The Many Symmetries of Hubbard Alternant Polyenes

Abstract

In the freeon unitary-group-formulation of quantum chemistry the relevant group is U(n) where n is the number of freeon orbitals. The Hamiltonian is a second degree polynomial in the U(n) generators so the Hilbert space of the Hamiltonian is the direct sum of the U(n) irreducible representation spaces (IRS). The Pauli principle is imposed by restricting the physically significant IRS to those labeled by the partitions [λ] = [2(N/2)-S,12S] where N is the number of electrons and S is the spin. The IRS have the following properties: i) For each IRS labeled by [λ] there exists a conjugate IRS labeled by [λ] = [2(N/2)-S,12S] where N = 2n-N is the number of holes in [λ]. ii) The dimension of the [λ]th IRS equals the dimension of the [λ]th IRS. iii) The symmetry-adaptation of the [λ]th IRS with respect to any group yields the same decomposition as does the symmetry-adaptation of the [λ]th IRS. iv) There is defined a selfconjugate Hamiltonian such that the [λ]th and the [λ]th spectra differ by only a constant energy shift, ΔE = ΔE°(n-N). v) For n = N the conjugate group Gk, is a group of the Hamiltonian and supplies the conjugation quantum number.