Complete treatment of a three-body break-up Coulomb problem with generalised Sturmian functions

J.M. RANDAZZO, D. MITNIK, L.U. ANCARANI, F.D. COLAVECCHIA, AND G. GASANEO

Hefei

Conferencia; International Symposium on (e,2e), Double Photo-Ionization and Related Topics; 2013

ICPEAC

In the last years, a new family of general Sturmians Functions was introduced [1]. These sets were shown to be optimal to deal both with bound and scattering states of few body Coulomb problems. The basis functions, generated by means of very accurate numerical procedures [2], are constructed in such a way that they (i) remove the inter particle kinetic energy and interaction potentials from the systems's Schrodinger equation, (ii) include the Kato cusp behaviour in the case of Coulomb potentials, and (iii) impose an externally xed boundary condition, which is then inherited by the expanded wave function. These properties can be employed to accelerate the convergence of the expansions of atomic and molecular states [1]. The functions can also been applied to evaluate scattering states which describes collision problems. In this case an outgoing (or incoming) ux behaviour is imposed to the whole set at a given inter particle distance (from that point up to innity the basis behaves like a pure Coulomb (or free) outgoing wave). The Galerkin method can be applied in order to solve the Schrodinger equation, where the radial projective integrals are evaluated on the whole interval [0,1) [3]. The Sturmian expansions has been successfully applied to the evaluation of atomic [1, 4] and molecular bound states [in preparation]. For collision problems, however, only results for s- wave model calculations have been performed. The principal reason of this being the attempt to maintain simplicity when dealing with the delicate topics such as the imposition of the correct condition which avoids numerical re ections, and to overcome the diculty of slow convergence of the cross sections associated to the long range character of the Coulomb interactions [3]. In this work we will present the rst Sturmian calculation of a fully correlated collision problem with Coulomb interactions: the double ionization of helium by single photon impact. Since only a single total angular momentum value is involved in the calculations, the formulation remains quite simple and then very adequate for our purposes. The calculation presented here is performed with a very simple set of Sturmian Functions which allow analytical expression for most of the matrix elements associated to the spectral representation. Both, the initial two electrons bound state, and the scattering wave function which describes the electron continuum, are evaluated with an adequate Sturmian expansion for each case. The large linear system is solved by means of the Conjugate Gradient Squared Method, applied trough a block-step schema which allow its implementation in a single desktop computer. Double photoionization of helium has been measured under dierent kinematical conditions, and several theoretical methods have reproduced the main features. Without surprises, our cross sections compare favorably with other theories, validating the eciency of the Sturmian approach.