Calculations are performed, in the one-particle-hole and projected Hartree-Fock (PHF) approximations, for the low-lying $2^{+}$ and $3^{-}$ excitations in ${\mathrm{C}}^{12}$ and ${\mathrm{O}}^{16}$, respectively. A new expression is derived for static moments of excited states in the random-phase approximation (RPA) and general one-particle-hole approximations. The purpose of this study was to investigate: the practical usefulness of the above approximations for nontrivial realistic problems; the validity of the one-particle-hole and PHF approximations for positive- as well as negative-parity excitations; the ability of the equations of motion and PHF methods to predict static moments of excited states; the magnitude of corrections to the RPA with explicit inclusion of ground-state correlations; and the validity of the inconsistent RPA procedure of including backward amplitudes at the expense of violating the Pauli principle. We conclude that the one-particle-hole approximation is particularly appropriate for negative-parity excitations and that the PHF approximation is appropriate for positive-parity excitations.

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