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Physical review. B./Physical review. B

Symmetrized density matrix renormalization group algorithm for low-lying excited states of conjugated carbon systems: Application to 1,12-benzoperylene and polychrysene

Suryoday Prodhan, S. Ramasesha

Published May 14, 2018• 12 citations
Publisher

Abstract

The symmetry adapted density matrix renormalization group (SDMRG) technique has been an efficient method for studying low-lying eigenstates in one- and quasi-one-dimensional electronic systems. However, the SDMRG method had bottlenecks involving the construction of linearly independent symmetry adapted basis states as the symmetry matrices in the DMRG basis were not sparse. We have developed a modified algorithm to overcome this bottleneck. The new method incorporates end-to-end interchange symmetry $({C}_{2})$, electron-hole symmetry $(J)$, and parity or spin-flip symmetry $(P)$ in these calculations. The one-to-one correspondence between direct-product basis states in the DMRG Hilbert space for these symmetry operations renders the symmetry matrices in the new basis with maximum sparseness, just one nonzero matrix element per row. Using methods similar to those employed in the exact diagonalization technique for Pariser-Parr-Pople (PPP) models, developed in the 1980s, it is possible to construct orthogonal SDMRG basis states while bypassing the slow step of the Gram-Schmidt orthonormalization procedure. The method together with the PPP model which incorporates long-range electronic correlations is employed to study the correlated excited-state spectra of 1,12-benzoperylene and a narrow mixed graphene nanoribbon with a chrysene molecule as the building unit, comprising both zigzag and cove-edge structures.

DOI
10.1103/physrevb.97.195125

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The symmetry adapted density matrix renormalization group (SDMRG) technique has been an efficient method for studying low-lying eigenstates in one- and quasi-one-dimensional electronic systems. However, the SDMRG method had bottlenecks involving the construction of linearly independent symmetry adapted basis states as the symmetry matrices in the DMRG basis were not sparse. We have developed a modified algorithm to overcome this bottleneck. The new method incorporates end-to-end interchange symmetry (C2)({C}_{2})(C2​), electron-hole symmetry (J)(J)(J), and parity or spin-flip symmetry (P)(P)(P) in these calculations. The one-to-one correspondence between direct-product basis states in the DMRG Hilbert space for these symmetry operations renders the symmetry matrices in the new basis with maximum sparseness, just one nonzero matrix element per row. Using methods similar to those employed in the exact diagonalization technique for Pariser-Parr-Pople (PPP) models, developed in the 1980s, it is possible to construct orthogonal SDMRG basis states while bypassing the slow step of the Gram-Schmidt orthonormalization procedure. The method together with the PPP model which incorporates long-range electronic correlations is employed to study the correlated excited-state spectra of 1,12-benzoperylene and a narrow mixed graphene nanoribbon with a chrysene molecule as the building unit, comprising both zigzag and cove-edge structures.
One-dimensional and quasi-one-dimensional correlated fermionic models can be studied efficiently using the Density Matrix Renormalization Group (DMRG) method with open boundary conditions. Although the implementation of the conventional DMRG technique in investigating a fermionic model with a periodic boundary condition (PBC) is still challenging due to the demand of high computational facility, this work reports the efficient use of the symmetrized DMRG (SDMRG) technique in studying the low-lying correlated excited-states of radially π-conjugated cycloparaphenylene ([6]CPP) and [n]cyclacene molecules within the Pariser-Parr-Pople (PPP) model Hamiltonian with adequate computational cost. The low-lying correlated singlet excited energies of [6]CPP calculated within the PPP model are in very good agreement with experiment. Compared to [6]CPP, the numerical accuracy of DMRG calculations for highly correlated [n]cyclacene depends on the number of DMEV basis (m). Such a study shows an efficient pathway to implement the SDMRG technique in studying fermionic systems with PBC in the future.
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.
Electronic excited state calculation consists of an important issue in theoretical and computational chemistry. Under the Pariser-Parr-Pople (PPP) model and taking polyacenes and polyenes as examples, we compared the accuracy and efficiency of the state specific method and the state average method for the excited state calculation in the density matrix renormalization group formalism. We found that the state specific method can correctly trace the low-lying excited states and can provide more accurate eigen energies than the state average method. Nevertheless, for higher-lying excited states, the state specific method could converge to wrong eigenstates, which should be highly cautioned.
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.
We extend the density matrix renormalization group method to exploit Parity, C2C_2C2​ (rotation by πππ) and electron-hole symmtries of model Hamiltonians. We demonstrate the power of this method by obtaining the lowest energy states in all the eight symmetry subspaces of Hubbard chains with upto 50 sites. The ground-state energy, optical gap and spin gap of regular U=4tU=4tU=4t and U=6tU=6tU=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.