Sturmian functions in a L2 basis: critical nuclear charge for n-electron atoms

A.L. FRAPICCINI , G. GASANEO , F. D. COLAVECCHIA, AND D. MITNIK

Roma

Conferencia; International Conference on Many particle spectroscopy of atoms, molecules, clusters and surfaces; 2007

The two particle Sturmian functions [1-2] have been successfully used in several atomic problems, to determine atomic energy levels [3-5] or as a basis to expand the Coulomb Green function [6]. This Sturmian basis set is chosen to satisfy certain boundary conditions, such as regularity at the origin and the correct asymptotic behavior according to the energy domain: exponential decay (bound states) for negative energy and outgoing (or incoming or standing wave). Although this functions are hard to compute, in particular for the positive energy case, they provide the correct boundary conditions to the solution of scattering equations [7]. In this report, we introduce a method to obtain Sturmian functions by solving the radial part of the Schrödinger equation. We seek for the solutions regular at the origin with outgoing wave asymptotic behavior. The method here proposed consist on expanding the Sturmian function in terms of a finite L2 Laguerre-type basis set. In order to check our results, two alternative methods have been developed: in the first one the Schrodinger equation, Eq. (1) is solved by direct diagonalization of the matrix representing the full Hamiltonian in a radial lattice basis. The new feature of our method, is that this matrix is modified (it becomes a complex matrix), in such a way that it includes the desired condition at the boundary. The second approach uses a Numerov shooting technique. We numerically solve the equation in the interior region and obtain the eigenvalues by matching the interior solution with the boundary condition at a distance R greater than the range of the potential V(r).We use this Sturmians in the negative energy case to solve the Schrödinger equation for the Hellmann potential [9] and study the behavior of the ground state energy for different values of γ and δ to obtain the value of the critical nuclear charge for which the atomic state becomes absorbed by the continuum [10].