Solvation Models

8/16/2019 Solvation Models

1/97

See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/7667183

Quantum Mechanical Continuum SolvationModels. Chemical Reviews, 105, 2999

ARTICLE in CHEMICAL REVIEWS · SEPTEMBER 2005

Impact Factor: 45.66 · DOI: 10.1021/cr9904009 · Source: PubMed

CITATIONS

3,943

3 AUTHORS:

Jacopo Tomasi

Università di Pisa

318 PUBLICATIONS 29,635 CITATIONS

SEE PROFILE

Benedetta Mennucci

Università di Pisa


SEE PROFILE

Roberto Cammi

Università degli studi di Parma


SEE PROFILE

Available from: Jacopo Tomasi

Retrieved on: 20 August 2015

http://www.researchgate.net/profile/Roberto_Cammi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_7http://www.researchgate.net/profile/Roberto_Cammi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_7http://www.researchgate.net/profile/Roberto_Cammi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_7http://www.researchgate.net/profile/Jacopo_Tomasi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_4http://www.researchgate.net/profile/Jacopo_Tomasi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_5http://www.researchgate.net/?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_1http://www.researchgate.net/profile/Roberto_Cammi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_7http://www.researchgate.net/institution/Universita_degli_studi_di_Parma?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_6http://www.researchgate.net/profile/Roberto_Cammi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_5http://www.researchgate.net/profile/Roberto_Cammi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_4http://www.researchgate.net/profile/Benedetta_Mennucci?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_7http://www.researchgate.net/institution/Universita_di_Pisa?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_6http://www.researchgate.net/profile/Benedetta_Mennucci?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_5http://www.researchgate.net/profile/Benedetta_Mennucci?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_4http://www.researchgate.net/profile/Jacopo_Tomasi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_7http://www.researchgate.net/institution/Universita_di_Pisa?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_6http://www.researchgate.net/profile/Jacopo_Tomasi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_5http://www.researchgate.net/profile/Jacopo_Tomasi?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_4http://www.researchgate.net/?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_1http://www.researchgate.net/publication/7667183_Quantum_Mechanical_Continuum_Solvation_Models._Chemical_Reviews_105_2999?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_3http://www.researchgate.net/publication/7667183_Quantum_Mechanical_Continuum_Solvation_Models._Chemical_Reviews_105_2999?enrichId=rgreq-3b21c279-5fd7-4109-bb01-42527a39e9a6&enrichSource=Y292ZXJQYWdlOzc2NjcxODM7QVM6OTg4MDgzMjMxMTcwNjdAMTQwMDU2OTE0ODczNQ%3D%3D&el=1_x_2


2/97

Quantum Mechanical Continuum Solvation Models

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

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

Received January 6, 2005

Contents

1. Introduction 30001.1. 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 3039

4.1. Dielectric Theory Including Nonlinear Effects 30394.2. Nonlocal Electrostatic Theories 30404.3. Nonuniformities around Small Ions 3041

4.3.1. (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 3044

5. Nonequilibrium in Time-Dependent Solvation 3046

5.1. Dynamic Polarization Response 3047

5.2. Vertical Electronic Transitions 3047

5.3. Solvation Dynamics 3049

5.4. Spectral Line Broadening and SolventFluctuations

3053

5.5. Excitation Energy Transfers 3054

5.6. Time-Dependent QM Problem for ContinuumSolvation Models

3056

6. Molecular Properties of Solvated Systems 3058

6.1. Energy Properties 3059

6.1.1. Geometrical Derivatives 3059

6.1.2. IR and Raman Intensities 3061

6.1.3. Surface-Enhanced IR and Raman 3062

6.2. Response Properties to Electric Fields 3063

6.2.1. QM Calculation of Polarizabilities ofSolvated Molecules

3064

6.2.2. Definition of Effective Properties 3064

6.3. Response Properties to Magnetic Fields 3066

6.3.1. Nuclear Shielding 3066

6.3.2. Indirect Spin−Spin Coupling 3067

6.3.3. EPR Parameters 3068

6.4. Properties of Chiral Systems 3069

6.4.1. Electronic Circular Dichroism (ECD) 3069

6.4.2. Optical Rotation (OR) 3069

6.4.3. VCD and VROA 3070

7. Continuum and Discrete Models 3071

7.1. Continuum Methods within MD and MCSimulations

3072

7.2. Continuum Methods within ab Initio MolecularDynamics

3074

7.3. Mixed Continuum/Discrete Descriptions 30757.3.1. Solvated Supermolecule 3076

7.3.2. QM/MM/Continuum: ASC Version 3076

7.3.3. ONIOM/Continuum 3077

7.3.4. (Direct) Reaction Field Model 3078

7.3.5. Langevin Dipole 30787.4. Other Methods 3079

7.4.1. ASEP-MD 30797.4.2. RISM-SCF 3080

8. Concluding Remarks 3081

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

2999Chem. Rev. 2005, 105, 2999−3093

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


3/97

9. Acknowledgment 308410. References 3084

1. Introduction

1.1. Generalities about This Review

This review on continuum solvation models hasbeen 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 of chemical 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 and

computational 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 of efficient 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 of

methods based on quantum mechanics for the de-scription of isolated molecules, which have now

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

Computers have also completely modified the waysof doing theoretical and computational studies of liquid 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 of condensed 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.


4/97

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 made

possible by the use of computers: the continuummodels.Continuum models were introduced more than a

century 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 of these three aspects is completely new, but only in thepast few years have they acquired the necessaryreliability and accuracy. In parallel, the extension of the 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 of future prospects will be done in the last section

dedicated 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 the

original wave function. According to the formaltheory, one-electron FMe (r) and two-electron FM

e (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


5/97

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. A single solute molecule in a dilute solution is just an

example, 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 of the 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 H ˆ R(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 H ˆ R(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 electric

field. 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, . The expression of 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 4

will be reconsidered, but here it is sufficient toremark that the whole set of solvent coordinates {r}is replaced by a function depending only on oneparameter, the position vector rb, or by a couple of position 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 large variety of proposals a third concept emerges for itsgenerality, the concept of layering.

Layering can be considered a generalization of focusing. The material components of the models arepartitioned into several parts, or layers, because often

these 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 of layering; 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.

H ˆ FR(f ,r) ) H ˆ F(f ) + H ˆ R(r) + H ˆ int(f ,r) (1)

H ˆ eff FR(f ,r) ) H ˆ F(f ) + H ˆ int(f ,r) (2)

H ˆ eff FR

(f ,r) ) H ˆF

(f ) + V ˆint

[f , Q(rb,rb′)] (3)

PB ) - 14π

EB (4)



6/97

2. Methodological Outlines of the Basic QM Continuum 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 Model

For the basic QM model we intend models with thefollowing characteristics:

(1) The solute is described at a homogeneous QMlevel. 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. Cavity

The cavity is a basic concept in all continuummodels. 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 but

existing 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 a

tradeoff between conflicting physical requirements.The shape of the cavity has also been the object of many 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)



7/97

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 others19-24is 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,25the 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 of potential 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 of the 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 chemical

literature is that defined by Bondi

26

(∼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 CRC Handbook.29

A third set of values has been inserted in the latest versions 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 radius values currently used in continuum models and theconsequent effect on the solvation energies deservea few words of comment. Chemical applications of molecular calculations generally involve trends, inparticular, comparisons of selected properties com-puted for different systems. In the particular case of solvation calculations, the comparisons may involveproperties of different solutes in the same solvent orof the same solute in different solvents. Absolute values 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 not

a 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 of surfaces 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 of problems, and they have been also used to compute



8/97

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 sphere

of 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 of spheres 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 QM

continuum 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 of spheres, 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 the

result 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 progressive

deformations 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 of the deformed polyhedron is then shifted along theline connecting the initial position with the origin of the 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 the

surface 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 of

the 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.



9/97

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 polarization

of 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, is the dielectric function (actually a constant) withinthe medium, and V is the sum of the electrostaticpotential V M generated by the charge distribution FMand the reaction potential V R 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 of boundary conditions, the first at infinity and thesecond on the cavity surface. At infinity we have

with finite values for R and . 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 valid

also 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 (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 Methods

From the jump condition (eq 15) one may derivean auxiliary quantity that defines all of the ASCmethods: an apparent surface charge σ (s) spread on

the 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 V R of eq 9. There are no approximations in this elabora-tion: the definition of V R given with eq 16 is exact

- ∇ B[(rb)∇ V B(rb)] ) 4π FM(rb) (6)

-∇ 2V (rb) ) 4π FM(rb) within C (7)

-∇ 2V (rb) ) 0 outside C (8)

V (rb) ) V M(rb) + V R(rb) (9)

limrf ∞

rV (r) ) R (10)

limrf ∞

r2V (r) ) (11)

[V ] ) 0 on Γ (12)

[∂V ] ) 0 on Γ (13)

[V ] ) V in - V out ) 0 (14)

[∂V ] )

(

∂V

∂n)in

-

(

∂V

∂n)out

) 0 (15)

V σ (rb) ) ∫Γσ (sb)

|rb - sb|d2s (16)



10/97

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 of

complex shape is computationally challenging. Thesolutions are generally based on a discretization of the 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 is

possible 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 iterative

procedure. 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 for

some 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 of nonoverlapping dielectric regions at different permit-tivity, constant within each region.

The reason for this versatility is in the use of the ASC 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 i 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 surface

pointing 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 i ) 1 and j ) 1 and that wehave computed the gradient on the internal (in) partof the surface, namely

where nb indicates the unit vector perpendicular to

the 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

V σ (rb) = ∑k

σ (sbk) Ak

|rb - sbk|) ∑

k

qk

|rb - sbk|(17)

PBi(rb) ) - i - 1

4π ∇ V B(rb) (18)

σ ij ) - ( PB j - PBi)‚nbij (19)

σ (s) ) - 14π

∂

∂nb(V M + V σ )in (20)



11/97

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 to

almost 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 to validate it by comparisons with other procedures:simulations of different type, other continuum mod-els. Among them we cite a comparison with two

methods 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 of popularizing 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 of

solvation 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 solvation, 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 is

relaxed 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 andSchüürmann,58 the dielectric constant of the mediumis changed from the specific finite value , charac-teristic of each solvent, to ) ∞. 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 on

the 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 of the medium, the ideal unscreened charge density, σ *,corresponding to ) ∞, is finally scaled by a properfunction of , namely

The scaling function, f (), 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,59,60and 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 ()σ *(s) (21)

f () ) - 1 + k

(22)



12/97

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 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 in

the 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 the

partners. 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 byCancès 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, ). 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 of solvation 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, V M, and the “reactionpotential”, V R.

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-∇ (∇ ), and GR( x, y) ) Gs( x, y) - G( x, y), the followingrelationships arise:

Exploiting some basic results of the theory of

integral equations (see ref 76 for details), it can beproved that the reaction potential, V R, 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 for external (i.e., outside the cavity) and internal (i.e.,

inside the cavity), respectively. These operators areformally defined for σ by

where a ) 1 for a ) i and a ) (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) d y

V M( x) ) ∫ R3 G( x, y)FM( y) d y (23)

V R( x) ) ∫ R3 G R( x, y)FM( y) d y

- ∇ 2V R ) 0 in C and outside C

[V R] ) 0 on Γ (24)

V R f 0 at infinity

V R( x) ) ∫Γσ ( y)

| x - y|d y ∀ x ∈ R3 (25)

Aσ ) - g (26)

A ) (2π - De) Si + Se(2π + Di/)

g ) (2π - De)V M + Se(∂V M / ∂n) (27)

( Saσ )( x) ) ∫Γ Ga( x, y)σ ( y) d y

( Daσ )( x) ) ∫Γ [(a∇ B yGa( x, y))‚nb( y)]σ ( y) d y (28)

( Da/σ )( x) ) ∫

Γ [(a∇ B xGa( x, y))‚nb( x)]σ ( y) d y



13/97

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 of the 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, Si Di/ ) Di Si.

Equation 26 may be further simplified using theequality (2π - Di)V M + Si(∂V M / ∂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 V M (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 because

the calculation of a single scalar function (the poten-tial) is computationally less demanding than theparallel calculation of both the potential and the vectorial 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 IEF ASC 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 reporting

the final equations for the IEF ASC in the case of isotropic solvents; these, in fact, will be useful in thefollowing when we shall compare IEF to other ASCmethods. Namely, using the relations, Se ) Si / , 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

/ ) Di Si. In the following, this version of IEF willbe denoted IEF(V) to indicate that only the soluteelectrostatic potential (V M) 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 [namely Ge( x, y) ) 1/(| 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 for

further 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 the

potential, in the opposite way so as to keep (∂V M / ∂n)instead. To recover COSMO ASC, on the other hand,we have to consider that f ∞, and thus all of theterms in eqs 27 and 29 involving the Se operator canbe neglected ( Se is in fact proportional to 1/ ).

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 apparent volume 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 was approximated bya collection of point charges located at various nodes

V R( x) ) V σ ( x) + V ( x)

V ( x) ) ∫ext ( y)| x - y| d y ) - ( - 1 )∫ext F

M( y)| x - y| d y

(32)

g ) [(2π - De) - Se Si-1(2π - Di)]V M (29)

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

g ) (1 - 1/ )(2π - Di)V M(30)

[2π

(

+ 1

- 1) - D

i] S

iσ ) - (2π - D

i)V

M (31)



14/97

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 discontinuous volume 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-dimensional

space.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 the volume 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 volume

charge 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 is

not 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 ( + 1)/( - 1) and to use the relationship Si

Di/ ) D i Si. 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 alternative ASC 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 section

2.3.1.3) remains in all cases. The late recognition of the equivalence of the two approaches is also reflectedin a further PCM approach proposed by Cossi et al.88In 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 its

27th meeting in 200591

), can be found on this verypowerful mathematical approach. As introduced at the beginning of section 2.3.1, the

definition of ASC elements uses the discretization of the 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 ASC

methods, 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 were

fully 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 of connected arc segments, and their surface areas can

be 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 tessellation94and symmetry-adapted tessellations.95,96 This ad- vanced version of GEPOL is available in GAMESS97,98and 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.

T σ SS(V)PE ) ( - 1 + 1)[

Di

2π - I ]V M

T ) Si - ( - 1 + 1) 14π

( Di Si + Si Di/)

(33)



15/97

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 is

then 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 of σ (s) on each of these tesserae times the corresponding

area. In doing this the electrostatic equation pre-sented in sections 2.3.1.1-2.3.1.4 for σ (s) within the various 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 of the 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 ) and the second for anisotropic solvents (i.e.,characterized by a constant but tensorial permittivity) 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;102here, 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 (2π + 1 - 1 A -1 - D*)-1

with Dij ) (sbi - sb j)‚nb j

|sbi - sb j|3 98

if jDij

/ En

COSMO [CPCM] S-1 with Sij ) 1

|sbi - sb j| V

IEFPCM(iso) [IVPCM]

{[2π

(

+ 1

- 1) A -1 - D

]S

}

-1[2π A -1 - D] V

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

2 ]}-1

[2π A -1 - D]

IEFPCM {[2π A -1 - De]S + Se[2π A -1 + D*]}-1{[2π A -1 - De] -

SeS-1[2π A -1 - D]}

V

aniso[ De]ij )

(sbi - sb j)‚nb j det E[(-1‚(sbi - sb j))‚(sbi - sb j)]

3/2

[ Se]ij ) 1

det E[(-1‚(sbi - sb j))‚(sbi - sb j)]1/2

ionic [ De]ij )

exp[-k|sbi - sb j|][1 + k|sbi - sb j|](sbi - sb j)‚nb j|sbi - sb j|3

[ Se]ij ) exp[-k| sbi - sb j|]

|sbi - sb j|

q ) -Kf (34)



16/97

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 which

k ) 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 R I 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 and

ionic 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 if compared 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 of ab 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, the

significant 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 of atoms.

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 m

Solvation Models

Documents

Transcript of Solvation Models