The instabilities of the hartree‐fock solutions for cyclic polyenes with respect to the spin and charge density fluctuations

The stability conditions which ensure that the Hartree‐Fock (HF) determinant minimizes locally the energy expectation value are formulated. These conditions are then specified for the closed‐shell electronic systems with spin independent Hamiltonian, which enables classification of the instabilities into the so‐called “singlet” and “non‐singlet” ones. In the case that the HF solution is unstable, another single determinantal HF solution with lower energy expectation value must exist. Moreover, if the original HF solution is singlet unstable, the new HF solution is again a pure singlet function, but has a lower spatial symmetry than the original Hamiltonian. This leads to the electron density waves, while the non‐singlet instability leads to the spin‐density waves in the system considered. Generalization of these results to simple open‐shell systems is briefly mentioned. The numerical results for pi‐electronic models of cyclic polyenes using the three different semiempirical parameterization schemes, which are commonly used, are briefly reviewed. It is shown that for cyclic polyenes, both singlet and non‐singlet instabilities occur if a large enough polyene is considered. The equations determining new HF solutions may be expressed in a form resembling BCS‐Bogoliubov equations of superconductivity. Further, the singlet instability may lead, in this case, to either the atomic‐charge alternating solutions or the bond‐order alternating solutions. It is shown that in cases where both solutions are possible, the latter have the lowest energy while the former is again singlet unstable. The implications of the existence of singlet instabilities for the occurrence of an energy gap and for the occurence of bond‐length alternation as well as for the applicability of some many body techniques are briefly discussed.