F. D. Colavecchia

Numerical evaluation of Appell's F1 hypergeometric function

In this work we present a numerical scheme to compute the two-variable hypergeometric function F1(α,β,β′,γ;x,y) of Appell for complex parameters α,β,β′ and γ, and real values of the variables x and y. We implement a set of analytic continuations that allow us to obtain the F1 function outside the region of convergence of the series definition. These continuations can be written in terms of the Horns G2 function, Appells F2 function related, and the F1 hypergeometric itself. The computation of the function inside the region of convergence is achieved by two complementary methods. The first one involves a single-index series expansion of the F1 function, while the second one makes use of a numerical integration of a third order ordinary differential equation that represents the system of partial differential equations of the F1 function. We briefly sketch the program and show some examples of the numerical computation.

Computational methods for Generalized Sturmians basis

Article history: The computational techniques needed to generate a two-body Generalized Sturmian basis are described. These basis are obtained as a solution of the Schrodinger equation, with two-point boundary conditions. This equation includes two central potentials: A general auxiliary potential and a short-range generating potential. The auxiliary potential is, in general, long-range and it determines the asymptotic behavior of all the basis elements. The short-range generating potential rules the dynamics of the inner region. The energy is considered a fixed parameter, while the eigenvalues are the generalized charges. Although the finite differences scheme leads to a generalized eigenvalue matrix system, it cannot be solved by standard computational linear algebra packages. Therefore, we developed computational routines to calculate the basis with high accuracy and low computational time. The precise charge eigenvalues with more than 12 significant figures along with the corresponding wave functions can be computed on a single processor within seconds.

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.

Hypergeometric integrals arising in atomic collisions physics

We introduce a method of obtaining volume integrals involving confluent hypergeometric functions. This method is based on the integral representation of these functions and enabled us to write a generalized Nordsieck integral in terms of hypergeometric functions of many variables. We explore some particular results that could be useful when calculating transition matrices in collision theories.

Double ionization of helium by fast electrons with the Generalized Sturmian Functions method

The double ionization of helium by high energy electron impact is studied. The corresponding four-body Schrodinger equation is transformed into a set of driven equations containing successive orders in the projectile–target interaction. The first order driven equation is solved with a generalized Sturmian functions approach. The transition amplitude, extracted from the asymptotic limit of the first order solution, is equivalent to the familiar first Born approximation. Fivefold differential cross sections are calculated for (e, 3e) processes within the high incident energy and small momentum transfer regimes. The results are compared with other numerical methods, and with the only absolute experimental data available. Our cross sections agree in shape and magnitude with those of the convergent close coupling method for the (10+10) eV and (4+4) eV emission energies. To date this had not been achieved by any two different numerical schemes when solving the three–body continuum problem for the fast projectile (e, 3e) process. Though agreement with the experimental data, in particular with respect to the magnitude, is not achieved, our findings partly clarify a long standing puzzle.

Electron-ion correlation effects in ion-atom single ionization

We study the effect of electron-ion correlation in single ionization processes of atoms by ion impact. We present a distorted wave model where the final state is represented by a correlated function solution of a non-separable three-body continuum Hamiltonian, that includes electron-ion correlation as coupling terms of the wave equation. A comparison of the electronic differential cross sections computed with this model with other theories and experimental data reveals that the influence of the electron-ion correlation is more significant for low energy emitted electrons.

Multivariable hypergeometric solutions for three charged particles

We present a new wavefunction which describes the ion - atom problem above the ionization threshold. This is an approximate solution of the Schrodinger equation for the three-body Coulomb problem that can be expressed in terms of a confluent hypergeometric function of two variables. The proposed wavefunction includes correlation among the motions of the three particles and verifies the correct Coulombic asymptotic behaviours.

Solving three-body-breakup problems with outgoing-flux asymptotic conditions

An analytically solvable three-body collision system (s wave) model is used to test two different theoretical methods. The first one is a configuration interaction expansion of the scattering wave function using a basis set of Generalized Sturmian Functions (GSF) with purely outgoing flux (CISF), introduced recently in A. L. Frapicinni, J. M. Randazzo, G. Gasaneo, and F. D. Colavecchia [J. Phys. B: At. Mol. Opt. Phys. 43, 101001 (2010)]. The second one is a finite element method (FEM) calculation performed with a commercial code. Both methods are employed to analyze different ways of modeling the asymptotic behavior of the wave function in finite computational domains. The asymptotes can be simulated very accurately by choosing hyperspherical or rectangular contours with the FEM software. In contrast, the CISF method can be defined both in an infinite domain or within a confined region in space. We found that the hyperspherical (rectangular) FEM calculation and the infinite domain (confined) CISF evaluation are equivalent. Finally, we apply these models to the Temkin-Poet approach of hydrogen ionization.

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.