Aca - Departamento de Física - Universidad Técnica Federico Santa ...

V Encuentro Sud Americano de 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 de Colisiones Atómicas, Centro atómico Bariloche, San Carlos de Bariloche, Río Negro, Argentina.
2 Depto. de Física, Universidad Nacional del Sur, Bahía Blanca, Buenos Aires, Argentina
3 Instituto de Astronomía y Física del Espacio and Departamento de Física, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires C.C. 67, Suc. 28, (C1428EGA) Buenos Aires, Argentina.
4 Laboratoire de Physique Moléculaire et des Collisions,Université Paul Verlaine-Metz, 57078 Metz, France.
5 Consejo Nacional de Investigaciones Científicas y Técnicas (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 depending only on the distances
between them. The most general Schrödinger
equation we will consider 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 considered
as a parameter. Constructing the basis in this
way enables us to set boundary conditions of
the complete problem in each Sturmian depending
on coordinates r 1 and r 2 : Kato cusp
conditions and Coulomb exponentially decaying
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 bounded atomic and molecular systems
and models. 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 degree of accuracy in the en-
15 Valparaíso, Chile