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International Journal of Quantum Chemistry

Exact results on homopolar singlet excitation energies of the hubbard model for linear polyenes

Katsufumi Hashimoto

Published Nov 1, 1986• 21 citations
Publisher

Abstract

Abstract The Lieb–Wu equations for homopolar singlet excited states of the Hubbard model of linear polyenes are formulated in a form tractable by numerical calculation. They are solved to obtain exact excitation energies of the lowest two excited states, corresponding to the 1B and 1E states of the benzene molecule, for systems with up to 50 and 30 electrons, respectively. These results may be particularly useful for checking accuracy of approximate methods for the homopolar states for which the conventional CI method is very difficult to apply. Calculation is also carried out for the 1B+ ionic excited state involving a spin excitation, which corresponds to the 1B state of the benzene molecule. The length dependence of the excitation energies is examined and shown to have no N−1 dependence for large N, where N is the number of electrons. This supplements the result on the 1E ionic state obtained in our previous paper.

DOI
10.1002/qua.560300507

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Abstract The Lieb–Wu equations for homopolar singlet excited states of the Hubbard model of linear polyenes are formulated in a form tractable by numerical calculation. They are solved to obtain exact excitation energies of the lowest two excited states, corresponding to the 1B and 1E states of the benzene molecule, for systems with up to 50 and 30 electrons, respectively. These results may be particularly useful for checking accuracy of approximate methods for the homopolar states for which the conventional CI method is very difficult to apply. Calculation is also carried out for the 1B+ ionic excited state involving a spin excitation, which corresponds to the 1B state of the benzene molecule. The length dependence of the excitation energies is examined and shown to have no N−1 dependence for large N, where N is the number of electrons. This supplements the result on the 1E ionic state obtained in our previous paper.
Abstract Some accurate results on the length dependence of the excitation energies from the ground state to ionic excited states in the Hubbard model of linear polyenes are obtained based on the method of Lieb and Wu. To this end, it is first shown that singly ionic excited states with “plus” alternancy symmetry in the Hubbard model are described by the wave functions in which the two electron operator [∑ (−)nC C ] is acted on (N − 2)‐electron covalent eigenstates. Then by solving the Lieb‐Wu equations the exact excitation energies of the lowest ionic state, which corresponds to the E state in this model, are calculated for systems with up to 50 electrons. The result, together with a correction for the end effect, indicates that the excitation energies do not decrease as 1/N but converge to the limiting value more rapidly when the number of electrons N becomes large.
We present density-matrix renormalization-group calculations of the Pariser-Parr-Pople-Peierls model of linear polyenes within the adiabatic approximation. We calculate the vertical and relaxed transition energies, and relaxed geometries for various excitations on long chains. The triplet (13Bu+)(1{}^{3}{B}_{u}^{+})(13Bu+​) and even-parity singlet (21Ag+)(2{}^{1}{A}_{g}^{+})(21Ag+​) states have a 2-soliton and 4-soliton forms, respectively, both with large relaxation energies. The dipole-allowed $(1{}^{1}{B}_{u}^{\ensuremath{-}})$ state forms an exciton-polaron, and has a very small relaxation energy. The relaxed energy of the 21Ag+2{}^{1}{A}_{g}^{+}21Ag+​ state lies below that of the $1{}^{1}{B}_{u}^{\ensuremath{-}}$ state. We observe an attraction between the soliton-antisoliton pairs in the 21Ag+2{}^{1}{A}_{g}^{+}21Ag+​ state. The calculated excitation energies agree well with the observed values for polyene oligomers; the agreement with polyacetylene thin films is less good, and we comment on the possible sources of the discrepancies. The photoinduced absorption is interpreted. The spin-spin correlation function shows that the unpaired spins coincide with the geometrical soliton positions. We study the roles of electron-electron interactions and electron-lattice coupling in determining the excitation energies and soliton structures. Electronic interactions play the key role in determining the ground-state dimerization and the excited-state transition energies.
The one-dimensional Hubbard model is studied for general electron densities and large U. On the basis of the coupled integral equations of Lieb and Wu exact asymptotic expansions are derived to fifth order in ${U}^{\mathrm{\ensuremath{-}}1}$ for the ground-state energy, the kinetic energy, and the number of doubly occupied sites. A closed-form expression for the distribution functions of charge and spin degrees of freedom is proposed which yields the same large-U expansion as that derived directly from the Lieb-Wu equations. Our analytical results are in excellent agreement with previous numerical studies.
We apply the Hubbard-Peierls Hamiltonian to study the low-lying excited states of several model conjugated polymers. The calculations employ a numerical renormalization-group method for chains of intermediate length, and a more approximate truncated configuration-interaction scheme to extrapolate the results to longer systems. In the range of on-site Coulomb repulsion strengths of physical interest, we find that a correlated even-parity singlet excited state competes with the odd-parity singlet excitation expected in the U=0 limit of the model, in agreement with previous studies on the Pariser-Parr-Pople model. The lowest electronic excitations in the model are identified as triplet excitations. Self-consistently-relaxed lattice configurations surrounding these electronic excitations are studied. In addition to the stable photoexcitations predicted in the U=0 limit of the theory, we find that the interelectronic repulsive potential leads to a rich spectrum of competing neutral structures, which are studied and characterized. Coulomb correlations provide a mechanism for production of long-lived spin-1/2 neutral excitations for degenerate ground-state systems, and spin-1 neutral excitations for nondegenerate ground-state systems.
An interpolation technique which is based on the strongly and weakly correlated cases of the one-dimensional Hubbard model is proposed. The input information consists of the perturbation expansions which are obtained from the Lieb–Wu equations in both limits. The Hubbard model is used to describe cyclic polyene rings, and for the case of N=6 sites, this would correspond to benzene. The technique has been applied to several symmetries of the model. It has been shown that the exact secular problem can be reconstructed for these symmetries, and the results for N=6 agree exactly with the results which have been obtained in a different way, that is, by using a full configuration interaction calculation.
Those excited states of a half-filled 1D Hubbard chain that are connected with electron pairs occupying the same site are studied. It is argued that these states should be described by solutions of the Lieb-Wu equations (1968) in which some of the wavenumbers are complex. Solutions of this type have been found that correspond to states in which the spin part is not excited. The energy-momentum dispersion is also calculated. The gap in the spectrum of the single excitations equals the discontinuity of the chemical potential calculated by Lieb and Wu.