Stationary Direct Perturbation Theory for Many-Electron Systems

Stationary Direct Perturbation Theory for Many-Electron Systems 

The energy expression is the usual (unrestricted) expression for a closed-shell state, 

This energy is to be expanded as a perturbation series in 1/c and the appropriate functionals varied subject to the unitary normalization conditions to arrive at stationarity conditions. The stationarity conditions naturally lead to the DHF equations, so we expect the variation of the perturbation functionals to involve an expansion of the DHF operators. 

To perform the variation of the energies at various orders, we need the expansion of the one- and two-electron modified Dirac operators in terms of the components, which are found in (17.84) and (15.43). From these we can define two-component direct and exchange operators. 

The superscripts give the order in perturbation theory of the spinor, the subscripts correspond to the superscripts on the operators and determine whether rif or ( ) is used for the density, and may take the values 0 and 2. The second and third of the exchange operators exchange the component types as well as the electron indices this is merely a convenient form of expression that reflects the fact that the spinors contributing to the density for any electron in the Dirac-Coulomb approximation must have the same component type. 

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