Perturbation formulation Brillouin-Wigner

Hubac and his co-workers222"231 have explored the use of Brillouin-Wigner perturbation theory in solving the coupled cluster equations. For the case of a single reference function, this approach is entirely equivalent to other formulations of the coupled cluster equations. However, for the multireference case, the Brillouin-Wigner coupled cluster theory shows some promise in that it appears to alleviate the intruder state problem. No doubt perturbative analysis will help to gain a deeper understanding of this approach. 

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. 

Many-body Brillouin-Wigner second-order perturbation theory using a multireference formulation an application to bond breaking in the diatomic hydrides BH and FH Molecular Physics 104, 2367 (2006)... 

Let us consider a />-state system and obtain an explicit formulation of the multireference Brillouin-Wigner perturbation theory for this case. In the -state case, we have a reference space spaimed by p functions, o, 4>i,. .., The projector onto this space is... 

In this section, we present Brillouin-Wigner perturbation theory in both its single reference and its multireference form. This will serve both to emphasize the similarity of single reference and multireference formulations of Brillouin-Wigner perturbation theory and to establish notation for later sections. In section 3.1, we define the basic concepts of any perturbation theory. The definition of single and multireference spaces is considered in section 3.2 and the model wave function is described in section 3.3. The Brillouin-Wigner expansion is developed in section 3.4. 

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... 

It should be noted that the wave operator 17 no longer depends on the exact energies and therefore represents a much more suitable formulation for practical calculations. Within the multi-reference Brillouin-Wigner perturbation theory, we have been able to construct a multi-root wave operator together with an effective Hamiltonian operator, Jfeff, which formally possess the same properties as those employed in the multi-reference theories based on the Bloch equation. For this reason, the adjective multi-root is clearly not necessary here. 

Single-root formulation of multi-reference Brillouin-Wigner perturbation theory... 

---