Implications of Brillouins Theorem

I mentioned the Brillouin theorem in earlier chapters if rpQ is a closed-shell HF wavefunction and represents a singly excited state, then 

If we were to seek an improvement on Pq using configuration interaction, we could write 

If we used perturbation theory to estimate the expansion coefficients c etc., then all the singly excited coefficients would be zero by Brillouin s theorem. This led authors to make statements that HF calculations of primary properties are correct to second order of perturbation theory , because substitution of the perturbed wavefunction into 

If we were to use CISD to calculate the expansion coefficients, then the need not be zero they couple in to the ground state through the higher-order excitations. 

In any case, the argument ignores the fact that molecular HF calculations are invariably done at the HF-LCAO level and the choice of basis set often turns out to be the dominant source of error. 

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