Dirac-Hartree-Fock Total Energy of Closed-Shell Atoms

Dirac-Hartree-Fock Total Energy of Closed-Shell Atoms 

In accordance with the equivalence restriction a shell is classified by one pair of quantum numbers n and k. A shell comprises 2 k atomic spinors and is therefore 2 x -fold degenerate. A closed shell is characterized by the fact that all of these degenerate spinors enter the single Slater determinant in Dirac-Hartree-Fock theory. 

In closed-shell atoms, the nij dependence of the structure factors vanishes so that much simpler equations are obtained. The expectation value of the total ground-state energy can be written in cases of atoms with closed shells for the electronic state A as (cf. Refs. [126] and [57, p. 152]), 

A simplified closed-shell expression for the Breit interaction can be derived as well [201]. For open-shell atoms, an average of configurations yields a similar total energy expression except for the intra-shell exchange energy [57, p. 164]. 

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