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Annual Review of Physical Chemistry

The Density Matrix Renormalization Group in Quantum Chemistry

Garnet Kin‐Lic Chan, Sandeep Sharma

Published Mar 2, 2011• 819 citations
Publisher

Abstract

The density matrix renormalization group is a method that is useful for describing molecules that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the density matrix renormalization group works, a survey of its existing applications to molecular problems, and some thoughts on the future of the method.

DOI
10.1146/annurev-physchem-032210-103338

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The density matrix renormalization group is a method that is useful for describing molecules that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the density matrix renormalization group works, a survey of its existing applications to molecular problems, and some thoughts on the future of the method.
This chapter is concerned with the problem of strongly correlated electrons in quantum chemistry. We describe how a technique known as the density matrix renormalization group (DMRG) can tackle complicated chemical problems of strong correlation by capturing the local nature of the correlations. We analyse the matrix product state structure of the DMRG wavefunction that encodes one-dimensional aspects of locality. We also discuss the connection to the traditional ideas of the renormalization group. We finish with a survey of applications of the DMRG, its strengths and weaknesses in chemical applications, and its recent promising generalization to tensor network states.
A major challenge in molecular electronic structure theory arises from chemical systems where the mean-field picture based upon a single electronic configuration fails to properly describe mechanics of many electrons. Such complex electronic states should be accounted for using a correlated superposition of multiple determinants and are referred to as multireference in quantum chemistry. Application domain of traditional multireference methods is severely limited to systems containing a small number of correlated electrons because of their exponential complexity. In recent years, much attention has been drawn to the density matrix renormalization group (DMRG) theory as a promising alternative. It has been introduced in ab initio quantum chemistry calculations and shown to be a highly-scalable multireference method that can handle full quantum degrees of freedom for the strongly-correlated wavefunction with unprecedented large configuration space. In this article, we provide an explicative introduction to the underlying theory and formalism built into the DMRG method and give a brief overview of our methodological extensions that combine DMRG and active-space correlation models.
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.
This paper is an introduction to a density matrix renormalization group method (DMRG) and its use in quantum chemistry. Firstly we define the term of strongly correlated systems and explain why the standard methods of quantum chemistry are not suitable for use with such systems. Then, motivated by the principle of locality, we introduce the wave function approximation in the form of matrix product state (MPS), its graphical representation and calculation of expectation values of operators. In the next section, we present the canonical form of MPS expansion and show its relationship to the algorithm of the DMRG method as a form of optimization of matrices in MPS in the sense of variational minimization of energy. Then, a brief overview is given to the properties of DMRG method, with its computational complexity and the resulting limitations, followed by a list of methods for inclusion of the remaining dynamical correlation. Finally, we provide example calculations on realistic, chemically interesting molecules.
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.