Time-dependent density matrix renormalization group quantum dynamics for realistic chemical systems

ML-MCTDH-Aid: An auxiliary package for multilayer multiconfiguration time-dependent Hartree calculations

The multilayer-multiconfiguration time-dependent Hartree (ML-MCTDH) method has garnered significant attention in the realm of theoretical chemistry owing to its powerful ability to perform numerically exact descriptions of multi-dimensional quantum dynamics and exhibit the remarkable performance in simulating the nonadiabatic dynamics of complex systems. Despite the availability of computational packages within the ML-MCTDH framework, executing these calculations seamlessly is not a straightforward task. Typically, substantial efforts are necessitated to configure the correct inputs for ML-MCTDH calculations, which require to correctly define several non-trivial parameters, to reasonably setup the optimal tree expansion of wavefunctions, and to properly select basis function numbers. To address these challenges, we have developed an auxiliary package named ML-MCTDH-Aid, which facilitates the setup of ML-MCTDH calculations using the Heidelberg MCTDH package in a user-friendly manner. This package is primarily tailored to handle the high-dimensional nonadiabatic dynamics governed by the Hamiltonian composed of several electronic states, several vibrational modes and their linear vibronic coupling terms. It automatically generates multiple essential input files, and all the calculations can be performed in an all-in-one black-box easy-to-use manner. To show the utility of the ML-MCTDH-Aid package, we provide a step-by-step tutorial that demonstrates running ML-MCTDH studies on three models. These examples illuminate how the utilization of the ML-MCTDH-Aid package significantly enhances the efficiency and effectiveness of ML-MCTDH calculations. This substantially boosts the accessibility of ML-MCTDH calculations in tackling the high-dimensional quantum dynamics of complex systems.

Many-body localization in the age of classical computing. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2025;88:026502. [PMID: 39591655 DOI: 10.1088/1361-6633/ad9756] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBLregimegive rise to an MBLphase, in which the thermalization does not occur even in theasymptoticlimit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.

Comparison of Matrix Product State and Multiconfiguration Time-Dependent Hartree Methods for Nonadiabatic Dynamics of Exciton Dissociation

The excited-state dynamics of organic molecules, molecular aggregates, and donor-acceptor clusters is typically governed by the interplay of electronic excitations and, due to their flexibility and soft bonding, by the interaction with their vibrations. This interaction in these systems can be characterized by a few relevant electronic states that are coupled to numerous vibrational normal modes, encompassing a vast configurational space of the molecules. The full quantum simulation of these type of systems has been long dominated by the multiconfiguration time-dependent Hartree (MCTDH) approach and its multilayer variants, which are considered the gold standard in the presence of electron-vibration coupling with a large number of modes. Recently, also the matrix product state ansatz (MPS) with appropriate time-evolution schemes has been applied to these types of Hamiltonians. In this article, we provide a numerical comparison of excited-state dynamics between the MCTDH and MPS approaches for two electron-vibration coupled systems. Notably, we consider two models for exciton dissociation at a P3HT:PCBM heterojunction, comprising two electronic states and 100 vibrational modes, and 26 electronic states and 113 vibrational modes, respectively. While both methods agree very well for the first model, more pronounced deviations are found for the second model. We trace back the divergence between the methods to the different way entanglement is treated.

Unsupervised Machine Learning in the Analysis of Nonadiabatic Molecular Dynamics Simulation

The all-atomic full-dimensional-level simulations of nonadiabatic molecular dynamics (NAMD) in large realistic systems has received high research interest in recent years. However, such NAMD simulations normally generate an enormous amount of time-dependent high-dimensional data, leading to a significant challenge in result analyses. Based on unsupervised machine learning (ML) methods, considerable efforts were devoted to developing novel and easy-to-use analysis tools for the identification of photoinduced reaction channels and the comprehensive understanding of complicated molecular motions in NAMD simulations. Here, we tried to survey recent advances in this field, particularly to focus on how to use unsupervised ML methods to analyze the trajectory-based NAMD simulation results. Our purpose is to offer a comprehensive discussion on several essential components of this analysis protocol, including the selection of ML methods, the construction of molecular descriptors, the establishment of analytical frameworks, their advantages and limitations, and persistent challenges.

Photo-induced dynamics with continuous and discrete quantum baths

The ultrafast quantum dynamics of photophysical processes in complex molecules is an extremely challenging computational problem with a broad variety of fascinating applications in quantum chemistry and biology. Inspired by recent developments in open quantum systems, we introduce a pure-state unraveled hybrid-bath method that describes a continuous environment via a set of discrete, effective bosonic degrees of freedom using a Markovian embedding. Our method is capable of describing both, a continuous spectral density and sharp peaks embedded into it. Thereby, we overcome the limitations of previous methods, which either capture long-time memory effects using the unitary dynamics of a set of discrete vibrational modes or use memoryless Markovian environments employing a Lindblad or Redfield master equation. We benchmark our method against two paradigmatic problems from quantum chemistry and biology. We demonstrate that compared to unitary descriptions, a significantly smaller number of bosonic modes suffices to describe the excitonic dynamics accurately, yielding a computational speed-up of nearly an order of magnitude. Furthermore, we take into account explicitly the effect of a δ-peak in the spectral density of a light-harvesting complex, demonstrating the strong impact of the long-time memory of the environment on the dynamics.

Optimal tree tensor network operators for tensor network simulations: Applications to open quantum systems

Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.

Electronic dynamics through conical intersections via non-Markovian stochastic Schrödinger equation with complex modes

Conical intersections (CIs) play a crucial role in photochemical reactions, offering an efficient channel for ultrafast non-adiabatic relaxation of excited states. This significantly influences the reaction pathways and the resulting products. In this work, we utilize the non-Markovian stochastic Schrödinger equation with complex modes method to explore the dynamics of electronic transitions through conical intersections (CIs) in pyrazine. The linear vibronic coupling model serves as the foundational framework, incorporating both intra-state and inter-state electron-vibrational interactions. The dynamics of the excited electronic transitions are analyzed across varying strengths of system-bath coupling and different bath relaxation times. The accuracy of this method is demonstrated by comparing its predictions with those from the hierarchical equations of motion method.

Kylin-V: An open-source package calculating the dynamic and spectroscopic properties of large systems

Quantum dynamics simulation and computational spectroscopy serve as indispensable tools for the theoretical understanding of various fundamental physical and chemical processes, ranging from charge transfer to photochemical reactions. When simulating realistic systems, the primary challenge stems from the overwhelming number of degrees of freedom and the pronounced many-body correlations. Here, we present Kylin-V, an innovative quantum dynamics package designed for accurate and efficient simulations of dynamics and spectroscopic properties of vibronic Hamiltonians for molecular systems and their aggregates. Kylin-V supports various quantum dynamics and computational spectroscopy methods, such as time-dependent density matrix renormalization group and our recently proposed single-site and hierarchical mapping approaches, as well as vibrational heat-bath configuration interaction. In this paper, we introduce the methodologies implemented in Kylin-V and illustrate their performances through a diverse collection of numerical examples.

TD-DMRG Study of Exciton Dynamics with both Thermal and Static Disorders for Fenna-Matthews-Olson Complex

Photosynthesis is a fundamental process that converts solar energy into chemical energy. Understanding the microscopic mechanisms of energy transfer in photosynthetic systems is crucial for the development of novel optoelectronic materials. Simulating these processes poses significant challenges due to the intricate interactions between electrons and phonons, compounded by static disorder. In this work, we present a numerically nearly exact study using the time-dependent density matrix renormalization group (TD-DMRG) method to simulate the quantum dynamics of the Fenna-Matthews-Olson (FMO) complex considering an eight-site model with both thermal and static disorders. We employ the thermo-field dynamics formalism for temperature effects. We merge all electronic interactions into one large matrix product state (MPS) site, boosting accuracy efficiently without increasing complexity. Previous combined experimental and computational studies indicated that the static disorders range from 30 to 90 cm-1 for different FMO sites. We employ a Gaussian distribution and the auxiliary bosonic operator approach to consider the static disorder in our TD-DMRG algorithm. We investigate the impact of different initial excitation sites, temperatures, and degrees of static disorder on the exciton dynamics and temporal coherence. It is found that under the influence of the experimentally determined static disorder strength, the exciton population evolution shows a non-negligible difference at zero temperature, while it is hardly affected at room temperature.

Ultrafast Excited-State Energy Transfer in Phenylene Ethynylene Dendrimer: Quantum Dynamics with the Tensor Network Method

Photoinduced excited-state energy transfer (EET) processes play an important role in solar energy conversions. Owing to their excellent photoharvesting and exciton-transport properties, phenylene ethynylene (PE) dendrimers display great potential for improving the efficiency of solar cells. In this work, we investigated the intramolecular EET dynamics in a dendrimer composed of two linear PE units (2-ring and 3-ring) using a fully quantum description based on the tensor network method. We first constructed a diabatic model Hamiltonian based on the electronic structure calculations. Using this diabatic vibronic coupling model, we tried to obtain the main features of the EET dynamics in terms of the several diabatic models with different numbers of vibrational modes (from 4 modes to 129 modes) and to explore the corresponding vibronic coupling interactions. The results show that the EET in this PE dendrimer is ultrafast. Four modes of A' symmetry play dominant roles in the dynamics; the remaining 86 modes of A' symmetry can dampen the electronic coherence; and the modes of A″ symmetry do not exhibit significant influence on the EET process. Overall, the first-order intrastate vibronic coupling terms show the dominant role in the EET dynamics, while the second-order intrastate vibronic coupling terms cause damping of the electronic coherence and slow down the overall EET process. This work provides a microscopic understanding of the EET dynamics in PE dendrimers.

Benchmarking various nonadiabatic semiclassical mapping dynamics methods with tensor-train thermo-field dynamics

Accurate quantum dynamics simulations of nonadiabatic processes are important for studies of electron transfer, energy transfer, and photochemical reactions in complex systems. In this comparative study, we benchmark various approximate nonadiabatic dynamics methods with mapping variables against numerically exact calculations based on the tensor-train (TT) representation of high-dimensional arrays, including TT-KSL for zero-temperature dynamics and TT-thermofield dynamics for finite-temperature dynamics. The approximate nonadiabatic dynamics methods investigated include mixed quantum-classical Ehrenfest mean-field and fewest-switches surface hopping, linearized semiclassical mapping dynamics, symmetrized quasiclassical dynamics, the spin-mapping method, and extended classical mapping models. Different model systems were evaluated, including the spin-boson model for nonadiabatic dynamics in the condensed phase, the linear vibronic coupling model for electronic transition through conical intersections, the photoisomerization model of retinal, and Tully's one-dimensional scattering models. Our calculations show that the optimal choice of approximate dynamical method is system-specific, and the accuracy is sensitively dependent on the zero-point-energy parameter and the initial sampling strategy for the mapping variables.

Time-Dependent Variational Principle with Controlled Bond Expansion for Matrix Product States

We present a controlled bond expansion (CBE) approach to simulate quantum dynamics based on the time-dependent variational principle (TDVP) for matrix product states. Our method alleviates the numerical difficulties of the standard, fixed-rank one-site TDVP integrator by increasing bond dimensions on the fly to reduce the projection error. This is achieved in an economical, local fashion, requiring only minor modifications of standard one-site TDVP implementations. We illustrate the performance and accuracy of CBE-TDVP with several numerical examples on finite quantum lattices, including new results on bipolaron formation in the Peierls-Hubbard model and spin pumping via adiabatic flux insertion in a chiral spin liquid.

Encoding a Many-Body Potential Energy Surface into a Grid-Based Matrix Product Operator

An efficient algorithm for compressing a given many-body potential energy surface (PES) of molecular systems into a grid-based matrix product operator (MPO) is proposed. The PES is once represented by a full-dimensional or truncated many-body expansion form, which is obtained by ab initio calculations at each grid mesh point, and then all terms in the expansion are compressed and merged into a single MPO while maintaining the bond dimension of the MPO as small as possible. It was shown that the ab initio PES of the H2CO was compressed by more than 2 orders of magnitude in the size of the site operators without loss of accuracy. By the use of grid basis, the tensor rank of the site operators of the MPO is reduced from four to three due to the diagonal nature of the position-dependent operators on grid basis, which significantly reduces the computational cost of the tensor contractions required in the real and imaginary time evolution of the matrix product state (MPS) wave functions with the grid-based MPO (Grid-MPO) Hamiltonian. Similar to other grid-based methods, Grid-MPO is easily applicable to any kinds of potentials of molecular systems, such as analytical empirical model potentials expressed by position operators and ab initio potentials, if the values at the grid points are available. Using the Grid-MPO combined with the MPS, we calculated the time correlation function of the Eigen cation H 3 O + ( H 2 O ) 3 to predict the infrared spectrum and compared with the experimental and the previous theoretical studies. The actual scaling with the size of systems was examined for the multidimensional Henon-Heiles Hamiltonian. It was shown that the method is considerably accelerated by the graphic processing unit (GPU) because the sizes of site operators were kept small and all tensors were able to be stored on the VRAM of a GPU.

Seeking a quantum advantage with trapped-ion quantum simulations of condensed-phase chemical dynamics

Simulating the quantum dynamics of molecules in the condensed phase represents a longstanding challenge in chemistry. Trapped-ion quantum systems may serve as a platform for the analog-quantum simulation of chemical dynamics that is beyond the reach of current classical-digital simulation. To identify a 'quantum advantage' for these simulations, performance analysis of both analog-quantum simulation on noisy hardware and classical-digital algorithms is needed. In this Review, we make a comparison between a noisy analog trapped-ion simulator and a few choice classical-digital methods on simulating the dynamics of a model molecular Hamiltonian with linear vibronic coupling. We describe several simple Hamiltonians that are commonly used to model molecular systems, which can be simulated with existing or emerging trapped-ion hardware. These Hamiltonians may serve as stepping stones towards the use of trapped-ion simulators for systems beyond the reach of classical-digital methods. Finally, we identify dynamical regimes in which classical-digital simulations seem to have the weakest performance with respect to analog-quantum simulations. These regimes may provide the lowest hanging fruit to make the most of potential quantum advantages.

Simultaneous Intra- and Intermolecular Singlet Fission in Bipentacene Macrocycle Aggregates

Singlet fission (SF) is a process where a singlet state splits into two triplet states, which is essential for enhancing optoelectronic devices. Macrocyclic structures allow for precise control of chromophore orientation and facilitate singlet fission in solutions. However, the behavior of these structures in thin films, crucial for solid-state device optimization, remains underexplored. This study examines the aggregation and singlet fission processes of bipentacene macrocycles (BPc) in thin films using molecular dynamics simulations and electronic structure calculations. Findings indicate that BPc aggregates more rapidly with less chloroform, aligning parallel to the substrate. Intramolecular singlet fission (iSF) rates are rarely changed during evaporation, but the efficiency of intermolecular singlet fission (xSF) improves due to the increase in packing domains, suggesting that orderly crystal domains are not necessary for device efficiency. This opens avenues for varied device designs and traditional solution-based methods for optimal device development.

Efficient time-dependent vibrational coupled cluster computations with time-dependent basis sets at the two-mode coupling level: Full and hybrid TDMVCC[2]

The computation of the nuclear quantum dynamics of molecules is challenging, requiring both accuracy and efficiency to be applicable to systems of interest. Recently, theories have been developed for employing time-dependent basis functions (denoted modals) with vibrational coupled cluster theory (TDMVCC). The TDMVCC method was introduced along with a pilot implementation, which illustrated good accuracy in benchmark computations. In this paper, we report an efficient implementation of TDMVCC, covering the case where the wave function and Hamiltonian contain up to two-mode couplings. After a careful regrouping of terms, the wave function can be propagated with a cubic computational scaling with respect to the number of degrees of freedom. We discuss the use of a restricted set of active one-mode basis functions for each mode, as well as two interesting limits: (i) the use of a full active basis where the variational modal determination amounts essentially to the variational determination of a time-dependent reference state for the cluster expansion; and (ii) the use of a single function as an active basis for some degrees of freedom. The latter case defines a hybrid TDMVCC/TDH (time-dependent Hartree) approach that can obtain even lower computational scaling. The resulting computational scaling for hybrid and full TDMVCC[2] is illustrated for polyaromatic hydrocarbons with up to 264 modes. Finally, computations on the internal vibrational redistribution of benzoic acid (39 modes) are used to show the faster convergence of TDMVCC/TDH hybrid computations towards TDMVCC compared to simple neglect of some degrees of freedom.

New Density Matrix Renormalization Group Approaches for Strongly Correlated Systems Coupled with Large Environments

Thanks to the high compression of the matrix product state (MPS) form of the wave function and the efficient site-by-site iterative sweeping optimization algorithm, the density matrix normalization group (DMRG) and its time-dependent variant (TD-DMRG) have been established as powerful computational tools in accurately simulating the electronic structure and quantum dynamics of strongly correlated molecules with a large number (10^1-2) of quantum degrees of freedom (active orbitals or vibrational modes). However, the quantitative characterization of the quantum many-body behaviors of realistic strongly correlated systems requires a further consideration of the interaction between the embedded active subsystem and the remaining correlated environment, e.g., a larger number (10^2-3) of external orbitals in electronic structure or infinite condensed-phase phononic modes in nucleus dynamics. To this end, we introduced three new post-DMRG and TD-DMRG approaches, namely (1) DMRG2sCI-MRCI and DMRG2sCI-ENPT by the reconstruction of selected configuration interaction (sCI) type of compact reference function from DMRG coefficients and the use of externally contracted MRCI (multireference configuration interaction) and Epstein-Nesbet perturbation theory (ENPT), without recourse to the expensive high order n-electron reduced density matrices (n-RDMs). (2) DMRG combined with RR-MRCI (renormalized residue-based MRCI), which improves the computational accuracy and efficiency of internally contracted (ic) MRCI by renormalizing the contracted bases with small-sized buffer environment(s) of a few external orbitals as probes based on quantum information theory. (3) HM (hierarchical mapping)-TD-DMRG in which a large environment is reduced to a small number of renormalized environmental modes (which accounts for the most vital system-environment interactions) through stepwise mapping transformation. These advances extend the efficacy of highly accurate DMRG/TD-DMRG computations to the quantitative characterization of the electronic structure and quantum dynamics in realistic strongly correlated systems coupled with large environments and are reviewed in this paper.

Dynamical Simulations of Carotenoid Photoexcited States Using Density Matrix Renormalization Group Techniques

We present a dynamical simulation scheme to model the highly correlated excited state dynamics of linear polyenes. We apply it to investigate the internal conversion processes of carotenoids following their photoexcitation. We use the extended Hubbard-Peierls model, H^UVP, to describe the π-electronic system coupled to nuclear degrees of freedom. This is supplemented by a Hamiltonian, H^ϵ, that explicitly breaks both the particle-hole and two-fold rotation symmetries of idealized carotenoid structures. The electronic degrees of freedom are treated quantum mechanically by solving the time-dependent Schrödinger equation using the adaptive time-dependent DMRG (tDMRG) method, while nuclear dynamics are treated via the Ehrenfest equations of motion. By defining adiabatic excited states as the eigenstates of the full Hamiltonian, H^=H^UVP+H^ϵ, and diabatic excited states as eigenstates of H^UVP, we present a computational framework to monitor the internal conversion process from the initial photoexcited 1¹B_u⁺ state to the singlet triplet-pair states of carotenoids. We further incorporate Lanczos-DMRG to the tDMRG-Ehrenfest method to calculate transient absorption spectra from the evolving photoexcited state. We describe in detail the accuracy and convergence criteria for DMRG, and show that this method accurately describes the dynamical processes of carotenoid excited states. We also discuss the effect of the symmetry-breaking term, H^ϵ, on the internal conversion process, and show that its effect on the extent of internal conversion can be described by a Landau-Zener-type transition. This methodological paper is a companion to our more explanatory discussion of carotenoid excited state dynamics in Manawadu, D.; Georges, T. N.; Barford, W. Photoexcited State Dynamics and Singlet Fission in Carotenoids. J. Phys. Chem. A 2023, 127, 1342.

Density Matrix via Few Dominant Observables for the Ultrafast Non-Radiative Decay in Pyrazine

Unraveling the density matrix of a non-stationary quantum state as an explicit function of a few observables provides a complementary view of quantum dynamics. We have recently developed a practical way to identify the minimal set of the dominant observables that govern the quantal dynamics even in the case of strong non-adiabatic effects and large anharmonicity [Komarova et al., J. Chem. Phys. 155, 204110 (2021)]. Fast convergence in the number of the dominant contributions is achieved when instead of the density matrix we describe the time-evolution of the surprisal, the logarithm of the density operator. In the present work, we illustrate the efficiency of the proposed approach using an example of the early time dynamics in pyrazine in a Hilbert space accounting for up to four vibrational normal modes, {Q10a, Q6a, Q1, and Q9a}, and two coupled electronic states, the optically dark B 1 3 u ( n π * ) and the bright B 1 2 u ( π π * ) states. Dynamics in four-dimensional (4D) configurational space involve 19,600 vibronic eigenstates. Our results reveal that the rate of the ultrafast population decay as well as the shape of the nuclear wave packets in 2D, accounting only for {Q10a,Q6a} normal modes, are accurately captured with only six dominant time-independent observables in the surprisal. Extension of the dynamics to 3D and 4D vibrational subspace requires only five additional constraints. The time-evolution of a quantum state in 4D vibrational space on two electronic states is thus compacted to only 11 time-dependent coefficients of these observables.

Hierarchical Mapping for Efficient Simulation of Strong System-Environment Interactions

Quantum dynamics (QD) simulation is a powerful tool for interpreting ultrafast spectroscopy experiments and unraveling their microscopic mechanism in out-of-equilibrium excited state behaviors in various chemical, biological, and material systems. Although state-of-the-art numerical QD approaches such as the time-dependent density matrix renormalization group (TD-DMRG) already greatly extended the solvable system size of general linearly coupled exciton-phonon models with up to a few hundred phonon modes, the accurate simulation of larger system sizes or strong system-environment interactions is still computationally highly challenging. Based on quantum information theory (QIT), in this work, we realize that only a small number of effective phonon modes couple to the excitonic system directly regardless of a large or even infinite number of modes in the condensed phase environment. On top of the identified small number of direct effective modes, we propose a hierarchical mapping (HM) approach through performing block Lanczos transformations on the remaining indirect modes, which transforms the Hamiltonian matrix to a nearly block-tridiagonal form and eliminates the long-range interactions. Numerical tests on model spin-boson systems and realistic singlet fission models in a rubrene crystal environment with up to 7000 modes and strong system-environment interactions indicate HM can reduce the system size by 1-2 orders of magnitude and accelerate the calculation by ∼80% without losing accuracy.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Photo-induced phase-transitions in complex solids

Photo-induced phase-transitions (PIPTs) driven by highly cooperative interactions are of fundamental interest as they offer a way to tune and control material properties on ultrafast timescales. Due to strong correlations and interactions, complex quantum materials host several fascinating PIPTs such as light-induced charge density waves and ferroelectricity and have become a desirable setting for studying these PIPTs. A central issue in this field is the proper understanding of the underlying mechanisms driving the PIPTs. As these PIPTs are highly nonlinear processes and often involve multiple time and length scales, different theoretical approaches are often needed to understand the underlying mechanisms. In this review, we present a brief overview of PIPTs realized in complex materials, followed by a discussion of the available theoretical methods with selected examples of recent progress in understanding of the nonequilibrium pathways of PIPTs.

Tensor-Train Split-Operator KSL (TT-SOKSL) Method for Quantum Dynamics Simulations

Numerically exact simulations of quantum reaction dynamics, including nonadiabatic effects in excited electronic states, are essential to gain fundamental insights into ultrafast chemical reactivity and rigorous interpretations of molecular spectroscopy. Here, we introduce the tensor-train split-operator KSL (TT-SOKSL) method for quantum simulations in tensor-train (TT)/matrix product state (MPS) representations. TT-SOKSL propagates the quantum state as a tensor train using the Trotter expansion of the time-evolution operator, as in the tensor-train split-operator Fourier transform (TT-SOFT) method. However, the exponential operators of the Trotter expansion are applied using a rank-adaptive TT-KSL scheme instead of using the scaling and squaring approach as in TT-SOFT. We demonstrate the accuracy and efficiency of TT-SOKSL as applied to simulations of the photoisomerization of the retinal chromophore in rhodopsin, including nonadiabatic dynamics at a conical intersection of potential energy surfaces. The quantum evolution is described in full dimensionality by a time-dependent wavepacket evolving according to a two-state 25-dimensional model Hamiltonian. We find that TT-SOKSL converges faster than TT-SOFT with respect to the maximally allowed memory requirement of the tensor-train representation and better preserves the norm of the time-evolving state. When compared to the corresponding simulations based on the TT-KSL method, TT-SOKSL has the advantage of avoiding the need to construct the matrix product state Laplacian by exploiting the linear scaling of multidimensional tensor-train Fourier transforms.

On the fly swapping algorithm for ordering of degrees of freedom in density matrix renormalization group

Density matrix renormalization group (DMRG) and its time-dependent variants have found widespread applications in quantum chemistry, includingab initioelectronic structure of complex bio-molecules, spectroscopy for molecular aggregates, and charge transport in bulk organic semiconductors. The underlying wavefunction ansatz for DMRG, matrix product state (MPS), requires mapping degrees of freedom (DOF) into a one-dimensional topology. DOF ordering becomes a crucial factor for DMRG accuracy. In this work, we propose swapping neighboring DOFs during the DMRG sweeps for DOF ordering, which we term 'on the fly swapping' (OFS) algorithm. We show that OFS is universal for both static and time-dependent DMRG with minimum computational overhead. Examples are given for one dimensional antiferromagnetic Heisenberg model,ab initioelectronic structure of N₂molecule, and the S₁/S₂internal conversion dynamics of pyrazine molecule. It is found that OFS can indeed improve accuracy by finding better DOF ordering in all cases.

Stochastic Adaptive Single-Site Time-Dependent Variational Principle

In recent years, the time-dependent variational principle (TDVP) method based on the matrix product state (MPS) wave function formulation has shown its great power in performing large-scale quantum dynamics simulations for realistic chemical systems with strong electron-vibration interactions. In this work, we propose a stochastic adaptive single-site TDVP (SA-1TDVP) scheme to evolve the bond-dimension adaptively, which can integrate the traditional advantages of both the high efficiency of the single-site TDVP (1TDVP) variant and the high accuracy of the two-site TDVP (2TDVP) variant. Based on the assumption that the level statistics of entanglement Hamiltonians, which originate from the reduced density matrices of the MPS method, follows a Poisson or Wigner distribution, as generically predicted by random-matrix theory, additional random singular values are generated to expand the bond-dimension automatically. Tests on simulating the vibrationally resolved quantum dynamics and absorption spectra in the pyrazine molecule and perylene bisimide (PBI) J-aggregate trimer as well as a spin-1/2 Heisenberg chain show that it can be automatic and as accurate as 2TDVP but reduce the computational time remarkably.

Variational Squeezed Davydov Ansatz for Realistic Chemical Systems with Nonlinear Vibronic Coupling

Chemical systems normally possess strong nonlinear vibronic couplings at both zero and finite temperature. For the lowest-order quadratic couplings, here, we introduce a squeezing operator into a variational coherent-state-based method, Davydov ansatz, to simulate the quantum dynamics and the respective spectroscopy. Two molecular systems, pyrazine and the 2-pyridone dimer, are taken as calculated model systems, both of which involve nontrivial quadratic vibronic couplings in high- and low-frequency regions, respectively. Upon a comparison with the benchmarks, the method manifests its advantage for nonlinear couplings. The squeezed bases are also proven to be applicable for the finite temperature by adapting with the thermofield dynamics.

A multisite decomposition of the tensor network path integrals

Tensor network decompositions of path integrals for simulating open quantum systems have recently been proven to be useful. However, these methods scale exponentially with the system size. This makes it challenging to simulate the non-equilibrium dynamics of extended quantum systems coupled with local dissipative environments. In this work, we extend the tensor network path integral (TNPI) framework to efficiently simulate such extended systems. The Feynman-Vernon influence functional is a popular approach used to account for the effect of environments on the dynamics of the system. In order to facilitate the incorporation of the influence functional into a multisite framework (MS-TNPI), we combine a matrix product state (MPS) decomposition of the reduced density tensor of the system along the sites with a corresponding tensor network representation of the time axis to construct an efficient 2D tensor network. The 2D MS-TNPI network, when contracted, yields the time-dependent reduced density tensor of the extended system as an MPS. The algorithm presented is independent of the system Hamiltonian. We outline an iteration scheme to take the simulation beyond the non-Markovian memory introduced by solvents. Applications to spin chains coupled to local harmonic baths are presented; we consider the Ising, XXZ, and Heisenberg models, demonstrating that the presence of local environments can often dissipate the entanglement between the sites. We discuss three factors causing the system to transition from a coherent oscillatory dynamics to a fully incoherent dynamics. The MS-TNPI method is useful for studying a variety of extended quantum systems coupled with solvents.

Comprehending radicals, diradicals and their bondings in aggregates of imide-fused polycyclic aromatic hydrocarbons

The structures and optical properties of imide-fused polycyclic aromatic hydrocarbon radicals, which exhibit room-temperature ferromagnetism as discovered recently and the strong pancake bondings in NDI radical aggregates are investigated.

Compacting the density matrix in quantum dynamics: Singular value decomposition of the surprisal and the dominant constraints for anharmonic systems

We introduce a practical method for compacting the time evolution of the quantum state of a closed physical system. The density matrix is specified as a function of a few time-independent observables where their coefficients are time-dependent. The key mathematical step is the vectorization of the surprisal, the logarithm of the density matrix, at each time point of interest. The time span used depends on the required spectral resolution. The entire course of the system evolution is represented as a matrix where each column is the vectorized surprisal at the given time point. Using the singular value decomposition (SVD) of this matrix, we generate realistic approximations for the time-independent observables and their respective time-dependent coefficients. This allows for a simplification of the algebraic procedure for determining the dominant constraints (the time-independent observables) in the sense of the maximal entropy approach. A non-stationary coherent initial state of a Morse oscillator is used to introduce the approach. We derive the analytical exact expression for the surprisal as a function of time, and this offers a benchmark for comparison with the accurate but approximate SVD results. We discuss two examples of a Morse potential of different anharmonicities, H₂ and I₂ molecules. We further demonstrate the approach for a two-coupled electronic state problem, the well-studied non-radiative decay of pyrazine from its bright state. Five constraints are found to be enough to capture the ultrafast electronic population exchange and to recover the dynamics of the wave packet in both electronic states.

Computational and data driven molecular material design assisted by low scaling quantum mechanics calculations and machine learning

Electronic structure methods based on quantum mechanics (QM) are widely employed in the computational predictions of the molecular properties and optoelectronic properties of molecular materials. The computational costs of these QM methods, ranging from density functional theory (DFT) or time-dependent DFT (TDDFT) to wave-function theory (WFT), usually increase sharply with the system size, causing the curse of dimensionality and hindering the QM calculations for large sized systems such as long polymer oligomers and complex molecular aggregates. In such cases, in recent years low scaling QM methods and machine learning (ML) techniques have been adopted to reduce the computational costs and thus assist computational and data driven molecular material design. In this review, we illustrated low scaling ground-state and excited-state QM approaches and their applications to long oligomers, self-assembled supramolecular complexes, stimuli-responsive materials, mechanically interlocked molecules, and excited state processes in molecular aggregates. Variable electrostatic parameters were also introduced in the modified force fields with the polarization model. On the basis of QM computational or experimental datasets, several ML algorithms, including explainable models, deep learning, and on-line learning methods, have been employed to predict the molecular energies, forces, electronic structure properties, and optical or electrical properties of materials. It can be conceived that low scaling algorithms with periodic boundary conditions are expected to be further applicable to functional materials, perhaps in combination with machine learning to fast predict the lattice energy, crystal structures, and spectroscopic properties of periodic functional materials.

Efficient quantum dynamics simulations of complex molecular systems: A unified treatment of dynamic and static disorder

We present a unified and highly numerically efficient formalism for the simulation of quantum dynamics of complex molecular systems, which takes into account both temperature effects and static disorder. The methodology is based on the thermo-field dynamics formalism, and Gaussian static disorder is included into simulations via auxiliary bosonic operators. This approach, combined with the tensor-train/matrix-product state representation of the thermalized stochastic wave function, is applied to study the effect of dynamic and static disorders in charge-transfer processes in model organic semiconductor chains employing the Su-Schrieffer-Heeger (Holstein-Peierls) model Hamiltonian.

Chebyshev Matrix Product States with Canonical Orthogonalization for Spectral Functions of Many-Body Systems

We propose a method to calculate the spectral functions of many-body systems by Chebyshev expansion in the framework of matrix product states coupled with canonical orthogonalization (coCheMPS). The canonical orthogonalization can improve the accuracy and efficiency significantly because the orthogonalized Chebyshev vectors can provide an ideal basis for constructing the effective Hamiltonian in which the exact recurrence relation can be retained. In addition, not only the spectral function but also the excited states and eigenenergies can be directly calculated, which is usually impossible for other MPS-based methods such as time-dependent formalism or correction vector. The remarkable accuracy and efficiency of coCheMPS over other methods are demonstrated by calculating the spectral functions of spin chain and ab initio hydrogen chain. For the first time we demonstrate that Chebyshev MPS can be used to deal with ab initio electronic Hamiltonian effectively. We emphasize the strength of coCheMPS to calculate the low excited states of systems with sparse discrete spectrum. We also caution the application for electron-phonon systems with dense density of states.

Simulation of Nonlinear Femtosecond Signals at Finite Temperature via a Thermo Field Dynamics-Tensor Train Method: General Theory and Application to Time- and Frequency-Resolved Fluorescence of the Fenna-Matthews-Olson Complex

Addressing needs of contemporary nonlinear femtosecond optical spectroscopy, we have developed a fully quantum, numerically accurate wave function-based approach for the calculation of third-order spectroscopic signals of polyatomic molecules and molecular aggregates at finite temperature. The systems are described by multimode nonadiabatic vibronic-coupling Hamiltonians, in which diagonal terms are treated in harmonic approximation, while off-diagonal interstate couplings are assumed to be coordinate independent. The approach is based on the Thermo Field Dynamics (TFD) representation of quantum mechanics and tensor-train (TT) machinery for efficient numerical simulation of quantum evolution of systems with many degrees of freedom. The developed TFD-TT approach is applied to the calculation of time- and frequency-resolved fluorescence spectra of the Fenna-Matthews-Olson (FMO) antenna complex at room temperature taking into account finite time-frequency resolution in fluorescence detection, orientational averaging, and static disorder.

Evaluating the anharmonicity contributions to the molecular excited state internal conversion rates with finite temperature TD-DMRG

In this work, we propose a new method to calculate molecular nonradiative electronic relaxation rates based on the numerically exact time-dependent density matrix renormalization group theory. This method could go beyond the existing frameworks under the harmonic approximation (HA) of the potential energy surface (PES) so that the anharmonic effect could be considered, which is of vital importance when the electronic energy gap is much larger than the vibrational frequency. We calculate the internal conversion (IC) rates in a two-mode model with Morse potential to investigate the validity of HA. We find that HA is unsatisfactory unless only the lowest several vibrational states of the lower electronic state are involved in the transition process when the adiabatic excitation energy is relatively low. As the excitation energy increases, HA first underestimates and then overestimates the IC rates when the excited state PES shifts toward the dissociative side of the ground state PES. On the contrary, HA slightly overestimates the IC rates when the excited state PES shifts toward the repulsive side. In both cases, a higher temperature enlarges the error of HA. As a real example to demonstrate the effectiveness and scalability of the method, we calculate the IC rates of azulene from S₁ to S₀ on the ab initio anharmonic PES approximated by the one-mode representation. The calculated IC rates of azulene under HA are consistent with the analytically exact results. The rates on the anharmonic PES are 30%-40% higher than the rates under HA.

Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors

We develop new methods to efficiently propagate the hierarchical equations of motion (HEOM) by using the Tucker and hierarchical Tucker (HT) tensors to represent the reduced density operator and auxiliary density operators. We first show that by employing the split operator method, the specific structure of the HEOM allows a simple propagation scheme using the Tucker tensor. When the number of effective modes in the HEOM increases and the Tucker representation becomes intractable, the split operator method is extended to the binary tree structure of the HT representation. It is found that to update the binary tree nodes related to a specific effective mode, we only need to propagate a short matrix product state constructed from these nodes. Numerical results show that by further employing the mode combination technique commonly used in the multi-configuration time-dependent Hartree approaches, the binary tree representation can be applied to study excitation energy transfer dynamics in a fairly large system including over 10⁴ effective modes. The new methods may thus provide a promising tool in simulating quantum dynamics in condensed phases.

Multi-layer Gaussian-based multi-configuration time-dependent Hartree (ML-GMCTDH) simulations of ultrafast charge separation in a donor-acceptor complex

We report on first applications of the Multi-Layer Gaussian-based Multi-Configuration Time-Dependent Hartree (ML-GMCTDH) method [Römer et al., J. Chem. Phys. 138, 064106 (2013)] beyond its basic two-layer variant. The ML-GMCTDH scheme provides an embedding of a variationally evolving Gaussian wavepacket basis into a hierarchical tensor representation of the wavefunction. A first-principles parameterized model Hamiltonian for ultrafast non-adiabatic dynamics in an oligothiophene-fullerene charge transfer complex is employed, relying on a two-state linear vibronic coupling model that combines a distribution of tuning type modes with an intermolecular coordinate that also modulates the electronic coupling. Efficient ML-GMCTDH simulations are carried out for up to 300 vibrational modes using an implementation within the QUANTICS program. Excellent agreement with reference ML-MCTDH calculations is obtained.

Analysis of bath motion in MM-SQC dynamics via dimensionality reduction approach: Principal component analysis

The system-plus-bath model is an important tool to understand the nonadiabatic dynamics of large molecular systems. Understanding the collective motion of a large number of bath modes is essential for revealing their key roles in the overall dynamics. Here, we applied principal component analysis (PCA) to investigate the bath motion in the basis of a large dataset generated from the symmetrical quasi-classical dynamics method based on the Meyer-Miller mapping Hamiltonian nonadiabatic dynamics for the excited-state energy transfer in the Frenkel-exciton model. The PCA method clearly elucidated that two types of bath modes, which either display strong vibronic coupling or have frequencies close to that of the electronic transition, are important to the nonadiabatic dynamics. These observations were fully consistent with the physical insights. The conclusions were based on the PCA of the trajectory data and did not involve significant pre-defined physical knowledge. The results show that the PCA approach, which is one of the simplest unsupervised machine learning dimensionality reduction methods, is a powerful one for analyzing complicated nonadiabatic dynamics in the condensed phase with many degrees of freedom.

Simulating Quantum Vibronic Dynamics at Finite Temperatures With Many Body Wave Functions at 0 K

For complex molecules, nuclear degrees of freedom can act as an environment for the electronic “system” variables, allowing the theory and concepts of open quantum systems to be applied. However, when molecular system-environment interactions are non-perturbative and non-Markovian, numerical simulations of the complete system-environment wave function become necessary. These many body dynamics can be very expensive to simulate, and extracting finite-temperature results—which require running and averaging over many such simulations—becomes especially challenging. Here, we present numerical simulations that exploit a recent theoretical result that allows dissipative environmental effects at finite temperature to be extracted efficiently from a single, zero-temperature wave function simulation. Using numerically exact time-dependent variational matrix product states, we verify that this approach can be applied to vibronic tunneling systems and provide insight into the practical problems lurking behind the elegance of the theory, such as the rapidly growing numerical demands that can appear for high temperatures over the length of computations.

A general automatic method for optimal construction of matrix product operators using bipartite graph theory

Constructing matrix product operators (MPOs) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For the DMRG to be conveniently used in different problems described by different Hamiltonians, in this work, we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) it is automatic in that only the definition of the operator is required; (ii) it is symbolic thus free of any numerical error; (iii) the complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; and (iv) the symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin-boson model and the Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with the sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.

Finite-Temperature TD-DMRG for the Carrier Mobility of Organic Semiconductors

A large number of nonadiabatic dynamical studies have been applied to reveal the nature of carrier transport in organic semiconductors with different approximations. We present here a "nearly exact" graphical-process-unit-based finite-temperature time-dependent density matrix renormalization group (TD-DMRG) method to evaluate the carrier mobility in organic semiconductors, as described by the electron-phonon model, in particular, in rubrene crystal, one of the prototypical organic semiconductors, with parameters derived from first-principles. We find that (i) TD-DMRG is a general and robust method that can bridge the gap between hopping and band pictures, covering a wide range of electronic coupling strengths and (ii) with realistic parameters, TD-DMRG is able to account for the experimentally observed "band-like" transport behavior (∂μ/∂T < 0) in rubrene. We further study the long-standing puzzle of the isotope effect for charge transport and unambiguously demonstrate that the negative isotope effect (∂μ/∂m < 0 where m is the atomic mass) should be universal.

Finite Temperature Dynamical Density Matrix Renormalization Group for Spectroscopy in Frequency Domain

We present a novel methodology through casting the dynamical density matrix renormalization group (DDMRG) into the matrix product state (MPS) formulation to calculate the spectroscopy at finite temperature for molecular aggregates. The frequency domain algorithm can avoid the time evolution accumulation of error and is naturally suitable for parallelization, in addition to facile graphic processing unit (GPU) acceleration. The high accuracy is demonstrated by simulating the optical spectra of vibronic model systems ranging from an exactly solvable dimer model to a more complex real-world perylene bisimide (PBI) J-aggregate. The relationship between the 0-0 emission strength and the exciton thermal coherent length is discussed for linearly stacked aggregates. The computing performance largely boosted by GPU demonstrates that DDMRG emerges as a promising method to study dynamical properties for complex systems.

Quantum Dynamics Simulation of Doublet Excitation and Magnetic Field Effect in Neutral Radical Materials

The photon absorption and the relevant magnetic field effect of a doublet in neutral radical materials are investigated by combining the density functional theory and the variational quantum dynamics with Davydov ansatz. The doublet is a perfect model system to study the full-quantum dynamics of a two-level system coupling to a realistic molecular vibrational environment. In this work, we simulate the optical absorption spectroscopy of the neutral radical material, (4-N-carbazolyl-2,6-dichlorophenyl)bis(2,4,6-trichlorophenyl)-methyl, and find a good agreement with experiments for both highest occupied molecular orbital-singly occupied molecular orbital (SOMO) and SOMO-lowest unoccupied molecular orbital transitions. The nontrivial role of the intramolecular vibronic couplings is comprehensively discussed with separate spectroscopy and population dynamics, suggesting different contributions and the long time scale effect stemming from the vibrations, according to different symmetries. On the basis of the model, an applied magnetic field is taken into account to qualitatively investigate its magnetic properties in a dynamics manner, leading to a result which can be described by a sum of Lorentzian functions.

Numerical assessment for accuracy and GPU acceleration of TD-DMRG time evolution schemes

The time dependent density matrix renormalization group (TD-DMRG) has become one of the cutting edge methods of quantum dynamics for complex systems. In this paper, we comparatively study the accuracy of three time evolution schemes in the TD-DMRG, the global propagation and compression method with the Runge-Kutta algorithm (P&C-RK), the time dependent variational principle based methods with the matrix unfolding algorithm (TDVP-MU), and with the projector-splitting algorithm (TDVP-PS), by performing benchmarks on the exciton dynamics of the Fenna-Matthews-Olson complex. We show that TDVP-MU and TDVP-PS yield the same result when the time step size is converged and they are more accurate than P&C-RK4, while TDVP-PS tolerates a larger time step size than TDVP-MU. We further adopt the graphical processing units to accelerate the heavy tensor contractions in the TD-DMRG, and it is able to speed up the TDVP-MU and TDVP-PS schemes by up to 73 times.

Study of the exciton dynamics in perylene bisimide (PBI) aggregates with symmetrical quasiclassical dynamics based on the Meyer–Miller mapping Hamiltonian

The exciton dynamics in one-dimensional stacked PBI (Perylene Bisimide) aggregates was studied with SQC-MM dynamics (Symmetrical Quasiclassical Dynamics based on the Meyer–Miller mapping Hamiltonian).