**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=[a*p*+b*m*+*V*(*r*)-*E*]Y
= l*Q*Y, (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.

Back to A. I. Sherstyuk.

Designed by A. Sergeev.