Resonances by the exterior-scaling method within the framework of the finite-basis-set approximation

Abstract

Simon's exterior-scaling procedure is applied to model systems within the framework of the finite-basis-set approximation. We show that if the basis-set functions are scaled when r )ro, a variational solution is obtained by adding the term , 5(r -ro)[d/drk][lexp( -2iB)] to the Hamiltonian, where k is the logarithmic derivative of the resonance wave function at ro. Two computational methods are proposed. One is an ab initio linear variational iterative procedure, whereas in the second one A, is semiclassically estimated during the iteration procedure and all the variational calculations are carried out for r ro and therefore are exactly 6 independent. An illustrative numerical application is presented.

Electron-impact ionization of atomic hydrogen

“…The motivation for tlds development was the desire to treat Harniltonians that have nonanaIyticities in the interior region, such as the BomOppenheimer Harniltonian whose electron-nuclear attraction terms are not ddatation analytic when viewed solely as a function of the electronic coordinates [15]. In computational applications, exterior complex scaling has been used mainly in direct numerical integration methods [16][17][18], although there have been a few attempts, in connection with resonance evaluations, to implement the method in a basis [19][20][21].…”

Exterior complex dilation for grid methods: Application to the cumulative reaction probability

“…Of relevance here is a comparison with the basis set exterior scaling results by Lipkin et al 19 and the grid complex dilation results by Museth and Leforestier. 15 Here the standard mapping ͑2.7͒ and the same approach as in Ref.…”

“…45 There are several possible choices of mappings, or complex contours. 24 The ''standard'' choice being [17][18][19] …”

“…This is a consequence of the discontinuity of the wave function at the point of rotation. 19,24 Thus ordinary complex dilation should be used to determine single resonances with high precision.…”

“…If the point of rotation x 0 is chosen such that the potential is constant after that point ͑the ''unphysical'' region͒ the potential energy matrix will not be rotated into the complex plane. 13,19 Note the similarity with the NIP: instead of adding a complex term to the potential energy we make the kinetic energy operator complex.…”