  Mixing single and triple fermions an exact killing operator of the Coupled
Cluster Doubles (CCD) wave function with good symmetry was found in
\cite{Tohy13}. Using these operators with the equation of motion (EOM) method
the so-called self-consistent odd particle number random phase approximation
(odd-RPA) was set up. Together with the stationarity condition of the two body
density matrix it is shown that the killing conditions allow to reduce the
order of correlation functions contained in the matrix elements of the odd-RPA
equations to a fully self consistent equation for the single particle
occupation numbers. Excellent results for the latter and the ground state
energies are obtained in an exactly solvable model from weak to strong
couplings.
