Perturbation theory, multi-reference Brillouin-Wigner

In this section we briefly survey the basic formalism of the multi-reference Brillouin-Wigner perturbation theory. This will serve to introduce our notation. 

We seek solutions of the time-independent Schrddinger equation 

Let us consider the projection of the exact wave function, a, onto the reference space, i.e. 

The exact wavefunction, f, and the model function, satisfy the following intermediate normalization conditions 

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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]) ... 

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... 

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... 

Multi-reference Brillouin-Wigner perturbation theory for limited configuration interaction... 

We turn, in this section, to the multi-reference Brillouin-Wigner perturbation theory. We divide our discussion into two parts. In Section 4.4.2.1, we survey the basic theoretical apparatus of multi-reference second-order Brillouin-Wigner perturbation theory. In Section 4.4.3, we describe an a posteriori correction to multi-reference Brillouin-Wigner perturbation theory. 

If we restrict the order of perturbation admitted in (4.241) then we realize a finite order multi-reference Brillouin-Wigner perturbation theory. Specifically, if we neglect terms of order A are higher, we are led immediately to the second-order theory for which the matrix elements of the effective Hamiltonian (4.239) take the form ... 

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