Approximate solutions to the one-particle Dirac equation: numerical results R. A. MOORE AND T. C. SCOTT Guelph-Waterloo Program for Graduate Work in Physics, Waterloo Campus, University of Waterloo, Waterloo, Ont., Canada N2L 3GI Received August 27, 1985 The zero-, first-, and second-order differential equations in a previously defined hierarchy of equations giving approximate solutions to the one-particle Dirac equation and the corresponding eigenvalue contributions are each written as power series in a, the fine structure constant, for an arbitrary, spherically symmetric potential. These equations are solved numerically for the hydrogen-atom potential to obtain wave functions to order a* and eigenvalues to order a4 for all states with n = 1-4, inclusive. The numerical solutions are then used to evaluate a number of matrix elements to order a'. A comparison with the exact expressions shows that the numerical values for the coefficients of the different powers of a have at least six significant figures in the eigenfunctions and eigenvalues and five in the matrix elements. Thus, the procedure is validated and can be applied with confidence to other atomic systems. Les Cquations diffkrentielles d'order zero, du premier ordre et du deuxieme ordre, dans une hiCrarchie, dCfinie anterieurement, d16quations donnant des solutions approximatives de l'tquation de Dirac pour une particule, sont Ccrites sous forme de stries de puissances de a, la constante de structure fine, pour un potentiel a symCtrie sphCrique arbitraire. On rCsout numkriquement ces Cquations, pour le potentiel de l'atome d'hydrogene, afin d'obtenir les fonctions d'onde a l'ordre a ' et les valeurs propres a l'ordre a4, pour tous les Ctats avec n = 1 a 4 inclusivement. Les solutions numkriques sont ensuite utilisCes pour calculer un certain nombre d'C1Cments de matrice a I'ordre a'. Une comparaison avec les expressions exactes montre que les valeurs numkriques pour les coefficients des differentes puissances de a sont correctes avec au moins six chiffres significatifs dans le cas des fonctions propres et des valeurs propres, et cinq chiffres significatifs dans le cas des ClCments de matrice. La procCdure est donc valide, et on peut l'appliquer avec confiance a d'autres systemes atomiques. [Traduit par le journal] Can. J. Phys. 64, 297 (1986) 1. Introduction A particular procedure was suggested previously (see refs. 1, 2, and 3; referred to as I, 11, and I11 hereafter, respectively), to obtain approximate solutions to the one-particle Dirac equation. This procedure was examined further by Moore and Lee (4, 5) (referred to as IV and V hereafter, respectively). One takes advantage of the fact that the wave function consists of a large spinor part and a small spinor part to decouple the two parts and set up a hierarchy of perturbative-like equations to which, essentially, nonrelativistic calculational methods can be ap- plied. Tests of the procedure were made in I, 11, 111, IV, and V by obtaining a number of analytic approximate solutions for the hydrogen atom and by comparing them with the exact results. To do so, since everything was done analytically, each quantity was expanded, at an appropriate point, in terms of a power series in a, a being the usual fine-structure constant, and a term-by-term comparison was made. Exact correspondence was used to validate the procedure. Of course, in any atomic system other than hydrogen, numerical techniques are required. Tomishima (6) pointed out that if the zero-order equation in the above-mentioned hierarchy is solved numerically to obtain the complete zero-order wave function and total zero-order energy, and if the first-order equation is now used to obtain the total first-order energy, then the sum of these values of the energy contributions could give quite erroneous results. A similar conclusion was found if only the zero-order wave functions were used to evaluate a variety of matrix elements. As indicated in IV and V, the reason for the above situation is that at each order, the wave functions and energies are functions of a. Thus, for example, the total zero-order energy contains contributions to all orders of a'", n = 0, 1, 2, . . . and the total first-order energy contributions for n = 1,2, . . . . although the sum gives the a0 and a2 contributions correctly, it does not, of course, give the a4 contribution correctly, which may also have the wrong sign. Thus, in the cases where the a4 contribution is significant, inaccurate numerical values result from a direct calculation. In general, truncating the hierarchy of equations at any point in such a direct calculation causes the next order contribution in a power series in a to be incompletely, or even improperly, included, hence leading to possible inaccurate numerical results. Papers IV and V have addressed this problem for the hydrogen atom and have shown that each equation in the hierarchy can be solved as a power series in a and that a term-by-term separation is exactly effected, analytically. One has left to demonstrate that the process can be handled to a high degree of accuracy by numerical means. In the present work, the equations to be solved and the expressions to be evaluated are rewritten as power series in a to apply to an arbitrary, spherically symmetric atomic potential. This means that one evaluates the coefficients independently and thus can expect reliable results for each contribution. The generalized differeniial equations for the wave functions and the expressions for the energies are detailed in Sect. 2, and the numerical procedures are specified. Numerical solutions are obtained for the hydrogen atom for all states with n = 1-4, inclusive. The numerical results are stepped through and compared with the analytic results at each step. In this case, generally, at least six significant-figure accuracy is obtained in the coefficients. In Sect. 3, the numerical solutions found in Sect. 2 are used to evaluate a number of matrix elements, namely, expectation values of r", matrix elements for the Zeeman splitting, matrix elements for the Stark effect (and hence in principle, oscillator strengths), and matrix elements for the hyperfine interaction. In all cases, the coefficients are reproduced numerically to at least five significant figures. Conclusions are made in Sect. 4. It appears that this procedure can be applied with confidence to more complex atomic systems. 2. Approximate solution; eigenvalues and eigenfunctions 2 .l. General structure and notation The notation, equations, and expressions to be evaluated are taken directly from I-V with the exception that atomic units are Can. J. Phys. Downloaded from www.nrcresearchpress.com by 23.91.232.172 on 07/25/15 For personal use only.