Inclusion of the Two-Electron Terms

For many-electron systems we wish to incorporate the two-electron terms of the Hamiltonian into the model. As for the one-electron terms, we may perform an unnormalized Foldy-Wouthuysen transformation on the two-electron integrals to obtain a transformed set of integrals. 

Naturally, we also need a new expression for X that includes the two-electron terms, and this we get from the Dirac-Hartree-Fock equations, which we represent here in unrestricted form. 

The definition we end up with is recursive because both and contain X if we replace AF with XA throughout. In an iterative scheme this would not be a serious problem because we could use the value of X from the previous iteration to construct the J and K matrices, and at convergence the input and output X matrices would be the same to the required numerical precision. 

With the expressions for the density, the direct and the exchange matrices, and the definition of the transformed integrals, we can write down the NESC SCF equation as 

We have so far only considered the Coulomb interaction, but it is straightforward to incorporate the Gaunt or Breit interactions. 

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