Multi-reference perturbation theory correlation

Brillouin Wigner based multi-reference perturbation theory for electronic correlation effects Journal of Chemical Physics 108,4714 (1998)... 

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). 

Electron correlation was treated by the CIPSI multi-reference perturbation algorithm ([24,25] and refs, therein). The Quasi Degenerate Perturbation Theory (QDPT) version of the method was employed, with symmetrisation of the effective hamiltonian [26], and the Maller-Plesset baricentric (MPB) partition of the C.I. hamiltonian. 

A number of types of calculations begin with a HF calculation and then correct for correlation. Some of these methods are Moller-Plesset perturbation theory (MPn, where n is the order of correction), the generalized valence bond (GVB) method, multi-conhgurational self-consistent held (MCSCF), conhgu-ration interaction (Cl), and coupled cluster theory (CC). As a group, these methods are referred to as correlated calculations. 

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. 

Subsequently, Kozlowski et al. [24] also revisited the Cope rearrangement with inclusion of dynamic correlation between the active and inactive electrons. However, they used Davidson s own version of multi-reference, second-order perturbation theory [25], which allows the coefficients of the configurations in the CASSCE wave function to be recalculated after inclusion of dynamic electron correlation. Kozlowski et al. found that the addition of dynamic correlation to the (6/6)CASSCE wave function for the Cope TS causes the weight of the RHE configuration to increase at the expense of the pair conhgurations that are necessary to describe the two diradical extremes in Eig. 30.1. Thus, without the inclusion of dynamic electron correlation in the wave function, (6/6) CASSCF overestimates the diradical character of the C2 wave function [24]. 

Given the remarkable progress in many-body theories, accurate descriptions of electron correlation in molecular systems of variable near-degeneracies are still challenging and remain an active area of research. One framework that has provided not only accurate results but also qualitative insight is effective Hamiltonian (EH), based on which various multi-reference (MR) or quasi-degenerate (QD) perturbation theories (PT) have been proposed [ 1-26] in the past (see Refs. [27] and [28] for careful comparisons). Yet, the premises of MRPTs and QDPTs that electron correlation can be separated into static and dynamic components and then that they can respectively be treated variationally and per-turbatively are not always justified. As a consequence, low-order MRPTs and QDPTs usually depend strongly on the... 

In spite of this progress, problems remain and the description of electron correlation in molecules will remain an active field of research in the years ahead. The most outstanding problem is the development of robust theoretical apparatus for handling multi-reference treatments. Methods based on Rayleigh-Schrodinger perturbation theory suffer from the so-called intruder state problem. In recent years, it has been recognized that Brillouin-Wigner perturbation theory shows promise as a robust technique for the multi-reference problem which avoids the intruder state problem. 

This situation may be met when aiming to describe dynamic electron correlation starting from a single, multi-determinantal reference vector. Correction schemes based on perturbation theory (PT) have been applying successive Gram-Schmidt orthogonahzation in such circumstances [5-7], occasionally combined with Lowdin s symmetrical [8] or... 

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