In Brillouin-Wigner form

We emphasize that the wave operator (A) is applied to the /xth state only. It is state specific. Equation 2.29 is termed the Bloch equation [7] in Brillouin-Wigner form. Having introduced the Brillouin-Wigner wave operator, we turn our attention now to the corresponding reaction operator (A). 

Combining (159) and (164) gives a Lippmann-Schwinger-Zife equation in Brillouin-Wigner form... 

The Bloch-like equation in Brillouin-Wigner form... 

Comparing equation (34), which defines the wave operator, with equation (32), which is the Brillouin-Wigner expansion for the exact wave function, we can write the wave operator in Brillouin-Wigner form as... 

The wave operator (34) depends on the exact energy which is not known a priori. To obtain the exact energy we begin by defining the reaction operator and then consider the Lippmann-Schwinger-like equation in Brillouin-Wigner form [122,123]. 

By combining equation (41), the Bloch-like equation in Brillouin-Wigner form, with equation (53), the definition of the reaction operator, a Lipp-mann-Schwinger-like equation [122,123] is obtained... 

Equation (55) is a Lippmann-Schwinger-like equation in Brillouin-Wigner form. Recursion of equation (55) gives... 

Solution of the Lippmann-Schwinger-like equation in Brillouin-Wigner form, equation (55), for the reaction operator followed by solution of the eigenvalue problem (49) for the effective hamiltonian given in equation (52) is entirely equivalent to the solution of the time-independent Schrodinger equation, equation (1), for the state a. Furthermore, although recursion leads to the expansion (56), equation (55) remains valid when the series expansion does not converge. Equation (55) can be written... 

Equations (91), (92) and (128) completely define the single reference, single and double replacement configuration interaction method, CISD, in Brillouin-Wigner form. We designate this method BWCISD. 

Equation (4.29) can be regarded as the analogue of the Bloch equation [39] in Brillouin-Wigner form. 

Finally, the two sets of equations given above for the wave operator (4.71) and (4.75), are entirely equivalent. Our first approach represented by the set of eqs. (4.71) may be regarded as a Bloch equation [85] in Brillouin-Wigner form. Similarly, in terms of perturbation theory, the generalized Bloch equation (4.77) may be viewed as a Bloch equation in the Rayleigh-Schrodinger form. 

Combining the Bloch-/ite equation (4.174) and the definition of the reaction operator, eq. (4.179), gives a Lippmann-Schwinger-ftte equation [160] in Brillouin-Wigner form ... 

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