A B-spline Galerkin method for the Dirac equation

The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y″=−λ2yy″=−λ2y, that can also be written as a pair of first-order equations y′=λzy′=λz, z′=−λyz′=−λy. Expanding both y(r)y(r) and z(r)z(r) in the BkBk basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r)y(r) is expanded in the BkBk basis and z(r)z(r) in the dBk/drdBk/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method (Bk,Bk±1Bk,Bk±1) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas–Reiche–Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.