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Zeitschrift für Physikalische Chemie

The Density Matrix Renormalization Group Algorithm in Quantum Chemistry

Konrad Heinrich Marti, Markus Reiher

Published Apr 1, 2010• 210 citations
Publisher

Abstract

Abstract In this work, we derive the density matrix renormalization group (DMRG) algorithm in the language of configuration interaction. Furthermore, the development of DMRG in quantum chemistry is reviewed and DMRG-specific peculiarities are discussed. Finally, we discuss new results for a dinuclear μ-oxo bridged copper cluster, which is an important active-site structure in transition-metal chemistry, an area in which we pioneered the application of DMRG.

DOI
10.1524/zpch.2010.6125

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Abstract In this work, we derive the density matrix renormalization group (DMRG) algorithm in the language of configuration interaction. Furthermore, the development of DMRG in quantum chemistry is reviewed and DMRG-specific peculiarities are discussed. Finally, we discuss new results for a dinuclear μ-oxo bridged copper cluster, which is an important active-site structure in transition-metal chemistry, an area in which we pioneered the application of DMRG.
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The ab-initio density matrix renormalization group (DMRG) is a tool that can be applied to a wide variety of interesting problems in quantum chemistry. Here, we examine the density matrix renormalization group from the vantage point of the quantum chemistry user. What kinds of problems is the DMRG well-suited to? What are the largest systems that can be treated at practical cost? What sort of accuracies can be obtained, and how do we reason about the computational difficulty in different molecules? By examining a diverse benchmark set of molecules: π-electron systems, benchmark main-group and transition metal dimers, and the Mn-oxo-salen and Fe-porphine organometallic compounds, we provide some answers to these questions, and show how the density matrix renormalization group is used in practice.
In the past two decades, the density matrix renormalization group (DMRG) has emerged as an innovative new method in quantum chemistry relying on a theoretical framework very different from that of traditional electronic structure approaches. The development of the quantum chemical DMRG has been remarkably fast: it has already become one of the reference approaches for large-scale multiconfigurational calculations. This perspective discusses the major features of DMRG, highlighting its strengths and weaknesses also in comparison with other novel approaches. The method is presented following its historical development, starting from its original formulation up to its most recent applications. Possible routes to recover dynamical correlation are discussed in detail. Emerging new fields of applications of DMRG are explored, such as its time-dependent formulation and the application to vibrational spectroscopy.
Abstract. The Density Matrix Renormalization Group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of onedimensional quantum lattices. The method, as originally introduced, was based on the iterative inclusion of sites on a real-space lattice. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be modified for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the subject, ranging from efforts to clarify the optimal means of implementing the algorithm to extensive applications in a variety of fields. In this article, we review these recent developments. Following a description of the realspace DMRG method, we discuss the key steps that were undertaken to modify it for use on finite Fermi systems and then describe its applications to Quantum Chemistry, ultrasmall superconducting grains, finite nuclei and two-dimensional electron systems. We also describe a recent development which permits symmetries to be taken into account consistently throughout the DMRG algorithm. We close with an outlook for future applications of the method.
Despite the success of modern quantum chemistry in predicting properties of organic molecules, the treatment of inorganic systems, which have many close lying states, remains out of quantitative reach for current methods. To treat non-dynamic correlation, we take advantage of the density matrix renormalization group (DMRG) method that has become very successful in the field of solid state physics. We present a detailed study of the DMRG method, and we pay special attention to the evolution of the understanding behind the mathematical structure of the DMRG wave function. Our primary goal is to develop a density matrix renormalization group self--consistent--field (DMRG-SCF) approach, analogous to the complete active space self--consistent field (CASSCF) method, but dealing with large active spaces that are too demanding for the full configuration interaction (FCI) method. \nAs a first step towards such a DMRG-SCF procedure, we present a spin-adapted DMRG algorithm designed to target spin- and spatial-symmetry states that are hard to obtain while using an unrestricted algorithm. \nOur next step is a modification of the DMRG algorithm to obtain decreasing energy at every step during the sweep. This monotonically convergent DMRG scheme lets us obtain the two-body density matrix as a by--product of the existing procedure without any additional cost in storage. Additionally, the two-body density matrix produced at convergence using this scheme is free from the N-representability problem that is present when the two--body density matrix is produced with the two-site DMRG scheme without additional storage cost. Finally, taking advantage of the modifications developed herein, we present results obtained from our DMRG-SCF method. Lastly, we discuss possible ways of incorporating dynamical correlation into the DMRG scheme, in order to obtain a modern multireference approach.