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Physical review. B, Condensed matter

Dimerization of polyacetylene using a distance-dependent Hubbard model and the density-matrix renormalization-group method

Marie-Bernadette Lepetit, G. M. Pastor

Published Aug 15, 1997• 21 citations
Publisher

Abstract

The dimerization of polyacetylene is studied using a distance-dependent Hubbard Hamiltonian for the valence $\ensuremath{\pi}$ electrons and approximating $\ensuremath{\sigma}$-electron contributions by a sum of pair interactions ${E}_{\ensuremath{\sigma}}{(r}_{\mathrm{ij}})$. The $\ensuremath{\pi}$-electron Coulomb repulsion $U$ is taken to be independent of the C-C distances ${r}_{\mathrm{ij}}$ and is varied systematically from the uncorrelated or H\"uckel limit all over to the strongly correlated or Heisenberg regime. For each value of $U$, ${E}_{\ensuremath{\sigma}}{(r}_{\mathrm{ij}})$, and the $\ensuremath{\pi}$-electron hopping integral ${t(r}_{\mathrm{ij}})$ are fitted to reproduce ab initio results on the ground-state and first-excited-state energy of the ethylene molecule. The one-dimensional many-body problem is solved numerically by means of the density-matrix renormalization-group (DMRG) method. The ground-state energy $E$ of the infinite open polymer is obtained by extrapolation considering chains of up to 300 atoms. The dimerization $\ensuremath{\delta}{=(r}_{i,i+1}\ensuremath{-}{r}_{i\ensuremath{-}1,i})/2$ and the average bond length ${R=(r}_{i,i+1}{+r}_{i\ensuremath{-}1,i})/2$ are determined by minimizing $E$. While $R$ is not significantly affected by the value of $U$, $\ensuremath{\delta}$ shows a remarkable nonmonotonic behavior. The differences between the delocalized regime, small $U/t$, and spin-Peierls behavior, large $U/t$, are discussed. The convergence properties of the DMRG algorithm are analyzed as a function of $U$, $\ensuremath{\delta}$, and the number states kept at each RG step. Criteria are examined in order to control the accuracy of the DMRG calculations in both weakly and strongly correlated situations.

DOI
10.1103/physrevb.56.4447

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The dimerized one-dimensional Hubbard model is studied in the framework of lattice density-functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable. The corresponding interaction-energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is defined by Levy's constrained search. Exact numerical results are obtained for $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23}),$ where ${\ensuremath{\gamma}}_{12}={\ensuremath{\gamma}}_{i,i+1}$ for odd i and ${\ensuremath{\gamma}}_{23}={\ensuremath{\gamma}}_{i,i+1}$ for even i are the nearest-neighbor density-matrix elements along the chain. The domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}}$ and the functional dependence of $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ are analyzed. A simple, explicit approximation to $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ is proposed, which is derived from scaling properties of W, exact dimer results, and known limits. Using this approximation, LDFT is applied to determine ground-state properties and charge-excitation gaps of finite and infinite dimerized chains as a function of the Coulomb-repulsion strength U/tU/tU/t and of the alternation $\ensuremath{\delta}t$ of the hopping integrals tij{t}_{\mathrm{ij}}tij​ ${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.
We apply the density matrix renormalization group method to the Pariser–Parr–Pople Hamiltonian and investigate the onset of dimerization. We deduce the parameters of the hopping term and the contribution of the σ bonds from ab initio calculations on ethylene. Denoting by Rij the C–C distances, we perform a variational optimization of the dimerization δ=(Ri,i+1−Ri−1,i)/2 and of the average bond length R0 for chains up to N=50 sites. The critical value of N at which the transition occurs is found to be between N=14 and N=18 for the present model. The asymptotic values for large N for R0 and δ are given by 1.408(3) Å and 0.036(0) Å.
Discrepancies between the single-particle band structure and the exact many-particle behavior are particularly striking in one-dimensional systems. The one-dimensional alternating Hubbard model is especially interesting as a nontrivial model for conjugated polymer chains, such as polyacetelyne. We study this model for chains of 60 sites using the density-matrix renormalization-group method. We obtain the first electronic and magnetic excited-state energies at half filling as a function of an alternating hopping parameter and on-site electron-electron interaction. We also study the effect of the interaction on the dimerization and discuss its relation to the charge gap and spin triplet gap. In addition, we calculate the local spin and charge densities of some excited states, which manifest a clear crossover from bandlike to strongly correlated behavior.
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