Numerical electronic structure calculations for atoms. II. Generalized variable transformation and relativistic calculations

The pairs of radial functions P-i and Q(i), which are part of the four-comp onent single-particle spinors in the relativistic description of the electr onic structure of bound states of atoms, are usually determined as solution s of eigenvalue problems. The latter constitute two-point boundary value pr oblems which involve coupled pairs of first-order ordinary radial different ial equations. To introduce a suitable notation, the theory involved in rel ativistic electronic structure calculations for atoms is briefly reviewed, including a general treatment of arbitrary transformations for the radial v ariable. Such variable transformations must be specified then, either expli citly (by a transformation function) or implicitly (by the solution method itself), for any actual calculation, though initially the variable transfor mation is almost completely independent of the numerical method to be appli ed. We consider suitable transformation functions out of which an actual ch oice can be made within wide limits. In our approach the resulting transfor med radial differential equations are discretized then with finite-differen ce methods. Since the accuracy and efficiency of these methods is increased when a constant step size h between contiguous points tin the transformed variable) is used, it is only here where the careful choice of the variable transformation is important. The resulting system of linear equations is s olved for the radial functions with standard linear algebra methods rather than "shooting" methods. The present work extends the general study of vari able transformations, given recently for nonrelativistic electronic structu re calculations in Part I, to the relativistic case, important results foll owing from the present work are (i) a discretization scheme for first-order ordinary differential equations similar to the well-known standard Numerov scheme used within the nonrelativistic framework, (ii) a consistent numeri cal algorithm with an overall truncation error of order h(4), (iii) a metho d for handling effective potentials behaving singularly like r(-1) at the o rigin las encountered, e.s., when the atomic nucleus is represented by a po intlike charge density distribution), and, in connection with this last poi nt, (iv) the avoidance of nonanalytic short-range behavior of the solutions to be obtained from the differential equations. (C) 2000 John Wiley & Sons , Inc.