Extension of the Effective Fragment Potential Method toMacromoleculesPradeep Kumar Gurunathan,†,# Atanu Acharya,‡,# Debashree Ghosh,‡,§ Dmytro Kosenkov,†,∥

Ilya Kaliman,†,‡ Yihan Shao,⊥ Anna I. Krylov,*,‡ and Lyudmila V. Slipchenko*,†

†Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States‡Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, United States§Physical and Materials Chemistry Division, CSIR-National Chemical Laboratory, Pune 411008, Maharashtra, India∥Department of Chemistry and Physics, Monmouth University, West Long Branch, New Jersey 07764, United States⊥Q-Chem Inc., 6601 Owens Drive, Suite 105 Pleasanton, California 94588, United States

*S Supporting Information

ABSTRACT: The effective fragment potential (EFP) approach, which can be described as a nonempirical polarizable force field,affords an accurate first-principles treatment of noncovalent interactions in extended systems. EFP can also describe the effect ofthe environment on the electronic properties (e.g., electronic excitation energies and ionization and electron-attachmentenergies) of a subsystem via the QM/EFP (quantum mechanics/EFP) polarizable embedding scheme. The original formulationof the method assumes that the system can be separated, without breaking covalent bonds, into closed-shell fragments, such assolvent and solute molecules. Here, we present an extension of the EFP method to macromolecules (mEFP). Several schemes forbreaking a large molecule into small fragments described by EFP are presented and benchmarked. We focus on the electronicproperties of molecules embedded into a protein environment and consider ionization, electron-attachment, and excitationenergies (single-point calculations only). The model systems include chromophores of green and red fluorescent proteinssurrounded by several nearby amino acid residues and phenolate bound to the T4 lysozyme. All mEFP schemes show robustperformance and accurately reproduce the reference full QM calculations. For further applications of mEFP, we recommendeither the scheme in which the peptide is cut along the Cα−C bond, giving rise to one fragment per amino acid, or the schemewith two cuts per amino acid, along the Cα−C and Cα−N bonds. While using these fragmentation schemes, the errors insolvatochromic shifts in electronic energy differences (excitation, ionization, electron detachment, or electron-attachment) do notexceed 0.1 eV. The largest error of QM/mEFP against QM/EFP (no fragmentation of the EFP part) is 0.06 eV (in most cases,the errors are 0.01−0.02 eV). The errors in the QM/molecular mechanics calculations with standard point charges can be aslarge as 0.3 eV.

1. INTRODUCTION

Electronic processes in complex environments are at the heartof numerous phenomena of fundamental and societalimportance, such as catalysis, solar energy harvesting, andphotovoltaics. Predictive computational modeling is instrumen-tal for advancing our mechanistic understanding of the redoxand photoinduced processes in a condensed phase; it requires acombination of quantum mechanical methods and anappropriate description of the environment (solvent, protein,

molecular solids, etc.). The effect of the environment ismultifaceted. First, it spatially confines the reacting species: forexample, the protein matrix controls the structures of reactioncenters in enzymes and restricts the range of motion ofchromophores in photoactive proteins. Second, the environ-

Received: April 25, 2016Revised: June 16, 2016Published: June 17, 2016

Article

pubs.acs.org/JPCB

© 2016 American Chemical Society 6562 DOI: 10.1021/acs.jpcb.6b04166J. Phys. Chem. B 2016, 120, 6562−6574

ment serves as a thermal bath. Third, the environment oftenstrongly perturbs the electronic structure of the system by localand long-range electric fields. Preferential stabilization of someelectronic states relative to others leads to solvatochromism,solvent-induced shifts of electronic excitation energies, whichcan be as large as 1 eV.1,2 The effect on ionization/electron-attachment energies (quantities determining the redoxpotentials) is even more pronounced; shifts of severalelectronvolts in polar solvents and in proteins are rathercommon.3−5 Fourth, the environment itself can be perturbedby the solute: the changes in the electronic structure of anactive center (e.g., ionization) induce both structural andelectronic response of the solvent. In sum, even when theelectronic processes are confined to a well-defined domain(such as solvated or protein-bound chromophores), electronicinteractions between the solute and solvent need to be carefullyaccounted for.Of course, full quantum mechanical treatment of the entire

solvent−solute system would correctly describe these effects.However, such a brute-force approach is impractical due tosteep computational scaling of electronic structure methods.6

For example, robust and reliable equation-of-motion coupled-cluster methods7,8 with single and double substitutions (EOM-CCSD) can be applied only to moderate-size systems (20−30heavy atoms), even when combined with efficient paralleliza-tion and other algorithmic enhancements.9,10 Low-cost time-dependent density functional theory (TD-DFT) and scaled-opposite-spin configuration interaction singles with doublescorrection (SOS-CIS(D)) methods11−13 can treat considerablylarger systems (a hundred of heavy atoms), but not sufficientlylarge to model, for example, bulk solvation or an entire protein.To overcome this hurdle, one can employ a more

approximate description of the environment while treatingthe solute quantum mechanically. Several strategies of variousdegrees of sophistication have been developed toward this end.The simplest one is to describe the solvent by a polarizablecontinuum model. Multiple flavors of these methods exist;14−16

some have been shown to be capable of capturingsolvatochromic effects17 and yielding reasonably accurateredox potentials18 in aqueous solutions. However, thedescription of specific solvent−solute interactions (such ashydrogen bonding) and inhomogeneous environments (such asproteins and interfaces) is beyond the reach of these modelsbecause of their implicit solvent nature. Thus, explicit solventmodels, which are based on more detailed representation of thesolvent, are needed to tackle proteins, interfaces, and complexmolecular solids. The popular quantum mechanics/molecularmechanics (QM/MM) approach19−22 allows one to combinean arbitrary complex QM treatment of the solute (reactioncenter) with an atomistic description of the environment. Themost common variants of QM/MM use standard (non-polarizable) force fields in which charges on the MM atomsare fixed. Thus, electronic interactions between the QM andMM parts are described as perturbation of the QM part by the(fixed) electrostatic field of the environment. The electronicresponse of the environment to the changes in the electronicstructure of the solute is neglected, which may introduceundesirable errors in relative state energies. Moreover, thecharges used in these force fields are known to be too large (tocompensate for the lack of polarization); this can lead, forexample, to overestimation of reorganization energies inelectron-transfer calculations.23 Finally, the empirical natureof MM force-fields limits their predictive power.

Effective fragment potential (EFP) approach24−31 has beendeveloped with an aim to address these limitations of QM/MM. EFP is based on a rigorous representation of differentcomponents of intermolecular interactions (electrostatics,polarization, dispersion, exchange-repulsion, and optionalcharge-transfer) based on perturbative expansion; it can bedescribed as a nonempirical parameter-free polarizable force-field. In QM/EFP, the interactions between the QM and EFPsubsystems include both electrostatics and mutual polarization.The extension of EFP to electronically excited and ionizedstates3,32−34 includes polarization response of the EFPenvironment to changes in the electronic distribution of thesolute. The existing implementations allow one to combineEFP with many popular electronic structure methods includingcoupled-cluster (CC), EOM-CC, TD-DFT, CIS(D), and SOS-CIS(D).EFP is similar to the polarizable embedding (PE)

approach.35−38 The description of electrostatics and polar-ization is nearly identical in EFP and PE; however, EFP featuresmore rigorous treatment of dispersion and exchange repulsion.For modeling intermolecular interactions, similar strategiesbased on multipolar expansion have been utilized.39−41

Recently, EFP was combined with the fragment molecularorbital (FMO) method42 giving rise to the EFMO methodcapable of describing both molecular aggregates and largemolecules43,44 (however, the EFMO method has not yet beenextended to excited states). The detailed reviews of variousflavors of fragment-based methods can be found in refs 45 and46.The original formulation of the EFP method assumes that

the system can be separated, without breaking covalent bonds,into closed-shell rigid fragments, such as solvent and solutemolecules. To extend EFP to macromolecules, one needs tofigure out how to break large molecules into fragments thatcould then be described by EFP. A similar problem often arisesin traditional QM/MM calculations when one needs toseparate the QM part from the rest of the system by breakingcovalent bonds. These broken bonds need to be saturated (orcapped) for subsequent calculations. Various approachesincluding hydrogen atom caps and frozen molecular orbitalshave been developed.20,47−50 In the context of fragmentation-based methods,46 schemes such as link atoms, the moleculartailoring approach,51 and molecular fractionation with con-jugate caps52−54 have been used. Flexible EFP schemes havealso been proposed.55−58

In this work, we break the protein into amino acid fragmentscapped by hydrogen link atoms.47 We benchmark severalschemes of breaking a macromolecule into effective fragments:breaking along (1) the peptide bond, (2) either the Cα−C orCα−N bond, or (3) both the Cα−C and Cα−N bonds.We focus on the electronic effects (such as excitation,

ionization, and electron-attachment energies) of moleculesembedded in the protein matrix. Here, our goal is to enablesingle-point calculations of electronic properties and energydifferences rather than structure optimization. The test casesinclude excitation, ionization, and electron-attachment energiesof common chromophores in the presence of a proteinbackbone.The structure of the article is as follows. The next section

presents a brief overview of the EFP scheme and thenintroduces different approaches for splitting macromoleculesinto effective fragments. Computational details are given inSection 3. Section 4 presents the results of benchmark

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calculations and discusses relative merit of different fragmenta-tion schemes. Our concluding remarks are given in Section 5.

2. METHODOLOGY2.1. EFP Scheme: QM−EFP Interactions. The EFP

method24−31 describes noncovalent interactions by usingperturbation theory starting from the noninteracting (un-perturbed) fragments. The total interaction energy (EEFP−EFP)between the effective fragments is broken down into fourcontributions: electrostatic (E elec), polarization (E pol), dis-persion (E disp), and exchange repulsion (E exrep)

= + + +−E E E E EEFP EFPelec pol disp exrep

(1)

The interactions between the QM part and effective fragmentsare computed by the polarization embedding approach inwhich the Coulomb and polarization parts of the EFPsubsystem contribute to the quantum Hamiltonian (H) viaone-electron terms

= + + †H H V Vp q p q0Coul pol

(2)

where H0 is an unperturbed Hamiltonian of the QM part, V Coul

and V pol are electrostatic and polarization perturbations,respectively, and p and q are the atomic orbitals in theQM part. The electrostatic and polarization terms areconsidered the most important ones, as far as the effect ofthe polarizable environment on the electronic properties of theQM solute are concerned.25 In the discussion below, we focuson these terms; the complete details of our EFP implementa-tion including the treatment of dispersion and exchange-repulsion contributions can be found in refs 27−31. Note thateq 1 can be further augmented by including the charge-transferterm,29 which is important for strongly interacting fragments(i.e., ionic liquids); in our implementation, this term is omitted.Both terms represent classic electrostatic interactions; V Coul

is a Coulomb potential due to the fragments’ nuclear chargesand their electron densities represented by the multipoleexpansion, whereas V pol describes an electrostatic field due tothe induced dipoles (thus, it depends on the polarizabilities ofthe fragments). The induced dipoles are found by an iterativeself-consistent procedure, such that the converged dipoles arefully consistent with each other and with the electronic wavefunction of the QM part.Neglecting the dispersion and exchange-repulsion contribu-

tion terms, the total ground-state (or, more precisely, reference-state) energy of the QM/EFP system is

= Φ + + Φ +

+

E H V V E

E

refQM/EFP

ref 0Coul

refpol

ref Coul

pol,ref (3)

where Φref is the reference-state wave function, VCoul and V ref

pol

are the Coulomb and polarization EFP contributions to theHamiltonian. The subscript ref indicates that the induceddipoles correspond to the electronic density of the referencestate, that is, the ground electronic state in excited-statecalculations, a closed-shell neutral state in the EOM-IPcalculations of radical cations, and so forth. ECoul is theelectrostatic fragment−fragment interaction energy and Epol,refis the self-consistent reference-state polarization energy of theQM/EFP system; it is computed using converged induceddipoles of the fragments and the fields due to the static

fragment multipoles and the nuclei and electrons of thequantum region. Note that the polarization contributionsappear in both the quantum Hamiltonian through V pol and theEFP energy as Epol,ref; this is because self-consistency precludesthe separation of the QM−EFP and EFP−EFP polarizationcontributions.24

Electrostatics is the leading term in the total interactionenergy in hydrogen-bonded and polar systems. E Coul betweenthe effective fragments consists of charge−charge, charge−dipole, charge−quadrupole, charge−octupole, dipole−dipole,dipole−quadrupole, and quadrupole−quadrupole terms. Atclose separation between the fragments (or between a fragmentand the QM region), the charge penetration may becomesignificant; to correct classical multipoles for possible chargepenetration, several types of damping functions can be used.59

Here, we employ an exponential damping in which the charge−charge interaction energy is damped using the followingequation60

= −−

− −−

−−fb

b aar

aa b

br1 exp( ) exp( )klch ch

2

2 2 kl

2

2 2 kl

(4)

where rkl is the distance between multipole points k and l anddamping parameters a and b are for the multipole points k andl, respectively.For QM−EFP interactions, the electrostatic potential of the

molecule (QM part) at point x is expressed by the multipoleexpansion around K points located at the atomic centers andbond midpoints of the fragments

∑ ∑

∑

μ= − + Θ

− Ω

V x q T r T r T r

T r

( ) ( ) ( )13

( )

115

( )

xa

x y z

a a xa b

x y z

a b a b x

a b c

x y z

a b c a b c x

kCoul

k k

, ,k

k,

, ,

,k

, k

, ,

, ,

, ,k

, , k(5)

where q, μ, Θ, and Ω are the net charge, dipole, quadrupole,and octupole located at K points, respectively. T are theelectrostatic tensors of ranks 0−3. Interaction of the QMelectronic density with multipole charges is augmented by aGaussian-type damping function, such that eq 5 becomes

∑ ∑

∑

α

μ

= + − −

− + Θ

− Ω

V x q q r T r

T r T r

T r

( ) ( (1 exp( )) ( ))

( )13

( )

115

( )

x x

a

x y z

a a xa b

x y z

a b a b x

a b c

x y z

a b c a b c x

kCoul

knuc

kele

k k2

k, ,

kk

,

, ,

,k

, k

, ,

, ,

, ,k

, , k(6)

where qknuc is the nuclear charge and qk

ele is the electronic chargeon multipole point k, respectively. Thus, only the electroniccharges are damped (smeared) by Gaussians. Dampingparameters a, b, and α in eqs 4 and 6 are determined byminimizing the difference between the electrostatic potentialsfrom the damped multipole expansion and the electronic wavefunction in the parameter-generating calculation for eachfragment.Polarization is a many-body term, which is computed self-

consistently because the induced dipoles of one fragmentdepend on the static electric field and induced dipoles of otherfragments. The polarization energy of the QM−EFP system iscomputed as

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∑ μ μ= − + + E F F F12

( ( ) )pol,refk

k mult,k nuc,k k ai,k

(7)

where μk and μk are the induced and conjugated induceddipoles at distributed point k (at the centers of the localizedmolecular orbitals (LMOs)); F mult,k is the external field due tothe static multipoles and nuclei of other fragments; F nuc,k andF ai,k are the fields due to the nuclei and electrons of the QMregion (F ai,k is computed using the reference-state density).The induced dipole at each polarizability point k is given by

∑μ α= + + +F F F F( )k

k

k mult,k ind,k nuc,k ai,k

(8)

where F ind,k is the field due to other induced dipoles. Themany-body contribution is solved iteratively, as the induceddipole on a particular fragment depends on the values of theinduced dipoles of all other fragments and the wave function ofthe QM region. The distributed polarizabilities (αk) are locatedat the centers of the LMOs.To avoid polarization collapse when effective fragments

approach each other (which is always the case in macro-molecules; see below), the polarization damping functions areapplied to the electric fields produced by multipoles andinduced dipoles in eqs 7 and 859

= − − +f c d r c d r1 exp( )(1 )klpol

k l kl2

k l kl2

(9)

where ck and dl are the damping parameters on polarizabilitypoints k and l, respectively. The default values of thepolarization damping parameters are 0.6 for noncovalentneutral fragments and 0.1 for small anions and cations suchas halide and alkali ions. Appropriate values of the polarizationdamping parameters for mEFP are explored in Section 4.2.2. Breaking Macromolecules into Fragments. In its

original formulation, the EFP method cannot be applied tomacromolecules such as biopolymers, peptides, proteins, lipids,and DNA because it was designed as a rigid-fragment model fortreating interactions between (small) molecules in clusters andliquids. The building block of EFP (the so-called fragment) is amolecule rather than an atom (as typical in classical force-fields). The EFP scheme does not include covalent interactionsbetween the fragments.To enable conformational flexibility of macromolecules while

keeping the calculations affordable, the polymer chain shouldbe split into small effective fragments, and the parameters foreach of these fragments need to be generated, similar to that inother fragment-based methods.42,51,61−63 We generated theparameters for individual fragments using the followingprocedure. First, we saturate the dangling bonds of eachfragment using a capping group, to obtain well-behaved closed-shell structures. In this work, we use hydrogens as the cappinggroups. Once the parameters for the capped closed-shellmolecule are generated, we remove the capping groups and theassociated parameters (multipoles and polarizabilities) from thefragment. This scheme enables calculation of the electrostaticand polarization energies in the macromolecule represented byeffective fragments as well as electrostatic PE using QM/EFP.The specific details of the application of this scheme topolypeptides (such as positions of the cuts) are discussedbelow. Schemes for other macromolecules will be developed infuture work, following similar strategies.There are different ways to break macromolecules such as

proteins and DNA into fragments, depending on the position

of the cut between two covalently bound residues.64 Forpolypeptides, we consider the following cutting schemes(shown in Figure 1):

• cutting along the peptide link;

• cutting along the Cα−C bonds;

• cutting along the Cα−N bonds; and

• cutting along both the Cα−C and Cα−N bonds.

In the first three schemes, each fragment consists of a singleamino acid, whereas the last one yields two fragments peramino acid, one fragment containing the peptide group andanother containing the residue; see Figure 1.The advantage of fractioning the protein along the peptide

bond is that it yields “symmetric” fragments. However, in thisscheme, highly polarized bonds are broken, which may lead tounphysical multipole and polarizability values. One may expectthat the fragments obtained by cutting either the Cα−C or Cα−N bonds produce a more accurate representation of multipolesand polarizabilities near the cuts. Furthermore, smallerfragments and cutting both the Cα−C and Cα−N bonds willaid future extension of mEFP to dynamics, as this schemeensures flexibility along the two most important degrees offreedom in proteins, dihedral ϕ and ψ angles defining theconformation.In all schemes, the capped fragments mimic the protein;

however, the neighboring fragments have duplicated points(overlapping area) due to the hydrogen caps. The multipoleexpansion points and polarizability expansion points areextended on each isolated capped fragment by the standardprocedure (see Section 3), and the multipoles and polar-izabilities at the duplicated expansion points are removed. Tomaintain the net integer charge on each fragment, themonopole expansion of each cap is redistributed to the nearestatom.65 This method is called expand−remove−redistribute(ERR; see Figure 2). The polarizability points located throughthe “cut” bonds are removed to avoid over-polarization of theneighboring fragments.We have implemented the mEFP approach in libefp,30 which

is interfaced with the Q-Chem package.66,67

Figure 1. Various cutting schemes for polypeptides.

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3. COMPUTATIONAL DETAILSThe benchmark set was chosen with the focus on electronicproperties. In particular, we consider the effect of the proteinscaffold on the electronically excited states of chromophoresand on their redox properties. First, we quantify the effect ofthe protein environment by comparing the properties of thebare chromophores (small QM) with the full QM calculations.Then, we test different fragmentation strategies and comparethe results of QM/mEFP against full QM calculations andagainst QM/EFP (when no breaking along covalent bonds inthe EFP part is performed). Finally, we consider QM/MMcalculations with fixed point charges using standard force-fields.68,69

While setting up such benchmark calculations, one shouldkeep in mind several important points. The QM/MM andQM/mEFP calculations can be compared with the full QMcalculation only when the target electronic properties can bedescribed as the electronic properties of the QM regionperturbed by the environment. In the calculations of electroni-cally excited states, this means that one should consider onlythe states in which both the initial and target molecular orbitalsare confined within the QM part. In calculations of ionizationor electron-attachment energies, the spin density of the targetstate needs to be localized in the QM region. Within Koopmans

theorem, the highest occupied molecular orbital (HOMO) ofthe initial state is representative of the density of the unpairedelectron; however, we found several examples where the shapeof the HOMO is quite different from the spin density of theionized/electron detached state. Because spin density repre-sents the actual shape of the hole (or the unpaired electron),one needs to always consider spin density, whereas the shape ofthe HOMO is relevant only for assessing the validity ofKoopmans’ description of ionization. In a similar vein, oneshould consider spin density while analyzing electron-attachedstates. In addition to the above consideration, one should alsoassess whether the states in question are bound or unboundwith respect to electron ejection. This is particularly relevant forelectron-attached states and excited states of anionic systems.The following model systems were used in the benchmark

calculations:

1. Green fluorescent protein (GFP) chromophore in itsanionic (deprotonated) and neutral forms surrounded byfour nearby amino acid residues (Figure 3).

2. The anionic form of the mPlum chromophore withnearby amino acids and one water molecule (Figure 3).

Figure 2. The ERR procedure for generating the parameters of theeffective fragments in a macromolecule.

Figure 3. Left: Protonated GFP chromophore and the VAL93, GLN94, GLU95, and ARG96 tetrapeptide strings; GLU95 is protonated. Thestructure of the model system with the anionic chromophore is the same. Right: Model mPlum system with a deprotonated extended chromophore.The protein is represented by the ARG88, VAL89, and MET90 tripeptide strings and one water molecule.

Figure 4. Phenolate embedded into the apolar cavity consisting of fouramino acids in the T4 lysozyme.

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3. A phenolate molecule with four amino acids from the T4lysozyme M102E/L99A mutant (Figure 4).

All relevant geometries and EFP parameters are provided in theSupporting Information (SI).3.1. Model Systems. 3.1.1. GFP Model Systems. We

constructed the model system as follows. We began from the X-ray structure of enhanced GFP (pdb id:1EMA). We addedhydrogens following the protonation states determined in ref70. Then, we optimized the structure using the CHARMM27force field.68 The parameters of the chromophore are from ref69. From the optimized structure, we extracted the modelsystem comprising the chromophore and four other amino acidresidues, VAL93, GLN94, GLU95, and ARG96. Because theseresidues constitute a single sequence, we capped only α-carbonsof VAL93 and ARG96 (these capping hydrogens were notincluded in the MM point charges in the QM/MMcalculations). Also, two capping hydrogens were added to thechromophore.

We constructed two model systems, one with the anionic(deprotonated) chromophore and another with the neutralchromophore. In the model system with the anionicchromophore, GLU95 is deprotonated and ARG96 isprotonated. The total charge of the model system is −1(negatively charged chromophore and GLU95 and positivelycharged ARG96). The model structure is shown in Figure 3.The model system with the neutral GFP chromophore was

prepared following the same protocol, except that GLU95 wasprotonated (this is necessary for finite-cluster calculations tosuppress the ionization and electronic excitation from the MOlocalized on GLU95). The total charge of the model systemwith the neutral GFP chromophore is +1.In small QM, QM/MM, and QM/EFP calculations, only the

chromophore constitutes the QM part. Vertical detachmentand ionization energies (VDEs and VIEs, respectively) werecalculated with ωB97X-D/aug-cc-pVDZ.71 We note that usinga range-separated functional is important for these charged

Figure 5. (a−e) Spin densities of open-shell states for four model systems. In all cases, the spin density is localized on the residue of interestconstituting the QM part.

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systems. Vertical excitation energies were computed with SOS-CIS(D)/aug-cc-pVDZ. To investigate basis-set effects, weperformed calculations also with a smaller basis set, 6-31G(d).3.1.2. mPlum Model System. The mPlum model system

features a larger anionic chromophore, so one can assesswhether the effect of the environment depends on the extent ofπ-conjugation in the chromophore. The system was con-structed from the PDB ID:2QLG structure of mPlum followingthe protocol from ref 72. The model system was extracted fromthe QM/MM-optimized structure (as described in ref 72). Themodel system comprises the extended conjugated anionicchromophore along with three amino acids (ARG88-VAL89-MET90) and one water molecule. The total charge of thesystem is 0. We rearranged the water molecule to facilitatehydrogen bonding between phenolate’s oxygen of thechromophore and water (this makes the radical-dianion statebound with respect to electron detachment). ARG89 andMET90 were capped with H-atoms at Cα.In small QM, QM/MM, and QM/EFP calculations, only the

chromophore constitutes the QM part. VDE and excitationenergies were calculated using the same methods as in the GFPmodel systems, and the attachment (VEA) energies werecomputed using ωB97X-D/aug-cc-pVDZ.3.1.3. T4 Lysozyme Model System. The model system

consists of the tyrosine residue in an apolar cavity in the T4lysozyme.73 The following residues were retained from thecrystal structure (3GUO): phenolate, ALA99, ILE100, GLU102(protonated at the side chain), and VAL103. We choosephenolate rather than phenol to ensure the spin density to belocalized in the QM region (phenolate). ALA99 and VAL103were capped with hydrogens at the peptide bond (C−N).Hydrogen positions were optimized by ωB97X-D/6-31+G(d).All residues except for the phenolate are neutral; the totalcharge of the model system is −1. In small QM, QM/MM, andQM/EFP calculations, only the phenolate constitutes the QMpart. We computed VDE using ωB97X-D/aug-cc-pVDZ.Figure 5a−e shows the spin densities of all open-shell cases

under study with full QM treatment for each system. We seethat the spin density is always localized on the small QM forrespective systems. This justifies the comparison among smallQM, QM/MM, QM/mEFP, and full QM.3.2. QM/MM and QM/mEFP Calculations. The EFP

parameters for the peptide residues were prepared following theprocedure outlined in Section 2.2. In addition to the fourfragmentation schemes described above, we also considered thesuperfragment scheme in which all covalently linked peptidesconstitute a single EFP fragment. That is, in the GFP andmPlum model systems, all residues form one superfragment; inthe T4 lysozyme model system, there are two superfragments.The EFP parameters for each isolated capped residue and

superfragments have been generated using the MAKEFP job(RUNTYP = MAKEFP) of the GAMESS program.74 Only theparameters responsible for electrostatic and polarization terms(multipole expansion, damping parameters, and static polar-izabilities) were computed at the Hartree−Fock level of theoryand with the 6-31G(d) basis set. The 6-311++G(3df,2p) basisset is usually recommended for computing polarization,dispersion, and exchange-repulsion terms.Exponential electrostatic damping of charges (SCREEN2) is

employed between the fragments;59 Gaussian-type damping ofcharges (SCREEN) is used to mitigate the charge-penetrationerrors in the QM−EFP interactions.25 Gaussian-type polar-ization damping controlled by a damping parameter is

employed between the fragments but not between the QMand EFP regions.59 The effects of electrostatic screening andpolarization damping on the QM/EFP energies are discussedbelow. The QM region is described at the same level of theoryas in the full QM and QM/MM calculations.

3.2.1. QM/MM Calculations. In the QM/MM calculations,we tested three different schemes. Following the same strategyas in QM/mEFP, we do not include the capping H-atom in theMM part. We use CHARMM’s force-field parameters for theMM point charges.68 This creates an additional charge (e.g.,+0.16 in the neutral GFP model system) in the MM part,making the total charge of the MM subsystem noninteger.There are several ways to handle this artifact of QM/MM. Forexample, we can add an extra −0.16 charge to the next atom(Cα in our case), or we can distribute that extra charge over thesame residue. In QM/mEFP, the charge is added to the nextcarbon (see Figure 1). We compared these two protocols andthe QM/MM calculations with noninteger charges for aprotonated GFP model system:

1. QM/MM; protocol 1: charge distributed over all atomsof the entire residue.

2. QM/MM; protocol 2: charge added to the next atom.3. QM/MM; protocol 3: no redistribution of charge;

noninteger total MM charge.

Table 1 shows that all QM/MM calculations overestimate VIEsrelative to full QM. Importantly, protocol 1 and protocol 2

produce very similar IEs, which indicates that IEs are not verysensitive to the details of charge redistribution, as long as thetotal charge is conserved. Because QM/mEFP uses a protocolsimilar to protocol 2, in all QM/MM calculations reportedbelow, we employed protocol 2.

4. RESULTS4.1. Electronic Properties of Model GFP Chromo-

phores. Table 2 summarizes the results for the GFP modelsystem with the anionic chromophore. The VDE of the barechromophore is 2.70 eV. The environment strongly stabilizesthe anionic chromophore leading to a significant blue shift(1.65 eV) in VDE. The VDE value in the full QM calculation is4.35 eV. As illustrated in Figure 5, the spin density in the fullQM calculations is localized on the chromophore, which meansthat QM/MM and QM/mEFP calculations with the QM partcomprising the chromophore can be directly compared to thefull QM and small QM calculations. The QM/MM calculationscapture the effect of the stabilization of the anionicchromophore reasonably well, yielding VDE of 4.39 eV,which is blue-shifted relative to the full QM value by 0.04eV. Similar to those for QM/MM, the QM/mEFP and QM/EFP VDEs and excitation energies are within 0.02−0.04 eVfrom the full QM results. Importantly, neither detachment nor

Table 1. VIE of the Neutral GFP Chromophore Surroundedby Four Nearby Amino Acid Residues Computed withωB97X-D/aug-cc-pVDZ Using Three Different QM/MMProtocols

calculation VIE, eV Koopmans IP (DFT), eV

full QM 9.887 10.095protocol 1 9.953 10.095protocol 2 9.943 10.068protocol 3 10.104 10.231

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excitation energies are sensitive to the choice of the mEFPfragmentation schemes.The analysis of the excited-state calculations requires care.

There are several types of excited states of chromophores in acondensed phase:4,75−77 (1) local excitations in which both theelectron and hole are confined to the chromophore; (2) charge-transfer states in which the electron is transferred between thechromophore and another neighboring group;76,77 (3) charge-transfer-to-solvent states in which the electron is transferredfrom the chromophore to a cavity, forming states resemblingthose of solvated electrons78,79 (in finite clusters, one oftenobserves surface states in which the target orbital correspondsto an electron on the surface). In addition to these physicalstates, one should also be aware of pseudocontinuum statescorresponding to the electron detached from the system. InCIS calculations, these artificial states will appear above theKoopmans IE/DE when the basis is sufficiently diffuse;80,81

they may spoil the description of the bright states that lie abovethe Koopmans onset.Obviously, in the QM/MM and QM/EFP calculations with

the QM part comprising the chromophore alone, only localexcitations corresponding to bound excited states can bedescribed correctly. Thus, it is important to carefully analyzethe character of the states in such calculations. Here, we focuson the bright ππ* state of the chromophore, which can beeasily identified by its large oscillator strength and by inspectingthe target MO.The Koopmans IE of the isolated anionic GFP chromophore

is 2.86 eV (HOMO energy, HF/aug-cc-pVDZ). Thus, all CISexcited states above this energy are embedded in the electron-detachment continuum, which spoils their description.81,82 FullQM CIS and SOS-CIS(D) calculations yield two lowest statesof different characters. One state is relatively dark and hascharge-transfer character, whereas the other state is bright andcorresponds to the local ππ* excitation on the chromophore.The leading MOs for the bright states are shown in SI. Wecarefully checked that the nature of the HOMO and the targetvirtual orbital involved in the bright transition remains the sameamong different schemes of calculation. Table 2 shows theenergies of the bright state. The energy of the second state,which cannot be correctly described in the QM/MM and QM/mEFP calculations with a small QM region, is given in the SI.Table 3 summarizes the results for the GFP model system

with the neutral chromophore. The VIE of the barechromophore is 7.40 eV. The interaction with the neighboringresidues leads to a blue shift of 2.5 eV (small QM vs full QM).

Relative to full QM, the QM/MM calculations overestimate theionization energy by 0.05 eV. QM/mEFP with the double-fragmentation scheme performs well, matching the blue shift inVIE. We also note that QM/mEFP with the Cα−Nfragmentation scheme yields the largest deviation (+0.14 eV)relative to full QM.In this system, the lowest excited state is the bright ππ* state

(the relevant orbitals are shown in the SI; as in the examplebefore, they are of the same character in different calculations).The effect of the environment on the excitation energies isnoticeable: the energy of the bright state is red-shifted by 0.24eV in the full QM calculation relative to that in the barechromophore (small QM). The QM/MM calculation over-estimates the excitation energy by 0.02 eV. The QM/EFPsuperfragment calculation underestimates the excitation energyby 0.01 eV. Different QM/mEFP schemes show similarperformance; they underestimate the full QM excitation energyby 0.01−0.02 eV.Using the anionic and neutral forms of the GFP, we

investigate the effects of electrostatic and polarization dampingon electronic properties. The results for VDEs are collected inTable 4. In Tables 2 and 3 discussed above, the QM/mEFPvalues were computed using polarization damping between theEFP fragments with the default parameter value of 0.6 andelectrostatic damping between the QM and EFP regions givenin eq 6 with parameters precomputed as described in Section 3.In Table 4, the results with the default polarization andelectrostatic damping are compared with the values obtainedwith (1) the reduced polarization damping parameter, whichcorresponds to a stronger damping of polarization energiesbetween the fragments, (2) polarization completely turned off,which corresponds to electrostatic embedding, and (3)electrostatic QM−EFP damping turned off. Although thedefault value of the polarization damping parameter is 0.6, wefind that smaller values of this parameter, that is, strongerdamping, are occasionally necessary for avoiding the polar-ization catastrophe. In particular, this often happens forstrongly interacting ionic species.83 Although in the systemsthat we considered here, we did not encounter difficultiesconverging polarization energies with the default dampingvalues, we explore the effect of polarization damping on IEs/DEs, in the case when the modification of damping parameteris required in more complex systems. Note that polarizationdamping is applied only to fragment−fragment polarization; theQM → EFP and EFP ← QM polarization interactions are notdamped.

Table 2. Electronic Properties of the Anionic GFPChromophore Surrounded by Four Nearby Amino AcidResiduesa

calculation VDE, eV Eex, eV ( f L)

small QM 2.70 2.76 (0.94)full QM 4.35 2.56 (1.52)QM/MM 4.39 2.57 (1.45)QM/EFP: superfragment 4.39 2.60 (1.45)QM/mEFP: Cα−C 4.39 2.60 (1.45)QM/mEFP: Cα−N 4.37 2.60 (1.45)QM/mEFP: C−N 4.38 2.60 (1.45)QM/mEFP: Cα−C and Cα−N 4.38 2.60 (1.45)

aVDEs computed using ωB97X-D/aug-cc-pVDZ. Excitation energiescomputed with SOS-CIS(D)/aug-cc-pVDZ; f L computed at the CISlevel. The lowest bright state is shown (see the text).

Table 3. Electronic Properties of the Neutral GFPChromophore Surrounded by Four Nearby Amino AcidResiduesa

calculation VIE, eV Eex, eV ( f L)

small QM 7.40 3.80 (1.05)full QM 9.89 3.56 (1.09)QM/MM 9.94 3.58 (1.08)QM/EFP: superfragment 9.90 3.55 (1.03)QM/mEFP: Cα−C 9.91 3.55 (1.03)QM/mEFP: Cα−N 10.03 3.54 (1.04)QM/mEFP: C−N 9.92 3.55 (1.03)QM/mEFP: Cα−C and Cα−N 9.89 3.55 (1.04)

aVIEs computed using ωB97X-D/aug-cc-pVDZ. Excitation energiescomputed with SOS-CIS(D)/aug-cc-pVDZ; f L computed at the CISlevel. The lowest bright state is shown (see the text).

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Comparing IEs computed with polarization turned offentirely or partially, we observe several interesting effects.The first observation is that polarization of the environmentplays a more significant role in the anionic than in the neutralchromophore, as illustrated by the VIE/VDE differencesbetween the default and polarization-off values in thesuperfragment calculations, which are 0.18 and 0.11 eV forthe anionic and neutral chromophores, respectively. Thesecond observation is that the polarization effect is smaller infragmented (mEFP) than in the superfragment (EFP)calculations. For example, in the anionic form, the change inVDEs due to polarization decreases from 0.18 eV in QM/EFPto 0.06−0.08 eV in QM/mEFP, and similarly in the neutralform. This is an interesting observation, which suggests thatneighboring fragments “depolarize” each other. This effectoccurs in QM/mEFP but not in superfragment calculations. (Insuperfragment calculations, the QM and EFP subsystemspolarize each other, but the EFP superfragment does notpolarize itself.) Finally, a moderate decrease in polarizationdamping to 0.3 does not affect the results within 0.01 eV.However, decreasing the damping parameter to 0.1 effectivelysuppresses polarization, i.e., these results become identical tothose with the polarization turned off.Electrostatic damping takes care of charge penetration

energy; it has been explored in detail for the EFP−EFPinteractions.59,60 The effect of electrostatic damping on theQM−EFP interactions is analyzed in this work for the firsttime. For both the anionic and neutral chromophores, theQM−EFP electrostatic damping shifts energy differences by0.01−0.05 eV, improving the agreement between the QM/EFPresults and the reference QM data. Similar shifts are observedfor different fragmentation schemes.To summarize, our analysis suggests that both electrostatic

damping and polarization interactions are important foraccurate description of VIEs/VDEs. For example, for theCα−C scheme in the anionic chromophore, the best (withdefault electrostatic and polarization damping) QM/mEFPcalculations underestimate VDE by 0.09 eV, whereas switchingoff electrostatic damping and polarization increases the errorsfurther, by 0.04 and 0.07 eV, respectively. The Cα−C scheme inthe neutral chromophore underestimates VDEs by 0.06 eV, andturning off electrostatic damping and polarization introducesadditional errors of 0.01 eV each.

4.2. mPlum Model System. The electronic properties forthe mPlum model system are collected in Table 5. The mPlum

chromophore features an extended π-system in which theconjugation extends beyond the imidazolinone ring and intothe side chain. The extended π-system is responsible for red-shifted absorption relative to the GFP chromophore.75 Theextended conjugation also leads to higher detachment energy ofmPlum, as compared to deprotonated GFP (Table 2); thereason for that was described by Ghosh et al.84 In this system,the chromophore is also anionic and the environment stronglystabilizes its ground state. Thus, we observe similar trends inTables 2 and 5.We begin by considering the trends in VDE. From Table 5,

we once again observe that the protein environment affects thedetachment energy significantly, resulting in 2.4 eV blue shift infull QM/MM relative to small QM. QM/MM captures theeffect of the environment well; VDE is overestimated by +0.04eV relative to full QM. QM/mEFP also performs very well. Wenote that within the same fragmentation scheme, QM/mEFP

Table 4. VDE and VIE (in eV) of the Anionic (Top) and Neutral (Bottom) GFP Chromophores Surrounded by Four NearbyAmino Acid Residues as a Function of Electrostatic and Polarization Damping in the EFP Regiona

calculation pol damp = 0.6 (default) pol damp = 0.3 pol damp = 0.1 pol off elec damp offb

anionic GFPc

QM/EFP: superfragment 3.93 3.75 3.88QM/mEFP: Cα−C 3.94 3.94 3.87 3.87 3.90QM/mEFP: Cα−N 3.92 3.92 3.86 3.86 3.87QM/mEFP: C−N 3.93 3.93 3.85 3.85 3.89QM/mEFP: Cα−C and Cα−N 3.93 3.93 3.87 3.87 3.88

neutral GFPd

QM/EFP: superfragment 9.67 9.56 9.66QM/mEFP: Cα−C 9.68 9.68 9.67 9.67 9.67QM/mEFP: Cα−N 9.81 9.81 9.79 9.79 9.80QM/mEFP: C−N 9.69 9.68 9.66 9.66 9.68QM/mEFP: Cα−C and Cα−N 9.66 9.66 9.65 9.65 9.65

aωB97X-D/6-31G(d) is used in all calculations. bDefault polarization (with damp = 0.6) is used. cFull QM: VDE = 4.03 eV, QM/MM: VDE = 3.97eV. dFull QM: VDE = 9.74 eV, QM/MM: VDE = 9.74 eV.

Table 5. Electronic Properties of the Anionic mPlumChromophore Surrounded by Four Nearby Amino AcidResidues and Watera

calculation VDE, eV Eex, eV ( f L) VEA, eV

small QM 3.20 2.30 (1.51) 2.03full QM 5.57 2.65 (0.91) −0.21

2.81 (0.64)QM/MM 5.61 2.53 (1.53) −0.20

QM/EFP: superfragment 5.60 2.53 (1.54) −0.22QM/mEFP: Cα−C 5.63 2.52 (1.54) −0.24pol off 5.51 2.51 (1.54) −0.08QM/mEFP: Cα−N 5.61 2.53 (1.54) −0.22pol off 5.52 2.51 (1.54) −0.07QM/mEFP: C−N 5.59 2.52 (1.54) −0.21pol off 5.52 2.51 (1.54) −0.08QM/mEFP: Cα−C and Cα−N 5.61 2.53 (1.54) −0.22pol off 5.51 2.51 (1.54) −0.07

aVDE and VEA computed with ωB97X-D/aug-cc-pVDZ. Excitationenergies computed with SOS-CIS(D)/aug-cc-pVDZ; f L computed atthe CIS level.

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overestimates VDE when polarization is on (highest deviation+0.06 eV compared to full QM) and underestimates VDE withpolarization switched off (the largest deviation from the fullQM result is −0.06 eV).In the bare chromophore (small QM), the lowest excited

state is the bright excited state of the ππ* character, and Eex =2.30 eV. In the full QM calculations, we observe two blue-shifted excited states of similar character (the oscillator strengthof the bright transition is distributed between the two). AllQM/MM and QM/mEFP calculations yield only one brightππ* state (which is the lowest excited state) carrying largeoscillator strength. The excitation energy is rather insensitive tothe fragmentation scheme (2.52−2.53 eV).In this model system, we also consider electron-attached

states. Although in the bare chromophore, such radical-dianionstate is not bound electronically (i.e., it has positive electronaffinity), it can be stabilized by interactions with the protein. Ina recent study of KillerRed (which has the same chromophoreas mPlum), such dianion-radical states were found to bestable.77 Figures 5d and 6a−d show spin densities for theelectron-attached states (the relevant MOs are shown in SI). Asone can see from the data in Table 5, this state has a similarcharacter across different calculation schemes. In Figure 6,inclusion of water in the QM part of small QM and QM/MMcalculations does not affect the spin density, thereby confirmingthe effect of water to be purely electrostatic.In this model system, the protein stabilizes the dianion state

leading to a change of 2.24 eV in VEA between the small QMand full QM calculations. The QM/MM and QM/mEFPresults are within 0.01 eV of the full QM value. The effect ofpolarization is more pronounced in the case of VEA: the effectof switching off polarization can be as large as 0.14 eV.4.3. T4 lysozyme Model System. Table 6 compares VDEs

computed with small QM, QM/MM, full QM, QM/EFP, and

QM/mEFP with different fragmentation schemes. In thissystem, we observe that QM/MM performs poorly andunderestimates VDE by 0.33 eV, as compared to full QM.On the other hand, the QM/mEFP schemes perform reallywell, with the largest error being −0.08 eV relative to full QM.It is also noteworthy that in QM/EFP and QM/mEFP,polarization plays a significant role here: when polarization isturned off completely, the QM/EFP and QM/mEFP resultschange by 0.2−0.3 eV and become very similar to the QM/MMdata. A significant contribution of polarization can berationalized by the charge density of the phenolate beinglargely localized on O−, which leads to strong polarizinginteractions with the GLU102 residue. Overall, the scheme withtwo fragments per amino acid (Cα−C and Cα−N) shows the

Figure 6. (a−d) Spin densities in the electron-attached dianionic mPlum chromophore model system in different calculations (see the text and Table5).

Table 6. VDEs (eV) of the Phenolate Bound to the T4Lysozyme System, ωB97X-D/aug-cc-pVDZ

system VDE, eV Koopmans IE (DFT)

small QM 2.12 2.14full QM 1.90 2.12QM/MM 1.57 1.58

QM/mEFP: superfragment 1.85 2.04pol off 1.57 1.58QM/mEFP: Cα−C 1.82 2.01pol off 1.51 1.52QM/mEFP: Cα−N 1.92 2.12pol off 1.64 1.63QM/mEFP: C−N 1.84 2.01pol off 1.61 1.61QM/mEFP: Cα−C and Cα−N 1.90 2.10pol off 1.60 1.61

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best performance for this system: it reproduces the full QMresult exactly.

5. CONCLUSIONSThe extension of the EFP method to polypeptides, calledmEFP, has been developed and validated by computingexcitation, ionization, and electron-attachment energies ofthree biologically relevant systems, GFP/mPlum chromophoresand phenolate in their natural surroundings. In the mEFPscheme, the polypeptide is split into smaller fragments, and theEFP parameters for each fragment are computed independentlyof other fragments. Four different fragmentation schemes havebeen tested; the most consistent results were obtained with theschemes in which the polypeptide is split either along the Cα−C bond or along both the Cα−C and Cα−N bonds.In all systems considered here, QM/mEFP accurately

reproduces excitation, ionization, and electron-attachmentenergies, provided that the electronic process is localized inthe QM subsystem. Discrepancies between the full QMcalculations and QM/mEFP calculations in all but one casedo not exceed 0.1 eV. Polarization interactions have anoticeable effect on the electronic properties of biologicalchromophores. Turning off polarization in QM/mEFPdeteriorates the accuracy and leads to additional errors of upto 0.3 eV. Short-range damping of electrostatic interactionsbetween the QM and EFP subsystems, which corrects classicalmultipole expansion for charge-penetration energy, also bringsthe QM/mEFP results into closer agreement with the referencefull QM values. In most cases, the errors of QM/mEFP againstQM/EFP (no fragmentation of the EFP part) are 0.01−0.02eV; the largest error is 0.06 eV.The QM/mEFP approach provides a rigorous way to

incorporate polarization embedding into studies of biologicalsystems. The developed mEFP algorithm can be generalized toother polymers and flexible molecules, which will be exploitedin future studies.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpcb.6b04166.

Computational details, relevant Cartesian geometries,molecular orbitals, and spin densities, as well asadditional results PDF

■ AUTHOR INFORMATIONCorresponding Authors*E-mail: krylov@usc.edu. Phone: +1(213)740-4929 (A.I.K.).*E-mail: lslipchenko@purdue.edu. Phone: +1(765)494-5255(L.V.S.).

Author Contributions#A.A. and P.K.G. contributed equally.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work has been supported by the U.S. National ScienceFoundation (CHE-1566428 to A.I.K. and CHE-1465154 toL.V.S.). D.G. thanks CSIR XIIth Five Year Plan on MultiscaleModeling and DAE-BRNS for funding.

■ REFERENCES(1) Reichardt, C. Solvents and Solvent Effects in Organic Chemistry, 2nded.; VCH: Weinheim, 1990.(2) DeFusco, A.; Minezawa, N.; Slipchenko, L. V.; Zahariev, F.;Gordon, M. S. Modeling solvent effects on electronic excited states. J.Phys. Chem. Lett. 2011, 2, 2184−2192.(3) Ghosh, D.; Isayev, O.; Slipchenko, L. V.; Krylov, A. I. The effectof solvation on vertical ionization energy of thymine: frommicrohydration to bulk. J. Phys. Chem. A 2011, 115, 6028−6038.(4) Bravaya, K. B.; Khrenova, M. G.; Grigorenko, B. L.; Nemukhin,A. V.; Krylov, A. I. Effect of protein environment on electronicallyexcited and ionized states of the green fluorescent proteinchromophore. J. Phys. Chem. B 2011, 115, 8296−8303.(5) Ghosh, D.; Roy, A.; Seidel, R.; Winter, B.; Bradforth, S.; Krylov,A. I. A first-principle protocol for calculating ionization energies andredox potentials of solvated molecules and ions: theory and applicationto aqueous phenol and phenolate. J. Phys. Chem. B 2012, 116, 7269−7280.(6) Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular ElectronicStructure Theory; Wiley & Sons: West Sussex, England, 2000.(7) Bartlett, R. J. Coupled-cluster theory and its equation-of-motionextensions. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2011, 2, 126−138.(8) Krylov, A. I. The spin-flip equation-of-motion coupled-clusterelectronic structure method for a description of excited states, bond-breaking, diradicals, and triradicals. Acc. Chem. Res. 2006, 39, 83−91.(9) Epifanovsky, E.; Wormit, M.; Kus, T.; Landau, A.; Zuev, D.;Khistyaev, K.; Manohar, P.; Kaliman, I.; Dreuw, A.; Krylov, A. I. Newimplementation of high-level correlated methods using a generalblock-tensor library for high-performance electronic structurecalculations. J. Comput. Chem. 2013, 34, 2293−2309.(10) Epifanovsky, E.; Zuev, D.; Feng, X.; Khistyaev, K.; Shao, Y.;Krylov, A. I. General implementation of resolition-of-identity andCholesky representations of electron-repulsion integrals withincoupled-cluster and equation-of-motion methods: theory and bench-marks. J. Chem Phys. 2013, 139, 134105.(11) Hirata, S.; Head-Gordon, M. Time-dependent density functionaltheory within the Tamm-Dancoff approximation. Chem. Phys. Lett.1999, 314, 291−299.(12) Rhee, Y. M.; Head-Gordon, M. Scaled second orderperturbation corrections to configuration interaction singles: efficientand reliable excitation energy methods. J. Phys. Chem. A 2007, 111,5314−5326.(13) Dreuw, A.; Head-Gordon, M. Single-reference ab initio methodsfor the calculation of excited states of large molecules. Chem. Rev.2005, 105, 4009−4037.(14) Tomasi, J.; Mennucci, B.; Cammi, R. Quantum mechanicalcontinuum solvation models. Chem. Rev. 2005, 105, 2999−3093.(15) Marenich, A. V.; Cramer, C. J.; Truhlar, D. G. Perspective onfoundations of solvation modeling: the electrostatic contribution to thefree energy of solvation. J. Chem. Theory Comput. 2008, 4, 877−887.(16) Cramer, C. J.; Truhlar, D. G. On the performance of continuumsolvation methods. A comment on universal approaches to solvationmodeling. Acc. Chem. Res. 2009, 42, 493−497.(17) Mewes, J. M.; You, Z. Q.; Wormit, M.; Kriesche, T.; Herbert, J.M.; Dreuw, A. Experimental benchmark data and systematic evaluationof two a posteriori, polarizable-continuum corrections for verticalexcitation energies in solution. J. Phys. Chem. A 2015, 119, 5446−5464.(18) Marenich, A. V.; Ho, J.; Coote, M. L.; Cramer, C. J.; Truhlar, D.G. Computational electrochemistry: prediction of liquid-phasereduction potentials. Phys. Chem. Chem. Phys. 2014, 16, 15068−15106.(19) Warshel, A.; Levitt, M. Theoretical studies of enzymaticreactions: dielectric electrostatic and steric stabilization of thecarbonium ion in the reaction of lysozyme. J. Mol. Biol. 1976, 103, 227.(20) Singh, U. C.; Kollman, P. A. A combined ab initio quantummechanical and molecular mechanical method for carrying outsimulations on complex molecular systems: applications to theCH3Cl+Cl

− exchange reaction and gas phase protonation ofpolyethers. J. Comput. Chem. 1986, 7, 718−730.

The Journal of Physical Chemistry B Article

DOI: 10.1021/acs.jpcb.6b04166J. Phys. Chem. B 2016, 120, 6562−6574

6572

(21) Monard, G.; Merz, K. M. Combined quantum mechanical/molecular mechanical methodologies applied to biomolecular systems.Acc. Chem. Res. 1999, 32, 904−911.(22) Lin, H.; Truhlar, D. G. QM/MM: what have we learned, whereare we, and where do we go from here? Theor. Chem. Acc. 2007, 117,185−199.(23) Sulpizi, M.; Raugei, S.; VandeVondele, J.; Carloni, P.; Spirk, M.Calculations of redox properties: understanding short- and long-rangeeffects in rubredoxin. J. Phys. Chem. B 2007, 111, 3969−3976.(24) Day, P. N.; Jensen, J. H.; Gordon, M. S.; Webb, S. P.; Stevens,W. J.; Krauss, M.; Garmer, D.; Basch, H.; Cohen, D. An effectivefragment method for modeling solvent effects in quantum mechanicalcalculations. J. Chem. Phys. 1996, 105, 1968−1986.(25) Gordon, M. S.; Freitag, M. A.; Bandyopadhyay, P.; Jensen, J. H.;Kairys, V.; Stevens, W. J. The effective fragment potential method: aQM-based MM approach to modeling environmental effects inchemistry. J. Phys. Chem. A 2001, 105, 293−307.(26) Gordon, M. S.; Slipchenko, L.; Li, H.; Jensen, J. H. In AnnualReports in Computational Chemistry; Spellmeyer, D. C., Wheeler, R.,Eds.; Elsevier: 2007; Vol. 3, pp 177−193.(27) Ghosh, D.; Kosenkov, D.; Vanovschi, V.; Williams, C.; Herbert,J.; Gordon, M. S.; Schmidt, M.; Slipchenko, L. V.; Krylov, A. I. Non-covalent interactions in extended systems described by the effectivefragment potential method: theory and application to nucleobaseoligomers. J. Phys. Chem. A 2010, 114, 12739−12745.(28) Ghosh, D.; Kosenkov, D.; Vanovschi, V.; Kaliman, J.; Flick, I.;Shao, Y.; Gilbert, A. T. B.; Slipchenko, L. V.; Krylov, A. I. Effectivefragment potential method in Q-Chem: a guide for users anddevelopers. J. Comput. Chem. 2013, 34, 1060−1070.(29) Gordon, M. S.; Smith, Q. A.; Xu, P.; Slipchenko, L. V. Accuratefirst principles model potentials for intermolecular interactions. Annu.Rev. Phys. Chem. 2013, 64, 553−578.(30) Kaliman, I. A.; Slipchenko, L. V. LIBEFP: a new parallelimplementation of the effective fragment potential method as aportable software library. J. Comput. Chem. 2013, 34, 2284−2292.(31) Kaliman, I. A.; Slipchenko, L. V. Hybrid MPI/OpenMPparallelization of the effective fragment potential method in the libefpsoftware library. J. Comput. Chem. 2015, 36, 129−135.(32) Arora, P.; Slipchenko, L. V.; Webb, S. P.; Defusco, A.; Gordon,M. S. Solvent induced frequency shifts: configuration interactionsingles combined with the effective fragment potential method. J. Phys.Chem. A 2010, 114, 6742−6750.(33) Slipchenko, L. V. Solvation of the excited states ofchromophores in polarizable environment: orbital relaxation versuspolarization. J. Phys. Chem. A 2010, 114, 8824−8830.(34) Kosenkov, D.; Slipchenko, L. V. Solvent effects on the electronictransitions of p-nitroaniline: a QM/EFP study. J. Phys. Chem. A 2011,115, 392−401.(35) Olsen, J. M.; Aidas, K.; Kongsted, J. Excited states in solutionthrough polarizable embedding. J. Chem. Theory Comput. 2010, 6,3721−3734.(36) Sneskov, K.; Schwabe, T.; Christiansen, O.; Kongsted, J.Scrutinizing the effects of polarization in QM/MM excited statecalculations. Phys. Chem. Chem. Phys. 2011, 13, 18551−18560.(37) Sneskov, K.; Schwabe, T.; Kongsted, J.; Christiansen, O. Thepolarizable embedding coupled cluster method. J. Chem. Phys. 2011,134, 104108.(38) Olsen, J. M. H.; Steinmann, C.; Ruud, K.; Kongsted, J.Polarizable density embedding: a new QM/QM/MM-based computa-tional strategy. J. Phys. Chem. A 2015, 119, 5344−5355.(39) Richard, R. M.; Herbert, J. M. A generalized many-bodyexpansion and a unified view of fragmented-based methods inelectronic structure. J. Chem. Phys. 2012, 137, 064113.(40) Richard, R. M.; Lao, K. U.; Herbert, J. M. Aiming for benchmarkaccuracy with the many-body expansion. Acc. Chem. Res. 2014, 47,2828−2836.(41) Lao, K. U.; Herbert, J. M. Accurate and efficient quantumchemistry calculations for noncovalent interactions in many-body

systems: the XSAPT family of methods. J. Phys. Chem. A 2015, 119,235−252.(42) Mullin, J. M.; Roskop, L. B.; Pruitt, S. R.; Collins, M. A.;Gordon, M. S. Systematic fragmentation method and the effectivefragment potential: an efficient method for capturing molecularenergies. J. Phys. Chem. A 2009, 113, 10040−10049.(43) Pruitt, S. R.; Steinmann, C.; Jensen, J. H.; Gordon, M. S. Fullyintegrated effective fragment molecular orbital method. J. Chem.Theory Comput. 2013, 9, 2235−2249.(44) Pruitt, S. R.; Bertoni, C.; Brorsen, K. R.; Gordon, M. S. Efficientand accurate fragmentation methods. Acc. Chem. Res. 2014, 47, 2786−2794.(45) Wen, S.; Nanda, K.; Huang, Y.; Beran, G. J. O. Practicalquantum mechanics-based fragment methods for predicting molecularcrystal properties. Phys. Chem. Chem. Phys. 2012, 14, 7578−7590.(46) Gordon, M. S.; Fedorov, D. G.; Pruitt, S. R.; Slipchenko, L. V.Fragmentation methods: a route to accurate calculations on largesystems. Chem. Rev. 2012, 112, 632−672.(47) Antes, I.; Thiel, W. Adjusted connection atoms for combinedquantum mechanical and molecular mechanical methods. J. Phys.Chem. A 1999, 103, 9290−9295.(48) Shao, Y.; Kong, J. YinYang atom: a simple combined ab initioquantum mechanical molecular mechanical model. J. Phys. Chem. A2007, 111, 3661−3671.(49) Assfeld, X.; Rivail, J.-L. Quantum chemical computations onparts of large molecules: the ab initio local self consistent field method.Chem. Phys. Lett. 1996, 263, 100−106.(50) Murphy, R. B.; Philipp, D. M.; Friesner, R. A. A mixed quantummechanics/molecular mechanics (QM/MM) method for large-scalemodeling of chemistry in protein environments. J. Comput. Chem.2000, 21, 1442−1457.(51) Sahu, N.; Gadre, S. R. Molecular tailoring approach: a route forab initio treatment of large clusters. Acc. Chem. Res. 2014, 47, 2739−2747.(52) Zhang, D. W.; Zhang, J. Z. H. Molecular fractionation withconjugate caps for full quantum mechanical calculation of protein-molecule interaction energy. J. Chem. Phys. 2003, 119, 3599−3605.(53) Zhang, D. W.; Chen, X. H.; Zhang, J. Z. H. Molecular caps forfull quantum mechanical computation of peptide-water interactionenergy. J. Comput. Chem. 2003, 24, 1846−1852.(54) Soderhjelm, P.; Ryde, U. How accurate can a force fieldbecome? A polarizable multipole model combined with fragment-wisequantum-mechanical calculations. J. Phys. Chem. A 2009, 113, 617−627.(55) Grigorenko, B. L.; Nemukhin, A. V.; Topol, I. A.; Burt, S. K.Modeling of biomolecular systems with the quantum mechanical andmolecular mechanical method based on the effective fragmentpotential technique: proposal of flexible fragments. J. Phys. Chem. A2002, 106, 10663−10672.(56) Minikis, R. M.; Kairys, V.; Jensen, J. H. Accurate intraproteinelectrostatics derived from first principles: an effective fragmentpotential method study of the proton affinities of lysine 55 andtyrosine 20 in turkey ovomucoid third domain. J. Phys. Chem. A 2001,105, 3829−3837.(57) Molina, P. A.; Li, H.; Jensen, J. H. Intraprotein electrostaticsderived from first principles: divide-and-conquer approaches for QM/MM calculations. J. Comput. Chem. 2003, 24, 1971−1979.(58) Steinmann, C.; Fedorov, D. G.; Jensen, J. H. The effectivefragment molecular orbital method for fragments connected bycovalent bonds. PLoS One 2012, 7, e41117.(59) Slipchenko, L. V.; Gordon, M. S. Damping functions in theeffective fragment potential method. Mol. Phys. 2009, 107, 999−1016.(60) Slipchenko, L. V.; Gordon, M. S. Electrostatic energy in theeffective fragment potential method: theory and application tobenzene dimer. J. Comput. Chem. 2007, 28, 276−291.(61) Steinmann, C.; Fedorov, D. G.; Jensen, J. H. Mapping enzymaticcatalysis using the effective fragment molecular orbital method:towards all ab initio biochemistry. PLoS One 2013, 8, e60602.

The Journal of Physical Chemistry B Article

DOI: 10.1021/acs.jpcb.6b04166J. Phys. Chem. B 2016, 120, 6562−6574

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(62) He, X.; Zhu, T.; Wang, X.; Liu, J.; Zhang, J. Z. H. Fragmentquantum mechanical calculation of proteins and its applications. Acc.Chem. Res. 2014, 47, 2748−2757.(63) Ganesh, V.; Dongare, R. K.; Balanarayan, P.; Gadre, S. R.Molecular tailoring approach for geometry optimization of largemolecules: energy evaluation and parallelization strategies. J. Chem.Phys. 2006, 125, 104109.(64) Other schemes exist such as the IUPAC definition with a theCα-C(O) bond cut or a definition splitting each amino acid into thebackbone and the side-chain moiety. This fragmentation leads toasymmetrical fragments, which are not convenient for computingDMA distribution because they result in delocalized multipoles.However, the cut is between atoms of same chemical nature. Naturalfragmentation solves the problem of asymmetry, but an extremelypolar bond is cut.(65) Other methods exist in the literature to guarantee a net integercharge; the expand-remove and scale (ERS) is similar to our methodup to the removal step. The monopoles located on all caps are thenscaled to reflect the net integer charge of the entire protein.56,57 Twoother methods developed by the same group are the remove andexpand (RE) method and the expand-collect-and-correct (ECC)method.57

(66) Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T. B.; Wormit, M.;Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X.; et al.Advances in molecular quantum chemistry contained in the Q-Chem 4program package. Mol. Phys. 2015, 113, 184−215.(67) Krylov, A. I.; Gill, P. M. W. Q-Chem: an engine for innovation.Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2013, 3, 317−326.(68) Foloppe, N.; MacKerell, A. D. All-atom empirical force field fornucleic acids: I. Parameter optimization based on small molecule andcondensed phase macromolecular target data. J. Comput. Chem. 2000,21, 86−104.(69) Reuter, N.; Lin, H.; Thiel, W. Green fluorescent proteins:empirical force field for the neutral and deprotonated forms of thechromophore. Molecular dynamics simulations of the wild type andS65T mutant. J. Phys. Chem. B 2002, 106, 6310−6321.(70) Bogdanov, A. M.; Acharya, A.; Titelmayer, A. V.; Mamontova, A.V.; Bravaya, K. B.; Kolomeisky, A. B.; Lukyanov, K. A.; Krylov, A. I.Turning on and off photoinduced electron transfer in fluorescentproteins by π-stacking, halide binding, and Tyr145 mutations. J. Am.Chem. Soc. 2016, 138, 4807−4817.(71) Chai, J.-D.; Head-Gordon, M. Long-range corrected hybriddensity functionals with damped atom-atom dispersion interactions.Phys. Chem. Chem. Phys. 2008, 10, 6615−6620.(72) Faraji, S.; Krylov, A. I. On the nature of an extended Stokes shiftin the mPlum fluorescent protein. J. Phys. Chem. B 2015, 119, 13052−13062.(73) Liu, L.; Baase, W. A.; Michael, M. M.; Matthews, B. W. Use ofstabilizing mutations to engineer a charged group within a ligand-binding hydrophobic cavity in t4 lysozyme. Biochemistry 2009, 48,8842−8851.(74) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.;Gordon, M. S.; Jensen, J. H.; Koseki, S.; Mastunaga, N.; Nguyen, K. A.;Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. General atomicand molecular electronic structure system. J. Comput. Chem. 1993, 14,1347−1363.(75) Bravaya, K.; Grigorenko, B. L.; Nemukhin, A. V.; Krylov, A. I.Quantum chemistry behind bioimaging: insights from ab initio studiesof fluorescent proteins and their chromophores. Acc. Chem. Res. 2012,45, 265−275.(76) Grigorenko, B. L.; Nemukhin, A. V.; Morozov, D. I.; Polyakov, I.V.; Bravaya, K. B.; Krylov, A. I. Toward molecular-level character-ization of photo-induced decarboxylation of the green fluorescentprotein: accessibility of the charge-transfer states. J. Chem. TheoryComput. 2012, 8, 1912−1920.(77) Vegh, R. B.; Bravaya, K. B.; Bloch, D. A.; Bommarius, A. S.;Tolbert, L. M.; Verkhovsky, M.; Krylov, A. I.; Solntsev, K. M.Chromophore photoreduction in red fluorescent proteins is

responsible for bleaching and phototoxicity. J. Phys. Chem. B 2014,118, 4527−4534.(78) Ballard, R. E.; Jones, J.; Sutherland, E.; Chun, B. L.Photoelectron spectra of anions in solutions and the ionization energyof the solvated electron. Chem. Phys. Lett. 1983, 97, 419−421.(79) Gorelsky, S. I.; Kotov, V. Yu; Lever, A. B. P. Vertical ionizationenergies and electron affinities of ions in solution from outer-spherecharge transfer transition energies. Inorg. Chem. 1998, 37, 4584−4588.(80) Epifanovsky, E.; Polyakov, I.; Grigorenko, B. L.; Nemukhin, A.V.; Krylov, A. I. Quantum chemical benchmark studies of theelectronic properties of the green fluorescent protein chromophore: I.Electronically excited and ionized states of the anionic chromophore inthe gas phase. J. Chem. Theory Comput. 2009, 5, 1895−1906.(81) Zuev, D.; Bravaya, K. B.; Crawford, T. D.; Lindh, R.; Krylov, A.I. Electronic structure of the two isomers of the anionic form of p-coumaric acid chromophore. J. Chem. Phys. 2011, 134, 034310.(82) Bravaya, K. B.; Zuev, D.; Epifanovsky, E.; Krylov, A. I. Complex-scaled equation-of-motion coupled-cluster method with single anddouble substitutions for autoionizing excited states: theory, imple-mentation, and examples. J. Chem. Phys. 2013, 138, 124106.(83) Rankin, B. M.; Hands, M. D.; Wilcox, D. S.; Fega, K. R.;Slipchenko, L. V.; Ben-Amotz, D. Interactions between halide anionsand a molecular hydrophobic interface. Faraday Discuss. 2013, 160,255−270.(84) Ghosh, D.; Acharya, A.; Tiwari, S. C.; Krylov, A. I. Towardsunderstanding the redox properties of model chromophores from thegreen fluorescent protein family: an interplay between conjugation,resonance stabilization, and solvent effects. J. Phys. Chem. B 2012, 116,12398−12405.

The Journal of Physical Chemistry B Article

DOI: 10.1021/acs.jpcb.6b04166J. Phys. Chem. B 2016, 120, 6562−6574

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