VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

12320

Efficient Generation of Flexible-Monomer Intermolecular Potential Energy Surfaces

Garold Murdachaew, Krzysztof Szalewicz, and Robert BukowskiDepartment of Physics and Astronomy, University of Delaware, Newark, Delaware 19716

(Received 24 September 2001; published 11 March 2002)

A new method of generating flexible-monomer intermolecular interaction potentials has been proposed.The method, based on symmetry-adapted perturbation theory, extends a rigid-monomer potential into aflexible-monomer one at a cost negligible compared to performing calculations on a full-dimensionalgrid (i.e., including internal degrees of freedom of monomers). The nonrigidity effects are accountedfor by density-overlap integrals and by asymptotic expansion coefficients. Results for a model system(Ar-HF) demonstrate that the method recovers a substantial portion of these effects.

DOI: 10.1103/PhysRevLett.88.123202 PACS numbers: 34.20.Gj

Molecular dynamics and Monte Carlo simulations ofcondensed phase and biomolecules are a thriving field ofcurrent research. Most of these simulations use empiri-cal intermolecular interaction potentials fitted within suchsimulations to reproduce a subset of experimental data.Clearly, first-principle simulations can provide a more reli-able and complete understanding of a wide range of physi-cal phenomena, but have been so far very limited. Thereare two possible approaches to performing first-principlesimulations. The first approach is to compute the ener-gies “on-the-fly,” i.e., at each step of a simulation. Sucha calculation involves all the molecules included in thesimulation and therefore can be done only with the mostinexpensive methods such as density functional theory,which are known to be unreliable for intermolecular in-teraction energies. The other approach is to build the totalpotential from two-body and possibly higher-body poten-tials computed a priori using a high-level ab initio methodof molecular structure calculation. If the monomers are as-sumed rigid, the pair potentials are at most six dimensionaland it is possible to calculate them reliably for fairly largesystems (like dimethylnitramine dimer [1]). However, ifthe rigidity restriction is removed, the dimer potential for asystem containing N atoms becomes 3N 2 6 dimensional.Since the number of grid points needed to determine a po-tential grows exponentially with the number of degrees offreedom, the 3N 2 6 dimensionality makes calculations ofsuch potentials impossible at present, except for diatomicmonomers. While the rigid-monomer approximation ap-pears to work reasonably well, flexible-monomer potentialsare needed for high-accuracy work on clusters and to inves-tigate observables depending explicitly on intramonomercoordinates such as bond lengths in solid phase or infraredspectra of condensed phase and molecular clusters.

With k points per dimension, calculation of a flexible-monomer potential energy surface for a dimer would re-quire k3N26 grid points. The difficulty can be illustratedby the example of the water dimer potential, whose dimen-sionality increases from 6D to 12D if the monomers areallowed to flex. Thus, to generate a flexible potential com-parable in accuracy to the existing rigid-water SAPT-5spotential [2] (fitted to 2510 ab initio points, i.e., k equal

2-1 0031-9007�02�88(12)�123202(4)$20.00

to about 4) would require the calculation of over 16 3 106

points. At the present state of computer technology (about0.5 h per point) this would take almost 1000 CPU ? yr, atask sure to remain challenging for some time despite rapidadvances in technology.

Most of the published flexible-monomer potentialswere obtained from rigid-monomer potentials by using the“atom-following” approach. If a rigid-monomer potentialis in the form of a site-site fit, changes of internal geometryof a monomer naturally modify the interaction energy.However, it has been shown recently by our group [3]that the atom-following approach performs poorly and infact may predict changes opposite to actual ones. Thepresent Letter develops a new method of extrapolating therigid-monomer potentials to flexible-monomer potentialsbased on limiting the expensive ab initio calculations toindividual monomers only, resulting in k3NA26 1 k3NB26

scaling, where NX is the number of atoms in monomer X.The main ingredients of the extension of rigid-monomer

potentials proposed here are (i) the use of asymptoticvan der Waals constants dependent on intramonomercoordinates and (ii) accounting for the overlap (chargepenetration) and exchange effects by density-overlapintegrals. Only asymptotically can the interaction energybe determined completely from knowledge of monomers’properties: multipole moments and static and dynamicpolarizabilities (which give the van der Waals constants).In the region of van der Waals minimum and for shorterintermolecular center of mass separations R the overlapeffects significantly modify the asymptotic predictions.The reduction of the 3N 2 6 dimensional problem takingplace asymptotically is not possible for the overlap effects,nor for the exchange energy dominant at very small R.However, there exists an approximate proportionality ofthe exchange energy to the overlap integral S of electrondensities of monomers A and B (see, e.g., Refs. [4–6]).This proportionality allowed Price, Wheatley, and Hodges[5,6] to compute (rigid-monomer) first-order exchangecomponents of intermolecular potentials from S with theproportionality constant treated as an adjustable parameter.Since the calculation of the densities is a problem whichscales as k3NA26 1 k3NB26, the method will be practically

© 2002 The American Physical Society 123202-1

VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

applicable if a calculation of the density-overlap integralrequires significantly less time than a regular calculationof a given component, which happens to be the case.

The approach presented here exploits properties ofsymmetry-adapted perturbation theory (SAPT) of inter-molecular interactions [7]. In SAPT the clamped-nucleielectronic Schrödinger equation for two arbitraryclosed-shell atoms or molecules is solved perturbativelywith the unperturbed problem defined by noninteractingmonomers. The interaction energy is computed as thesum of the four fundamental components: electrostatic,exchange, induction, and dispersion. This property iscrucial for the present approach which would not havebeen possible with the supermolecular method that calcu-lates the interaction energy as a difference of the dimer’sand monomers’ energies. The currently programmedlevel of SAPT is approximately equivalent to the super-molecular fourth-order many-body perturbation theory,but computationally SAPT is somewhat faster. SAPTis a well-established method that has been used [7] to

123202-2

determine interaction energies of many weakly boundcomplexes at high levels of electron correlation, and itsconvergence properties have been extensively studied [8].

The details of the method proposed here are as follows.Consider the SAPT exchange energy Eexch. We define the“scaling factor” Kexch as the ratio

Kexch�R,vA,vB;JA,JB� �Eexch�R,vA,vB;JA,JB�

S�R,vA,vB;JA,JB�,

(1)

where

S�R,vA,vB;JA,JB� �Z

rA�r;JA�rB�r;JB� dr

(2)

is the density-overlap integral. In these expressions vX de-notes the Euler angles defining the orientation of monomerX in a dimer-embedded coordinate system, JX is the setof internal coordinates of each monomer, and the electrondensity for monomer X is defined as

rX�r1;JX� � NXe

Xs

Z· · ·

ZjCX �x1, x2, . . . , xNX

e;JX�j2 dx2 · · · dxN , (3)

1

where NXe is the number of electrons in monomer X, xi

denote the space, ri , and spin coordinates of the ith elec-tron, the integrations include spin summations, the explicitsummation is over the spin of electron one, and CX is thewave function of monomer X. The densities rA and rB

in Eq. (2) are placed relative to each other at positions de-fined by the coordinates R, vA, and vB. If Kexch were toa good approximation constant (i.e., independent of all co-ordinates), calculations of Eexch at all points but one couldbe replaced by much simpler calculations of S (one pointis necessary to determine the value of the scaling factor)[4–6]. We propose to utilize the approximate constancyof Kexch in a somewhat weaker way, assuming only thatKexch is approximately independent of JX . Thus, the en-ergies Eexch and the integrals S are computed for all dimerconfigurations keeping monomers rigid, i.e., taking JX inEq. (1) equal to the equilibrium geometries of monomersJX,e (at most a 6D problem), which gives a set of numbersKexch�R,vA,vB;JA,e,JB,e�. These numbers are then ap-plied to predict exchange energies for distorted monomersusing overlap integrals S calculated on a full-dimensionalgrid:

Eexch�R, vA,vB;JA,JB� � Kexch�R,vA,vB;JA,eJB,e�3 S�R,vA,vB;JA,JB� ,

(4)

where the tilde is used to indicate the approximate char-acter of the quantity defined in this way. Figure 1 showsthe excellent performance of this approximation for theAr-HF dimer at the van der Waals minimum separationR � 6.55 bohrs. We have chosen Ar-HF as our test sys-tem since it is one of the very few dimers for which a fullyflexible potential exists [9]. The exchange energy shown is

the sum of the first-order exchange correction E�1�exch con-

sidered in Refs. [10] and [9] and a small term denotedby dEHF

int which collects induction and exchange-induction

FIG. 1. Dependence of the first-order exchange energy ofAr-HF on the angle u between the HF axis and the vectorconnecting Ar with the center of mass of HF (u is equal tozero for the Ar-H-F configuration) for intermonomer separationR � 6.55 bohrs. The solid line is the ab initio computedexchange energy. The dash-dotted line is the exchange energyfor the equilibrium rHF. The dotted line is the prediction fromEq. (4).

123202-2

VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

effects of the order higher than second. (The second-orderexchange corrections were grouped with the induction anddispersion corrections, as explained below.) In additionto the ab initio energies computed on the 3D grid (solidlines), the energies for rHF � rHF,e � 1.7328 bohrs, themonomer equilibrium distance, are shown (as dash-dottedlines) to indicate the size of the nonrigidity effects. Thepanels differ by the values of rHF which are close to theclassical turning points for y � 0 and y � 1 vibrations ofHF. Similarly good results were obtained for all other val-ues of R considered in Ref. [9], even for R � 5.5 bohrs.

An analogous procedure has been applied to the overlapcomponent of the dispersion energy defined as

O�2�disp � E

�2�disp 2

NmaxXn�6

Cn,disp�Rn 1 E�2�exch-disp . (5)

The upper limit of the summation was Nmax � 12. Theinduction energy was treated similarly except that theasymptotic induction was damped using the standard Tang-Toennies damping function with site-site damping pa-rameters (and site-site asymptotic coefficients) obtainedfrom the induction energy site-site fit of Ref. [3]. It wasfound that including the second-order exchange energycorrections, E

�2�exch-disp and E

�2�exch-ind, in the definitions of

O�2�disp and O

�2�ind, respectively, was helpful in canceling the

“spherical” overlap term which is particularly large inthe short-range induction energy. The O�2� componentsextrapolated in the same way as the first-order exchangeenergy were added to the flexible asymptotic expansions(center of mass expansion [9] for dispersion and site-siteexpansion [3] for induction) to get 3D second-order ener-gies. Figure 2 shows the predictions of the induction plusexchange-induction energy at R � 6.55 bohrs. Again,the predictions are fairly accurate and the dispersion plusexchange-dispersion energies (not shown) are predictedeven better. Only at rHF � 2.1032 bohrs and u � 0±,where H gets closest to Ar, does the absolute error becomenon-negligible.

The electrostatic component is relatively simple andcan be computed very inexpensively ab initio on a full-dimensional grid. The expression for the electrostatic en-ergy is [7]

E�1�elst�R,vA,vB;JA,JB� �

ZZrtot

A �r1;JA� jr1 2 r2j21

3 rtotB �r2;JB� d3r1 d3r2 ,

(6)

where the total charge density of molecule X with nucleiof charges Za at positions Ra is defined as r

totX �r;JX � �P

a Zad�r 2 Ra� 2 rX�r;JX�.For calculations of the electrostatic energies and the

density-overlap integrals we used the program GMUL [11]which expands the densities that are bilinear functions oforbitals into a set of Cartesian Gaussian functions dis-tributed at several sites within a molecule. This leads to

123202-3

FIG. 2. Dependence of the induction plus exchange-inductionenergy of Ar-HF on u for R � 6.55 bohrs. The solid and dash-dotted lines are analogous to those described in the captionof Fig. 1. The dotted line is the prediction made using the Sintegrals and rHF-dependent van der Waals constants.

significant reductions of computer time in calculations ofthe S integrals and the electrostatic energies. The depen-dence of the results on the length of this expansion and onthe number of sites has been investigated and will be pub-lished separately. The electrostatic energies computed byGMUL were compared with those computed by the SAPTprogram in the same basis set, reaching essentially exactagreement (with fairly compact GMUL expansions).

Next we had to address the problem of computing theelectrostatic energies and the density-overlap integrals us-ing monomer-centered basis sets (MCBS) only. Such ba-sis sets suffer from too fast decay of the tails of wavefunctions at large distances from a molecule, includingdistances corresponding to the van der Waals minima sepa-rations. This problem was circumvented in the 1970s byapplying dimer-centered basis sets (DCBS) which repre-sent a monomer wave function using both the basis set ofthis monomer and of its interacting partner [12]. How-ever, DCBS obviously cannot be used in our approachsince we want the densities to be calculated only once foreach monomer. To achieve reliable results in an MCBS,the size of the basis set on the HF monomer had to beincreased compared to the basis used in ab initio SAPTcalculations (no increase was needed for Ar). About a fac-tor of 2 increase in the size of the MCBS reduced the dif-ference in zeroth-order electrostatic energies between theMCBS and the original DCBS value to a mere 2%. Anincrease of the MCBS by a factor of 2 results — for iden-tical monomers — in a time of calculation identical to thatin the DCBS.

123202-3

VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

FIG. 3. Dependence of the total interaction energy of Ar-HFon u for R � 6.55 bohrs. The solid and dash-dotted lines areanalogous to those described in the caption of Fig. 1. The dottedline is the total energy predicted as the sum of the extrapolatedcomponents.

Figure 3 shows the total interaction energies at R �6.55 bohrs. The agreement of the predicted interactionenergy (from extrapolated components) with the referenceresult is very good. Clearly, the major part of the nonrigid-ity effect is recovered by our procedure. Only at rHF �2.1032 bohrs and u � 0± is the performance less satisfac-tory, due to the inaccuracy of the induction component.Even at that point about 60% of the nonrigidity effect isreproduced.

The agreement between the predicted interaction ener-gies and the ab initio values is even better for larger R. Forsmaller R it gradually worsens since at those separationsthere occurs a significant cancellation between the indi-vidual components of the interaction energy that becomemuch larger in magnitude than the interaction energy it-self. Thus, although the relative errors of individual com-ponents remain small as R decreases, the absolute errorsincrease and their sum may become a significant fractionof the interaction energy. To improve the prediction in thatregion, the interaction energy should be computed ab initioat some number of grid points with varying monomer ge-ometries and these points should be used to adjust thepotential.

Using only the ab initio energies computed on the 2D,rHF,e grid and the proposed extrapolation method, we gen-erated points on a 3D grid [9] and obtained an analyticflexible potential by fitting these points as in Ref. [9]. This

123202-4

potential was tested by computing the rovibrational ener-gies of Ar-HF which were found to be in reasonable agree-ment with those obtained from the reference 3D potential[9]. In particular, over 60% of the y � 0 ! 1 and y �0 ! 2 redshifts were recovered.

In summary, we have proposed a method of predict-ing flexible-monomer potentials from rigid-monomer po-tentials which involves negligible computational effort pergrid point compared to a conventional ab initio calculation.In contrast to the popular atom-following approach, theproposed method recovers a major fraction of the nonrigid-ity effect for deformations corresponding to the ground andlowest excited vibrational states of monomers. For the wa-ter dimer example, the reduction in CPU time relative to aconventional ab initio calculation is about 3 orders of mag-nitude. Moreover, the time of calculations of the density-overlap integrals increases very slowly with the size ofmonomers, so that the reduction will be even more signifi-cant for larger systems. With a single grid point calculationtaking of the order of 1 CPU ? sec, the calculation for the16 3 106 points needed would require abut half a year ofCPU time, a reasonable task on a modestly large multipro-cessor system. Work in progress on the water dimer givesencouraging results.

This research was supported by NSF Grant No. CHE-9982134. We thank Bogumil Jeziorski for reading andcommenting on the manuscript.

[1] R. Bukowski, K. Szalewicz, and C. F. Chabalowski, J. Phys.Chem. A 103, 7322 (1999).

[2] G. C. Groenenboom, E. M. Mas, R. Bukowski,K. Szalewicz, P. E. S. Wormer, and A. van der Avoird,Phys. Rev. Lett. 84, 4072 (2000).

[3] G. Murdachaew and K. Szalewicz, Faraday Discuss. 118,121 (2001).

[4] S. Kita, K. Noda, and H. Inouye, J. Chem. Phys. 64, 3446(1976).

[5] S. L. Price and R. J. Wheatley, Mol. Phys. 69, 507 (1990).[6] M. P. Hodges and R. J. Wheatley, Chem. Phys. Lett. 326,

263 (2000).[7] B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev.

94, 1887 (1994); B. Jeziorski and K. Szalewicz, in En-cyclopedia of Computational Chemistry, edited by P. vonRague Schleyer et al. (Wiley, New York, 1998), p. 1376.

[8] K. Patkowski, B. Jeziorski, and K. Szalewicz, J. Mol.Struct. (THEOCHEM) 547, 293 (2001).

[9] M. Jeziorska, P. Jankowski, K. Szalewicz, and B. Jeziorski,J. Chem. Phys. 113, 2957 (2000).

[10] V. F. Lotrich, H. L. Williams, K. Szalewicz, B. Jeziorski,R. Moszynski, P. E. S. Wormer, and A. van der Avoird,J. Chem. Phys. 103, 6076 (1995).

[11] R. J. Wheatley and J. B. O. Mitchell, J. Comput. Chem. 15,1187 (1994).

[12] H. L. Williams, E. M. Mas, K. Szalewicz, and B. Jeziorski,J. Chem. Phys. 103, 7374 (1995).

123202-4