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

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

We wish to approximate the solutions of the time-independent Schrodinger equation 

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

The Pa ars, in general, non-orthogonal but are assumed to be linearly independent. The exact wavefunction. Pa, is expanded as follows  

We now introduce the effective hamiltonian which acts in the reference space CP 

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