Systematic relations between collective energies from the GCM and RPA methods

It has previously been found numerically that GCM and RPA methods using the same forces can give almost the same collective energies. It is pointed out that there are systematic reasons for this. When the GCM approach is used with the usual Gaussian Overlap Approximation, the energy has a lower limit in terms of RPA energies. If oscillator properties are assigned to the orbitals, the inequalities are converted into equalities. It is shown that some formulations of the GCM approach imply violation of sum-rules, so should be used with caution. (Auth.)