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Wiley Interdisciplinary Reviews Computational Molecular Science

Time‐dependent density matrix renormalization group method for quantum dynamics in complex systems

Jiajun Ren, Weitang Li, Tong Jiang, Yuanheng Wang, Zhigang Shuai

Published Mar 21, 2022• 89 citations
Publisher

Abstract

Abstract The simulations of spectroscopy and quantum dynamics are of vital importance to the understanding of the electronic processes in complex systems, including the radiative/radiationless electronic relaxation relevant for optical emission, charge/energy transfer in molecular aggregates related to carrier mobility in organic materials, as well as photovoltaic and thermoelectric conversion, light‐harvesting and spin transport, and so forth. In recent years, time‐dependent density matrix renormalization group (TD‐DMRG) has emerged as a general, numerically accurate and efficient method for high‐dimensional full‐quantum dynamics. This review will cover the fundamental algorithms of TD‐DMRG in the modern framework of matrix product states (MPS) and matrix product operators (MPO), including the basic algebra with respect to MPS and MPO, the novel time evolution schemes to propagate MPS, and the automated MPO construction algorithm to encode generic Hamiltonian. Most importantly, the proposed method can handle the mixed state density matrix at finite temperature, enabling quantum statistical description for molecular aggregates. We demonstrate the performance of TD‐DMRG by benchmarking with the current state‐of‐the‐art methods for simulating quantum dynamics of the spin‐boson model and the Frenkel–Holstein(–Peierls) model. As applications of TD‐DMRG to real‐world problems, we present theoretical investigations of carrier mobility and spectral function of rubrene crystal, and the radiationless decay rate of azulene with an anharmonic potential energy surface. This article is categorized under: Theoretical and Physical Chemistry > Statistical Mechanics Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Software > Simulation Methods

DOI
10.1002/wcms.1614

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Abstract The simulations of spectroscopy and quantum dynamics are of vital importance to the understanding of the electronic processes in complex systems, including the radiative/radiationless electronic relaxation relevant for optical emission, charge/energy transfer in molecular aggregates related to carrier mobility in organic materials, as well as photovoltaic and thermoelectric conversion, light‐harvesting and spin transport, and so forth. In recent years, time‐dependent density matrix renormalization group (TD‐DMRG) has emerged as a general, numerically accurate and efficient method for high‐dimensional full‐quantum dynamics. This review will cover the fundamental algorithms of TD‐DMRG in the modern framework of matrix product states (MPS) and matrix product operators (MPO), including the basic algebra with respect to MPS and MPO, the novel time evolution schemes to propagate MPS, and the automated MPO construction algorithm to encode generic Hamiltonian. Most importantly, the proposed method can handle the mixed state density matrix at finite temperature, enabling quantum statistical description for molecular aggregates. We demonstrate the performance of TD‐DMRG by benchmarking with the current state‐of‐the‐art methods for simulating quantum dynamics of the spin‐boson model and the Frenkel–Holstein(–Peierls) model. As applications of TD‐DMRG to real‐world problems, we present theoretical investigations of carrier mobility and spectral function of rubrene crystal, and the radiationless decay rate of azulene with an anharmonic potential energy surface. This article is categorized under: Theoretical and Physical Chemistry > Statistical Mechanics Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Software > Simulation Methods
Recent advances in experimental techniques have enabled precise characterization of fundamental nonadiabatic and even electronic dynamics in molecules, on ultrafast time scales reaching femtosecond to attosecond resolution. However, accurate theoretical simulation of these ultrafast chemical dynamics processes in large systems remains challenging, largely due to the overwhelming number of degrees of freedom (DoFs) and the pronounced many-body correlations. In recent years, by leveraging efficient decomposition schemes for high-dimensional wave function and operator tensors, the time-dependent density matrix renormalization group (TD-DMRG) has emerged as a powerful and accurate quantum dynamics method for simulating nonadiabatic and electron dynamics in large chemical systems, in conjunction with realistic electron/exciton-vibration/phonon models or ab initio quantum chemistry many-electron Hamiltonians. This review outlines the fundamentals of TD-DMRG for chemical dynamics, covering matrix product state/operator (MPS/MPO) frameworks and algorithms from ground-state calculations to time evolution. We discuss thermal/environmental effects and compare TD-DMRG with other tensor network methods such as multiconfiguration time-dependent Hartree (MCTDH) and multilayer MCTDH (ML-MCTDH). Demonstrated applications include simulations of pyrazine absorption, singlet fission in rubrene crystal, and charge migration in chloroacetylene cation. These show TD-DMRG’s capability for modeling complex ultrafast processes from femtoseconds to attoseconds with controlled accuracy.
Matrix product state (MPS) is one efficient way to compress quantum states and reduce computational cost. This chapter presents the application of MPS framework as well as time-dependent density matrix renormalization group (tDMRG) variants in chemical systems. The elementary principles of MPS approaches are provided, including the basic concept of MPS and matrix product operator, the density matrix renormalization group method and tDMRG approaches based on the MPS/matrix product operator framework to solve the time-independent and dependent Schrödinger equation, respectively. Then, several examples of tDMRG applications in chemical systems investigated by the authors, including charge carrier dynamics in polymer chain, charge transfer and dissociation in organic systems, nonadiabatic dynamics of excited states S1/S2 in pyrazine molecule and singlet fission processes in organic systems, are displayed.
In this work, we simulate the electron dynamics in molecular systems with the time-dependent density matrix renormalization group (TD-DMRG) algorithm. We leverage the generality of the so-called tangent-space TD-DMRG formulation and design a computational framework in which the dynamics is driven by the exact nonrelativistic electronic Hamiltonian. We show that by parametrizing the wave function as a matrix product state, we can accurately simulate the dynamics of systems including up to 20 electrons and 32 orbitals. We apply the TD-DMRG algorithm to three problems that are hardly targeted by time-independent methods: the calculation of molecular (hyper)polarizabilities, the simulation of electronic absorption spectra, and the study of ultrafast ionization dynamics.
The density matrix renormalization group (DMRG) has become a powerful numerical method that can be applied to low-dimensional strongly correlated fermionic and bosonic systems. It allows for a very precise calculation of static, dynamic and thermodynamic properties. Its field of applicability has now extended beyond condensed matter, and is successfully used in quantum chemistry, statistical mechanics, quantum information theory, and nuclear and high-energy physics as well. In this article, we briefly review the main aspects of the method and present some of the most relevant applications so as to give an overview of the scope and possibilities of DMRG. We focus on the most important extensions of the method such as the calculation of dynamical properties, the application to classical systems, finite-temperature simulations, phonons and disorder, field theory, time-dependent properties and the ab initio calculation of electronic states in molecules. The recent quantum information interpretation, the development of highly accurate time-dependent algorithms and the possibility of using the DMRG as the impurity-solver of the dynamical mean field method (DMFT) give new insights into its present and potential uses. We review the numerous very recent applications of these techniques where the DMRG has shown to be one of the most reliable and versatile methods in modern computational physics.
Dynamical electronic- and vibrational-structure theories have received a growing interest in the past few years due to their ability to simulate spectra recorded with ultrafast experimental techniques. The exact time evolution of a molecular system can, in principle, be obtained from the time-dependent version of full configuration interaction. Such an approach is, however, limited to few-atom systems due to the exponential increase of its cost with the system dimension. In the present work, we overcome this unfavorable scaling by employing the time-dependent density matrix renormalization group (TD-DMRG) which parametrizes the time-dependent wave function as a matrix product state. The time-dependent Schrödinger equation is then integrated with a sweep-based algorithm, as in standard time-independent DMRG. Unlike other TD-DMRG approaches, the one presented here leads to a set of coupled equations that can be integrated exactly. The resulting theory enables us to study real- and imaginary-time evolutions of Hamiltonians comprising more than 20 degrees of freedom that are challenging for current state-of-the-art quantum dynamics algorithms. We apply our algorithm to the simulation of quantum dynamics of models of increasing complexity, ranging from simple excitonic Hamiltonians to more complex ab initio vibronic ones.
We implement and apply time-dependent density matrix renormalization group (TD-DMRG) algorithms at zero and finite temperature to compute the linear absorption and fluorescence spectra of molecular aggregates. Our implementation is within a matrix product state/operator framework with an explicit treatment of the excitonic and vibrational degrees of freedom, and it uses the locality of the Hamiltonian in the zero-exciton space to improve the efficiency and accuracy of the calculations. We demonstrate the power of the method by calculations on several molecular aggregate models, comparing our results against those from multilayer multiconfiguration time-dependent Hartree and n-particle approximations. We find that TD-DMRG provides an accurate and efficient route to calculate the spectrum of molecular aggregates.