Generalized Brillouin-Wigner perturbation theory

One of the drawbacks of Brillouin-Wigner perturbation theory is that the expressions for the energy components in second order and beyond contain the exact energy in the denominator factors. The equations must therefore be solved iteratively until self-consistency is achieved. The generalized Brillouin-Wigner perturbation theory [21] has the advantage that the denominators can be factored from the sum-over-states formulae. 

Generalized Brillouin-Wigner perturbation theory is based on a resolvent of the form... 

Single state coupled cluster expansions and multireference coupled cluster expansions based on the generalized Brillouin-Wigner pertm bation theory have been described elsewhere [19]. The generalized Brillouin-Wigner perturbation theory can also be applied to the configuration interaction problem. 

This leads to what Lowdin [6] has called generalized Brillouin-Wigner perturbation theory. 

Equation (2.133) is the Bloch equation [11]. Together with eq. (2.131), it provides the fundamental equation of the generalized Brillouin-Wigner perturbation theory. 

The evaluation of the Wp is non-iterative. The determination of the generalized Brillouin-Wigner perturbation theory energy in order k requires the determination of the lowest root of the k order polynomial ... 

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... 

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. ... 

Studies in Perturbation Theory. II. Generalization of the Brillouin-Wigner Formalism... 

Finally, the two sets of equations given above for the wave operator (4.71) and (4.75), are entirely equivalent. Our first approach represented by the set of eqs. (4.71) may be regarded as a Bloch equation [85] in Brillouin-Wigner form. Similarly, in terms of perturbation theory, the generalized Bloch equation (4.77) may be viewed as a Bloch equation in the Rayleigh-Schrodinger form. 

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