V Encuentro Sud Americano <strong>de</strong> Colisiones Inelásticas en la Materia 14 Valparaíso, Chile
V Encuentro Sud Americano <strong>de</strong> Colisiones Inelásticas en la Materia Ab-Initio Sturmian method for three-body quantum mechanical problems: Atomic and molecular bound states J. M. Randazzo 1 , 5 , A. L. Frapiccini 1 , 5 , G. Gasaneo 2 , 5 , F. D. Colavecchia 1 , 5 , D. M. Mitnik 3 , 5 and L. U. Ancarani 4 1 División <strong>de</strong> Colisiones Atómicas, Centro atómico Bariloche, San Carlos <strong>de</strong> Bariloche, Río Negro, Argentina. 2 Depto. <strong>de</strong> <strong>Física</strong>, <strong>Universidad</strong> Nacional <strong>de</strong>l Sur, Bahía Blanca, Buenos Aires, Argentina 3 Instituto <strong>de</strong> Astronomía y <strong>Física</strong> <strong>de</strong>l Espacio and <strong>Departamento</strong> <strong>de</strong> <strong>Física</strong>, Facultad <strong>de</strong> Ciencias Exactas y Naturales, <strong>Universidad</strong> <strong>de</strong> Buenos Aires C.C. 67, Suc. 28, (C1428EGA) Buenos Aires, Argentina. 4 Laboratoire <strong>de</strong> Physique Moléculaire et <strong>de</strong>s Collisions,Université Paul Verlaine-Metz, 57078 Metz, France. 5 Consejo Nacional <strong>de</strong> Investigaciones Científicas y <strong>Técnica</strong>s (CONICET). email address corresponding author: randazzo@cab.cnea.gov.ar In this work we review a recently introduced methodology to solve the Schrödinger equation of three particles. We assume that the particles interact through potentials <strong>de</strong>pending only on the distances between them. The most general Schrödinger equation we will consi<strong>de</strong>r reads: together with the boundary conditions: and Where U 1 , U 2 and U 12 can be any well behaved atomic potentials. We also assume that U 12 admits a simple partial wave expansion, such as Coulomb, Yukawa, armonic potentials, etc. Because of the symmetries of the Eq. (1), the wave function can be evaluated separately for each L, M, S and Π (total angular momentum, its projection along the z axis, the spin symmetry and parity, respectively). We then propose a partial wave expansion in terms of the bi-spherical harmonics, and obtain a coupled set of two-dimensional equations in the radial coordinates r 1 and r 2 . The set of coupled equations is solved by means of a Sturmian expansion (one Sturmian set for each coordinate)[1]. The Generalized Sturmian functions are solutions of the Sturm-Liouville equation: where V is a short range generating potential, β is the eigenvalue and E is consi<strong>de</strong>red as a parameter. Constructing the basis in this way enables us to set boundary conditions of the complete problem in each Sturmian <strong>de</strong>pending on coordinates r 1 and r 2 : Kato cusp conditions and Coulomb exponentially <strong>de</strong>caying behaviour for negative energies, or Coulomb outgoing wave conditions for positive ones[2]. In this work we will show some results of the application of the Sturmian expansion to the solution of equation (1) for a variety of three body boun<strong>de</strong>d atomic and molecular systems and mo<strong>de</strong>ls. We choose here using negative energy Sturmian functions, and compute ground as well as the different manifolds of excited states. We also analyze different choices for the generating potential to achieve a high <strong>de</strong>gree of accuracy in the en- 15 Valparaíso, Chile