Single-reference Brillouin-Wigner perturbation theory

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. 

Multi-reference Brillouin-Wigner theory overcomes the intruder state problem because the exact energy is contained in the denominator factors. Calculations are therefore state specific , that is they are performed for one state at a time. This is in contrast to multi-reference Rayleigh-Schrddinger perturbation theory which is applied to a manifold of states simultaneously. Multi-reference Brillouin-Wigner perturbation theory is applied to a single state. Wenzel and Steiner [105] write (see also [106]) ... 

We turn now to the Brillouin-Wigner perturbation theory for a system described in zero order by a multi-reference function. The multi-reference formalism closely parallels that given in the previous section for the case of a single-reference function. Let us begin by defining a reference space V. Let... 

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. 

Brillouin-Wigner perturbation theory can be developed for both the single reference function case and the multireference function case using a common formalism. This contrasts with the situation for Rayleigh-Schrodinger perturbation theory. We shall, therefore, consider the single reference and multireference formalisms together. 

BWPT = Brillouin-Wigner perturbation theory EN = Epstein-Nesbet FCI = full Cl MBPT = many-body perturbation theory MRS PT = multireference state perturbation theory PT = perturbation theory RSPT = Rayleigh-SchrSdinger perturbation theory SRS PT = single-reference state perturbation theory. 

We have included the parameter A in eq. (1.13) which is set equal to unity in order to recover the perturbed problem. Equation (1.13) is the basic formula of the Brillouin-Wigner perturbation theory for a single-reference function. 

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... 

By following procedures similar to those employed in the case of a single-reference function, the exact wave function f o, for a = 1,2,. .., d, in the Brillouin-Wigner perturbation theory can be written as the expansion... 

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

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... 

The relationship between single-reference Brillouin-Wigner perturbation theory and its Rayleigh-Schrodinger counterpart is well known, but for completeness we include a brief account of the single-reference case in Section 4.4.1 before turning to the multi-reference case in Section 4.4.2. 

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