Single-root formulation of multi-reference Brillouin-Wigner perturbation theory

Single-root formulation of multi-reference Brillouin-Wigner perturbation theory 

Equation (4.94) shows that 17 transforms the projection of the ground state wave function on to the model space, back into the exact ground state wave function. In contrast to the multi-root wave operator,J7, which was introduced in the previous section, our single-root wave operator, 17, is a state-specific wave operator, (see eq. (4.57)). Q does not transform projections of other exact wave functions CP Z a on to the model space into the wave functions for the corresponding exact states. To avoid any confusion we shall use a tilde to distinguish the single-root wave operator. 

If we denote the action of the wave operator 17 on the model space configurations as i.e. 

The inverse operator may exist even in the case when the corresponding power series in terms of Bo i does not converge. In the following, we will, therefore, start from eq. (4.92) instead of eq. (4.91).  

we can introduce the state-specific effective Hamiltonian, 9feff acting within the model space, in the same way as in the multi-root theory, i.e. 

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