Theory and Applications of Computational Chemistry the First Forty Years

Appeared in

Theory and Applications of Computational Chemistry The First Forty Years

Editors clifford e. dykstra Department of Chemistry Indiana University-Purdue University Indianapolis (IUPUI) Indianapolis, IN, U.S.A. gernot frenking Fachbereich Chemie Phillips-Universit¨at Marburg Marburg, Germany kwang s. kim Department of Chemistry Pohang University of Science and Technology Pohang, Republic of Korea gustavo e. scuseria Department of Chemistry Rice University Houston, TX, U.S.A.

2005 Contents

1 Introduction 1047

2 Prehistory: Before Computers 1048

3 Antiquity: the Sixties 1049 3.1 SupermolecularMethods ...... 1050 3.2 PerturbationMethods ...... 1052

4 The Middle Ages: Era of Mainframes 1054 4.1 UnexpandedDispersion...... 1055 4.2 Multipole-Expanded Dispersion ...... 1058 4.3 Applications...... 1060

5 Modern Times: Revolution and Democracy 1062 5.1 TheSAPTMethod ...... 1063 5.2 The Coupled Cluster Method ...... 1066 5.3 Latestdevelopments ...... 1068

References 1073

A Relationship between dispersion and EMP2AB 1087 1047

chapter 37

Forty years of ab initio calculations on intermolecular forces

Paul E. S. Wormer and Ad van der Avoird Institute 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 of intermolecular forces; emphasis is placed on dispersion forces. The last forty years, which saw the birth, growth, and maturation of ab initio methods, are reviewed.

1 Introduction

Intermolecular forces, sometimes called non-covalent interactions, are caused by Coulomb interactions between the electrons and nuclei of the molecules. Several contributions may be distinguished: electrostatic, induction, dispersion, exchange, that originate from different mechanisms by which the Coulomb interactions can lead to either repulsive or attractive forces between the molecules. This review deals with the ab initio calculation of complete intermolecular potential surfaces, or force fields, but we focus on dispersion forces since it turned out that this (relatively weak, but important) contribution took longest to understand and still is the most problematic in computations. Dispersion forces are the only attractive forces that play a role in the interaction between closed-shell (1S) atoms. We will see how the understanding of these forces developed, from complete puzzlement about 1048 their origin, to a situation in which accurate quantitative predictions are possible. The subfield of quantum chemistry concerned with the computation of intermolecular forces has always depended very much on computer technology, 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 two closed-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 intermolecular forces. Then the infancy of computers and computational quantum chemistry will be reviewed, followed by the era dominated by mainframes. We will end with the present democratic times in which every household has at its disposal the power of a 1980 supercomputer and ordinary research groups possess farms of powerful computers.

2 Prehistory: Before Computers

When on the 10th of July 1908 Kamerling Onnes and his coworkers saw the meniscus of liquid helium in their apparatus,1 it was proved to them that two helium atoms attract each other—so that a liquid can be formed—but also that 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 the van der Waals law of corresponding states, which gave them a fair idea of the temperature and pressure at which the liquefaction of helium would take place. In deriving his law van der Waals (1873) had to assume the existence of 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 introduction of the Schrödinger 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 found 6 an attractive potential with as leading term −C6/R . A few years later Eisenschitz and London (E&L)3 systematized this work by introducing a perturbation formalism in which the Pauli principle is consistently included. They showed that the intermolecular antisymmetrization of the electronic wave function (electron exchange), which is required by the Pauli principle, can give rise to repulsion. This is why the intermolecular repulsion is often referred to as exchange (or Pauli) repulsion. Considering distances long enough that intermolecular differential overlap 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 to the oscillator strengths arising in the classical theory of the dispersion of light (associated with the names of Drude and Lorentz) and in the quantum mechanical dispersion theory of Kramers and Heisenberg. Oscillator strengths, being simply proportional to squares of transition moments, are known experimentally, enabling E&L to give reasonable estimates of C6. In 1930 London4 published another paper in which he coined the name “dispersion effect” for the attraction between S state atoms, which is why it is common today to refer to these attractive long-range forces as “London” or “dispersion forces”. Apropos of nomenclature: the forces between closed-shell molecules (exchange repulsion, electrostatics, induction, and dispersion) are nowadays usually 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 was broken until computers and software had matured enough to try fresh attacks. In the meantime the study of intermolecular forces was mainly pursued by thermodynamicists who fitted model potentials, often of the Lennard-Jones form: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 the authoritative monograph of Hirschfelder, Curtiss, and Bird,11 who, incidentally, also give a good account of the relationship of Drude’s classical work to that of London.

3 Antiquity: the Sixties

Around 1960 the computer began to enter quantum chemistry. This was the beginning of a very optimistic era; expectations of the new tool were tremen- dous. All over the western world quantum chemists were appointed in the belief that many of the problems in chemistry could be solved by computation within a decade or so. However, computers and ab initio methods were not received with this great enthusiasm by everyone. Coulson,12 one of the outstanding quantum chemists of his days, stated in his after-dinner closing speech of the June 1959 Conference on Molecular Quantum Mechanics in Boulder, 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 its present predicament.” 1050

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 chemistry groups of the time, announced a research program13 on the computation of properties of diatomic molecules. With the benefit of hindsight one can say that his program was overambitious and far too optimistic, because he intended to use self-consistent field (SCF) methods with atomic orbital (AO) minimum basis sets, albeit of Slater type. The fourth paper14 of this research program was devoted to He2. Here Ransil considered the dispersion-bound dimer as a molecule amenable to ordinary molecular computational methods. Nowadays this method is referred to as a “supermolecule” approach. Ransil writes in his abstract that “remarkable good agreement with the available experimental data is obtained for distances greater than 1.5 A”.˚ We now know that his van der Waals minimum was spurious and solely due to the so-called basis set superposition error (BSSE). This BSSE is the lowering of the energy of monomer A, caused by the distance dependent improvement of the basis by the approaching AOs on B, and vice versa: the basis of B is improved by basis functions on A. How much the difficulties of ab initio calculations on intermolecular forces were still underestimated is witnessed by 15 another paper on He2, also originating from the Chicago group. Phillipson, 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 results on 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 completely repulsive curves by the SCF method. According to Kestner in 1968: “it is generally believed, but nobody has proved, that this should be the case for two closed-shell atoms”. This statement exhibits the great advance made in understanding ab initio results in the early sixties. Since Kestner used the Chicago computer codes and thanks Chicago staff members (Roothaan, Ran- sil, Cade, and Wahl) for guidance and support, it is clear that the Chicago work 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 was proposed by Boys and Bernardi17 in 1970. A similar correction was already applied 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 sets that are so large that the SCF counterpoise correction is essentially zero. In 1051 correlated supermolecular methods the counterpoise correction is usually still needed. 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 dimer basis set by the same computational method. Although the sum of the monomer 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 calculation) that the SCF method applied to He2 gives a purely repulsive interaction. Recall that by Löwdin’s definition23 the SCF energy serves as the zero of electron correlation, or in other words, the SCF method does not give any correlation. Sinano˘glu24 was the first to observe that interatomic sp pair correlation yields London R−6 dispersion. By the converse of this finding it seems plausible that without interatomic correlation dispersion effects are not accounted for. Since these effects contribute so significantly to the attraction of S-state atoms, one may conjecture that for non-polar systems there is no binding without inclusion of interatomic correlation. And indeed, we will show this below. As a matter of fact, Pauli repulsion is now usually taken for granted and attention is focused usually on the explanation of observed minima in intermolecular potentials. The fact that interatomic pair correlation gives dispersion was semi- 25 quantitatively confirmed in a calculation on He2. This Kestner-Sinano˘glu −1 work on He2 gave a well depth of 4.32 K = 3.00 cm , which is about 2.5 times lower than the presently accepted value. The discrepancy is due to an inadequate AO basis. Sinano˘glu’s method was later improved26 by adaptation of the pair func- 2 27 tions to the spin-operator S . The He2 potential was recomputed by this method with the use of a much larger AO basis. This paper, and a simultaneous paper published in the same issue of Phys. Rev. Lett. by Bertoncini and Wahl28 describing MCSCF calculations, are the first that report within one consistent supermolecule formalism a complete van der Waals curve that shows a physically meaningful well. In Ref. 27 the depth of this well is De = 12.0 K at the equilibrium distance Re = 2.96 A˚ and in 29, 30 Ref. 28 De = 11.4 K at Re =2.99 A.˚ The presently accepted values are De = 11.008 ± 0.008 K and Re = 5.6 bohr = 2.963 A.˚ The choice of configurations in the MCSCF calculation was inspired by the London long-range theory. So, while the 1960s started with the belief that SCF could give a complete potential curve for closed-shell atoms, at the end of the decade it was known that the inclusion of interatomic correlation is essential for obtaining the dispersion attraction. The new decade saw the light with the two papers 1052 just mentioned27, 28 proving this quantitatively.

3.2 Perturbation Methods Independent of the work on coding ab initio programs, other workers in the 1960s carried further the torch of London. In the first place methods were improved to compute better long-range C6 coefficients and the corresponding higher coefficients C8, C10, etc., in the expansion of the interaction energy ∞ −n − n=6 CnR . On the other hand, people took a closer look at the symmetrized pertur- bationP theory of Eisenschitz and London, which in principle can give a full potential energy surface, not just the long range of it. A variety of techniques has been employed for the estimation of dispersion coefficients. Good reviews are those by Dalgarno and Davison31 and 32 Dalgarno. The semi-empirical methods based on oscillator strengths fs were refined by using sum rules for Cauchy moments. A Cauchy moment S(k) is defined by the following sum over monomer states ψs with energies Es k S(k)= fs (Es − E0) , s>0 X where the oscillator strength fs is given as a squared matrix element of the dipole operator µ 2 f = (E − E ) h ψ | µ | ψ ih ψ | µ | ψ i. s 3 s 0 0 i s s i 0 i=x,y,z X The even Cauchy moments arise in the expansion of a frequency-dependent polarizability α(ω), ∞ f α(ω)= s = S(−2k − 2)ω2k. (1) (E − E )2 − ω2 s>0 s 0 X Xk=0 The coefficient C6 in the long-range interaction energy between two molecules A B A and B is given in terms of the oscillator strengths fs and fs by A B 3 fs fs′ C6 = A A B B A A B B . 2 ′ (Es − E0 )(Es′ − E0 )(Es − E0 + Es′ − E0 ) Xss One can factorize the denominator of this expression by invoking the identity (∆X > 0 is an excitation energy on X): 1 2 ∞ 1 1 = 2 2 2 2 dω. (2) ∆A∆B(∆A + ∆B) π 0 ∆A + ω ∆B + ω Z 1053

Casimir and Polder33 have shown that this identity, which can be proved easily by contour integration, can be used to express the long-range coefficient C6 in terms of frequency-dependent monomer polarizabilities 3 ∞ C = αA(iω)αB(iω)dω. 6 π Z0 Many values of fs are known empirically; their reliability can be checked by the sum rules: S(−2) = α(0), S(0) = N (number of electrons) and 2 S(−1) = 3 h ψ0 | µ · µ | ψ0 i. Effectively summing the power series in Eq. (1) by means of different Padéapproximants makes it possible to give upper and lower bounds on the C6 values. Much work in the 1960s was done on calculating such bounds, see Ref. 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 coupled TDCHF method.36 Intermolecular electron exchange does not play a role in the long-range, since all integrals that would arise by intermolecular antisymmetrization vanish by virtue of vanishing intermolecular differential overlap. However, for shorter distances where this overlap may not be neglected, the electrons on the monomers can no longer be distinguished and the wave functions must be antisymmetric under permutations of all electrons. As we saw earlier, Eisenschitz and London considered this problem as early as 1930 and it was revived in the late 1960s by Murrell, Randic and Williams,37 Hirschfelder and Silbey,38 Hirschfelder,39, 40 van der Avoird,41–44 Murrell and Shaw,45 and Musher 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 function in the perturbation theory of intermolecular interactions. Here AAB is A B the usual intermolecular antisymmetrization operator and Φ0 = Φ0 Φ0 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., AX Φ0 = Φ0 for X = A, B. Unfortunately, since intermolecular permutations (0) (0) do not commute with H , [AAB,H ] =6 0, it follows that AABΦ0 is not an eigenfunction of H(0). This has the consequence that conventional Rayleigh- Schrödinger (RS) perturbation theory is not applicable for those intermolecular distances where the effect of AAB is non-negligible. The RS perturbation treatment must be adapted to permutation symmetry. The workers just quoted proposed procedures to achieve this symmetrization. From a practical 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 forcing the symmetry operators enter the perturbation equations, significantly complicating their solution. In weak symmetry forcing the perturbed wave functions 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 in the second half of the 1960s, numerical applications were scarce and restricted + to H2 and H2. See Ref. 48 for a review of the exchange perturbation theories up to the beginning of the 1970s.

4 The Middle Ages: Era of Mainframes

Around 1970 every self-respecting university possessed a mainframe computer (in the majority of cases an IBM 360, sometimes a CDC 6600). This was usually placed in a stronghold well defended by brave knights (the computer center staff). A scientist who wanted access to the machine had to master a strange and difficult tongue (job control language) and to cross swords with computer personnel to conquer CPU cycles, RAM, tapes, and disk space. This medieval state of affairs lasted until workstations arrived at the end of the 1980s. The development of ab initio methods, such as speeding up the computation of Gausssian integrals, improving convergence of SCF procedures, and theory and programming of correlation methods was vigorously pursued on mainframes. Electron correlation can be included by configuration interaction (CI) or by coupled cluster (CC) methods. Especially the work on the development of CC methods proved later to be significant for the study of intermolecular forces, because the CC method, in contrast to the configuration interaction (CI) method, is size-extensive. Size-extensivity in the thermodynamic sense of this word (energy linear in amount of substance) implies that in the limit of zero density the energy of a system of molecules converges to the sum of energies of the individual molecules. Much work on the coupled cluster doubles (CCD) method (a supermolecule correlation method) was performed in the 1960s and 1970s by Paldus and Cˇíˇzek and their coworkers49–51 and from the late 1970s onward by Bartlett and coworkers,52, 53 who added single excitations to the method. Also Pople54 recognized the importance of the CC method at a rather early stage.∗ However, in the 1970s the supermolecule correlation methods—and the computers on which they ran—were not yet powerful enough to have much significance 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. 1055

like He2. It was already known that dimer SCF gives a fair description of electrostatic interactions (dipole-dipole, dipole-quadrupole, etc.), of induction forces (dipole-induced-dipole etc.), and also of Pauli repulsion, but not of 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. Dispersion is known to be affected by exchange and so for shorter distance the dispersion has to be damped.55–57 When the multipole-expanded form of the dispersion is used, this damping must also correct for the divergent character of the multipole series. Instead of computing converged dimer SCF energies, one often stopped the SCF procedure after the first cycle. This makes sense when the start orbital set is the direct sum of the sets of occupied MOs of the monomers. In that case the first SCF cycle gives the expecta- A B A B AB A B tion value NhAABΦ0 Φ0 | H + H + V |AABΦ0 Φ0 i, where N is the corresponding normalization constant and the intermolecular interaction operator V AB = H − HA − HB contains the Coulomb interactions between the electrons and nuclei of different molecules. This expectation value accounts for exchange repulsion and electrostatic interaction, albeit without any intramolecular correlation. Induction effects are obtained by cycling the dimer SCF, but because this cycling introduces BSSE and the counterpoise correction was deemed fairly expensive, as it requires three calculations for each geometry of the dimer, the neglect of induction was either accepted, or induction was added later in the multipole expanded form.

4.1 Unexpanded Dispersion Dispersion energy can be computed without invoking the multipole expansion. This was done in the 1970s by, among others, Kochanski,58, 59 Jeziorski and van Hemert,60 and by van Duijneveldt and coworkers.61 It is natural to assume that the supermolecular second-order Møller-Plesset62 energy AB EMP2 also accounts for dispersion energy. And, indeed, in their study on the connection between Møller-Plesset energies and the perturbation theory of intermolecular forces Cha lasi´nski and Szcz¸e´sniak63 showed that for large R the supermolecule second-order (MP2) energy becomes equal to a sum of two monomer MP2 energies plus an uncoupled HF dispersion term. They showed further that for two monomers possessing permanent multipoles the long-range MP2 energy also contains a correlation contribution to the electrostatic energy. In the Appendix we provide a short alternative derivation AB of the asymptotic (large R) limit of EMP2 for the special case of two S-state atoms. Figure 1 gives a diagrammatic representation of the connection be- 1056

AB Figure 1: Goldstone diagrams depicting the large-R behavior of EMP2. See text for definition of orbital labels. Particle orbitals run upward, hole orbitals downward. Each dashed line represents a two-electron integral. Closed lines are summed over. The first row shows Coulomb and the second row exchange interactions. Diagrams in the first row have h = 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+h2lw, where 2l is from spin integration. All diagrams, except w 1 the one with the bar in the middle, have weight = 2 (because of a vertical symmetry plane). An imaginary horizontal line in each diagram gives the energy denominator. The diagrams 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 integrals containing differential overlap between A and B vanish. tween dispersion and MP2 energy. The diagrams in this figure are similar to those of Ref. 64, which was the first work to apply many-body diagrammatic techniques to symmetry-adapted perturbation theory. Note that the disper- AB sion energy and the Coulombic part of the supermolecule energy EMP2 are represented 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 energies. This fact, which holds for S-state systems, is proved explicitly in the Appendix. The Appendix gives the following equation for the unexpanded dispersion energy,

|h ρ(1)σ(2) | r−1 | r(1)s(2) i|2 E =4 12 , (3) disp ǫ + ǫ − ǫ − ǫ ρ,r,σ,s ρ σ r s X 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 the corresponding orbital energy differences. This equation can also be extracted from the fourth diagram in the first row of Fig. 1 by application of the diagrammatic rules. 65, 66 In the mid 1970s valence bond studies on He2 and (C2H4)2 were performed. This work was based on valence bond structures (configurations) 1057 that account for most of the important dispersion effects. The VB structures were constructed from pure monomer MOs, which are orthogonal on each monomer but have intermolecular overlap. The exact, unexpanded, Hamiltonian was used and a secular problem was solved. Because of the non-orthogonality of the orbitals, the main drawback of the VB method is that only a relatively small subset of the electrons can participate in the binding, while the majority of electrons reside in closed shells. The all-electron 65 VB work on He2 brought to light very clearly the considerable size of basis set superposition errors (BSSE), not only in VB results, but also in full configuration interaction results. Until that time BSSE was mainly discussed at the SCF level. Two ab initio methods, which were well-known and much discussed in the 1970s and 1980s, were the pair natural orbital CI (PNO-CI) method and the coupled electron pair approximation (CEPA) method. They were proposed by W. Meyer67 in 1973 and two years later improved by Ahlrichs and coworkers.68 In 1983 Burton and Senff69 applied the method of Ahlrichs et al. to an analysis of the anisotropy of (H2)2 interaction near the minimum in the van der Waals interaction energy. In Eq. (3) we find orbital energy differences in the denominator. This is due 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 operator is due to Epstein70 saved from oblivion by Nesbet.71 This operator can be written as72 H(0) = | I ih I | H | I ih I |, I X where | I i is an excited Slater determinant consisting of localized HF orbitals. A simplification, compatible with Eq. (3), is obtained by restrict- ing the summation to determinants that are singly excited on each of the monomers. Working out the energy denominators h 0 | H | 0 i−h I | H | I i we find orbital energy differences, as in Eq. (3), but shifted by a few additional 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 the French school58, 59, 73–75 strongly preferred EN over MP partitioning, although later French work72 criticizes the use of EN partitioning with delocalized orbitals. See Kelly76 for a proof that EN partitioning gives an (infinite) number of diagonal ladder diagrams in addition the diagrams accounted for by the MP partitioning. Today EN partitioning is rarely applied, mainly for prag- matic reasons, because most standard ab initio packages only have the MP option. 1058

4.2 Multipole-Expanded Dispersion Before 1970 the multipole expansion (by which we mean the expansion in powers of 1/R) of the interaction operator V AB was usually truncated after the R−3 dipole-dipole term, so that the only dispersion interaction term −6 was −C6R . Around 1970 it became clear that this approximation was not suﬃcient and that more terms were needed. However, the straightforward application of the Taylor expansion, and its natural formulation in terms of Cartesian tensors,77 soon becomes cumbersome. Nineteenth century potential theory78, 79 came to the rescue. In this theory the multipole series is rephrased in terms of associated Legendre functions, which enables a closed form of it. Multipole operators are deﬁned as

NX lX lX lX Qm = ZξSm (~rXξ) − Sm (~rXi) with X = A, B ∈ i=1 Xξ X X where ~rXξ and ~rXi are the coordinates of the nuclei ξ, with charges Zξ, and the electrons i of molecule X with respect to a frame with its origin on the l l l nuclear center of mass of X. The function Sm(~r) ≡ r Cm(ˆr) is a regular solid l harmonic; Cm(ˆr) is a Racah normalized spherical harmonic. The intermolecular interaction operator is

N N N N A Z B Z A B 1 Z Z V AB = − β − α + + α β (4) riβ rjα rij rαβ i=1 β∈B j=1 α∈A i=1 j=1 α∈A X X X X X X Xβ∈B

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

~riβ = −~rAi + R~ AB + ~rBβ

~rjα = −~rBj − R~ AB + ~rAα

~rij = −~rAi + R~ AB + ~rBj

~rαβ = −~rAα + R~ AB + ~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, we get the multipole expansion

1/2 2l +2l l +l AB lB +m A B lA+lB ~ lA lB A B V = (−1) Υ−m (R) Q ⊗ Q m , 2lB lAlBm X 1059

~ ~ lA+lB ~ where R = RAB and we recall from the Appendix that Υ−m (R) is proportional 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 of transition matrix elements on A and B. They are of the type

A lA A B lB B lA+lB ′ h Φ0 | Q | Φn i⊗ h Φ0 | Q | Φn i m . This coupling is not convenient. As will become clear below it is better to ﬁrst couple the transition moments on each center. The dispersion energy becomes a sum of which the summand can be expressed with the use of Eq. (2) as a Casimir-Polder integral

∞ ′ ′ (lAlA)LA (lB lB )LB αMA (iω) αMB (iω)dω Z0 over a product of irreducible frequency-dependent polarizabilities. The latter are given by

′ 2(EX − EX ) α(lX lX )LX (iω) ≡ n 0 MX (EX − EX )2 + ω2 n>0 n 0 X ′ LX X lX X X lX A × h Φ0 | Q | Φn i⊗ h Φn | Q | Φ0 i . MX h i The recoupling of the transition moments requires a 9j-symbol, see Refs. 80– 82. Later the same recoupling was performed in Ref. 83. These references give the expression without introduction of the Casimir-Polder integral. See Refs. 84–88 for the recoupled expression containing the Casimir-Polder integral. We have now a closed form of all the terms in the multipole expansion of the dispersion energy. This energy can be computed once the irreducible monomer polarizabilities αLA (iω) and αLB (iω) are known. Unfortunately, this series does not converge; in fact it is divergent and “therefore we may be able to do something with it” [O. Heaviside (1899), as quoted in Ref. 89]. However, the series is asymptotic90–92 in the sense of Poincaré. A very similar equation holds for the multipole-expanded induction energy. The difference is that the polarizabilities are static, so that there is no Casimir-Polder integral, and that one of the irreducible polarizabilities is replaced by a Clebsch-Gordan coupled product of permanent multipole moments. The problem of computing dispersion energies is reduced to the computation of polarizabilities for a sufficient number of frequencies, so that the Casimir-Polder integral can be obtained by numerical quadrature.93 An 1060 alternative to this quadrature is the substitution of the product of the polarizabilities 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 obtained from time-dependent coupled Hartree-Fock calculations,85, 96–98 or from CI calculations.99 The CI calculations of Ref. 99 were at the single and double excitation level. They gave very good results for the frequency-dependent polarizabilities of He and H2—where SDCI is equivalent to full CI—and very poor results for N2, O2 and the neon atom. The failure of the SDCI method for the response properties of more-than-two-electron systems was shown to be caused by unlinked clusters.100 Addition of triply excited states removes the most important unlinked clusters and was shown for Ne2 to improve the results considerably.100 Doran101 was the first to apply Goldstone diagrammatic techniques to the computation of frequency-dependent polarizabilities and dispersion coefficients. He applied his method to Ne2 and heavier noble gases, but owing to an inadequate basis, got results of fairly poor quality. Later Wormer and coworkers87, 93, 102 derived and programmed all polarizability diagrams through second-order of intra-molecular correlation, so that dispersion (by definition second-order in V AB) is completely correlated to second-order on each monomer. Their programs are in practice hardly limited by the rank of the multipoles: up to l = 63 can be computed.

4.3 Applications At the beginning of the 1980s quantum chemical methods and computer hardware had developed to a stage that the computation of properties depending on potential energy surfaces (PESs) of systems larger than two atoms could be contemplated. Examples are thermodynamic properties, such as virial coeﬃcients11, 103 and moments of collision-induced infrared spectral densi- ties.104, 105 The computation of spectroscopic properties of van der Waals molecules came into reach106–111 and also of molecular crystals.112 Intramonomer vibrations have in general a much higher energy than intermolecular vibrations, i.e., the intramolecular motions are much “faster” than the intermolecular motions, so that an adiabatic separation of the two motions is reasonable. In practice this means that we can consider the monomers to be frozen in their vibrationally averaged geometry and that it 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 coordinates R, θ for an atom-diatom system, while for a system consisting of two rigid diatoms R (the distance between the respective mass centers), θA and θB 1061

(the colatitude angles of the diatomics) and φ (the dihedral angle) are very common. An early computation of a full (i.e., depending on all intermolecular coordinates) PES of two diatomics is the work by Berns and van der Avoird 113 on (N2)2. Their approach is in essence the one sketched above: one- cycle SCF to account for first-order exchange and electrostatics (including charge overlap effects) plus multipole-expanded dispersion. The dispersion coefficients were taken from Ref. 114. At that time this was a formidable calculation. It was performed on an IBM 370/158, which was not a supercomputer, but nevertheless a respectable mainframe. A basis of 144 Gaussian type orbitals (GTOs) was used; for restart purposes integrals were stored on tape, requiring two tapes of 170 Mbyte for one point on the potential energy surface. About 150 points were computed, so that 300 tapes were needed. A tape reel had a diameter of 10.5 inch and, including its case, was about 1 inch wide, so that a rack of about 7.5 m long and 30 × 30 cm wide had to be used to store the 50 Gbyte of information. One point on the surface took 2.5 to 3.5 h CPU time and since the whole university—from sociology to solid state physics—used the same mainframe for time-sharing during the day, at most one point per night could be done. The computational part of the project, therefore, took about half a year. Two fits of the PES were made: one in terms of products of spherical harmonics (coupled to a rotational invariant) and one as an atom-atom potential

AB qaqb Cab −Babrab ∆E = − 6 + Aabe . ∈ ∈ rab rab Xa A Xb B 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, cohesion energy and frequencies of translational phonon modes agreed well with experimental values. The frequencies of the librational modes were less well reproduced, but this turned out to be a shortcoming of the self-consistent phonon method. When, later,116, 117 a method was developed to deal properly with the large amplitude librational motions, also the librational frequencies agreed well with experiment. Ten years later van der Pol et al.118 published similar calculations of the CO–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 integrals. The dispersion, computed in the multipole expansion at the MP2 level 1062 of intramonomer correlation,87 was damped by the Tang-Toennies57 damp function. Notice, parenthetically, that the decrease in computer time from the calculations on (N2)2 to (CO)2, a decade apart, was certainly not revo- lutionary. Judging by these calculations the speed of mainframes improved less than an order of magnitude; Moores law119 (doubling of speed every 18 months) predicts two orders of magnitude. The (CO)2 potential of van der Pol was applied120 to the computation of properties of solid CO and gave good agreement with experimental values. A later application121 to the rotation-vibration spectrum of the dimer showed, however, that the potential was not of spectroscopic accuracy.

5 Modern Times: Revolution and Democ- racy

Around 1990 the advent of workstations initiated a revolution in scientific computing. Until that time batch processing was the norm for longer running jobs. The user estimated an upper limit for the CPU time that his/her computation would take, submitted the job, prayed that it contained no trivial errors causing an immediate crash, and then settled down to wait until there was room on the central computer to run the computation. When the workstations arrived, which had the computational speed of mainframes and were cheap enough that research groups could afford one or more, the mode of operation was revolutionized. In the first place, jobs went into execution immediately, so that the user had the chance to weed out trivial errors instantaneously. In the second place, there was no longer a need to chop 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 Sanden 122 123 et al. with the use of an ab initio calculated Ar–NH3 potential. The program Hibridon124 was used to compute the elastic and rotationally inelastic scattering cross sections and the probability that the collisions with Ar invert the ammonia umbrella. A single (one collisional energy) coupled channel calculation on para NH3 colliding with argon took 241 CPU hours and was finished in about two weeks. On a mainframe this would have been a matter of months. The increase of computer power made it possible not only to employ the most refined ab initio methods in the computation of potentials, but also to solve the nuclear motion problem sufficiently often to tune the ab initio potentials to the experimental results. This was the Leitmotif of the past 1063 decade: compute the best possible PES, fit it, solve the appropriate nuclear motion Schrödinger equation for the corresponding van der Waals complex, and compare with experiment. The remaining discrepancies between theory and experiment may be removed by scaling one or more of the parameters in the analytical fit of the potential surface. This procedure has helped in disentangling complicated spectra, for instance the ν3 (asymmetric stretch) 125, 126 spectrum of CH4 in interaction with the argon atom and the ν4 (asymmetric bend) spectrum of the same system.127, 128 At the same time, this provided an assessment of the quality of the ab initio Ar–CH4 potential. An ab initio calculated water pair potential129 was tested and improved130, 131 by the calculation of vibrational-rotational-tunneling spectra of the water dimer and comparison with experimental high-resolution spectra.132, 133 Again, the calculated energy levels and transition intensities134 could be used to assign the bands in the measured spectrum to specific intermolecular vibrations. For the computation of the interaction between two closed-shell monomers there are at present two excellent computational methods, both implemented in black box programs. The first is based on symmetry-adapted perturbation theory135 (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 quantum chemistry group. Jeziorski and his former supervisor Kolos,47, 138 believing in the prospects of SAPT, continued and extended the work of Refs. 37,40– 46, 48; later Szalewicz139 joined forces in this development. These workers came to the conclusion that symmetrized Rayleigh-Schrödinger theory (weak symmetry forcing—see above) was the most viable of the different variants of SAPT. We saw earlier that a very simple form of the dispersion energy is obtained from frequency-dependent polarizabilities at the so-called uncoupled Hartree-Fock level. The sum over states appearing in second order RS perturbation theory is simply a sum over (occupied and virtual) orbitals. A first improvement of this simple model is obtained by including apparent correlation,140 i.e., by using frequency-dependent polarizabilities obtained from the time-dependent coupled Hartree-Fock (TDCHF) method.36, 141 This method was initially proposed in the context of the multipole expansion, but could be generalized142–146 to charge density susceptibility functions (or polarization propagators), which avoids the use of the multipole expansion. It is possible to graft intramonomer correlation corrections onto TDCHF theory, but this is not the road taken by the Warsaw group. Instead they worked top down, 1064 from an exact formulation to (approximate) equations in terms of one- and two-electron integrals that are coded in the SAPT program. We now present the basic philosophy of symmetrized RSPT as implemented 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 Schrödinger (0) AB (0) equation (H + V )Ψpol = EΨpol. Projection with the eigenfunction Ψpol of H(0) with eigenvalue E(0) gives the following exact expression for the interaction energy

(0) AB (0) h Ψpol | V | Ψpol i E − E ≡ Epol = (0) . h Ψpol | Ψpol i

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

(0) AB h Ψpol | V |AABΨpol i ESRS ≡ (0) . (5) h Ψpol |AABΨpol i

If we introduce the intramonomer correlation W A, cf. Eq. (12), multiplied by the perturbation parameter µ, the Schr¨odinger equation for monomer A becomes

HA(µ)ΦA(µ)=(F A + µW A)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. Multiplying V AB with the perturbation parameter λ, we may expand the eigenfunction of H = F A + F B + µW A + νW B + λV AB

∞ i j k ijk Ψpol(λ,µ,ν)= λ µ ν Ψpol. i,j,kX=0

Here the subscript pol indicates that Ψpol(λ,µ,ν) is obtained from PT equations that do not contain intermolecular exchange. Observing that Ψpol ≡ (0) Ψpol(1, 1, 1) and Ψpol ≡ Ψpol(0, 1, 1), we may analytically continue Eq. (5) by 1065

substituting Ψpol(1, 1, 1) → Ψpol(λ,µ,ν) and Ψpol(0, 1, 1) → Ψpol(0,µ,ν) into this 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 by collecting the powers of λ, µ and ν arising from numerator and denominator, ESRS gets the form ∞ i j k ijk ESRS(λ,µ,ν)= λ µ ν ESRS. (6) i,j,kX=0

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, because ijk Epol can be expanded in the same RSPT manner, leading to terms Epol . The ijk ijk difference ESRS − Epol is the exchange contribution to the (i, j, k) term. The terms linear in λ are electrostatic terms and those quadratic in λ can be divided in induction and dispersion terms (including their exchange corrections). 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 forced to introduce a multitude of approximations. In practice, i is restricted to the values 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 to the level of one- and two-electron integrals is a far from trivial job. This has been done in a long series of papers that use techniques from coupled cluster theory and many-body PT; see Refs. 147 and 148 for references to this work and a concise summary of the formulas resulting from it. Some of the earliest potentials computed by the SRS variant of SAPT 149 150, 151 were for Ar–H2 and for He–HF. An application of the latter potential in a calculation of differential scattering cross sections152 and comparison with experiment shows that this potential is very accurate, also in the repulsive re- 153 154 155 gion. Some other SAPT results are for Ar–HF, Ne–HCN, CO2 dimer, and for the water dimer.129, 156 The accuracy of the water pair potential was tested130, 131 by a calculation of the various tunneling splittings caused by hydrogen bond rearrangement processes in the water dimer and comparison with high resolution spectroscopic data.132, 133 Other complexes studied are He–CO,157, 158 and Ne–CO.159 The pair potentials of He–CO and Ne–CO were applied in calculations of the rotationally resolved infrared spectra of these complexes measured in Refs. 160 and 161. They were employed162–165 in theoretical and experimental studies of the state-to-state rotationally inelastic He–CO and Ne–CO collision cross sections and rate constants. It was 1066 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). For 126, 166, 167 He–C2H2, Ne–C2H2, and Ar–CH4 the SAPT potentials were applied in ab initio calculations of the infrared spectra of these complexes. A typical feature of all these potentials for weakly interacting systems is that their shape is determined by a subtle balance between the geometry dependence of the repulsive short range interactions and that of the long range forces, which mostly are attractive. All these results demonstrate that the pair potentials from ab initio SAPT calculations are accurate. Another, more global, comparison with experiment, confirming this finding, was made by computations of the (mixed) second virial coefficients of most of these dimers over 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. For details about this development and a survey of its applications we refer to the 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 by Cˇ´iˇzek and Paldus.49–51 The coupled cluster method may be viewed as a consistent 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) in the Appendix] and coupled cluster theory. Both are supermolecule methods that 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 calculations were widely applied to weakly bound complexes, but soon it was discov- ered that a full MP4 computation (including terms that include sums over triply excited states) is hardly cheaper than a CCSD(T) computation. Since the latter is in general more reliable, MP4 lately lost much ground to the CCSD(T) method. We can be very brief about the coupled cluster method since the chapters by 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

T1+T2+···+TN | Ψ i = e | Φ0 i, (7) 1067

where Φ0 is a closed-shell Hartree-Fock reference function. In most appli- i a ij ab cations T3, T4,... are neglected and only T1 = taEi and T2 = tabEij (sum- a ab mation convention is used here) are included. Here Ei and Eij are orbital replacement operators, where orbital i and j are occupied in Φ0 and a, b refer i ij to virtual orbitals. The cluster amplitudes ta and tab are obtained from the ij i solution of equations that are quadratic in the tab and fourth order in ta. For the closed-shell (spin singlet) case the projection of the CC equation on dou- bly excited states (the CCD method) yields coupled equations of dimension K(K +1)/2 where K is the product of the number nocc of occupied and the number nvir of virtual orbitals. Naively one could expect that the solution of these equations scales as O(K4). For if one linearizes the equations according to the Newton-Raphson method, a set of O(K2) linear equations must be repeatedly solved, which takes O(K4) operations per solution. Fortunately, the scaling is not that bad. In the ﬁrst place the sums in the equations do not run over all four ab orbital labels of tij simultaneously, but at most over two. In the second place a quasi-Newton method, in which the linear equations are approximated by a partially diagonal form, usually converges well. See for more details about the 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 and a corresponding decomposition of the dimer energy into a sum of monomer energies. 176 2 4 In total, the solution of the CCSD equation scales as noccnvirNit, where Nit is the number of quasi-Newton iterations needed. Once the amplitudes a ab ti and tij have been solved, they can be used to compute additional perturbation terms that include triply excited states177 [a non-iterative O(n7) process], which are not accounted for in the CCSD method; their inclusion is indicated by (T) in CCSD(T). We will end this section by mentioning a dozen or so illustrative examples of modern supermolecule calculations on dispersion-bound complexes. Of course, it is hopeless to strive for completeness, almost daily new calculations are 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 applications 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 on 181 182 183 Ar–H2 and Ar–HCl, on N2–HF, and for CO–CO. A few examples of recent CCSD(T) computations on intermolecular potentials are by Cybulski and coworkers, who computed potentials of the noble gas 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 with He, Ne, Ar, Kr,186 and Ar–CO.187 Computational and experimental studies of intermolecular states and forces in the benzene–He complex were reported in Ref. 188. A thorough CCSD(T) study on benzene–Ar is by Koch et al.189 and on Ne–HCl by Fern´andez et al.190

5.3 Latest developments Lately two completely different topics in the field of intermolecular forces have drawn attention and are now actively being studied. In the first place there is the possible application of density functional theory (DFT) to van der Waals molecules. The second topic concerns van der Waals molecules of which the electronic state of one or more of the monomers is spatially degenerate. Density functional theory in the standard Kohn-Sham (KS) formulation has its limitations in application to dispersion forces. Standard local and gradient-corrected functionals are not appropriate for the description of dispersion, which is inherently a non-local correlation effect [there is no such thing as a dispersion potential Vdisp(~r )]. Despite the search for functionals 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 an approach to obtaining non-expanded dispersion by the Casimir-Polder integration of a product of two polarization propagators. This approach can easily and seamlessly be interwoven with DFT,192, 193 because DFT is known to give accurate response properties, provided functionals with correct asymp- totics are used.194–197 Complete intermolecular interaction potentials can be obtained from the so-called DFT-SAPT method that substitutes KS orbitals and exchange-correlation kernels into the SAPT expressions for the interaction energies with j = k = 0, cf. Eqs. (5) and (6). The evaluation of these expressions is computationally much cheaper than the inclusion of monomer correlation effects by calculation of the SAPT terms with j + k > 0. In Ref. 193 various non-hybrid and hybrid exchange-correlation potentials 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 effects of intramonomer electron correlation on the dispersion energy are most accurately reproduced with an asymptotically corrected197 version of the exchange-correlation potential of Perdew and coworkers.198 In Ref. 199 the importance of asymptotically correct exchange-correlation potentials in DFT- SAPT was emphasized particularly. In Ref. 192 dispersion energies of He, 1069

Ne, and H2O dimers were obtained by the DFT-SAPT approach to within 3% or better. Earlier we also discussed the uncoupled HF approach to dispersion, where the sum over states is performed at the orbital level. Of course, this approach can 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 asymptotic correction of the exchange-correlation potential. The DFT-SAPT approach has been very recently applied200 to the noto- 183 riously difficult case of the CO-dimer. Earlier computations of the (CO)2 PES by means of MP4 and CCSD(T) methods encountered some unexpected complications. It was shown that high-order correlation effects are important and that both CCSD(T) and CCSDT formally do not have a correct asymptotic (large R) behavior. Later201, 202 it was pointed out that on top of this problem also very large basis sets are needed for an accurate description of the CO–CO potential energy surface. Notwithstanding this problem, a full 4-dimensional PES (rigid monomers) was computed in Ref. 203 by the CCSD(T) method as a springboard for further refining. The potential was fitted in terms of analytic functions, and the fitted potential was used to compute the lowest rovibrational states of the dimer. It gave semi-quantitative agreement with the experimental infrared and millimeter wave spectra of McKellar, Winnewisser and coworkers.204–210 Application of a fit of the recent DFT-SAPT potential200 gave rovibrational results that differed somewhat from the CCSD(T) data, and were also in semi-quantitative agreement with the measured spectra. It was decided to combine the two potentials, CCSD(T) and DFT-SAPT, and it was shown that a weighted average of the DFT-SAPT (30%) and the CCSD(T) potential (70%) gives results that 12 are in very good agreement with experimental data, for both ( CO)2 and 13 ( CO)2. The second topic of recent interest—dimers that dissociate into a degenerate open-shell monomer and a non-degenerate closed-shell monomer or into two open-shell monomers—has two intrinsic difficulties that are both due to spatial degeneracy. The dimer is an open-shell system in such cases and it has multiple potential energy surfaces that become degenerate for large intermolecular separations and in many cases also for other geometries. In the first place, it is fair to say that at present there are no generally applicable size-extensive electronic structure methods for open-shell, spatially degenerate, systems. From a theoretical point of view, the complete active space multi-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 a reliable account of dynamic correlation effects like dispersion. The CASSCF method has been extended to CAS perturbation theory (CASPT) in order to include dynamical correlation effects.212–217 The CASPT approach is almost size-extensive when the CASSCF reference function is dominated by a single determinant. However, for reference wave functions in which several determinants 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 configuration interaction (MRCI) method. This method is plagued by unlinked diagrams, the presence of which break the size-extensivity of the MRCI energy. Often MRCI results are corrected by a simple formula introduced 30 years ago by Langhoff and Davidson,218 who derived it by inspection of a CI wave function consisting of all double excitations (DCI) from a single Slater determinant. One can look upon this “Davidson correction” as an approximate formula for the unlinked diagram that enters the DCI energy. Paldus, elsewhere in this book, discusses that there is as yet no generally applicable, open-shell, size-extensive, coupled cluster method, and the same holds for open-shell SAPT methods. Therefore, for the computation of potentials of open-shell van der Waals molecules one has the choice between CASSCF followed by a Davidson-corrected MRCI calculation of the interaction energy, or the single reference, high spin, method RCCSD(T). When the ground state of the open-shell monomer is indeed a high spin state, then RCCSD(T) is the method of choice. With regard to the latter method we recall that a major difficulty in open-shell systems is the adaptation of the wave function to the total spin operator S2; for the CCSD method a partial spin adaptation was published by Knowles et al.,219, 220 who refer to their method as “partially spin restricted”. When non-iterative triple corrections221 are included, the spin restricted CCSD(T) method, RCCSD(T), is obtained. Even when free monomers are in degenerate states, the RCCSD(T) method is 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) is formally applicable. But it can be applied only to the lowest state of a given symmetry, while one needs to know also the potential surfaces of the higher dimer states that become asymptotically degenerate with the ground state. Moreover, it is clear that the method fails for points on the PES that have symmetry higher than Abelian and states that belong to more-dimensional representations of the non-Abelian point group. The second problem that often occurs in open-shell van der Waals molecules is the breakdown of the Born-Oppenheimer (BO) approximation. As 1071 is well known, the BO approximation can be trusted when the potential energy surfaces are well separated in energy. However, when certain points on the PES are degenerate this condition is not fulfilled, not in the degenerate points themselves, but also not in nearby points. This breakdown of the BO approximation can be shown as follows. Let us write R for the collection of nuclear coordinates and r for the electron coordinates. Indicating electronic and nuclear interactions by subscripts e and n, respectively, the Schrödinger equation 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, α α α Tn = α Pn Pn /(2Mα) with the nuclear momentum Pn = −i∂/∂Rα. The wave function is expanded in eigenfunctions χk(r; R) of He ≡ Te + Vee + Vne P Ψ(R, r)= χk(r; R)φk(R) Xk ′ ′ with h χk (r; R) | χk(r; R) i(Ö ) = δk k and where the subscript (r) indicates that the integration is over electronic coordinates only. By deﬁnition

′ H ′

r R r R Ö R R h χk ( ; ) | He | χk( ; ) i( ) = e( ) k′k = δk kEk( ), and we assume that χk(r; R) is real (invariant under time-reversal). After multiplication by χk′ (r; R) and integration over r the Schr¨odinger equation is turned into a set of coupled equations depending on nuclear coordinates only [Hn(R)+ He(R)] φ(R)= Eφ(R), where the column vector φ(R) has elements φk(R). The matrix He(R) is diagonal and

H ′ ′

R r R r R Ö n( ) k′k = h χk ( ; ) | Tn | χk( ; ) i( ) + δk kVnn. Suppressing the coordinates in the notation, we can write the matrix elements of Tn as 1

′ ′ ′ α α Ö h χ | T | χ i Ö = δ T + h χ | P χ i P k n k ( ) k k n M k n k ( ) n α α X ′

+h χk | Tnχk i(Ö ) (8) ′ α

The diagonal (k = k) matrix elements h χk | Pn χk i(Ö ) of the operator P α vanish, because this operator is Hermitian and odd with respect to time n reversal. The oﬀ-diagonal matrix elements satisfy

′ α h χk | Pn ,He | χk i(Ö ) ′ α h χk | Pn χk i(Ö ) = . R ′ R Ek( ) − Ek ( ) 1072

We see that whenever two surfaces come close, Ek(R) ≈ Ek′ (R), the nuclear momentum coupling term is no longer negligible. Conversely, if all surfaces are well separated, all off-diagonal terms can be neglected and hence the α whole matrix of Pn is effectively zero. The third term on the right hand side α of Eq. (8) can be written as the matrix of Pn squared and, accordingly, is then 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 approximation equations. Let us, for the sake of argument, assume now that only two surfaces 1 and 2 approach each other and that all other surfaces are well separated; the argument is easily generalized to more surfaces. We then have to solve a set of two coupled nuclear Schrödinger equations with non-negligible coupling

element h χ1(r; R) | Tn | χ2(r; R) i(Ö ). Deﬁne two new orthonormal states by a 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”. ′ α Transformation of the matrix of nuclear momentum h χk | Pn χk i(Ö ) for k′,k =1, 2 gives α

h ϕk | Pn ϕk i(Ö ) =0 for k =1, 2, i.e., the diagonal matrix elements remain zero, and

α α α Ö h ϕ2 | Pn ϕ1 i(Ö ) = Pn γ(R) −h χ2 | Pn χ1 i( ). We search for a γ(R), such that to a good approximation

α α

Pn γ(R) −h χ2 | Pn χ1 i(Ö ) ≈ 0 (10) i.e., ϕ1 and ϕ2 diagonalize the 2 × 2 matrix of the nuclear momentum. By 222 the definition of F. T. Smith ϕ1 and ϕ2 are diabatic states. Smith was the first to define this concept. (Earlier the term “diabatic” was used somewhat loosely by Lichten223). The nuclear motion problem takes the following “generalized Born-Oppenheimer” form

E1(R)+ E2(R) Tn + Vnn + 0 2 φ(R)  E1(R)+ E2(R) 0 Tn + Vnn +  2   E (R) − E (R) cos2γ sin 2γ  + 2 1 φ(R)= Eφ(R). (11) 2 sin 2γ − cos2γ 1073

The surfaces E1(R) and E2(R) are BO energies obtained from electronic structure 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 the remaining 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 α h χ1 | Pn χ2 i(Ö ) by finite difference techniques, which gives the derivative of γ, cf. Eq. (10). Another is by supposing that the diabatic states ϕ and 1 ϕ2 are states of the free monomers and by using Eq. (9) backwards. This is obviously only possible when the adiabatic states χk and χk′ are (almost) pure linear combinations of the two monomer states. This approximation can be made at the orbital level or at the N-electron level (or at both levels simultaneously). Also mixing matrix elements of molecular properties over adiabatic states may be used. We will end this section by mentioning some recent representative calculations on van der Waals molecules consisting of a closed- and open-shell monomer. The simplest closed-shell monomer is of course the ground state helium 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) complex the potential was applied in computing the spectrum of the bound com- plex227 and in photodissociation processes.228 The He–CaH(2Σ+) interaction was employed in the study of collisions at cold and ultracold temperatures,231 and the He–NH(X3Σ−) potential was used in calculations on low temperature collisions 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 diabatic 2 235 intermolecular potential and bound states of the H2–F( P ) complex.

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AB A Relationship between dispersion and EMP2 Often dispersion energy is described as the interaction between mutually induced dipoles, one on each atom. One can see this as a “correlation” between two dipoles. It is not obvious how this “correlation” is related to L¨owdin’s “beyond-Hartree-Fock-correlation”.23 In this Appendix it is shown how the latter correlation and dispersion are interrelated. Earlier this connection was shown63 in a somewhat diﬀerent manner. The MP2 energy, the simplest correlation correction, is obtained from RS perturbation theory with the perturbation

W ≡ H − F −h Φ0 | H − F | Φ0 i, (12) where the Slater determinant Φ0 is the lowest eigenfunction of the Fock N N/2 operator F = k=1 f(k) with eigenvalue 2 i=1 ǫi. The Fock operator serves as the unperturbed (zeroth-order) operator. Since the ﬁrst order MP energy P P h Φ0 | W | Φ0 i is obviously zero, the lowest order MP energy appears in second 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-Condon rules for the simpliﬁcation of N-electron matrix elements and integrating out spin, it becomes

AB −1 EMP2 = h φi(1)φj(2) | r12 | φa(1)φb(2) i (13) i,j,a,bX 2h φ (1)φ (2) | r−1 | φ (1)φ (2) i−h φ (1)φ (2) | r−1 | φ (1)φ (2) i × a b 12 i j a b 12 j i , ǫi + ǫj − ǫa − ǫb where φi and φj are occupied and φa and φb are virtual orbitals of the dimer A–B. We consider the limit of this expression for R large enough that the diﬀerential overlap between wave functions of A and B can be neglected. We recall that we can localize SCF orbitals and write | ρ i and | r i for the occupied and virtual spatial orbitals localized on A and | σ i and | s i for the occupied and virtual orbital localized on B. These orbitals are expressed in the dimer basis. The Fock operator is invariant under unitary localization of the {φi}, i.e.,

(N +N )/2 A B 2 − P h φ (2) | 12 | φ (2) i = i r i i=1 12 X N /2 N /2 A 2 − P B 2 − P h ρ(2) | 12 | ρ(2) i + h σ(2) | 12 | σ(2) i. r r ρ=1 12 σ=1 12 X X 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 applicable here, become equal to the orbital energies of monomers A and B. Let us consider two S-state atoms and the action of the dimer Fock operator on, for instance, | ρ(1) i

1 Z 2 − P f AB(1)| ρ(1) i = − ∇2 − A + h ρ(2) | 12 | ρ(2) i | ρ(1) i 2 r r " A1 ρ 12 # X Z 2 − P + − B + h σ(2) | 12 | σ(2) i | ρ(1) i. (14) r r " B1 σ 12 # X Because of zero diﬀerential overlap the P12 contribution can be dropped in the second term of Eq. (14). The terms that remain in the second expression between large square brackets cancel each other. This is because the elec- 2 tronic charge distribution Q(~rB2) ≡ 2 σ |σ(2)| is spherically symmetric and screens completely the nucleus of B. P 1089

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

∞ 1 l = (−1)mΥl (R~)Sl (~r) (15) ~ −m m |R − ~r| − Xl=0 mX= l l 1/2 2l l Sl (~r − ~r ) = (−1)L Sl−L(~r ) ⊗ SL(~r ) . (16) m 1 2 2L 1 2 m L=0 X l ~ l−1 l ˆ l Here Υm(R) ≡ R Cm(R) is an irregular solid harmonic function and Sm(~r) ≡ l l l r Cm(ˆr) is a regular solid harmonic function. The function Cm(ˆr) is a spherical harmonic function normalized to 4π/(2l +1) (Racah normalization). The expression between square brackets in Eq. (16) is a Clebsch-Gordan coupled product. We write ~r12 = −~rA1 + R~ AB + ~rB2, and ﬁnd, assuming that |R~ AB| > |~rB2 − ~rA1|, Z 1 Z − B +2 h σ(2) | | σ(2) i = − B (17) r r r B1 σ 12 B1 X 1/2 2l l +2 (−1)L+m Υl (R~ ) Sl−L(~r ) ⊗h σ | SL(~r ) | σ i . 2L −m AB A1 B2 m L,l,m σ X X L L The expression h SM i ≡ 2 σh σ | SM (~rB2) | σ i is the Hartree-Fock expectation value of the (L, M) multipole moment of the S-state atom B. When P L the charge distribution Q(~rB2) is spherical symmetric around B h SM i = NBδL0δM0. Equation (17) becomes under this condition

ZB m l ~ l ZB NB − + NB (−1) Υ−m(RAB)Sm(~rA1)= − + . rB1 rB1 rB1 Xl,m The simpliﬁcation of this result follows from Eq. (15). Since for neutral atoms NB = ZB the second term of Eq. (14) indeed vanishes. It follows that the dimer Fock operator, when it acts on orbital | ρ i localized on monomer A, is equivalent to the atomic Fock operator of A

f AB| ρ i = f A| ρ i. (18)

Under the conditions of our derivation, i.e., S-state atoms A and B with vanishing diﬀerential overlap, we can show that the localized orbitals | ρ i and | σ i are identical (apart from mixing possibly degenerate orbitals) to 1090 the orbitals obtained by solving the monomer Hartree-Fock equations (in the dimer basis)

A B f | ρ i = ǫρ| ρ i and f | σ i = ǫσ| σ i.

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

If ǫρ =6 ǫk it follows that Ukρ = 0, so that, in general, | ρ i is a linear combi- nation of degenerate dimer orbitals | k i with orbital energy ǫk = ǫρ. If there is no degeneracy, then | ρ i is identical to | k i. The same argument may be applied to the other localized dimer orbitals | r i, | σ i and | s i. In other words, we can solve the monomer HF equations in the dimer basis and get the same orbital energies as from the solution of the dimer HF equations. When we now replace the sums over the canonical orbitals by sums over localized orbitals and the dimer orbital energies by monomer orbital energies in Eq. (13), we obtain

−1 2 AB A B |h ρ(1)σ(2) | r12 | r(1)s(2) i| lim EMP2 = EMP2 + EMP2 +4 . R→∞ ǫ + ǫ − ǫ − ǫ ρ,r,σ,s ρ σ r s X See Fig. 1 for a diagrammatic representation of this limit. The third term on the right hand side is the non-expanded “Hartree-Fock” expression59 for dispersion. Incidentally, this equation shows that the MP2 method is size- extensive. That is, when the distance RAB between A and B is so large that the interaction term vanishes, the dimer MP2 energy becomes the sum of the monomer MP2 energies. Although this statement sounds obvious, it is not. The singles and doubles conﬁguration interaction method forms a counter example. 1091

The equivalence between the interaction energy from dimer MP2 calculations and the simple expression for the dispersion interaction does not hold when the interacting systems are molecules or non-S-state atoms. The second term of Eq. (14) does not vanish in that case, because of non-vanishing multipole moments contributing to the expansion in Eq. (17). Even for large distances R, where all diﬀerential overlap between A and B vanishes and the dimer orbitals can be localized, these orbitals are not equal to the unperturbed monomer orbitals. This is due to the polarization of each monomer, induced by the multipole moments of the other monomer. This gives long range electrostatic and induction interactions, which thus are accounted for by the supermolecule HF method. Conversely, for spherically symmetric systems the HF method does not give any interaction at distances where diﬀerential overlap is negligible.