A B-spline Galerkin method for the Dirac equation

The B-spline methods Johnson and Sapirstein (1, 2) introduced into relativistic many-body perturbation the- ory have produced results of unprecedented accuracy (3). Essentially, the local non-orthogonal B-spline basis was transformed to an orthogonal orbital basis by the ap- plication of the Galerkin method to the Dirac equation over a finite interval (4). The resulting basis was fi- nite and effectively complete. Though the low-energy bound states were good approximations to solutions of the Dirac equation, no physical interpretation was impor- tant for continuum states. Rapidly oscillating solutions were observed but played a negligible role in the sum- mation over states in their applications (2). However, these spurious states perturbed the spectrum and slowed the convergence of quantum electrodynamic (QED) cal- culations. This led Shabaev et al. (5) to propose a dual kinetic balance basis similar to the basis Quiney et al. (6) employed with analytic Slater type functions. Bound- ary conditions were for the case of a finite nuclear-charge distribution, with the point nucleus considered as a lim- iting case. Different boundary conditions at the origin were proposed for positive and negative values ofand both large and small components were set to zero at the large r boundary. Recently Igarashi (7) investigated a variety of methods and boundary conditions. He pointed out that the four boundary conditions used by Froese Fischer and Parpia (8) were excessive and explored the use of B-splines of different order, kp and kq, as a way of avoiding spurious solutions. In a subsequent paper he concluded that kinetic balance also provided a good basis (9). No best method was identified. All his methods em- ployed analytic weighting factors to B-spline expansions in order to control the asymptotic properties of large and small components. R-matrix methods (see Ref.'s (10, 11) for recent re- views) differ from the applications considered by the above authors in that zero boundary conditions at large r, such as proposed by Shabaev et al. (5), cannot be used. R-matrix theory assumes an inner region r a where exchange with an outer electron can be neglected. What is needed is a basis for the inner region that satisfies certain conditions at the r = a boundary. B-splines were very successfully employed in the non-relativistic R-matrix calculations (12), however, they cannot be used in the Dirac-based calculations when spurious states in the continuum spectrum are present. At the same time, the kinetically balanced bases lead to extensive compu- tational difficultes in many-electron calculations. Spline methods are based on approximation theory. The grid that is selected along with boundary condi- tions determine a piecewise polynomial space with a finite basis. The unique B-spline basis has many advantages (12, 13), but there are many possible bases. The trans- formation from a non-orthogonal basis to an orthogonal orbital basis depends on how the Galerkin method is ap- plied. In this letter we propose a simple method for the Dirac matrix equation and apply it to the calculation of the R-matrix boundary condition. All calculations are for a point nucleus so that results can be compared with exact solutions. Special attention is given to the bound- ary conditions. We also show the relationship between kinetic balance and the use of splines of different order. At large values of r, the non-relativistic Schrodinger equation has the same form as