Appeared in

Theory and Applicationsof Computational ChemistryThe First Forty Years

Editors

clifford e. dykstra

Department of ChemistryIndiana University-Purdue University Indianapolis (IUPUI)Indianapolis, IN, U.S.A.

gernot frenking

Fachbereich ChemiePhillips-Universitat MarburgMarburg, Germany

kwang s. kim

Department of ChemistryPohang University of Science and TechnologyPohang, Republic of Korea

gustavo e. scuseria

Department of ChemistryRice UniversityHouston, TX, U.S.A.

2005

Contents

1 Introduction 1047

2 Prehistory: Before Computers 1048

3 Antiquity: the Sixties 1049

3.1 Supermolecular Methods . . . . . . . . . . . . . . . . . . . . . 10503.2 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . 1052

4 The Middle Ages: Era of Mainframes 1054

4.1 Unexpanded Dispersion . . . . . . . . . . . . . . . . . . . . . . 10554.2 Multipole-Expanded Dispersion . . . . . . . . . . . . . . . . . 10584.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

5 Modern Times: Revolution and Democracy 1062

5.1 The SAPT Method . . . . . . . . . . . . . . . . . . . . . . . . 10635.2 The Coupled Cluster Method . . . . . . . . . . . . . . . . . . 10665.3 Latest developments . . . . . . . . . . . . . . . . . . . . . . . 1068

References 1073

A Relationship between dispersion and EABMP2

1087

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chapter 37

Forty years of ab initio calculations

on intermolecular forces

Paul E. S. Wormer and Ad van der AvoirdInstitute of Theoretical Chemistry, IMM,

University of Nijmegen, Toernooiveld 1,

6525 ED Nijmegen, The Netherlands.

Abstract

This review sketches the development of methods for the computation ofintermolecular forces; emphasis is placed on dispersion forces. The last fortyyears, which saw the birth, growth, and maturation of ab initio methods, arereviewed.

1 Introduction

Intermolecular forces, sometimes called non-covalent interactions, are causedby Coulomb interactions between the electrons and nuclei of the molecules.Several contributions may be distinguished: electrostatic, induction, dis-persion, exchange, that originate from different mechanisms by which theCoulomb interactions can lead to either repulsive or attractive forces be-tween the molecules. This review deals with the ab initio calculation ofcomplete intermolecular potential surfaces, or force fields, but we focus ondispersion forces since it turned out that this (relatively weak, but important)contribution took longest to understand and still is the most problematic incomputations. Dispersion forces are the only attractive forces that play arole in the interaction between closed-shell (1S) atoms. We will see how theunderstanding of these forces developed, from complete puzzlement about

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their origin, to a situation in which accurate quantitative predictions arepossible.

The subfield of quantum chemistry concerned with the computation ofintermolecular forces has always depended very much on computer technol-ogy, not unlike most of the other subfields. Because of this strong influence,we will divide the following history along the lines of hardware development.The first attempt of an ab initio calculation on the interaction between twoclosed-shell atoms was made in 1961. Rather than let the story begin there,we will first review briefly the precomputer era of the theory of intermolecularforces. Then the infancy of computers and computational quantum chem-istry will be reviewed, followed by the era dominated by mainframes. Wewill end with the present democratic times in which every household has atits disposal the power of a 1980 supercomputer and ordinary research groupspossess farms of powerful computers.

2 Prehistory: Before Computers

When on the 10th of July 1908 Kamerling Onnes and his coworkers saw themeniscus of liquid helium in their apparatus,1 it was proved to them that twohelium atoms attract each other—so that a liquid can be formed—but alsothat they repel each other—so that the liquid does not implode. Of course,this is what they had known all along, because their work was guided by thevan der Waals law of corresponding states, which gave them a fair idea ofthe temperature and pressure at which the liquefaction of helium would takeplace. In deriving his law van der Waals (1873) had to assume the existenceof attractive and repulsive forces.

Until the advent of quantum mechanics it was an enigma why two S-state atoms would repel or attract each other. Shortly after the introductionof the Schrodinger equation in 1926, Wang2 solved this equation perturba-tively for two ground-state hydrogen atoms at large interatomic distance R.Approximating the electronic interaction by a Taylor series in 1/R he foundan attractive potential with as leading term −C6/R

6. A few years laterEisenschitz and London (E&L)3 systematized this work by introducing aperturbation formalism in which the Pauli principle is consistently included.They showed that the intermolecular antisymmetrization of the electronicwave function (electron exchange), which is required by the Pauli principle,can give rise to repulsion. This is why the intermolecular repulsion is oftenreferred to as exchange (or Pauli) repulsion.

Considering distances long enough that intermolecular differential over-lap and exchange can be neglected (the so-called long-range regime) Wang

1049

and E&L showed that the same dipole matrix elements that give rise to tran-sitions in the monomer spectrum, appear in the equations for the interaction.E&L pointed out that these transition dipole moments are closely related tothe oscillator strengths arising in the classical theory of the dispersion oflight (associated with the names of Drude and Lorentz) and in the quan-tum mechanical dispersion theory of Kramers and Heisenberg. Oscillatorstrengths, being simply proportional to squares of transition moments, areknown experimentally, enabling E&L to give reasonable estimates of C6. In1930 London4 published another paper in which he coined the name “dis-persion effect” for the attraction between S state atoms, which is why it iscommon today to refer to these attractive long-range forces as “London” or“dispersion forces”.

Apropos of nomenclature: the forces between closed-shell molecules (ex-change repulsion, electrostatics, induction, and dispersion) are nowadays usu-ally referred to as van der Waals forces. A stable cluster consisting of closed-shell molecules bound by these forces is called a van der Waals molecule.This terminology was introduced in the early 1970s, see Refs. 5–9.

After the pioneering quantum mechanical work not much new ground wasbroken until computers and software had matured enough to try fresh attacks.In the meantime the study of intermolecular forces was mainly pursued bythermodynamicists who fitted model potentials, often of the Lennard-Jonesform:10 4ǫ[(σ/R)12 − (σ/R)6], to quantities like second virial coefficients, vis-cosity and diffusion coefficients, etc. Much of this work is described in theauthoritative monograph of Hirschfelder, Curtiss, and Bird,11 who, inciden-tally, also give a good account of the relationship of Drude’s classical workto that of London.

3 Antiquity: the Sixties

Around 1960 the computer began to enter quantum chemistry. This was thebeginning of a very optimistic era; expectations of the new tool were tremen-dous. All over the western world quantum chemists were appointed in thebelief that many of the problems in chemistry could be solved by computa-tion within a decade or so. However, computers and ab initio methods werenot received with this great enthusiasm by everyone. Coulson,12 one of theoutstanding quantum chemists of his days, stated in his after-dinner closingspeech of the June 1959 Conference on Molecular Quantum Mechanics inBoulder, Colorado: “It is in no small measure due to the success of these[Coulson here refers to ab initio] programs that quantum chemistry is in itspresent predicament.”

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3.1 Supermolecular Methods

At the same 1959 Boulder conference Ransil, working in the Chicago Labora-tory of Molecular Structure and Spectra, one of the leading quantum chem-istry groups of the time, announced a research program13 on the computationof properties of diatomic molecules. With the benefit of hindsight one cansay that his program was overambitious and far too optimistic, because heintended to use self-consistent field (SCF) methods with atomic orbital (AO)minimum basis sets, albeit of Slater type. The fourth paper14 of this researchprogram was devoted to He2. Here Ransil considered the dispersion-bounddimer as a molecule amenable to ordinary molecular computational methods.Nowadays this method is referred to as a “supermolecule” approach. Ransilwrites in his abstract that “remarkable good agreement with the availableexperimental data is obtained for distances greater than 1.5 A”. We nowknow that his van der Waals minimum was spurious and solely due to theso-called basis set superposition error (BSSE). This BSSE is the lowering ofthe energy of monomer A, caused by the distance dependent improvementof the basis by the approaching AOs on B, and vice versa: the basis of B isimproved by basis functions on A. How much the difficulties of ab initio cal-culations on intermolecular forces were still underestimated is witnessed byanother paper on He2, also originating from the Chicago group. Phillipson,15

attributing the deviation of the energy for R < 1.5 A to correlation effects,introduces configuration interaction (CI) including 10 to 64 configurations,but still uses a minimum basis set and does not correct for the BSSE.

Six years later Kestner16 published a paper, containing SCF-MO resultson He2, in which he stresses the importance of the choice of AO basis sets,even for systems as small as He2. Using large basis sets he finds completelyrepulsive curves by the SCF method. According to Kestner in 1968: “it isgenerally believed, but nobody has proved, that this should be the case fortwo closed-shell atoms”. This statement exhibits the great advance made inunderstanding ab initio results in the early sixties. Since Kestner used theChicago computer codes and thanks Chicago staff members (Roothaan, Ran-sil, Cade, and Wahl) for guidance and support, it is clear that the Chicagowork on programming ab initio codes for diatomics was instrumental in gain-ing this understanding, in contrast to Coulson’s doubts.

A correction of the BSSE appearing in supermolecular calculations wasproposed by Boys and Bernardi17 in 1970. A similar correction was alreadyapplied somewhat earlier by Jansen and Ros.18 At present, 35 years later,the “counterpoise” procedure of Boys and Bernardi is still regularly applied,although—especially for smaller systems—we now can afford AO basis setsthat are so large that the SCF counterpoise correction is essentially zero. In

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correlated supermolecular methods the counterpoise correction is usually stillneeded. In essence, Boys and Bernardi proposed to perform all calculations(energy of monomer A and B and energy of dimer A–B) in the same dimerbasis set by the same computational method. Although the sum of themonomer energies, which serves as zero point, becomes distance dependent,vast experience has shown19–22 that this procedure yields the most reliable(basis set independent) results.

As stated above, it was already known in 1968 (and confirmed by cal-culation) that the SCF method applied to He2 gives a purely repulsive in-teraction. Recall that by Lowdin’s definition23 the SCF energy serves as thezero of electron correlation, or in other words, the SCF method does not giveany correlation. Sinanoglu24 was the first to observe that interatomic sp paircorrelation yields London R−6 dispersion. By the converse of this findingit seems plausible that without interatomic correlation dispersion effects arenot accounted for. Since these effects contribute so significantly to the attrac-tion of S-state atoms, one may conjecture that for non-polar systems there isno binding without inclusion of interatomic correlation. And indeed, we willshow this below. As a matter of fact, Pauli repulsion is now usually takenfor granted and attention is focused usually on the explanation of observedminima in intermolecular potentials.

The fact that interatomic pair correlation gives dispersion was semi-quantitatively confirmed in a calculation25 on He2. This Kestner-Sinanogluwork on He2 gave a well depth of 4.32 K = 3.00 cm−1, which is about 2.5times lower than the presently accepted value. The discrepancy is due to aninadequate AO basis.

Sinanoglu’s method was later improved26 by adaptation of the pair func-tions to the spin-operator S2. The He2 potential was recomputed27 bythis method with the use of a much larger AO basis. This paper, anda simultaneous paper published in the same issue of Phys. Rev. Lett. byBertoncini and Wahl28 describing MCSCF calculations, are the first thatreport within one consistent supermolecule formalism a complete van derWaals curve that shows a physically meaningful well. In Ref. 27 the depthof this well is De = 12.0 K at the equilibrium distance Re = 2.96 A and inRef. 28 De = 11.4 K at Re = 2.99 A. The presently accepted values29,30 areDe = 11.008 ± 0.008 K and Re = 5.6 bohr = 2.963 A. The choice of config-urations in the MCSCF calculation was inspired by the London long-rangetheory.

So, while the 1960s started with the belief that SCF could give a completepotential curve for closed-shell atoms, at the end of the decade it was knownthat the inclusion of interatomic correlation is essential for obtaining thedispersion attraction. The new decade saw the light with the two papers

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just mentioned27,28 proving this quantitatively.

3.2 Perturbation Methods

Independent of the work on coding ab initio programs, other workers in the1960s carried further the torch of London. In the first place methods wereimproved to compute better long-range C6 coefficients and the correspondinghigher coefficients C8, C10, etc., in the expansion of the interaction energy−

∑

∞

n=6CnR−n.

On the other hand, people took a closer look at the symmetrized pertur-bation theory of Eisenschitz and London, which in principle can give a fullpotential energy surface, not just the long range of it.

A variety of techniques has been employed for the estimation of disper-sion coefficients. Good reviews are those by Dalgarno and Davison31 andDalgarno.32 The semi-empirical methods based on oscillator strengths fs

were refined by using sum rules for Cauchy moments. A Cauchy momentS(k) is defined by the following sum over monomer states ψs with energiesEs

S(k) =∑

s>0

fs (Es −E0)k,

where the oscillator strength fs is given as a squared matrix element of thedipole operator µ

fs =2

3(Es −E0)

∑

i=x,y,z

〈 ψ0 | µi | ψs 〉〈 ψs | µi | ψ0 〉.

The even Cauchy moments arise in the expansion of a frequency-dependentpolarizability α(ω),

α(ω) =∑

s>0

fs

(Es − E0)2 − ω2=

∞∑

k=0

S(−2k − 2)ω2k. (1)

The coefficient C6 in the long-range interaction energy between two moleculesA and B is given in terms of the oscillator strengths fA

s and fBs by

C6 =3

2

∑

ss′

fAs f

Bs′

(EAs − EA

0 )(EBs′ − EB

0 )(EAs − EA

0 + EBs′ −EB

0 ).

One can factorize the denominator of this expression by invoking the identity(∆X > 0 is an excitation energy on X):

1

∆A∆B(∆A + ∆B)=

2

π

∫

∞

0

( 1

∆2A + ω2

)( 1

∆2B + ω2

)

dω. (2)

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Casimir and Polder33 have shown that this identity, which can be provedeasily by contour integration, can be used to express the long-range coefficientC6 in terms of frequency-dependent monomer polarizabilities

C6 =3

π

∫

∞

0

αA(iω)αB(iω)dω.

Many values of fs are known empirically; their reliability can be checkedby the sum rules: S(−2) = α(0), S(0) = N (number of electrons) andS(−1) = 2

3〈 ψ0 | µ · µ | ψ0 〉.

Effectively summing the power series in Eq. (1) by means of differentPade approximants makes it possible to give upper and lower bounds on theC6 values. Much work in the 1960s was done on calculating such bounds, seeRef. 34 for more on this.

The 1960s also saw the first ab initio calculations of α(ω) by the time-dependent uncoupled Hartree-Fock (TDUHF) method35 and by the coupledTDCHF method.36

Intermolecular electron exchange does not play a role in the long-range,since all integrals that would arise by intermolecular antisymmetrization van-ish by virtue of vanishing intermolecular differential overlap. However, forshorter distances where this overlap may not be neglected, the electrons onthe monomers can no longer be distinguished and the wave functions mustbe antisymmetric under permutations of all electrons. As we saw earlier,Eisenschitz and London considered this problem as early as 1930 and it wasrevived in the late 1960s by Murrell, Randic and Williams,37 Hirschfelderand Silbey,38 Hirschfelder,39,40 van der Avoird,41–44 Murrell and Shaw,45 andMusher and Amos.46

From the Pauli principle follows that the projected function AABΦ0,rather than Φ0, should be considered as the correct zeroth-order wave func-tion in the perturbation theory of intermolecular interactions. Here AAB isthe usual intermolecular antisymmetrization operator and Φ0 = ΦA

0 ΦB0 is (the

lowest) eigenfunction of H(0) ≡ HA+HB, the sum of monomer Hamiltonians.We assume here that Φ0 is antisymmetric under monomer permutations, i.e.,AXΦ0 = Φ0 for X = A,B. Unfortunately, since intermolecular permutationsdo not commute with H(0), [AAB, H

(0)] 6= 0, it follows that AABΦ0 is not aneigenfunction of H(0). This has the consequence that conventional Rayleigh-Schrodinger (RS) perturbation theory is not applicable for those intermolec-ular distances where the effect of AAB is non-negligible. The RS pertur-bation treatment must be adapted to permutation symmetry. The workersjust quoted proposed procedures to achieve this symmetrization. From apractical point of view their theories can be divided into two categories:47

first project with AAB then perturb (“strong symmetry forcing”), or perturb

1054

first and project later (“weak symmetry forcing”). In strong symmetry forc-ing the symmetry operators enter the perturbation equations, significantlycomplicating their solution. In weak symmetry forcing the perturbed wavefunctions are obtained by minor modifications of RS perturbation theory:the operator AAB only enters overlap terms and perturbation energies.

Although symmetry-adapted perturbation theories were well studied inthe second half of the 1960s, numerical applications were scarce and restrictedto H+

2 and H2. See Ref. 48 for a review of the exchange perturbation theoriesup to the beginning of the 1970s.

4 The Middle Ages: Era of Mainframes

Around 1970 every self-respecting university possessed a mainframe com-puter (in the majority of cases an IBM 360, sometimes a CDC 6600). Thiswas usually placed in a stronghold well defended by brave knights (the com-puter center staff). A scientist who wanted access to the machine had tomaster a strange and difficult tongue (job control language) and to crossswords with computer personnel to conquer CPU cycles, RAM, tapes, anddisk space. This medieval state of affairs lasted until workstations arrived atthe end of the 1980s.

The development of ab initio methods, such as speeding up the com-putation of Gausssian integrals, improving convergence of SCF procedures,and theory and programming of correlation methods was vigorously pursuedon mainframes. Electron correlation can be included by configuration in-teraction (CI) or by coupled cluster (CC) methods. Especially the work onthe development of CC methods proved later to be significant for the studyof intermolecular forces, because the CC method, in contrast to the config-uration interaction (CI) method, is size-extensive. Size-extensivity in thethermodynamic sense of this word (energy linear in amount of substance)implies that in the limit of zero density the energy of a system of moleculesconverges to the sum of energies of the individual molecules. Much workon the coupled cluster doubles (CCD) method (a supermolecule correlationmethod) was performed in the 1960s and 1970s by Paldus and Cizek andtheir coworkers49–51 and from the late 1970s onward by Bartlett and cowork-ers,52,53 who added single excitations to the method. Also Pople54 recognizedthe importance of the CC method at a rather early stage.∗

However, in the 1970s the supermolecule correlation methods—and thecomputers on which they ran—were not yet powerful enough to have muchsignificance in the field of intermolecular forces, except for small systems

∗See for more on the development of the CC method Chapter 7 by Paldus in this book.

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like He2. It was already known that dimer SCF gives a fair description ofelectrostatic interactions (dipole-dipole, dipole-quadrupole, etc.), of induc-tion forces (dipole-induced-dipole etc.), and also of Pauli repulsion, but notof dispersion. So, for systems where dispersion was expected to be domi-nant, other paths than SCF supermolecule computations had to be followed.A well-known procedure was separate computation of SCF and perturba-tive dispersion (without exchange effects) and to add the two. Dispersionis known to be affected by exchange and so for shorter distance the dis-persion has to be damped.55–57 When the multipole-expanded form of thedispersion is used, this damping must also correct for the divergent characterof the multipole series. Instead of computing converged dimer SCF ener-gies, one often stopped the SCF procedure after the first cycle. This makessense when the start orbital set is the direct sum of the sets of occupiedMOs of the monomers. In that case the first SCF cycle gives the expecta-tion value N〈 AABΦA

0 ΦB0 | HA + HB + V AB | AABΦA

0 ΦB0 〉, where N is the

corresponding normalization constant and the intermolecular interaction op-erator V AB = H − HA − HB contains the Coulomb interactions betweenthe electrons and nuclei of different molecules. This expectation value ac-counts for exchange repulsion and electrostatic interaction, albeit withoutany intramolecular correlation. Induction effects are obtained by cycling thedimer SCF, but because this cycling introduces BSSE and the counterpoisecorrection was deemed fairly expensive, as it requires three calculations foreach geometry of the dimer, the neglect of induction was either accepted, orinduction was added later in the multipole expanded form.

4.1 Unexpanded Dispersion

Dispersion energy can be computed without invoking the multipole expan-sion. This was done in the 1970s by, among others, Kochanski,58,59 Jeziorskiand van Hemert,60 and by van Duijneveldt and coworkers.61 It is natu-ral to assume that the supermolecular second-order Møller-Plesset62 energyEAB

MP2 also accounts for dispersion energy. And, indeed, in their study onthe connection between Møller-Plesset energies and the perturbation theoryof intermolecular forces Cha lasinski and Szczesniak63 showed that for largeR the supermolecule second-order (MP2) energy becomes equal to a sum oftwo monomer MP2 energies plus an uncoupled HF dispersion term. Theyshowed further that for two monomers possessing permanent multipoles thelong-range MP2 energy also contains a correlation contribution to the elec-trostatic energy. In the Appendix we provide a short alternative derivationof the asymptotic (large R) limit of EAB

MP2 for the special case of two S-stateatoms. Figure 1 gives a diagrammatic representation of the connection be-

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Figure 1: Goldstone diagrams depicting the large-R behavior of EAB

MP2. See text fordefinition of orbital labels. Particle orbitals run upward, hole orbitals downward. Eachdashed line represents a two-electron integral. Closed lines are summed over. The first rowshows Coulomb and the second row exchange interactions. Diagrams in the first row haveh = 2 (two hole lines) l = 2 (two loops); diagrams in the second row have h = 2 and l = 1.The overall factor is (−1)l+h2l

w, where 2l is from spin integration. All diagrams, exceptthe one with the bar in the middle, have weight w = 1

2(because of a vertical symmetry

plane). An imaginary horizontal line in each diagram gives the energy denominator. Thediagrams in the first three columns give the MP2 energy of A–B, A, and B, respectively.The fourth column gives the London dispersion energy. In the large R limit integralscontaining differential overlap between A and B vanish.

tween dispersion and MP2 energy. The diagrams in this figure are similar tothose of Ref. 64, which was the first work to apply many-body diagrammatictechniques to symmetry-adapted perturbation theory. Note that the disper-sion energy and the Coulombic part of the supermolecule energy EAB

MP2 arerepresented by diagrams that are topologically the same, so that diagram-matically their relationship seems obvious. It is tacitly assumed, however,that for large R the dimer orbital energies are equal to monomer orbital en-ergies. This fact, which holds for S-state systems, is proved explicitly in theAppendix. The Appendix gives the following equation for the unexpandeddispersion energy,

Edisp = 4∑

ρ,r,σ,s

|〈 ρ(1)σ(2) | r−112 | r(1)s(2) 〉|2

ǫρ + ǫσ − ǫr − ǫs, (3)

where ρ is an occupied (‘hole’) and r a virtual (‘particle’) spatial orbital on A.The definition of σ and s on B is analogous. The denominator contains thecorresponding orbital energy differences. This equation can also be extractedfrom the fourth diagram in the first row of Fig. 1 by application of thediagrammatic rules.

In the mid 1970s valence bond studies65,66 on He2 and (C2H4)2 were per-formed. This work was based on valence bond structures (configurations)

1057

that account for most of the important dispersion effects. The VB struc-tures were constructed from pure monomer MOs, which are orthogonal oneach monomer but have intermolecular overlap. The exact, unexpanded,Hamiltonian was used and a secular problem was solved. Because of thenon-orthogonality of the orbitals, the main drawback of the VB method isthat only a relatively small subset of the electrons can participate in the bind-ing, while the majority of electrons reside in closed shells. The all-electronVB work65 on He2 brought to light very clearly the considerable size of basisset superposition errors (BSSE), not only in VB results, but also in full con-figuration interaction results. Until that time BSSE was mainly discussed atthe SCF level.

Two ab initio methods, which were well-known and much discussed inthe 1970s and 1980s, were the pair natural orbital CI (PNO-CI) methodand the coupled electron pair approximation (CEPA) method. They wereproposed by W. Meyer67 in 1973 and two years later improved by Ahlrichsand coworkers.68 In 1983 Burton and Senff69 applied the method of Ahlrichset al. to an analysis of the anisotropy of (H2)2 interaction near the minimumin the van der Waals interaction energy.

In Eq. (3) we find orbital energy differences in the denominator. This isdue to the fact that we took the Fock operator as the zeroth-order operator,as did Møller and Plesset62 in 1934. An alternative zeroth-order operatoris due to Epstein70 saved from oblivion by Nesbet.71 This operator can bewritten as72

H(0) =∑

I

| I 〉〈 I | H | I 〉〈 I |,

where | I 〉 is an excited Slater determinant consisting of localized HF or-bitals. A simplification, compatible with Eq. (3), is obtained by restrict-ing the summation to determinants that are singly excited on each of themonomers. Working out the energy denominators 〈 0 | H | 0 〉 − 〈 I | H | I 〉we find orbital energy differences, as in Eq. (3), but shifted by a few addi-tional Coulomb and exchange integrals.

The pros and cons of Møller-Plesset (MP) versus Epstein-Nesbet (EN)partitioning were of some interest all through the 1970s. Especially theFrench school58,59, 73–75 strongly preferred EN over MP partitioning, althoughlater French work72 criticizes the use of EN partitioning with delocalized or-bitals. See Kelly76 for a proof that EN partitioning gives an (infinite) numberof diagonal ladder diagrams in addition the diagrams accounted for by theMP partitioning. Today EN partitioning is rarely applied, mainly for prag-matic reasons, because most standard ab initio packages only have the MPoption.

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4.2 Multipole-Expanded Dispersion

Before 1970 the multipole expansion (by which we mean the expansion inpowers of 1/R) of the interaction operator V AB was usually truncated af-ter the R−3 dipole-dipole term, so that the only dispersion interaction termwas −C6R

−6. Around 1970 it became clear that this approximation was notsufficient and that more terms were needed. However, the straightforwardapplication of the Taylor expansion, and its natural formulation in terms ofCartesian tensors,77 soon becomes cumbersome. Nineteenth century poten-tial theory78,79 came to the rescue. In this theory the multipole series isrephrased in terms of associated Legendre functions, which enables a closedform of it. Multipole operators are defined as

QlXm =

∑

ξ∈X

ZξSlXm (~rXξ) −

NX∑

i=1

SlXm (~rXi) with X = A,B

where ~rXξ and ~rXi are the coordinates of the nuclei ξ, with charges Zξ, andthe electrons i of molecule X with respect to a frame with its origin on thenuclear center of mass of X. The function Sl

m(~r) ≡ rlC lm(r) is a regular solid

harmonic; C lm(r) is a Racah normalized spherical harmonic.

The intermolecular interaction operator is

V AB = −

NA∑

i=1

∑

β∈B

Zβ

riβ−

NB∑

j=1

∑

α∈A

Zα

rjα+

NA∑

i=1

NB∑

j=1

1

rij+

∑

α∈A

β∈B

ZαZβ

rαβ(4)

Introducing the notation ~rPQ for a vector pointing from P to Q we substituteinto Eq. (4)

~riβ = −~rAi + ~RAB + ~rBβ

~rjα = −~rBj − ~RAB + ~rAα

~rij = −~rAi + ~RAB + ~rBj

~rαβ = −~rAα + ~RAB + ~rBβ ,

where A and B are the nuclear centers of mass of the respective monomers.Upon using the expansions given in Eqs. (15) and (16) of the Appendix, weget the multipole expansion

V AB =∑

lAlBm

(−1)lB+m

(

2lA + 2lB2lB

)1/2

ΥlA+lB−m (~R)

[

QlA ⊗ QlB]lA+lB

m,

1059

where ~R = ~RAB and we recall from the Appendix that ΥlA+lB−m (~R) is propor-

tional to R−(lA+lB+1).When we substitute this expansion into the RS second-order expression,

we get a numerator that contains two Clebsch-Gordan coupled products oftransition matrix elements on A and B. They are of the type

[

〈 ΦA0 | QlA | ΦA

n 〉 ⊗ 〈 ΦB0 | QlB | ΦB

n′ 〉]lA+lB

m.

This coupling is not convenient. As will become clear below it is better tofirst couple the transition moments on each center. The dispersion energybecomes a sum of which the summand can be expressed with the use ofEq. (2) as a Casimir-Polder integral

∫

∞

0

α(lAl′

A)LA

MA(iω) α

(lB l′B

)LB

MB(iω)dω

over a product of irreducible frequency-dependent polarizabilities. The latterare given by

α(lX l′X)LX

MX(iω) ≡

∑

n>0

2(EXn − EX

0 )

(EXn −EX

0 )2 + ω2

×[

〈 ΦX0 | QlX | ΦX

n 〉 ⊗ 〈 ΦXn | Ql′X | ΦA

0 〉]LX

MX

.

The recoupling of the transition moments requires a 9j-symbol, see Refs. 80–82. Later the same recoupling was performed in Ref. 83. These referencesgive the expression without introduction of the Casimir-Polder integral. SeeRefs. 84–88 for the recoupled expression containing the Casimir-Polder inte-gral. We have now a closed form of all the terms in the multipole expansionof the dispersion energy. This energy can be computed once the irreduciblemonomer polarizabilities αLA(iω) and αLB(iω) are known. Unfortunately,this series does not converge; in fact it is divergent and “therefore we maybe able to do something with it” [O. Heaviside (1899), as quoted in Ref. 89].However, the series is asymptotic90–92 in the sense of Poincare.

A very similar equation holds for the multipole-expanded induction en-ergy. The difference is that the polarizabilities are static, so that there isno Casimir-Polder integral, and that one of the irreducible polarizabilitiesis replaced by a Clebsch-Gordan coupled product of permanent multipolemoments.

The problem of computing dispersion energies is reduced to the com-putation of polarizabilities for a sufficient number of frequencies, so thatthe Casimir-Polder integral can be obtained by numerical quadrature.93 An

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alternative to this quadrature is the substitution of the product of the po-larizabilities by a sum over Hartree-Fock orbitals,94,95 or a sum over effective(pseudo) states of the monomers.85,96 The pseudo states can be obtainedfrom time-dependent coupled Hartree-Fock calculations,85,96–98 or from CIcalculations.99 The CI calculations of Ref. 99 were at the single and doubleexcitation level. They gave very good results for the frequency-dependentpolarizabilities of He and H2—where SDCI is equivalent to full CI—and verypoor results for N2, O2 and the neon atom. The failure of the SDCI methodfor the response properties of more-than-two-electron systems was shown tobe caused by unlinked clusters.100 Addition of triply excited states removesthe most important unlinked clusters and was shown for Ne2 to improve theresults considerably.100

Doran101 was the first to apply Goldstone diagrammatic techniques tothe computation of frequency-dependent polarizabilities and dispersion coef-ficients. He applied his method to Ne2 and heavier noble gases, but owingto an inadequate basis, got results of fairly poor quality. Later Wormerand coworkers87,93, 102 derived and programmed all polarizability diagramsthrough second-order of intra-molecular correlation, so that dispersion (bydefinition second-order in V AB) is completely correlated to second-order oneach monomer. Their programs are in practice hardly limited by the rank ofthe multipoles: up to l = 63 can be computed.

4.3 Applications

At the beginning of the 1980s quantum chemical methods and computer hard-ware had developed to a stage that the computation of properties dependingon potential energy surfaces (PESs) of systems larger than two atoms couldbe contemplated. Examples are thermodynamic properties, such as virialcoefficients11,103 and moments of collision-induced infrared spectral densi-ties.104,105 The computation of spectroscopic properties of van der Waalsmolecules came into reach106–111 and also of molecular crystals.112

Intramonomer vibrations have in general a much higher energy than in-termolecular vibrations, i.e., the intramolecular motions are much “faster”than the intermolecular motions, so that an adiabatic separation of the twomotions is reasonable. In practice this means that we can consider themonomers to be frozen in their vibrationally averaged geometry and thatit is a good approximation to consider the interaction energy as a function(referred to as PES) of the relative coordinates of the rigid monomers. Ex-amples of intermolecular coordinates are the well-known Jacobi coordinatesR, θ for an atom-diatom system, while for a system consisting of two rigiddiatoms R (the distance between the respective mass centers), θA and θB

1061

(the colatitude angles of the diatomics) and φ (the dihedral angle) are verycommon.

An early computation of a full (i.e., depending on all intermolecularcoordinates) PES of two diatomics is the work by Berns and van der Avoirdon (N2)2.113 Their approach is in essence the one sketched above: one-cycle SCF to account for first-order exchange and electrostatics (includingcharge overlap effects) plus multipole-expanded dispersion. The dispersioncoefficients were taken from Ref. 114.

At that time this was a formidable calculation. It was performed on anIBM 370/158, which was not a supercomputer, but nevertheless a respectablemainframe. A basis of 144 Gaussian type orbitals (GTOs) was used; forrestart purposes integrals were stored on tape, requiring two tapes of 170Mbyte for one point on the potential energy surface. About 150 points werecomputed, so that 300 tapes were needed. A tape reel had a diameter of10.5 inch and, including its case, was about 1 inch wide, so that a rack ofabout 7.5 m long and 30 × 30 cm wide had to be used to store the 50 Gbyteof information. One point on the surface took 2.5 to 3.5 h CPU time andsince the whole university—from sociology to solid state physics—used thesame mainframe for time-sharing during the day, at most one point per nightcould be done. The computational part of the project, therefore, took abouthalf a year.

Two fits of the PES were made: one in terms of products of spheri-cal harmonics (coupled to a rotational invariant) and one as an atom-atompotential

∆EAB =∑

a∈A

∑

b∈B

[

qaqbrab

−Cab

r6ab

+ Aabe−Babrab

]

.

This potential was subsequently used in self-consistent phonon lattice dynam-ics calculations115 for α and γ nitrogen crystals. And although the potential—and its fit—were crude by present day standards, lattice constants, cohesionenergy and frequencies of translational phonon modes agreed well with ex-perimental values. The frequencies of the librational modes were less wellreproduced, but this turned out to be a shortcoming of the self-consistentphonon method. When, later,116,117 a method was developed to deal properlywith the large amplitude librational motions, also the librational frequenciesagreed well with experiment.

Ten years later van der Pol et al.118 published similar calculations of theCO–CO interaction potential, also performed on a mainframe (NAS 9160).The GTO basis was of dimension 148; 315 points on the PES were computed.One point took 30 minutes CPU time so that there was no need to save inte-grals. The dispersion, computed in the multipole expansion at the MP2 level

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of intramonomer correlation,87 was damped by the Tang-Toennies57 dampfunction. Notice, parenthetically, that the decrease in computer time fromthe calculations on (N2)2 to (CO)2, a decade apart, was certainly not revo-lutionary. Judging by these calculations the speed of mainframes improvedless than an order of magnitude; Moores law119 (doubling of speed every18 months) predicts two orders of magnitude. The (CO)2 potential of vander Pol was applied120 to the computation of properties of solid CO andgave good agreement with experimental values. A later application121 to therotation-vibration spectrum of the dimer showed, however, that the potentialwas not of spectroscopic accuracy.

5 Modern Times: Revolution and Democ-

racy

Around 1990 the advent of workstations initiated a revolution in scientificcomputing. Until that time batch processing was the norm for longer runningjobs. The user estimated an upper limit for the CPU time that his/hercomputation would take, submitted the job, prayed that it contained notrivial errors causing an immediate crash, and then settled down to waituntil there was room on the central computer to run the computation. Whenthe workstations arrived, which had the computational speed of mainframesand were cheap enough that research groups could afford one or more, themode of operation was revolutionized. In the first place, jobs went intoexecution immediately, so that the user had the chance to weed out trivialerrors instantaneously. In the second place, there was no longer a need tochop the calculations into chunks of a few hours CPU time.

An example of a calculation, performed on an IBM RS/6000-320 work-station, is the study of the collisions of argon and NH3 by van der Sandenet al.122 with the use of an ab initio calculated Ar–NH3 potential.123 Theprogram Hibridon124 was used to compute the elastic and rotationally in-elastic scattering cross sections and the probability that the collisions withAr invert the ammonia umbrella. A single (one collisional energy) coupledchannel calculation on para NH3 colliding with argon took 241 CPU hoursand was finished in about two weeks. On a mainframe this would have beena matter of months.

The increase of computer power made it possible not only to employ themost refined ab initio methods in the computation of potentials, but alsoto solve the nuclear motion problem sufficiently often to tune the ab initiopotentials to the experimental results. This was the Leitmotif of the past

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decade: compute the best possible PES, fit it, solve the appropriate nuclearmotion Schrodinger equation for the corresponding van der Waals complex,and compare with experiment. The remaining discrepancies between theoryand experiment may be removed by scaling one or more of the parametersin the analytical fit of the potential surface. This procedure has helped indisentangling complicated spectra, for instance the ν3 (asymmetric stretch)spectrum of CH4 in interaction with the argon atom125,126 and the ν4 (asym-metric bend) spectrum of the same system.127,128 At the same time, thisprovided an assessment of the quality of the ab initio Ar–CH4 potential. Anab initio calculated water pair potential129 was tested and improved130,131 bythe calculation of vibrational-rotational-tunneling spectra of the water dimerand comparison with experimental high-resolution spectra.132,133 Again, thecalculated energy levels and transition intensities134 could be used to assignthe bands in the measured spectrum to specific intermolecular vibrations.

For the computation of the interaction between two closed-shell monomersthere are at present two excellent computational methods, both implementedin black box programs. The first is based on symmetry-adapted perturbationtheory135 (SAPT) and the second is the supermolecule CCSD method136,137

with triply excited terms added in a non-iterative fashion.

5.1 The SAPT Method

The SAPT method was mainly developed by workers in the Warsaw quantumchemistry group. Jeziorski and his former supervisor Kolos,47,138 believingin the prospects of SAPT, continued and extended the work of Refs. 37,40–46, 48; later Szalewicz139 joined forces in this development. These workerscame to the conclusion that symmetrized Rayleigh-Schrodinger theory (weaksymmetry forcing—see above) was the most viable of the different variantsof SAPT.

We saw earlier that a very simple form of the dispersion energy is ob-tained from frequency-dependent polarizabilities at the so-called uncoupledHartree-Fock level. The sum over states appearing in second order RS per-turbation theory is simply a sum over (occupied and virtual) orbitals. A firstimprovement of this simple model is obtained by including apparent correla-tion,140 i.e., by using frequency-dependent polarizabilities obtained from thetime-dependent coupled Hartree-Fock (TDCHF) method.36,141 This methodwas initially proposed in the context of the multipole expansion, but could begeneralized142–146 to charge density susceptibility functions (or polarizationpropagators), which avoids the use of the multipole expansion. It is possibleto graft intramonomer correlation corrections onto TDCHF theory, but thisis not the road taken by the Warsaw group. Instead they worked top down,

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from an exact formulation to (approximate) equations in terms of one- andtwo-electron integrals that are coded in the SAPT program.

We now present the basic philosophy of symmetrized RSPT as imple-mented in the SAPT program,135 see for more details Refs. 88 and 147.Referring to Eq. (4) for the definition of V AB, we rewrite the Schrodinger

equation (H(0) + V AB)Ψpol = EΨpol. Projection with the eigenfunction Ψ(0)pol

of H(0) with eigenvalue E(0) gives the following exact expression for the in-teraction energy

E − E(0) ≡ Epol =〈 Ψ

(0)pol | V

AB | Ψpol 〉

〈 Ψ(0)pol | Ψpol 〉

.

The subscript pol (polarization40) indicates that no intermolecular antisym-metry has been introduced, or, in other words, that Ψpol is expanded inproducts of monomer wave functions. See Ref. 88 about the convergencecharacteristics of this expansion. The convergence to a state satisfying thePauli principle is greatly improved by introducing the intermolecular anti-symmetrizer AAB. Hence we define, in the spirit of weak symmetry forcing,the energy expression

ESRS ≡〈 Ψ

(0)pol | V

AB | AABΨpol 〉

〈 Ψ(0)pol | AABΨpol 〉

. (5)

If we introduce the intramonomer correlation WA, cf. Eq. (12), multipliedby the perturbation parameter µ, the Schrodinger equation for monomer Abecomes

HA(µ)ΦA(µ) = (FA + µWA)ΦA(µ) = EA(µ)ΦA(µ).

Clearly, ΦA(0) is the Hartree-Fock function of monomer A with energy EA(0)(a sum of orbital energies) and ΦA(µ) can be developed in a power series in µ.Analogously we assume for monomer B a power expansion in ν. MultiplyingV AB with the perturbation parameter λ, we may expand the eigenfunctionof H = FA + FB + µWA + νWB + λV AB

Ψpol(λ, µ, ν) =

∞∑

i,j,k=0

λiµjνkΨijkpol.

Here the subscript pol indicates that Ψpol(λ, µ, ν) is obtained from PT equa-tions that do not contain intermolecular exchange. Observing that Ψpol ≡

Ψpol(1, 1, 1) and Ψ(0)pol ≡ Ψpol(0, 1, 1), we may analytically continue Eq. (5) by

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substituting Ψpol(1, 1, 1) → Ψpol(λ, µ, ν) and Ψpol(0, 1, 1) → Ψpol(0, µ, ν) intothis equation. The resulting expression of ESRS is a function of λ, µ and ν.After expanding also the denominator in powers of λ, µ and ν, followed bycollecting the powers of λ, µ and ν arising from numerator and denominator,ESRS gets the form

ESRS(λ, µ, ν) =∞

∑

i,j,k=0

λiµjνkEijkSRS. (6)

Obviously, the exact antisymmetrized interaction is equal to ESRS(1, 1, 1).This is the basis of SRS theory—a weak-symmetry-forcing variant of SAPT.

Exchange effects have a clear operational definition in SAPT, becauseEpol can be expanded in the same RSPT manner, leading to terms Eijk

pol . The

difference EijkSRS −Eijk

pol is the exchange contribution to the (i, j, k) term. Theterms linear in λ are electrostatic terms and those quadratic in λ can bedivided in induction and dispersion terms (including their exchange correc-tions).

As we stated above, SAPT is formulated in a “top down” manner. Equa-tion (6) then forms the top; going down to workable equations, one is forcedto introduce a multitude of approximations. In practice, i is restricted to thevalues 1 and 2: interactions of first and second order in V AB. Different trun-cation levels for j + k are applied, depending on the importance of the term(and the degree of complexity of the formula). Working out the equations tothe level of one- and two-electron integrals is a far from trivial job. This hasbeen done in a long series of papers that use techniques from coupled clustertheory and many-body PT; see Refs. 147 and 148 for references to this workand a concise summary of the formulas resulting from it.

Some of the earliest potentials computed by the SRS variant of SAPTwere for Ar–H2

149 and for He–HF.150,151 An application of the latter potentialin a calculation of differential scattering cross sections152 and comparison withexperiment shows that this potential is very accurate, also in the repulsive re-gion. Some other SAPT results are for Ar–HF,153 Ne–HCN,154 CO2 dimer,155

and for the water dimer.129,156 The accuracy of the water pair potential wastested130,131 by a calculation of the various tunneling splittings caused byhydrogen bond rearrangement processes in the water dimer and compari-son with high resolution spectroscopic data.132,133 Other complexes studiedare He–CO,157,158 and Ne–CO.159 The pair potentials of He–CO and Ne–COwere applied in calculations of the rotationally resolved infrared spectra ofthese complexes measured in Refs. 160 and 161. They were employed162–165

in theoretical and experimental studies of the state-to-state rotationally in-elastic He–CO and Ne–CO collision cross sections and rate constants. It was

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reaffirmed that both potentials are accurate, especially the one for He–CO.Small organic molecules in interaction with noble gases were studied in

Refs. 166 (He–C2H2), 167 (Ne–C2H2), 125 (Ar–CH4), and 168 (He–CO2). ForHe–C2H2, Ne–C2H2, and Ar–CH4 the SAPT potentials were applied126,166, 167

in ab initio calculations of the infrared spectra of these complexes. A typicalfeature of all these potentials for weakly interacting systems is that theirshape is determined by a subtle balance between the geometry dependenceof the repulsive short range interactions and that of the long range forces,which mostly are attractive. All these results demonstrate that the pairpotentials from ab initio SAPT calculations are accurate. Another, moreglobal, comparison with experiment, confirming this finding, was made bycomputations of the (mixed) second virial coefficients of most of these dimersover a wide range of temperatures.169

An extension of SAPT that includes also third-order interactions170–174

permits the explicit calculation and analysis of three-body interactions. Fordetails about this development and a survey of its applications we refer tothe chapter in this book on many-body interactions written by Szalewicz,Bukowski, and Jeziorski.

5.2 The Coupled Cluster Method

Above we referred to the development of the coupled cluster method byCizek and Paldus.49–51 The coupled cluster method may be viewed as aconsistent summation to infinite order of certain type of linked correlation(MBPT, MP) diagrams. Thus, there is a clear relationship between many-body perturbation theory [based on the Møller-Plesset operator of Eq. (12) inthe Appendix] and coupled cluster theory. Both are supermolecule methodsthat give size-extensive energies.

Around 1980 MP calculations at second-order of perturbation (MP2)came within computational reach, while around 1990 third- (MP3) and fourth-order (MP4) calculations became feasible. For some time MP4 calculationswere widely applied to weakly bound complexes, but soon it was discov-ered that a full MP4 computation (including terms that include sums overtriply excited states) is hardly cheaper than a CCSD(T) computation. Sincethe latter is in general more reliable, MP4 lately lost much ground to theCCSD(T) method.

We can be very brief about the coupled cluster method since the chaptersby Paldus and others in this book give in-depth treatments of it. As is well-known, the exact N -electron wave function Ψ is written as

| Ψ 〉 = eT1+T2+···+TN | Φ0 〉, (7)

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where Φ0 is a closed-shell Hartree-Fock reference function. In most appli-cations T3, T4, . . . are neglected and only T1 = tiaE

ai and T2 = tijabE

abij (sum-

mation convention is used here) are included. Here Eai and Eab

ij are orbitalreplacement operators, where orbital i and j are occupied in Φ0 and a, b referto virtual orbitals. The cluster amplitudes tia and tijab are obtained from thesolution of equations that are quadratic in the tijab and fourth order in tia. Forthe closed-shell (spin singlet) case the projection of the CC equation on dou-bly excited states (the CCD method) yields coupled equations of dimensionK(K + 1)/2 where K is the product of the number nocc of occupied and thenumber nvir of virtual orbitals.

Naively one could expect that the solution of these equations scales asO(K4). For if one linearizes the equations according to the Newton-Raphsonmethod, a set of O(K2) linear equations must be repeatedly solved, whichtakes O(K4) operations per solution. Fortunately, the scaling is not thatbad. In the first place the sums in the equations do not run over all fourorbital labels of tab

ij simultaneously, but at most over two. In the second placea quasi-Newton method, in which the linear equations are approximated bya partially diagonal form, usually converges well. See for more details aboutthe computational aspects of the CCSD method the recent book by Helgaker,Jørgensen, and Olsen.175 This book also shows that the exponential ansatz,Eq. (7), leads in the long-range to a factorization of the wave function anda corresponding decomposition of the dimer energy into a sum of monomerenergies.

In total,176 the solution of the CCSD equation scales as n2occn

4virNit, where

Nit is the number of quasi-Newton iterations needed. Once the amplitudestai and tab

ij have been solved, they can be used to compute additional per-turbation terms that include triply excited states177 [a non-iterative O(n7)process], which are not accounted for in the CCSD method; their inclusionis indicated by (T) in CCSD(T).

We will end this section by mentioning a dozen or so illustrative examplesof modern supermolecule calculations on dispersion-bound complexes. Ofcourse, it is hopeless to strive for completeness, almost daily new calculationsare published, and hence the following list of references is far from exhaustive.

As stated above, around 1990 many workers used the MP4 method, see,for instance, Ref. 178 for the potential of CH4–H2O, Ref. 179 for MP4 appli-cations to CO2–Ar, and Ref. 180 for argon in interaction with Cl2 and ClF.Later the MP4 and CCSD(T) methods were compared, in calculations onAr–H2 and Ar–HCl,181 on N2–HF,182 and for CO–CO.183

A few examples of recent CCSD(T) computations on intermolecular po-tentials are by Cybulski and coworkers, who computed potentials of the noblegas dimers He2, Ne2, Ar2, He–Ne, He–Ar, and Ne–Ar (Ref. 184) and Ne–Kr,

1068

Ar–Kr, and Kr2 (Ref. 185). Further they considered HCN in interaction withHe, Ne, Ar, Kr,186 and Ar–CO.187 Computational and experimental studiesof intermolecular states and forces in the benzene–He complex were reportedin Ref. 188. A thorough CCSD(T) study on benzene–Ar is by Koch et al.189

and on Ne–HCl by Fernandez et al.190

5.3 Latest developments

Lately two completely different topics in the field of intermolecular forceshave drawn attention and are now actively being studied. In the first placethere is the possible application of density functional theory (DFT) to vander Waals molecules. The second topic concerns van der Waals moleculesof which the electronic state of one or more of the monomers is spatiallydegenerate.

Density functional theory in the standard Kohn-Sham (KS) formulationhas its limitations in application to dispersion forces. Standard local andgradient-corrected functionals are not appropriate for the description of dis-persion, which is inherently a non-local correlation effect [there is no suchthing as a dispersion potential Vdisp(~r )]. Despite the search for function-als capable of describing London forces (cf. Ref. 191 and references therein),there still is no generally applied solution in the framework of KS-DFT. How-ever, via a detour DFT can play an important role. Earlier we discussed anapproach to obtaining non-expanded dispersion by the Casimir-Polder inte-gration of a product of two polarization propagators. This approach can eas-ily and seamlessly be interwoven with DFT,192,193 because DFT is known togive accurate response properties, provided functionals with correct asymp-totics are used.194–197 Complete intermolecular interaction potentials can beobtained from the so-called DFT-SAPT method that substitutes KS orbitalsand exchange-correlation kernels into the SAPT expressions for the interac-tion energies with j = k = 0, cf. Eqs. (5) and (6). The evaluation of theseexpressions is computationally much cheaper than the inclusion of monomercorrelation effects by calculation of the SAPT terms with j + k > 0.

In Ref. 193 various non-hybrid and hybrid exchange-correlation poten-tials and suitable adiabatic local density approximations for the exchange-correlation kernel were compared for the dimers He2, Ne2, Ar2, NeAr, NeHF,ArHF, (H2)2, (HF)2, and (H2O)2. This comparison showed that the ef-fects of intramonomer electron correlation on the dispersion energy are mostaccurately reproduced with an asymptotically corrected197 version of theexchange-correlation potential of Perdew and coworkers.198 In Ref. 199 theimportance of asymptotically correct exchange-correlation potentials in DFT-SAPT was emphasized particularly. In Ref. 192 dispersion energies of He,

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Ne, and H2O dimers were obtained by the DFT-SAPT approach to within3% or better.

Earlier we also discussed the uncoupled HF approach to dispersion, wherethe sum over states is performed at the orbital level. Of course, this approachcan also be applied with KS orbitals. However, Heßelmann and Jansen193

found that the uncoupled sum-over-states approximation yields unaccept-able errors. These are mainly due to neglect of the Coulomb and exchange-correlation kernels and are not substantially improved through an asymptoticcorrection of the exchange-correlation potential.

The DFT-SAPT approach has been very recently applied200 to the noto-riously difficult case of the CO-dimer. Earlier computations183 of the (CO)2

PES by means of MP4 and CCSD(T) methods encountered some unexpectedcomplications. It was shown that high-order correlation effects are impor-tant and that both CCSD(T) and CCSDT formally do not have a correctasymptotic (large R) behavior. Later201,202 it was pointed out that on top ofthis problem also very large basis sets are needed for an accurate descriptionof the CO–CO potential energy surface. Notwithstanding this problem, afull 4-dimensional PES (rigid monomers) was computed in Ref. 203 by theCCSD(T) method as a springboard for further refining. The potential wasfitted in terms of analytic functions, and the fitted potential was used to com-pute the lowest rovibrational states of the dimer. It gave semi-quantitativeagreement with the experimental infrared and millimeter wave spectra ofMcKellar, Winnewisser and coworkers.204–210 Application of a fit of the re-cent DFT-SAPT potential200 gave rovibrational results that differed some-what from the CCSD(T) data, and were also in semi-quantitative agreementwith the measured spectra. It was decided to combine the two potentials,CCSD(T) and DFT-SAPT, and it was shown that a weighted average ofthe DFT-SAPT (30%) and the CCSD(T) potential (70%) gives results thatare in very good agreement with experimental data, for both (12CO)2 and(13CO)2.

The second topic of recent interest—dimers that dissociate into a degen-erate open-shell monomer and a non-degenerate closed-shell monomer or intotwo open-shell monomers—has two intrinsic difficulties that are both due tospatial degeneracy. The dimer is an open-shell system in such cases and ithas multiple potential energy surfaces that become degenerate for large in-termolecular separations and in many cases also for other geometries. In thefirst place, it is fair to say that at present there are no generally applicablesize-extensive electronic structure methods for open-shell, spatially degener-ate, systems. From a theoretical point of view, the complete active spacemulti-configuration SCF (CASSCF) method211 is probably the most satis-

1070

factory, as it handles electron spin correctly and is size-extensive. However,the active spaces that can be handled in practice are too small to give areliable account of dynamic correlation effects like dispersion. The CASSCFmethod has been extended to CAS perturbation theory (CASPT) in order toinclude dynamical correlation effects.212–217 The CASPT approach is almostsize-extensive when the CASSCF reference function is dominated by a singledeterminant. However, for reference wave functions in which several determi-nants have large weights, as is the case for spatially degenerate open-shells,size-extensivity is broken.175

An alternative electron-correlation method is the multi-reference config-uration interaction (MRCI) method. This method is plagued by unlinkeddiagrams, the presence of which break the size-extensivity of the MRCI en-ergy. Often MRCI results are corrected by a simple formula introduced 30years ago by Langhoff and Davidson,218 who derived it by inspection of aCI wave function consisting of all double excitations (DCI) from a singleSlater determinant. One can look upon this “Davidson correction” as anapproximate formula for the unlinked diagram that enters the DCI energy.

Paldus, elsewhere in this book, discusses that there is as yet no gener-ally applicable, open-shell, size-extensive, coupled cluster method, and thesame holds for open-shell SAPT methods. Therefore, for the computation ofpotentials of open-shell van der Waals molecules one has the choice betweenCASSCF followed by a Davidson-corrected MRCI calculation of the inter-action energy, or the single reference, high spin, method RCCSD(T). Whenthe ground state of the open-shell monomer is indeed a high spin state, thenRCCSD(T) is the method of choice. With regard to the latter method we re-call that a major difficulty in open-shell systems is the adaptation of the wavefunction to the total spin operator S2; for the CCSD method a partial spinadaptation was published by Knowles et al.,219,220 who refer to their methodas “partially spin restricted”. When non-iterative triple corrections221 areincluded, the spin restricted CCSD(T) method, RCCSD(T), is obtained.

Even when free monomers are in degenerate states, the RCCSD(T) methodis often employed, because for most points on the PES the symmetry is low-ered to Abelian symmetry, so that degeneracies are lifted and RCCSD(T) isformally applicable. But it can be applied only to the lowest state of a givensymmetry, while one needs to know also the potential surfaces of the higherdimer states that become asymptotically degenerate with the ground state.Moreover, it is clear that the method fails for points on the PES that havesymmetry higher than Abelian and states that belong to more-dimensionalrepresentations of the non-Abelian point group.

The second problem that often occurs in open-shell van der Waals mol-ecules is the breakdown of the Born-Oppenheimer (BO) approximation. As

1071

is well known, the BO approximation can be trusted when the potentialenergy surfaces are well separated in energy. However, when certain pointson the PES are degenerate this condition is not fulfilled, not in the degeneratepoints themselves, but also not in nearby points. This breakdown of the BOapproximation can be shown as follows. Let us write R for the collection ofnuclear coordinates and r for the electron coordinates. Indicating electronicand nuclear interactions by subscripts e and n, respectively, the Schrodingerequation takes the form

(Tn + Te + Vnn + Vne + Vee)Ψ(R, r) = EΨ(R, r),

where the kinetic energy terms Tn and Te have the usual form. In particular,Tn =

∑

α Pαn P

αn /(2Mα) with the nuclear momentum P α

n = −i∂/∂Rα. Thewave function is expanded in eigenfunctions χk(r; R) of He ≡ Te + Vee + Vne

Ψ(R, r) =∑

k

χk(r; R)φk(R)

with 〈 χk′(r; R) | χk(r; R) 〉(r) = δk′k and where the subscript (r) indicatesthat the integration is over electronic coordinates only. By definition

〈 χk′(r; R) | He | χk(r; R) 〉(r) =(

He(R))

k′k= δk′kEk(R),

and we assume that χk(r; R) is real (invariant under time-reversal).After multiplication by χk′(r; R) and integration over r the Schrodinger

equation is turned into a set of coupled equations depending on nuclear co-ordinates only

[Hn(R) + He(R)] φ(R) = Eφ(R),

where the column vector φ(R) has elements φk(R). The matrix He(R) isdiagonal and

(

Hn(R))

k′k= 〈 χk′(r; R) | Tn | χk(r; R) 〉(r) + δk′kVnn.

Suppressing the coordinates in the notation, we can write the matrix elementsof Tn as

〈 χk′ | Tn | χk 〉(r) = δk′kTn +∑

α

1

Mα〈 χk′ |

(

P αn χk

)

〉(r)P αn

+〈 χk′ |(

Tnχk

)

〉(r) (8)

The diagonal (k′ = k) matrix elements 〈 χk |(

P αn χk

)

〉(r) of the operatorP α

n vanish, because this operator is Hermitian and odd with respect to timereversal. The off-diagonal matrix elements satisfy

〈 χk′ |(

P αn χk

)

〉(r) =〈 χk′ |

[

P αn , He

]

| χk 〉(r)Ek(R) − Ek′(R)

.

1072

We see that whenever two surfaces come close, Ek(R) ≈ Ek′(R), the nuclearmomentum coupling term is no longer negligible. Conversely, if all surfacesare well separated, all off-diagonal terms can be neglected and hence thewhole matrix of P α

n is effectively zero. The third term on the right hand sideof Eq. (8) can be written as the matrix of P α

n squared and, accordingly, isthen negligible also. Only the first (diagonal) kinetic energy term in Eq. (8)survives and a diagonal, uncoupled, set of nuclear motion equations results.These are the normal second-step of the Born-Oppenheimer approximationequations.

Let us, for the sake of argument, assume now that only two surfaces 1and 2 approach each other and that all other surfaces are well separated; theargument is easily generalized to more surfaces. We then have to solve a setof two coupled nuclear Schrodinger equations with non-negligible couplingelement 〈 χ1(r; R) | Tn | χ2(r; R) 〉(r). Define two new orthonormal states bya rotation of χ1 and χ2 (for clarity reasons we suppress the coordinates)

(ϕ1, ϕ2) = (χ1, χ2) R(γ), (9)

where R(γ(R)) is a 2 × 2 rotation matrix and γ(R) is the “diabatic angle”.Transformation of the matrix of nuclear momentum 〈 χk′ |

(

P αn χk

)

〉(r) fork′, k = 1, 2 gives

〈 ϕk |(

P αn ϕk

)

〉(r) = 0 for k = 1, 2,

i.e., the diagonal matrix elements remain zero, and

〈 ϕ2 |(

P αn ϕ1

)

〉(r) =(

P αn γ(R)

)

− 〈 χ2 |(

P αn χ1

)

〉(r).We search for a γ(R), such that to a good approximation

(

P αn γ(R)

)

− 〈 χ2 |(

P αn χ1

)

〉(r) ≈ 0 (10)

i.e., ϕ1 and ϕ2 diagonalize the 2 × 2 matrix of the nuclear momentum. Bythe definition of F. T. Smith222 ϕ1 and ϕ2 are diabatic states. Smith wasthe first to define this concept. (Earlier the term “diabatic” was used some-what loosely by Lichten223). The nuclear motion problem takes the following“generalized Born-Oppenheimer” form

Tn + Vnn +E1(R) + E2(R)

20

0 Tn + Vnn +E1(R) + E2(R)

2

φ(R)

+E2(R) −E1(R)

2

(

cos 2γ sin 2γsin 2γ − cos 2γ

)

φ(R) = Eφ(R). (11)

1073

The surfaces E1(R) and E2(R) are BO energies obtained from electronicstructure calculations and Tn is the first term of Eq. (8). The (transformed)third term in this equation is neglected. The determination of γ(R) is theremaining problem before a solution of Eq. (11) can be attempted.

Several methods for the determination of γ(R) have been proposed.224,225

One is the direct computation of the non-adiabatic coupling matrix element〈 χ1 |

(

P αn χ2

)

〉(r) by finite difference techniques, which gives the derivativeof γ, cf. Eq. (10). Another is by supposing that the diabatic states ϕ1 andϕ2 are states of the free monomers and by using Eq. (9) backwards. Thisis obviously only possible when the adiabatic states χk and χk′ are (almost)pure linear combinations of the two monomer states. This approximationcan be made at the orbital level or at the N -electron level (or at both levelssimultaneously). Also mixing matrix elements of molecular properties overadiabatic states may be used.

We will end this section by mentioning some recent representative cal-culations on van der Waals molecules consisting of a closed- and open-shellmonomer. The simplest closed-shell monomer is of course the ground statehelium atom. Its interaction with NO(X2Π),226 CO(a3Π),227,228 CaH(2Σ+),229

and NH(X3Σ−)230 was studied recently. In the case of the He–CO(3Π) com-plex the potential was applied in computing the spectrum of the bound com-plex227 and in photodissociation processes.228 The He–CaH(2Σ+) interactionwas employed in the study of collisions at cold and ultracold temperatures,231

and the He–NH(X3Σ−) potential was used in calculations on low temperaturecollisions in the presence of a magnetic field.232 Further work is on Cl(2P )–HCl233 and its bound states.234 Finally, we refer to the work on the diabaticintermolecular potential and bound states of the H2–F(2P ) complex.235

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A Relationship between dispersion and EABMP2

Often dispersion energy is described as the interaction between mutually in-duced dipoles, one on each atom. One can see this as a “correlation” betweentwo dipoles. It is not obvious how this “correlation” is related to Lowdin’s“beyond-Hartree-Fock-correlation”.23 In this Appendix it is shown how thelatter correlation and dispersion are interrelated. Earlier this connection wasshown63 in a somewhat different manner.

The MP2 energy, the simplest correlation correction, is obtained fromRS perturbation theory with the perturbation

W ≡ H − F − 〈 Φ0 | H − F | Φ0 〉, (12)

where the Slater determinant Φ0 is the lowest eigenfunction of the Fockoperator F =

∑Nk=1 f(k) with eigenvalue 2

∑N/2i=1 ǫi. The Fock operator serves

as the unperturbed (zeroth-order) operator. Since the first order MP energy〈 Φ0 | W | Φ0 〉 is obviously zero, the lowest order MP energy appears insecond order. We write the MP2 energy formula for a supermolecule A–B

1088

with closed-shell monomers A and B. After application of the Slater-Condonrules for the simplification of N -electron matrix elements and integrating outspin, it becomes

EABMP2 =

∑

i,j,a,b

〈 φi(1)φj(2) | r−112 | φa(1)φb(2) 〉 (13)

×2〈 φa(1)φb(2) | r−1

12 | φi(1)φj(2) 〉 − 〈 φa(1)φb(2) | r−112 | φj(1)φi(2) 〉

ǫi + ǫj − ǫa − ǫb,

where φi and φj are occupied and φa and φb are virtual orbitals of the dimerA–B. We consider the limit of this expression for R large enough that thedifferential overlap between wave functions of A and B can be neglected.We recall that we can localize SCF orbitals and write | ρ 〉 and | r 〉 for theoccupied and virtual spatial orbitals localized on A and | σ 〉 and | s 〉 for theoccupied and virtual orbital localized on B. These orbitals are expressed inthe dimer basis. The Fock operator is invariant under unitary localization ofthe {φi}, i.e.,

(NA+NB)/2∑

i=1

〈 φi(2) |2 − P12

r12| φi(2) 〉 =

NA/2∑

ρ=1

〈 ρ(2) |2 − P12

r12| ρ(2) 〉 +

NB/2∑

σ=1

〈 σ(2) |2 − P12

r12| σ(2) 〉.

In general, the Fock operator is no longer diagonal when the orbitals {φi}are localized, but we will show below that we can still use its eigenvalues,i.e., the dimer orbital energies ǫi, which under specific conditions applicablehere, become equal to the orbital energies of monomers A and B.

Let us consider two S-state atoms and the action of the dimer Fockoperator on, for instance, | ρ(1) 〉

fAB(1)| ρ(1) 〉 =

[

−1

2∇2 −

ZA

rA1+

∑

ρ

〈 ρ(2) |2 − P12

r12| ρ(2) 〉

]

| ρ(1) 〉

+

[

−ZB

rB1+

∑

σ

〈 σ(2) |2 − P12

r12| σ(2) 〉

]

| ρ(1) 〉. (14)

Because of zero differential overlap the P12 contribution can be dropped inthe second term of Eq. (14). The terms that remain in the second expressionbetween large square brackets cancel each other. This is because the elec-tronic charge distribution Q(~rB2) ≡ 2

∑

σ |σ(2)|2 is spherically symmetricand screens completely the nucleus of B.

1089

We will prove this intuitive statement and to that end we need the follow-ing two expansions, dating back to the 19th century,78,79 (see for a modernversion, e.g., Appendix VI of Ref. 236). Together they give the multipoleexpansion of 1/r12 (for R > r)

1

|~R− ~r|=

∞∑

l=0

l∑

m=−l

(−1)mΥl−m(~R)Sl

m(~r) (15)

Slm(~r1 − ~r2) =

l∑

L=0

(−1)L

(

2l

2L

)1/2[

Sl−L(~r1) ⊗ SL(~r2)]l

m. (16)

Here Υlm(~R) ≡ Rl−1C l

m(R) is an irregular solid harmonic function and Slm(~r) ≡

rlC lm(r) is a regular solid harmonic function. The function C l

m(r) is a spheri-cal harmonic function normalized to 4π/(2l+1) (Racah normalization). Theexpression between square brackets in Eq. (16) is a Clebsch-Gordan cou-

pled product. We write ~r12 = −~rA1 + ~RAB + ~rB2, and find, assuming that|~RAB| > |~rB2 − ~rA1|,

−ZB

rB1

+ 2∑

σ

〈 σ(2) |1

r12| σ(2) 〉 = −

ZB

rB1

(17)

+2∑

L,l,m

(−1)L+m

(

2l

2L

)1/2

Υl−m(~RAB)

∑

σ

[

Sl−L(~rA1) ⊗ 〈 σ | SL(~rB2) | σ 〉]l

m.

The expression 〈 SLM 〉 ≡ 2

∑

σ〈 σ | SLM(~rB2) | σ 〉 is the Hartree-Fock expec-

tation value of the (L,M) multipole moment of the S-state atom B. Whenthe charge distribution Q(~rB2) is spherical symmetric around B 〈 SL

M 〉 =NBδL0δM0. Equation (17) becomes under this condition

−ZB

rB1+NB

∑

l,m

(−1)mΥl−m(~RAB)Sl

m(~rA1) = −ZB

rB1+NB

rB1.

The simplification of this result follows from Eq. (15). Since for neutral atomsNB = ZB the second term of Eq. (14) indeed vanishes. It follows that thedimer Fock operator, when it acts on orbital | ρ 〉 localized on monomer A,is equivalent to the atomic Fock operator of A

fAB| ρ 〉 = fA| ρ 〉. (18)

Under the conditions of our derivation, i.e., S-state atoms A and B withvanishing differential overlap, we can show that the localized orbitals | ρ 〉and | σ 〉 are identical (apart from mixing possibly degenerate orbitals) to

1090

the orbitals obtained by solving the monomer Hartree-Fock equations (inthe dimer basis)

fA| ρ 〉 = ǫρ| ρ 〉 and fB| σ 〉 = ǫσ| σ 〉.

These Fock equations yield solutions for A and B with corresponding chargedistributions that are spherically symmetric around A and B, respectively[i.e., the solutions span irreps of SO(3)]. Hence the spherical symmetry ofS-state atom A is not disturbed by the presence of S-state atom B and viceversa, so that Eq. (18) holds. Expand the solution of A in dimer MOs | k 〉

| ρ 〉 =∑

k

| k 〉Ukρ with fAB| k 〉 = ǫk| k 〉.

Thenǫρ| ρ 〉 = ǫρ

∑

k

| k 〉Ukρ = fAB| ρ 〉 =∑

k

ǫk| k 〉Ukρ

so thatǫρUkρ = ǫkUkρ.

If ǫρ 6= ǫk it follows that Ukρ = 0, so that, in general, | ρ 〉 is a linear combi-nation of degenerate dimer orbitals | k 〉 with orbital energy ǫk = ǫρ. If thereis no degeneracy, then | ρ 〉 is identical to | k 〉.

The same argument may be applied to the other localized dimer orbitals| r 〉, | σ 〉 and | s 〉. In other words, we can solve the monomer HF equationsin the dimer basis and get the same orbital energies as from the solution ofthe dimer HF equations.

When we now replace the sums over the canonical orbitals by sums overlocalized orbitals and the dimer orbital energies by monomer orbital energiesin Eq. (13), we obtain

limR→∞

EABMP2 = EA

MP2 + EBMP2 + 4

∑

ρ,r,σ,s

|〈 ρ(1)σ(2) | r−112 | r(1)s(2) 〉|2

ǫρ + ǫσ − ǫr − ǫs.

See Fig. 1 for a diagrammatic representation of this limit. The third termon the right hand side is the non-expanded “Hartree-Fock” expression59 fordispersion. Incidentally, this equation shows that the MP2 method is size-extensive. That is, when the distance RAB between A and B is so large thatthe interaction term vanishes, the dimer MP2 energy becomes the sum of themonomer MP2 energies. Although this statement sounds obvious, it is not.The singles and doubles configuration interaction method forms a counterexample.

1091

The equivalence between the interaction energy from dimer MP2 calcu-lations and the simple expression for the dispersion interaction does not holdwhen the interacting systems are molecules or non-S-state atoms. The sec-ond term of Eq. (14) does not vanish in that case, because of non-vanishingmultipole moments contributing to the expansion in Eq. (17). Even for largedistances R, where all differential overlap between A and B vanishes and thedimer orbitals can be localized, these orbitals are not equal to the unper-turbed monomer orbitals. This is due to the polarization of each monomer,induced by the multipole moments of the other monomer. This gives longrange electrostatic and induction interactions, which thus are accounted forby the supermolecule HF method. Conversely, for spherically symmetricsystems the HF method does not give any interaction at distances wheredifferential overlap is negligible.