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New Interaction, Eq.()

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


where tex2html_wrap_inline5891 is the spin of the tex2html_wrap_inline6451 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 tex2html_wrap_inline8069 in Eq.(gif) with respect to the tex2html_wrap_inline8071 are


Therefore the eigenvalue, tex2html_wrap_inline8073 , is given by


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


where tex2html_wrap_inline8075 is the total energy of the system with an extra electron added into the tex2html_wrap_inline8077 orbital. Note that if tex2html_wrap_inline8079 is replaced with the standard tex2html_wrap_inline8081 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.(gif) is given by


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

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996