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International Journal of Quantum Chemistry

On the importance of orbital localization in QC‐DMRG calculations

Alexander O. Mitrushchenkov, G. Fano, Roberto Linguerri, Paolo Palmieri

Published Dec 29, 2011• 26 citations
Publisher

Abstract

Abstract We investigate the importance of orbital localization in the application of the Density Matrix Renormalization Group (DMRG) technique to ab initio studies of molecular electronic structure. Our previous implementation of DMRG has been generalized to allow for the use of localized nonorthogonal orbitals. Simple cycles of equidistant hydrogen atoms, which are good examples of one dimensional metal‐like lattices with fully delocalized electronic structures, have been taken as test models. In this study, the efficiency of the DMRG method and the importance of orbital localization for the generation of the DMRG building blocks are confirmed. However, it is found that the convergence of the procedure based on nonorthogonal orbitals is slower and requires more DMRG components than the standard orthogonal formulation. Symmetrically orthonormalized atomic orbitals are shown to be a good compromise solution: they satisfy the requirement of orbital localization for the generation of the DMRG blocks and improve the convergence, reducing the number of components of the DMRG expansion. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

DOI
10.1002/qua.23173

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Abstract We investigate the importance of orbital localization in the application of the Density Matrix Renormalization Group (DMRG) technique to ab initio studies of molecular electronic structure. Our previous implementation of DMRG has been generalized to allow for the use of localized nonorthogonal orbitals. Simple cycles of equidistant hydrogen atoms, which are good examples of one dimensional metal‐like lattices with fully delocalized electronic structures, have been taken as test models. In this study, the efficiency of the DMRG method and the importance of orbital localization for the generation of the DMRG building blocks are confirmed. However, it is found that the convergence of the procedure based on nonorthogonal orbitals is slower and requires more DMRG components than the standard orthogonal formulation. Symmetrically orthonormalized atomic orbitals are shown to be a good compromise solution: they satisfy the requirement of orbital localization for the generation of the DMRG blocks and improve the convergence, reducing the number of components of the DMRG expansion. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012
We describe the theory and implementation of two extensions to the density-matrix renormalization-group (DMRG) algorithm in quantum chemistry: (i) to work with an underlying nonorthogonal one-particle basis (using a biorthogonal formulation) and (ii) to use non-Hermitian and complex operators and complex wave functions, which occur naturally in biorthogonal formulations. Using these developments, we carry out ground-state calculations on ethene, butadiene, and hexatriene, in a polarized atomic-orbital basis. The description of correlation in these systems using a localized nonorthogonal basis is improved over molecular-orbital DMRG calculations, and comparable to or better than coupled-cluster calculations, although we encountered numerical problems associated with non-Hermiticity. We believe that the non-Hermitian DMRG algorithm may further become useful in conjunction with other non-Hermitian Hamiltonians, for example, similarity-transformed coupled-cluster Hamiltonians.
The application of the quantum-chemical density-matrix renormalization group (DMRG) algorithm is cumbersome for complex electronic structures with many active orbitals. The high computational cost is mainly due to the poor convergence of standard DMRG calculations. A factor which affects the convergence behavior of the calculations is the choice of the start-up procedure. In this start-up step matrix representations of operators have to be calculated in a guessed many-electron basis of the DMRG environment block. Different possibilities for the construction of these basis states exist, and we first compare four procedures to approximate the environment states using Slater determinants explicitly. These start-up procedures are applied to DMRG calculations on a sophisticated test system: the chromium dimer. It is found that the converged energies and the rate of convergence depend significantly on the choice of the start-up procedure. However, since already the most simple start-up procedure, which uses only the Hartree-Fock determinant, is comparatively good, Slater determinants, in general, appear not to be a good choice as approximate environment basis states for convergence acceleration. Based on extensive test calculations it is demonstrated that the computational cost can be significantly reduced if the number of total states m is successively increased. This is done in such a way that the environment states are built up stepwise from system states of previous truncated DMRG sweeps for slowly increasing m values.
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.
We have compared different strategies for ab initio quantum chemistry density matrix renormalization group treatments. The two starting orbital blocks include all valence and active orbitals of the reference complete active space self consistent field wave function. To generate the remaining blocks, we propose following the order of the contributions to the correlation energy: a posteriori using approximate occupation numbers or a priori, using a Møller–Plesset type of arguments, by explicit evaluation of second-order interactions. We have compared two different schemes for orbital localization to identify the important and less important orbital interactions and simplify the generation of the orbital blocks. To truncate the expansion we have compared two approaches, keeping constant the number m of components or the threshold λ to fix the residue of the expansion at each step. The extrapolation of the energies is found to provide accurate estimates of the full configuration interaction energy, making the expansion independent on the actual values of the two parameters m and λ. We propose to generate the factors for the two blocks from ground and excited eigenvectors of the Hamiltonian matrix.
The density-matrix renormalization group algorithm has emerged as a promising new method in ab initio quantum chemistry. However, many problems still need to be solved before this method can be applied routinely. At the start of such a calculation, the orbitals originating from a preceding quantum chemical calculation must be placed in a specific order on a one-dimensional lattice. This ordering affects the convergence of the density-matrix renormalization group iterations significantly. In this paper, we present two approaches to obtain optimized orderings of the orbitals. First, we use a genetic algorithm to optimize the ordering with respect to a low total electronic energy obtained at a predefined stage of the density-matrix renormalization group algorithm with a given number of total states kept. In addition to that, we derive orderings from the one- and two-electron integrals of our test system. This test molecule is the chromium dimer, which is known to possess a complicated electronic structure. For this molecule, we have carried out calculations for the various orbital orderings obtained. The convergence behavior of the density-matrix renormalization group iterations is discussed in detail.
We derive an automatic procedure for generating a set of highly localized, nonorthogonal orbitals for linear scaling quantum Monte Carlo (QMC) calculations. We demonstrate the advantage of these orbitals for calculating the total energy of both semiconducting and metallic systems by studying bulk silicon and the homogeneous electron gas. For silicon, the improved localization of these orbitals reduces the computational time by a factor of 5 and the memory by a factor of 6 compared to localized, orthogonal orbitals. For jellium at typical metallic densities, we demonstrate that the total energy is converged to 3 meV per electron for orbitals truncated within spheres with radii 7rs7{r}_{s}7rs​, opening the possibility of linear scaling QMC calculations for realistic metallic systems.