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DMRG for multicomponent quantum chemistry

Mikuláš Matoušek

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

One of the more modern directions in quantum chemistry, which is mostly unexplored, is using Hamiltonians which include not only the electrons of the molecule, but also other quantum particles. These are also termed multicomponent Hamiltonians. The Density Matrix Renormalization Group method has become one of the mainline methods in multireference quantum chemistry. In this work we describe a way to extend the DMRG algorithm to use multicomponent Hamiltonians. We start by a description of the DMRG algorithm, showcasing the usage of quantum chemical DMRG on multiple challenging systems. We include a brief description how to extend the DMRG algorithm to also treat dynamical correlation with the Adiabatic Connection method. We then describe the extension of DMRG to multicomponent Hamiltonians which has been implemented by the author of this work, and finally show how DMRG can be useful for polaritonic chemistry, where the molecule interacts with the quantized states of light in an optical cavity.

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One of the more modern directions in quantum chemistry, which is mostly unexplored, is using Hamiltonians which include not only the electrons of the molecule, but also other quantum particles. These are also termed multicomponent Hamiltonians. The Density Matrix Renormalization Group method has become one of the mainline methods in multireference quantum chemistry. In this work we describe a way to extend the DMRG algorithm to use multicomponent Hamiltonians. We start by a description of the DMRG algorithm, showcasing the usage of quantum chemical DMRG on multiple challenging systems. We include a brief description how to extend the DMRG algorithm to also treat dynamical correlation with the Adiabatic Connection method. We then describe the extension of DMRG to multicomponent Hamiltonians which has been implemented by the author of this work, and finally show how DMRG can be useful for polaritonic chemistry, where the molecule interacts with the quantized states of light in an optical cavity.
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.
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.
Recent advances in high-resolution spectroscopy and cavity quantum technologies have driven growing interest in multicomponent quantum chemistry, where electronic degrees of freedom are treated on an equal footing with other quantum particles. Here, we present a unified symmetry-aware density matrix renormalization group (DMRG) framework for multicomponent quantum chemistry. By decoupling physical model specification from tensor-network representation and numerical optimization, new multicomponent models are introduced through symmetry quantum numbers, local basis, and operators, without modifying the underlying tensor algorithms. Built upon a mature tensor infrastructure, the framework provides a reusable symmetry-aware numerical engine, allowing new models to be implemented through concise Python-level specifications while fully reusing the optimized computational backend. The framework is demonstrated through symmetry-adapted nuclear-electronic orbital (NEO) and cavity quantum electrodynamics (CQED) DMRG calculations, yielding accurate electronic structures, excited-state energies, and polaritonic properties. This work establishes a reusable and extensible computational framework for symmetry-aware multicomponent quantum chemistry.
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) method has been well-established and has become one of the most accurate numerical methods for the precise electronic structure solution of large active spaces. In the past few years, to capture the missing dynamic correlation, various post-DMRG approaches have been proposed through the combination of DMRG and multireference quantum chemical methods or density functional theory. With this in mind, this work provides a brief overview of ab initio DMRG principles and the new developments within post-DMRG methods. For clarity, post-DMRG methods are classified into two main categories depending on whether high-order n -electron reduced density matrices are used, and their merits and disadvantages are properly discussed. Finally, we conclude by discussing unsolved bottlenecks and giving development perspectives of post-DMRG approaches, which are expected to yield quantitative descriptions of complex electronic structures in large strongly correlated molecules and materials.
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.