CalculableR-matrix method for the Dirac equation

Abstract

An efficient version of the calculable R-matrix method, a technique of determination of scattering and bound-state properties, is extended to the Dirac equation. The configuration space is divided into internal and external regions at the channel radius. In both regions, the introduction of a Bloch operator allows restoring the Hermiticity. The most general Bloch operator contains three free parameters. With a basis without constraint at the channel radius in the internal region, the phase shifts converge to the same value for any choice of these parameters. Nevertheless, some choices provide a faster convergence than others. The determination of the bound-state energies is performed with an extension of the method using a second set of basis functions in the external region. Neither the knowledge of asymptotic expressions nor a large channel radius are required. These R-matrix methods are particularly simple and very accurate when combined with the Lagrange-mesh method. No analytical or numerical evaluation of matrix elements is then necessary. Very accurate phase shifts are obtained with a Legendre mesh for various short-range potentials. A combination of Legendre and Laguerre meshes provides accurate energies for the bound states even for potentials with a Coulomb-like asymptotic behavior.

Confinement of hydrogen atom with Dirac equation

“…The method developed in ref. for bound states is devised in such a way that the analytical form of the solution for r > R does not need to be known. In that reference, while the internal wave function is expanded over some basis as usual, an expansion in a set of appropriate basis functions is also employed in the external region.…”

“…The combination of the Gauss quadrature (21) and the Lagrange property (25) leads to the main advantage of the Lagrange-mesh method, that is, the simple approximation of matrix elements…”

“…These expressions are formally identical to equations (60) and (61) of ref. [21] but with zeros of arbitrary shifted Jacobi polynomials in place of zeros of shifted Legendre polynomials.…”

“…Remarkably, this method gives excellent results for the Dirac equation as well for bound-state properties, that is, energies and wave functions, [18] polarizabilities, [19] and one-photon and two-photon transition probabilities, [20] as for phase shifts. [21] It is thus expected to be also efficient for the confinement problem.…”

“…Þdx is computed at the Gauss-quadrature approximation(21), V(x) is some infinitely differentiable function of x, and F i (x) and G i (x) represent any Lagrange function amongf i (x),f i x ð Þ, orf i (x). Matrix elements of a function V(x),such as a local potential, are diagonal at this approximation and the diagonal elements are given by values of this function at mesh points.…”