Multi-reference Brillouin-Wigner correction

Whereas the multi-reference Rayleigh-Schrodinger perturbation theory approximates a manifold of states simultaneously, the multi-reference Brillouin-Wigner perturbation theory approach is applied to a single state - it is said to be state-specific . The multi-reference Brillouin-Wigner perturbation theory avoids the intruder state problem. If a particular Brillouin-Wigner-based formulation is not a valid many-body method, then a posteriori correction can be applied. This correction is designed to restore the extensivity of the method. This extensivity may be restored approximately... 

In Brillouin-Wigner coupled cluster theory, the simple a posteriori correction described above is exact in the case of the single-reference formalism. In the state-specific multi-reference Brillouin-Wigner coupled cluster theory, the simple a posteriori correction is approximate. An iterative correction for lack of extensivity has been studied by Kttner [38], but this reintroduces the intruder state problem. 

We turn, in this section, to the multi-reference Brillouin-Wigner perturbation theory. We divide our discussion into two parts. In Section 4.4.2.1, we survey the basic theoretical apparatus of multi-reference second-order Brillouin-Wigner perturbation theory. In Section 4.4.3, we describe an a posteriori correction to multi-reference Brillouin-Wigner perturbation theory. 

A posteriori correction to multi-reference Brillouin-Wigner perturbation theory... 

A posteriori Brillouin-Wigner correction to limited multi-reference configuration interaction... 

A posteriori corrections can be developed for calculations performed by using the Brillouin-Wigner perturbation expansion. These a posteriori corrections can be obtained for the Brillouin-Wigner perturbation theory itself and, more importantly, for methods, such as limited configuration interaction or multi-reference coupled cluster theory, which can be formulated within the framework of a Brillouin-Wigner perturbation expansion. 

We shall consider in turn each of these widely used approximation methods in this chapter. We shall provide an account of the Brillouin-Wigner formulation of each of these methods in a self-contained manner so that extensive cross referencing can be avoided. We shall establish the value of the Brillouin-Wigner method in the study of problems requiring a multi-reference formalism for a broad range of theoretical approaches. In this way, any problems associated with intruder states can be avoided. A posteriori corrections can be introduced to remove terms which scale in a non linear fashion with particle number. We shall not, for example, consider in any detail hybrid... 

If j) is a determinant related to one of the reference determinants by a double replacement, then k) involves, at most, quadruple replacements with respect to 1 ) in eq. (4.193). Repeated application of the Lippmann-Schwinger-file equation [160] leads to higher order replacements. If we restrict the degree of replacement admitted in (4.193) then we realize a limited multi-reference configuration interaction method. It is this realization of the multi-reference limited configuration interaction method that we use to obtain an a posteriori correction based on Brillouin-Wigner perturbation theory. 

We are now in a position to develop an a posteriori correction to state-specific limited multi-reference configuration interaction in the case of a p-state reference function. The following identity relates the Brillouin-Wigner and the Rayleigh-Schrodinger denominators [18,76,77] for the ground state (a = 0) ... 

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