A p-state system

Let us consider a p-state system and obtain an explicit formulation of the multireference Brillouin-Wigner perturbation theory for this case. In the p-state case, we have a reference space spanned byp functions, , The projector onto 

The secular equation for the lowest state takes the form  

Multi-reference configuration interaction in Brillouin-Wignerform 

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. 

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. 

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