  We study the spectra of the molecular orbital Hessian (stability matrix) and
random-phase approximation Hamiltonian of broken-symmetry Hartree-Fock
solutions, focusing on zero eigenvalue modes. After all negative eigenvalues
are removed from the Hessian by following their eigenvectors downhill, one is
left with only positive and zero eigenvalues. Zero modes correspond to orbital
rotations with no restoring force. These rotations determine states in the
Goldstone manifold, which originates from a spontaneously broken continuous
symmetry in the wave function. Zero modes can be classified as improper or
proper according to their different mathematical and physical properties.
Improper modes arise from symmetry breaking and their restoration always lowers
the energy. Proper modes, on the other hand, correspond to degeneracies of the
wave function, and their symmetry restoration does not necessarily lower the
energy. We discuss how the RPA Hamiltonian distinguishes between proper and
improper modes by doubling the number of zero eigenvalues associated with the
latter. Proper modes in the Hessian always appear in pairs which do not double
in RPA. We present several pedagogical cases exemplifying the above statements.
The relevance of these results for projected Hartree-Fock methods is also
addressed.
