Lippmann-Schwinger method

The above procedure resembles the Lippmann-Schwinger method [106] for constructing the incoming state in scattering theory. 

This method simply involves the solution of the Lippmann—Schwinger equations (6.73) or (6.87) with the potential matrix elements (7.35). The states i) are not eigenstates of the target Hamiltonian. They are configuration-interaction states or pseudostates obtained by diagonalising the target Hamiltonian in a square-integrable basis as described in section 5.6. 

The Lippmann—Schwinger equations (6.73) are written formally in terms of a discrete notation i) for the complete set of target states, which includes the ionisation continuum. For a numerical solution it is necessary to have a finite set of coupled integral equations. We formulate the coupled-channels-optical equations that describe reactions in a channel subspace, called P space. This is projected from the chaimel space by an operator P that includes only a finite set of target states. The entrance channel 0ko) is included in P space. The method was first discussed by Feshbach (1962). Its application to the momentum-space formulation of electron—atom scattering was introduced by McCarthy and Stelbovics... 

V (z) describes a decreasing in time quasi-stationary state. Contrary to the Lippmann-Schwinger equation, which requires scattering boundary conditions, V (z) does require outgoing boundary conditions commensurate with the Gammow-Siegert method. It is inherent in the complex technique and defined in a nonambiguous manner as a continued wavefunction in the second Riemann sheet. 

During the past few years we have observed an intensive development of many-channel approaches to the collision problem. In particular, the coupled-channels method is based on an expansion of the total wave fmiction in internal states of reactants and products and a numerical solution of the coupled-channels equations.This method was applied in the usual way to the atom-diatom reaction A + BC by MOR-TENSBN and GUCWA /86/, MILLER /102/, WOLKEN and KARPLUS /103/, and EL-KOWITZ and WYATT /101b/. Operator techniques based on the Lippmann-Schwinger equation (46.II) or on the transition operator (38 II) has also been used, for instance, by BAER and KIJORI /104/ The effective Hamiltonian approach( opacity and optical-potential models) and the statistical approach (phase space models, transition state models, information theory) provide other relatively simple ways for a solution of the collision problem in the framework of the many-channel method /89/<. 

Size consistency of the Brillouin-Wigner perturbation theory is studied using the Lippmann-Schwinger equation and an exponential ansatz for the wave function. Relation of this theory to the coupled cluster method is studied and a comparison through the effective Hamiltonian method is also provided. 

We can see that this condition will be satisfied if, and only t/the wavefimction/ satisfies the lippmann-Schwinger equation, so that (l-G V) f = lj>. This, we feel, is where the power of a variational method over a nonvariadond one originates. The requirement that the functional (5) be stationary with respect to small changes in/ sets a level of accuracy with which the equation (1-G V) > = ls> must be satisfied, which in turn forces an accurate solution for/ A comparison of Rg.l to Figs. 2 and 3 supports this argument,... 

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... 

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. 

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