Overview: Accurate Description of Low-Lying Electronic States and

Chapter 1

Overview: Accurate Description of Low-Lying Electronic States and Potential Energy Surfaces 1

Mark R. Hoffmann and Kenneth G. Dyall

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Chemistry Department, University of North Dakota, Grand Forks, ND 58202-9024 Eloret Corporation, 690 West Fremont Avenue, Sunnyvale, CA 94087

Downloaded by 80.82.77.83 on April 20, 2017 | http://pubs.acs.org Publication Date: August 14, 2002 | doi: 10.1021/bk-2002-0828.ch001

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This paper provides an overview of recent trends in the development of electronic structure theory for the accurate characterization of all, or large regions, of ground and excited potential energy surfaces. Topics include the treatment of dynamical and nondynamical correlation and the calculation of nonadiabatic coupling matrix elements, as arising from spinorbit coupling and from nuclear motion.

The theoretical study of accurate potential energy surfaces (PESs) has seen some essential progress in the last decade. Much of this progress can be attributed, at least in broad terms, to advancements in the ability to include nondynamical electron correlation equitably with dynamical electron correlation. Perhaps this point can be underscored by noting the tremendous response of the greater chemistry community to the CASPT2 (1) functionality in the widely

© 2002 American Chemical Society

Hoffmann and Dyall; Low-Lying Potential Energy Surfaces ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

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distributed G A U S S I A N (2) software package. Moreover, C A S P T 2 is but one of many such efforts to achieve a useful balance between the different types of electron correlation. There has also been progress in other areas than that of correlation, such as nonadiabatic couplings via the spin-orbit interaction and nuclear motion. In this chapter we give an overview of some of the important theoretical developments. It should be stated at the outset that this overview is not intended to be encyclopedic, and more complete bibliographies are to be found in the individual chapters of this volume. Rather, this overview represents our attempts to indicate important trends in the field and therefore the motivations in organizing the symposium that served as a basis of this book.

Dynamical vs. Nondynamical Electron Correlation While the venerable

multireference

configuration interaction

method,

including single and double excitations (MR-CISD), remains the approach of choice for some problems, alternatives have been proposed and are showing their usefulness.

One can describe M R - C I S D as the variational calculation

within a specified reference, or model, space, c=/o> and the spaces generated by all single excitations relative to the specified model space, J!\, and likewise the double excitations, J! . 2

Provided that the model space is sufficiently large to

describe the nondynamic electron correlation adequately, M R - C I S D provides a well-balanced and unbiased description of both nondynamic and dynamic electron correlation. It is precisely in the caveat in which lies the problem with MR-CISD. Since the size of the excitation spaces c^i and Jit grow rapidly with the size of the dimension of the model space, «/o, one is often faced with the unpleasant task of restricting J!Q to a size smaller than physically justified. O f course, if such restriction is not required for the problem at hand, M R - C I S D is a reliable and robust method and, so, continues to see modern usage. It is also worth noting that MR-CISD suffers from a lack of size-extensivity, and, although this is usually not a serious problem, relative to other sources of error, for the mapping of PESs of small molecular systems, there is need for theoretical advancements. In situations in which physically well-motivated M R - C I S D calculations are not computationally feasible, the need for adequate approximations of nondynamical and dynamical electron correlation must be balanced. One useful way of categorizing alternatives to M R - C I S D is to focus on the sequence of treating nondynamical and dynamical correlation. One could address dynamical correlation for all, or part, of the model space first and, then, proceed to address the nondynamical correlation. This approach gives rise to an effective



3 Hamiltonian. Indeed, this approach, with many variations, finds expression in the current coupled cluster equation of motion (EOM) or similarity transform E O M (STEOM) procedures (3-5) and also in the multireference (or quasidegenerate) perturbation theory approaches (MRPT or QDPT) (6-10). In fact, the CCSD(T) method (11), which was recently referred to as the "gold standard" of modern quantum chemistry (12), also addresses dynamic correlation prior to nondynamic correlation. However, since CCSD(T) is generally only applicable to the ground electronic state (or, possibly, the lowest state of a particular symmetry) and the emphasis of this volume is on multiple PESs, this approach is not considered in any detail herein. Alternatively, one could address the nondynamical correlation first and then only consider the dynamical correlation. State selective methods, including "diagonalize-then-perturb" multireference perturbation theories (e.g., CASPT2) and internally-contracted CI methods (13), are of this philosophy. Taking into account that both general approaches are currently being used successfully, and continue to be developed further, it must be clear that both have merit.

Multiple Potential Energy Surfaces One criterion for recommending effective Hamiltonian methods over stateselective methods is whether multiple PESs are of simultaneous interest. Stateselective methods are in general capable of determining excited PESs, even of the same symmetry as the ground state (or lowest state of a given symmetry), but, by their nature, use essentially different representations of the multiple surfaces. Conversely, precisely because state-selective approaches need consider only one state in a particular calculation, they can often achieve quite high computational efficiencies. In many of the state-selective procedures, a multiconfiguration self-consistent (MCSCF) calculation is performed to determine the nondynamical electron correlation, after which the dynamical electron correlation method of choice is applied. So, for different states, different molecular orbitals will be used. This complicates, but not hopelessly, the calculation of matrix elements that couple surfaces. Such matrix elements arise when considering nonadiabatic effects, i.e., in consideration of spin-orbit coupling or nuclear derivative coupling. Multiple PESs may be of simultaneous interest based not only on physical reasons, as emphasized in the previous paragraph, but also for mathematical or computational reasons. Consider the basic paradigm of state-selective methods: the nondynamical electron correlation for a specific state is calculated within a model space and then the dynamical electron correlation is calculated. The implicit assumption is that the zero-order model space many-electron basis functions (MEBFs) (e.g., M C S C F functions and M C S C F complementary space



4 functions) are sufficiently close to the correlated (i.e., true) model space M E B F s that the mixing between the correlated space M E B F s can be made on the basis of the zero-order model space M E B F s . In fact, for well-separated states, and a physically realistic model space, this assumption is well founded. However, in cases in which the states are in close proximity the dynamic electron correlation can significantly alter the mixing between zero-order fonctions of the model space. Avoided crossings represent just such physical conditions. In these cases, state-selective methods are presented with severe mathematical challenges. The reader is directed to the chapter by Mukherjee in this volume in which recent progress on this problem has been made. On the other hand, it is precisely in such situations that multireference or quasidegenerate methods, which form effective Hamiltonians, are most appropriate. The dynamic correlation is considered prior to the nondynamic correlation that mixes the two close lying states. However, effective hamiltonians can be mathematically unstable in precisely the situations in which state selective methods are most adept: widely separated energy levels. The problem is the appearance of intruder states, whether physical (or infinite order) or zero-order (i.e., many-electron basis functions). Intruder states stymied the development of quasidegenerate perturbation theories for many years and, although there are now known remedies, intruder states continue to require care when formulating new variants of QDPT. In essence, one compromises on the goal of treating the effect of dynamical electron correlation on the entire model space prior to consideration of nondynamical correlation. As pioneered by Malrieu and coworkers and by Kirtman, one contents oneself with describing only part of the model space well. Recent work in the group of Hoffmann has demonstrated that it is feasible to remove the arbitrariness in selecting the part of the model space that will be treated well (14). In fact, both state-selective methods and effective Hamiltonian methods are faced with the challenge of describing variable quasidegeneracy equitably over entire potential energy surfaces.

Nonadiabatic Coupling Matrix Elements Several other issues are relevant to the discussion of multiple PESs. As previously mentioned, multiple PESs are required for the description of any nonadiabatic effects. Such processes arise in considering relativistic effects and nuclear dynamics. The situation here is more complex than with just the calculation of multiple PESs. Indeed, the accuracy and effort of the calculation of nonadiabatic matrix elements must be assessed against the reliability of the PESs themselves. Furthermore, the issue of coupling of adiabatic surfaces



5 places additional emphasis on the criterion of uniform accuracy of PESs. Whereas for individual PES calculations it might not be really essential to maintain uniform accuracy, provided, for example, that each PES had its critical points (e.g., local minima and transition states) calculated at comparable accuracies, for nonadiabatic effects it may be regions of PESs far removed from these relatively benign points which require uniform accuracy. It is precisely for this reason that the majority of studies involving nonadiabatic dynamics and relativistic descriptions of stationary points have utilized MR-CISD wavefunctions. However, as previously noted, computational considerations recommend the development of alternatives to MR-CISD. The development of methods for inclusion of relativistic effects on PESs is at a much earlier stage than that of electron correlation effects. Granted, scalar relativistic effects can be - and have been - routinely incorporated into nonrelativistic calculations, either using effective core potentials or ab initio model potentials, or one of the scalar all-electron methods such as the DouglasKroll-Hess (15,16) method. The incorporation of spin-orbit coupling is more difficult because it breaks both the spatial and the spin symmetry of the nonrelativistic wavefunction. But it is precisely this phenomenon that makes it so interesting and important in the consideration of multiple potential energy surfaces. Spin-orbit mediated avoided crossings can radically change reaction dynamics, for example, and change the nature of conical intersections, as shown earlier (17), and addressed further in this symposium, by Yarkony. Furthermore, one must consider where to include spin-orbit effects in relation to both dynamical and nondynamical correlation. The most widely used approach is to construct an effective Hamiltonian for spin-orbit coupling from M R C I wavefunctions that are built from a common orbital set. The small set of wavefunctions is often sufficient to describe spin-orbit coupling between the nonrelativistic surfaces, and was used to good effect by Werner in the reaction dynamics calculations presented in this symposium. Another approach is to obtain a set of natural orbitals from M R C I calculations and use these in a CI calculation that includes spin-orbit interaction. In this way dynamical correlation, nondynamical correlation and spin-orbit interaction are treated on the same footing, albeit with some compromise on the first of these three. The more rigorous approaches are still in their infancy due largely to technical difficulties and the magnitude of the calculations resulting from the lowering of the symmetry. These approaches include M R C I methods based on nonrelativistic wave functions such as that of Yabushita and coworkers (18) and Rakowitz and Marian (19) that include the spin-orbit interaction on an equal footing with the Coulomb (or scalar relativistic) interaction, and methods that start from spin-coupled wavefunctions, either at a 2-component or 4-component level (20). Such methods are necessary to address the challenges like those


6 presented by high-accuracy experimental PESs for the halogen monoxides measured by Miller and coworkers (21 ).


Nondynamical (or Static) Correlation Another issue which is relevant to the accurate description of PESs is the question of the treatment of nondynamical correlation. Above, we accepted, without comment, the conventional paradigm of description of nondynamical electron correlation by an M C S C F calculation, in the case of antecedent nondynamical electron correlation, or of a variational calculation within the model space in the case of subsequent nondynamical electron correlation. Although this is the modus operandi, which usually takes specific form in a C A S S C F calculation and/or a state-averaged variant (SA-MCSCF), this can hardly be considered a wholly resolved issue. Indeed, it has been shown that coupled cluster based alternatives to variational calculations within a model space have merit. Within this volume, Head-Gordon describes some very recent advances. Another related issue concerns the molecular orbitals themselves. Earlier, Freed demonstrated that high-lying valence molecular orbitals obtained from M C S C F calculations are not particularly desirable M O s for describing excited states (22), as required in an effective hamiltonian calculation. Although the situation may be expected to be improved by increasingly large stateaveraged calculations, such calculations can of themselves become computationally resource significant. Moreover, such calculations are quite prone to optimization to one of many, and possibly myriad, local minima; thus, seriously raising the question of reproducibility. In this volume, Freed and coworkers give further account of efforts to circumvent the low occupation M O problem by a novel improved virtual orbital scheme.

Concluding Remarks One of the driving forces for the development of theoretical methods to describe accurately ground and excited PESs, and their couplings, is the study of chemical reaction dynamics [for a recent overview, with many references, see Ref. (23)]. Although many reactions can be reasonably well described as proceeding on a single adiabatic surface, there are at least two regimes in which this conceptualization is wanting. First, near dissociation limits PESs that are well isolated from each other in other geometrical regions can become quasidegenerate or even exactly degenerate. In such situations, couplings, such as spin-orbit, Coriolis and nuclear derivative, can give rise to an effective PES that is different than the adiabatic surface. As mentioned previously, the



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challenge to electronic structure theory is to describe surfaces equitably under conditions of variable quasidegeneracy. Second, even in the interior, or nonfragment, regions of PESs, it may be the case that two surfaces couple relatively strongly in a small geometrical region. Using a classical description, reaction paths approaching or passing through such regions can lead to qualitatively different results than in the absence of the couplings. Conical intersections are a dramatic example. In this symposium, several speakers were asked to describe advances and challenges in chemical reaction dynamics. Several of the speakers, complemented by contributions from outside the symposium, also contributed to this volume. Although this volume is even less comprehensive with respect to advances in chemical dynamics than it is with respect to the description of PESs, it was our intention that the interplay between electronic structure theory and the study of chemical reactions be represented herein.

References 1.

Andersson, K . ; Malmqvist, P.-Å; Roos, B . O. J. Chem. Phys. 1992, 96, 1218. 2. Gaussian 98; Frisch, M. J., et al.; Gaussian, Inc.: Pittsburgh, PA, 1998. 3. Koch, H ; Jørgensen, P. J. Chem. Phys. 1990, 93, 3333. 4. Stanton, J. F.; Bartlett, R. J. J. Chem. Phys. 1993, 98, 7029. 5. Nooijen, M.; Bartlett, R. J. J. Chem. Phys. 1997, 106, 6441. 6. Sheppard, M. G ; Freed, K . F. J. Chem. Phys. 1981, 75, 4507. 7. Hirao, Κ Chem. Phys. Lett. 1992, 190, 374. 8. Hoffmann, M. R. Chem. Phys. Lett. 1992, 195, 127. 9. Nakano, H J. Chem. Phys. 1993, 99, 7983. 10. Kozłowski, P. M.; Davidson, E . R. J. Chem. Phys. 1994, 100, 3672. Raghavachari, K ; Trucks, G . W.; Pople, J. Α.; Head-Gordon,M.Chem.

Phys. Lett. 1989, 157, 479. 12. Head-Gordon, M . Presented at the 10th American Conference on Theoretical Chemistry, Boulder, CO, 1999. 13. Werner, H.-J.; Knowles, P. J. J. Chem. Phys. 1988, 89, 5803. 14. Khait, Y . G.; Hoffmann, M. R. J. Chem. Phys. 1998, 108, 8317. 15. Douglas, M.; Kroll, Ν. M. Ann. Phys. (N.Y.) 1974, 82, 89. 16. Hess, B . A . Phys. Rev. A 1985, 32, 756; 1986, 33, 3742. 17. Matsika, S.; Yarkony, D. R. J. Chem. Phys. 2001, 115, 2038-2050, 50665075. 18. Yabushita, S.; Zhang, Z. Y.; Pitzer, R. M. J. Phys. Chem. A 1999, 103, 5791. 19. Rakowitz, F, Marian, C. Chem. Phys. 1997, 225, 223.


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20. Visscher, L . ; Visser, Ο.; Aerts, P. J. C.; Merenga, H . ; Nieuwpoort, W. C. Comp. Phys. Commun. 1994, 81, 120. 21. Miller, C. E.; Drouin, B. J. J. Mol. Spectrosc. 2001, 205, 128; 2001, 205, 312. 22. Finley, J. P; Freed, K . F. J. Chem. Phys. 1995, 102, 1306. 23. Truhlar, D. G. Molecular-Scale Modeling of Reactions and Solvation. In First International Conference on Foundations of Molecular Modeling an Simulation; Cummings, P. T.; Westmoreland, P. R.; Carnahan, B, Eds.; AIChE Symposium Series V o l . 97; American Institute of Chemical Engineers: New York, 2001; pp 71-83.


Overview: Accurate Description of Low-Lying Electronic States and

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