STATIONARY DIRECT PERTURBATION-THEORY OF RELATIVISTIC CORRECTIONS

The stationary variant of direct perturbation theory of relativistic e ffects is presented. In this variant neither the unperturbed (nonrelat ivistic) equation nor the equations for the relativistic corrections a re solved exactly, but each of them is replaced by the condition that a certain functional becomes stationary. Let psi(0)=(phi(0),chi(0)) be the four-component spinor with modified metric in the nonrelativistic limit and <(psi)over bar>(2)=(phi(2),chi(2)) the leading relativistic correction of O(c(-2)), then one can define functionals F-0(phi(2),ch i(2)) and F-4(phi(2),chi(2)) called respectively the Levy-Leblond and the Rutkowski-Hylleraas functional, such that stationarity of F-0 with respect to variation of phi(0) and chi(0) determines phi(0) and chi(0 ), and stationarity of F-4 with respect to variation of phi(2) and chi (2) determines phi(2) and chi(2). The unperturbed (i.e., nonrelativist ic) energy E(0) as well as the leading relativistic correction c(-2)E( 2) are expressible through phi(0) and chi(0) while for the next higher corrections c(-4)E(4) and c(-6)E(6), phi(2) and chi(2) are also neede d. Either of the two functionals F-0 and F-4 can be decomposed into tw o contributions, the error of one of which is greater than or equal to 0 while that of the other is less than or equal to 0. An upper-bound property is obtained if the error of the second part vanishes. A stric t variation perturbation theory requires that the approximate <(phi)ov er tilde>(2) and <(chi)over tilde>(2) reproduce the behavior of the ex act phi(2) and chi(2) near a nucleus, which implies terms in ln r. If one regularizes <(phi)over tilde>(2) one must also regularize <(chi)ov er tilde>(2); otherwise E(6) diverges. If one regularizes both phi(2) and chi(2) in the sense of a kinetic balance, one gets regular results for E(4) and E(6), but one loses the strict upper-bound property. The Breit-Pauli expression for E(2) is shown to be correct only if the no nrelativistic wave equation has been solved exactly. Otherwise there i s an extra term. Finally the question as to which extent some of the s ingularities in the perturbation theory of relativistic effects might be artifacts due to the unphysical assumption of a point nucleus is di scussed. It is shown, however, that these singularities are not remove d if one uses realistic extended nuclei. For all atoms, the critical r adius r(c) inside of which the nuclear attraction energy is larger tha n the rest energy of the electron is larger than the extension of the nucleus.