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The Journal of Chemical Physics

Bond configurations in the Peierls–Hubbard model of small ring-shaped molecules

Emilia Liviotti, Francesco Ubertini, P. Erdös

Published Aug 8, 1999• 4 citations
Publisher

Abstract

The Peierls–Hubbard model applied to N-membered ring-shaped molecules is investigated numerically and analytically. The bond configurations minimizing the total energy at half-filling of the electronic states are determined by a self-consistent method for N⩽8 in a wide range of the electron–lattice and electron–electron coupling parameters. In the even-N case, only dimerized and homogeneous configurations have been found to be stable. Odd-N rings show three types of bond configurations depending on the coupling parameters. Two of these configurations have reflection symmetry whereas the third one is irregular.

DOI
10.1063/1.479516

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•Jan 1, 2001•0 citations
The Peierls–Hubbard model applied to N-membered ring-shaped molecules is investigated numerically and analytically. The bond configurations minimizing the total energy at half-filling of the electronic states are determined by a self-consistent method for N⩽8 in a wide range of the electron–lattice and electron–electron coupling parameters. In the even-N case, only dimerized and homogeneous configurations have been found to be stable. Odd-N rings show three types of bond configurations depending on the coupling parameters. Two of these configurations have reflection symmetry whereas the third one is irregular.
The stationary states of a half-filled Peierls-Hubbard chain are calculated. To describe electron correlations, the wave functions as antisymmetrized products of strongly orthogonal geminals are used; wave functions and bond lengths are determined self-consistently by the variational method in an N=50 ring. The neutral soliton pair S0{S}^{0}S0S0{S}^{0}S0 is obtained as corresponding to a local total-energy minimum above the perfectly dimerized ground state. The charged soliton pair S+{S}^{+}S+${S}^{\mathrm{\ensuremath{-}}}$ refers to the minimum in the subspace of wave functions which are odd with respect to charge conjugation. The creation energies for each of the pairs are estimated as functions of Hubbard parameter U and compared with the exact excitation energies of the uniform Hubbard chain. The electron correlation is shown to make the in-gap S0{S}^{0}S0S0{S}^{0}S0 state stable even for a moderate electron-lattice coupling; the soliton width turns out to be much smaller than that in the U=0 model and comparable with the lattice constant. Applicability of the results to polyacetylene is discussed.
We investigate the conventional tight-binding model of L \ensuremath{\pi} electrons on a ring-shaped molecule of L atoms with nearest-neighbor hopping. The hopping amplitudes t(w) depend on the atomic spacings w with an associated distortion energy V(w). A Hubbard-type on-site interaction as well as nearest-neighbor repulsive potentials can also be included. We prove that when L=4k+2 the minimum energy E occurs either for equal spacing or for alternating spacings (dimerization); nothing more chaotic can occur. In particular, this statement is true for the Peierls-Hubbard Hamiltonian, which is the case of linear t(w) and quadratic V(w), i.e., t(w)=t0{\mathit{t}}_{0}t0​-\ensuremath{\alpha}w and V(w)=k(w-a)2{)}^{2})2, but our results hold for any choice of couplings or functions t(w) and V(w). When L=4k we prove that more chaotic minima can occur, as we show in an explicit example, but the alternating state is always asymptotically exact in the limit L\ensuremath{\rightarrow}\ensuremath{\infty}. Our analysis suggests three interesting conjectures about how dimerization stabilizes for large systems. We also treat the spin-Peierls problem and prove that nothing more chaotic than dimerization occurs for L=4k+2 and L=4k.
The single-band Hubbard model (both with and without first-neighbor interactions) is studied by exact diagonalization on several one- and two-dimensional clusters, ranging from four to nine sites, including rings and arrangements of squares. Occupancies include the half-filled band (one electron per site) and configurations with one and two holes in the half-filled band. Ground-state properties are determined for a wide range of parameters. We discuss the phase diagram in the (U,V) plane and consider spin, charge, and pairing correlations. The binding of a pair of holes is investigated. Circumstances in which binding and pairing are enhanced by first-neighbor interactions are described.
To the conventional Peierls-Hubbard model, involving both on-site (U) and nearest-neighbor (V) Coulomb repulsions, we add ``off-diagonal'' terms, not expressible purely in terms of site densities, representing bond-bond (W) and mixed bond-site (X) electron-electron repulsive interactions involving nearest neighbors. We review earlier analyses of these interactions and discuss relative magnitudes of the parameters in applications to real materials. As a specific illustration, we investigate the effects of the off-diagonal W and X terms on dimerization in the one-dimensional, half-filled-band Peierls-Hubbard models, which have been widely applied to conjugated polymers (such as trans-polyacetylene) and to related quasi-one-dimensional charge-density wave (CDW) systems. Using both weak- and strong-coupling perturbation theory for large systems and exact diagonalizations of small systems, we investigate thoroughly the nature of the ground state of the model. For a broad range of the site-diagonal Hubbard parameters (U,V), including the values believed to be relevant to trans-polyacetylene, we find that the off-diagonal terms (W,X) initially enhance dimerization, thereby stabilizing the dimerized [or bond-order-wave (BOW)] ground state. For (unphysically) large values of W relative to U and V, dimerization is destroyed, and the BOW ground state goes over to a ferromagnetic ground state or a CDW ground state, depending on the relative sizes of U, V, and W. We conclude with a general discussion of the applicability of the Peierls-Hubbard models to quasi-one-dimensional materials, including the potential importance of the breaking of charge conjugation (``particle-hole'') symmetry by the X term.
"We have studied the extended Hubbard and the Peierls-extended Hubbard models in one dimension using primarily computational methods. We discuss the derivation of the models and the computational methods used to study them, focusing primarily on our extensions over existing numerical methods. In particular, we present the construction of ""operator-loop"" updates for spin-1/2 fermions and the inclusion of electron-phonon interactions in the Stochastic Series Expansion Quantum Monte Carlo method and the use of Quantum Parallel Tempering to improve the quality of the numerical data. For the extended Hubbard model at half-filling, we have confirmed the existence of a recently reported bond-order wave phase for the ground state in a small region of parameter space between the well-known charge- and spin-density wave phases. We have studied in detail the nature of the associated quantum phase transitions between all the phases. Upon doping away from half-filling, the doubly-degenerate, spin and charge gapped long-range-ordered ground states of the model are shown to evolve to a Luther-Emery liquid with no charge gap but a finite spin gap. For the Peierls-extended Hubbard model, which includes both electron-phonon and electron-electron interactions, we have studied the dimerization transition of the half-filled system in the presence of fully quantum mechanical phonons. We show that the transition to a dimerized state occurs only when the electron-phonon coupling exceeds a finite critical value that depends on the bare phonon frequency. This is in contrast to previous studies in similar models with adiabatic phonons that had predicted a dimerized state for arbitrarily small values of the electron-phonon coupling."