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arXiv:cond-mat/9802216 · Physical review. B, Condensed matter

Identification of excitons in conjugated polymers: A density-matrix renormalization-group study

Magnus Boman, Robert J. Bursill

Published Jun 15, 1998• 25 citations
arXivHTMLPDF

Abstract

This work addresses the question of whether low-lying excitations in conjugated polymers are comprised of free charge carriers or excitons. States are characterized as bound or unbound according to the scaling of the average particle-hole separation with system size. We critically examine other criteria commonly used to characterize states. The polymer is described by an extended Hubbard model with alternating transfer integrals. The model is solved by exact diagonalization and the density-matrix renormalization-group (DMRG) method. We demonstrate that the DMRG accurately determines excitation energies, transition dipole moments and particle-hole separations of a number of dipole-forbidden ${(A}_{g})$ and dipole-allowed ${(B}_{u})$ states. Within a parameter regime considered reasonable for polymers such as polyacetylene, it is found that the charge gap, often used to define the exciton binding energy, is not a good criterion by which to decide whether a state is bound or unbound. The essential nonlinear optical state ${\mathrm{mA}}_{g}$ is found to mark the onset of unbound excitations in the ${A}_{g}$ symmetry sector. In the ${B}_{u}$ symmetry sector, on the other hand, it is found that all low-lying states are unbound and that there is no well-defined ${\mathrm{nB}}_{u}$ state. That is, the ${1B}_{u}$ state marks the onset of unbound excitations in this sector.

DOI
10.1103/physrevb.57.15167

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Comment on “Density-matrix renormalization-group method for excited states”

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Excited states in polydiacetylene chains: A density matrix renormalization group study

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We study theoretically polydiacetylene chains diluted in their monomer matrix. We employ the density matrix renormalization group method on finite chains to calculate the ground state and low-lying excitations of the corresponding Peierls--Hubbard-Ohno Hamiltonian which is characterized by the electron transfer amplitude ${t}_{0}$ between nearest neighbors, by the electron-phonon coupling constant $\ensuremath{\alpha}$, by the Hubbard interaction $U$, and by the long-range interaction $V$. We treat the lattice relaxation in the adiabatic limit, i.e., we calculate the polaronic lattice distortions for each excited state. Using chains with up to 102 lattice sites, we can safely perform the extrapolation to the thermodynamic limit for the ground-state energy and conformation, the single-particle gap, and the energies of the singlet exciton, the triplet ground state, and the optical excitation of the triplet ground state. The corresponding gaps are known with high precision from experiments. We determine a coherent parameter set $({t}_{0}^{*}=2.4\phantom{\rule{0.16em}{0ex}}\mathrm{eV},{\ensuremath{\alpha}}^{*}=3.4\phantom{\rule{0.16em}{0ex}}\mathrm{eV}/\text{\AA{}},{U}^{*}=6\phantom{\rule{0.16em}{0ex}}\mathrm{eV},{V}^{*}=3\phantom{\rule{0.16em}{0ex}}\mathrm{eV})$ from a fit of the experimental gap energies to the theoretical values which we obtain for 81 parameter points in the four-dimensional search space $({t}_{0},\ensuremath{\alpha},U,V)$. We identify dark in-gap states in the singlet and triplet sectors as seen in experiments. Using a fairly stiff spring constant, the length of our unit cell is about 1$%$ larger than its experimental value.

arXiv:1303.2492•Physical Review B•Jun 18, 2013•21 citations
This work addresses the question of whether low-lying excitations in conjugated polymers are comprised of free charge carriers or excitons. States are characterized as bound or unbound according to the scaling of the average particle-hole separation with system size. We critically examine other criteria commonly used to characterize states. The polymer is described by an extended Hubbard model with alternating transfer integrals. The model is solved by exact diagonalization and the density-matrix renormalization-group (DMRG) method. We demonstrate that the DMRG accurately determines excitation energies, transition dipole moments and particle-hole separations of a number of dipole-forbidden (Ag){(A}_{g})(Ag​) and dipole-allowed (Bu){(B}_{u})(Bu​) states. Within a parameter regime considered reasonable for polymers such as polyacetylene, it is found that the charge gap, often used to define the exciton binding energy, is not a good criterion by which to decide whether a state is bound or unbound. The essential nonlinear optical state mAg{\mathrm{mA}}_{g}mAg​ is found to mark the onset of unbound excitations in the Ag{A}_{g}Ag​ symmetry sector. In the Bu{B}_{u}Bu​ symmetry sector, on the other hand, it is found that all low-lying states are unbound and that there is no well-defined nBu{\mathrm{nB}}_{u}nBu​ state. That is, the 1Bu{1B}_{u}1Bu​ state marks the onset of unbound excitations in this sector.
In a recent paper [Phys. Rev. B 59,59,59, 9699 (1999)], Chandross and Hicks claim to present a new density-matrix renormalization group (DMRG) method for dealing with excited-states of quantum lattice models. The proposed improvement to the DMRG---the inclusion of excited-state wave functions in addition to the ground state in the density matrix when calculating excitations---is in fact standard practice, is clearly stated in White's original papers, and has been used repeatedly by many groups to study excited states. The authors apply the method to the extended, dimerized Hubbard model for conjugated polymers. The criteria for determining whether states are bound or not are assessed. The authors claim that their results show that the optically important ‘‘1Bu′′``{1B}_{u}''‘‘1Bu′′​ state is bound (excitonic), in contrast to a previous study. However, the discussion is qualitative, and the authors arrive at conclusions on the basis of results for one lattice size only. We show that when the criterion of Chandross and Hicks is developed into a quantitative definition of particle-hole separation, with the finite-size dependence analyzed, the implication is that the 1Bu{1B}_{u}1Bu​ state is unbound, at least in the sense of the density-density correlation function, in keeping with the conclusions of a previous study.
Using a symmetrized density-matrix renormalization-group formulation, we have investigated the electronic binding energy of the lowest optically allowed exciton (1Bu{1\mathrm{B}}_{\mathrm{u}}1Bu​) within an extended Hubbard-Peierls model (with parameters U, V, and \ensuremath{\delta}) for conjugated chains with N=80 sites. Three symmetries, the C2{\mathrm{C}}_{2}C2​, spin parity, and electron-hole symmetries, have been applied to construct the projector operator. Analysis of our results on the ratios between exciton binding energies and exciton energies, sheds light on the current experimental and theoretical controversy on exciton binding for polyacetylene and polyparaphenylene vinylene. We show that in the absence of dimerization, the exciton binding energy is vanishingly small for U/t up to 5 and V/t up to 2.3; for a finite dimerization, it is only when V is large enough that the exciton gets to be significantly bound. This result questions the applicability to conjugated polymers of the strong correlation picture, in which the exciton binding energy is equal to V.
We have carried out symmetrized density-matrix renormalization-group calculations to study the nature of excited states of long polyacene oligomers within a Pariser-Parr-Pople Hamiltonian. We have used the C2{C}_{2}C2​ symmetry, the electron-hole symmetry, and the spin parity of the system in our calculations. We find that there is a crossover in the lowest dipole forbidden two-photon state and the lowest dipole allowed excited state with size of the oligomer. In the long system limit, the two-photon state lies below the lowest dipole allowed excited state. The triplet state lies well below the two-photon state and energetically does not correspond to its description as being made up of two triplets. These results are in agreement with the general trends in linear conjugated polymers. However, unlike in linear polyenes wherein the two-photon state is a localized excitation, we find that in polyacenes, the two-photon excitation is spread out over the system. We have doped the systems with a hole and an electron and have calculated the charge excitation gap. Using the charge gap and the optical gap, we estimate the binding energy of the ${1}^{1}{B}^{\ensuremath{-}}$ exciton to be 2.09 eV. We have also studied doubly doped polyacenes and find that the bipolaron in these systems, to be composed of two separated polarons, as indicated by the calculated charge-density profile and charge-charge correlation function. We have studied bond orders in various states in order to get an idea of the excited state geometry of the system. We find that the ground state, the triplet state, the dipole allowed state, and the polaron excitations correspond to lengthening of the rung bonds in the interior of the oligomer while the two-photon excitation corresponds to the rung bond lengths having two maxima in the system.
We study the elementary excitations of a model Hamiltonian for the $\ensuremath{\pi}$ electrons in polydiacetylene chains. In these materials, the bare band gap is only half the size of the observed single-particle gap and the binding energy of the exciton of 0.5 eV amounts to 20% of the single-particle gap. Therefore, exchange and correlations due to the long-range Coulomb interaction require a numerically exact treatment which we carry out using the density-matrix renormalization group method. Employing both the Hubbard-Ohno potential and the screened potential in one dimension, we reproduce the experimental results for the binding energy of the singlet exciton and its polarizability. Our results indicate that there are optically dark states below the singlet exciton, in agreement with experiment. In addition, we find a weakly bound second exciton with a binding energy of 0.1 eV. The energies in the triplet sector do not match the experimental data quantitatively, probably because we do not include polaronic relaxation effects.
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.
We study theoretically polydiacetylene chains diluted in their monomer matrix. We employ the density matrix renormalization group method on finite chains to calculate the ground state and low-lying excitations of the corresponding Peierls--Hubbard-Ohno Hamiltonian which is characterized by the electron transfer amplitude t0{t}_{0}t0​ between nearest neighbors, by the electron-phonon coupling constant $\ensuremath{\alpha}$, by the Hubbard interaction UUU, and by the long-range interaction VVV. We treat the lattice relaxation in the adiabatic limit, i.e., we calculate the polaronic lattice distortions for each excited state. Using chains with up to 102 lattice sites, we can safely perform the extrapolation to the thermodynamic limit for the ground-state energy and conformation, the single-particle gap, and the energies of the singlet exciton, the triplet ground state, and the optical excitation of the triplet ground state. The corresponding gaps are known with high precision from experiments. We determine a coherent parameter set $({t}_{0}^{*}=2.4\phantom{\rule{0.16em}{0ex}}\mathrm{eV},{\ensuremath{\alpha}}^{*}=3.4\phantom{\rule{0.16em}{0ex}}\mathrm{eV}/\text{\AA{}},{U}^{*}=6\phantom{\rule{0.16em}{0ex}}\mathrm{eV},{V}^{*}=3\phantom{\rule{0.16em}{0ex}}\mathrm{eV})$ from a fit of the experimental gap energies to the theoretical values which we obtain for 81 parameter points in the four-dimensional search space $({t}_{0},\ensuremath{\alpha},U,V)$. We identify dark in-gap states in the singlet and triplet sectors as seen in experiments. Using a fairly stiff spring constant, the length of our unit cell is about 1% larger than its experimental value.