A L Frapiccini

Chapter 7 – Three-Body Coulomb Problems with Generalized Sturmian Functions

Abstract The study of structure and collision processes of three- and four-body problems has seen an extraordinary progress in the last decades. This progress has been in part associated to the incredible fast growth of the computer capabilities. However, the tools used to solve structure problems are different from those corresponding to the treatment of collision processes. In this review, we provide the theoretical framework and a selection of results for both structure as well as collision problems using only one technique that we have developed in the last few years, based on the use of Generalized Sturmian functions. We present results obtained in structure studies of isolated and confined two-electron atoms, and exotic and molecular systems. The same technique is applied to the study of various benchmark problems for the single ionization of hydrogen and the double ionization of helium by electron impact. In this way, we demonstrate that the Generalized Sturmian method can be successfully applied to the treatment of both types of problems.

A boundary adapted spectral approach for breakup problems

We present a spectral method to study three-body fragmentation processes. The basis set explicitly includes continuum asymptotic boundary conditions, and it is built upon generalized Sturmian functions. These functions are eigenvectors of a two-body problem where the magnitude of a potential is assumed as the eigenvalue. Comparison with a simple solvable analytical model demonstrates that our approach rapidly converges to the exact results, with basis sizes much smaller than other previous calculations. Preliminary calculations of H ionization by electron impact in the L = 0 approximation suggest that these convergence properties also apply to long-range Coulomb problems.

Theory of Hyperspherical Sturmians for Three-Body Reactions †

In this paper we present a theory to describe three-body reactions. Fragmentation processes are studied by means of the Schrodinger equation in hyperspherical coordinates. The three-body wave function is written as a sum of two terms. The first one defines the initial channel of the collision while the second one describes the scattered wave, which contains all the information about the collision process. The dynamics is ruled by an nonhomogeneous equation with a driven term related to the initial channel and to the three-body interactions. A basis set of functions with outgoing behavior at large values of hyperradius is introduced as products of angular and radial hyperspherical Sturmian functions. The scattered wave is expanded on this basis and the nonhomogeneous equation is transformed into an algebraic problem that can be solved by standard matrix methods. To be able to deal with general systems, discretization schemes are proposed to solve the angular and radial Sturmian equations. This procedure allows these discrete functions to be connected with the hyperquatization algorithm. Finally, the fragmentation transition amplitude is derived from the asymptotic limit of the scattered wave function.

On the numerical treatment of Coulomb forces in scattering problems

We investigate the limiting procedures to obtain Coulomb interactions from short-range potentials. The application of standard techniques used for the two-body case (exponential and sharp cutoff) to the three-body break-up problem is illustrated numerically by considering the Temkin-Poet (TP) model of e-H processes.

Explicitly correlated Sturmian functions for structure and collision three-body problems

We introduce explicitly correlated Sturmian functions with appropriate boundary conditions to deal with both bound and scattering states of three-body systems.

Hyperspherical versus spherical treatment of asymptotic conditions for three-body scattering problems

We consider two different Sturmian Function (SF) expansions to describe three-body break-up wave functions having outgoing flux behavior: one in hyper-spherical coordinates (ρ and α) and one in spherical coordinates (r1 and r2). We compare the convergence rates of the two treatment, and relate the analysis to geometry aspects of the problem.

Use of uncorrelated Sturmian basis functions for three-body systems of finite masses

The Configuration Interaction method with uncorrelated Sturmian basis functions is applied to describe a variety of atomic and molecular three-body systems.

B-splines approach of the Sturmian basis: Atomic and scattering calculations.

Synopsis We test a new methodology to obtain a Sturmian basis set using B-spline expansion of arbitrary order and imposing the correct boundary conditions for both negative and positive energy domains. The Sturmians can be applied to solve both atomic systems, which are transformed into an eigenvalue problem, and scattering processes, which leads to a complex system of linear equations. We show how our methodology converges rapidly and give accurate results compared with other techniques

Single and double ionization of He by electron-impact

Our aim is to study the ionization-excitation and (e,3e) processes of the He atom. The high energy first Born-approximation transition amplitudes are obtained from the asymptotic limit of the exact three-body wave function. This wave function is obtained by solving a driven Schr?dinger equation which contains the double bound He wave function. The resulting equation is solved using Generalized Sturmian functions.

An analytically solvable model to test the hyperspherical Sturmian approach for break up processes

An analytically solvable model for three particles break up processes is presented. The scattering process is represented by a non-homogeneous Schrodinger equation where the driven term is given by a Yukawa-like interaction multiplied by the product of a continuum wave function and a bound state in the particles coordinates. A closed form solution is derived in hyperspherical coordinates (ρ,α). This leads to an analytic expression for the transition amplitude associated to the scattering process. This model is used to test calculations performed within the frame of an hyperspherical Sturmian approach.