Brillouin-Wigner perturbation theory equations

Our interest in Brillouin-Wigner perturbation theory was stimulated by our finding [44] that this theoretical tool proved very useful in the scattering theory. The fundamental equation, known in the scattering theory as Lippmann-Schwinger equation, expresses the scattering operator as... 

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. 

The Brillouin-Wigner perturbation theory can be employed to solve the equations associated with an explicit many-body method. 

Size consistency of the Brillouin-Wigner perturbation theory is studied using the Lippmann-Schwinger equation and an exponential ansatz for the wave function. Relation of this theory to the coupled cluster method is studied and a comparison through the effective Hamiltonian method is also provided. 

Let us go back to our problem we want to have Eq on the left-hand side of the last equation, while - for the time being - Eq occurs on the right-hand sides of both equations. To exit the situation we will treat Eq occurring on the right-hand side as a parameter manipulated in such a way as to obtain equality in both above equations. We may do it in two ways. One leads to Brillouin-Wigner perturbation theory, the other to Rayleigh-Schrodinger 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. 

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. 

HubaC and Neogrady [68] have explored the use of Brillouin-Wigner perturbation theory in solving the equations of coupled cluster theory. In a paper published in The Physical Review in 1994, entitled Size-consistent Brillouin-Wigner perturbation... 

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. 

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. 

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... 

Brillouin-Wigner perturbation theory and limited configuration interaction Let us write the exact Schrodinger equation as... 

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. 

These two equations together are the basic equations of non-degenerate (singlereference) Brillouin-Wigner coupled cluster (Bwcc) theory. We emphasize that these equations are obtained directly from Brillouin-Wigner perturbation expansion. In particular, we have not used the linked cluster theorem and neither have we employed... 

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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