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arXiv:1303.0616 · The Journal of Chemical Physics

Assessment of various natural orbitals as the basis of large active space density-matrix renormalization group calculations

Yingjin Ma, Haibo Ma

Published Jun 12, 2013• 46 citations
arXivHTMLPDF

Abstract

It is well-known that not only the orbital ordering but also the choice of the orbitals itself as the basis may significantly influence the computational efficiency of density-matrix renormalization group (DMRG) calculations. In this study, for assessing the efficiency of using various natural orbitals (NOs) as the DMRG basis, we performed benchmark DMRG calculations with different bases, which included the NOs obtained by various traditional electron correlation methods, as well as NOs acquired from preliminary moderate DMRG calculations (e.g., preserved states less than 500). The tested systems included N2, transition metal Cr2 systems, as well as 1D hydrogen polyradical chain systems under equilibrium and dissociation conditions and 2D hydrogen aggregates. The results indicate that a good compromise between the requirement for low computational costs of acquiring NOs and the demand for high efficiency of NOs as the basis of DMRG calculations may be very dependent on the studied systems' diverse electron correlation characteristics and the size of the active space. It is also shown that a DMRG-complete active space configuration interaction (DMRG-CASCI) calculation in a basis of carefully chosen NOs can provide a less expensive alternative to the standard DMRG-complete active space self-consistent field (DMRG-CASSCF) calculation and avoid the convergence difficulties of orbital optimization for large active spaces. The effect of different NO ordering schemes on DMRG-CASCI calculations is also discussed.

DOI
10.1063/1.4809682

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It is well-known that not only the orbital ordering but also the choice of the orbitals itself as the basis may significantly influence the computational efficiency of density-matrix renormalization group (DMRG) calculations. In this study, for assessing the efficiency of using various natural orbitals (NOs) as the DMRG basis, we performed benchmark DMRG calculations with different bases, which included the NOs obtained by various traditional electron correlation methods, as well as NOs acquired from preliminary moderate DMRG calculations (e.g., preserved states less than 500). The tested systems included N2, transition metal Cr2 systems, as well as 1D hydrogen polyradical chain systems under equilibrium and dissociation conditions and 2D hydrogen aggregates. The results indicate that a good compromise between the requirement for low computational costs of acquiring NOs and the demand for high efficiency of NOs as the basis of DMRG calculations may be very dependent on the studied systems' diverse electron correlation characteristics and the size of the active space. It is also shown that a DMRG-complete active space configuration interaction (DMRG-CASCI) calculation in a basis of carefully chosen NOs can provide a less expensive alternative to the standard DMRG-complete active space self-consistent field (DMRG-CASSCF) calculation and avoid the convergence difficulties of orbital optimization for large active spaces. The effect of different NO ordering schemes on DMRG-CASCI calculations is also discussed.
It is well-known that not only the orbital ordering but also the choice of the orbitals themselves as the basis may significantly influence the computational efficiency of density-matrix renormalization group (DMRG) calculations. In this study, for assessing the efficiency of using various natural orbitals (NOs) as the DMRG basis, we performed benchmark DMRG calculations with different bases, which included the NOs obtained by various traditional electron correlation methods, as well as NOs acquired from preliminary moderate DMRG calculations (e.g., preserved states less than 500). The tested systems included N2_22​, transition metal Cr2_22​ systems, as well as 1-D hydrogen polyradical chain systems under equilibrium and dissociation conditions and 2-D hydrogen aggregates. The results indicate that a good compromise between the requirement for low computational costs of acquiring NOs and the demand for high efficiency of NOs as the basis of DMRG calculations may be very dependent on the studied systems' diverse electron correlation characteristics and the size of the active space. It is also shown that a DMRG-complete active space configuration interaction (DMRG-CASCI) calculation in a basis of carefully chosen NOs can provide a less expensive alternative to the standard DMRG-CASSCF calculation and avoid the convergence difficulties of orbital optimization for large active spaces. The effect of different NO ordering schemes on DMRG-CASCI calculations is also discussed.
We present the density matrix renormalization group self-consistent field (DMRG-SCF) approach that is analogous to the complete active space self-consisted field (CASSCF) method but instead of using for the description of the active space the full configuration interaction (FCI) method, the DMRG-SCF uses the density matrix renormalization group (DMRG) method. The DMRG-SCF approach, similarly to CASSCF, properly describes the multiconfigurational character of the wave function but avoids the exponential scaling of the FCI method and replaces it with a polynomial scaling. Hence, calculations for a larger number of orbitals and electrons in the active space are possible since the DMRG method provides an efficient tool to automatically select from the full Hilbert space the many-body contracted basis states that are the most important for the description of the wave function.
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.
This paper describes an interface between the density matrix renormalization group (DMRG) method and the complete active-space self-consistent field (CASSCF) method and its analytical gradient, as well as an extension to the second-order perturbation theory (CASPT2) method. This interfacing allows large active-space multi-reference computations to be easily performed. The interface and its extension are both implemented in terms of reduced density matrices (RDMs) which can be efficiently computed via the DMRG sweep algorithm. We also present benchmark results showing that, in practice, the DMRG-CASSCF calculations scale with active-space size in a polynomial manner in the case of quasi-1D systems. Geometry optimization of a binuclear iron-sulfur cluster using the DMRG-CASSCF analytical gradient is demonstrated, indicating that the inclusion of the valence p-orbitals of sulfur and double-shell d-orbitals of iron lead to non-negligible changes in the geometry compared to the results of small active-space calculations. With the exception of the selection of M values, many computational settings in these practical DMRG calculations have been tuned and black-boxed in our interface, and so the resulting DMRG-CASSCF and DMRG-CASPT2 calculations are now available to novice users as a common tool to compute strongly correlated electronic wavefunctions.
The density-matrix renormalization group (DMRG) method, which can deal with a large active space composed of tens of orbitals, is nowadays widely used as an efficient addition to traditional complete active space (CAS)-based approaches. In this paper, we present the DMRG algorithm with a multi-level (ML) control of the active space based on chemical intuition-based hierarchical orbital ordering, which is called as ML-DMRG with its self-consistent field (SCF) variant ML-DMRG-SCF. Ground and excited state calculations of H2O, N2, indole, and Cr2 with comparisons to DMRG references using fixed number of kept states (M) illustrate that ML-type DMRG calculations can obtain noticeable efficiency gains. It is also shown that the orbital re-ordering based on hierarchical multiple active subspaces may be beneficial for reducing computational time for not only ML-DMRG calculations but also DMRG ones with fixed M values.
In previous work we have shown that the density matrix renormalization group (DMRG) enables near-exact calculations in active spaces much larger than are possible with traditional complete active space algorithms. Here, we implement orbital optimization with the DMRG to further allow the self-consistent improvement of the active orbitals, as is done in the complete active space self-consistent field (CASSCF) method. We use our resulting DMRG-CASSCF method to study the low-lying excited states of the all-trans polyenes up to C24H26 as well as beta-carotene, correlating with near-exact accuracy the optimized complete pi-valence space with up to 24 active electrons and orbitals, and analyze our results in the light of the recent discovery from resonance Raman experiments of new optically dark states in the spectrum.