Perturbation expansions Brillouin-Wigner

Eq. (3.12) represents the Brillouin-Wigner (BW) perturbation expansion of the exact wave function, whereas the corresponding expression for the... 

The Lennard-Jones Brillouin Wigner perturbation expansion is a simple geometric series. However, it contains the unknown exact energy within the denominators. This expansion is, therefore, not a simple power series in the perturbation. 

Rayleigh-Schrodinger Perturbation Theory.—In Rayleigh-Schrodinger perturbation theory the unknown energy in the denominators of the Lennard-Jones Brillouin Wigner expansion is avoided. This enables a size-consistent theory to be derived. 

The Lennard-Jones, Brillouin, Wigner perturbation expansion is not a simple power series in A since each depends on the exact energy, S. Each energy coefficient in the Rayleigh-Schrodinger perturbation expansion consists of a principal term of the form... 

Two prominent perturbation schemes namely the Brillouin-Wigner (BW) and Rayleigh-Schrbdinger (RS) expansions are often used in... 

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 1967, Brandow [69] used the Brillouin-Wigner expansion in his derivation of a multireference many-body (Rayleigh-Schrodinger) perturbation theory. In the abstract to his paper entitled Tinked-Cluster expansions for the nuclear many-body problem Brandow writes ... 

T/ie Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms AE are expanded out of the BW energy denominators, and the series is then rearranged to obtained the linked-cluster result. ... 

There are two basic differences in [his] approach which permit all orders to be treated at once. First, the starting point is the Brillouin-Wigner BW) perturbation theory, whose formal structure is much simpler than that of the RS expansion. Secondly, we use a factorization theorem , which expresses the required energy-denominator identities in a simple and general form. ... 

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. 

In Brillouin-Wigner perturbation theory a power series expansion is made for the wave function (A) while in the more familiar Rayleigh-Schrodinger perturbation theory a power series expansion is made for both the wave function and the energy a (A). Furthermore, we require that for A 7 0, lintruder states arise. 

We are now in a position to develop the Brillouin-Wigner perturbation expansion for the exact wave function. 

In the previous section, we have given the Brillouin-Wigner perturbation expansion for the exact wave function for state a developed with respect to some single reference or multireference model function In this section, we define the Brillouin-Wigner wave operator and the corresponding Bloch-like equation [64]. 

---