Hierarchical Equations Of Motion Method Research Articles

Hierarchical Equations of Motion for Quantum Chemical Dynamics: Recent Methodology Developments and Applications.

ConspectusQuantum effects are critical to understanding many chemical dynamical processes in condensed phases, where interactions between molecules and their environment are usually strong and non-Markovian. In this Account, we review recent progress from our group in development and application of the hierarchical equations of motion (HEOM) method, highlighting its ability to address some challenging problems in quantum chemical dynamics.In the HEOM method, the bath degrees of freedom are represented using effective modes from exponential decomposition of the bath correlation function. Complex spectral densities and low temperature simulations often require a larger number of modes, making the simulations very expensive. Recent advances, such as the barycentric spectral decomposition (BSD) technique, can significantly reduce the number of effective modes, allowing to handle complex spectral densities and enabling simulations at very low temperatures, including near-zero temperature dynamics.Another key improvement in the computational efficiency is the use of tensor network methods like matrix product states and hierarchical tensor networks. These techniques allow for efficient HEOM propagation with thousands of effective modes, crucial for simulating large molecular systems interacting with multiple baths. This combination enables simulations of excitation energy transfer (EET) in systems like the Fenna-Matthews-Olson (FMO) complex and even larger systems with experimentally determined spectral densities.The versatility of the HEOM method is demonstrated through applications to a wide range of chemical dynamics problems. Simulations of EET and related ultrafast spectroscopy are first briefly covered. Applications of the HEOM to quantum tunneling effects in proton transfer reactions are then presented. Early works have studied the non-Kramers dependence of the rate constant as a function of bath friction due to deep tunneling and revealed vibrationally nonadiabatic dynamics within the so-called nontraditional view of proton transfer reactions. A recent work on the large kinetic isotope effects in soybean lipoxygenase also indicated that many quantum correction approximations to classical transition-state theory may fall short in describing deep tunneling effects.Charge transport and separation dynamics in organic semiconductors are another area where the HEOM method has been instrumental. We first demonstrate that the HEOM provides a unified description of both band-like and thermally assisted charge carrier transport in organic materials. The effect of non-nearest neighbor transitions is then investigated by combining generalized master equations with exact memory kernels. The HEOM method also enables simulation of charge separation in organic photovoltaics (OPVs) and reveals how factors such as external electric fields, entropy, and charge delocalization influence the charge separation barrier and dynamics.Moreover, HEOM has been applied to investigate hydrogen atom scattering on the Au(111) surface and vibrational energy relaxation at molecule-metal interfaces. These studies provide deeper insights into how electron-hole pair excitations and temporary charge transfer states influence the nuclear motion, offering a new framework for simulating nonadiabatic dynamics on metal surfaces.In summary, the HEOM method has developed into a robust tool for simulating quantum effects in condensed phases. Future developments in algorithm efficiency and computational power will likely expand its applicability to even more complex systems.

A stochastic Schrödinger equation and matrix product state approach to carrier transport in organic semiconductors with nonlocal electron-phonon interaction.

Evaluation of the charge transport property of organic semiconductors requires exact quantum dynamics simulation of large systems. We present a numerically nearly exact approach to investigate carrier transport dynamics in organic semiconductors by extending the non-Markovian stochastic Schrödinger equation with complex frequency modes to a forward-backward scheme and by solving it using the matrix product state (MPS) approach. By utilizing the forward-backward formalism for noise generation, the bath correlation function can be effectively treated as a temperature-independent imaginary part, enabling a more accurate decomposition with fewer complex frequency modes. Using this approach, we study the carrier transport and mobility in the one-dimensional Peierls model, where the nonlocal electron-phonon interaction is taken into account. The reliability of this approach was validated by comparing carrier diffusion motion with those obtained from the hierarchical equations of motion method across various parameter regimes of the phonon bath. The efficiency was demonstrated by the modest virtual bond dimensions of MPS and the low scaling of the computational time with the system size.

Simulation of the Pump-Probe Spectra and Excitation Energy Relaxation of the B850 Band of the LH2 Complex in Purple Bacteria.

Ultrafast spectroscopic techniques have been vital in studying excitation energy transfer (EET) in photosynthetic light harvesting complexes. In this paper, we simulate the pump-probe spectra of the B850 band of the light harvesting complex 2 (LH2) of purple bacteria, by using the hierarchical equation of motion method and the optical response function approach. The ground state bleach, stimulated emission, and excited state absorption components of the pump-probe spectra are analyzed in detail. The laser pulse-induced population dynamics are also simulated to help understand the main features of the pump-probe spectra and the EET process. It is shown that the excitation energy relaxation is an ultrafast process with multiple time scales. The first 40 fs of the pump-probe spectra is dominated by the relaxation of the k = ±1 states to both the k = 0 and higher energy states. Dynamics on a longer time scale around 200 fs reflects the relaxation of higher energy states to the k = 0 state.

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.

Is there a finite mobility for the one vibrational mode Holstein model? Implications from real time simulations.

The question of whether there exists a finite mobility in the standard Holstein model with one vibrational mode on each site remains unclear. In this Communication, we approach this problem by employing the hierarchical equation of motion method to simulate model systems where the vibrational modes are dissipative. It is found that, as the friction becomes smaller, the charge carrier mobility increases significantly and a friction-free limit cannot be obtained. The current autocorrelation functions are also calculated for the friction-free Holstein model, and converged results cannot be obtained with an increase in the number of sites. Based on these observations, we conclude that a finite mobility cannot be defined for the standard Holstein model in the parameter regime explored in this work.

An efficient Julia framework for hierarchical equations of motion in open quantum systems

The hierarchical equations of motion (HEOM) approach can describe the reduced dynamics of a system simultaneously coupled to multiple bosonic and fermionic environments. The complexity of exactly describing the system-environment interaction with the HEOM method usually results in time-consuming calculations and a large memory cost. Here, we introduce an open-source software package called HierarchicalEOM.jl: a Julia framework integrating the HEOM approach. HierarchicalEOM.jl features a collection of methods to compute bosonic and fermionic spectra, stationary states, and the full dynamics in the extended space of all auxiliary density operators (ADOs). The required handling of the ADOs multi-indexes is achieved through a user-friendly interface. We exemplify the functionalities of the package by analyzing a single impurity Anderson model, and an ultra-strongly coupled charge-cavity system interacting with bosonic and fermionic reservoirs. HierarchicalEOM.jl achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.

Effects of electronic-vibrational resonance on the absorption and two-dimensional rephasing spectra of the Fenna–Matthews–Olson complex

The absorption and two-dimensional rephasing spectra of the Fenna–Matthews–Olson (FMO) complex at 77 K have been simulated using the hierarchical equations of motion method. Three cases of vibrational coupling have been studied in the FMO complex model. In the first case, no specific vibrational mode is coupled. In the second and third cases, low-energy and high-energy electronic-vibrational resonance dimers are considered in the model, respectively. It has been observed that the vibrational mode significantly weakens the intensity of the absorption peak of resonant high-energy electronic transitions. We have also confirmed that long-lived quantum beatings originate from vibration, while short-lived electronic coherence still persists. Furthermore, although the electronic resonance mode speeds up the rephasing dynamics, it is not directly proportional to the coupling strength. The low-energy electronic-vibrational resonance dimer exhibits a greater speed-up of rephasing dynamics.

Theoretical study of nonadiabatic hydrogen atom scattering dynamics on metal surfaces using the hierarchical equations of motion method.

Hydrogen atom scattering on metal surfaces is investigated based on a simplified Newns-Anderson model. Both the nuclear and electronic degrees of freedom are treated quantum mechanically. By partitioning all the surface electronic states as the bath, the hierarchical equations of motion method for the fermionic bath is employed to simulate the scattering dynamics. It is found that, with a reasonable set of parameters, the main features of the recent experimental studies of hydrogen atom scattering on metal surfaces can be reproduced. Vibrational states on the chemisorption state whose energies are close to the incident energy are found to play an important role, and the scattering process is dominated by a single-pass electronic transition forth and back between the diabatic physisorption and chemisorption states. Further study on the effects of the atom-surface coupling strength reveals that, upon increasing the atom-surface coupling strength, the scattering mechanism changes from typical nonadiabatic transitions to dynamics in the electronic friction regime.

Tree tensor network state approach for solving hierarchical equations of motion.

The hierarchical equations of motion (HEOM) method is a numerically exact open quantum system dynamics approach. The method is rooted in an exponential expansion of the bath correlation function, which in essence strategically reshapes a continuous environment into a set of effective bath modes that allow for more efficient cutoff at finite temperatures. Based on this understanding, one can map the HEOM method into a Schrödinger-like equation, with a non-Hermitian super-Hamiltonian for an extended wave function being the tensor product of the central system wave function and the Fock state of these effective bath modes. In this work, we explore the possibility of representing the extended wave function as a tree tensor network state (TTNS) and the super-Hamiltonian as a tree tensor network operator of the same structure as the TTNS, as well as the application of a time propagation algorithm using the time-dependent variational principle. Our benchmark calculations based on the spin-boson model with a slow-relaxing bath show that the proposed HEOM+TTNS approach yields consistent results with those of the conventional HEOM method, while the computation is considerably sped up. In addition, the simulation with a genuine TTNS is four times faster than a one-dimensional matrix product state decomposition scheme.

Efficient low-temperature simulations for fermionic reservoirs with the hierarchical equations of motion method: Application to the Anderson impurity model

The hierarchical equations of motion (HEOM) approach is an accurate method to simulate open system quantum dynamics, which allows for systematic convergence to numerically exact results. To represent effects of the bath, the reservoir correlation functions are usually decomposed into summation of multiple exponential terms in the HEOM method. Since the reservoir correlation functions become highly non-Markovian at low temperatures or when the bath has complex band structures, a present challenge is to obtain accurate exponential decompositions that allow efficient simulation with the HEOM. In this work, we employ the barycentric representation to approximate the Fermi function and hybridization functions in the frequency domain. This method, by approximating these functions with optimized rational decomposition, greatly reduces the number of basis functions in decomposing the reservoir correlation functions, which further allows the HEOM method to be applied to ultralow temperature and general band structures. We demonstrate the efficiency, accuracy, and long-time stability of this decomposition scheme by applying it to the Anderson impurity model in the low-temperature regime with the Lorentzian and tight-binding hybridization functions.

Temperature Dependence of Entanglement and Coherence in Fenna-Matthews-Olson Complex

The high efficiency in excitation energy transfer observed in the Fenna-Matthews-Olson light-harvesting complex in green sulfur bacteria is due to its arrange of photo-pigments, called bacteriochlorophylls. They are central for the photosynthetic process of those bacteria being controversially associated to long-lived coherence. The study of this protein complex and its energy dynamics continues, trying to understand the environmental factors affecting it. This work explores the temperature effects in the behaviour of entanglement and coherence among bacteriochlorophyll excitation energy transfer within the Fenna-Matthews-Olson complex, considering the latest pigment-protein monomer arrangement of 8 bacteriochlorophylls. An analysis for the system evolution using the Hierarchical Equations of Motion method, a non-Markovian approach, is performed to get the global and semi-local entanglement, as well as the coherence in the system.

Hierarchical equations of motion approach for accurate characterization of spin excitations in quantum impurity systems.

Recent technological advancement in scanning tunneling microscopes has enabled the measurement of spin-field and spin-spin interactions in single atomic or molecular junctions with an unprecedentedly high resolution. Theoretically, although the fermionic hierarchical equations of motion (HEOM) method has been widely applied to investigate the strongly correlated Kondo states in these junctions, the existence of low-energy spin excitations presents new challenges to numerical simulations. These include the quest for a more accurate and efficient decomposition for the non-Markovian memory of low-temperature environments and a more careful handling of errors caused by the truncation of the hierarchy. In this work, we propose several new algorithms, which significantly enhance the performance of the HEOM method, as exemplified by the calculations on systems involving various types of low-energy spin excitations. Being able to characterize both the Kondo effect and spin excitation accurately, the HEOM method offers a sophisticated and versatile theoretical tool, which is valuable for the understanding and even prediction of the fascinating quantum phenomena explored in cutting-edge experiments.

Recent advances in fermionic hierarchical equations of motion method for strongly correlated quantum impurity systems

Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.

On the practical truncation tier of fermionic hierarchical equations of motion.

The fermionic hierarchical equations of motion (HEOM) approach has found wide application in the exploration of open quantum systems, and extensive efforts have been committed to improving its efficiency and accuracy in practical calculations. In this work, by scrutinizing the stationary-state and dynamic properties of Kondo-correlated quantum impurity systems, we show that the strength of Kondo correlation induced by the system-environment entanglement primarily determines the converged hierarchical truncation tier of the HEOM method. This complements the rule of thumb regarding the positive correlation between the height of hierarchy and system-environment coupling strength. These insights will provide useful guidelines for developing a more sophisticated fermionic HEOM method for the investigation of many-body open quantum systems.

Electronic absorption spectra from off-diagonal quantum master equations.

Quantum master equations (QMEs) provide a general framework for describing electronic dynamics within a complex molecular system. Off-diagonal QMEs (OD-QMEs) correspond to a family of QMEs that describe the electronic dynamics in the interaction picture based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. The fact that OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. A key experimental quantity that relies on the ability to obtain accurate Schrödinger picture electronic coherences is the absorption spectrum. In this paper, we propose using a recently introduced procedure for extracting Schrödinger picture electronic coherences from interaction picture inputs to calculate electronic absorption spectra from the electronic dynamics generated by OD-QMEs. The accuracy of the absorption spectra obtained this way is studied in the context of a biexciton benchmark model, by comparing spectra calculated based on time-local and time-nonlocal OD-QMEs to spectra calculated based on a Redfield-type QME and the non-perturbative and quantum-mechanically exact hierarchical equations of motion method.

Spectral Functions of the Holstein Polaron: Exact and Approximate Solutions.

It is generally accepted that the dynamical mean field theory gives a good solution of the Holstein model, but only in dimensions greater than two. Here, we show that this theory, which becomes exact in the weak coupling and in the atomic limit, provides an excellent, numerically cheap, approximate solution for the spectral function of the Holstein model in the whole range of parameters, even in one dimension. To establish this, we make a detailed comparison with the spectral functions that we obtain using the newly developed momentum-space numerically exact hierarchical equations of motion method, which yields electronic correlation functions directly in real time. We crosscheck these conclusions with our path integral quantum MonteCarlo and exact diagonalization results, as well as with the available numerically exact results from the literature.

Competition between Spin Excitation and Kondo Correlation in Magnetic Molecular Junctions: Theoretical Insight from First-principles-based Simulations

Abstract: In magnetic molecular junctions, the interactions between the local spin state at the transition- metal center and the conduction electrons from the electrodes or substrates can bring about many interesting strong correlation effects. Spin excitation and the Kondo effect are two representative phenomena, where the spin-unpaired d or f electrons plays the key role in forming these manybody states. This paper reviews the recent developments and applications of several first-principles methods in conjunction with the hierarchical equations of motion (HEOM) approach for the accurate simulation of magnetic molecular systems. The large-scale electrodes and substrates are treated by the density functional theory (DFT), while the properties of the magnetic center are studied by using the high-level complete active space self-consistent field method. The competition between the spin excitation and the Kondo effect are scrutinized by the HEOM approach. This combined DFT+HEOM method has proven to be useful for the accurate characterization of strongly-correlated magnetic molecular systems.

Application of the imaginary time hierarchical equations of motion method to calculate real time correlation functions.

We investigate the application of the imaginary time hierarchical equations of motion method to calculate real time quantum correlation functions. By starting from the path integral expression for the correlated system-bath equilibrium state, we first derive a new set of equations that decouple the imaginary time propagation and the calculation of auxiliary density operators. The new equations, thus, greatly simplify the calculation of the equilibrium correlated initial state that is subsequently used in the real time propagation to obtain the quantum correlation functions. It is also shown that a periodic decomposition of the bath imaginary time correlation function is no longer necessary in the new equations such that different decomposition schemes can be explored. The applicability of the new method is demonstrated in several numerical examples, including the spin-Boson model, the Holstein model, and the double-well model for proton transfer reaction.

Effects of counter-rotating-wave terms on the noisy frequency estimation

We investigate the problem of estimating the tunneling frequency of a two-level atomic system embedded in a dissipative environment by employing a numerically rigorous hierarchical equations of motion method. The effect of counter-rotating-wave terms on the attainable precision of the noisy quantum metrology is systematically studied beyond the usual framework of perturbative treatments. We find the counter-rotating-wave terms are able to boost the noisy quantum metrological performance in the intermediate and strong coupling regimes, whether the dissipative environment is composed of bosons or fermions. The result presented in this paper may pave a guideline to design a high-precision quantum estimation scenario under practical decoherence.

Two-dimensional vibronic spectroscopy with semiclassical thermofield dynamics.

Thermofield dynamics is an exactly correct formulation of quantum mechanics at finite temperature in which a wavefunction is governed by an effective temperature-dependent quantum Hamiltonian. The optimized mean trajectory (OMT) approximation allows the calculation of spectroscopic response functions from trajectories produced by the classical limit of a mapping Hamiltonian that includes physical nuclear degrees of freedom and other effective degrees of freedom representing discrete vibronic states. Here, we develop a thermofield OMT (TF-OMT) approach in which the OMT procedure is applied to a temperature-dependent classical Hamiltonian determined from the thermofield-transformed quantum mapping Hamiltonian. Initial conditions for bath nuclear degrees of freedom are sampled from a zero-temperature distribution. Calculations of two-dimensional electronic spectra and two-dimensional vibrational-electronic spectra are performed for models that include excitonically coupled electronic states. The TF-OMT calculations agree very closely with the corresponding OMT results, which, in turn, represent well benchmark calculations with the hierarchical equations of motion method.