Excited states multi-reference perturbation theory

The renewal of interest in Brillouin-Wigner perturbation theory for many-body systems seen in recent years, is driven by the need to develop a robust multi-reference theory. Multi-reference formalisms are an important prerequisite for theoretical descriptions of dissociative phenomena and of many electronically excited states. Brillouin-Wigner perturbation theory is seen as a remedy to a problem which plagues multi-reference Rayleigh-Schrodinger perturbation theory the so-called intruder state problem. 

Unfortunately, in many situation of interest, both static and dynamic electronic correlation need to be taken into account. This is particularly true for the study of processes involving excited electronic states as in UV spectroscopy or in photochemistry. In this case, methods capable of taking into account the dynamic electronic correlation on top of a multi-determinantal wavefunction of the MCSCF type are needed. These methods are usually called multi-reference methods. The two standard methods that are able to account for both the static and dynamic electronic correlation are the multi-reference configuration interaction (MRCI) and several variants of second-order multi-reference perturbation theory (MRPT). 

The investigation of the electronic structure of excited states of polyenes has a long history which cannot be reviewed here. It suffices to say that modern methods of electronic-structure theory, in particular multi-reference configuration-interaction and multi-reference perturbation methods, allow the determination of the vertical excitation energies of valence and Rydberg... 

However, if this is not the case, the perturbations are large and perturbation theory is no longer appropriate. In other words, perturbation methods based on single-determinant wavefunctions cannot be used to recover non-dynamic correlation effects in cases where more than one configuration is needed to obtain a reasonable approximation to the true many-electron wavefunction. This represents a serious impediment to the calculation of well-correlated wavefunctions for excited states which is only possible by means of cumbersome and computationally expensive multi-reference Cl methods. 

We continue this section by pointing out that calculations exist that combine (PT) and MC methods. For example, methods such as complete active space, second order perturbation theory (CASPT2) and multi-configuration quasidegenerate perturbation theory MCODPT use a MC wave function as the reference. PT, generally to second order, is used to estimate the contribution from excited states that arise from excitation outside the original active space of the MC calculation. 

The main importance of Cl is that the FCI calculations provide results that are used as benchmarks for testing other post-Hartree-Fock methods. Less important seems to be the use of Cl as a post-Hartree-Fock method in routine chemical applications, because results of about the same accuracy may be obtained more economically by other methods. The size inconsistency of CI-SD may also be a drawback in some applications. Still, the recent progress in the development of Cl programs indicates that Cl might regain its importance even in this field. The traditional domain of Cl has been in electronic spectroscopy and excited electronic states in general. This is still true for semiempirical calculations. For ab initio calculations, however, it may be preferable to use multi-reference second-order perturbation theory, SAC-CI, or the equation-of-motion CC approach. 

In Section 4.2.3.2, we presented the basic equations of single-root (state-specific) multi-reference Brillouin-Wigner coupled cluster theory. We derived these equations from the single-root (state-specific) multi-reference Brillouin-Wigner perturbation theory presented in Section 4.2.3.1. In this section, we turn our attention to the coupled cluster single- and double-excitations approximation, ccsd. We present... 

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