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Abstract

This paper describes a relativistic generalization of the nonrelativistic Z-dependent theory of many-electron atoms recently given by the senior author. The present theory affords a unified description of relativistic and nonrelativistic effects in the structure and spectra of many electron atoms that is valid over the entire range of the coupling parameter χ ≡ α2Z3 and allows fully for all electronic contributions to the energy (expressed in units of mc2) of order ϵk and ϵkZ−1, k ≧ 1, ϵ ≡ (αZ)2. As in the nonrelativistic theory, the grouping of states into complexes, characterized by a definite set of principal quantum numbers and a definite parity, plays a central role. Relativistic and nonrelativistic contributions to the energy that involve interaction elements connecting different complexes are of order Z−2 as compared, respectively, with the leading relativistic and nonrelativistic contributions to the energy. On the other hand, all interactions connecting states of the same complex contribute quantities of order ϵk or ϵkZ−1, k ≧ 1, to the energy.Three expansion parameters—Z−1, ϵ, and χ—figure in the relativistic theory. In the nonrelativistic approximation, ϵ and χ drop out, and the energy can be expanded in powers of Z−1. In general, the energy can be expressed as a double power series in Z−1 and ϵ whose coefficients normally depend upon χ; for χ ⪡ 1, they can be expanded in powers of χ, for χ ⪢ 1 in powers of χ−1.In the past, two distinct forms of the relativistic interaction between electrons (both due to Breit) have been used, one valid for ϵ ⪡ 1, the other for χ ⪢ 1. In this paper we derive a formula for the interaction that is valid over the entire range of Z, and show that it differs from Breit's formula for ϵ ⪡ 1 by quantities that are smaller than the Lamb shift.In many practical applications it is a sufficiently good approximation to retain only the portion of the relativistic two-electron interaction that is of order ϵ2Z−1. The calculations required for a consistent application of this approximation differ in several respects from conventional calculations based on the Pauli approximation: They employ relativistic hydrogenic (Dirae) wave functions (instead of nonrelativistic variational wave functions); they include certain effects of configuration mixing within a given complex; and they can normally be performed more readily in a representation based on Dirac spinors, with the Breit interaction in its “unreduced” form, than in one based on Pauli spinors, with the Breit interaction in its “reduced” form.We evaluate the relativistic level shift for states belonging to the configurations 1s2, 1s2s, 1s3s, 1s22s, and 1s23s, and find excellent agreement between theory and experiment.