Essay

Valence Bond Theory for Chemical Dynamics DONALD G. TRUHLAR

Department of Chemistry and Supercomputing Institute, University of Minnesota, 207 Pleasant Street S.E., Minneapolis, Minnesota 55455-0431 Received 25 April 2006; Revised 30 June 2006; Accepted 4 July 2006 DOI 10.1002/jcc.20529 Published online 20 October 2006 in Wiley InterScience (www.interscience.wiley.com).

Abstract: This essay provides a perspective on several issues in valence bond theory: the physical significance of semilocal bonding orbitals, the capability of valence bond concepts to explain systems with multireferences character, the use of valence bond theory to provide analytical representations of potential energy surfaces for chemical dynamics by the method of semiempirical valence bond potential energy surfaces (an early example of specific reaction parameters), by multiconfiguration molecular mechanics, by the combined valence bond-molecular mechanics method, and by the use of valence bond states as coupled diabatic states for describing electronically nonadiabatic processes (photochemistry). The essay includes both ab initio and semiempirical approaches. q 2006 Wiley Periodicals, Inc.

J Comput Chem 28: 73–86, 2007

Key words: bonding; electronically excited states; multireference character; potential energy surfaces; reactivity

Introduction I am grateful to the organizers of this special issue for the invitation to write one of the essays on chemical bonding, and I have chosen to focus on valence bond (VB) theory. Valence bond theory1 is of special interest to chemical reaction dynamicists, dating back to the early work of London, Eyring, and Polanyi2,3 on potential energy surfaces for reactive systems, and this essay will be especially flavored by my experiences in chemical dynamics research. An important lesson learned in the early days of computational quantum chemistry is the slow convergence of configuration interaction treatments of dynamical correlation energy when the configuration state functions are expressed in hydrogen-like atomic orbitals.4 This led to one of the early reasons for emphasizing the molecular orbital (MO) approach, namely that use of orthogonal MOs (as compared to nonorthognal valence bond orbitals) is very convenient for large configuration interaction calculations, and the MOs that are self-consistently optimized by Hartree–Fock theory are very close to the natural orbitals that lead to optimum convergence of the configuration interaction expansion.5–7 Many years later, as correlated wave function approaches began to be applied to larger and larger systems, it became clear that localized orbitals have important advantages for such calculations8–13 because they allow one to concentrate the correlation energy in a smaller number of terms,8 and they allow one to treat the correlation in only the part of the molecule, sometimes called the active part, that is directly involved in the process of interest, or in only a subset of bonds.12 The

correlation energy of the rest of the molecule, sometimes called the spectator part,8 tends to be more constant, and a constant shift in the energy is usually of no interest. The localized orbitals needed to consider correlation in only part of a molecule are usually obtained by localization of delocalized MOs,14–18 but they can also be obtained by generalized valence bond theory,12 and they are most easily understood in valence bond language. Thus localized orbitals are now an important component of quantitative computational methodologies, which provides another reason for studying them, along with their long recognized utility for qualitative discussions of reactivity19–29 and suggesting functional forms for potential energy surfaces.2,3,30,31 The present essay will start in next section with some remarks on multireference systems and on localized and delocalized treatments of chemical bonding. This discussion is based both on calculations from the early days of quantum mechanics1,32–34 and quantum chemistry35,36 and also on modern computational valence bond theory,13,37–113 which provides insight into the question of how localized and semilocalized orbitals arise in the treatment of so-called multireference systems. The section after that presents remarks on the use of VB formalisms and concepts for semiempirical treatments of potential energy surfaces for electronically adiabatic reactions. The last section

Correspondence to: D. G. Truhlar; e-mail: [email protected] Contract/grant sponsor: National Science Foundation; contract/grant number: CHE03-49122

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discusses the use of valence bond concepts in the treatment of photochemical reactions. In keeping with the essay character of this contribution, references are selected as representative examples of relevant work and are specifically not intended to be exhaustive.

Quantum Mechanical Description of Chemical Bonding The introduction mentions ‘‘the process of interest.’’ Chemical reactions are processes involving rearrangements of chemical bonds, and that provides one reason why chemical bonding is emphasized here. It is interesting to stop and consider properties that are not processes, e.g., dipole moments. One-electron properties such as dipole moments depend on the one-electron density matrix, which is correct to first order for methods, whether MO or VB, that are based on self-consistently optimized orbitals.38,114,115 Many other properties, though, are not as static as they might at first seem. For example, bond distances may be considered as the equilibrium position along a bond-breaking axis, and bond stretching force constants describe the change in energy for small-amplitude vibrations along bond-dissociated coordinates; if they are calculated by a method that is not adequate for bond breaking, they should show systematic errors.116,117 Similarly polarizabilities are response properties corresponding to the process of changing an external electric field. Thus it is important to treat molecular systems by methods that show uniform accuracy as a progress coordinate is varied. Electronic correlation energy is defined as the difference in energy between a fully correlated (‘‘exact’’) electronic wave function (complete configuration interaction) and an uncorrelated (or mean-field) reference electronic wave function. Consideration of the change in correlation energy along a reaction coordinate such as a bond dissociation coordinate brings up the question of static correlation. Although there is no distinct border between different kinds of correlation energy, it is convenient to divide electron correlation energy into two basically different kinds of effects, dynamical correlation and static correlation; these are sometimes called external and internal correlation, respectively, and another very useful name for static correlation is neardegeneracy correlation.118–121 Static correlation is also sometimes called left–right correlation, to distinguish it from in–out and angular correlation,122–127 where in–out is dynamic and angular can be either. Dynamic correlation is usually associated with correlation between ‘‘local fluctuations’’ in the charge density and is therefore sometimes called instantaneous correlation, which is a source of confusion when one recalls that one that is describing the wave function of a stationary state. Thus this language should be understood as fluctuations in Hilbert space, not as fluctuations in time. For the purposes of the present essay, though, static correlation energy is of more immediate interest. For closed shell singlets, an uncorrelated reference wave function is given unambiguously by the restricted Hartree– Fock128 method. For open-shell systems the precise definition of correlation energy is sometimes slightly ambiguous, but for most purposes it can be taken as the difference in energy between a complete configuration interaction wave function and the sim-

plest linear combination of Slater determinants that is correctly symmetry adapted. The latter is called a single-configuration reference state. When there are no near-degeneracy effects, the weight of the single-configuration reference state in the complete configuration interaction wave function is close to unity. In such a case there is no significant amount of static correlation. Sometimes though the weight of the leading symmetry-adapted configuration defined in terms of Hartree–Fock orbitals is much smaller, e.g., 95%129 or 50%.130 In general, correlated wave function methods for electronic structure calculations involve two steps. In the first step one optimizes the orbitals for a reference wave function, which is a zero-order wave function for the system. Some methods129,131 base excited-state calculations on the ground-state reference function, whereas other methods132,133 use a zero-order excitedstate wave function as the reference wave function. In the second step one calculates the electron correlation energy in terms of the orbitals from the first step, in particular by considering excitations (or de-excitations) of one, two, three, or more electrons from occupied orbitals in the reference function to various unoccupied orbitals. A key issue in such calculations is whether the reference function consists of a single configuration state function129,131–133 (CSF) or a linear combination of two or more CSFs,13,43,84,91,120,129,134–140 where a CSF is an antisymmetric many-electron trial wave function composed (usually) of the minimum number of Slater determinants required to obtain the correct spatial and spin symmetry. Calculations in which the reference functions are single CSFs (such as Hartree–Fock wave functions) are called single-reference methods,129,131–133 and calculations in which the reference functions consist of multiple CSFs are called multi-reference methods.13,43,84,91,120,129,134–136 The portion of the correlation energy contributed by the nearly degenerate configurations with large weight is the static correlation energy, and the rest is dynamical correlation energy. The portion of the configuration interaction wave function consisting of the dominant nearly degenerate configurations is a better zero-order wave function than the most important single configuration, and this is why a multi-configuration part of the full wave function is sometimes taken as a reference for perturbation theory or configuration interaction calculations, e.g., one can define a configuration interaction calculation to include all single and double excitations out of a multi-configuration reference wave function rather than out of a single configuration. Any system for which a multi-reference treatment is qualitatively or significantly better than the single-reference treatment is called a multi-reference system. This whole paragraph is intentionally filled with vague words like ‘‘nearly degenerate,’’ ‘‘very large,’’ and ‘‘qualitatively or significantly’’ because the border between single-reference and multi-reference systems is not precisely defined. One can, however, make the general observation that multi-reference effects are usually important for systems with
bonds (especially radicals with
bonds), for systems containing transition metals, for homopolar dissociation, for some transition states, and for most electronically excited states. Sometimes the multi-reference character of a system is dramatic. For example, the single-configuration restricted Hartree– Fock reference state for nearly dissociated H2 is 6.8 eV higher in energy than the best two-configuration reference state, and

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single-reference perturbation theory fails abysmally in this case (and for most processes involving bond breaking). However, multi-reference effects can also be subtle. For example, one often scales stretching vibrationally frequencies obtained by single-reference perturbation theory by a few percent to account for systematic overestimation. This may be considered a multi-reference effect in that bond stretching is incipient bond dissociation,116,117 and single-reference restricted Hartree–Fock wave functions always improperly dissociate to a non-ground-state asymptote for odd-electron fragments; thus they yield too high of an energy in that limit, which leads to too steep of a potential curve near equilibrium geometries, and hence an overestimation of stretching frequencies. One key motivation for some of the interest in valence bond theory is the need to devise approximate electronic structure methods that do not suffer from large quantitative errors when applied to multireference systems. An alternative to valence bond theory for this purpose is to use multiconfiguration Hartree–Fock theory120,134,141–144 (usually called multi-configuration self-consistent-field, but here we distinguish it from valence bond self-consistent-field which can also be multiconfigurational or multistructural). Complete active space self-consistent-field (CASSCF) theory120,143 (which was originally called fully optimized reaction space theory141) is a particularly attractive form of multi-configuration Hartree–Fock theory because it is very systematic (especially for full-valence active spaces) and shows an attractive invariance to orbital transformations. But the number of configurations in CASSCF theory grows very rapidly with system size. Valence bond theory provides a way to cap this growth by using chemical concepts to select the important configurations or structures in the variational wave function. This is an advantage in terms of affordability and manageability of the calculation, but there is also a disadvantage in that an unsystematic human element is involved in limiting the variational wave function, and that makes it hard to define a theoretical model chemistry. A theoretical model chemistry is ‘‘an approximate but well defined mathematical procedure of simulation. . .. The approximate mathematical procedure must be precisely formulated,’’145 Both CASSCF and VB theory can be applied in a systematic way, but eliminating the chemical intuition element often leads (in either method) to the inclusion of insignificant configurations and makes the calculations impractical for many systems of moderate size (or larger). The difficulty of specifying in a general way the precise form of a practical valence bond wave function for an arbitrary system has been a major stumbling block in providing a systematic approach to using valence bond theory that can be widely tested in the same way as post-Hartree–Fock model chemistries,145 which are discussed briefly in next section; although progress has been made84 using valence bond theory with perfect pairing, which is introduced in the next paragraph. Already in the early days of quantum mechanics, valence bond theory was a good candidate for an electronic structure method that does not suffer from improper dissociation. Heitler and London proposed the simplest version of valence bond theory for H2.146,147 It involves a single-configuration wave function based on what is now known as a perfect-pairing41,57 (covalent) valence bond structure (a valence bond structure148 is a particular type of CSF corresponding to a single set of orbitals

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with a spin coupling corresponding to a particular covalent or ionic bonding scheme). The orbitals in the Heitler–London valence bond wave function are nonorthogonal unpolarized atomic eigenfunctions. This treatment predicts a binding energy (De) for H2 of 3.16 eV,146 which is only 67% of the correct value of 4.75 eV. Thus the quantitative accuracy of this simple approach is not good. It was quickly realized that this treatment can be improved by allowing the atomic orbitals to scale32 and polarize.33 Scaling was accomplished by optimizing the orbital exponent, although the modern way would be to optimize a linear combination of fixed s-type basis functions (by using so-called double zeta, triple zeta,. . . or split valence basis sets). Scaling improves De to 3.78 eV. Polarization (which for H involves mixing p character into the atomic s orbitals) is also called hybridization, and the combination of breathing and polarization improves the predicted binding energy to 4.04 eV.33 An alternative procedure to improve the perfect pairing valence bond wave function is to mix in ionic valence bond structures, in this case corresponding to HþH
and H
Hþ. (In valence bond theory, most workers refer to zwitterionic states of neutral systems as ‘‘ionic,’’ and I will follow that practice.) Ionic configurations plus scaling give De ¼ 4.02 eV,34 and ionic terms plus scaling and polarization yield De ¼ 4.12 eV.34 Since 4.04 eV corresponds to 85% of the full binding energy and 4.12 eV to 87% of the full binding energy, we conclude that ionic terms provide only a small improvement in the treatment of H2 if one uses optimized atomic orbitals. However, the ionic valence bond structures can make much more important contributions to other nonpolar covalent bonds (like the F

F bond in F2).62,66 Such bonds are sometimes called charge shift bonds.108 More generally it is necessary to allow the orbitals to delocalize onto neighboring centers to obtain accurate results without ionic terms.61–65,87,95 In modern work the combination of scaling and polarization is sometimes called breathing, and modern breathing orbital valence bond calculations provide useful quantitative accuracy as well as a more consistent picture of the relative contribution of covalent and ionic structures.83,96 Usually, breathing orbitals are localized to a single center or fragment, in keeping with the original Heitler–London model. A key element of breathing orbital valence bond theory is the use of more than a minimum set of orbitals.96 (A language sometimes used is that the VB orbitals ‘‘breathe’’ because the coefficients of the atomic orbitals in the VB orbitals are allowed to change. Thus the method is sometimes called breathing orbital VB or BOVB.83,96) Thus, for example, the VB orbitals on F in a valence bond ionic structure containing F
are not restricted to be the same as the VB orbitals on F in a valence bond structure containing neutral F. An alternative strategy that can also obtain good accuracy is to maintain a minimum set of orbitals but allow the orbitals to delocalize while optimizing them self-consistently along with the spin coupling.40,68 If one uses a physical valence bond structure or structures and a physical basis set, the optimized nonorthogonal orbitals, although they do delocalize to an important extent, tend to be much more localized than the orthogonal canonical Hartree–Fock molecular orbitals. For example, for CH4 one obtains a doubly occupied core orbital on C and four bond-

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ing pairs of nonorthogonal singly occupied orbitals corresponding to four C

H bonds. One can orthogonalize the orbitals in each pair to those in the other pairs with little loss of accuracy (the so called ‘‘strong orthogonality’’ approximation46), but the nonorthogonality to each other of the two orbitals in a given bond pair is the hallmark of valence bond theory. Allowing for some polarization and delocalization allows these orbitals to overlap better and give a better description of covalent bonding. Understanding the relative contributions of covalent and ionic structures is important for understanding the nature of chemical bonding, as for example in distinguishing ‘‘charge shift’’ bonding from conventional ‘‘polar covalent’’ bonds.83,108 Therefore, it is important to recognize, as already mentioned, that if sufficient variational flexibility is not present in the VB orbitals, one sometimes requires unrealistically high weights on ionic terms to accomplish the same purpose, but for conventionally covalently bonded polyatomic molecules, including large contributions from ionic terms, makes the interpretation of the bonding less conceptually appealing. (It also greatly increases the cost and complexity of the calculations.) This leads to a key methodological choice in all valence bond calculations, namely whether or not to let the distorted atomic orbitals delocalize. For attempts to use valence bond theory for quantitative numerical calculations of potential energy surfaces, one usually wants to allow delocalization to reduce or eliminate the very large number of ionic structures that must be considered. For semiempirical calculations it is also usually even more advantageous to reduce the number of valence bond structures to the minimum necessary. However, for interpretations of chemical bonding and reactivity, it is often desirable to include the ionic structures explicitly.23,24,28,80,96,108 Even though the decision is more important for some molecules than others, the choice of whether to delocalize the orbitals is the first one to make in starting a calculation and the first issue to understand in studying a valence bond calculation in the literature. The decision as to whether to use a minimal orbital set (same orbitals in all structures or different orbitals in different structures) is also conceptually and quantitatively important. Thus valence bond states are not unique. Making general statements about contributions of various valence bond structures or the magnitudes of energy lowering due to resonance (configuration interaction) between various valence bond structures without specifying these choices is almost meaningless. The optimized orbitals of valence bond theory that delocalize into bonding regions are sometimes called deformed atomic orbitals, overlap-enhanced atomic orbitals, or bond-distorted AOs. The significance of these singly occupied, nonorthogonal, semilocal orbitals was first discussed by Coulson and Fischer35 and Mueller and Eyring.36 They can be used to explain not only simple covalent bonds like those in H2, CH4, and F2 but also more complicated bonds like those in conjugated
systems58,100,106 (including aromatic systems) or metals.55 A key difference between alkanes and other insulators (on the one hand) and polyenes and metals (on the other) is that the latter have too few electrons to allow a two-electron bond between each pair of neighboring atoms.55,149,150 The locally optimized and semilocal optimized orbitals of modern valence bond theory are the ones that must underlie

physical bonding explanations and semiempirical uses of valence bond theory. It is interesting to contrast them to the canonical ‘‘molecular orbitals’’ of Hartree–Fock theory. The Hartree–Fock orbitals are orthogonal and are usually delocalized. But this delocalization cannot tell us anything about the physical nature of the bonding since the many-electron Hartree–Fock wave function is invariant under orthogonal transformations of the orbitals,151 including localizing transformations. Localized orbitals obtained by such transformations are therefore consequences of an ad hoc localizing algorithm14–16 rather than of the variational principle37,39 or of maximizing overlap with a more general configuration interaction wave function.52,54,73,74,78 (The latter approach is sometimes called CASVB.73–75,77,78 There are two kinds of CASVB. One of these73,78 is mathematically equivalent to CASSCF, and the orbitals are maximally localized so that they are not as arbitrary as in CASSCF. In the other74–77 the VB component is projected from a CASSCF calculation by maximizing the overlap of the VB and CASSCF wave functions.) The character of the canonical molecular orbitals may be explained38,44 by using Koopmans’ theorem152; the delocalized orbitals are approximations to the Dyson orbitals153,154 and may be interpreted in terms of the difference between the molecule and the states of its ionized counterpart. An illuminating perspective on the difference between the delocalized Hartree–Fock orbitals and the semilocal valence bond ones is that the restricted form of the Hartree–Fock wave function forces each molecular orbital to share the symmetry of the entire molecule, whereas valence bond wave functions allow the components of the total wave function to have a lower symmetry than the full wave function.37,44 In fact it can be proved37 that a Hartree–Fock wave function is a special case of a spincoupled valence bond wave function60 or a generalized valence bond perfect pairing wave function,42,44 and the localization of self-consistent valence bond orbitals (as compared to delocalized Hartree–Fock symmetry orbitals) is driven by the variational principle. When this variationally driven localization results in a significant lowering of the energy or a significant change in the expectation values computed from the many-electron wave function, the molecule is a multi-reference system (defined earlier). Generalized valence bond theory with fully optimized orbitals and fully optimized spin coupling is the most general wave function that retains an independent-particle interpretation for the orbitals.40 (The Weinbaum34 wave function mentioned earlier is the special case of this kind of wave function for a minimum basis set and a two-electron system.) In general though one can restrict the allowed spin couplings on physical grounds,40 and the perfect pairing approximation (mentioned earlier) in which all pairs of electrons are treated as singlet coupled (except for the last electron in old-electron systems) is often adequate. A further simplification is to treat core and nonbonding pairs and perhaps even spectator bonds (bonds that are neither formed nor broken in the process of interest and for which neither bonding atom undergoes a change of hybridization) at the Hartree–Fock level and to correlate only N active pairs of electrons with a single perfect pairing spin coupling and orthogonality between orbital spaces corresponding to different pairs. The efficiency of such a calculation is demonstrated by

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noting that this single-structural generalized-valence-bond perfect-pairing (GVB-PP) wave function41,46 is equivalent to a multi-configuration Hartree–Fock wave function with 2N configurations. Relaxation of the perfect pairing restriction may be accomplished in more than one way, e.g., by allowing for (and optimizing simultaneously with the orbitals) more general spin couplings,40,68 by including more than 2N configurations in a configuration mixing calculation based on orthogonal orbitals obtained by rewriting the GVB-PP wave function in the natural orbital representation48,56,85 (or other configuration mixing methods98,102) or by a resonance superposition of two (or more) GVB-PP single-structural wave functions.49 A fourth possibility is to reoptimize the orbitals for the resonating superposition.50 When applied to chemical reactions, the fourth of these possibilities allows the reactant and the product to each be described by a GVB-PP wave function, and this method constitutes the fundamental ab initio approach underlying most semiempirical applications of valence bond theory to chemical reactivity, as discussed in next section. The fourth possibility also provides the fundamental underpinning of traditional discussions of aromaticity and conjugation effects in polyenes. Generalized valence bond calculations provide a minimal set of orbitals with an independent-particle interpretation38 while also including some of the static correlation needed to provide a qualitatively correct wave function for multi-reference systems.37,39,40,42,44,47 Kohn–Sham density functional theory (usually just called DFT since the Kohn–Sham version is the only widely applied version of DFT) provides an alternative approach that uses a minimal set of orbitals and that, like generalized valence bond theory, goes beyond the Hartree–Fock model in that it includes electron correlation.155–157 The correlation potential derived from the exchange-correlation functional includes dynamical correlation, and—which is particularly relevant in the present context—the ‘‘exchange potential’’ derived from the exchange-correlation functional includes some static correlation.126,127,157–164 This is a strength of DFT in that the inclusion of some static correlation makes DFT more applicable than popular Hartree–Fock-based wave function theories for many multireference situations, most notably bonds involving transition metals.160,165,166 A disadvantage of this approach is that the portion of the static correlation included in the exchange potential is incomplete and uncontrolled. Various approaches are under development to deal with this problem. This activity is marked in part by a rapidly growing body of research167–205 on many issues of combining DFT with multi-configurational energies or densities and on the closely related subject of orbitally dependent density functionals. One required area of progress is developing correlation functionals that provide accurate results when used with a high percentage of Hartree–Fock exchange, rather than requiring a cancellation of errors between exchange and correlation functionals.206 Another important related subject is developing new diagnostics for multireference character.166,207–209 Just as Hartree–Fock theory suffers from the restricted form of the wave function and the consequential delocalized orbitals, standard versions of DFT also suffer from this limitation.196 One can also use the classification of static and dynamic correlation to distinguish certain VB methods. In the process of doing this, we will define the VBSCF and VBCI notation. In a

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VBSCF wave function,53 all the active orbitals are localized on an atom or fragment and are optimized as a common orbital set, corresponding to the electrons moving in a common mean field. Thus a VBSCF wave function lacks dynamical correlation. At the BOVB level, different VB orbital sets are used for each VB structure, which corresponds to including some portion of the dynamic correlation.83 A way to enable a more complete accounting for dynamic correlation is called VBCI.98,102 In VBCI, one localizes the unoccupied orbital such that the excited VB structures may be interpreted as providing dynamic correlation to a reference structure with a specific spin pairing and charge distribution. In fact, one may contract each VBSCF structure and its associated excited ones into a single contracted function; this allows one to improve the quantitative accuracy while maintaining the simplicity of the original VBSCF structure set.98,102,108

Potential Energy Surfaces Valence bond theory has been used as a framework for modeling chemical reaction dynamics since the early days of quantum mechanics.2,3,20–24,28,29,40,41,45,48,50,51,67,72,79,81,86,87,89,90,92,95–99,101–104,106,112,210–286 In part this may be explained by the fact that valence bond theory often provides a concise description of multireference character, and transition states are more likely than stable molecules to exhibit multireference character. Furthermore, valence bond theory provides a natural framework for modeling reactive potential energy surfaces because it allows one to build on one’s knowledge of the reactants and products in a straightforward way. Early work was mainly semiempirical plus a few (far from converged) ab initio calculations on the H þ H2 reaction.235 This section is primarily concerned, at least explicitly, with gasphase reactions, but the same issues occur in describing a solute in the condensed phase. In fact, valence bond theory is well suited to describing complex processes in solution. Both ab initio prediction and semiempirical modeling play important roles in chemical dynamics, and fortunately we do not have to choose between them. On the one hand, it is important to extend the size of the system for which we can obtain reliable results without parameterizing, and ab initio wave function methods (including valence-bond-based wave function theory) are continually being refined to treat larger systems more accurately.287 Density functional theory (which almost always contains parameters) is also being improved.206 For most problems, the required accuracy can only be obtained with parameters. Methods for parameterization have been extensively developed. Many years ago, semiempirical methods of parameterizing potential energy surfaces were divided into two classes: those in which the parameters are adjusted to kinetics, for example to activation energies, and those in which the parameters are adjusted only to nonkinetics data, such as thermochemistry and spectroscopy.288 These distinctions still hold; for example, G3SX289 was parameterized without using data from kinetics, whereas BB1K290 was parameterized using, among other data, barrier heights extracted from kinetics experiments. However, there is another distinction that seems more important in judging modern practices, namely, the parameters drawn from the reaction being modeled (for example, when a potential energy surface has parameters adjusted to reproduce the activation energy

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for the reaction being modeled) or from a broad set of training data. One can also ask whether the parameters apply only to a single reaction (specific reaction parameters) or to a restricted range of reactions (specific range parameters) or whether they apply to all or most reactants (general parameters). General parameterizations have been widely used with post-Hartree–Fock model chemistries (e.g., G2,291 MCG2,292 G3,145,293 G3SX,289 G3SX (MP3),289 MCG3/3,294 BMC-CCSD295), and often these are the most reliable affordable methods. General parameterizations are also widely employed in semiempirical molecular orbital theory (e.g., AM1296) and DFT (e.g., B3PW91,297 B3LYP,298 MPW1K,299 BB1K,290 M05-2X206), but are less well developed, although not totally neglected,80 for valence bond theory. In contrast, the use of specific reaction parameters has a long history in valence bond theory3,31,235 and has only more recently been extended to molecular orbital theory300,301 and density functional theory.302 Orbital-based semiempirical methods based on valence bond theory have been presented by Cullen303,304 and Wu et al.169,171,175,177 In addition to the applications-driven practical motivations for obtaining useful models by parameterization, the use of parameterized potential energy surfaces can also be useful to aid fundamental understanding. Sometimes305 we can gain considerable understanding of the dynamics most simply with a parameterized model,224 even if the numerical energetics are not quantitatively accurate—and when one can do that, it is wasteful and less elegant to do more expensive calculations. Thus, it may be useful to distinguish two motivations for parameterizing: to get accurate numbers or to set up a qualitatively correct description. Both of these uses should be distinguished from a third class of applications, namely fitting. When a few parameters are adjusted, for example to the reaction exothermicity and barrier height and perhaps to the saddle point geometry, we call the resulting analytical surface a semiempirical potential energy surface31,231 or an empirical valence bond surface.238 When the functional form is flexibilized and adjusted even more finely to provide a analytical representation of higher level calculations of the global or semiglobal potential energy surface, we call the result a fitted surface.306–311 However, the precise location of the division between semiempirical parameterization and fitting of higher-level potential surfaces is often arbitrary, and other classifications are possible. For example, one can use many (>30) parameters to adjust a semiempirical valence bond surface to experimental information as well as to high-level calculations.284 Next we discuss the multiconfiguration molecular mechanics (MCMM) method,279,312–314 which is based on simple valence bond theory and which provides an efficient and systematic method for fitting reactive potential energy surfaces. The formulation of the MCMM method, like all methods based on molecular mechanics, intrinsically involves the definition of valence coordinates. Valence coordinates consist of bond stretches, bond angle bends, torsions, and improper torsions. These coordinates change when the bonding pattern changes. A conventional molecular mechanics potential315–318 is a sum of valence-coordinate,319 vibrational potentials (often harmonic), and van der Waals and Coulomb interactions between nongeminal atoms (sometimes these are also omitted or scaled down for vicinal atoms; in other words the van der Waals and Coulomb

interactions are omitted when they duplicate terms already included in the valence force field). A conventional molecular mechanics potential corresponds to a specific arrangement of bonds and involves a single set of valence coordinates. The MCMM method involves two (or more) arrangements of bonds (corresponding to reactants and products), and therefore it involves two (or more) sets of valence coordinates. MCMM begins by writing a 2  2 Hamiltonian matrix H (the method can be generalized to include more states320) in what is nominally a valence bond basis:  H ¼

h

1 jHj 1 i

H12

H12 h 2 jHj

 2i

(1)

where 1 is the reactant valence bond structure, and 2 is the product valence bond structure. These valence bond structures are (as mentioned earlier) configuration state functions, and they also correspond to molecular mechanics configurations identified with specific arrangements of the bonds. In the diatomics-in-molecules method,222,223,236,239,242,244 the matrix elements of eq. (1) are evaluated in terms of diatomic fragment data, but ‘‘the extreme assumptions involved prevent this approach from being very fruitful. . .. The method is very sensitive to its assumptions.’’235 Thus in MCMM we take a different approach. Following Warshel and Weiss,238 the diagonal elements are equated to molecular mechanics315–318 energy expressions. As is well known, molecular mechanics energy expressions become inaccurate for large deviations from equilibrium geometries, and in particular they do not describe bond dissociations or chemical reactions. This behavior of MM energies is precisely analogous to the diagonal valence bond matrix elements in eq. (1) in that h 1|H| 1i should be very high when evaluated at the equilibrium geometry of state 2, and h 2|H| 2i should be very high when evaluated at the equilibrium geometry of state 1. Therefore, in the vicinity of these two equilibrium geometries, the lowest eigenvalue will be equal (to a very good approximation, unless H12 is unphysically large) to the corresponding molecular mechanics energy, which is accurate enough for many purposes. The essence of the method is then to fit H12 in the transition state region between the two equilibrium structures such that the lowest eigenvalue reproduces an accurate (or high-level) calculation of the (ground-electronic-state) potential energy surface in that region. This requires much less accurate (or high-level) data than directly fitting the potential energy surface for two reasons. First, H12 is much more smooth and slowly varying than the potential energy surface, which has a saddle point and also has a steep repulsive wall on one side of the reaction path, which is itself located in a curved reaction valley. Second, one expects H11 and H22 to show very similar variations along spectator coordinates. The eigenvalue of H will reproduce this behavior even if H11 and H22 are steep and even if the variation of H12 in these coordinates is inaccurate. In fact, H12 needs to be accurate only when both H11 and H22 are low in energy, and to some extent it can be decoupled from the spectator coordinates in that its variation needs to be accurate only in coordinates for which H11 and H22 vary differently. Alternatively, the dependence of H12 on spectator-atom coordinates can be fitted

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using one of the VB structures and is in principle transferable to the other(s). Although a variety of methods could be used to fit H12, we use Shepard interpolation307,321,322 of Hessians in curvilinear internal coordinates. At a given geometry, the Hessians of H12 are computed in Cartesian coordinates by the method of Chang and coworkers, 251,255 and are transformed to curvilinear coordinates by the method of Wilson et al.319 The method has proved to be a very efficient way to fit potential energy surfaces,312,313 and it can be made even more efficient by using different amounts of data to interpolate different elements of the Hessian of the resonance integral314 and by adjusting the diagonal valence-bond matrix elements as well as the off-diagonal ones.320 Thus, the lessons learned from valence bond theory1–3,13,20–112,210–286 can be imported into MCMM to provide improved representations of reactive potential energy surfaces. It is often stated that molecular mechanics is not applicable to transition states. That statement follows from the association of a conventional molecular mechanics force field315–318 with a specific bonding arrangement. The empirical valence bond method of Warshel and Weiss238,260,263,274,277 and the MCMM method312 reviewed earlier provide a natural way to extend molecular mechanics to chemical reactions by equating a molecular mechanics energy expression for the entire system to a valence bond energy. (The earlier method238,260,263,274,277 is similar to MCMM but has a different focus and makes the assumption that H12 may be calculated for an active site in one environment and used in another, as a way to study solvent and enzyme-environment effects by only perturbing H11 and H22; this assumption requires further study.) If the resonance integral (H12) is set equal to zero, one obtains an even simpler method;323 although the saddle point mountain pass is replaced by a cuspidal ridge when the resonance integral is neglected, one can still obtain useful approximations to transition state structures by energy minimization along the ridge.323 An even simpler method is to treat the transition state as a minimum in a force field.324–326 In the rest of this section, we discuss an alternative way to combine valence bond theory with molecular mechanics for modeling reactive potential energy surfaces; unlike MCMM, this involves treating only part of the system by valence bond theory. An important development in the use of molecular-orbitalbased methods to calculate potential energy surfaces for large and complex systems has been the combined quantum mechanical/molecular mechanical (QM/MM) method.327–342 In this method an active subset of atoms or orbitals is treated by QM molecular orbital theory (e.g., AM1,296 B3LYP, . . .298), and the remainder of the system is treated by molecular mechanics. This approach can also be applied where the QM region is treated by valence bond theory,122,232,247,248,253,261,262,264,265,276,282,283 and we call this a combined valence bond/molecular mechanical (CVBMM) method. This method is particularly well balanced when one uses semiempirical analytic valence bond potential energy functions for the active part of the systems232,247,248,261,264,271,276,283 because then both the valence bond part and the molecular mechanics part are analytic and semiempirical. However, the use of ab initio VB for the QM part112 has two advantages: (i) in principle it eliminates the need

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for specific reaction parameters when the ab initio level is high enough, and (ii) it allows for active region polarization in the presence of the MM system by adding a classical interaction term to each of the diagonal elements of the VB Hamiltonian,112 a procedure that was used earlier in empirical valence bond theory.238,256 Because CVBMM is in its infancy, we can expect further development to turn it into an even more generally useful tool; initial results are promising. When one considers the problem of reactive potentials for complex systems in a general way, it seems clear that making use of generally parameterized molecular mechanics methods for the unreactive parts of reactive systems is very efficient as compared to remapping the functional dependence of the potential on spectator coordinates for each new system studied. Another way to combine valence bond theory and molecular mechanics for large, complex systems is to combine342,343 MCMM with MM in the same way that QM is combined with MM in MCMM schemes; we call this MCMM/MM or QMMCMM/MM. (This has some computational advantages compared to using QM/MM to replace QM in calibrating the resonance integrals; we call that strategy QM/MM-MCMM.342,343 QM/MM-MCMM has the advantage of being able to use well established QM/MM methods to include the electrostatic effects of the MM system in the QM Hamiltonian.342,343) Another important area of research in using valence bond formalisms for modeling reactive potential energy surfaces is the use of multistate empirical valence bond methods for modeling proton solvation and proton transport in condensed phase.344–357 A key new element that arises in some cases is the need to identify the significantly contributing valence bond states as a function of the changing location of the proton.

Electronically Excited States In previous sections I emphasized the importance of treating multireference character in electronic ground states. This problem can be even more severe in electronically excited states, and as a general rule, multireference character is the rule rather than the exception in excited states. For example the leading configuration, beyond the Hartree–Fock one, in the ground state of ethylene contributes 3%,143 which is already higher than for a typical -bonded molecule. But the leading MO configuration of the
?
* state of ethylene contributes only 87% in a CASSCF calculation143,357 (or 76% in a configuration interaction calculation based on Hartree–Fock orbitals132 or 92% in a configuration interaction calculation based on approximate natural orbitals132). As for ground electronic states, valence bond theory can help to identify the key contributing structures in multireference cases. Valence bond theory also provides a useful framework for discussing broadly representative prototype problems in electronically nonadiabatic chemistry. Three key examples are  The properties of conical intersections,358–362  The Landau-Zener picture of electronically nonadiabatic transitions,363–365 which implicitly uses diabatic states that

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are much closer to valence bond structures than to molecular orbital configurations,  The curve crossing or avoided crossing of ionic and covalent curves.19,366–377 All of these phenomena are much easier to explain with valence bond structures than with configuration state functions composed of Hartree–Fock molecular orbitals. For example, valence bond structures provide a very natural way to discuss the dissociation of diatomic cesium bromide; in a shock tube, where some fraction of these species dissociate to ions because the ionic and covalent VB potentials are weakly coupled.378 Similarly, the MO wave functions of alkali halides and alkali hydrides change character in a rather localized region whose location is easily predicted from VB potentials, which provide a very natural way to think about ionic–covalent configuration state mixing.19,43,349,373– 375,377 The same issues also occur in polyatomics.244,379–383 In addition to these uses of valence bond states for qualitative interpretation of multielectronic-state phenomena, valencebond and valence-bond-like wave functions may serve as (approximate) diabatic states in dynamical simulations of electronically nonadiabatic phenomena.379,381 Electronically nonadiabatic processes are an example of non-Born–Oppenheimer dynamics361–365,384–405 since in the Born–Oppenheimer approximation401,406,407 all molecular processes would conserve the electronic quantum numbers, i.e., would be electronically adiabatic. Electronically adiabatic processes in which virtually excited electronic states or electronically excited intermediates play a role are also examples of non-Born–Oppenheimer dynamics. Theoretical simulations of non-Born–Oppenheimer dynamics may use the electronically adiabatic representation401,406 in which the electronic Hamiltonian (defined as the total molecular Hamiltonian minus the nuclear kinetic energy) is diagonal or they may use a nonadiabatic representation in which it is not diagonal. In the adiabatic representation, coupling between the electronic states is due to off-diagonal electronic matrix elements of nuclear momentum operator and the nuclear kinetic energy operator.401,406,407 The latter are usually neglected in semiclassical treatments because they become relatively less important as nuclear kinetic energy increases.384 The former matrix elements are sometimes called nonadiabatic coupling terms (NACTs). The gradient corresponding to the nuclear momentum is an anti-Hermitean vector operator, and so, for M electronic states and N atoms, there are 12MðM
1Þ unique NACTs for each of the 3N components of the nuclear momentum. It is impossible to make all 3N components of a NACT vanish over a finite region of space.408–411 It is, however, possible to find representations in which the NACTs are just as negligible as when the Born–Oppenheimer approximation is valid.411 Nonadiabatic electronic wave function representations in which the effects of the NACTs are negligible compared to effects of the nonzero off-diagonal matrix elements of the electronic Hamiltonian are called diabatic representations.369,375,379,381,384,387,401,412–425 The off-diagonal elements of the electronic Hamiltonian are called diabatic couplings. In principle one could also use intermediate representations in which both the NACTs and the off-diagonal elements of the electronic Hamiltonian are non-negligible,426 but this is seldom done. Nevertheless there is still flexibility because

although the electronically adiabatic states are unique, the diabatic states, like valence bond states, are not unique. If one solved the molecular dynamics problem by converged quantum mechanics without approximations, one would obtain the same results by using either an adiabatic or diabatic representation. This is also true when coupled electron-nuclear dynamics are treated by the semiclassical Ehrenfest method.387 However, converged quantum dynamics is usually impractical,401 and the Ehrenfest approximation is not very accurate.401–405 One would expect that approximate dynamical methods would be most accurate if carried out in whichever representation is less coupled, and indeed, with an appropriate definition of ‘‘less coupled,’’ this is usually found to be the case.397 In real applications it is not always clear whether the results obtained with one or the other representation is more accurate, and the difference between the two sets of results can be disconcertingly large. Furthermore in polyatomic systems it is possible that an adiabatic representation is more appropriate in some regions of phase space, and a diabatic representation is more appropriate in others, and so it is useful to develop dynamics methods that give equally accurate results with both representations or are at least less sensitive to the choice of representation. With this as one motivation, we have developed a more reliable method for simulating non-Born–Oppenheimer dynamics that is much less sensitive to the choice of representation than the popular trajectory surface hopping methods. The new method403 is called coherent switches with decay of mixing, and it is also more accurate than the Ehrenfest method.402–405 A second motivation for developing dynamical methods that are accurate in both adiabatic and diabatic representations is that most electronic structure calculations of excited states are carried out in the adiabatic representation, where the variational principle can be used to improve the calculations. But the adiabatic approximation presents several difficulties when it is used in dynamics calculations. First of all, the NACTs are difficult to work with—they are multidimensional (3N components), not slowly varying or smooth (in fact singular at conical intersections), and they have formal difficulties427 that make them origin dependent with spurious behavior at large interfragment distances. In contrast the diabatic couplings are scalar functions and are expected to be well-behaved and nonsingular. Therefore, for dynamics calculations, it is useful to be able to generate diabatic representations as well as the more common adiabatic representations. Individual valence bond structures may be considered to be diabatic representations,379,381 but they do not span an optimized portion of electronic state space. Although one can systematically converge a valence bond calculation by adding more structures, calculations that are converged by such a method involve more valence bond structures than the number of adiabatic states important in the process. For this reason we favor a method that uses the variational principle to optimize the relevant space (which must be done in an adiabatic representation) and then transforms to a diabatic representation that spans the same portion of electronic state space. One could use valence bond concepts directly for this transformation as in CASVB,73–75,77,78,144 in the bonding scheme428 method, or in a method based on VB structures obtained using the spin-exchange density computed

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with orbitals localized by the method of Boys.14,429 Instead we have developed a procedure, called the fourfold way,375,423,424 based on diabatic molecular orbitals375 and configurational uniformity.421 The diabatic molecular orbitals are determined by the criterion of molecular orbital uniformity, which means that they should vary smoothly along continuous paths through nuclear coordinate space. We found that molecular oribtal uniformity can be achieved under quite general conditions by maximizing a threefold functional of the electronic density matrix and enforcing maximum-overlap reference molecular orbital criterion.375,423 Although CASVB, the bonding scheme method, and the spin-exchange density method have so far involved the transformation of a CASSCF wave function into valence bond structures, the fourfold way has been implemented not only for CASSCF calculations, which contain static correlation, but also for MC-QDPT calculations,144,430,431 which add dynamical correlation to a CASSCF reference wave function. Thus, the fourfold way allows for better quantitative accuracy.432

Acknowledgments I am grateful to John Cullen, Joaquin Espinosa-Garcia, Jiabo Li, Hai Lin, Sason Shaik, Avital Shurki, and Wei Wu for comments on the original version of this essay.

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Journal of Computational Chemistry

DOI 10.1002/jcc