B-spline methods for radial Dirac equations

Although B-spline techniques have been used to solve two-point boundary value problems with Dirac Hamiltonians for more than 20 years, the treatment of boundary conditions is still a matter of controversy. Spurious, non-physical, solutions are endemic when boundary conditions are not handled correctly. These pathological problems are absent when traditional finite difference methods are used as in computer packages such as GRASP. Accurate approximation using both finite differences and B-splines depends on controlling local approximation errors, and this common property suggests no a priori reason to suppose that B-spline algorithms should be more prone to generate spurious solutions. The relativistic Bloch operators of [24], when added to the Dirac differential operator, permit the construction of a self-adjoint differential operator for the two-point boundary value problem on a finite interval. Approximate solution of this problem exploiting the properties of B-splines in variational or Galerkin schemes is then straightforward. Although the analysis is presented primarily in terms of the Dirac R-matrix method for electron–atom scattering, it is also relevant to other applications such as relativistic atomic and molecular structure calculations, many-body perturbation calculations of structure and properties and relativistic dynamics.