Some Consequences of the Nonlinearity of the Hartree-Fock Approach, Demonstrated on the Example of the PPP Model for Closed Shell Alternant Hydrocarbons

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Abstract
It is shown that the nonuniqueness of the H – F solutions of the PPP model for alternant hydrocarbons is generally a consequence of the fact that the part E 2 of the energy expectation value E = E 1 + E 2 , connected with two electron operators in the Hamiltonian, can exhibit various minima. Existence conditions for minima of E 2 are given. These minima have bond orders corresponding to distribution of pure single and pure double bonds as in classical chemical structural formulas (Kekulé and Dewar formulas). There are cases for which the superposition of the part E 1 does not erase the minima corresponding to Kekulé and Dewar formulas. By the direct minimization method, two distinct electronic H – F distributions are obtained for realistic or nearly realistic parametrizations and geometries in the case of annulenes and long polyenes. Cases of different H – F solutions with the same symmetry as the symmetry of the molecule itself are found. A conjecture is made that there may be two stable geometric configurations for very long polyenes. The possible connection is discussed between the strange features of the Hartree‐Fock approach for long linear polyenes and their instability.