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Enhanced Version from Eq.()

Let us now repeat the above analysis using the enhanced expression for the electron-electron interaction of Eq.(gif), designed to remove the effect of the unwanted periodic array of additional electrons. The HF equivalent of this energy expression is


The resultant HF equations obtained from minimising tex2html_wrap_inline8083 in Eq.(gif) with respect to the tex2html_wrap_inline8071 are exactly the same as Eq.(gif), obtained from the original energy expression. The eigenvalues, tex2html_wrap_inline8073 , are therefore also exactly the same as those given in Eq.(gif). However, if one considers the change in the total energy on adding an electron to state k, using the energy expression in Eq.(gif), one obtains a similar expression to Eq.(gif) but without the term arising from the interaction between the tex2html_wrap_inline8077 electron and its images, i.e.


Therefore, when using the enhanced version of the electron-electron interaction from Eq.(gif), one recovers the standard version of Koopmans' theorem where the eigenstates of the HF equations correspond to the excitation energies of the system,


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


The comparison of the two energy expressions, Eq.(gif) and Eq.(gif), within HF theory can therefore be summarised as follows. The enhanced version of the electron-electron interaction, Eq.(gif), improves over the original expression in two ways, (i) it removes the self-term due to the interaction of an electron and its images, (ii) it recovers a proper version of Koopmans' theorem. Both these results provide additional support for the use of the enhanced interaction, Eq.(gif), in the following QMC calculations.

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