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arXiv:cond-mat/9908066 · Physical review. B, Condensed matter

Matrix-product approach to conjugated polymers

Miguel Angel Martin-Delgado, Germán Sierra, S. Pleutin, Eric Jeckelmann

Published Jan 15, 2000• 3 citations
arXivHTMLPDF

Abstract

The matrix product method (MPM) has been used in the past to generate variational Ans\"atze of the ground state (GS) of spin chains and ladders. In this paper we apply the MPM to study the GS of conjugated polymers in the valence bond basis, exploiting the charge and spin conservation as well as the electron-hole and spin-parity symmetries. We employ the U-$V$-$\ensuremath{\delta}$ Hamiltonian, which is a simplified version of the Pariser-Parr-Pople Hamiltonian. For several coupling constants U and V and dimerizations $\ensuremath{\delta},$ we compute the GS energy per monomer, which agrees within a 2--4 % accuracy with the density-matrix renormalization group results. We also show the evolution of the MP-variational parameters in the weak and strong dimerization regimes.

DOI
10.1103/physrevb.61.1841

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The matrix product method (MPM) has been used in the past to generate variational Ans"atze of the ground state (GS) of spin chains and ladders. In this paper we apply the MPM to study the GS of conjugated polymers in the valence bond basis, exploiting the charge and spin conservation as well as the electron-hole and spin-parity symmetries. We employ the U-VVV-\ensuremathδ\ensuremath{\delta}\ensuremathδ Hamiltonian, which is a simplified version of the Pariser-Parr-Pople Hamiltonian. For several coupling constants U and V and dimerizations \ensuremathδ,\ensuremath{\delta},\ensuremathδ, we compute the GS energy per monomer, which agrees within a 2--4 % accuracy with the density-matrix renormalization group results. We also show the evolution of the MP-variational parameters in the weak and strong dimerization regimes.
We have carried out density-matrix renormalization group calculations on the ground state of long polyacene oligomers within a Pariser-Parr-Pople (PPP) Hamiltonian. The PPP model includes long-range electron correlations which are required for physically realistic modeling of conjugated polymers. We have obtained the ground-state energy as a function of the dimerization $\ensuremath{\delta}$ and various correlation functions and structure factors for $\ensuremath{\delta}=0$. From energetics, we find that the nature of the Peierls' instability in polyacene is conditional and strong electron correlations enhance the dimerization. The cis form of the distortion is favored over the trans form. However, from the analysis of correlation functions and associated structure factors, we find that polyacene is not susceptible to the formation of a bond order wave, spin-density wave, or a charge-density wave in the ground state.
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.
We extend the density-matrix renormalization-group (DMRG) method to exploit parity, C2{\mathit{C}}_{2}C2​ (rotation by \ensuremath{\pi}), and electron-hole symmetries of model Hamiltonians. We demonstrate the power of this method by obtaining the lowest-energy states in all eight symmetry subspaces of Hubbard chains with up to 50 sites. The ground-state energy, optical gap, and spin gap of regular U=4t and U=6t Hubbard chains agree very well with exact results. This development extends the scope of the DMRG method and allows future applications to study of optical properties of low-dimensional conjugated polymeric systems. \textcopyright{} 1996 The American Physical Society.
The density matrix renormalization group (DMRG) method introduced by White for the study of strongly interacting electron systems is reviewed; the method is variational and considers a system of localized electrons as the union of two adjacent fragments A,B. A density matrix ρ is introduced, whose eigenvectors corresponding to the largest eigenvalues are the most significant, the most probable states of A in the presence of B; these states are retained, while states corresponding to small eigenvalues of ρ are neglected. It is conjectured that the decreasing behavior of the eigenvalues is Gaussian. The DMRG method is tested on the Pariser-Parr-Pople Hamiltonian of a cyclic polyene (CH)N up to N=34. A Hilbert space of dimension 5.×1018 is explored. The ground state energy is 10−3 eV within the full CI value in the case N=18. The DMRG method compares favorably also with coupled cluster approximations. The unrestricted Hartree-Fock solution (which presents spin density waves) is briefly reviewed, and a comparison is made with the DMRG energy values. Finally, the spin-spin and density-density correlation functions are computed; the results suggest that the antiferromagnetic order of the exact solution does not extend up to large distances but exists locally. No charge density waves are present.
The Pariser-Parr-Pople model of pi-conjugated electrons is solved by a three-block, symmetry-adapted density matrix renormalization group (DMRG) method for the light emitting polymer, poly(para-phenylene vinylene). The energies of the primary excited states are calculated. There is excellent agreement between theory and experiment when solid state screening is incorporated into the model parameters, enabling us to make an identification of the origin of the key spectroscopic features. Appendices describe important technical aspects of the three-block DMRG approach: Local Hilbert space efficiency and its relation to the matrix product formulation of the DMRG; an efficient computational procedure for constructing symmetry-adapted states for DMRG calculations; and correct superblock state targeting to ensure good convergence of the method.