Brillouin-Wigner perturbation theory posteriori corrections

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. 

These a posteriori corrections are based on a very simple idea which is suggested by the work of Brandow [10]. Brandow used the Brillouin-Wigner perturbation theory as a starting point for a derivation of the Goldstone linked diagram expansion by elementary time-independent methods . At a NATO Advanced Study Institute held in 1991, Wilson wrote [112] ... 

In the work of Brandow [10], Brillouin-Wigner perturbation theory is used as a step in the theoretical development of first Rayleigh-Schrodinger perturbation theory and then the many-body perturbation theory. In the a posteriori correction developed by the present authors in a paper entitled On the use of Brillouin-Wigner... 

Rayleigh-Schrodinger and Brillouin-Wigner Perturbation Theories and A Posteriori Many-Body Corrections... 

A posteriori corrections to second order multireference Brillouin-Wigner perturbation theory based on the identity ... 

Unlike the BWCCSD method, the MR-BWCCSD theory, the BWCISD theory and the MR-BWCISD theory do not support energies which scale linearly with the number of electrons in the system. In the multireference cases, that is MR-BWCCSD and MR-BWCISD theories, this lack of ex-tensivity arise both from nonlinear terms in the matrix elements of the effective hamiltonian and from diagonalization of the effective hamiltonian when an incomplete model space is employed. By exploiting the known relation between the Brillouin-Wigner and the Rayleigh-Schrddinger denominators it is possible to devise a posteriori corrections to methods formulated within the framework of Brillouin-Wigner perturbation theory. 

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... 

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 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... 

Now it is well known that Brillouin-Wigner perturbation theory is not, in general, a many-body theory, in that it contains terms which scale non-Unearly with the number of electrons in the system. However, it has been shown that a posteriori corrections to Brillouin-Wigner perturbation theory can be made based on the identity... 

We shall provide an overview of the applications that have been made over the period being review which demonstrate the many-body Brillouin-Wigner approach for each of these methods. By using Brillouin-Wigner methods, 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 methods such as the widely used ccsd(t) which employs ccsd theory together with a perturbative estimate of the triple excitation component of the correlation energy. 

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