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

Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration