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The Non-Abelian Density Matrix Renormalization Group Algorithm

Ian P. McCulloch, Miklós Gulácsi

Published Jan 1, 2000• 120 citations
OpenAlex

Abstract

Abstract. – We describe here the extension of the density matrix renormalization group algorithm to the case where the Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original density matrix renormalization group basis. We demonstrate the usefulness of the construction via the the one-dimensional Hubbard model as the symmetry group is enlarged from U(1) × U(1), up to SU(2) × SU(2). In past years, the density matrix renormalization group (DMRG) method [1] has been extensively used to study one and two dimensional strongly correlated electron systems [2]. This method became very popular when it was realized that it enabled a level of numerical accuracy for one dimensional systems that was not possible using other methods [3]. One major drawback of DMRG is that calculations are performed in a subspace of purely Abelian symmetries, such as the U(1) symmetries of total particle number and the z component of the total spin. Thus one can only obtain a few states in different total particle

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This article presents an efficient and parallelized implementation of the density matrix renormalization group (DMRG) algorithm for quantum chemistry calculations. The DMRG method as a large-scale multireference electronic structure model is by nature particularly efficient for one-dimensionally correlated systems, while the present development is oriented toward applications for polynuclear transition metal compounds, in which the macroscopic one-dimensional structure of electron correlation is absent. A straightforward extension of the DMRG algorithm is proposed with further improvements and aggressive optimizations to allow its application with large multireference active space, which is often demanded for metal compound calculations. Special efficiency is achieved by making better use of sparsity and symmetry in the operator and wave function representations. By accomplishing computationally intensive DMRG calculations, the authors have found that a large number of renormalized basis states are required to represent high entanglement of the electron correlation for metal compound applications, and it is crucial to adopt auxiliary perturbative correction to the projected density matrix during the DMRG sweep optimization in order to attain proper convergence to the solution. Potential energy curve calculations for the Cr(2) molecule near the known equilibrium precisely predicted the full configuration interaction energies with a correlation space of 24 electrons in 30 orbitals [denoted by (24e,30o)]. The energies are demonstrated to be accurate to 0.6mE(h) (the error from the extrapolated best value) when as many as 10,000 renormalized basis states are employed for the left and right DMRG block representations. The relative energy curves for [Cu(2)O(2)](2+) along the isomerization coordinate were obtained from DMRG and other correlated calculations, for which a fairly large orbital space (32e,62o) is modeled as a full correlation space. The DMRG prediction nearly overlaps with the energy curve from the coupled cluster with singles, doubles, and perturbative triple [CCSD(T)] calculations, while the multireference complete active space self-consistent field (CASSCF) calculations with the small reference configuration (8e,8o) are found to overestimate the biradical character of the electronic state of [Cu(2)O(2)](2+) according to the one-electron density matrix analysis.

The Journal of Chemical Physics•Jun 19, 2009•193 citations
Abstract. – We describe here the extension of the density matrix renormalization group algorithm to the case where the Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original density matrix renormalization group basis. We demonstrate the usefulness of the construction via the the one-dimensional Hubbard model as the symmetry group is enlarged from U(1) × U(1), up to SU(2) × SU(2). In past years, the density matrix renormalization group (DMRG) method [1] has been extensively used to study one and two dimensional strongly correlated electron systems [2]. This method became very popular when it was realized that it enabled a level of numerical accuracy for one dimensional systems that was not possible using other methods [3]. One major drawback of DMRG is that calculations are performed in a subspace of purely Abelian symmetries, such as the U(1) symmetries of total particle number and the z component of the total spin. Thus one can only obtain a few states in different total particle
We describe here the extension of the density matrix renormalization group algorithm to the case where the Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original density matrix renormalization group basis. We demonstrate the usefulness of the construction via the one-dimensional Hubbard model as the symmetry group is enlarged from U (1) × U (1), up to SU (2) × SU (2).
In this paper we give an introduction to the numerical density matrix renormalization group (DMRG) algorithm, from the perspective of the more general matrix product state (MPS) formulation. We cover in detail the differences between the original DMRG formulation and the MPS approach, demonstrating the additional flexibility that arises from constructing both the wavefunction and the Hamiltonian in MPS form. We also show how to make use of global symmetries, for both the Abelian and non-Abelian cases.
We extend our previous work [S. Sharma and G. K.-L. Chan, J. Chem. Phys. 136, 124121 (2012)], which described a spin-adapted (SU(2) symmetry) density matrix renormalization group algorithm, to additionally utilize general non-Abelian point group symmetries. A key strength of the present formulation is that the requisite tensor operators are not hard-coded for each symmetry group, but are instead generated on the fly using the appropriate Clebsch-Gordan coefficients. This allows our single implementation to easily enable (or disable) any non-Abelian point group symmetry (including SU(2) spin symmetry). We use our implementation to compute the ground state potential energy curve of the C2 dimer in the cc-pVQZ basis set (with a frozen-core), corresponding to a Hilbert space dimension of 10(12) many-body states. While our calculated energy lies within the 0.3 mEh error bound of previous initiator full configuration interaction quantum Monte Carlo and correlation energy extrapolation by intrinsic scaling calculations, our estimated residual error is only 0.01 mEh, much more accurate than these previous estimates. Due to the additional efficiency afforded by the algorithm, the excitation energies (Te) of eight lowest lying excited states: a(3)Πu, b(3)Σg (-), A(1)Πu, c(3)Σu (+), B(1)Δg, B(') (1)Σg (+), d(3)Πg, and C(1)Πg are calculated, which agree with experimentally derived values to better than 0.06 eV. In addition, we also compute the potential energy curves of twelve states: the three lowest levels for each of the irreducible representations (1)Σg (+), (1)Σu (+), (1)Σg (-), and (1)Σu (-), to an estimated accuracy of 0.1 mEh of the exact result in this basis.
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.
This article presents an efficient and parallelized implementation of the density matrix renormalization group (DMRG) algorithm for quantum chemistry calculations. The DMRG method as a large-scale multireference electronic structure model is by nature particularly efficient for one-dimensionally correlated systems, while the present development is oriented toward applications for polynuclear transition metal compounds, in which the macroscopic one-dimensional structure of electron correlation is absent. A straightforward extension of the DMRG algorithm is proposed with further improvements and aggressive optimizations to allow its application with large multireference active space, which is often demanded for metal compound calculations. Special efficiency is achieved by making better use of sparsity and symmetry in the operator and wave function representations. By accomplishing computationally intensive DMRG calculations, the authors have found that a large number of renormalized basis states are required to represent high entanglement of the electron correlation for metal compound applications, and it is crucial to adopt auxiliary perturbative correction to the projected density matrix during the DMRG sweep optimization in order to attain proper convergence to the solution. Potential energy curve calculations for the Cr(2) molecule near the known equilibrium precisely predicted the full configuration interaction energies with a correlation space of 24 electrons in 30 orbitals [denoted by (24e,30o)]. The energies are demonstrated to be accurate to 0.6mE(h) (the error from the extrapolated best value) when as many as 10,000 renormalized basis states are employed for the left and right DMRG block representations. The relative energy curves for [Cu(2)O(2)](2+) along the isomerization coordinate were obtained from DMRG and other correlated calculations, for which a fairly large orbital space (32e,62o) is modeled as a full correlation space. The DMRG prediction nearly overlaps with the energy curve from the coupled cluster with singles, doubles, and perturbative triple [CCSD(T)] calculations, while the multireference complete active space self-consistent field (CASSCF) calculations with the small reference configuration (8e,8o) are found to overestimate the biradical character of the electronic state of [Cu(2)O(2)](2+) according to the one-electron density matrix analysis.