Lippmann-Schwinger-like equation

Like the wave operator, the reaction operator Vj (A) is state-specific. It satisfies a Lippmann-Schwinger-like equation for multi-reference Brillouin-Wigner theory ... 

BRILLOUIN-WIGNER EXPANSIONS IN QUANTUM CHEMISTRY BLOCH-LIKE AND LIPPMANN-SCHWINGER-LIKE EQUATIONS... 

The Lippmann-Schwinger-like equation in Brillouin-Wigner form... 

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]. 

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... 

Using the sum-over-states form of the Brillouin-Wigner propagator, equation (33), the Lippmann-Schwinger-like equation, equation (55), can be written in matrix form as... 

Given that in equation (92) is obtained by, at most, a double replacement with respect to the first application of (93) can result in involving quadruple replacements. Repeated application of the Lippmann-Schwinger-like equation (93) leads to successively higher order levels of replacement with respect to... 

The Brillouin-Wigner Lippmann-Schwinger-like equation can be written... 

Brillouin-Wigner expansions in quantum chemistry Bloch-like and Lippmann-Schwinger-like equations 71... 

We do not propose to describe here the details of specific applications of Brillouin-Wigner methods to many-body systems in chemistry and physics. Such details can be found in our article in the Encyclopedia of Computational Chemistry [1] and in our review entitled Brillouin-Wigner expansions in quantum chemistry Bloch-like and Lippmann-Schwinger-like equations [36]. We have established a website at... 

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