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IMPERIAL COLLEGE PRESS eBooks

Solving Problems with Strong Correlation Using the Density Matrix Renormalization Group (DMRG)

Garnet Kin‐Lic Chan, Sandeep Sharma

Published Aug 1, 2011• 2 citations
Publisher

Abstract

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.

DOI
10.1142/9781848167254_0003

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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.
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.
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.
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.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
This chapter starts with an introductory survey on the physical background and historical events that lead to the emergence of the density matrix renormalization group (DMRG) and its tensor network generalization. We then briefly overview the major progress on the renormalization group methods of tensor networks and their applications in the past three decades. The tensor network renormalization was initially developed to solve quantum many-body problems, but its application field has grown constantly. It has now become an irreplaceable tool for investigating strongly correlated problems, statistical physics, quantum information, quantum chemistry, and artificial intelligence.