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 ${\mathrm{\psi }}_0$=(${\mathrm{\Phi }}_0$,${\mathrm{\chi }}_0$) be the four-component spinor with modified metric in the nonrelativistic limit and ψ${\mathrm{¯}}_2$=(${\mathrm{\Phi }}_2$,${\mathrm{\chi }}_2$) the leading relativistic correction of O(${\mathit{c}}^{\mathrm{-}2}$), then one can define functionals ${\mathit{F}}_0$(${\mathrm{\Phi }}_0$,${\mathrm{\chi }}_0$) and ${\mathit{F}}_4$(${\mathrm{\Phi }}_2$,${\mathrm{\chi }}_2$) called respectively the Lévy-Leblond and the Rutkowski-Hylleraas functional, such that stationarity of ${\mathit{F}}_0$ with respect to variation of ${\mathrm{\Phi }}_0$ and ${\mathrm{\chi }}_0$ determines ${\mathrm{\Phi }}_0$ and ${\mathrm{\chi }}_0$, and stationarity of ${\mathit{F}}_4$ with respect to variation of ${\mathrm{\Phi }}_2$ and ${\mathrm{\chi }}_2$ determines ${\mathrm{\Phi }}_2$ and ${\mathrm{\chi }}_2$. The unperturbed (i.e., nonrelativistic) energy ${\mathit{E}}_0$ as well as the leading relativistic correction ${\mathit{c}}^{\mathrm{-}2}$ ${\mathit{E}}_2$ are expressible through ${\mathrm{\Phi }}_0$ and ${\mathrm{\chi }}_0$ while for the next higher corrections ${\mathit{c}}^{\mathrm{-}4}$ ${\mathit{E}}_4$ and ${\mathit{c}}^{\mathrm{-}6}$ ${\mathit{E}}_6$, ${\mathrm{\Phi }}_2$ and ${\mathrm{\chi }}_2$ are also needed. Either of the two functionals ${\mathit{F}}_0$ and ${\mathit{F}}_4$ can be decomposed into two contributions, the error of one of which is ≥0 while that of the other is ≤0. An upper-bound property is obtained if the error of the second part vanishes. A strict variation perturbation theory requires that the approximate Φ${\mathrm{̃}}_2$ and χ${\mathrm{̃}}_2$ reproduce the behavior of the exact ${\mathrm{\Phi }}_2$ and ${\mathrm{\chi }}_2$ near a nucleus, which implies terms in ln r. If one regularizes Φ${\mathrm{̃}}_2$ one must also regularize χ${\mathrm{̃}}_2$; otherwise ${\mathit{E}}_6$ diverges. If one regularizes both ${\mathrm{\Phi }}_2$ and ${\mathrm{\chi }}_2$ in the sense of a kinetic balance, one gets regular results for ${\mathit{E}}_4$ and ${\mathit{E}}_6$, but one loses the strict upper-bound property. The Breit-Pauli expression for ${\mathit{E}}_2$ 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 ${\mathit{r}}_{\mathit{c}}$ inside of which the nuclear attraction energy is larger than the rest energy of the electron is larger than the extension of the nucleus. © 1996 The American Physical Society.