A finite element implementation of exterior complex scaling for the accurate determination of resonance energies

Abstract

An implementation of exterior complex scaling using the finite elements method with high degree polynomials is presented. We apply the method to find the resonances of the potential 7.5r 2e−r and of a phenomenological coupled channel model of the CaH molecule. In both cases the method is quickly convergent and extremely stable numerically. Convergence could be pushed to the point where the real parts of most resonance energies were independent of the complex scaling angle and of the exterior scaling radius within machine precision (14 significant digits). All imaginary parts were stable to at least eight significant digits. Several resonances of CaH which had evaded searches with a finite difference method could be located.

Infinite-range exterior complex scaling as a perfect absorber in time-dependent problems

“…For this we closely follow earlier work, Ref. [13]. In one dimension, exterior complex scaling consists in continuing the coordinate outside a "scaling radius" R 0 into the complex plane…”

“…We only give here plausibility arguments and refer the reader to Ref. [13] for a more extensive discussion and references to mathematical literature.…”

Electron-impact ionization of atomic hydrogen

“…The~~~can be written as a finite sum of products of Yl,m, u shown in l@atiOn F.4, where the (J112m1m21LM) are the well-known Clebsch-Gordan coefficients. Procedures for calculating Clebsch-Gordan coefficients can be found in standard references such as Messiah [21] and Zare [39].…”

“…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].…”

Perfect absorption in Schrödinger-like problems using non-equidistant complex grids

“…It is well known that for a correct definition of the discretization of the differential operators i∂ x and ∂ 2 x it is sufficient to ensure that the Ψ(x) are continuous at x n , if one secures that all operators are implemented in a manifestly symmetric form. The correct symmetric form is typically obtained by a formal partial integration where boundary terms are dropped (see, e.g., [13] for a more detailed derivation). For example…”