Stability Conditions for the Solutions of the Hartree-Fock Equations for Atomic and Molecular Systems. V.The Nonanalytic Behavior of the Broken-Symmetry Solutions at the Branching Point

It is shown that both the energy and the wave function of the stable Hartree-Fock solution are nonanalytic functions of the coupling constant at the branching point, dividing the symmetry-adapted and the broken-symmetry solutions. The $\pi$-electronic model of benzene is used to illustrate this fact for the case of the Hartree-Fock singlet instability. It is indicated that the same type of nonanalytical behavior is found in other types of instabilities, i.e., nonsinglet (triplet) instabilities for closed shells and doublet instabilities for simple open shells. This nonanalyticity persists even after the projection on the totally symmetric subspace is carried out, and can only be avoided by projecting before the variational principle is applied.