Sturmian bases for two-electron systems in hyperspherical coordinates

Abstract

We give a detailed account of an ab initio spectral approach for the calculation of energy spectraof two active electron atoms in a system of hyperspherical coordinates. In this system ofcoordinates, the Hamiltonian has the same structure as the one of atomic hydrogen with theCoulomb potential expressed in terms of a hyperradius and the nuclear charge replaced by anangle dependent effective charge. The simplest spectral approach consists in expanding thehyperangular wave function in a basis of hyperspherical harmonics. This expansion however, isknown to be very slowly converging. Instead, we introduce new hyperangular Sturmianfunctions. These functions do not have an analytical expression but they treat the first term of themultipole expansion of the electron?electron interaction potential, namely the radial electroncorrelation, exactly. The properties of these new functions are discussed in detail. For the basisfunctions of the hyperradius, several choices are possible. In the present case, we use Coulomb?Sturmian functions of half integer angular momentum. We show that, in the case of H−, theaccuracy of the energy and the width of the resonance states obtained through a singlediagonalization of the Hamiltonian, is comparable to the values given by state-of-the-art methodswhile using a much smaller basis set. In addition, we show that precise values of the electricdipoleoscillator strengths for S P transitions in helium are obtained thereby confirming theaccuracy of the bound state wave functions generated with the present method.