Faraday Symp.Chem. Soc. 1984 19 7-15 General Introduction Computational Quantum Chemistry -1984 BY ERNESTR. DAVIDSON Indiana University Bloomington Indiana 47405 U.S.A. Received 12th December 1984 Over the course of the last three decades theoretical chemists have devised a quantitative model of the chemical bond which in spite of its increasing applicability is still often disregarded by experimentalists. Application of this model differs from the pre-quantum tradition of interp- olating between established experimental facts using human reasoning alone in that it relies on digital computers to extract the implications of the model for a particular molecule. While the improved accuracy associated with this model has resulted in increasingly complicated highly correlated wavefunctions a qualitative understanding of the model’s predictions can usually be obtained from a consideration of the simpler zeroth-order wavefunction.G. H. Hardy,l in an essay entitled ‘Apologies of a Pure Mathematician’ explained his motivation for wasting his life in such a useless activity as pure mathematics. Basically he sought in the theorems he created the same kinds of beauty and permanence which are usually attributed to great works of art or music. Many chemists also regard computational quantum chemistry as essentially useless. Worse the results obtained by computation are somehow regarded as uglier and less creative than those obtained by pure reasoning.** Certainly computational results are less permanent as improved calculations on simple molecules seem to appear monthly.In reply to these criticisms I would claim that through computation quantum chemists have created a quantitative model of the chemical bond which is beautiful to those who understand it and which is likely to be permanent. Further this develop- ment would never have been possible by reasoning alone. In this introductory lecture I want to address what I feel are the essential differences between this model and the traditional chemical models of bonding. I also wish to illustrate the range of validity of the model by giving examples of its successes and failures. The present quantitative model of the chemical bond is the result of 30 years of hard work by numerous chemists and applied mathematicians.In spite of its sweeping importance to chemistry the model itself is little appreciated by many chemists who continue to interpret their experimental results in terms of quite crude older models. Partly this is because the literature produced by computational chemists is so specialized and technical that most chemists are unaware that a true revolution has taken place. However it is also true that this new model requires use of computers to extract its implications for any given molecule so experimental chemists are less likely to adopt this model when analysing their data. From the beginnings of the atomic theory of matter chemists have been concerned with what held atoms together in molecules. The pre-quantum theory culminated in the work of G. N. Lewis and others which recognized concepts such as the shared electron pair the octet rule electronegativity partially ionic bonds etc.These models 7 GENERAL INTRODUCTION and the known periodic trends of the elements allowed qualitative predictions and interpretations of experimental data based on pure human reasoning. After the discovery of the Schrodinger equation the discovery of the electron spin and the formulation of the Pauli principle in terms of Slater determinants these pre-quantum ideas were transcribed into statements about wavefunctions. This approach was set forth most elegantly by Pauling in his book The Nature of the Chemical Bond,4 which became the Bible of a generation of inorganic chemists. This work continued the grand tradition of making qualitative predictions about chemical behaviour by using only human reasoning to interpolate between established experi- mental facts.It also made semiquantitative predictions of heats of reaction bond lengths etc. using simple formulae with a vague quantum-mechanical justification to interpolate between known quantitative data from other molecules. During this era of 1930-1950 two other models of the chemical bond were introduced. Crystal-field theory5 was developed to explain the perturbation of atomic energy levels by a crystal environment. This model led naturally to the ligand-field model of the perturbation of the d-electron energies in a transition-metal complex. Unlike the models discussed above this model required at least a calculator to solve the resultant eigenvalue problem.The resultant Tanabe-Sugano diagrams are still widely used to interpret spectra and ligand-metal bond energies. Another spin-off from the physicists’ approach to crystal structure was the molecular-orbital model of the chemical bond advocated by Mullikeq6 Huckel,’ WalshS and others. As a qualitative tool this model also allowed prediction of bond length and energy trends. It was more successful than the valence-bond method in describing molecular spectra but probably less successful in qualitatively accounting for the shape of molecules. With the advent of computers attempts were made to extend the applicability of the molecular-orbital model and to improve its reliability. The valence-boqd model was more difficult to reduce to a simpler computer algorithm so it was largely abandoned in quantitative applications.The present extended Huckel theory used by H~ffmann,~ CNDO developed by PoplelO and MNDO developed by Dewarl’ are the results of this refinement of the empirical molecular-orbital model. Embedded in these computer programs is a model of the chemical bond and a set of parameters derived by comparing results with experimental data. The model however is now complex enough that semiquantitative predictions require the use of a computer to work out the predictions of this model for any particular molecule. From the conception of the Schrodinger equation,12 there has been another group of chemists and physicists who were interested in really solving this equation.Heitler and London13 quickly carried out calculations for the hydrogen molecule. Hartree14 developed numerical SCF methods for atoms. Hylleraas15 and James and Coolidge16 carried out very-large-scale calculations (as measured by the number of hours of computing machine time used) on the helium atom and the hydrogen molecule. These calculations served mostly to establish that the Schrodinger equation unlike the Bohr theory which preceded it was probably ‘correct’ to at least one part in ten thousand for these two-electron systems. Application to larger systems was cut short by lack of adequate computational tools and the intervention of the Depression and World War 11. With access to computers in the 1950s a few chemists returned to this effort to accurately solve the Schrodinger equation and began developing computer programs to carry out ab initio calculations.As an aside let me pause at this point to say that I want to use ab initio in a very narrow sense in this lecture. For the purpose of this talk ab initio will mean ‘from the beginning’ i.e. without any input from past E. R. DAVIDSON Table 1. Results using atomic orbitalsav reaction AE,,,,/kJ mol-l AE,,,,/kJ mol-l H +2H 280 456 HF+H+F 209 569 CH +C+4H 1105 1636 CH +CH,+H 364 43 1 a Full CI experimental geometry minimum basis of near Hartree-Fock atomic orbitals. For H an SCF calculation in this basis gives AE = 209 kJ mol-l. MCSCF gives an extra 71 kJ mol-1 and distortion of the atomic orbitals gives 117 kJ mol-l.Correlation effects not describable with two distorted atomic orbitals give the remaining 59 kJ mol-l. experience or any parameters chosen because they worked for other atoms or molecules. With this definition true ab initio calculations are always exploratory and of interest mainly to other quantum chemists. If successful they may lead to improvements which can be incorporated into widely used ‘calibrated ab initio’ calculations carried out by applied quantum chemists. Like semiempirical methods these calculations assume transferability of parameters and methods between molecules and do not check the convergence of every result with respect to quality of wavefunction. The first ab initio calculations merely evaluated exactly the energies associated with the model wavefunctions of the semiempirical models.These calculations were disasters and led to a major schism between pure and empirical theorists2 which still has not healed. To oversimplify the situation these calculations gave energies which were in serious disagreement with experiment. The empiricists regarded this as proof that the ab initio approach was a waste of time. The purists on the other hand felt that their calculations proved that the empirical models were without fundamental justification and hence were basically ‘wrong’ even if they did produce useful predictions. Table 1 shows this dichotomy. While the binding in simple molecules like H, HF and CH is usually ‘explained’ using atomic orbitals actual ab initio calculations with Hartree-Fock atomic orbitals only account for 40-60% of the bond energy.It is recognized today of course that models like MNDO which seem to use atomic orbitals actually are parametrized to account for the effect of a more extended basis set. During the years from 1961 to 1970 a few theorists with access to computers persisted in their efforts to develop an accurate method based firmly on the Schrodinger equation for making predictions of chemical facts. The hoped-for method was required to give not only ‘right’ answers for ‘right’ reasons but was also required to be computationally tractable conceptually simple and widely applicable. By and large the poor results produced by many of the suggested methods led to this whole enterprise and the chemists associated with it being held in ill repute by the rest of the chemical community.Accurate results were slow in coming and when available gave little chemical insight. The requirement of tractibility was a major bottleneck in early calculations. As the power of computers has increased and the costs have gone down this requirement has eased considerably. Anyone carrying out today an MCSF gradient optimization of a transition-state geometry for a reaction like the Cope rearrangement” must be aware that most of the ideas required for this calculation were in place by 1969 but GENERAL INTRODUCTION Table 2. Approximate characteristics of some typical computers computer speed memory costa cost/speed IBM 650 1 103 1O6 1O6 CDC 1604 102 104 107 105 IBM 7094 104 104 107 103 CDC 7600 106 105 107 10 IBM 370 1O6 1O6 107 10 CRAY 1 107 1O6 107 1 VAX 11/780 105 1O6 105 1 FPS-164 106 1O6 105 lo-’ CRAY-XMP 1O8 107 107 10-1 a Cost new in 1984 U.S.dollars. such a calculation was unthinkable at that time because of the computer limitations. Table 2 summarizes this trend in computer costs. The major price break associated with the super-mini computers and the class VI computers has spurred the very ambitious calculations of the last decade. Wide applicability is also a limiting consideration. It is relatively easy to construct good basis sets for atoms and diatomic molecules which cannot be used for polyatomics. Use of elliptical orbitals for example gave excellent resultsl89 l9 for H and LiH but were not even applicable to heavier diatomics.Explicit use of the interelectron distance in the wavefunction also gave excellent resultsl5? l6 for H and He but is not widely useful for polyatomic molecules. Wide applicability requires an open-ended method which is capable at least in principle of arbitrary accuracy for any molecule. Conceptual simplicity was a major consideration in the early search for accurate methods. Much time and effort went into testing new types of wavefunctions such as GVB,,O strongly orthogonal geminals,21 valence-space MCSCF2 etc. These wavefunctions were regarded as conceptually simple extensions of SCF wave function^.^^ Direct calculation of configuration interaction wavef~nctions~~ with non-orthogonal basis functions was abandoned because it was feasible only for a few electrons and also because it led to wavefunctions which could not be interpreted.At present configuration interaction with configurations built from orthonormal orbitals as advocated by Boys,25 has been adopted as the only tractable open-ended scheme.26.27 Use of MCSFC28 or natural orbitalsl9$ 29 in the CI reduces the number of configurations needed but this method is still criticized for the lack of conceptual simplicity of the resultant wavefunctions. This criticism is very much to the point when an elaborate calculation is carried out as I recently did for a molecule such as water at its ground-state equilibrium geometry.30 Most chemists will concede that the Schrodinger equation is correct and that the exact solution will reproduce the expeximental binding energy dipole moment etc.Hence a calculation must do more than merely agree with the experimental facts without offering an interpretation of why the dipole moment or binding energy has that particular value. On the other hand correlation effects are not simple. It is in fact a contradiction to ask for a wavefunction which is both simple and accurate. Calibration tests of computational schemes on molecules like water are important to establish credibility of a new method before it is applied to molecules where the experimental facts are in question. In doing elaborate calculations of properties of molecules however it E. R. DAVIDSON must always be kept in mind that the experimentalist who measures these properties does so in order to learn about the electronic structure of the molecule.A calculation should not only reproduce the experimental properties but it should also offer the correct interpretation of why the property has that particular value. ATOMIC ORBITALS The new model of the chemical bond which developed during the 1960s was a logical extension of the LCAO-MO-SCF model which had previously failed to provide quantitative results. The new model is still built at least conceptually on near Hartree-Fock atomic orbitals. In order to obtain quantitative results it is essential to add to this basis additional functions which can account for the change in size and shape of the valence orbitals in the molecular environment.Hence the simplest basis set for a semiquantitative model is the split-valence plus polarization basis introduced by Ne~bet.~l Chemists who are used to thinking in terms of loosely defined 'atomic orbitals' find it hard to interpret the results based on this extended basis set. The fact that d orbitals are essential to get the correct bond angles in hydrogen peroxide3' or the correct bond length in oxygen33 goes beyond the crude model based on unmodified atomic orbitals. Because integrals are easier for Gaussian basis functions these orbitals are usually approximated as linear combinations of Cartesian Gaussian orbitals. For computa- tional simplicity a linear transformation of this basis is often made to a new basis spanning the same vector space but requiring fewer Gaussians per basis function.Also core orbitals are sometimes represented in lower accuracy in the hope that any errors introduced will cancel in computing energy differences. This contracted-orbital approach was pioneered by Whitten34 and is used in almost all molecule calculations. Split-valence plus polarization Gaussian basis sets were deveioped independently by P~ple~~ and MOLECULAR ORBITALS In the model of 1970 molecular orbitals were constructed from the atomic basis by solving SCF equations for the electronic state of intere~t.'~ The virtual orbitals from SCF calculations proved useless for describing electron correlation so they were transformed to natural orbitals by an iterative process.lg A different set of molecular orbitals was used for each state of the molecule.During the decade following 1970 some refinements of the method for constructing molecular orbitals was made. The GVB method of Goddard2* and Wahl's limited MCSCF calculations2s were expanded to include more general MCSCF wavefunctions as methodological improvements (and faster computers) made this feasible. These more general wavefunctions incorporate the virtues of valence-bond wavefunctions without the computational difficulties of dealing with non-orthogonal basis functions. They still require different orbitals for different states and some modification of virtual orbitals to make them useful for describing electron c~rrelation.~~ These generalized molecular orbitals do not give orbital energies which can be used to form a molecular-orbital energy diagram of the type popular in inorganic texts.The interpretation of the orbitals used to describe electron correlation is difficult because there is usually no electronic state of the molecule in which these orbitals play a dominant role. GENERAL INTRODUCTION WAVEFUNCTIONS In 1970 the correlation effect on the energy and properties was estimated by carrying out a variational calculation using the double excitations which enter the wavefunction in first-order perturbation theory. Single excitations were also included because they affect properties other than the energy in the same order of perturbation theory as the double excitation^.^^ Peyerimhoff Bender and Schaeffer to name a few investigated the ground and excited states of several m01ecules.~~ Following 1970 configuration interaction was enlarged to include a more general approximation based on including all configurations with large coefficients as part of the zeroth-order wavefunction plus all configurations mixing with these in first order.26 Recognition of the fundamental inability of CI to describe correlation effects in large led to an increased interest in perturbation theory.Unfortunately the simplest form of perturbation theory proved inadequate because it could not handle zeroth-order wavefunctions requiring several configuration^.^^ However perturbation theory showed that triple and quadruple excitations were more important than previous CI calculations had assumed.The standard ‘state-of-the-art ’ ab initio model of 1980 certainly includes all single and double excitations from all the important zeroth-order configurations. Configuration interaction and perturbation theory are merging into a more unified approach based on linked-cluster experiments unlinked-cluster corrections and variation-perturbation theory. Accurate wavefunctions are complicated. Fortunately one can usually understand the zeroth-order wavefunction. The other terms merely describe the fact that electrons avoid each other and are needed for quantitative but not qualitative results. Use of charge densities bond orders etc. to interpret complex wavefunctions greatly facilitates understanding them. 42 APPLICATIONS Many calibrated ab initio packages such as GAUSSIAN 80 are now available which allow any chemist access to the predictions of this modern model of the chemical bond.Interest in the results has been spurred by the technological advances which make geometry optimization easier. As an outgrowth of Pulay’s ideas,43 optimization based on analytical gradients is now routine for both SCF and MCSCF calculations. One application of these methods to electronic structure has been the prediction of the singlet-triplet energy difference of methylene. Over the years both theory and experiment have given a wide variety of results for this molecule. Since 1970 however the theoretical results have agreed on the bond angles and relative energies.44 Table 3 shows that the energy difference does converge with improved calculations.As in the previous examples atomic orbitals give the wrong results but with allowance for distortion in size and shape the correct result is easily obtained. Because the correct result differed from some experiments much more elaborate calculations have been done to verify the convergence. The ability to compute derivatives of the potential energy has made it possible to enlarge the studies of molecular spectra to include vibrational frequencies and intensities in addition to electronic excitation energies. Some examples of this are presented in this Symposium by Peyerimh~ff~~ and S~haeffer.~~ The ideas introduced by Woodward and Hoffmann4’ to explain chemical reactivity have also popularized calculations among organic chemists.The ability to predict the transition-state geometry and to follow the Fuk~i~~ reaction path features not accessible to experiment has increased interest in the theoretical results. A typical E. R. DAVIDSON Table 3. Convergence of the singlet-triplet energy difference of methylene (in kJ mol-l)a* basis SCF TCSCF CI minimum 167 139 DZ 134 96 100 DZP 109 54 63 extended 105 46 46 a Double-zeta basis (DZ) allows for change in size of atomic orbitals polarization (DZP) allows also for change in shape. Extended basis sets allow for better description of these effects as well as better description of electron correlation. The TCSCF results are based on using a single spin-restricted SCF wavefunction for the triplet and a two-configuration description for the singlet.Two configurations are needed for the singlet in a zeroth-order description at larger bond angles. example of reactions which have been studied is the methylenecyclopane rearrangement.49 This molecule may undergo either stepwise or concerted ring opening to trimethylene methane. Calculations located three transition states and agreed with experiment that the non-concerted path was lowest. Another reaction illustrating this same question of step-wise versus concerted reactions is the decomposition of 2-carbena- 1,3-dioxolane to ethylene and carbon dioxide. Even though the concerted reaction is Woodward-Hoffmann allowed calculations indicate the reaction proceeds step-wise with a diradical intermediate.50 The paper by Bernardi et al.in this Symposium deals with a similar reaction.51 More recently the Cope rearrangement has been studied. In this case the calculations indicate the reaction is concerted and passes through a symmetrical transition statel’ rather than proceeding via a diradical intermediate. Now that an adequate model exists for predicting the electronic structure of small gaseous organic molecules over a wide range of geometries and excitation levels one might wonder whether there are problems left requiring a truly ab initio approach. The answer is definitely yes. Calculations on metallo-organic and metal-metal bonds are still unsatisfactory and not r0utine.~~9 53 Little progress has been made on studying reactions in s01ution.~~9 Properties like the spin density56 and field gradient30 are 55 inadequately predicted by standard basis sets.The standard model has not even been applied to such a simple problem as predicting the structure of NaCl (solid) so it is uncertain whether it will predict the correct crystal structure. Calculations on the structure of ice indicate that it is very difficult to predict the correct 0-0 and j8 0-H distances and proton field gradients in a hydrogen-bonded On a more mundane level it should be noted that even the electron affinity of carbon and oxygen atoms are difficult to compute to better than 10 kJ mol-1 even with expanded basis sets and CI.59Thus there remains ample challenge to those wishing to do ab initio calculations. At the same time we can look with satisfaction at the very useful model which has developed over the past 20 years.This model should be introduced into freshman texts and other descriptive works as a refinement of the common LCAO-MO-SCF and VB models. The model itself is actually fairly simple even though a computer is required to work out the implication of the model for any particular molecule. 14 GENERAL INTRODUCTION G. H. Hardy A Mathematician’s Apology (Cambridge University Press Cambridge 1969). C. A. Coulson Rev. Mod. Phys. 1960 33 170. a J. 0.Hirschfelder Annu. Rev. Phys. Chem. 1983 34 1. L. Pauling The Nature of the Chemical Bond (Cornell University Press Ithaca NY 1940). Y. Tanabe and S. Sugano J. Phys. Soc. Jpn 1954 9 753. R. S. Mulliken Phys.Reti. 1936 50 1017. E. Huckel Z. Phys. 1931 70 204. A. D. Walsh J. Chem. Soc. 1953 2260. R. Hoffmann J. Chem. Phys. 1963 39 1397. ” J. A. Pople and G. A. Segal J. Chem. Phys. 1966 44 3289. M. J. S. Dewar and W. Thiel J. Am. Chem. Soc. 1977,99 4899. l2 E. Schrodinger Ann. Phys. 1926 79 361. l3 W. Heitler and F. London 2. Phys. 1927,44 455. l4 D. R. Hartree Proc. Cambridge Phil. Soc. 1928 24 89. l5 E. Hylleraas Z. Phys. 1930 65 209. l6 H. M. James and A. S. Coolidge J. Chem. Phys. 1933 1 825. l7 Y. Osamura S. Kato K. Morokuma D. Feller E. R. Davidson and W. T. Borden J. Am. Chem. Soc. 1984 106 3362. l8 S. Rothenberg and E. R. Davidson J. Chem. Phys. 1966 45 2560. l9 C. Bender and E. R. Davidson J. Phys. Chem. 1966,70 2675. 2o W. J. Hunt P.J. Hay and W. A. Goddard 111 J. Chem. Phys. 1972 57 738. *l A. C. Hurley J. E. Lennard-Jones and J. A. Pople Proc. R. Soc. London Ser. A 1953 220,446. 22 G. Das and A. C. Wahl J. Chem. Phys. 1966,44 87. 23 C. C. J. Roothaan Rev. Mod. Phys. 1951 23 69. 24 J. C. Browne and F. A. Matsen Phys. Rev. 1964 135 A1227. 25 S. F. Boys Proc. R. Soc. London Ser. A 1950 200 542. 26 R. J. Buenker S. D. Peyerimhoff and W. Butocher Mol. Phys. 1978,35 771. 27 I. Shavitt in Methods of Electronic Structure Theory ed. H. F. Schaeffer I11 (Plenum Press New York 1977). 28 A. C.Wahl and G. Das in Methods of Electronic Structure Theory ed. H. F. Schaeffer I11 (Plenum Press New York 1977). 29 P. 0.Lowdin Phys. Rev. 1955 97 1474. 30 D. Feller and E. R. Davidson Chem. Phys.Lett. 1984 104 54. 31 R. K. Nesbet Phys. Rev. 1968 175 2. 32 T. H. Dunning and N. W. Winter Chem. Phys. Lett. 1971 11 194. 33 R. A. Whiteside M. J. Frisch J. S. Binkley D. J. DeFrees H. B. Schlegel K. Raghawahri and J. A. Pople Carnegie-Mellon Quantum Chemistry Archive (Carnegie-Mellon University Pittsburgh PA 2 vol edn 1981). 34 J. L. Whitten J. Chem. Phys. 1963 39 349. 35 P. C. Hariharan and J. A. Pople Theor. Chim. Acta 1973 28 213. 36 T. H. Dunning Jr and P. J. Hay in Methods of Electronic Structure Theory ed. H. F. Schaeffer I11 (Plenum Press New York 1977). 37 D. Feller and E. R. Davidson J. Chem. Phys. 1981 74 3977. 38 E. R. Davidson Reduced Density Matrices in Quantum Chemistry (Academic Press New York 1976). 39 E. R. Davidson and L.E. McMurchie in Excited States ed. E. C. Lim (Academic Press New York 1982) vol. 5. 40 E. R. Davidson and D. W. Silver Chem. Phys. Lett. 1977 52 403. 41 R. J. Bartlett Annu. Rev. Phys. Chem. 1981 32 359. 42 E. R. Davidson J. Chem. Phys. 1967 46 3319. 43 P. Pulay Mol. Phys. 1969 17 197. 44 E. R. Davidson in Diradicals ed. W. T. Borden (Wiley New York 1982). 45 S. D. Peyerimhoff Faraday Symp. Chem. Soc. 1984 19 63. 46 M. E. Colvin and H. F. Schaeffer 111 Faraday Symp. Chem. Soc. 1984 19 39. 47 R. B. Woodward and R. Hoffmann The Conservation of Orbital Symmetry (Academic Press New York 1971). 48 K. Fukui J. Phys. Chem. 1970 74 4161. 49 D. Feller K. Tanaka E. R. Davidson and W. T. Borden J. Am. Chem. Soc. 1982 104 967. 50 D. Feller E. R. Davidson and W.T. Borden J. Am. Chem. Soc. 1981 103 2558. j1 F. Bernardi A. Bottoni J. J. W. McDouall M. A. Robb and H. B. Schlegel Faraday Symp. Chem. Soc. 1984 19 137. A. D. McLean and B. Liu Chem. Phys. Lett. 1983 101 144. j3 P. E. M. Siegbahn Faraday Symp. Chem. Soc. 1984 19,97. E. R. DAVIDSON 54 J. Chandrasekahn S. F. Smith and W. L. Jorgensen J. Am. Chem. Soc. 1984 106 3049. 55 K. Morokuma K. Ohta N. Koga S. Obara and E. R. Davidson Faraday Symp. Chem. SOC.,1984. 19 49. 56 D. Feller and E. R. Davidson J. Chem. Phys. 1984 80 1006. 57 E. R. Davidson and K. Morokuma J. Chem. Phys. 1984,81 3741. 58 E. R. Davidson and K. Morokuma Chem. Phys. Lett. 1984 111 7. 59 D. Feller L. E. McMurchie W. T. Borden and E. R. Davidson J. Chem. Phys. 1982 77 6134.