Multi-reference Brillouin-Wigner formulation

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... 

These two possible formulations of multi-reference Brillouin-Wigner coupled cluster theory are discussed further below. In Section 4.2.2.1, we present a multiroot formulation of Brillouin-Wigner theory. This formalism is employed in Section 4.2.2.2 to develop a multi-root, multi-reference Brillouin-Wigner coupled cluster theory, using a Hilbert space approach. In Section 4.2.2.3, we discuss the basic approximations employed in the multi-reference Brillouin-Wigner coupled cluster method. 

I. Multi-root formulation of multi-reference Brillouin-Wigner coupled cluster 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. 

The application of the Brillouin-Wigner coupled cluster theory to the multireference function electron correlation problem yields two distinct approaches (i) the multi-root formalism which was discussed in Section 4.2.2 and (ii) the single-root formalism described in the previous subsections of this section. Section 4.2.3. The multiroot multi-reference Brillouin-Wigner coupled cluster formalism reveals insights into other formulations of the multi-reference coupled cluster problem which often suffer from the intruder state problem which, and in practice, may lead to spurious... 

We shall consider in turn each of these widely used approximation methods in this chapter. We shall provide an account of the Brillouin-Wigner formulation of each of these methods in a self-contained manner so that extensive cross referencing can be avoided. We shall establish the value of the Brillouin-Wigner method in the study of problems requiring a multi-reference formalism for a broad range of theoretical approaches. In this way, any problems associated with intruder states can be avoided. A posteriori corrections can be introduced to remove terms which scale in a non linear fashion with particle number. We shall not, for example, consider in any detail hybrid... 

Equation (4.190) has p roots of which we take only one. The exact energy, q, occurs in the denominator factors in eqs. (4.191) and (4.193). The eq. (4.190) must, therefore, be solved iteratively until self-consistency is achieved. These are the basic equations of the multi-reference configuration interaction method in a Brillouin-Wigner formulation in the case of a p state reference. 

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. 

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