and DFT levels have been developed,215,527,597,599,600

from which linear and nonlinear response functionsfor the solute can be obtained. Further details willbe given in the next section. In the time-independentlimit eq 146 reduces to the variational equation forthe basic model (section 2.4).

The generalized Frenkel stationary condition, eq146, has been subsequently reformulated by Chris-tiansen and Mikkelsen140,141 to derive the responsefunctions for solutes described at the coupled cluster(CC) level within the framework of spherical MPEsolvation model.

Some comments on the Frenkel variation principleare here of interest. The time-dependent variationalprinciple for the linear Hamiltonian has been thesubject of several formulations,601-603 which gener-ated some confusion and ambiguity in the literature.A careful analysis by van Leuven604-606 and by otherauthors200,602,603,607-613 has shown that the equivalenceof the various formulations is satisfied in a numberof important cases for electronic structure calcula-tions including also the nonlinear Hamiltonian caseand thus continuum solvation models.

We conclude the analysis of TD-QM approaches forsolvated systems by reporting a brief description ofthe linear response (LR) theory cited many times inthis and in the previous subsections. We make herethis short digression, at the end of the section devotedto TD solvation and immediately before the sectionfocused on molecular properties, as LR is becomingone of the most important methodologies both in thestudy of time-dependent phenomena and in thedetermination of the response to applied fields.

The starting point is again the Frenkel stationarycondition (eq 146); if we express the TD variationalwave function Ψ(t) in terms of the time-independent

unperturbed variational wave function

and we limit the time-dependent parameter d to itslinear term, we get200

where Q is the project operator and ∇∇G and ∇G∇collect the Hessian components of the free energyfunctional with respect to the wave function varia-tional parameters.

Instead of working in terms of time, we mayconsider an oscillatory perturbation and express W(t)by its single Fourier component A exp(-iωt). In thisframework, the linear term in the parameter as-sumes the form d ) [X exp(-iωt) + Y exp(iωt)]/2 andeq 148 becomes

where

is the inverse of the linear response matrix for themolecular solute. This response matrix depends onlyon intrinsic characteristics of the solute-solventsystem, and it permits one to obtain linear responseproperties of a solute with respect to any appliedperturbation in a unifying and general way. Forexample, the time-dependent change in the expecta-tion value of any quantity B (with associated opera-tor) as the result of the perturbation containing theoperator A can be expressed in terms of the corre-sponding linear response function as

In addition, the poles ((ωn) of the response function[Ω - ω∆]-1 give an approximation of the transitionenergies of the molecules in solution (see section 5.2for more comments); these are obtained as eigenval-ues of the system

where (Xn Yn) are the corresponding transition eigen-vectors.

This general theory can be made more specific byintroducing the explicit form of the wave function;in such a way, by using an HF description, we obtainthe random phase approximation (RPA) (or TD-HF).In this case the expansion (eq 147) becomes527,614

where we have used the standard convention in the

Ψ(t) ) Ψ0 + ∇Ψ0d + ... (147)

∇W + ∇∇Gd* + ∇G∇d ) i∇Q∇d4 (148)

(∇AA∇ ) + [Ω - ω∆](XY ) ) 0 (149)

[Ω - ω∆] ) (∇G∇ ∇∇GG∇∇ ∇G∇* ) - ω(∇Q∇ 0

0 -∇Q∇ )(150)

⟨⟨B;A⟩⟩ω ) (∇BB∇ )[Ω - ω∆]-1(∇A

A∇ ) (151)

[Ω - ωn∆](XnYn

) ) 0 (152)

Ψ(t) ) Ψ0 + ∑i,m

pmiΨ(ifm) +

∑i*j,m*n

pmipnjΨ(ifm,jfn) + ...

ddt

⟨B⟩ ) ⟨∂B∂t ⟩ - i⟨[B,Heff]⟩ (145)

δ⟨Ψ(t)|G - i ∂

∂t||Ψ(t)⟩ + i ∂

∂t⟨δΨ(t)|Ψ(t)⟩ ) 0 (146)


labeling of molecular orbitals, that is, (i, j, ...) foroccupied and (m, n, ...) for virtual orbitals, andindicated with Ψ(ifm) and Ψ(ifm,jfn) the func-tions obtained from the unperturbed Ψ0 by substitut-ing one or two occupied orbital(s), respectively.

In parallel, the free energy Hessian terms yield

where ⟨mn|ij⟩ indicates two-electron repulsion inte-grals and εr orbital energies.

In the definitions (eqs 153) the effect of the solventacts into ways, indirectly by modifying the molecularorbitals and the corresponding orbital energies (theyare in fact solutions of the Fock equations includingsolvent reaction terms, see section 2.4.2) and explic-itly through the perturbation term Bmi,nj. This termcan be described as the electrostatic interactionbetween the charge distribution ψm

/ ψi and the dy-namic contribution to the solvent reaction potentialinduced by the charge distribution ψn

/ψj (see section5.2).

A parallel theory can be presented also for a DFTdescription; in this case the term TDDFT584-589,615-617

is generally used as we have already noted in section5.5.

In the TDDFT approach, the starting expansion ison the electronic density the first-order variation ofwhich, δF, is conveniently written in the hole-particle,particle-hole formalism (within the frequency do-main) as

where δPmi(ω) is the linear response matrix in thereference of unperturbed molecular orbitals ψ.

Within this formalism an analogue of eq 149 isfound by using

where now the indices (i,j) and (m,n) refer to occupiedand virtual Kohn-Sham orbitals and the two-electron repulsion integrals have been substituted bythe coupling matrix Kmi,nj containing the Coulombintegrals and the proper “exchange repulsion” inte-grals determined by the functional used.

We note that the explicit solvent term Bmi,nj hasexactly the same meaning (and the same form) of theequivalent one defined in the HF method.583

6. Molecular Properties of Solvated Systems

The characterization and interpretation of molec-ular properties have always been important subjects

in quantum chemistry since the beginning of thediscipline. The extremely rapid evolution of thesestudies in recent years witnesses the combination oftwo factors, the advances in experimental techniques,leading to more accurate determinations and also tothe discovery of new properties, and the progress inthe formulation of new, or renewed, methodologicalmethods for the theoretical and computational analy-sis.

QM solvation models have followed this evolution,examining in recent years more properties at a moreaccurate level and reformulating for the specific caseof molecules in a medium a good number of the mostrecent advances in the methodology. To summarizethese developments we have to consider that thesubject is so large and the related physical phenom-ena so numerous that we shall necessarily limit ourexposition to some specific aspects according to thefollowing classification:

For both sets of properties we shall give first anintroduction that summarizes some aspects of thegeneral methodologies developed so far for solvatedsystems and then the description of the applicationof these methodologies to the study of specific proper-ties. In particular, for the first set, here called energyproperties, we shall consider properties that can beobtained from the potential free energy surface G(R),that is, the free energy functional as a function ofthe nuclear coordinates (see section 3.2.6), whereasfor the second set we shall consider response proper-ties that describe the effects of any externally appliedfield on a molecular system such as dipole (hyper)-polarizabilities, both static and frequency dependent,and magnetic and chiro-optic properties.

In the following analysis of both energy andresponse properties we shall limit our exposition tomolecules in their electronic ground state. We must,however, not neglect to say that in recent years,thanks to the evolution of both QM and solvationmodels, increasing attention has been paid to thestudy of the same properties for solvated systems inelectronic excited states. The first examples618-621 ofthese studies often used incomplete descriptions ofthe solvent effects (for example, neglecting the ex-plicit effects on the excited state geometries), whereasin the past few years more complete studies havestarted to appear. Many of these more recent ap-plications of QM continuum models622-627 are basedon the extension of the PCM model to the CI single(or CIS) approximation for the calculation of solventeffects on excited state geometries.614

More accurate QM methods such as CASPT2 andMRCI (see also section 5.2) have also been appliedto the study of the photophysics of solvated sys-

(∇∇G)mi,nj ) ⟨mn||ij⟩ + Bmi,nj

(∇G∇)mi,nj ) δmnδij(εm - εi) + ⟨mj|in⟩ + Bmi,nj

(153)

δF(ω) ) ∑mi

δPmi(ω)ψmψi/ + ∑

miδPim(ω)ψiψm

/ (154)

(∇∇G)mi,nj ) Kmi,jn + Bmi,nj

(∇G∇)mi,nj ) δmnδij(εm - εi) + Kmi,nj + Bmi,nj (155)

I. energy properties(a) geometrical derivatives(b) IR/Raman intensities(c) surface-enhanced IR and Raman

II. external or response properties(a) to electric fields(b) to magnetic fields(c) of chiral molecules


tems;624,625,628 in these cases, as no analytical deriva-tives are available, studies of geometry effects havebeen limited to a few important motions such astwisting and wagging modes in push-pull aromaticmolecular systems.629 In parallel, linear and nonlin-ear response methods (see section 5.6) have been usedto calculate solvent effects on excited state propertiessuch as dipole moments and (hyper)polariza-bilities.215,630-632

6.1. Energy Properties

Modern quantum chemistry has been revolution-ized by the ability to calculate analytical derivativesof the wave function and of the energy with respectto various parameters. A reference book by Yama-gouchi et al., published in 1994,633 is rightly titled ANew Dimension to Quantum Chemistry. AnalyticalDerivative Methods in ab Initio Molecular ElectronicStructure Theory. The appreciation of the role ofanalytical derivatives expressed in this title wasspeculative when formulated in 1993, but now maybe considered a reality. Nowadays, practically all abinitio methods are provided with analytical deriva-tives, and this availability has had a remarkableimpact on computational strategies.

In the past decade there has been a similarevolution in QM solvation models, and in this sectionwe describe how the methodologies originally formu-lated for isolated systems have been generalized toevaluate energy properties such as IR and Ramanspectra.

6.1.1. Geometrical Derivatives

The characterization of the free energy hypersur-face of solvated molecules, G(R), is based on freeenergy gradients, refined by second-order derivativeswith respect to the nuclear coordinates. The defini-tion of the equilibrium geometry is based on thesearch of critical points corresponding to local minimaon G(R), the definition of molecular motions leading,for example, to chemical reactions are again basedon the examination of the gradient, supplemented bythe examination of the sign of the local Hessianmatrix to characterize transition states and minima.The diagonalization of the Hessian is also necessaryto have the elements to characterize the molecularvibrations, and thus to study all of the properties thatdepend in some way on nuclear motions.

These geometrical derivatives can be computed asfinite differences as currently done in classical andsemiempirical continuum models, in which repeatedcalculations of the energy are not excessively costly.For ab initio methods the use of numerical deriva-tives strongly reduces the size of the systems sub-jected to scrutiny and the quality of the QM formu-lation.

The elaboration of analytic QM methods to com-pute derivatives of the free energy with respect tonuclear coordinates in continuum solvation modelsstarted at the beginning of the 1990s. The evolutionof this elaboration, which started from methodologies

developed in the 1970s for ab initio methods in vacuo,has encountered additional specific problems. Mostof these problems have been solved, but the develop-ment in the algorithms is still in progress.

Some among the first codes for gradients in con-tinuum methods adopted the approximation of keep-ing the cavity rigid (see, for example, ref 634). Thisis a minor approximation for crude cavities, but itrepresents a strong approximation when the cavityis accurately modeled in terms of spheres centeredon nuclei as done in the most recent versions of thecontinuum methods. The infinitesimal displacementof a nucleus, considered in a given partial derivative,entails an analogous infinitesimal motion in thecorresponding atomic sphere, which in turn producesan infinitesimal change in the intersection of thissphere with the others. The neglect of this infinitesi-mal motion of the cavity gives origin to apparentforces acting on the surface, which can have animportant effect, as it results from the comparisonof the derivatives computed with or without thisapproximation.

These considerations hold for all methods using acavity modeled on the molecular volume. Methodsbelonging to the ASC family, or making otherwiseuse of a discretization of the cavity (see section2.3.1.5), have other specific problems. For example,using a cavity formed by interlocking spheres, surfaceelements at the intersection of two spheres in generalhave an irregular form, depending of the position ofthe intersection circle with respect to the mesh. Acorrect procedure for analytical derivatives must beable to give a satisfactory solution to this problem.In addition, in some of the algorithms used to buildthe cavity (for example, the GEPOL approach43-45,92

used in the PCM class of models) an additionalproblem appears. To avoid crevices in which solventcannot enter, additional spheres not centered on thenuclei can be automatically introduced by the algo-rithm; the positions of these added spheres dependon the infinitesimal displacements of nuclei, and thustheir contribution to the analytical derivatives mustbe correctly taken into account. Analogous consider-ations hold for cavities making use of re-entrantportions of the surface (the Connolly surface, forexample).

Another point deserving mention is the use ofderivatives for solutes of large size. Methods foranalytical derivatives in molecules now scale linearlywith the size of the system. Linear scaling in con-tinuum methods is being actively pursued, and thesolutions to this problem require notable changes inthe formulation of analytical derivatives. Some ex-amples in the direction of more efficient computa-tional strategies to compute geometrical derivativesof large solvated systems have recently been pre-sented within the framework of PCM.62,635

Following the formalism used in section 2.4.2,analytical derivatives for solvated systems can beobtained by directly differentiating the expression ofthe free energy G given in eq 75, taking into accountthe SCF stationarity constraints. In this way we


obtain for the first and second energy deriva-tives636,637

where R and â are nuclear coordinates.In eqs 156 and 157 we have introduced the short-

ened notation GR ) ∂G/∂R and GRâ ∂2G/∂R∂â, omittedthe superscript 0 for the matrices h0 and G0, andadded superscripts on the matrices to denote deriva-tives of the corresponding integrals over the basis set.

The solvent-induced terms, hRRXR(P) and XRâ(P),

are obtained by differentiating once or twice thematrices of eqs 64-67, for example

where the second term in brackets involves deriva-tives of the apparent charges.

In eqs 156 and 157 we have introduced the ad-ditional matrix, W ) PFP, and its derivatives, forinstance

where F is the Fock matrix including solvent terms(see eq 63) and F(â) its derivative, namely

We note that the derivative of the density matrixP is not necessary for the first derivative GR becausethe SCF stationary condition leads to the equivalencetr PRF ) tr SRPFP as for the isolated molecule. Incontrast, density matrix derivatives are necessary forthe second- and higher order derivatives. In particu-lar, the second derivatives require the first deriva-tives of P (one for each parameter), which can beobtained by solving the appropriate coupled per-turbed equations. These equations can be put in thegeneral form

with the usual orthonormality condition:

The elements of the matrices hRR,XR(P) and

hRRâ,XRâ(P) contain the derivatives of the surface

apparent charges with respect to nuclear displace-ments. These derivatives depend on the specificequation used to compute q (see eq 34), but in anycase they require that we differentiate all of thegeometrical factors which define the ASC matrix K.93

Cances and Mennucci638,639 proposed an alternativeway to proceed within the IEFPCM framework; sucha procedure avoids computing any geometrical de-rivative of the surface charges. Their approachconsists of differentiating first the basic electrostaticequation

where λ indicates a solute nuclear coordinate and ε-(λ) ) 1 inside the cavity and ε(λ) ) ε outside, andobtaining ∂V/∂λ as a solution of the differentiatedequation. This quantity is then inserted in theformula of the derivatives of the solute-solventinteraction energy. Within this approach, the onlyterm coming from the dependence of the cavity onthe motions of the nuclear coordinates is reduced toa simple sum of the apparent charges q(sk) lying onthe moving part Γ of the cavity itself, namely

The function UΓ assumes a very simple form forstandard cavities given as unions of spheres, each ofthem centered on a solute nucleus. In this case wecan write

when λ is the Rth coordinate of the lth nucleus. Inthe aforementioned expression, (e1, e2, e3) is theorthonormal basis of the real space and Γl is the partof the cavity belonging to the sphere centered on thelth nucleus.

Going back to the QM expressions, the free energyderivative (eq 156) reduces now to

where the fifth and sixth terms are those collectingthe explicit solvent terms. The cavity-dependent termτ is shown in eq 164, whereas D, which denotes theinteraction energy between the variation of the solutecharge distribution ∂F/∂λ and the apparent charges,is expressed as

From eqs 164-167 it appears evident that noderivatives of the apparent charges q are required.The same method has been extended to secondderivatives;640 also in this case no derivatives of thecharges are required.

GR ) tr PhR + 12

tr PGR(P) + tr PhRR +

12

tr PXRR(P) - tr SRW + Vnn

R + 12

UnnR (156)

GRâ ) tr PhRâ + 12

tr PGRâ(P) + tr PhRRâ +

12

tr PXRâ(P) - tr SRâW + 12

UnnRâ + Vnn

Râ +

tr PâhR + tr PâGR(P) + tr PâhRR + tr PâXR(P) -

tr SRWâ (157)

XµνR (P) ) ∑

k[∂Vµν(sk)

∂Rqe(sk) + Vµν(sk)

∂qe(sk)

∂R ] (158)

Wâ ) PâFP + PF(â)P + P[G(Pâ) + XR(Pâ)]P +

PFPâ (159)

Fâ ) hâ + hRâP + Gâ(P) + XR

â (P) (160)

FPR + FRP - PRF - PFR + FPSR - SRPF ) 0(161)

PPR + PRP + PSRP ) PR (162)

-div[ε(λ)∇V(λ)] ) 4πF(λ) (163)

τ )4πε

ε - 1∑k

q2(sk)

ak

[UΓ(sk)‚n(sk)] (164)

UΓ(sk) ) 0 if sk ∉ ΓleR if sk ∈ Γl

(165)

GR ) tr PhR + 12

PGR(P) - tr SRW + VnnR +

D(∂F/∂λ,q) + 12

τ (166)

D(∂F/∂R,q) ) ∑k

∂V

∂R|sk

q(sk) (167)


Equations 166 and 167 represent an importantcomputational gain with respect to the standardmethod described above in which the solvent matriceshR and XR are explicitly differentiated. This approach,however, is much more sensitive to the quality of thecavity surface as the cavity term τ of eq 164 can bestrongly affected by the singular nature of the surfacein the regions where two or more spheres intersect.As a matter of fact, numerical tests have shown that,with GEPOL-like cavities, this approach gives largernumerical instabilities than the standard one. At themoment, these numerical difficulties prevent themethod from being competitive, but it is surely worthcontinuing to check other forms of cavities withsufficient smoothness as this approach could repre-sent a computationally powerful approach, especiallyfor large solutes.

We finish this section by noting that the equationshave been presented in the PCM framework; asimilar formalism, however, has been also developedfor other ASC methods such as COSMO,641,642 whereasa different approach is required for continuum mod-els not belonging to the ASC class. For MPE methods,Wiberg and co-workers634 presented the formulationwithin the Onsager formalism for spherical fixedcavities. Mikkelsen et al.136 presented multipolar one-center formulations still for spherical fixed cavities,and Rivail and co-workers presented a theory for bothspherical and ellipsoid cavities with one-center mul-tipolar expansion and molecular cavities with mul-ticenter multipolar expansions.145,643 Within the GBapproach, analytical energy gradients have beenformulated by Cramer and Truhlar and co-workersusing a SMx solvation model based on CM2 atomiccharges.171

For some of these models derivatives for post-HFmethods have also been presented both for varia-tional (see refs 96, 212, and 644 for MCSCF) and fornonvariational (MP2234,236 and CC140) methods. Theformal apparatus looks more complex, but actuallyit is not much different from that used for analogousderivatives in vacuo: for variational methods theformalism resembles that of Hartree-Fock, whereasnonvariational methods have required the extensionof the relaxed density matrix approaches to thecontinuum solvation methods.

6.1.2. IR and Raman Intensities

Theoretical modeling of solvent effects on IR andRaman intensities has been a matter of study formany years within semiclassical continuummodels.645-651 The various formulations were basedon the consideration that when the light interactswith a molecule in condensed phase, the “local field”that is experienced by the molecule is different fromthe Maxwell field in the medium.

Local field effects, which are also important in thestudy of other properties, such as linear and nonlin-ear optical properties (see section 6.2.2) and opticaland vibrational properties of chiral molecules (seesection 6.4), can be separated into two terms. Thefirst one, the “reaction field”, is connected with theresponse (polarization) of the medium to the moleculecharge distribution. The second term (the “cavity

field”, Ec) depends on the polarization of the mediuminduced by the externally applied electric field oncethe cavity, which hosts the molecule, has beencreated.

A further aspect introduced by some of thesemodels concerns nonequilibrium solvation effectsassociated with nuclear vibrational motions.

Only recently, both local field and nonequilibriumeffects have been reconsidered within the frameworkof QM solvation models to study vibrational proper-ties. We present some details about these solventeffects on the IR intensities, but similar consider-ations can be applied to Raman intensities.

We shall start from the analysis of “local field”effects and follow the approach used in refs 652-654within the framework of the PCM solvation model.

6.1.2.1. Cavity Field Effects. The starting pointis the expression for the transition probability of anisolated system in the dipolar approximation betweentwo vibrational states 0 and 1 induced by radiationof frequency ω

In eq 168 E1 and E0 are the energies of the vibrationalstates 1 and 0, respectively, µb is the dipole moment,and EBgas is the electric field acting on the molecule.

Passing to molecules in solution, the transitionprobability can be expressed in two alternative butequivalent ways. We introduce the cavity field Ec,which interacts with the dipole moment µb of thesolvated molecule, or we use the (Maxwell) electricfield of the radiation in the medium, EB, whichinteracts with the so-called external dipole moment.Following Onsager, this quantity is defined as thesum of the molecular dipole moment and the dipolemoment µ arising from the polarization induced bythe molecule on the dielectric. The resulting expres-sion for the transition probability in solution yields

and the corresponding integrated absorption becomes

where NA is Avogadro’s number, c the velocity of lightin vacuo, and ns the refractive index of the solventunder study. In eq 170 Q collects the solute massweighted normal coordinates associated with thevibrational mode.

The computational expression for µ, within thePCM framework,652 is given in terms of apparentsurface charges, qex, analogous to those defined insection 2.3.1.5, but having the external Maxwell fieldas a source of polarization.655,656 The final expressionis

where the sum runs over the number of tesserae, V

W1r0gas ) 2π

pδ(E1 - E0 - pω)|⟨1|µb‚EBgas|0⟩|2 (168)

W1r0sol ) 2π

pδ(E1 - E0 - pω)|⟨1|(µb + µ)‚EB|0⟩|2

(169)

Asol )πNA

3nsc2(∂(µb + µ)

∂Q )2(170)

µ ) -(∑kV(sk)∇BEB[qex(sk)]) (171)


is the potential generated by the solute on the sametesserae, and the symbol ∇EB indicates the differentia-tion of the apparent charges with respect to thecomponents of the external Maxwell field.

6.1.2.2 Nonequilibrium Effects. Looking at typi-cal vibration times, namely, 10-14-10-12 s, it isevident that the orientational component of thesolvent polarization is not able to follow the oscillat-ing charge distribution of the solute and a nonequi-librium polarization (comparable to that presentedin section 5.2 for electronic motions) must be consid-ered.

In common practice, the polarization is partitionedinto two terms: one term accounts for all motionsthat are slower than those involved in the molecularvibrations and the other includes the faster contribu-tions. The next assumption usually made is that onlythe latter are instantaneously equilibrated to themomentary molecule charge distribution, whereasthe former cannot readjust.

In the case of vibrations of solvated molecules theslow term contains contributions arising from themotions of the solvent molecules as a whole (transla-tion and rotations), whereas the fast term takes intoaccount the internal molecular motions (electronicand vibrational). Such a division is reasonable whenthe solute vibration under study is sufficiently farin the frequency from those of the solvent; whennecessary, this distinction may be modified.

The first attempt to introduce nonequilibriumeffects into the QM vibrational studies with con-tinuum models is due to Olivares del Valle et al.using the DPCM formulation.657,658 They introduceda set of models spanning from a complete decouplingof the vibrational charge distribution and the solventpolarization to a complete equilibrium coupling,passing through a nonequilibrium model (see ref 1for more details) to compute both vibrational fre-quencies and intensities.

Also, Rivail and co-workers659 elaborated a non-equilibrium scheme (in the framework of the MPEmethod using an ellipsoidal cavity) but limited to thestudy of the frequencies. Their formulation is the firstto exploit analytical first and second derivatives ofG(R). The numerical applications they presentedshowed that nonequilibrium solvation has little effecton the vibrational frequencies. We remark that Rivailused a nonequilibrium polarization scheme based onthe partition approach described in section 5.2.

More recently, Cappelli et al.653 elaborated a moresophisticated nonequilibrium formulation for thePCM scheme. Their method allows for analyticalderivatives of the nonequilibrium free energy Gneq-(R) to compute IR frequencies and intensities. In thismodel a rigid cavity is exploited consistently with anonequilibrium scheme in which the orientationaland translational motions of the solvent moleculesare not able to follow the vibrating charge distribu-tion of the solute. The computational scheme is basedon the partition of the polarization into inertial anddynamic terms and on the use of the correspondingexpression for the free energy functional (see section5.2). In this model, the nonequilibrium is due to theoscillating motion of both electrons and nuclei. The

analytical derivatives of Gneq(R) were computed byexploiting the same techniques used for the corre-sponding derivatives of the equilibrium free energyfunctional (see eq 157). However, this time the cavityis kept fixed, and thus the evaluation of the geo-metrical derivatives is less computationally demand-ing. Numerical applications to carbonyl frequenciesin ketones confirm the previous finding of Rivail thatnonequilibrium solvation has little effect on thefrequencies, whereas they found that nonequilibriumpolarization has large effects on the IR intensities.

Cappelli et al. have subsequently extended theirnonequilibrium formulation to the study of Ramanscattering factors for solvated molecules.505,654,660 Thestarting point of this formulation was the consider-ation of the Raman scattering as the result of themodulation in the electric-field-induced oscillatingdipole moment due to vibrational motions. Thedynamic aspects of Raman scattering are to bedescribed in terms of two time scales, one connectedto the vibrational motions of the nuclei and the otherto the oscillation of the incoming electric field; bothgive origin to oscillations in the solute electronicdensity). In the presence of a solvent, both time scalesoriginate nonequilibrium effects in the solvent re-sponse to the fast oscillations in the solute electronicdensity. The proposed model, which includes such anonequilibrium effect, has been applied to the studyof Raman intensities of simple molecules in varioussolvents.

6.1.3. Surface-Enhanced IR and Raman

Molecules adsorbed on metal surfaces exhibit aninfrared light absorption (due to vibrational transi-tions), which is amplified by a factor of 102-103

compared to the molecule without the metal [thisphenomenon is called surface-enhanced infraredabsorption (SEIRA)].661,662 In parallel, molecules ad-sorbed on specially prepared metal surfaces or onmetal colloids exhibit Raman spectra with somebands enhanced by a factor of up to 1010-1014 incomparison with the spectra of nonadsorbed mol-ecules, enabling the detection of a single molecule.This phenomenon is generally known as surface-enhanced Raman scattering (SERS).663-667 The met-als that promote SERS more readily are Ag, Au, andCu (coinage metals), although SERS has been re-ported for a large variety of metals.

When an electromagnetic field is applied to a metalspecimen having a curved surface (such as a roughsurface, a metal nanoparticle deposited on a surface,or a metallic colloids), it can excite collective reso-nances of the electron gas in the metal (surfaceplasmons). Such resonances can create an evanescentelectromagnetic field of high intensity, localized inproximity of the metal surface. A molecule close tosuch a metallic specimen feels a local field much moreintensely than the incident one, so its effectiveresponse properties will be greatly amplified. Inparticular, local enhancements of the field give originto so-called “hot spots”, that is, points at which thechromophore, interacting with the resulting totalfield, shows an enhanced IR/Raman signal. Theenhancement factors tentatively considered are that


of the incident electric field due to the whole metalaggregate, that due to the interaction of the moleculewith the local hot spot combined with the effect ofinteraction of the molecule with the liquid phase (ifpresent), that due to the interaction of the lightemitted by the molecule plus hot spot system withthe rest of the metal cluster, and that due to chemicalinteraction of the molecule with the metal.

The effects on electromagnetic fields due to thepresence of metal bodies represent a unifying elementin the origin of the SEIRA and SERS phenomenaeven if each phenomenon has its own features.Because SERS has been the most studied, here weshall focus on that only.

A considerable amount of theoretical work on thissubject has been done since the discovery of SERS,but most of these, although quite refined in thedescription of the metal,668 represent the moleculejust as a polarizable point dipole. Another approachused on some occasions669,670 consists of a whole abinitio description of the molecule interacting with ametal cluster of a few atoms. This model accountsalso for chemisorption effects, but the small metalcluster clearly cannot reproduce the behavior of, forinstance, a nanosized metal particle, which is com-posed of thousands of atoms.

If we limit ourselves to the study of moleculesphysisorbed on a metal, that is, molecules which arenot chemically bound to the metal surface, theproblem of a proper description of such systemsstrongly resembles that for solvated molecules. Infact, a molecule in solution weakly interacts with agreat number of other electrons and nuclei, feelingall of the fields of external as well as internal origin,as a molecule physisorbed on a metal surface does.Following the theoretical lines of the PCM approach,an extension of such a model to treat molecules closeto metal particles (which, in turn, can be immersedin a dielectric medium) has been presented79,504,507

(see also section 5.5). In such an extension, the metalis described as a continuous body characterized byelectric response properties only, which behaves asa perfect conductor for static fields and as a dielectricfor time-dependent fields. Infinite planar metal sur-faces, complex shaped nanoparticles, and aggregatesof nanoparticles have been considered, but in anycase the only interactions considered among thevarious portions of the complex system (the molecule,the metal, and, possibly, the solvent) are electrostaticin origin.

The basic idea underlying this model is that it isnot necessary to treat the whole system (metalparticle plus molecule) to the same degree of ac-curacy, but it is better to focus the attention on theportion of the system of which we want to study theproperties.

Subsequently, the model has been improved507 bypartitioning the system into three “layers,” describedat different levels of accuracy. The whole metalcluster (the first layer) is treated as a collection ofspherical metallic particles described as polarizabledipoles to be able to recognize where the hot spotsare at a given excitation wavelength and how strongthe electric field locally acting on them is. The metal

particles in a given hot spot (the second layer) canbe considered as a unique complex shaped particleand treated by using the original model, whichneglects the detailed atomic and electronic structureof the metal but which takes into account effects onthe electric field due to the complex shape (union ofinterlocking spheres). Finally, the molecule (the thirdlayer) is described at the ab initio level, taking intoaccount electrostatic interaction with the metal par-ticle and properly treating the electric field acting onit. By exploiting this model, huge enhancementfactors have been found for Raman spectra of mol-ecules close to different metal particle aggregates,which compare well with experiments.

6.2. Response Properties to Electric FieldsThe investigation of linear and nonlinear optical

(NLO) properties of solvated molecules and liquidshas been one of the most important developments ofthe QM continuum solvation models in recent years.Such progress has been prompted by the success ofmodern quantum chemical tools in predicting re-sponse properties for molecules in the gas phase.

The experience gained for isolated molecules hasin fact represented a fundamental background asconcerns theoretical issues as well as more physicalaspects. Following these studies, for example, con-tinuum solvation models have been extended to treatnot only the leading electronic term of the opticalproperties but also (even if in a less widespread way)the vibrational contributions. Due to the far largernumber of studies present in the literature, in thefollowing sections we shall focus on the electroniccontribution only. However, it is interesting to addsome notes on the vibrational part also becausestudies appeared so far seem to show that vibrationaleffects can become very important (if not dominant)for NLO properties of solvated molecules.671,672

Vibrational contributions are generally divided intotwo components,673-676 the “curvature” related to thefield dependency of the vibrational frequencies (i.e.,the changes in the potential energy surface in thepresence of the external field) and including the zero-point vibrational correction and the “nuclear relax-ation” arising from the field-induced nuclear relax-ation (i.e., the modification of the equilibrium geometryin the presence of the external field).

The nuclear relaxation is generally the dominantcontribution, especially when in the presence ofapplied static fields. The most largely used methodto account for these effects is based on series expan-sion of the energy and of the electric dipole withrespect to both the nuclear coordinates and theexternal field components. As shown in refs 671, 672,and 677, the same procedure can be applied tosolvated systems when described through continuumsolvation models; this equivalence is made possibleif expansions of the free energy functional are usedand the related dipole and polarizability derivativeswith respect to normal coordinates are computed inthe presence of the solvent.

The equivalence between condensed phase and gasphase systems concerning the calculation of opticalproperties, however, is not complete as the presence


of the environment introduces more critical issuesthan for isolated systems, but it also offers thepossibility to compare the computed microscopicproperties with experimental macroscopic quantities.Formal correspondences and physical differences willbe analyzed in the following two subsections, respec-tively.

6.2.1. QM Calculation of Polarizabilities of SolvatedMolecules

In section 5.6 we have presented the general time-dependent QM theory for solvated systems; here, wecome back to such a theory but specialize it to thecase of a perturbation represented by the combina-tion of a static (E0) and an oscillating electric field(Eω). In this case, the operator W(t) in eq 143 becomes

where µ is the electronic dipole moment operator ofthe solute.

By using the Frenkel variational principle (eq 146)within the restriction of a one-determinant wavefunction with orbital expansion over a finite atomicbasis set, we obtain the following time-dependentHartree-Fock (or Kohn-Sham) equation

with the proper orthonormality condition; S, C, andε represent the overlap, the MO coefficients, and theorbital energy matrices, respectively.

In eq 173 the prime on the Fock matrix indicatesthat terms accounting for the solvent effects areincluded, that is

where the first four terms on the right-hand side ofthe equation represent the Fock operator for thesolvated molecule (see section 2.4.2 for more details)and m represents the matrix containing the dipoleintegrals; we note that in eq 174 we have discardedthe superscript “0” used in the previous sections toindicate the Hamiltonian terms of the isolated mol-ecule (here the same superscript is used to indicatestatic fields) and the superscript R for the solvent-induced terms, j and X(P).

The solution of the time-dependent HF or KS eq173 can be obtained within a time-dependent coupledHF (or KS approach). We first expand eq 174 to itsFourier components, and subsequently, each compo-nent is expanded in terms of the components of theexternal field. The separation by orders leads to aset of CPHF equations for which the Fock matricescan be written in the form

where a,b, ... indicate the Cartesian components ofthe field and ωx are the frequencies related to theexternal fields (eventually static and thus ωx ) 0).We note that the elements of the solvent-reactionmatrices XωT (see eqs 67 and 68) depend twice on thefrequency-dependent nature of the field, in theperturbed density matrices Pab...(ω1,ω2,...), and in thevalue of the solvent dielectric permittivity ε(ωT) atthe resulting frequency ωT ) Σωx (see section 5.2).

Once solved, the proper CPHF equations, thedynamic polarizabilities of interest, can be expressedin the following forms:

We note that an alternative (but equivalent) methodcan be used to get Rab(ω); namely, we can resort tothe linear response theory presented at the end ofsection 5.6 and use eq 151 with B and A being thedipole moment operators, that is, ⟨⟨µa;µb⟩⟩ω (to get theanalogue equation for â and γ a quadratic and a cubicresponse theory have to be invoked).

Applications of the QM theory for the calcula-tion of linear and nonlinear optical properties ofsolvated molecules have been carried out byWillets and Rice,678 Yu and Zerner,679 Mikkelsenand co-workers,139,516,680-684 and the PCMgroup,599,655,656,671,685-689 all applying different solventmodels as well as different levels of the quantumtheory.

6.2.2. Definition of Effective PropertiesA key factor in the study of response properties is

the relationship between the molecular and thematerial properties, that is, the distinction betweenthe microscopic field at a molecule and the macro-scopic field applied to the material.690 The macro-scopic field is determined directly from the experi-mental conditions, but the microscopic field has tobe calculated using a suitable theory.

The classical approach is to use the Onsager-Lorentz model and write the measured susceptibili-ties (the macroscopic equivalent of the linear andnonlinear optical, NLO, molecular properties) interms of the gas phase molecular values multipliedby so-called local field factors (see also section 6.1.2).In recent years different models have been proposedto improve such a description. Most of them are basedon the concept of a solute in a cavity in a dielectriccontinuum provided by the rest of the material.Within the framework of semiclassical descriptions,important generalizations of the Onsager-Lorentzmodel for NLO properties have been made byWortmann and Bishop691 and by A° gren and co-workers,677,692-695 whereas a QM approach has beenformulated within the PCM approach described insection 2.3.655,656,689

The PCM extension to get macroscopic susceptibili-ties that can be directly compared with experimentshas been achieved by introducing so-called effective

W(t) ) µ‚[EBω(eiωt + e-iωt) + EB0] (172)

F′C - i ∂

∂tSC ) SCε (173)

F′ ) h + G(P) + j + X(P) +m‚[Eω(eiωt + e-iωt) + E0] (174)

F′a(ω) ) ma + mω + G(Pa(ω)) + Xω(Pa(ω))

F′ab...(-ωT;ω1,ω2,...) ) G(Pab(-ωT;ω1,ω2,...)) +

XωT(Pab(-ωT;ω1,ω2,...))

(175)

Rab(-ω;ω) ) -tr[maPb(ω)]

âabc(-ωT;ω1,ω2) ) -tr[maPbc(-ωT;ω1,ω2)] (176)

γabcd(-ωT;ω1,ω2,ω3) ) -tr[maPbcd(-ωT;ω1,ω2,ω3)]


properties: the properties describing the change ofmolecular polarization with respect to the externalapplied fields. Here we briefly describe the mainaspects of this extension emphasizing when usefulthe connection with semiclassical approaches.

The response of the medium to a macroscopic fieldEB(t) generated by the superposition of a static andan optical component is represented by the dielectricpolarization vector (dipole moment per volume) PB(t),which in terms of Fourier components yields

Exactly as for the molecular dipole expansion usuallyintroduced for isolated molecules, each Fourier am-plitude can be rewritten as a power series withrespect to the applied macroscopic (or Maxwell) fieldin the medium, for example

where the tensorial coefficients are the nth-ordermacroscopic susceptibilities ø(n). They are tensor ofrank n + 1 with 3(n+1) components. The argumentin parentheses describes the kind of interactingwaves; in all cases the frequency of the resultingwave is stated first, then the frequency of the incidentinteracting wave(s) (two in a first-order process, threein the second-order analogue, and four in the third-order case). The different susceptibilities involved inthe various Fourier amplitudes correspond to experi-ments in linear and nonlinear optics; in eq 178, forexample, ø(1)(-ω;ω) is the linear optical susceptibilityrelated to the refractive index n at frequency ω andthe Pockels susceptibility ø(2)(-ω;ω,0) describes thechange of the refractive index induced by an exter-nally applied static field.

If we consider as macroscopic sample a liquidsolution of different molecular components, each ata concentration cJ (moles per volume), the globalmeasured response becomes696-699

where the effects of the individual components havebeen assumed to be additive and úJ

(n) are the nth-order molar polarizabilities of the constituent J. Thevalues of each úJ

(n) can be extracted from measure-ments of ø(n) at different concentrations. The calcula-tion of the molar polarizabilities úJ

(n) involves statis-tical mechanical averaging over orientational distribu-tions of the molecules. In particular, by introducingthe Z space-fixed axes of the laboratory, we obtainfor a general nth-order molar polarizability

where NA is Avogadro’s number and m is the effectivedipole.656

We note that in the case of a pure liquid, thedefinition (eq 179) becomes700

where F is the number density of the liquid.The derivation of the expressions for the molar

polarizabilities in terms of the proper effective prop-erties can be found in ref 656. We only stress that asthe molar polarizabilities represent an easily avail-able “experimental” set of data, these expressionsbecome important for the theoretical evaluation ofmolecular response properties. In fact, they representthe most direct quantities to compare with thecomputed molecular effective properties.

The theory for the calculation of the effectivepolarizabilities has been presented both at semiclas-sical691 and quantum mechanical level.656 The QMtheory is parallel to that presented in section 6.2.1;now, however, we have to introduce a further per-turbing term for a correct comparison with experi-mental results. As a result, the operator W(t) in eq172 becomes

In eq 182 “external” ASCs, qex, have been introducedexactly as in eq 171; these charges can be describedas the response of the solvent to the external field(static or oscillating) when the volume representingthe molecular cavity has been created in the bulk ofthe solvent. This effect must be summed to thestandard reaction field in order to fully consider theeffective response of the solvent to the combinedaction of the internal (due to the solute) and theexternal fields.655,656 We note that the effects of qex

in the limit of a spherical cavity coincide with thatof the cavity field factors historically introduced totake into account the changes induced by the solventmolecules on the average macroscopic field at eachlocal position inside the medium.

Due to the presence of the additional term also, theFock matrix defined in eq 174 changes to

where the last two terms, mω and m0, are thematrices related to the apparent charge qex inducedby the external oscillating and static field, respec-tively, namely (see eq 171)

where V(sk) is the matrix containing the solutepotential atomic integrals computed on the surfacecavity.

By solving the new time-dependent coupled HF (orKS) equation, the dynamic effective polarizabilitiesof interest can be expressed exactly as in eq 176; the

PB(t) ) PB0 + PBω cos(ωt) + PB2ω cos(2ωt) + ... (177)

PBω ) ø(1)(-ω;ω)‚EBω + 2ø(2)(-ω;ω,0)‚EBωEB0 + ...(178)

ø(n) ) ∑J

úJ(n)cJ (179)

úZZ...(n) (nω) ) NA(∂n⟨µjZ(nω)⟩

∂EZ∂EZ... )Ef0

(180)

ø(n) ) cúZZ...(n) (nω) ) F(∂n⟨µjZ(nω)⟩

∂EZ∂EZ... )Ef0

(181)

W(t) ) µ‚[EBω(eiωt + e-iωt) + EB0] +

∑k

Vk(∇BEBx [qωex(k)]‚EBω(eiωt + e-iωt) +

∇BEB0[q0ex(k)]‚EB0) (182)

F′ ) h + G(P) + m‚[Eω(eiωt + e-iωt) + E0] +j + X(P) + mω‚Eω(eiωt + e-iωt) + m0‚E0 (183)

mx ) -(∑k V(sk)∇ebx[qex(sk)]) with x ) 0,ω (184)


only difference is that now perturbed electron densityPa(ω) [or Pab(ω)] is different due to the presence ofthe new perturbing term (eq 184) in the related Fockmatrices.

We conclude by noting that to extend the QMcalculations of effective polarizabilities to pure liq-uids, a further element has to be introduced. In thatcase, the optical radiation at the fundamental fre-quency ω produces in the liquid a macroscopicpolarization density at the output frequency, whichacts as source of an additional perturbing term.700

6.3. Response Properties to Magnetic FieldsIn this section, we present a research field rela-

tively new for QM continuum solvation models,namely, that of the study of solvent effects onmagnetic molecular properties, which determinenuclear magnetic resonance (NMR) and electronicparamagnetic resonance (EPR) phenomena. Due tothis “young existence”, many aspects of both formaland physical nature have yet to be deeply understood,but the research is very active in many laboratories.To date, almost all of the studies in this field haveinvolved two classes of continuum models, that basedon a multipolar expansion (MPE) of the reaction fieldand that introducing an apparent surface charge(ASC). In particular, we note that greatest progresshas been achieved within the ASC approach [in itsdifferent formulations, CPCM, IEFPCM, and SS(V)-PE]. On the one hand, generalizations to includemore detailed aspects of solvation, such as specificsolute-solvent interactions, through couplings withdiscrete pictures have been presented using solute-solvent clusters obtained through QM minimizationmethods or taken from MD simulations. On the otherhand, extensions to more complex environments havebeen also realized; we recall here, as an example, therecent extension of the IEFPCM for anisotropicdielectrics to the study of NMR properties of solutesimmersed in liquid crystalline solvents.102,701 Both ofthese extensions indicate new perspectives of ap-plications of continuum models toward systems ofincreasing complexity, heretofore considered as com-pletely prohibitive for a continuum description.

To better quantify the importance that QM studiesof solvent effects on magnetic properties are rapidlyachieving, in Table 3 we try to summarize recentpapers in which continuum solvation models havebeen used.

6.3.1. Nuclear ShieldingThe effects of solvent on nuclear magnetic shielding

parameters derived from NMR spectroscopy havebeen of great interest for a long time. In 1960Buckingham et al.743 suggested a possible classifica-tion in terms of various additive corrections to theshielding arising from (i) the bulk magnetic suscep-tibility of the solvent, (ii) the magnetic anisotropy ofthe solvent molecules, (iii) van der Waals interac-tions, and (iv) long-range electrostatic interactions.In the original scheme, strong specific interactions,such as those acting in intermolecular hydrogenbonds, were not specifically dealt with but justmentioned as a possible extreme form of the electro-

static, or, more generally “polar”, effect; in thenumerous applications that followed Buckingham’sclassification, however, this further effect has beenalways included as a separate contribution.

On the basis of such a classification an empiricalapproach based on the so-called solvent empiricalparameters was formulated to evaluate solvent ef-fects on nuclear shieldings. In brief, this approach,originally proposed by Kamlet, Taft, and co-work-ers,744 does not involve QM or other types of calcula-tions but introduces a numerical treatment of ex-perimental data obtained for a given reference systemto obtain an estimate of solvent effects on variousproperties. Its generalization to the study of thesolvent effect on nitrogen nuclear shielding in variouscompounds has been proposed by Witanowski etal.745-749

In parallel to these semiclassical analyses, recentlywe have observed growing attention to an explicitdescription of the electronic aspects of the solventeffects on NMR properties and in particular on thenuclear shielding. This change of perspective hasbeen made possible by the large development of QMsolvation models we have described in section 2,which have been coupled to QM methodologies ini-tially formulated for isolated systems.

The theory of chemical shielding was originallydeveloped many years ago,750,751 but only later haveab initio methods752-756 and density functional theo-ries (DFT)757,758 been reliably used for the predictionof chemical shielding for molecular systems.

Because these recent developments greatly im-proved the accuracy of the calculation of the nuclearshieldings, the following step has been the inclusionof solvent-induced shifts. This has been achieved byextending the cited theoretical methods to continuumsolvation models. Such extension began by redefiningthe components of the shielding tensor σ, as secondderivatives of the free energy functional defined insection 2.4712

where Ba and Mb (a,b ) x, y, z) are the Cartesiancomponents of the external magnetic field B and of

Table 3. NMR and EPR Properties (NuclearShielding, Magnetizability, Spin-Spin Coupling,EPR) Calculated with Continuum Solvation Models(or Their Hybrid Discrete/ContinuumGeneralizations)

solvation model authora refs

MPE Mikkelsen 702-708Pecul 709, 710Pennanen 711

ASC Cammi/Mennucci 102, 701, 712-722(CPCM, Barone/Cossi 723-734IEFPCM, SS(V)PE) Manalo 735, 736

Chipman 104Cremer 737, 738Zaccari 739, 740Ciofini 741Rinkevicius 742

a Reference name, not necessarily the first author.

σabX ) ∂

2G∂Ba∂Mb

X(185)


the nuclear magnetic moment of nucleus X, respec-tively. In this way, the general description of thenuclear magnetic shielding for an isolated moleculein terms of the influence on the total energy of themolecule of the nuclear magnetic moment and of theapplied uniform magnetic field has been translatedto molecular solutes in the presence of solventinteractions.705,712

We recall here that in order to reliably calculatethe nuclear shielding in ab initio calculations, thegauge invariance problem has to be faced. Gaugefactors are generally introduced into the atomicorbitals of the basis set by using gauge invariantatomic orbitals (GIAO).752,753 The GIAO method isused in conjunction with analytical derivative theory(see section 6.1.1). In this approach the magnetic fieldperturbation is treated in an analogous way to theperturbation produced by changes in the nuclearcoordinates (see eq 157), and the components of thenuclear magnetic shielding tensor are obtained as

where PB is the derivative of the density matrix withrespect to the magnetic field. Matrices hM and hB,M

contain the first derivative of the one-electron termof the Hamiltonian with respect to the nuclearmagnetic moment and the second derivative withrespect to the magnetic field and the nuclear mag-netic moment, respectively. Neither matrix containsexplicit solvent-induced contributions as the latterdo not depend on the nuclear magnetic moment ofthe solute, and thus the corresponding derivativesare zero. On the contrary, solvent effects act on thefirst derivative of the density matrix PB, which canbe obtained as solution of the corresponding coupledperturbed first-order HF or KS equations (see eqs 161and 162).

As shown by the literature reported in Table 3, theapplications of continuum models to the QM studyof nuclear shieldings are numerous and they involvemany different aspects; here, we comment on two ofthem.

The first aspect concerns the analysis of the solventeffects on the chemical shielding in terms of twodistinct contributions:713 a direct one due to theperturbation of the solvent on the electronic wavefunction of the solute held at the geometry optimizedin vacuo, and an indirect one due to the relaxationof this geometry under the influence of the solvent.The studies reported in Table 3 have shown theimportance of a proper description of the solvent-induced changes on the geometry of solvated systems,in particular for polar and flexible molecules.

The second aspect, already introduced above interms of the Buckingham and Kamlet-Taft analyses,concerns the importance that both short-range andlong-range solute-solvent interactions have in de-termining the solvent effect on the nuclear shieldings.As will be further discussed in the following sections,the currently held idea is that short-range inter-actions can be effectively handled by supermole-cule calculations involving a solute surrounded bya number of explicitly treated solvent mole-

cules,713,719,759-762 whereas reaction field (or con-tinuum) methods generally provide an effective al-ternative to describe long-range electrostatic inter-actions. As a result, the combination of the twoapproaches when coupled to accurate QM methodsgives an effective computational tool to includesolvent effects into nuclear shielding calcula-tions.716,717,720-722,729,732,733 It must also be noted that,with respect to other properties of optical origin, thenuclear shielding presents a further specificity to beaccounted for when the comparison with experimen-tal data is sought. The NMR spectroscopy is in factcharacterized by medium-long times of measure; asa consequence, anytime explicit solvent molecules areused, it becomes compulsory to correctly account forthe statistical picture inherent in the nature of thesolvation shells for either weakly or strongly inter-acting solute-solvent systems.721,722

6.3.2. Indirect Spin−Spin CouplingSolvent effects on indirect nuclear spin-spin cou-

pling constants J have their origins in intermolecularinteractions, repulsion, dispersion, electrostatic, andinduction forces, as well as hydrogen bonding effects.As for nuclear shieldings, also for J two mainapproaches are used in the recent literature toaccount for these effects: the continuum approachand the supermolecule method.

Many applications on J mediated through hydro-gen bonds763-768 can be considered to fall into thesupermolecule category, although the goal there isin calculating actual intermolecular couplings.

Until 2003 only one implementation of a dielectriccontinuum model for studying solvent effects onspin-spin coupling constants was present in theliterature.706 It was initially used to study solventeffects on the spin-spin couplings of H2Se, H2S, andHCN,706,707 but has in recent years been used forstudies on larger molecules, often together withsupermolecular models to account for the effects ofspecific solute-solvent interactions.710 The dielectriccontinuum model presented in this first applicationwas the MPE approach described in section 2.3.2.

In 2003 a new approach for calculating solventeffects on indirect spin coupling constants based onthe IEFPCM model described in section 2.3.1.3 waspresented.718 The implementation is based on theextensions of the PCM model to the singlet linearresponse functions we have described in section5.6,215,527 but it can include triplet linear responsefunctions as well. In this approach, based on DFTreference states, the indirect spin-spin couplingparameters are determined as second derivatives ofthe electronic free energy functional G defined insection 2.4.4 with respect to the nuclear magneticmoments MB K

where γK are the nuclear magnetogyric ratios, thatis, the ratio between the nuclear magnetic momentsand the nuclear spins. The evaluation of the spin-spin coupling constants as second derivatives of the

σab ) tr[PhBaMbX

+ PBahMbX

] (186)

JabKL ) h

γK

2πγL

2πd2G

dMaK dMb

L(187)


free-energy functional G requires a procedure basedon the linear response theory for molecular solutes,that is, a variational-perturbative scheme whichensures the stationary condition of the free energyfunctional at first order with respect to any value ofthe nuclear magnetic moments. Ignoring the detailsof the parametrization, we denote the free energy asG(MK, λS, λT), where λS and λT are two sets ofvariational parameters associated with singlet andtriplet variations of the electronic state, respec-tively.769 For the optimized free energy state, thesecond derivatives of eq 187 may then be calculatedas

The derivatives of the variational parameters λS andλT with respect to MB L are obtained by solving theresponse equations

where the second derivatives on the left-hand sidesare, respectively, the singlet and triplet electronicHessians. The solution of the singlet linear responseequation 189 gives the perturbing effect of theimaginary singlet paramagnetic spin-orbit (PSO)operator, whereas the solution of the triplet linearresponse equation 190 represents the perturbingeffect of the real triplet Fermi contact (FC) and ofthe spin-dipole (SD) operator. The perturbing effectof the second-order real singlet diamagnetic spin-orbit (DSO) operators enters the reduced couplingconstant via the first term of eq 188 and does notrequire solution of any response equation.

6.3.3. EPR Parameters

Electronparamagnetic resonance (EPR) is the lead-ing magnetic technique for obtaining geometrical andelectronic information about systems with unpairedelectrons. EPR experimental results are commonlyanalyzed using a two-term spin Hamiltonian: thefirst term is the Zeeman term describing the interac-tion between the electronic spin SB and the externalmagnetic field BB, through the Bohr magneton µB andthe g tensor, whereas the second term describes thehyperfine interaction between SB and the nuclear spinI through the hyperfine coupling (hc) tensor A.

Compared to the state of the calculation of NMRproperties, much less has been done on the first-principles calculation of EPR parameters. As a mat-ter of fact, the treatment of EPR hyperfine couplingconstants does have an appreciable history of first-principles theoretical treatments,770,771 even if quan-

titative calculations of electronic g tensors havebecome possible only very recently, especially thanksto the developments of density functionaltheory.742,772-775

Between the g tensor and the hc tensor, the formeris far more complex to evaluate through QM ap-proaches. The g tensor is defined as a second-orderderivative of the electronic energy E with respect tothe external magnetic field and the electronic spin:

As the g tensors may be affected significantly bysmall changes in electron density distribution, theyare important probes of the environment around themolecule. Generally, environmental effects may in-fluence the g tensor in two different ways: (a)indirectly, by modifying the structure of the radical(e.g., bond lengths or conformations), and (b) directly,by polarizing the electron density distribution andaltering the ground state wave function at a givengeometrical structure.

In the case of specific solvent-solute interactions,such as hydrogen bonding, supermolecule modelsmay provide the most appropriate description evenif the question remains as to what extent more long-range dielectric effects contribute in these cases. Incontrast, nonspecific solvent-solute interactions, forexample, in aprotic environments, might be describedmore efficiently by dielectric continuum models.

The first application of dielectric continuum modelsin the context of electronic g tensors has appearedonly recently.741 In this work, the performance ofPCM within the framework of uncoupled DFT (UDFT)calculations is investigated for semiquinone radicalanions in various solvents. Because the calculationsinvolved a generalized gradient approximation func-tional and thus a UDFT approach, as well as acommon gauge origin, no coupling terms due to thecontinuum have been computed during the perturba-tion treatment. In other words, the g tensor wascomputed in approximate way using a formula refer-ring to isolated systems but with the Kohn-Shamwave function obtained in self-consistent reactionfield (SCRF) calculations for a given solvent.

In contrast, a rigorous treatment of solvent effectsshould account for explicit solvent terms in theHessian of the solvated molecule free energy func-tional and, thus, in the linear response equationsdetermining the perturbed wave function exactly asdone for nuclear shielding and/or spin-spin couplingconstant. This has been achieved by Rinkevicius etal.742 still within the PCM solvation approach.

Hyperfine splittings of free radicals are determinedby the electron spin density at, or near, the positionof any magnetic nucleus, and they are thus affectedby the presence of the solvent. This has stimulatedsome attempts to apply a continuum solvation modelto the calculation of the corresponding hyperfinetensor A, which in liquid solutions reduces to theisotropic component only.702,723-728,731 For each nucleusN of the solute molecule located at rN , the isotropiccomponent of the hyperfine interaction tensor, a(N),

d2GdMa

K dMbL

) ∂2G

∂MaK∂Mb

L+ ∂

2G∂Ma

K∂λS

∂λS

∂MbL

+ ∂2G

∂MaK∂λT

∂λT

∂MbL

(188)

∂2G

∂λS∂λS

∂λS

∂MaL

) - ∂2G

∂MaL∂λS

(189)

∂2G

∂λT∂λT

∂λT

∂MaL

) - ∂2G

∂MaL∂λT

(190)

gab ) 1µB

∂2E

∂Ba∂Sb(191)


is related to the local spin density through

where âe, âN, and γN are the electronic and nuclearmagnetons and the nuclear magnetogiric ratio, re-spectively, the indices µ and ν run over the basisfunctions, P is the difference between the densitymatrices of spin R and spin â electrons, and δ(r -rN) is the Dirac delta. From the definition (eq 192),it appears evident that the evaluation of a(N) doesnot require any response equation to be solved butonly the knowledge of the density matrices fordifferent spins. These are easily obtained from un-restricted Hartree-Fock (UHF) or Kohn-Sham (UKS)calculations.

6.4. Properties of Chiral Systems

In this section we present extensions of continuummodels to describe chiral properties of solvatedsystems using QM approaches. These extensions arequite recent, whereas the problem of theoreticallypredicting chiral properties in solution is an old issue.Many empirical rules have been formulated to ac-count for solvent effects on optical rotation (OR) andelectronic circular dichroism (ECD) spectra, andclassical continuum models have been used as well.Historically, solvent effects on properties related tooptical activity have been described by includingLorentz local field factors in the expression for thespecific property (see section 6.1.2). These local fieldfactors are of the type

where n(ω) is the refractive index of the solvent atthe frequency ω of the external perturbation. How-ever, the inclusion of such factors accounts for onlythe difference in the external and microscopic (thefield acting on the molecule in solution) fields. It doesnot account for the effect of changes in the electronicwave function and in the property due to the presenceof the solvent. In the following sections, we shallpresent models that overcome this limitation andintroduce direct solvent effects on the property at aquantum mechanical level.

6.4.1. Electronic Circular Dichroism (ECD)

Circular dichroism (CD) is the most widely usedform of chiroptical spectroscopy. The phenomenon ofCD consists of the differential absorption of left- andright-circularly polarized light by a chiral molecule.CD due to electronic transitions is generally referredto as CD (also as ECD) and that due to vibrationaltransitions as VCD (see section 6.4.3).

The integrated intensity of a CD band gives ameasure of the strength of CD, called the rotationalstrength. Rotational strength is defined theoretically,following Rosenfeld’s treatment,776 as the imaginarypart of the scalar product of the electric and magneticdipole transition moments of an electronic transition;

for a given transition 0 f a it thus becomes777

where µ is the electric dipole transition momentoperator, a measure of the linear displacement ofcharge upon excitation, and m is the magnetic dipoletransition moment operator, a measure of the circulardisplacement of electron density upon excitation. Theproduct of µ and m results in a helical displacementof charge, which interacts differently with left- andright-circularly polarized light.

As previously noted for other spectroscopic proper-ties, the presence of the solvent leads to differentimportant effects on CD. First, solvent induces a localmodification on the applied radiation, exactly as forIR and Raman spectra (see section 6.1.2). In addition,the solvent reaction field changes the electronic wavefunction and thereby the rotatory strengths. Finally,solvent may induce geometrical distortions in themolecules and indirectly change the rotatory strengths.In any case, even for rigid molecules the solventdependence of CD spectra is found to vary signifi-cantly in different solvents, and a method thatdescribes the direct solvent effects on the rotatorystrength is therefore crucial, in particular whentheory is invoked to assign absolute configurationsof chiral molecules.

Despite this important role of solvation, only veryrecently have QM continuum methods been appliedto the evaluation of solvents effects on ECD (althougha first tentative attempt was presented by Ruiz-Lopezand Rinaldi778 in the 1980s using a semiempiricalapproach). This delay was also due to the fact thatthe calculations of CD require large basis sets andelectronic structure methods which account for elec-tron correlation, especially dynamical correlation,779-781

and thus only when continuum models have beenextended to accurate QM approaches have applica-tions to CD become possible. We can expect thatmany studies will appear soon as QM methods forECD based on DFT are now becoming available incomputational packages such as Gaussian,103 Dal-ton,135 TURBOMOLE,135 or ADF.64,65

This first example in the literature of the applica-tion of QM continuum models to the calculation ofrotational strengths is represented by the MPEmethod of Mikkelsen and collaborators (see section2.3.2). The generalization of this method (both in theequilibrium and in the nonequilibrium version) to CDhas been done using a coupled-cluster approach.782,783

In this study, the solvent effects were found to havea significant influence on the transition properties,leading, in some cases, to a sign change of therotatory strengths. In addition, the inadequacy of theLorentz local field factors (see eq 193) to properlydescribe the solvent influence on this specific prop-erty is also shown. A subsequent example784 of theuse of continuum models to simulate CD spectra insolution has been presented by Pecul et al. using thePCM solvation model (see section 2.3.1).

6.4.2. Optical Rotation (OR)The optical rotation (OR) and its variation with

wavelength (optical rotatory dispersion, ORD) in

a(N) )8π

3âeâNγN ∑

µνPµν

R-â⟨æµ|δ(r - rN)|æν⟩ (192)

f ) (n2(ω) + 2)/3 (193)

R0a ) Im[⟨0|µ|a⟩‚⟨a|m|0⟩] (194)


organic compounds in solutions were first observedby Biot, at the beginning of the 19th century. Biotalso introduced the modern definition of “specificrotatory power” or “specific rotation,” [R], of a liquidas R/Fl, where R is the measured optical rotation indegrees, l is the optical path length (or sample cellthickness) in decimeters, and F is the density ofliquid. For an isotropic dilute solution of a chiralmolecule, the specific rotation [R] (usually measuredat the sodium D line of 589.3 nm) is generally writtenas785-787

where k is a numerical constant, M is the molar massin g mol-1, and ν is in cm-1. In eq 195, f(ν) indicatesthe local field factor of eq 193 calculated at the properfrequency ν, and â(ν) is equal to one-third the traceof the frequency-dependent electric dipole-magneticdipole polarizability tensor having components de-fined as

In eq 196, 0 and k label ground and excited electronicstates and µel and µmag are the electronic electric andmagnetic dipole moment operators. This sum-over-states has been cast in the framework of time-dependent linear response theory for SCF (either HFor DFT), multiconfigurational SCF, and coupledcluster wave functions. For SCF wave functions, theexplicit evaluation of the sum-over-states can beavoided by rewriting âRâ in terms of electric andmagnetic field derivatives of the ground state elec-tronic wave function, namely788-792

where ∂Ψ0/∂E and ∂Ψ0/∂H are frequency-dependentderivatives of the ground state electronic wave func-tion with respect to the electric and magnetic fields,E and H, respectively.

In eq 195, the Lorentz local field factor f(ν) of eq193 has been introduced to account for the solvent-induced changes in the microscopic (or local) electricfield acting on the chiral molecule with respect to themacroscopic electric field of the light wave; â(ν) alsodepends on the solvent, exactly in the same way wehave outlined in the previous section for the CDrotatory strength. Once more, to properly account forsolvent effects on â(ν), a QM description is required.

The first QM study of the solvent effects on â(ν)has been presented by Mennucci et al.600,793 withinthe framework of the IEF version of the PCMsolvation model (see section 2.3.1.3). In this study, afrequency-dependent DFT/GIAO response approachhas been used to compute the electric dipole-magnetic dipole polarizability: the mixed nature ofâ(ν), however, requires two response procedurescontaining an electric and a magnetic perturbation,respectively. For the electric perturbation, the inclu-sion of the additional solvent terms in the coupled

perturbed equations is exactly equivalent to whatwas seen in the case of polarizabilities (see section6.2.2). Due to the imaginary nature of the magneticperturbation, solvent-induced terms do not appearexplicitly but only contribute to the first-order expan-sion term of the Fock operator exactly as for thenuclear shielding (see section 6.3.1). In this study,solvent contributions to the electric perturbation wereobtained in terms of solvent charges calculated usingthe value of the dielectric constant at the frequencyof the external field (nonequilibrium solvation). Inthe general case, this is the sodium D line frequency,and thus the value for ε(ν) coincides with the so-calledoptical dielectric constant, εopt, defined as the squareof the refractive index.

The model was first applied to some conformation-ally rigid chiral organic molecules for a set of polarand apolar solvents. The predicted variation inspecific rotation for polar solvents was found to bein excellent agreement with experiment for all of themolecules. For apolar solvents the agreement wasmuch poorer and thus, because only electrostaticsolute-solvent interactions were included in themodel, it was concluded that nonelectrostatic effectsmay have some importance in determining specificrotation, at least for aromatic and chlorinated sol-vents such as carbon tetrachloride, benzene, andchloroform. As already noted for the ECD, the varia-tions predicted by the classical Lorentz factor (eq 193)were found to be inconsistent, both qualitatively andquantitatively, with those calculated in terms of theQM IEF model. In subsequent papers, the samemodel was successfully applied to the study of opticalrotation of more complex and flexible molecules suchas glucose in aqueous solution794 and paraconic acidin methanol;795 in both cases it becomes fundamentalto account for the solvent effects both in the propertyand in the relative population of the various conform-ers that contribute to the final OR value.

An alternative methodology to account for solventeffects on the OR has been presented by Mikkelsenand co-workers796 using a coupled-cluster approachcombined with a nonequilibrium version of the single-center multipole based on the spherical cavity dielec-tric continuum model described in section 2.3.2.

6.4.3. VCD and VROAChiral molecules exhibit optical activities also in

their vibrational spectra. Specifically, vibrations as-sociated with the chiral molecular structure producespectral intensities that are dependent on the circularpolarization direction of the excitation light and thedetection optics. This optical activity can be observedin infrared spectra, as vibrational circular dichroism(VCD), and in Raman spectra, where it is referredto as Raman optical activity (ROA).

Both VCD and ROA were discovered in the earlyto mid-1970s (see refs 797, 798 and 799, 800, respec-tively), and since then, they have evolved into maturefields of research and application. Both can be usedto determine the absolute configuration and solutionconformation of chiral molecules. Each has advan-tages relative to the other that parallel the relativeadvantages of ordinary infrared absorption (IR) andRaman scattering.

[R] ) kf(ν)â(ν)ν2M-1 (195)

âab(ν) )c

3πhIm[∑k*0

⟨0|µael|k⟩⟨k|µb

mag|0⟩

νk02 - ν2 ] (196)

âab ) hc3π

Im[⟨∂Ψ0

∂Ea|∂Ψ0

∂Hb⟩] (197)


Instrumentation for measuring VCD spectra andsoftware for calculating VCD spectra from QM firstprinciples have been commercially available since1997. In contrast, commercial instrumentation formeasuring ROA spectra became available only someyears later, and user-friendly software for calculatingROA spectra is not yet available, even if it willbecome so very soon. As a result, there are manymore publications describing the use of VCD for thedetermination of absolute configuration than thereare for ROA and, therefore, this section will mainlyfocus on VCD.

Application of VCD to the assignment of absoluteconfiguration and identification of dominant solutionconformations entail comparison of observed spectrawith spectra calculated for a specific configurationand conformation of the molecule or a suitablefragment of the molecule. Thanks to the work ofStephens and co-workers,801-803 these calculations,which are typically obtained at the DFT level, arenowadays routinely performed for isolated moleculesand exhibit an accuracy similar to that of otherspectroscopic quantities (see, for example, ref 804).Less studied is the prediction of VCD spectra forsolvated systems: only a few papers are present inthe literature. These resort to the IEFPCM714,715,805

of section 2.3.1.3 or to the simple Onsager continuummodel (eventually with explicit solvent molecules)806,807

to treat the solute-solvent electrostatic interaction.To better understand how solvation models have

been applied to the study of solvent effects on VCDwe briefly present the general aspects of their QMformulation.

VCD intensity is proportional to the rotational orrotatory strength, the scalar product of the electricdipole and magnetic dipole transition moments. Inthe harmonic approximation, the rotational strengthR is proportional to the scalar product between thederivative of the electric dipole moment of themolecule with respect to normal mode displacement(Qa) and the derivative of the magnetic dipole mo-ment of the molecule with respect to the nuclearvelocities of the normal mode, expressed in terms ofthe conjugate momentum (Pa), namely

Quantum chemistry programs calculate these in-tensities by taking derivatives of the energy of themolecule with respect to electric field or magneticfield and vibrational normal mode displacement ormomentum. The calculations first require optimiza-tion of the geometry and determination of the normalmodes of vibration. Coupled perturbed Hartree-Fockor Kohn-Sham methods are used to obtain theenergy derivatives. In the case of the magnetic dipolederivatives, the method that is incorporated intocommercial software such as Gaussian17 utilizes themagnetic field perturbation method of Stephens etal.,801 with GIAO orbitals to eliminate the origindependence in the magnetic field derivatives.808

VCD spectra are affected by the solvent in differentways.

First, solvent can affect geometries and thus vi-brational frequencies: for this effect the same com-ments reported in section 6.1.2 for IR spectra arevalid here. Then, solvent can modify VCD intensities.The calculation of VCD intensities in the presenceof a solvent still relies on the same equationsformulated for isolated molecules, but with somerefinements. As already remarked with regard toinfrared intensities for molecules in solution (seesection 6.1.2), the electric dipole in eq 198 has to bereplaced by the sum of the dipole moment of themolecule and the dipole moment arising from thepolarization induced by the molecule on the solvent.The latter takes into account effects due to the fieldgenerated from the solvent response to the probingfield once the cavity has been created (the so-called“cavity field”). In principle, the magnetic dipoleshould be similarly reformulated. However, by as-suming the response of the solvent to magneticperturbations to be described only in terms of itsmagnetic permittivity (which is usually close tounity), it is reasonable to consider that the magneticanalogue of the electric “cavity field” gives minorcontributions to the rotational strength. A formula-tion of this solvent-specific term has been given inref 805 within the IEF version of the PCM model:in this case an additional set of charges spread onthe solute cavity surface is introduced in addition tothe PCM apparent charges representing the reactionfield. These additional charges are those produced bythe (Maxwell) electric field of the radiation in themedium, exactly as in the PCM treatment of the IRand Raman intensities shown in section 6.1.2.

A further important solvent effect on VD spectrais implicit, that is, through the possible changes inthe relative energies of different conformers of thesolute molecule when passing from gas phase tosolution. In fact, if the solute can exist in differentconformations that are differently stabilized by thesolvent, the VCD spectra must be obtained by com-bining population-weighted spectra of all conformerswhere the weights are evaluated in solution.805

7. Continuum and Discrete ModelsIn section 1 we have anticipated one of the most

important extensions that continuum solvation mod-els have presented in recent years. This extension,which we denoted layering, has been propelled by twoopposing reasons: namely, the necessity to reducethe complexity of calculations using discrete ap-proaches on systems composed by a large number ofmolecules and the increasing accuracy required in thestudy of solvated systems (especially their molecularproperties).

The layering can be considered a generalization offocusing. The material component of the modelspartitioned into several parts, or layers, and eachlayer is defined at a given level of accuracy in thedescription of the material system and with anappropriate reduction of the degrees of freedom.There are many possibilities for layering, which canbe simply abbreviated as, for example, QM/QM/Contor QM/MM/Cont for a couple of three-layer modelsin which the inner layer is treated at a given QM

Ra ) p2( ∂µb

∂Qa)‚(∂mb

∂Pa) (198)


level, the second to a lower QM or at a molecularmechanics level, respectively, and the third using acontinuum approach.

In this section we present some examples of layer-ing.

7.1. Continuum Methods within MD and MCSimulations

Continuum solvation models have been largelyexploited in classical simulations, such as moleculardynamics (MD) or Monte Carlo (MC), to reducecomputational efforts and to simplify two of the maindifficulties of this kind of approach: the correctrepresentation of the damping (or screening) of theelectrostatic interactions between charges of thesolute due to polarization of the medium, and thetreatment of the solvent boundary conditions.

The most straightforward way to include effects ofscreening is to use a Coulomb law with a distance-dependent dielectric constant (see section 4.5). For agiven interatomic distance, this model does notdistinguish between atom pairs at the surface of themolecular system (generally a protein), where thesolvent screening effect is large, and atom pairs inthe core of the system, where the screening effect issmall. A more physically correct model represents thesolvent as a polarizable dielectric continuum sur-rounding the solute van der Waals surface, and theelectrostatic potential is obtained by solving thePoisson-Boltzmann (PB) equation (see sections 2.3.4and 2.3.5 and the references cited therein). However,for solving problems such as long-term protein dy-namics or even protein folding, if PB electrostaticsis used, even the fastest methods to solve the PBequation are too slow. To overcome this problem, ageneralized Born approach such as those presentedin section 2.3.3 can be used. For example, Schaeferand Karplus151 developed the so-called analyticalcontinuum electrostatics (ACE) potential, which ap-proximates the potential given by solving the Poissonequation but needs much less computational effort.By adding a nonpolar free energy of solvation term,ACE has been extended to the analytical continuumsolvent (ACS).809

Another continuum solvation method for use withMC and MD simulations is that known as “screenedCoulomb potentials-based implicit solvent model” orSCP-ISM.470,810,811 This model is based on the theoryof polar liquids, originated in the works by Debye,Lorentz, and Sack and incorporates Onsager’s reac-tion field effects. The SCP-ISM is derived usingscreening functions of sigmoidal behavior. The modelyields an expression for the electric potential in thesystem (i.e., protein plus solvent treated as a com-bined, complex solvent medium with complex dielec-tric characteristics), an expression for the electro-static energy of the solvated protein (that containsthe free energy of the solvent), and an expression forthe solvation free energy of the protein.

With regard to the boundary, several differenttreatments exist. The use of one instead of anotherdepends strongly on the type of problem the simula-tion is to address. Periodic boundary conditions (PBC)enable a simulation to be performed using a relatively

small number of particles in such a way that theparticles experience forces as though they were in abulk solution. In this scheme, a microscopic compu-tational box is periodically replicated to form aninfinite periodic system, which serves as a model forthe macroscopic system. For all interactions with arange of action shorter than the size of the compu-tational box (e.g., covalent and van der Waals inter-actions), artificial periodicity is not likely to be aproblem. However, electrostatic interactions are ef-fective at distances that typically go beyond the sizeof the computational box. There are two alternativepractical approaches to handle electrostatic interac-tions in molecular simulations, which represent ap-proximations of a different kind for the descriptionof dilute solutions. The first approximation is toenforce the exact periodicity of the system in thecontext of Coulombic interactions, that is, to includeall Coulombic interactions between any atom in thereference computational box and all atoms in theinfinite periodic system. Methods relying on thisapproximation are called lattice-sum methods andinclude the Ewald summation and computationallymore efficient grid-based methods.812,813 The secondapproximation is to partially break the periodicity bytruncating electrostatic interactions beyond a con-venient cutoff distance, that is, include only interac-tions between any atom in the reference computa-tional box and the closest (i.e., within the cutoffsphere) atoms in the infinite periodic system. Meth-ods relying on this approximation are called cutoff-based methods and include (i) straight truncation ofCoulombic interactions,814,815 (ii) smooth truncationof Coulombic interactions by using a switching orshifting function,815,816 and (iii) straight truncationof an effective interaction function including a Cou-lombic term and a reaction-field correction.816-818

The reaction-field approach has been used both forfixed boundary systems, as a correction for theabsence of solvent outside a nonperiodic system offinite size,193,819,820 and for multiple moving boundarysystems, as a correction to the truncation of theCoulomb interaction beyond a given cutoff dis-tance.821,822 In the fixed boundary case, one considersa region around the system (formed by a distributionof charge possibly inside a solvent of permittivity ε1)to behave as a dielectric continuum of permittivityε2. In the simple case of a spherical boundary, thecorrection to be applied will depend on the multipoleexpansion describing the charge distribution in thesystem (see section 2.2). Because molecular dynamicssimulation requires only knowledge of interactionforces and possibly energies, only the reaction electricfield and potential at any location r inside theboundary need to be considered. In the multiplemoving boundary case, each charge, or charge group,is surrounded by a region (where the Coulombinteraction is calculated explicitly) of permittivity ε1;in turn this region is embedded in a dielectriccontinuum of permittivity ε2, which is assumed tocompensate for the mean effect of the neglectedinteractions. For simplicity, the boundary to thecontinuum is approximated by a sphere around thecentral charge. Because each charge ‘‘sees” a different


part of space as a continuum, the electric reactionpotential and field caused by the polarization of thecontinuum seen by a given charge i bear only aphysical relationship to this particular charge. Asshown by Hunenberger and van Gunsteren,818 thereaction-field boundary conditions are simpler toimplement and generally numerically faster than theEwald sum method. They do not impose artificialperiodicity on systems that are not inherently peri-odic and constitute a possible alternative to latticesum methods, provided that their accuracy is com-parable.

Despite the substantial use of concepts and toolsof continuum models in MD/MC simulations we havementioned so far, only recently have “real” continuumMD (or MC) approaches been formulated. Here weshall focus on a few.

In the method developed by Borgis and co-work-ers,823 the interaction between the solvent and thesolute degrees of freedom is described by means of apolarization density free energy functional which isminimum at electrostatic equilibrium. The approachis based on the electrostatic free energy functionalfor a charge distribution, the solute, in a polarizablemedium, namely

where NR is the number of smooth spherical chargesdefining the solute charge density F(r) and generatingthe in vacuo potential φ0 and PB(rb) is the polarizationdensity field. The minimization of the functional ineq 199 with respect to PB(rb) for a fixed chargeddistribution leads to the constitutive equation ofelectrostatics

with EB(rb) the Maxwell electrostatic field defined as

At its minimum the excess functional Gex takes thevalue

which is the usual expression for the electrostaticenergy of a dielectric in an external electric field

It is easy to show that eqs 200 and 201 areequivalent to the Poisson equation and, thus, thatminimizing Gel with respect to PB(rb) provides analternative way of solving the Poisson differentialequation with respect to the techniques shown insection 2. The advantage of the functional formula-tion is that an analogy can be drawn with respect toDFT electronic structure calculations, and it can be

used in a dynamical context using “on-the-fly” Car-Parrinello-like minimization techniques (see section7.2).

To solve numerically the electrostatic problem fora given charge distribution and a macroscopic dielec-tric, one has to expand the polarization field coordi-nates in some basis set. This will transform thefunctional in eq 199 into an ordinary function of theexpansion coefficients which, in turn, become thevariational parameters of a standard multidimen-sional optimization problem. This transformation isaccomplished by using a Fourier pseudospectral (PS)approximation for the field variables, PB(rb). In thiscontext, it is crucial that the position-dependentdielectric constant, ε(rb), be continuous. To define sucha continuous function, a model system is introducedwith an internal permittivity, εi, in the proximity ofeach atom surrounded by a dielectric continuum ofpermittivity εs. Then, a computationally suitable formfor the position dependent dielectric permittivity canbe written as

where H(r) is a proper volume exclusion functionparametrized for each atomic radius.

The coupling of this electrostatic framework to themolecular dynamics method is achieved by treatingthe variational parameters PB as virtual dynamicalvariables. A virtual mass of appropriate units isassigned to the PBs, and a fictitious polarizationkinetic energy is included in the system Hamiltonian.The virtual dynamic variables PBs are treated in thesame spirit as the electronic coordinates in the Car-Parrinello method. Unlike quantum systems in whichelectrons are always much faster than nuclei, thetime scales of solvent and solute overlap in reality,and this allows us to choose a mass yielding a virtualdynamics much slower than that of electrons. Duringthe simulation, the temperature of PB must also below enough for the particles to move adiabatically ona minimum of the surface of the polarization func-tional Gel. If this condition is satisfied, at each timestep of a MD run the value of the computed func-tional Gel will fall very close to that obtained byreoptimization of the PB for the instantaneous atomiccoordinates.

The method has been programmed as an indepen-dent set of Fortran 90 routines and implemented inthe MD program ORAC.824

The second example of continuum MD methods weanalyze here is that proposed by Basilevsky and co-workers.825-827 The main new feature is that themethod combines the dielectric continuum approxi-mation for treating fast electronic (inertialess) po-larization effects arising from solvent electronicpolarization modes and a MD simulation for the slow(inertial) polarization component, including orienta-tional and translational solvent modes. This approachexploits the same idea of the previously cited “fre-quency-resolved cavity model” (FRCM)412 in whichthe inertialess response of the medium was spatiallydistinguished from the inertial one (see section 5.3).According to this model, the solute is surrounded by

Gel[PB(rb),riNR] ) 1

2 ∫ F(rb)φ0 drb + Gex[PB(rb),riNR]

(199)

PB(rb) )ε(rb) - 1

4πEB(rb) (200)

EB(rb) ) -∇Bφ0(rb) + ∇B ∫ ∇′‚PB(rb)|rb - rb′| drb′ (201)

Gex[PB(rb)min,riNR] ) - 1

2 ∫ PB(rb)EB0(rb) drb (202)

EB0(rb) ) -∇φ0(rb)

1ε(rb)

) 1εi

- (1εi

- 1εs

)H(rb) (203)


two surfaces, each of which is constructed as acollection of overlapping atomic spheres similar to thePCM method. For the first cavity, contained withinthe inner surface and representing the space occupiedby the solute, the dielectric constant is taken as ε )1. Between the two surfaces the medium is repre-sented by the inertialess dielectric constant ε ) ε∞.This layer corresponds roughly to the first solvationshell. Outside the outer cavity the static dielectricconstant is applied (ε ) ε0). The new aspect intro-duced in the method is that explicit solvent particleswith mean dipoles and associated mean point chargesassigned to the selected sites of the solvent moleculesare immersed in the medium with ε ) ε∞. The soluteparticle with charges qi (taken as nonpolarizablevacuum charges) is placed at the center of the MDcell, and the electrostatic forces between solute andsolvent charges as well as between solvent chargesare scaled by the factor 1/ε∞. In combination withLennard-Jones forces they form the effective forcefield for an MD simulation producing the equilibriumensemble of solvent particles. We note that thissimplistic screening of all electrostatic interactionsby 1/ε∞ is not fully consistent with the dielectric modelin which the dielectric constant in the solute regionis not ε∞, but unity; however, the authors found thatthe consequences of this discrepancy are negligible.This approach yields an ensemble of equilibriumsolvent configurations adjusted to the electric fieldcreated by a charged or strongly polar solute. Theelectrostatic solvent response field is found as thesolution of the Poisson equation including both soluteand explicit solvent charges, with an accurate ac-count of electrostatic boundary conditions at thesurfaces separating spatial regions with differentdielectric permittivities. This leads us to define twosurface apparent charge densities spreading on thesurfaces of the inner and outer layers.

This combined molecular-continuum approachcomprises a contracted analogue of a conventionalpolarizable MD simulation. Despite obvious limita-tions, the present approach seems to provide severalcomputational as well as conceptual advantages. Anessentially related feature is that the MD-FRCMmethod is formulated in terms of “dressed” chargesof solvent particles, whereas vacuum charges areused in conventional polarizable theories. This “dress-ing” makes an electronic contribution to the inertialresponse of the solvent particles. The mean chargescomprise input parameters of the MD-FRCM scheme,and their explicit values are available either byrescaling the effective charges of some nonpolarizablesolvent model or directly from a conventional polariz-able computation of bulk solvent. Both equilibriumand nonequilibrium solvation effects can be studiedby means of this model, and their inertial andinertialess contributions are naturally separated. Themethodology has been coupled with the GROMOS96MD program,828 and it has been applied to computecharge transfer reorganization energies in a modeltwo-site dipolar system in the water solvent.

We conclude this section by citing a further ex-ample of the coupling between ASC approaches andstatistical methods. This algorithm has been devel-

oped by Colominas, Luque, and Orozco,52,829 by com-bining Monte Carlo (MC) calculation with a quasi-classical version of the DPCM (here called MST, seesection 2.3.1.1). In the MC-MST approach, configura-tions are accepted or rejected following the Metropolisrules applied to an energy functional including thesolvent electrostatic, van der Waals, and cavitationterms described in sections 2 and 3, respectively. Inparticular, the electrostatic term is obtained byassuming that solute moves in a fully equilibratedsolvent represented by apparent surface chargesdetermined by solute atomic charges following froma fitting of the electrostatic potential of a preliminaryQM MST calculation. The method is especially usefulin conformational studies of large molecules.830

7.2. Continuum Methods within ab InitioMolecular Dynamics

The term ab initio molecular dynamics (AIMD) isused to refer to a class of methods831-834 that com-bines finite temperature dynamics with forces ob-tained from electronic structure calculations per-formed “on the fly” as the MD simulation. Currently,the most commonly used electronic structure theoryin AIMD is the Kohn-Sham formulation of densityfunctional theory (DFT), in which the electronicorbitals are expanded in a plane-wave basis set. Thisprotocol provides a reasonably accurate descriptionof the electronic structure for many types of chemicalenvironment while maintaining an acceptable com-putational overhead and constitutes the originalCar-Parrinello (CP) formulation of the method.831 Inthis approach, the parameters in the electronic wavefunction are treated as dynamical variables (electrondynamics) and the electronic structure problem issolved by the application of the steepest descentmethod to the classical Newtonian equations ofmotions. The evolution of the electronic wave functionand the forces acting on the atoms can be computedsimultaneously, provided that the electronic state ofthe system is conserved (i.e., remains on the Born-Oppenheimer surface). This is achieved by the ex-ploitation of the quantum mechanical adiabatic timescale separation of fast electronic and slow nuclearmotion by transforming the electronic problem intoclassical mechanical adiabatic energy scale separa-tion in the framework of dynamical systems theory.Hence, both the electronic structure problem and thedynamics of the atoms are solved concurrently by aset of Newton’s equations. In contrast to the tradi-tional approach, this theoretical breakthrough allowscalculations of the fully dynamic time evolution of astructure (molecular dynamics) without resorting toa predefined potential energy surface.

The AIMD methods have been used to study a widevariety of chemically interesting and important prob-lems in areas such as liquid structure, acid-basechemistry, industrial and biological catalysis, andmaterials (see, for example, refs 835 and 836). Forcondensed phases, AIMD methods have been ex-tended to QM/MM836,837 and combined (ONIOM-like)approaches838,839 as well to continuum solvationmodels.


In particular, three different models to include acontinuum model within the framework of AIMD areworth reviewing here.

An approach to introduce a continuum descriptionwithin the framework of CP ab initio moleculardynamics has been presented by Ziegler and co-workers100 using the conductor-like screening model(COSMO) described in section 2.3.1.2. In this imple-mentation, the COSMO surface charges are treatedas fictitious dynamic variables (with associated ficti-tious masses mq) within the extended Lagrangianapproach, solving the electrostatic problem determin-ing the charges on the fly as the system evolves intime. The Lagrangian is also augmented by thefictitious kinetic energy of these charges and theadditional potential energy terms due to solvation.The value of the fictitious masses mq is chosen smallenough that the charges are adiabatically decoupledfrom the other degrees of freedom.

From the Lagrangian, it is in principle straight-forward to obtain the corresponding equations ofmotion that propagate the surface charges togetherwith the nuclei and the wave functions. However, inpractice this approach is nontrivial because of theimplicit dependence of the solution energy terms onthe nuclear positions. Both the electrostatic Gdiel andthe nonpolar Gnp parts depend on the surface, whichchanges in area and shape as the nuclei move, andhence depend on the nuclear positions. This gives riseto additional forces acting on the nuclei, which needto be evaluated as derivatives of Gdiel and Gnp,respectively, with respect to the coordinates. Werecall that in this approach Gnp, which collects allnonelectrostatic contributions, notably the formationof the cavity in the solvent, in which the solute isplaced, and the dispersion and exchange repulsioninteractions between solute and solvent, is approxi-mated as a linear function of the surface area of thecavity.

To perform energy-conserving dynamics, consis-tent, accurate forces are a fundamental prerequisite.To obtain such accurate forces, a steep but continuoussurface function that effectively switches the surfacecharges off when they are not exposed on the molec-ular surface is used (see section 2.2. for more details).

The second example of coupling between con-tinuum models and AIMD techniques we report inthis section is that proposed by Fattebert and Gy-gi.840,841 To carry out microcanonical total energy-conserving molecular dynamics simulation with acontinuum solvation model, a density-based molec-ular cavity definition and a smooth solute-solventtransition are used. In this model the Poisson equa-tion is solved directly in the form

where the dielectric medium is now represented bya function ε of the electronic density, F, of the solute,and thus its effects can be easily integrated into adensity functional. Using eq 204, one finds that thevariation of electrostatic interaction energy withrespect to F leads to the usual Hartree potential plus

an additional term

which is included in the Kohn-Sham equations tobe solved.

Numerically, a finite difference (FD) approach (asthat described in section 2.3.5) is exploited, and thePoisson equation is discretized by finite differenceson a regular grid, which is the same used to computethe electronic density in terms of plane waves. Forthe dielectric function, a local model depending onthree parameters is used, namely

where ε∞ is the asymptotic value of ε for F f 0, F0denotes the critical density in the middle of theinterface solute-solvent, and â is related to the widthof the interface. The parameters F0 and â have beendetermined so that the continuum solvent effect onthe total energy of a water molecule is the same asthe average binding energy of an explicit solventenvironment.

The introduction of the smooth dielectric functionallows us to run constant-energy molecular dynamicsthat include the effects of a continuum solvent, andthis is achieved by implementing the model in a CPmolecular dynamics code.

7.3. Mixed Continuum/Discrete DescriptionsThe large variety of techniques to represent the

solvent by a macroscopic (polarizable) continuumsurrounding the solute as well as those involving theexplicit incorporation of many solvent molecules eachpresent their particular weaknesses. For example,not all of the electronic aspects of hydrogen bondingbetween solvent and solute are correctly describedby continuum models, whereas explicit descriptions,which are always size-limited, cannot fully take intoaccount long-range (or bulk) effects. It follows thatmany of the approaches complement each other, andit is therefore worthwhile to investigate the possibil-ity of combining different techniques in the samecalculation. This has indeed been the topic of anumber of recent studies in which the analysis isfocused on the combination of complementary rep-resentations of the solvent, namely, explicitly by asmall number of solvent molecules (microsolvation)and implicitly by a continuum.

This kind of generalized continuum model in whichthe solute includes a few explicit solvent molecules(namely, those belonging to the first solvation shell/s) will be here called “solvated supermolecules” tomaintain a correspondence with the standard super-molecule approach in which gas phase clusters ofinteracting molecules are studied to account for theintermolecular interactions. It is important to specifythat in this approach we include all methods in whichboth parts of the supermolecule (solute and the nsolvent molecules) are described at the same level ofquantum mechanical theory: possible hybrid or

-∇B‚(ε[F]∇BV) ) 4πF (204)

Vε(rb) ) - 18π

|∇BV(rb)|2 δε

δF(rb) (205)

ε(F(rb)) ) 1 +ε∞ - 1

2 (1 +1 - (F(rb)/F0)

2â

1 + (F(rb)/F0)2â) (206)


combined approaches mixing different QM or QM andMM descriptions will be reviewed in the next sec-tions.

7.3.1. Solvated Supermolecule

The coupling of the supermolecule approach witha continuum is an attractive technique that hasproved to be extremely useful in studying solventeffects on electronic and vibrational spectra andresponse molecular properties as well in studyingreaction mechanisms and energetics in solution; insection 6 we reported several references to which weadd here a few more.842-848

One of the main reasons to explicitly treat somesolvent molecules is to get an accurate descriptionof strong specific interactions (e.g., hydrogen bond-ing); however, such an approach is not straightfor-ward, and its correct realization requires a detailedanalysis on the nature and the strength of suchinteractions.

In the case of very strong solute-solvent interac-tions such as solutes with strong H-bond acceptorcenters in water or in alcohols, a sufficient descriptionis often obtained by including all of the solventmolecules necessary to saturate such strong acceptorcenters (in general, a few solvent molecules areenough) in the QM optimization procedure. Theresulting structure is then used to compute thecorresponding vibrational spectrum or the propertiesof interest or just to compute the thermal free energynecessary to get the energetics of a reaction. In all ofthe steps of this strategy an external continuum isused to account for long-range (or bulk) effects.

When we are in the presence of weaker interac-tions, however, a representation of the supermoleculein terms of a single rigid structure obtained as theminimum of the potential energy surface of thecluster cannot be adopted. The real situation is infact dynamic and a variety of different representativestructures can and do occur. A possible way to getsuch a picture is to consider structures derived fromeither classical or ab initio MD shots taken atdifferent simulation times: for each of these selectedstructures the number of solvent molecules to be usedin the supermolecule calculation is determined by athreshold imposed in the distance between the soluteH-bond acceptor/donor center and the correspondingdonor/acceptor center in the solvent molecules. Eachof the resulting clusters (generally involving morethan the solvent molecules really needed to saturateH-bonding centers) is then embedded in the externalcontinuum and such a solvated supermolecule isstudied at the proper QM level. In the most recentexamples, the PCM method (either in the IEF orCPCM version, see section 2.3.1) is exploited toinclude bulk effects,716,720-722,729,731,733 whereas olderstudies849-852 adopted an image reaction field plusexclusion model to represent the polarization of thedielectric material embedding the clusters.853

In this approach, no geometry optimizations of theclusters are required; this of course makes theprocedure easier from a computational point of viewbut not necessarily faster as an average among allof the selected structures has to be done to get a

reliable description. A disadvantage of this procedureis that it cannot be used to directly compute vibra-tional spectra; more complex analyses are necessaryas the structures do not correspond to real minima.

This approach combining (but not coupling) MDsimulations and averaged QM calculations on se-lected clusters is not limited to the examples citedabove, and they do not necessarily include an exter-nal continuum; indeed, it has been used in manydifferent studies as an alternative to continuummodels. The main difference is that without includingthe bulk effects through a continuum dielectric, verylarge clusters are required to get the complete pictureof solvation, and thus the QM level of the theory isusually limited to semiempirical approaches or low-level ab initio methods.854-862

7.3.2. QM/MM/Continuum: ASC VersionAnother example involving discrete and continuum

approaches describes the solute at QM level, the firstsolvation shells with MM solvents, and the rest witha dielectric continuum. In such a way, both the firstsolvation shell effects (such as short-range repulsionand dispersion interactions and solvent entropicterms related to hydrophobic interactions) and long-range electrostatic effects in the bulk can be takeninto account with modest computational cost. Thesmall number of explicit solvent molecules makes thecalculations of quantities that require configurationalsamplings much faster compared to a full-scalemicroscopic simulation.

In principle, all of the continuum models we havedescribed in section 2.3 can be coupled with a QM/MM system. Here, however, we shall limit theexposition to those belonging to the ASC class ofcontinuum models.

An example of this approach is the model developedby Cui,863 in which the solute and a number of solventmolecules (“solute-solvent cluster”) are describedwith QM/MM and the bulk solvent is treated withthe IEFPCM model (see section 2.3.1.3). In thismodel, both the MM atoms, represented by fixedpartial charges, and the QM atoms are contained ina generalized cavity embedded in the continuum.Following the same formalism used in section 2.4,the total electrostatic free energy of the “solvated”QM/MM system thus becomes

where h ) h + J + Y, G ) G + X, and J, Y, and Xare the PCM matrices described in section 2.4.3 (seeeqs 64-67). The terms JQM/MM, YQM/MM, UNN

QM/MM,UNN

MM/QM, and UNNMM represent the additional electro-

static interactions between the QM/MM part and theinduced surface charges. There are four types ofinteractions, namely, between QM electrons and MM-induced surface charges, between MM and QMelectron-induced surface charges, between QM nuclei

Gel ) tr P[h + hQM/MM + 12

(JQM/MM + YQM/MM)] +

12

tr PG + (VNN + EelMM + VNN

QM/MM + 12

UNNQM/MM +

12

UNNMM/MM + 1

2UNN

MM) (207)


and MM-induced surface charges, between MM andQM nuclei-induced surface charges, and between MMand MM-induced surface charges.

This QM/MM/PCM method has been implementedinto the simulation package CHARMM,864 which waspreviously interfaced with GAMESS for ab initio QM/MM calculations,865 and it can be used for single-pointsolvation free energy calculations, Monte Carlo simu-lations, and geometry optimization as well as MDcalculations for reactions in solution.

Another QM/MM/Cont approach is that of couplingcontinuum models with the QM/MM model developedby Gordon and co-workers, known as the effectivefragment potential (EFP) method.426,866,867 The basicidea behind this method is to replace the chemicallyinert part of a system by EFPs while performing aregular ab initio calculation on the chemically activepart. A simple example of an active region might bea solute molecule, with a surrounding spectatorregion of solvent molecules represented by fragments.The charge distribution of the fragments is repre-sented by an arbitrary number of charges, dipoles,quadrupoles, and octupoles, which interact with theab initio Hamiltonian as well as with multipoles onother fragments. An arbitrary number of dipolepolarizability tensors can be used to calculate theinduced dipole on a fragment due to the electric fieldof the ab initio system as well as all other fragments.These induced dipoles interact with the ab initiosystem as well as the other EFPs, in turn changingtheir electric fields. All induced dipoles are thereforeiterated to self-consistency. The combination of theEFP method with continuum solvation initially usedan Onsager-like model868 but subsequently has beenextended to the IEF-PCM model.869 Using the sameformalism presented in section 2.4.2, the electrostaticcomponent of the free energy of the solute (ab initio+ EFP) continuum system can be written

where for h, G(P), VNN, an UNN we have used thesame notation as that in eq 207. All of the otherterms are due to the coupling between the EFP andPCM methods. In particular, (Jstat

EFP)µν ) ∑qstatEFP Vµν

e

and (JpolEFP)µν ) ∑qpol

EFPVµνe , where Vµν

e are the electro-static potential integrals in the atomic orbital basisand qstat

EFP) and qpolEFP are the ASC due to the static

multipoles and the induced dipoles of the fragments,respectively. In the remaining terms, VEFP denotesthe electrostatic potential induced by the EFP frag-ments and the subscript indicates the source of suchpotential, namely, the static multipoles (stat) and theinduced dipoles (pol).

Subsequently, the EFP/PCM approach has beenreformulated870 using a simultaneous iterative solu-tion of the QM self-consistent field (SCF) and of the

electrostatic equations (see also section 2.4.3): in thisway, bulk solvation of large solutes can be efficientlymodeled.

7.3.3. ONIOM/Continuum

An alternative formulation of the coupling of QM,MM, and continuum approaches is given by energysubtraction methods. The latter are a very generalclass of methods in which calculations are done onvarious regions of the molecule with various levelsof theory, and the energies obtained at each level arefinally added and subtracted to give suitable correc-tions. The introduction of continuum solvation intothis kind of method has been realized by coupling theIEFPCM model we have described in section 2.3.1.3with the class of subtraction methods developed byMorokuma and co-workers871-875 and generally knownby the acronym ONIOM. This acronym actuallycovers different techniques, combining either two (ormore) different orbital-based techniques or orbital-based technique with an MM technique.

The concept of the ONIOM methods is extremelysimple. The target calculation is the high levelcalculation for a large real system, E(high, real),which is prohibitively expensive. Instead of doingsuch a calculation, an inexpensive low-level calcula-tion leading to the energy E(low, real) is performedtogether with two calculations (one at accurate highlevel and the other at the same low level) for asmaller part of the system, usually indicated as themodel system, leading to the energies E(high, model)and E(low, model), respectively. Starting from E(low,model), if one assumes the correction for the highlevel, E(high, model) - E(low, model), and the cor-rection for the real system, E(low, real) - E(low,model), to be additive, the energy of the real systemat the high level can be estimated from three inde-pendent calculations as

Clearly this approach is completely general, and itcan be straightforwardly extended to more than twolayers. In addition, it has the advantage of notrequiring a parametrized expression to describe theinteraction of various regions. Any systematic errorsin the way that the lower levels of theory describethe inner regions will be canceled out.

The combination of ONIOM with IEFPCM hasbeen achieved in four alternative ways.876 The result-ing computational schemes differ mainly with respectto the level of coupling between the solute chargedistribution and the continuum dielectric, which hasimportant consequences for the computational ef-ficiency. All of the schemes share the fact that onlyone cavity is defined, based on the geometry of thereal system, which is subsequently used for all threesubcalculations. Three of the four schemes can beplaced in a hierarchy. In the highest method in theclass (called ONIOM-PCM/A in the reference paper),all three subcalculations exploit the reaction fieldthat is obtained self-consistently by employing theONIOM integrated density. Obviously this is the

E ) E(low, real) + E(high, model) -E(low, model) (209)

G ) [tr Ph + 12PG(P) + VNN + 1

2UNN] +

tr P(JstatEFP + Jpol

EFP) + 12 ∑qstat

EFP VstatEFP +

12∑qpol

EFP VpolEFP + ∑qstat

EFP VpolEFP + ∑qNVstat

EFP +

∑qNVpolEFP (208)


“correct” method, consistent with the philosophy of“integration” of the ONIOM method. The secondmethod in the hierarchy (ONIOM-PCM/B) differsfrom the first one in the reaction field, which is nowobtained from the (nonintegrated) density for the realsystem at the low level. In the lowest ONIOM-PCMscheme in this class (ONIOM-PCM/C), the solvationeffect is included in only the low-level calculation onthe real system, whereas the model system calcula-tions are carried out in vacuo. The fourth method(ONIOM-PCM/X) is an approximation to the correctONIOM-PCM/A but it does not fit in the hierarchy.The reaction field is in fact determined independentlyfor each of the three subcalculations.

The ONIOM/PCM has been successfully applied tothe study of magnetic properties of solvated sys-tems,716,730 and it has been used to describe solvatedsupermolecules (see section 7.3.1), being the modelsystem represented by the solute.716

7.3.4. (Direct) Reaction Field Model

A further QM/MM approach is that developed byvan Duijnen and co-workers,876-878 known as directreaction field (DRF).

The DRF model is an embedding technique en-abling the computation of the interaction between aquantum mechanically described molecule and itsclassically described surroundings. The classical sur-roundings may be modeled in a number of ways: (i)by point charges to model the electrostatic field dueto the surroundings, (ii) by polarizabilities to modelthe (electronic) response of the surroundings, (iii) byan enveloping dielectric to model bulk response of thesurroundings, and (iv) by an enveloping ionic solu-tion, characterized by its Debye screening length. Thefour representations may be combined freely.

The QM system may be described by a HF, DFT,GVB, CI, or MCSCF wave function. The static mul-tipole moments of the MM part (charge, dipole,quadrupole, etc.) are represented by a point chargedistribution at selected points.879 The response mo-ments (dipole polarizability only at present) arerepresented by a local (anisotropic) polarizabilitytensor located at the selected point or by distributedinteracting atomic polarizabilities. Finally, the en-veloping dielectric is assumed to be isotropic andhomogeneous and the related Poisson-Boltzmannequation is solved numerically by defining a dis-cretized surface that forms the boundary between thesource and the dielectric. Solution of the equationprovides a response potential inside the boundary.In DRF the surface is adaptable to the shape of theinterior and local moments are used as the source,rather than the overall dipole moment.

These different partitions may be coupled self-consistently, enabling the computation of the energyof the entire system. To this end, potential and fieldoperators are defined between the systems. In theDRF approach the self-consistency is not achieved byresorting to an iterative scheme but by writing outthe coupling equations between the induced momentsand deriving a coupling (or relay) matrix that, wheninverted, solves the self-consistency problem formallyand for any source field. This has great advantages

in combining classical descriptions of matter with QMelectronic structure calculations.

The DRF model has been also extended to treatsolvent nonequilibrium response,880 by separatingorientational and electronic components in the con-tinuum dielectric. As described in section 5.2, thisfeature is important for spectroscopic applications,where the orientational response is not able to followthe electronic transition, so that the excited state isto be calculated in the orientational reaction field ofthe ground state, whereas the electronic response isallowed to adapt to the excited state. Recently, theDRF approach has been applied to the study ofoptical response properties (polarizabilities and hy-perpolarizabilities) of solvated molecules.881-883

The DFR model has been implemented in differentcomputational packages: the HONDO8.1,884 GAMESS(UK),885 ZINDO,886 and a local version of the ADF.64,65

7.3.5. Langevin Dipole

The Langevin dipole (LD) solvation model devel-oped by Warshel and co-workers887-889 is based on theevaluation of interactions between the electrostaticfield of the solute and point dipoles placed on a gridaround the solute. This grid of dipoles is thensurrounded by a dielectric continuum (the outersolvent region). The solute electrostatic field is gener-ated from the point charges placed at the atomicnuclei. The total electrostatic field at a given dipoleand the magnitude of this dipole are related via theLangevin function that exhibits saturation for largefields. The resulting electrostatic part of the solvationenergy (∆GES) is augmented by the solvation contri-bution from the outer continuum dielectric (∆Gbulk),van der Waals (∆GvdW), hydrophobic (∆Gphob), andsolute polarization (∆Grelax) terms to give the solva-tion free energy, ∆Gsolv. The contributions (∆Gbulk) tothe solvation free energy from the outer solventregion are evaluated by a continuum approximation.This is done using Born’s and Onsager’s formulas forcharges and dipoles. The hydrophobic free energy,∆Gphob, is assumed to be related to the magnitude ofthe nonpolar molecular surface, which is proportionalto the number of Langevin dipoles (grid points) thatlie within 1.5 Å of the vdW surface of the solute. Thevan der Waals energy, ∆GvdW, is calculated as thesum of r-9 repulsion and London dispersion terms.

Because ∆GES is evaluated from the gas phasecharges, some correction is required to account forthe increase in the solvation free energy due to thepolarization of the solute electron density as itinteracts with Langevin dipoles. This correction,which is denoted here ∆Grelax, is evaluated within theframework of the linear response approximation.Within this approximation, the energy invested inpolarizing the solute is -1/2 the energy gained by theinteraction between the polarizing potential and theinduced changes in solute atomic charges (∆qi). Insome versions the factor of 1/2 is replaced by anadjustable constant krelax.

In the most recent implementation,890-892 atomiccharges are determined by fitting to the ab initioelectrostatic potential. Because the atomic chargesare not empirically adjustable (in contrast to the


previous versions of the model), other parameters ofthe LD solvation model had also to be reoptimized.These changes include introduction of the field-dependent grid points and surface-constrained di-poles on the outer solvent surface, the new relation-ship between magnitudes of dipoles placed at theinner and outer grid points, reparametrized van derWaals and hydrophobic terms, increased extent ofdipole-dipole interactions, and a newly added solute-polarization term.

The LD model has been extended to study proteins,and the corresponding version of the model is knownas protein dipoles Langevin dipoles (PDLD).446,893

The computational code of the LD model (calledChemSol894) has been also implemented in the Qchemquantum package.894,895

7.4. Other MethodsThe electrostatic solvation methods examined in

section 2 are all based on averaged solvent polariza-tion, expressed in terms of a response function inwhich no use is made of the detailed microscopicdescription of the solvent. This definition of thesolvent mean field greatly reduces the computationaltask but neglects the possible presence of specificsolute-solvent interactions.

We pass now to examine other approaches in whichsome elements of the discreteness of solvent mol-ecules are considered.

We shall limit our attention to two approaches thatstill use the mean solvent field approximations, butfrom a mean field obtained from a thermally aver-aged discrete description of the solvent. The twoapproaches are based on different formulations of thisaveraged description and will be separately treated.They will be indicated with the acronyms ASEP(averaged solvent electrostatic potential) and RISM-SCF (reference site model in a SCF version), butactually different versions of the two approaches havebeen elaborated, and we shall review all of thepublished versions to give a fuller idea of the workin progress in this specific field.

The problem both approaches are addressing is offinding a procedure able to reduce the strong com-putational burden due to the exceedingly largenumber of solvent freedom degrees a standard dis-crete description presents.

7.4.1. ASEP-MDThe method we shall consider here has been

elaborated by Olivares del Valle and Aguilar, withco-workers, in Badajoz.

The starting point is a standard MD calculation:the solute and solvent coordinates were dumped atevery N (with N properly defined) steps for furtheranalysis. The resulting configurations are then trans-lated and rotated in such a way that all of the solventcoordinates can be referred to a reference systemcentered on the solute mass center with the coordi-nate axis lying along the principal axes of inertia ofthe solute. This process is necessary to define a gridof points equal for all configurations.

For each stored configuration only those solventmolecules that are within a sphere centered on the

solute molecule and radius equal to half the size ofthe simulation box are selected. The solute and theselected solvent molecules are then included in adielectric continuum. The Laplace equation that givesthe polarization of the continuum is solved numeri-cally, making use of apparent charges (i.e., throughthe DPCM approach presented in section 2.3.1.1).The total electrostatic potential due to the permanentsolvent charges, induced dipoles, and the surfacecharge density is calculated at each point of the griddefined inside the volume occupied by the solutemolecule, and the averaged value of this solventelectrostatic potential (ASEP) is obtained by averag-ing over the configurations. The potential obtainedin this way is a numerical potential and hence is notvery suitable for use in quantum calculations; in-stead, a set of charges chosen in such a way that theyreproduce the value of the ASEP inside the solutevolume is introduced. The charges are usually ar-ranged on spherical shells centered on the solutemass center (a solvent diameter apart), but shellsadapted to the molecular shape of the solute anddefined in terms of intersecting spheres centered onthe solute atoms have also been used. Charges placedon two concentric shells (spherical or molecularlyshaped) are generally sufficient to represent the effectof the total solvent electrostatic potential. The energyand the wave function of the solute molecule in thepresence of this averaged solvent configuration arethus obtained by solving the Schrodinger equationassociated with the perturbed Hamiltonian, namely

where HQM is the usual vacuum Hamiltonian of thesolute molecule. The interaction term, HQM-MM, takesthe form

where F is the charge density operator associatedwith the solute molecule and ⟨V(r)⟩ is the value ofASEP at the point r.

There are different versions of the ASEP procedure.In the early versions a single QM calculation isperformed with unpolarized MM descriptions of thesolvent molecules896 or with MM molecules includingpolarization.897 More recent versions introduce itera-tions on the QM part, allowing solute polarization.898

These versions have been called by the authors“coupled methods”. There are two versions: thepartial coupling (PC) in which the MM solventmolecules are kept unpolarized, and the completecoupling (CC) in which all of the molecules of theliquid specimen are polarized at the same level. TheCC version is at present available for pure liquidsonly.

The computational efficiency of ASEP has beenchecked by comparisons with QM/MM codes;899 theASEP method turns out to be more efficient by someorders of magnitude. The most important advantageof ASEP-MD when compared with previous QM/MMcalculations is that the number of quantum calcula-tions is considerably less: around four to eight with

(HQM + HQM-MM)|Ψ⟩ + E|Ψ⟩ (210)

HQM-MM ) ∫ dr F⟨V(r)⟩ (211)


ASEP-MD versus several hundreds or even thou-sands with QM/MM. Although typical QM/MM meth-ods have to make a quantum calculation for eachsolvent configuration selected, the number of calcula-tions in the ASEP-MD method is independent of thenumber of solvent configurations. It depends on onlythe number of cycles necessary to find convergencein the process that couples the solute charge distri-bution and the solvent structure around it.

The robustness of ASEP has been demonstrated bythe satisfactory results obtained in a sizable numberof examples covering different types of solvationeffects. We cite here a few: solvent shifts in electronicspectra,900,901 anomeric equilibria,902 multiple confor-mational equilibria,903 structure and properties ofpure liquids,903,904 and Stark component of the inter-action energy.905

More recently, the ASEP approach has been ex-tended to study saddle points of reactions in solu-tion906 by using the free energy gradient (FEG)method,907 in which the effective forces on the nucleiare taken as the average force over a set of equilib-rium solvent-solute configurations. Once the molec-ular dynamics simulation has been made and theASEP obtained, a geometry optimization calculationis performed. In this way, a new solute geometry isobtained, which is introduced into the next moleculardynamics simulation. The process is repeated untilconvergence. The gradient and Hessian, needed forthe optimization process, are given by the quantumcalculation program, but the appropriate van derWaals components have to be added. These compo-nents are calculated as an average over all of thesolute-solvent configurations considered.

The ASEP method has been implemented by cou-pling the MOLDY code908 for the MD part andGAUSSIAN103 or HONDO909 for the quantum calcu-lations. A detailed description of the ASEP-MD codeis also available.910

7.4.2. RISM-SCF

As we have pointed out, continuum solvationapproaches cannot properly account for the averagesolvent structure caused by the granularity, packing,and hydrogen bonding. On the contrary, statisticalmechanical theories based on distribution functionsand integral equations can provide a rigorous frame-work for incorporating such effects into a descriptionof solvation (see section 1.2).

One important class of integral equation theoriesis based on the reference interaction site model(RISM) proposed by Chandler and Andersen in1972.911 The theory is a natural extension of theOrnstein-Zernike equation of simple liquids to amixture of atoms but with strong intramolecularcorrelation, which represents chemical bonds. Thetheory takes account of one of the two importantchemical aspects of molecules, geometry, in terms ofthe site-site interactions between atoms. In itsoriginal form, however, it does not handle the otherchemical aspect of molecules, electrostatics. Thecharge distribution in a molecule, which is a classicalmanifestation of the electronic structure, plays a

dominant role in determining the chemical specificityof the molecule. Therefore, without including thecharge distribution in molecules, the description isincomplete in terms of chemical specificity. A com-plete chemical characterization of molecular liquidsbecame possible at the beginning of 1980s due to theappearance of the extended RISM theory (also knownas XRISM), which takes the charge distribution aswell as the molecular geometry into account.912,913

The extended RISM method was successfully usedin the calculation of structural and thermodynamicalproperties of various chemical and biological systemsand solutions,914 but it was soon recognized that itpoorly describes dielectric properties of polar liquids.At infinite dilution, a correction for a phenomenologi-cal dielectric constant can be introduced to theextended RISM integral equations, which allows oneto obtain sensible results for ionic pairs in a polarmolecular solvent and water. However, at finite saltconcentrations the extended RISM theory fails dueto an inherent dielectric inconsistency. To improvedielectric predictions of the RISM theory, Perkynsand Pettitt915,916 introduced consistent correctionsthat are similar to a bridge function of moleculartheories. This dielectrically consistent approach(DRISM) yielded accurate thermodynamics and struc-ture in 1/1 salt aqueous solutions at finite concentra-tions.

A further improvement of the method was done byKovalenko et al.917-919 to introduce spatial distribu-tions of liquid rather than radial correlation functionsproduced by the site-site RISM approach. Suchgeneralization is known as three-dimensional RISM(3D-RISM) and it gives a detailed solvation structurein the form of three-dimensional correlation functionsof interaction sites of solvent molecules near a soluteparticle of arbitrary shape.

More than 10 years after the original formulationof the extended RISM, Hirata’s group proposed anextension to the ab initio electronic structure theory(SCF); such an extension was named RISM-SCF.920-922

In the RISM-SCF theory, the total free energy Aincluding solute and solvent is defined as

where Esolute represents the electronic energy of asolute molecule including the nuclear repulsion en-ergy and ∆µ is the excess chemical potential (solva-tion free energy) calculated from the RISM equationwith the hypernetted-chain (HNC) closure.913,922,923

A can be seen as a functional of the site-densitycorrelation functions between solute and solvent aswell as of the wave function of the solute molecule.By solving these coupled equations, the mutuallyconverged Esolute and ∆µ give energies in the equilib-rium state that include coupling of the solute electronand solvent.

As recently done by Sato and Sakaki,924 it isinteresting to directly compare standard continuummodels (here in particular the PCM approach) andthe RISM-SCF with regard to the terms to be addedto the Fock operator so as to account for the solventreaction field effects on the solute wave function. Insection 2.3.1 we have shown that in the PCM

A ) Esolute + ∆µ (212)


framework the operator has the form

where the sum runs over the solvent apparentcharges. In the RISM-SCF approach the paralleloperator has the form

where F is the number density, bλ is a populationoperator, gλR(r) is a pair correlation function betweenλ (solute) and R (solvent), and qR is the partial chargeon the solvent site R.

The main difference between the two operators(besides the obvious differences in the form) is thatthe RISM potential (also indicated as microscopicmean field VRISM) includes molecular level informa-tion as well as bulk properties of the solvent. Tobetter understand this point, we briefly summarizethe RISM-SCF main computational strategy, whichcan be profitably presented as a sequence of steps.

In advance of the RISM-SCF cycle, the solventcorrelation function is prepared by solving the RISMintegral equations through a standard iterativescheme with a set of fixed intermolecular potentialfunctions and with guess partial charges of the solutemolecule as variables. Then using such a correlationfunction, the solute-solvent pair correlation func-tions are obtained. Immediately after the RISMconvergence has been attained, the electrostaticsolvent operator is calculated from eq 214 andinserted in the solvated Fock operator to calculatethe solute wave function. Then the electrostaticcontributions due to the nucleus and electronic cloudof the molecule are optimized independently withthose due to partial charges assigned on each inter-action site, using the least-squares fitting technique.Normalization constraints to preserve the total chargefor each contribution are imposed during the fittingprocedure. The RISM-SCF iteration is continueduntil both the electronic and solvent structuresbecome self-consistent within given convergence cri-teria.

The RISM-SCF method has been extended toinclude analytical gradients for geometry optimiza-tions and to the multiconfigurational self-consistent-field (MCSCF) method,922 which can be used forexploring the excited state of a molecule in solution.

Sato et al.925 have also reformulated the RISM-SCF/MCSCF for the 3D-RISM formalism to properlyinclude the three-dimensional picture of the solvationstructure necessary for complex solutes. The authorsfound that this reformulation allows one to reliablyresolve locations and directions of hydrogen bondingin the hydration shells. At the same time, however,they also found that the results from the originalRISM-SCF/MCSCF method are reasonably similar tothose following from the 3D-RISM/SCF approachafter reduction of the orientational dependence.

In recent years, applications of RISM-SCF to thecalculation of NMR chemical shifts of solvated mol-ecules have also been presented926,927 as well asstudies on the solvent reorganization energy associ-ated with a charging process of organic com-pounds.928,929 In the latter case both nonequilibriumsolvation and nonelectrostatic effects were included.

In all of the above RISM-SCF versions the short-range interactions are described by site-site Len-nard-Jones potentials, neglecting the solute-solventshort-range QM effects such as the exchange repul-sion coming from the Pauli exclusion principle andthe charge transfer interaction. To overcome theselimitations, Yoshida and Kato930 proposed an elec-tronic structure theory in solution using the molec-ular Ornstein-Zernike equation, which is referredto as the MOZ-SCF method. In their treatment, theexchange repulsion/charge transfer interactions areincorporated for calculating the solute-solvent in-teractions by introducing an effective potential lo-cated on solvent molecule. Comparing the MOZ-SCFresults with those from the RISM-SCF and PCMmethods, it was found that the MOZ-SCF exerts anintermediate effect on the solute electronic statebetween the RISM-SCF and PCM.

Subsequently, Sato et al.931 proposed a new ap-proach to account for the quantum nature in theshort-range interaction between solute and solventin solution. Unlike the RISM-SCF theory, this ap-proach describes not only the electronic structure ofsolute but also solute-solvent interactions in termsof a kind of model-potential method based on theHartree-Fock frozen density formulation. In thetreatment, the quantum effect due to solvent, includ-ing exchange repulsion, is projected onto the soluteHamiltonian using the spectral representation method,whereas the integral equation theory of liquids isemployed to calculate the solvent distribution aroundsolute.

8. Concluding Remarks

To conclude this review we express some consid-erations about the present state of the art of con-tinuum models and about perspectives for the future.

The introductory remarks of section 1 represent abackground for the whole review, which is centeredon QM versions of the continuum solvation models.We shall comment later what this background meansfor the perspectives of development.

In section 2 and 3 we have examined, with abun-dant details, the elaboration of the models for solva-tion effects in very dilute solutions (discarding non-linearities and time-dependent effects). Within thisrestrictive definition of solvation, we may considerthe elaboration (computational as well as structural)of models given in these two sections as havingreached maturity. By comparing the present exposi-tion with that done in the 1994 review,1 one noticesa remarkable increase in the number of methods andcomputational codes, now far more efficient andaccurate. Several new original procedures have alsobeen formulated, almost all completely elaboratedand introduced in distributed codes.

V(rb) ) ∑i

qi

|rb - sbi|(213)

VRISM(r) ) F ∑λ,R

bλqR ∫ 4πr2gλ,R(r)

rdr (214)


Most of the published applications of these methods(that we have only partially reported in this review)are about the energy of the focused subsystem andrelated quantities, such as solvent effects on thegeometry. These studies address a large area ofchemical applications: conformational changes, reac-tion mechanisms, dissociation/association equilibria,and solvent partition functions. The number of suchapplications, generally claiming success, is quitelarge; we are not able to give an estimate of theirnumber, but we know of more than 1000 papers, andsurely there are many more.

In contrast, less attention has been paid to extract-ing from the QM calculations elements for analysisand interpretation of solvent effects. Theoreticalchemistry has elaborated a large number of conceptsand computational tools in terms of submolecularunits for the analysis and interpretation of structuresand phenomena for systems in the gas phase; thesetools are easily extended to solvated systems withina continuum representation. The inclusion of solventeffects adds a new dimension to the understandingof chemical structures and processes, which is richin possible suggestions. We give an example drawnfrom a topic not treated in this review: the study ofreaction mechanisms. Differential solvation forcesrelated to the specific nature of chemical groups mayaccelerate or retard a chemical reaction. This hasbeen shown in one of the first examples of a detailedab initio study of an organic reaction,932 in which theforces exerted by solvation on the reacting chemicalgroups (expressed in terms of localized orbitals) wereexplicitly derived and used to give a rationale fornuclear geometry and electron distribution changesnear the transition state. This example, however, hasnot been followed by other researchers, perhapsbecause the necessary codes were not collected intoa distributed package. The elaboration and distribu-tion of easy-to-use interpretation packages is one ofthe aspects of conventional energy-based solvationmethods for which a more decisive effort is called forand on which we encourage other researchers tointervene.

Section 4 collects under the heading “Nonunifor-mities in the Continuum Model” topics of differentorigins for which there is still the need of furthermethodological elaboration. For this reason we havegiven bibliographic references also related to the“history” of the subject. To do further developmentin an already considered field, it is in fact advisableto examine the sources; some ideas given in seminalpapers are often neglected in the immediately fol-lowing extensions and applications, but they canbecome valuable after some years. The spatial non-uniformities considered in that section are of localnature, occurring at some boundaries in the system.These characteristics prompted the extension of thereview from the dielectric nonuniformities aroundions, to which the “historical” exposition mainlyrefers, to other boundaries, enlarging the field ofsystems where continuum methods may be applied.Some among them are related to systems to whichcontinuum methods, mostly in classical versions,have long been applied such as polyions and molec-

ular polymers in general; others involve heteroge-neous systems in which the boundary involves aphase with a structure more stable in time thannormal liquids.

We have given a few examples of successful ap-plications of QM continuum methods to systems ofthis type, but clearly these examples simply indicatethe beginning of methodological studies that stillhave to be developed. We have also indicated anotherfield in which local nonuniformities may play a role,namely, that of the complex liquids, characterized bya mesoscopic length scale. This is a novel field towhich the application of continuum models has notyet attempted, an attempt which seems to us possibleand promising. In the same line of referring tomaterial systems to which some basic methodologicalissues could be applied, we cite here the ionic liquids,and the isotropic solutions at finite rather thannegligible concentration.

In conclusion, it may be said that for the topics ofsection 4 the state of the art indicates a situation inprogress, not yet assessed, but with brilliant perspec-tives.

Time-dependent (TD) phenomena and responseproperties have been considered in sections 5 and 6,respectively. Remarkable advances have been regis-tered in both areas in recent years. In particular, forTD phenomena, continuum methods have started torepresent a valid alternative to simulation methods(historically the best suited for this kind of studies),and for response properties they are slowly butprogressively acquiring the reliability of gas phasecalculations. In both cases, an important discriminat-ing factor has been the easy insertion of high-levelQM descriptions of a molecule in continuum models.The computational advantage of a continuum ap-proach with respect to simulation methods must beemphasized again: simulations have to dedicate aconsiderable portion of computational time to de-scribe solvent degrees of freedom, and this, forexample, makes almost impossible a serious studyof response properties.

Concerning the description of molecular propertiesin condensed phases (both in continuum and indiscrete approaches) some problems arise. Quantummechanics gives a complete (in principle) descriptionof the studied system which satisfies well-establishedrules: completeness of the set of eigenfunctions,complete orthogonality, expansion coefficients overthe states directly related to the probability of findingthat state in a measurement experiment, etc. Theseformal properties, however, are no longer valid whenthe molecule interacts with others. In other words,the molecular wave function has a different statusin the presence or absence of molecular interactions.This fact is well-known to people trying to derive thevalue of a property of one of the molecular partnersfrom calculations performed on a supermolecule. Toobtain this value there is the need of additionalassumptions, varying from case to case, because auniversal protocol has never been established. It maybe said that this protocol exists for continuummodels, being the Hartree separation between thefocused molecule and the continuum in general use.


This model can be good or bad, but in any case it isa common starting point on which additional refine-ments can be based.336

This, however, is not the only important issue toaddress. In condensed systems exhibiting rapid mo-tions of the molecular components, such as liquids,other complications arise. There is, in fact, the needof an appropriate thermal average, which may changefrom property to property. For example, the calcula-tion of electronic excitation energies in the simplemodel of vertical transitions gives poor results ifbased on the excited states of a molecule in equilib-rium with the solvent. The inclusion in the model ofchanges in the fast component of the solvent polar-ization leads to excited states of better quality.

Other examples could be done, all indicating thatthe description of properties in condensed systemsis more complex than for isolated molecules. This facthas been known for a long time, and experimentalvalues of properties are always published with em-pirical corrections addressing the effect of the sur-rounding medium (namely, the local field factorsintroduced in section 6.2). These corrections, elabo-rated many years ago, are all based on quite simplecontinuum models. As shown in section 6.2.2, re-cently more precise expressions of these corrections,tailored on the single molecule, have been elaboratedwith the aid of modern solvation methods.

In parallel to this evolution, the number of proper-ties for which there are available detailed and testedcomputational strategies has greatly increased, as acomparison of this review with the preceding onesshows. This rate of increase shows no tendency tolevel off, because there are many properties not yetexamined in detail and because their number in-creases thanks to the ingenuity of experimentalistswho continually devise new ways of probing themolecular matter.

The expansion of continuum methods in the de-scriptions of properties has now passed the dividebetween simple systems (such as homogeneous solu-tions) and complex systems, in which the wholesystem gives responses that cannot be reduced tothose of the separate components. Even in this newfield the positive results are numerous and giveconfidence in further rapid progresses. The realm ofcomplex systems is quite interesting, being at thefrontier of physical and chemical research and, forits variety, presenting a challenging variety of fea-tures each requiring an attentive analysis for itsmodeling.

We do not enter into an analysis of the role of thecalculations of properties in the intrinsic context oftheoretical chemistry and of its support to applica-tions in academic and technological research, becauseour views have already been expressed in severalpapers.336,933-935 For the scope of this section, it issufficient to say that, in our opinion, the state of theart and the prospects for the topics considered in thesection 6 are decidedly positive. The basic method-ological elements have been established, the compu-tational methods (now well documented) have beenvalidated, and they can be safely employed, even ifthere is still a lot of work to refine.

Section 7 is focused on another aspect of theextension of continuum methods which is quiteimportant and in progress, namely, the combinationof continuum and discrete representations of themedium.

The admixture of continuum and discrete models(using a solute-solvent supermolecule) has been inuse for a long time in different studies including thedescription of solute conformations and the analysisof the direct role of solvent molecules in chemicalreaction mechanisms. We explicitly cite a paper936

published the same year as the first PCM paper8 tounderscore that the use of supermolecules has beenconsidered in QM continuum methods since theirvery beginning, and it has continued to the present.In that paper, a supermolecule approach plus con-tinuum was used to compute solvent shifts in elec-tronic absorption spectra. Nowadays, it is clear thatsmall solute-solvent clusters can represent an im-portant evolution of solvation methods to accuratelydescribe spectroscopic properties (as also discussedin section 6). There is, howevet, the need to ac-company this approach with others in which impor-tant aspects of the material model, such as theaverages on the motions, are better described. Thereis a sizable variety of computational strategies ad-dressing this mixing of continuum and discretemethods, and some protocols of use are in the processof refinement. In parallel, discrete simulations, intheir struggle against computational costs, tend toexploit more extensively the computational advan-tages of continuum descriptions. This is an exampleof admixture of approaches from which our advance-ment in the description of matter may derive greatbenefits.

The perspective of a stronger admixture betweencontinuum and discrete models is in line with whatwas pointed out in section 1.2 about distributionfunctions. In the present case, the main continuumdistribution in question is the density distribution ofmassive particles, the molecules, interacting amongthemselves via the forces described by the intermo-lecular interaction theories. The dielectric functionis one of the ancillary quantities related to themolecular density distribution; other functions, ofdifferent physical origin, exist, but they do not playa role in the present continuum solvation methods.The discrete molecule distribution can be replaced,partially or totally with continuum functions, as wehave discussed in several points of this review. Thischange of description leads us to neglect a portion ofthe information carried by a discrete description. Oneneeds to evaluate, problem by problem, whether thisreduction in the information compromises the de-scription of the phenomenon or whether the reductionin the computational burden is not accompanied bya loss of important details. In the case of solvationmethods, as we have shown, the advantages of acontinuum description are in general greater thanthe disadvantages. The topics considered in section7 to a large extent address the area in whichcontinuum methods require discrete corrections.

In section 1.2 we have considered another distribu-tion, that of electrons in molecules interacting via


electrostatic forces. The discrete description of thesingle elements, given by the molecular orbitaltheory, may be replaced by a continuum distributionas it is given by the density functional theory, anapproach that presents considerable advantages withrespect to the basic MO theory (and some minorshortcomings). DFT has introduced some new con-cepts and techniques in the QM description of themolecules, some of which are now exploited in QMcontinuum solvent calculations.

Continuum solvation methods can be considered toreside at the separation between the realm of theelectronic and molecular density distribution, receiv-ing inputs and information from both sides. Othertypes of continuum/discrete pairs of descriptions existfor other aspects of the physical and chemical sci-ences. For some among them, describing the motionof electrons and of more massive particles both boundto some substrate (metal, dielectric solid body), aconnection with the methods analyzed in this reviewhas been already initiated. They actually representthe beginning of another chapter of continuum meth-ods, in which the solvent is not compulsorily present.

To conclude this section, and the review, we maystate that the part of the realm of applications ofcontinuum methods in chemistry we have examinedis vast and complex, with important sections arrivedat maturity, others in rapid progress, and others stillat the initial steps.

9. AcknowledgmentWe thank Dr. Susanna Monti at the IPCF-CNR

(Pisa, Italy) for valuable help in the realization of thecover picture.

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