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arXiv:cond-mat/9803071 · The Journal of Chemical Physics

The density matrix renormalization group method: Application to the PPP model of a cyclic polyene chain

G. Fano, F. Ortolani, L. Ziosi

Published Jun 8, 1998• 79 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.476379

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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 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.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
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.
An extension of the density-matrix renormalization-group (DMRG) method is presented. Besides the two groups or classes of block states considered in White's formulation, the retained m states, and the neglected ones, we introduce an intermediate group of block states having the following p largest eigenvalues ${\ensuremath{\lambda}}_{i}$ of the reduced density matrix: ${\ensuremath{\lambda}}_{1}>~\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}>~{\ensuremath{\lambda}}_{m}>~{\ensuremath{\lambda}}_{m+1}>~\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}>~{\ensuremath{\lambda}}_{m+p}.$ These states are taken into account when they contribute to intrablock transitions but are neglected when they participate in more delocalized interblock fluctuations. Applications to one-dimensional models (Heisenberg, Hubbard, and dimerized tight binding) show that in this way the involved computer resources can be reduced without significant loss of accuracy. The efficiency and accuracy of the method is analyzed by varying m and p and by comparison with standard DMRG calculations. A Hamiltonian-independent scheme for choosing m and p and for extrapolating to the limit where m and p are infinite is provided. Finally, an extension of the 3-classes approach is outlined, which incorporates the fluctuations between the p states of different blocks as a self-consistent dressing of the block interactions among the retained m states.
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.
The similarities between Hartree-Fock (HF) theory and the density matrix renormalization group (DMRG) are explored. Both methods can be formulated as the variational optimization of a wave-function Ansatz. Linearization of the time-dependent variational principle near a variational minimum allows to derive the random phase approximation (RPA). We show that the nonredundant parameterization of the matrix product state (MPS) tangent space [J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pi\ifmmode \check{z}\else \v{z}\fi{}orn, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for MPS, i.e., an explicit nonredundant parameterization of the entire MPS manifold, starting from a specific MPS reference. Excitation operators are identified, which extends the analogy between HF and DMRG to the Tamm-Dancoff approximation (TDA), the configuration interaction (CI) expansion, and coupled cluster theory. For a small one-dimensional Hubbard chain, we use a CI-MPS Ansatz with single and double excitations to improve on the ground state and to calculate low-lying excitation energies. For a symmetry-broken ground state of this model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also discuss calculations of the RPA-MPS correlation energy. With the long-range quantum chemical Pariser-Parr-Pople Hamiltonian, low-lying TDA-MPS and RPA-MPS excitation energies for polyenes are obtained.