Matrix Dirac-Hartree-Fock Equations in a 2-Spinor Basis

Matrix Dirac-Hartree-Fock Equations in a 2-Spinor Basis 

The Dirac-Hartree-Fock operator for a single-determinant wave function in terms of atomic spinors has been derived previously (section 8.1), and appears as 

While the expansion of hpq follows readily from the development for the Dirac operator above, the electron-electron interaction integrals must be considered separately. We also want to develop a Kramers-restricted Dirac-Hartree-Fock (KR-DHF) theory, but first we develop expressions for the general, Kramers-unrestricted case. In the developments below we follow the practice of giving only the basis function index in the integrals. 

For the Coulomb contribution to the Fock operator we have to consider the interaction between charge distributions involving two 4-spinors, for example, rlfplrlr. If this distribution is decomposed into 2-spinor form, we get 

On substituting this expression into the two-electron Coulomb integral, we get 

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