Within the HF approximation the interaction of Eq.() leads to the following expression for the total energy;

where is the spin of the electron. The two charge densities in the second (Hartree) term have been chosen to be the same for simplicity. In the case of HF calculations performed using fixed LDA orbitals, this corresponds to using the LDA charge density as the `input' charge density to the interaction in the same way as is done for VMC calculations.

The resultant HF equations obtained from minimising in Eq.() with respect to the are

Therefore the eigenvalue, , is given by

Eqs.() and () yield an analogue of Koopmans' theorem for adding an electron

where is the total energy of the system with an extra electron added into the orbital. Note that if is replaced with the standard we retrieve the standard Koopmans' theorem.

The equivalent expression for removing an electron is

and therefore the HF energy gap, obtained using the expression for the electron-electron interaction of Eq.() is given by

Koopmans' theorem has therefore been modified. The interaction is, in a sense, including self-interaction like terms.

Tue Nov 19 17:11:34 GMT 1996