B-splines in variational atomic structure calculations

Many of the problems associated with the use of finite differences for the solution of variational Hartree-Fock or Dirac-Hartree-Fock equations are related to the orthogonality requirement and the need for node counting to control the solution of the two-point boundary value problem. By expanding radial functions in a B-spline basis, the differential equations are replaced by non-linear systems of equations. Hartree-Fock orbitals become solutions of generalized eigenvalue problems where orthogonality requirements can be dealt with through projection operators. When expressed as banded systems of equations, all orbitals may be improved simultaneously using singular value decomposition or the Newton-Raphson method for faster convergence. Computational procedures are described for non-relativistic multiconfiguration Hartree-Fock variational methods and extensions to the calculation of Rydberg series. The effective completeness of spline orbitals can be used to combine variational methods with many-body perturbation theory or first-order configuration interaction. Many options are available for improving atomic structure calculations. Although spline methods are discussed only in connection with non-relativistic theory, they apply equally to Dirac-Hartree-Fock theory.