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, dis- persion, exchange, that originate from different mechanisms by which the Coulomb interactions can lead to either repulsive or attractive forces be- tween 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 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 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 chem- istry 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 Schr¨odinger 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 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 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 quan- tum 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 “dis- persion 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 (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 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, inciden- tally, 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 computa- tion 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 chem- istry 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 cal- culations 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.