Development of additive isotropic site potentialfor exchange-repulsion energy, based onintermolecular perturbation theory

Daisuke Yokogawa, Hirofumi Sato, Sergey Gusarov, and Andriy Kovalenko

Abstract: We have developed an additive spherical site potential for exchange-repulsion energy by applying the local den-sity approximation in Hilbert space, the local-site approximation, and the s-type auxiliary basis set to the equation derivedfrom intermolecular perturbation theory. The method efficiently addresses the decomposition of molecular interactionsderived from quantum chemistry into additive spherical site potentials, required as force field input in a statistical-mechanical, reference interaction site model (RISM and 3D-RISM), molecular theory of solvation. The present method re-produces the exchange-repulsion energy between simple molecules obtained from quantum chemical calculations.

Key words: exchange-repulsive potential, intermolecular perturbation theory, quantum chemical calculation, force field,reference interaction site model.

Resume : On a developpe un nouveau potentiel de site spherique et additif pour l’energie de repulsion d’echange qui estobtenu en appliquant l’approximation de densite locale de l’espace de Hilbert, l’approximation de site local, ainsi que l’en-semble de base auxiliaire de type s a l’equation derivee de la theorie de la perturbation intermoleculaire. La methode tientcompte exactement de la decomposition des interactions moleculaires derivees de la chimie quantique en potentiels de sitespheriques additifs requis comme donnee du champ de force en mecanique statistique, le modele de site d’interaction dereference (MSIR et MSIR-3D), theorie moleculaire de solvatation. La presente methode reproduit l’energie de repulsiond’echange entre les molecules simples obtenues a l’aide de calculs de la chimie quantique.

Mots-cles : potentiel de repulsion d’echange, theorie de la perturbation intermoleculaire, calcul de chimie quantique,champ de force, modele du site d’interaction de reference.

[Traduit par la Redaction]

IntroductionA large number of studies to develop force fields for pre-

dicting the interactions between molecules have been per-formed over the last three decades.1 In a standard forcefield, interactions are split up into terms classified, for ex-ample, as electrostatic interaction, dispersion interaction,and exchange interaction. Expressions for these interactionsare constructed under the assumption that they can betreated as a sum of additive spherical potentials dependenton the distance R between the constituent atoms or interac-tion sites but not on the molecular orientations. This type offorce field makes it possible to study a large system such asa biosystem by molecular simulation with reasonable com-

putational cost. Nowadays, many parameter sets have beenproposed.2–4

The most important term in the force field at long rangeis the electrostatic interaction when a molecule is polarized.This interaction has been widely studied for a long time.Fortunately, this interaction is sufficiently well-approxi-mated with point charges assigned on each atom (pointcharge model) and described as a sum of additive isotropic,Coulomb potentials 1/R. Nowadays, atomic charges areeasily assessed from quantum chemical calculations.5,6

At short range, the exchange interaction becomes impor-tant. In a standard force field, this interaction is modelledwith additive isotropic potentials as well, such as 1/R12.

Received 9 July 2009. Accepted 21 August 2009. Published on the NRC Research Press Web site at canjchem.nrc.ca on 28 November2009.

We dedicate this paper to Professor Tom Ziegler, our dear collaborator, with compliments and best wishes of health, prosperity, andscientific joy and further achievements.

D. Yokogawa and H. Sato. Department of Molecular Engineering, Graduate School of Engineering, Kyoto University, Nishikyo-ku,Kyoto 615-8510, Japan.S. Gusarov. National Institute for Nanotechnology, National Research Council of Canada, 11421 Saskatchewan Drive, Edmonton,AB T6G 2M9, Canada.A. Kovalenko.1 National Institute for Nanotechnology, National Research Council of Canada, 11421 Saskatchewan Drive, Edmonton,AB T6G 2M9, Canada; Department of Mechanical Engineering, University of Alberta, 4-9 Mechanical Engineering Building,Edmonton, AB T6G 2G8, Canada.

1Corresponding author (e-mail: andriy.kovalenko@nrc-cnrc.gc.ca).

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However, determination of its parameters is much more in-volved than for the electrostatics. One reason is that the in-teraction originates from the Pauli exclusion principle,which requires an accurate nonlocal, multideterminant repre-sentation of the wave function and thus makes it very diffi-cult to derive an additive isotropic potential directly fromquantum mechanics. Therefore, parameter sets for exchangeinteraction are usually empirically adjusted to fit somechemical properties. Although this approach works well formany systems, it is not easy to improve a force field system-atically. Theoretical methods to evaluate the exchange po-tential based on ab initio calculations have also beenproposed.7–11 One of the well established theories is inter-molecular perturbation theory (IMPT).1,7,8 The resultingmethod can evaluate the energy very accurately.9,10 How-ever, this is computationally expensive because of the neces-sity to compute two-electron integrals of the total system.

There have been some approaches to derive the exchange-repulsive potential in a less empirical way, based on ab ini-tio treatment.12–16 In the effective fragment potential (EFP)method, a very simple equation was derived based onIMPT.17,18 The potential between molecules is composed offundamental intermolecular interactions, similar to theSIBFA (sum of interactions between fragments ab initiocomputed) method.19 The parameter of each interactionterm is generated in a single ab initio calculation; therefore,the integrals required in the calculation are greatly simpli-fied, compared to those in IMPT.

In this work, we derived a simple exchange-repulsionenergy potential based on IMPT. This method has the ad-vantage of being fully ab initio, without any semiempiricalparameters. The derivation focuses on the following threegoals: (1) construct the total potential from additive atomicinteraction potentials; (2) have the atomic potentials spheri-cally symmetric; (3) tabulate the site potentials on a spatialgrid directly from the matrix elements of the operatorsprojected onto the augmented spherical basis set, ratherthan calculate the site contributions by brute-force orienta-tional averaging of the six-dimensional intermolecularpotential obtained from symmetry-adapted perturbation

theory or its derivatives. Such site potentials are typicallyrequired in a self-consistent field coupling of quantumchemical methods with statistical-mechanical, moleculartheory of solvation,20 such as KS-DFT/3D-RISM-KH21,22

and CASSCF/3D-RISM-KH.23 As classical force fields typ-ically use terms constructed along the lines of require-ments (1) and (2), the present strategy will be helpful tosystematically improve the classical force field, based onab initio calculations.

Method

Under the condition of orthogonality of the orbitals insidethe isolated molecules A and B, exchange-repulsion energybetween the molecules can be expressed with intermolecularoverlap integrals of atomic orbitals (AOs),7,8,18

½1�

Eexch ¼ � 1

2

Xmd

Xng

PAmdP

Bnghmnjdgi �

1

2

Xmd

Xng

PAmdP

Bnghmjni VA

dg þ GAdg þ VB

dg þ GBdg

h iþ 1

4

Xmh

Xng

PAmhPB

nghmjniXdq

PAdqhdjgiðVB

hq þ JBhqÞ þ

Xdq

PBdqhhjdiðVA

qg þ JAqgÞ

" #

� 1

8

Xmh

Xng

Xdq

X3u

PAmhPB

ngPAdqP

B3uhmjnihdj3ihhqjugi

� 1

8

Xmd

Xgh

PAmdP

Bghhdjgi

Xnq

PAnqhhjqiKB

mn þXnq

PBnqhmjqiKA

hn

" #

þ 1

2

Xmh

Xnq

Xgu

Xd3

PAmhPB

nqPAguPB

d3hhjqihuj3ihmnjgdi

¼ EexchS2 þ Eexch

S4;P4

Fig. 1. Exchange-repulsion energy of the model system, calculatedat levels Eexch

S2 , EexchS2 þ Eexch

S4;P4, and EXR. The exponent of ABS is 2.0.

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where PX is the density matrix for molecule X (X = A orB), <m|n> is the overlap matrix, VXmn is the nucleus attractionintegral in the field of molecule X, and <mn|dg> is the elec-tron repulsion integral. JX and KX are the Coulomb and ex-change matrices and GX is defined as

½2� GXmn ¼ JX

mn �1

2KXmn

The exchange-repulsion energy EexchS2 to the second order

of <m|n> has a familiar form obtained in previous work.7

EexchS4;P4 is the exchange-repulsion energy to the fourth order

of <m|n> and to the fourth order of PX.

Equation [1] can be simplified by density fitting. Theelectron density is expressed with AOs as follows:

½3� rXðrÞ ¼Xmn

PXmnfmðrÞfnðrÞ

This value is fitted by the auxiliary basis set ~f with coef-ficients d,

½4� ~r XðrÞ ¼Xn

dXn

~f nðrÞ ~f nðrÞ

Fitting r(r) with ~rðrÞ using proper auxiliary basis set(ABS) ~f can reproduce the effect of considerable off-diago-nal elements of the density matrix. The use of d and ~f leadsto the following equation:

Fig. 2. Exchange-repulsion energies between H2O and molecule X (X = H2O, NH3, CH4, Cl–, and Na+) calculated by the present method(solid line), EFP (dashed line), KM (dotted line), and LJ (dash-dotted line). (a) water oxygen site approaching molecule X (approach I);(b) water hydrogen sites approaching molecule X (approach II).

Yokogawa et al. 1729

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½5�

Eexch ¼ � 1

2

Xm

Xn

dAmdB

n h ~m ~n j ~m ~n i � 1

2

Xm

Xn

dAmdB

n h ~m j~n i VAmn þ bGA

mn þ VBmn þ bGB

mn

h iþ 1

4

Xm

Xn

dAmdB

n h ~m j~n iXd

dAd h~d j~n iðVB

md þ bJB

mdÞ þXd

dBd h ~m j~d iðVA

dn þ bJA

dnÞ" #

� 1

8

Xm

Xn

Xd

X3

dAmdB

n dAd dB

3 h ~m j~n ih~d j~3 ih ~m ~d j~3 ~n i

� 1

8

Xm

Xg

dAmdB

g h ~m j ~g iXn

dAn h ~g j~n ibK B

mn þXn

dBn h ~m j~n ibK A

gn

" #

þ 1

2

Xm

Xn

Xg

Xd

dAmdB

n dAg dB

d h ~m j~n ih ~g j~d ih ~m ~n j ~g ~d i

The matrices bGX, bJX, and bKX

are given by

½6� bGX

mn ¼ bJX

mn �1

2bK X

mn

½7� bJX

mn ¼Xg

dXg h ~m ~n j ~g ~g i

½8� bK X

mn ¼Xg

dXg h ~m ~g j ~g ~n i

Our first goal is to derive the equation for the decomposi-tion of the exchange-repulsion energy

½9� Eexch ¼XI2A

XJ2B

EexchIJ

where I and J refer to atoms belonging to molecules A andB, respectively. Based on eq. [5], the component Eexch

IJ is de-fined as

½10�

EexchIJ � � 1

2

Xm2I

Xn2J

dAmdB

n hmnjmni �1

2

Xm2I

Xn2J

dAmdB

n hmjni VðmÞmn þ GðmÞmn þ V ðnÞmn þ GðnÞmn

h iþ 1

4

Xm2I

Xn2J

dAmdB

n hmjniXd2I

dAd hdjniðV ðnÞmd þ JðnÞmd Þ þ

Xd2I

dBd hmjdi V

ðmÞdn þ J

ðmÞdn

� �" #

� 1

8

Xm2I

Xn2J

Xd2I

X32J

dAmdB

n dAd dB

3 hmjnihdj3ihmdj3ni �1

8

Xm2I

Xg2J

dAmdB

g hmjgiXn2I

dAn hgjniKðgÞmn þ

Xn2J

dBn hmjniKðmÞgn

" #

þ 1

2

Xm2I

Xn2J

Xg2I

Xd2J

dAmdB

n dAg dB

d hmjnihgjdihmnjgdi

The matrix elements V ðnÞmg , GðnÞmg , JðnÞmg , and KðnÞmg are given by

½11� V ðnÞmg ¼ �ZIn m1

jr� RIn j

���� ����n� �

½12� GðnÞmg ¼ JðnÞmg �1

2KðnÞmg

½13� JðnÞmg ¼Xh

dInh hmgjhhi

½14� KðnÞmg ¼Xh

dInh hmhjhgi

where In is the atom to which AO cn belongs, ZI is the

atomic number of atom I, and fdInh g is the density coeffi-cient which is assigned to atom In.

To achieve our second goal, EexchIJ defined in eq. [10] has

to be evaluated only in terms of the distance between atomsI and J. Applying s-type ABS makes each component inEexchIJ very simple. For example, the two-electron integralhmnjgdi in the last term of eq. [10] is expressed as

½15� hmnjgdi ¼ffiffiffiffiffiffiffiffiffiffiaman

pam þ an

24 353

2ffiffiffiffiffiffiffiffiffiffiagad

pag þ ad

24 353

2

� 8

RABerf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðam þ anÞðag þ adÞam þ an þ ag þ ad

vuut24 35

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where orbitals fn and fm are assigned to atom I, fg and fd

to atom J, and ac (c = n, m, g, and d) is the index of theexponent of fc. The strategy for preparing such an ABSand determining the d coefficients was described in detailin the previous work.24,25

In statistical-mechanical studies of classical systems,many kinds of empirical potentials for exchange-repulsionenergy are employed. Interestingly, even a hard-sphere po-tential, infinite if atom cores overlap and zero otherwise,can account for some chemical properties of liquid sys-tems.1,26 This means that the atomic radius (the positionwhere the potential becomes very large) is a very importantpoint in the exchange-repulsion energy. In the following dis-cussion, we focus on the shape and radius evaluated by thepresent method rather than on the numerical values, andcompare them with those obtained from ab initio calcula-tions and empirical potentials.

Application to water and simple moleculesand ions

Before applying eq. [9] to the molecular system, the ex-change-repulsive potential in a model system is evaluated.In this system, two atoms with Z = 2 are prepared and thenone Gaussian AO is employed for each atom,

½16� fðrÞ ¼ 2a

p

� 34

expð � aðr� RÞ2Þ

where R is the position of each atom. The number of elec-trons for each atom is two (each atom is neutral). With thebasis set, eqs. [1] and [9] reduce to the same equation. Fig-ure 1 presents the exchange-repulsive potentials for thecomponent Eexch

S2 and EexchS2 þ Eexch

S4;P4 versus the interatomicdistance r for a = 2.0. For comparison, the exchange-

repulsion energies without the truncation of the AO overlap(EXR, eq. [13] in 18) are also calculated and shown in Fig. 1.The exchange-repulsion energy calculated as Eexch

S2 becomeszero at r = 0. Because the higher term, larger than the sec-ond order of AO overlap, is truncated in Eexch

S2 , it gets farfrom EXR in the region where the AO overlap is large. How-ever, the component Eexch

S4;P4 improves the potential in that re-gion.

The exchange-repulsion energies between H2O and mole-cules H2O, NH3, CH4, Cl–, and Na+ were calculated fromeq. [9]. The geometry optimization of H2O, NH3, and CH4,and calculation of the exchange-repulsion energy were per-formed at the HF/6–311G* level. For comparison, the ex-change-repulsion energy was also evaluated by using theKitaura–Morokuma (KM) analysis,27 the EFP,12 and the Len-nard-Jones potential. The KM analysis is one of the methodsused in decomposing the total intermolecular interactionenergy into specific energy components. Because the KManalysis is widely employed to analyze intermolecular inter-actions and its definition is very clear, we took the KM ex-change-repulsion energy as a reference. It should be notedthat the surfaces evaluated by EFP show some areas of neg-ative values in the panels of water–water and water–NH3 inFig. 2. This effect is probably due to the fact that the equa-tion employed in EFP is constructed from many interactionterms,18 and the values of the exchange-repulsion potentialcome from a cancellation of the large positive and negativeterms. The reasonable balance might have been lost in theregions where the value evaluated by the EFP implementa-tion we used is negative. The repulsive part of the Lennard-Jones potential was calculated using the SPC-like potential,28

with a correction concerning the Lennard-Jones parametersof the hydrogen sites (s = 1.0 A, 3 = 0.056 kcal/mol) forH2O, and the OPLS-AA all-atom models29 for NH3, CH4,Cl–, and Na+. All calculations were performed by using ourin-house code implemented in the GAMESS software pack-age.30

The exchange-repulsion energy surfaces were calculatedfor H2O approaching molecule X in two directions: the oxy-gen site approaching molecule X (approach I), and the hy-drogen site approaching molecule X (approach II).Figures 2a and 2b present the corresponding energy surfa-ces; the reaction coordinates for approaches I and II arealso depicted in the right upper panel of each figure. Exceptfor Na+�H2O in approach II, the shapes calculated by thepresent method are close to those evaluated by using quan-tum chemical calculation and standard classical potentials.The strategy of MK analysis is summarized well by Mo etal.31 This also shows that the atomic radii evaluated withthese methods are almost the same. Although the presentmethod somewhat overestimates the repulsion for Na+�H2Oin approach II, this does not lead to severe problems, be-cause the electrostatic interaction between Na+ and the posi-tively charged H site of H2O is dominant, compared to theexchange-repulsion energy.

To check the dependence of the results of the presentmethod on the basis set, we calculated the exchange-repulsion energy between water molecules in approach II atthe HF/(6–31G*, 6–311+G*, cc-pVDZ, cc-pVTZ, and aug-cc-pVTZ) levels. The resulting energy surfaces are shownin Fig. 3, including the energy at r = 2.6 A evaluated by the

Fig. 3. Basis set dependence of the exchange-repulsion energy be-tween water molecules in approach II, evaluated by the presentmethod. Also shown are the values of the energy at r = 2.6 A eval-uated by MK with cc-pVDZ, cc-pVTZ, and aug-cc-pVTZ, frombottom to top, respectively. The reaction coordinate is the same asin Fig. 2b for water molecules.

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KM analysis with cc-pVDZ, cc-pVTZ, and aug-cc-pVTZ.The calculation using the basis sets without diffuse functions(6–31G*, 6–311G*, cc-pVDZ, and cc-pVTZ) gave almostthe same shape for the energy surface. The energies, whichbecome very large (r = 2.6 A), are very close to those calcu-lated by MK with aug-cc-pVTZ. On the other hand, the en-ergy surfaces evaluated with 6–311+G* and aug-cc-pVTZare somewhat repulsive. This is because the electron distri-bution determined by the present density-fitting approach isbroad, so the overlap of electrons between two waters be-comes large.

Conclusions

We have proposed an ab initio IMPT-based scheme to de-rive the exchange-repulsive intermolecular energy, brokendown into additive spherical-site potentials. This strategy iscapable of becoming an important bridge in a multiscalescheme that will produce classical force fields for molecularmodeling with high accuracy from ab initio calculations.

AcknowledgmentsWe gratefully acknowledge the financial support from the

Grant-in Aid for ‘‘Molecular Theory for Real Systems’’(461), from the Global COE Program ‘‘International Centerfor Integrated Research and Advanced Education in Materi-als Science’’ (No. B-09), and from the Ministry of Educa-tion, Culture, Sports, Science and Technology (MEXT) inJapan. D.Y. acknowledges the Grant-in Aid for JSPS Fel-lows, and thanks the National Institute for Nanotechnology(NINT), Alberta, Canada, for the hospitality during his visitthere. S.G. and A.K. are supported by the National ResearchCouncil (NRC) of Canada.

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This article has been cited by:

1. D. Yokogawa. 2012. Development of the isotropic site-site potential for exchange repulsion energy and combination with theisotropic site-site potential for electrostatic part. The Journal of Chemical Physics 137:20, 204101. [CrossRef]

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