Discrete Basis for Relativistic Calculations of Many-Electron Atoms

Aleksey I. Sherstyuk

S.I.Vavilov State Optical Institute, Tuchkov per. 1, St.Petersburg, 199034 Russia

In the nonrelativistic perurbation theory calculations of many-electron atoms, a considerable progress has been attained with the use of expansions on the pure discrete basis of eigenfunctions of the generalized eigenvalue problem based on the Hartree - Fock equations [1]. Such a basis in the case of hydrogen-like atoms is reduced to the usual Sturmian basis [2, 3]. In this paper the relativistic generalization of above- mentioned approach is proposed for non-Coulomb potential term V(r) in the eigenvalue problem:

(D-E)Y=[ap+bm+V(r)-E]Y = lQY, (1)

where a, b are the Dirac matrices, h=e=c=1. The problem is to obtain such a form of Q that the spectrum of l to be real and a pure discrete one, and the set of eigenfunctions {Yn} to be a full system in the space of square-integrable four-component vectors. It is shown that for 0<E<m such a system can be constructed as a straight sum of the subspaces determined by the equations

(D-E)Yn(i) = ln(i) Qi Yn(i) (2)

where i=1, 2,

One can see that the eigenvalues of b are connected with the eigenvalues of k according to the inverse Zhukovsky transformation. So, for thr full Green's function of D we have:

It has been shown by the numerical calculations with the use of (3) that the convergence if corresponding expansions for high-order atomic parameters is extremely rapid, as well as in nonrelativistic case.

[1] P. F. Grusdev, G. S. Solovyova, A. I. Sherstyuk, Opt. and Spectr. 64 (1988) 976.

[2] V. A. Fock, Elements of quantum mechanics, Leningrad, 1932.

[3] A. I. Sherstyuk, Soviet Physics JETP 35 (1972) 655.

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Designed by A. Sergeev.