Discrete sets of many body Coulomb Sturmians

J. M. RANDAZZO, V. Y. GONZÁLEZ, A. L. FRAPICCINI, F. D. COLAVECCHIA AND G. GASANEO

Rosario, Argentina

Conferencia; XXIV ICPEAC, International Conference on Photonic, Electronic and Atomic Collisions; 2005

    The two-particle Sturmians functions have been successfully used in one-electron atom theory. Avery [1] proposed a method for constructing many-electron Sturmians basis sets in terms of products of two-particle Sturmians, by solving the independent N -electron atom. The basis proved to be highly efficient to determine the many electron atom energies and bound states [1]. However, Szmytkowski [2] showed that the set of functions proposed by Avery is not complete; the negative energy basis elements in whichpositive energies is associated with one electron, where not considered, and the charge spectrum obtained by including them are not discrete but continuum (negative energy pseudostates embedded in the continua).In this work, we propose an alternative N-electron sturmian basis set being discrete for all energies. For this purpose, we use a finiteL2 Sturmian basis, solution of the two-body Coulomb hamiltonian in which the energy is fixed and the charge is the eingenvalue. As a consequence of the confinement method, the eingencharges are discrete for all energies and satisfy an equation of the form f (Zn , E) = 0. The discrete functions can be obtained by diagonalizing the two body hamiltonian in a finite Laguerre type basis (LB). In that case, f correspond to a Pollakzec polynomial, being the determinant of a tridiagonal matrix arising from the system of equations for the coefficients of the expansion [3]. Alternative to the ”Laguerre” discretiation the solution of the Coulomb problem in a box may be used (BB). This correspond to the set of functions obtained by imposing the wave function to vanish at r0 , in which case the function f is the Coulomb state evaluated at that point.Then, the problem is reduced to find those energies Ei for which the eingenvalues Zni are all the same. Since the LB and BB basis sets are complete when M −→ ∞, and r0 −→ ∞ respectively, the N-electron basis procedure is complete in that limit. As an example we discuss the simple case of N = 2, and use the basis to find the energy spectrum and eingenfunctions for the He atom. Then we diagonalize the interelectronic potential, the problem being reduced to a linear system of P algebraic equations.