Momentum space numerical integration

One Important aspect of the supercomputer revolution that must be emphasized Is the hope that not only will It allow bigger calculations by existing methods, but also that It will actually stimulate the development of new approaches. A recent example of work along these lines Involves the solution of the Hartree-Fock equations by numerical Integration In momentum space rather than by expansion In a basis set In coordinate space (2.). Such calculations require too many fioatlng point operations and too much memory to be performed In a reasonable way on minicomputers, but once they are begun on supercomputers they open up several new lines of thinking. 

For maximum numerical efficiency, all derivative terms of the AWF are kept in momentum space so that one FFT is saved for each of their evaluations. For example, during the evaluation of the following general double integral, the first FFT can be avoided ... 

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... 

Recently Elster, Liu and Thaler (ELT) [El 91] proposed a novel method for dealing with the momentum space Coulomb problem, which is, in principle, exact and may be less prone to numerical difficulties than the VP method. Their approach is based on the separation of the optical potential in eq. (3.63) and employs the two-potential formula [Ro 67] to express the full scattering amplitude as a sum of the point Coulomb amplitude and the point Coulomb distorted nuclear amplitude. The latter is obtained by numerically solving an integral equation represented in terms of Coulomb wave function basis states rather than the usual plane wave states. 

The required integration over deformation histories is accomplished by integrating numerically microscopic particle trajectories for large global ensembles simultaneously with the macroscopic equations of mass and momentum conservation. The term trajectories in the previous sentence refers to both real space trajectories, i.e., positions r, t) and to configurational phase space trajeetories, i.e., in the case of a dumbbell model, connector vector Q. 

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