Feshbach partitioning

The coupled Schrodinger equations can be projected onto the fa fa subspace by Feshbach partitioning, giving an equation for the coefficient function Xd(q) in the component faxdiq) of the total wave function. The effective Hamiltonian in this equation is tn + Vd(q) + Vopt, which contains an optical potential that is nonlocal in the <7-space. This operator is defined by its kernel in the fa - fa subspace,... 

In order to simplify the time-consuming inversion of the complex matrix, it is better to separate out the real and imaginary part in (2.37). This has been done with Feshbach partitioning [23] and results in the following formulation ... 

The problem of unimolecular decomposition into one or several continua via simultaneous direct and resonance-mediated routes is conveniently formulated within Feshbach s partitioning framework [56]. Following Refs. 29 and 31, we partition the scattering eigenstate into its bound and continuum projections as... 

Feshbach s partitioning network, two-pathway excitation, coherence spectroscopy 160-163... 

A central problem in physics and chemistry has always been the solution of the Schrodinger equation (SE) for stationary states. Such stationary states may relate to electronic structure problems, in which case one is primarily interested in bound states, or to scattering problems, in which case the stationary solutions are continuum states. In both cases, one of the most powerful tools in the theoretical arsenal for solving such problems is the partitioning technique (PT), which has been developed in a series of papers prominently by Per-Olov Lowdin [1-6] and Herman Feshbach [7-9]. 

Using the projection-operator formalism of Feshbach [ 115,116], an implicit variational solution for the coefficients cIJiS in can be incorporated into an equivalent partitioned equation for the channel orbital functions. This is a multichannel variant of the logic used to derive the correlation potential operator vc in orbital-functional theory. Define a projection operator Q such that... 

Molecular optical potentials for non-reactive processes may be rigorously defined by means of partitioning techniques (see e.g. Feshbach, 1962), which are based on the classification of scattering channels in two groups the first one includes states which are asymptotically selected or detected, and is characterized by a projection operator P the second one includes all other states (in practice those to which flux is lost) and is characterized by the projector Q. An optical potential operator VH may then be constructed as... 

The dynamic part DJ E), which can be found in the literature [8], possesses poles related to resonances of two-particle-hole and higher excitations and a branch cut above the ionisation threshold. It has an analytic structure similar to the dynamic self energy (42) from the formal point of view. This is not surprising since the projection method for deriving the Feshbach Hamiltonian is formally similar to the derivation of Dyson s equation of Sec. Ill or in Refs. [26,9]. There, a partitioning was performed yielding a projection to the primary space spanned by the orthonormal T-states while Feshbach, on the other hand, projected onto the space spanned by the non-orthonormal states aj 5 ). 

In SRS-PT, the Feshbach-Ldwdin Hamiltonian Heft, which is obtained by partitioning the exact Hamiltonian with the help of the projection operators P and Q, is best known. 

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