Cramer C.J. Essentials of Computational Chemistry Theories and Models

HH

X

H H

multiconfigurational character in the singlets. When X = C, the two orbitals belong to the b1u and ag irreps of the D2h point group, respectively, and thus a single excitation from the former to the latter cannot contribute to the singlet ground state that has overall Ag symmetry; only a double excitation contributes. When X = NH+ , however, both orbitals are of a type symmetry within the Cs point group, so that a single excitation can (and does) make a large contribution to the singlet ground state that has overall A symmetry. The latter situation can contribute to instability in estimating the energetic effects of triples substitutions in ‘(T)’ methods based on a single-determinantal reference

photoelectron spectroscopy has established the singlet – triplet (S-T) splitting to be −3.8 ± 0.5 kcal mol−1 (Wenthold et al. 1998). This corresponds to an energy splitting of −4.2 ± 0.5 kcal mol−1 (i.e., differences in zero-point vibrational energy have been removed). Highlevel calculations suggest, not surprisingly, that the S-T splitting in the N -protonated pyridyne system should be very nearly the same (Debbert and Cramer 2000). Table 7.3 illustrates the results from a variety of different levels of electronic structure theory applied to computing the S-T splitting using the cc-pVDZ basis set.

Notice, first, how spectacularly wrong the HF results are, this being indicative of significant multireference character for the singlets, which should really be described as about 60:40 mixtures of the two determinants corresponding to double occupation of the antibonding and bonding combinations of the σ orbitals. In the p-benzyne case, the MP2 calculation correctly predicts the singlet to be the preferred state, but drastically overshoots in doing so. As is typical, MP3 oscillates back to the HF prediction (triplet ground state) but with a smaller margin of error, and then MP4 corrects back again in the proper direction, with a somewhat smaller overestimation than was observed for MP2. Clearly, however, one could not with confidence extrapolate to an infinite-order perturbation theory result from these four points. The situation is much the same for the 2,5-pyridynium ion, except that the MP2 result is very close to experiment. Such fortuitous agreement is obviously entirely coincidental, as the perturbation series is wildly oscillating.

Note also the importance of triple excitations in correcting for the multideterminantal character. The change in the S-T splitting from inclusion of the triples at the MP4 level is as large as or larger than the change in going from MP3 to MP4SDQ. A similar effect is seen with the QCISD and CCSD formalisms – both incorrectly predict triplet ground

Table 7.3 Singlet – triplet splittings (kcal mol−1) for p-benzyne and N -protonated 2,5-pyridynea

a Basis set cc-pVDZ unless otherwise indicated; geometries from BPW91/cc-pVDZ density functional calculations (see Chapter 8).

states, but inclusion of triples via the (T) formalism gives for the most part rather good results. A significant exception is the CCSD(T) result for the 2,5-pyridynium ion, where the triples correction drastically overcorrects. This is an example of instability arising from large singles amplitudes in the CCSD expansion. In p-benzyne, the symmetry of the bonding and antibonding combinations of the σ orbitals is different, so a single excitation from one orbital to the other cannot contribute to the closed-shell wave function. In the less symmetric 2,5-pyridynium ion, however, these orbitals are the same symmetry, so such excitations are allowed and are major contributors to the wave function (Figure 7.6). In such instances, BD(T) calculations are to be preferred over CCSD(T), and indeed, the BD(T) level of theory performs very nicely for this problem (in this particular case the QCISD(T) level also seems to be more robust than CCSD(T) with respect to sensitivity to singles, but this is the reverse of the situation that normally obtains).

As for basis-set convergence, triple-ζ calculations at the MP2 and CCSD levels are provided for comparison to the double-ζ results. For this particular property, the results for p-benzyne are not terribly sensitive to improvements in the flexibility of the basis set. In the pyridynium ion case, the CCSD results are also not very sensitive, but a large effect is seen at the MP2 level. This has more to do with the instability of the perturbation expansion than any intrinsic difference between the isoelectronic arynes.

Note that when multiconfigurational character is explicitly accounted for, by an MCSCF calculation using a complete active space including the relevant σ orbitals and electrons as well as the six π orbitals and electrons, the results even without accounting for dynamical electron correlation are fairly good. Including dynamical correlation at the CASPT2 level improves them to the point where they are quite good.

Insofar as CASPT2 uses a multiconfigurational reference, one might expect it to be less prone than MP2 to instability. This is entirely true, so long as the MCSCF reference is adequate. If the MCSCF has converged to a spurious solution, perturbation theory is often successful in identifying this because a very large contribution from one or more excitations will be observed. Alternatively, if the MCSCF failed to include one or more critical orbitals, again, large contributions will be obtained for corresponding excitations. From a formal standpoint, it is better to include those orbitals in the active space than it is to rely on CASPT2 to correct for both dynamical and non-dynamical behavior, even though in some instances it seems the latter approach can give good results.

An alternative way to account for multiconfigurational character in the singlet is to generate it using SF-CIS(D) from the triplet reference. Slipchenko and Krylov (2002) have done this for p-benzyne using the 6-31G(d) basis set and computed a S−T splitting of −2.2 kcal mol−1 . The SF-CIS(D) model is thus nearly as accurate as those including estimates for triple excitations even though it is substantially less computationally expensive.

A separate issue that can contribute to instability in correlated calculations is spin contamination. As noted in Chapter 6, spin contamination refers to the inclusion in the wave function of contributions from states of higher spin that mix in when unrestricted methods permit α and β spin orbitals to localize in different regions of space. As a rough rule, the sensitivity of different methods to spin contamination is about what it is to multiconfigurational character: MPn methods are to be avoided and inclusion of triples in CCSD or QCISD (or BD) is important. So-called projected MPn methods attempt to correct for spin contamination after the fact by projecting out states of higher spin from the correlated wave function (see Appendix B), and these methods tend to be helpful in cases where spin contamination is relatively small, say no more than 10% (Chen and Schlegel 1994). Unfortunately, analytic gradients are not available for spin-projected methods, so they must be applied to geometries the optimization of which may have taken place at a considerably less reliable level.

An issue related to spin contamination is so-called Hartree – Fock instability. Various wave functions can exhibit different kinds of instabilities, often, but not always, associated with trying to describe a multiconfigurational system with a single-determinant approach. Thus, for instance, RHF solutions may be unstable with respect to breaking the identical character of the α and β orbitals – a so-called RHF → UHF instability (the UHF singlet wave function is usually highly contaminated with triplet character after reoptimization). UHF wave functions for symmetric systems can also be unstable, in this case with respect to spatial symmetry breaking of the individual orbitals. The MOs, if allowed to relax, fail to fall into the irreps of the molecular point group, adopting instead lower symmetry shapes even if the molecular framework is held fixed so as to continue to belong to the higher symmetry point group. Such instability tends to be associated with systems having delocalized spin – the allyl radical is a classical example. All of these cases prove very problematic for perturbation theory, but are handled with somewhat greater success by other correlated methods. In certain very highly symmetric systems, the wave function can also be unstable to using complex instead of real MOs, but this situation is rare. Most modern electronic-structure programs allow one to check the stability of the HF wave function with respect to these

various phenomena, and such steps are warranted in cases having narrow frontier orbital separations and/or delocalized spin. Resort to MCSCF wave functions can be required in particularly problematic systems.

7.6.3Price/Performance Summary

For a typical equilibrium structure, the HF level of theory predicts bond lengths that are usually a little too short. It is simple to rationalize this using Eq. (7.1). To the extent that correlated methods include excited configurations in the wave function expansion, and to the extent that the orbitals into which excitations occur typically have some antibonding character, this tends to increase bond lengths in order to lower the energy. As a rule, the MP2 level is an excellent choice for geometry optimizations of minima that include correlation energy, and significant improvements can be obtained at fairly reasonable cost. Scheiner et al. (1997) examined a large number of bond lengths in 108 molecules containing from two to eight atoms and found that, with the 6-31G(d,p) basis set, the average error in bond length at the MP2 level was 0.015 A,˚ which may be compared to an error at the HF level of 0.021 A˚ . An improvement of roughly the same order was obtained by Feller and Peterson (1998) in a separate investigation of 184 small molecules using the aug-cc-pVnZ basis sets. Bond angles are already sufficiently accurate at the HF level that little improvement is observed at the MP2 level.

While analytic derivatives are available for several more highly correlated levels of theory, geometric improvements beyond the MP2 level tend to be so small for equilibrium structures that they are not worth the cost. This is not necessarily the case for TS structures, where the accurate description of a partial bond may well require correlation beyond the MP2 level. As a rough rule, if one observes a large change in some geometric property on going from the HF to the MP2 level, it is probably worthwhile to investigate the predictions from still higher levels of theory, since clearly the perturbations are large, and there is good reason to believe MP2 does not provide convergence in the property of interest. In some instances, convergence can be hard to achieve using perturbation theory (especially for cases of multiple bonds involving heteroatoms) and coupled-cluster methods are to be preferred, but this typically applies only to calculations seeking the most demanding accuracy (He and Cremer 2000c).

With respect to energetics, MP2 must again be considered a very efficient level of theory for energy differences between minima. In many instances, one finds that the error in such differences is reduced by 25 – 50% on going from the HF level to the MP2 level. For instance, Hehre reports a sample of 45 isomerizations where errors in isomer energies were reduced from 2.9 to 1.9 kcal mol−1 on going from HF/6-31G(d) to MP2/6-31G(d). For the 11 glucose conformers discussed in Chapters 5 and 6, the average error in conformational energy is reduced from 0.6 to 0.4 kcal mol−1 on going from HF/cc-pVTZ//MP2/cc-pVDZ to MP2/cc-pVTZ//MP2/cc-pVDZ (Barrows et al. 1998). Note, though, that for the same glucose conformers, the error increases from 0.1 to 1.0 kcal mol−1 on going from HF/ccpVDZ to MP2/cc-pVDZ, illustrating the degree to which errors in basis set and correlation approximations can sometimes offset one another.

However, the generally robust nature of MP2 in the above examples simply reflects the degree to which most minima are already fairly well described by HF wave functions. When this is not the case, e.g., in TS structures, there are no hard and fast rules that can be cited with respect to the expected quality of any level of theory. Instead, one is thrown back on the twin responsibilities of demonstrating either (a) agreement with known experimental data of one kind or another in the same or related systems or (b) convergence with respect to treatment of electron correlation. A rough quality ordering that is often observed is

Eq. (7.59) using data provided by Bartlett (1995) for the absolute errors in various levels of theory compared to full CI for HB, H2O, and HF using a polarized double-ζ basis set. In this case, calculations were carried out both at the equilibrium geometries, and also at geometries where the X – H bonds were stretched by 50% and 100%; correlation should become more important in describing these higher energy species (see also Dutta and Sherrill 2003). The ordering of the levels in the table is approximately that listed in Eq. (7.59), although CCD seems to do fortuitously well. As expected, the lower levels of correlation treatment degrade markedly compared to the higher levels when the bonds are stretched. A few levels not generally available in most electronic structure packages are included in the table for completeness, including MP5, MP6, and levels having full inclusion of triples, represented by ‘T’ instead of ‘(T)’. The heroically expensive CCSDTQ, which takes full account of all

Table 7.4 Average errors in correlation energies (kcal mol−1) compared to full CI for various methods applied to HB, H2O, and HF at both equilibrium and bond-stretched geometries

triple and quadruple excitations, is also included, and shows the extraordinarily high accuracy one might expect for so complete a treatment, albeit one that can only be applied to the smallest of molecules.

To further judge what level may be appropriate for a given problem, it is critical that cost be taken into account. The scaling behavior of the various levels in Eq. (7.59) varies widely, as indicated in Table 7.5. Given the price/performance ratios implied by comparing Eq. (7.59) with Table 7.5, there is, for instance, usually little point in doing an MP3 or CISD calculation when superior MP4SDQ or CCSD calculations may typically be accomplished at roughly similar cost (note that scaling similarity is not the same as overall time similarity, since the times for the benchmark ‘one-basis-function’ calculations may differ, but for small to moderately sized molecules, the overall times do not tend to be terribly dissimilar). It should also be recalled that in the large molecule limit, all scaling behaviors tend to reduce because prescreening techniques can avoid the calculation of many negligible integrals. In addition, progress continues to be made on linear scaling formalisms, for example, local coupled cluster theory (Li, Ma, and Jiang 2002; Schutz¨ 2002), so that increasingly sophisticated treatments of larger and larger molecules are becoming more and more accessible.

7.7 Parameterized Methods

Having ascended to the heights of theoretical rigor, it is perhaps time for a brief respite and a timely recapitulation of, of all things, the philosophy underlying the development of semiempirical methods: wouldn’t it be nice to get the right answer for any problem in general? Although methods like full CI and CCSDTQ, when used in conjunction with large and flexible basis sets, are breathtakingly accurate as solutions of the Schrodinger¨ equation, the bottom line is that they simply cannot be applied to more than the smallest fraction of chemically interesting systems because of their computational expense. And, with scaling behaviors on the order of N 10, this situation is unlikely to change anytime soon. As a result, particularly within the last decade, practitioners of ab initio MO theory have returned to the idea of introducing parameters to improve predictive accuracy, albeit with a considerably lighter touch than that associated with a full-blown semiempirical method. This section describes a variety of different approaches to improving the results from calculations including electron correlation.

7.7.1 Scaling Correlation Energies

The premise behind correlation scaling is particularly simple. Because of basis-set limitations and approximations in the correlation treatment, one is very rarely able to compute the full correlation energy. However, with a given choice of basis set and level of theory, the fraction that is calculated is often quite consistent over a fairly large range of structure. Thus, we might define an improved electronic energy as

where ‘e.c.m.’ is a particular electron correlation method, A is an empirical scale factor typically less than one, and thus all of the correlation energy, computed as the difference between Eecm and EHF, is scaled by the constant factor of A−1. SAC emphasizes this ‘scaling all correlation’ energy assumption. Note that the difference between SAC and the extrapolation schemes of section 7.6.1 is that the latter extrapolate the correlation energy associated with a given electronic structure model to an infinite basis set, but SAC attempts to estimate all of the correlation energy.

As first proposed by Gordon and Truhlar (1986), typically one would go about selecting A by comparison to known experimental data in a system of interest and/or systems related to it. For example, if the subject of interest is the PES for the reaction of the hydroxyl radical with ethyl chloride, and if the overall energies of reaction are known for the abstraction of the α and β hydrogen atoms (to make water and the corresponding alkyl radicals), then A would be selected for a given electron correlation method (say, MP2) in order to make ESAC−MP2 agree with experiment as closely as possible for those particular data points. This same value of A would then be used for any point on the PES. Of course, the more experimental details that can be included in the choice of A, the better the parameterization (and the better able one is to judge the utility of Eq. (7.60) by examination of the errors in a one-parameter fit).

Note that one particularly attractive feature of Eq. (7.60) is that if the particular electron correlation method has available analytic derivatives, so too must ESAC−e.c.m., since derivatives for the latter will be simply determined as appropriately scaled sums of the e.c.m. and HF derivatives. Geometry optimization, and indeed the entire calculation, can essentially be carried out for exactly the cost of the e.c.m.

While one might imagine that values of A might best be determined individually within any given system, Siegbahn and co-workers have examined a large number of primarily small inorganic systems and suggested that, for the modified coupled-pair functional (MCPF) treatment of correlation (which is analogous in spirit to coupled cluster) with a polarized double-ζ basis set, a value of 0.80 has broad applicability, and they name this choice PCI80 (Siegbahn, Blomberg, and Svensson 1994; Blomberg and Siegbahn 1998). A summary of the utility of this level of theory for inorganic systems including comparison to density functional theory (DFT) can be found in Table 8.2.

Gordon and Truhlar (1986) have emphasized that variations on the theme of Eq. (7.60) can be useful in different circumstances. For instance, one might imagine carrying out multireference calculations and assuming two different scale factors, one applying to the non-dynamical

one step further, and assume a similar behavior for the correlation energy. For instance, the QM energies for glucose conformers that have served as a benchmark for comparison with lower levels of theory in preceding chapters were computed at a composite (C) level as

E(C) =E(MP2/cc-pVTZ//MP2/cc-pVDZ)

+{E(CCSD/6-31G(d)//MP2/6-31G(d))

− E(MP2/6-31G(d))}

+{E(HF/cc-pTVQZ//MP2/cc-pVDZ)

Thus, triple-ζ MP2 energies at double-ζ MP2 geometries are augmented with a correction for doubles contributions beyond second order (line 2 on the r.h.s. of Eq. (7.61)) and a correction for basis set size increase beyond triple-ζ (line 3 on the r.h.s. of Eq. (7.61) where the ‘T’ superscript in the first basis set implies that polarization functions from cc-pVTZ were used in conjunction with valence functions from cc-pVQZ).

While such ad hoc multilevel methods have been employed for rather a long time, only in the late 1980s were efforts undertaken to systematize the approach so as to define a model chemistry having broad applicability. The first such effort was the so-called G1 theory of Pople and co-workers, which was followed very rapidly by an improved modification called G2 theory, so that the former may be considered to be obsolete (Curtiss et al. 1991). The steps involved in a G2 calculation are detailed in Table 7.6. In this instance, the goal of the calculation is accurate thermochemistry, so some of the steps are devoted to computing thermal contributions to the enthalpy, as opposed to the electronic energy, as described in more detail in Chapter 10. Although the philosophy of the remaining steps is essentially the same as that predicating Eq. (7.61), there is considerably more attention to detail in specific aspects of the basis-set problem and the accounting for electron correlation. There is also a completely empirical correction procedure (step 8) to, inter alia, account for core-valence correlation and to improve the performance of the model over systems having different numbers of unpaired spins. Note that ‘full’ following a correlation acronym implies that core electrons were included in the correlation treatment, as opposed to the more typical choice of freezing them to excitation. Over a test set of 148 enthalpies of formation, the average error of G2 theory is 1.6 kcal mol−1.

Over time, many different groups have suggested minor modifications of G2 theory (each spawning a new acronym). Some trade accuracy for computational efficiency in order to permit application to larger systems; the most popular of these has been G2(MP2), which avoids the costly MP4 calculations in G2 at the expense of increasing the error over the test set to 1.8 kcal mol−1 (Curtiss, Raghavachari, and Pople 1993). Others emphasize alternative methods for obtaining molecular geometries, or attempt to correct for other deficiencies in G2 applied to specific classes of molecules (G2 does poorly on perfluorinated species, for instance).