Stationary direct perturbation theory of relativistic corrections

The stationary variant of direct perturbation theory of relativistic effects is presented. In this variant neither the unperturbed (nonrelativistic) equation nor the equations for the relativistic corrections are 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),chi(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., nonrelativistic) 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 needed. Either of the two functionals F-0 and F-4 can be decomposed into two 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 strict variation perturbation theory requires that the approximate <(phi)over tilde>(2) and <(chi)over tilde>(2) reproduce the behavior of the exact 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)over 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 nonrelativistic wave equation has been solved exactly. Otherwise there is an extra term. Finally the question as to which extent some of the singularities in the perturbation theory of relativistic effects might be artifacts due to the unphysical assumption of a point nucleus is discussed. It is shown, however, that these singularities are not removed if one uses realistic extended nuclei. For all atoms, the critical radius r(c) inside of which the nuclear attraction energy is larger than the rest energy of the electron is larger than the extension of the nucleus.