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International Journal of Quantum Chemistry

Length dependence of the optical transition energies of the exactly solvable Hubbard model for linear polyenes

Katsufumi Hashimoto

Published Nov 1, 1985• 12 citations
Publisher

Abstract

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.

DOI
10.1002/qua.560280505

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Canonical BCS approximation for the attractive Hubbard model

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arXiv:cond-mat/9902285•AIP conference proceedings•Jan 1, 1999•0 citations
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.
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.
Exact ground states, charge densities, and excitation energies are found using valence bond methods for N-site modified Hubbard models with uniform spacing. At the neutral-ionic transition (NIT), the ground state has a symmetry crossover in 4n4n4n and 4n+24n+24n+2 rings with periodic and antiperiodic boundary conditions, respectively. Large site energies $\ensuremath{\Delta}$ stabilize a paired state of the half-filled chain, while large U stabilizes a covalent state. Finite-transfer integrals t shift the NIT to the covalent side of $U\ensuremath{-}2\ensuremath{\Delta}.$ Exact results to N=16N=16N=16 in the full basis and to N=22N=22N=22 in a restricted basis for large U and $\ensuremath{\Delta}$ are extrapolated to obtain the crossover and charge density of extended chains. The modified Hubbard model has a continuous NIT between a diamagnetic band insulator on the paired side and a paramagnetic Mott insulator on the covalent side. The singlet-triplet (ST), singlet-singlet (SS), and charge gaps for finite N indicate that the ST and SS gaps close at the NIT with increasing U, and that the charge gap vanishes only there. Finite-N excitations constrain all singularities to $\ifmmode\pm\else\textpm\fi{}0.1t$ of the symmetry crossover. The NIT is interpreted as a localized ground state (GS) with finite gaps on the paired side and an extended GS with vanishing ST and SS gaps on the covalent side. The charge gap and charge stiffness indicate a metallic GS at the transition that, however, is unconditionally unstable to dimerization. Finite $\ensuremath{\Delta}$ breaks electron-hole $(e\ensuremath{-}h)$ symmetry, but the modified Hubbard model has an extended $e\ensuremath{-}h$ symmetry, and a strong mixing of spin and charge excitations is limited to a few t's about the NIT. Exact finite-size results complement other approaches to valence or ferroelectric transitions in organic charge-transfer salts or in inorganic oxides, and to electron-vibration coupling and structural instabilities in one-dimensional systems.
We have applied a numerical renormalization-group technique to study the Hubbard-Peierls model for (CH)x{)}_{\mathrm{x}})x​. The method provides accurate many-body wave functions and energies for the ground and low-lying excited states of finite polyenes. We report tests of the method for finite chains (N=16) which show that the technique accurately reproduces known results in the noninteracting particle limit (U=0) and in the highly correlated limit (U/4t\ensuremath{\gg}1) and agrees well with Monte Carlo data where available for intermediate U. The model is applied to study correlation effects on the ground-state dimerization amplitude and to the ordering of the low-lying excited states of the rigid-lattice spectrum. Extension of the method to even longer systems appears to be quite practical.
We solve a model of interacting electrons coupled to longitudinal phonons using the density matrix renormalisation group method. The model is parametrised for polyenes. We calculate the ground state, and first excited odd-parity singlet and triplet states; and we investigate their energies, and bond length changes and fluctuations for up to 30 sites. The transition energy and the soliton width of the triplet state show deviations from the adiabatic approximation for chain lengths larger than the classical soliton size, because of de-pinning by the quantised lattice fluctuations.
Abstract The perturbation expansion coefficients for the ground‐state energy of the half‐filled one‐dimensional Hubbard model with N = 4ν + 2, (ν = 1, 2,…) sites and satisfying cyclic boundary conditions were calculated in the Hückel limit (U/β → 0), where β designates the one‐electron hopping or resonance integral, and U, the two‐electron on‐site Coulomb integral. This was achieved by solving—order by order—the Lieb–Wu equations, a system of transcendental equations that determines the set of quasi‐momenta {ki} and spin variables {τα} for this model. The exact values for these quantities were found for the N = 6 member ring up to the 20th order in terms of the coupling constant B = U/2β, as well as numerically for 10 ⩽ N ⩽ 50, and the N = 6 Lieb–Wu system was reduced to a system of sextic algebraic equations. These results are compared with those of the infinite system derived analytically by Misurkin and Ovchinnikov [Teor. Mat. Fiz. 11, 127 (1972)]. It is further shown how the results of this article can be used for the interpolation by the root of a polynomial. It is demonstrated that a polynomial of relatively small degree provides a very good approximation for the energy in the intermediate coupling region. © 1995 John Wiley & Sons, Inc.
We test the canonical BCS wave functions for fixed number of electrons for the attractive Hubbard model. We present results in one dimension for various chain lengths, electron densities, and coupling strengths. The ground-state energy and energy gap to the first excited state are compared with the exact solutions obtained by the Bethe ansatz as well as with the results from the conventional grand-canonical BCS approximations. While the canonical and grand-canonical BCS results are both in very good agreement with the exact results for the ground state energies, improvements due to conserving the electron number in finite systems are manifest in the energy gap. As the system size is increased, the canonical results converge to the grand-canonical ones. The ``parity'' effect that arises from the number parity (even or odd) of electrons are studied with our canonical scheme.