A new self-consistent-field valence bond method. II. The effective spin Hamiltonian

Abstract

The effective spin Hamiltonian is introduced as a practical tool of valence bond (VB) calculation on nonorthogonal orbitals. The effective spin Hamiltonian obtained in the SCF VB method proposed previously is examined in detail for conjugated π systems (linear polyenes and monocyclic systems) within the Pariser–Parr–Pople (PPP) model. It is found that the Heisenberg spin Hamiltonian with only the nearest-neighbor exchanges is a reasonable first approximation; this means that the well-known classical VB treatment, the simplified VB treatment developed by Slater, Pauling, and others, is justified theoretically as far as the PPP model systems are concerned. The next-nearest-neighbor exchanges are found to be not negligible from the quantitative standpoint. In the case of monocyclic systems, the ring (or cyclic) permutations, the permutations making a full circle round the ring, emerge with a coefficient a little larger than those of the next-nearest-neighbor exchanges; the VB calculation taking them into account yields the ground-state energy exhibiting the Hückel rule. The SCF VB calculation with a Heisenberg spin Hamiltonian was further carried out on linear polyenes (all-trans form, PPP model) of up to 14 π electrons. The calculated ground-state energy, which is expected to be fairly accurate in this specific case, becomes lower than that of the single and double excitations CI (SDCI) calculation for polyenes of more than ten π electrons. The first excited 1Ag state was also calculated and the calculated excitation energy decreases rather steadily with increasing polyene-chain length in contrast to the SDCI results and in agreement with observations and more accurate CI results.