Theoretical study of excitation energy transfer and nonlinear spectroscopy of photosynthetic light‐harvesting complexes using the nonperturbative reduced dynamics method

Role of Quantum Information in HEOM Trajectories

Open quantum systems often operate in the non-Markovian regime where a finite history of a trajectory is intrinsic to its evolution. The degree of non-Markovianity for a trajectory may be measured in terms of the amount of information flowing from the bath back into the system. In this study, we consider how information flows through the auxiliary density operators (ADOs) in the hierarchical equations of motion. We consider three cases for a range of baths, underdamped, intermediate, and overdamped. By understanding how information flows, we are able to determine the relative importance of different ADOs within the hierarchy. We show that ADOs sharing a common Matsubara axis behave similarly, while ADOs on different Matsubara axes behave differently. Using this knowledge, we are able to truncate hierarchies significantly, thus reducing the computation time, while obtaining qualitatively similar results. This is illustrated by comparing 2D electronic spectra for a molecule with an underdamped vibration subsumed into the bath spectral density.

The photodissociation dynamics and ultrafast electron diffraction image of cyclobutanone from the surface hopping dynamics simulation

The comprehension of nonadiabatic dynamics in polyatomic systems relies heavily on the simultaneous advancements in theoretical and experimental domains. The gas-phase ultrafast electron diffraction (UED) technique has attracted significant attention as a unique tool for monitoring photochemical and photophysical processes at the all-atomic level with high temporal and spatial resolutions. In this work, we simulate the UED spectra of cyclobutanone using the trajectory surface hopping method at the extended multi-state complete active space second order perturbation theory (XMS-CASPT2) level and thereby predict the results of the upcoming UED experiments in the Stanford Linear Accelerator Laboratory. The simulated results demonstrate that a few pathways, including the C2 and C3 dissociation channels, as well as the ring opening channel, play important roles in the nonadiabatic reactions of cyclobutanone. We demonstrate that the simulated UED signal can be directly interpreted in terms of atomic motions, which provides a unique way of monitoring the evolution of the molecular structure in real time. Our work not only provides numerical data that help to determine the accuracy of the well-known surface hopping dynamics at the high XMS-CASPT2 electronic-structure level but also facilitates the understanding of the microscopic mechanisms of the photoinduced reactions in cyclobutanone.

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.

Excitation energy transfer and vibronic relaxation through light-harvesting dendrimer building blocks: A nonadiabatic perspective

The light-harvesting excitonic properties of poly(phenylene ethynylene) (PPE) extended dendrimers (tree-like π-conjugated macromolecules) involve a directional cascade of local excitation energy transfer (EET) processes occurring from the "leaves" (shortest branches) to the "trunk" (longest branch), which can be viewed from a vibronic perspective as a sequence of internal conversions occurring among a connected graph of nonadiabatically coupled locally excited electronic states via conical intersections. The smallest PPE building block that is able to exhibit EET, the asymmetrically meta-substituted PPE oligomer with one acetylenic bond on one side and two parallel ones on the other side (hence, 2-ring and 3-ring para-substituted pseudo-fragments), is a prototype and the focus of the present work. From linear-response time-dependent density functional theory electronic-structure calculations of the molecule as regards its first two nonadiabatically coupled, optically active, singlet excited states, we built a (1 + 2)-state-8-dimensional vibronic-coupling Hamiltonian model for running subsequent multiconfiguration time-dependent Hartree wavepacket relaxations and propagations, yielding both steady-state absorption and emission spectra as well as real-time dynamics. The EET process from the shortest branch to the longest one occurs quite efficiently (about 80% quantum yield) within the first 25 fs after light excitation and is mediated vibrationally through acetylenic and quinoidal bond-stretching modes together with a particular role given to the central-ring anti-quinoidal rock-bending mode. Electronic and vibrational energy relaxations, together with redistributions of quantum populations and coherences, are interpreted herein through the lens of a nonadiabatic perspective, showing some interesting segregation among the foremost photoactive degrees of freedom as regards spectroscopy and reactivity.

Simulating the operation of a quantum computer in a dissipative environment

The operations of current quantum computers are still significantly affected by decoherence caused by interaction with the environment. In this work, we employ the non-perturbative hierarchical equations of motion (HEOM) method to simulate the operation of model quantum computers and reveal the effects of dissipation on the entangled quantum states and on the performance of well-known quantum algorithms. Multi-qubit entangled states in Shor's factorizing algorithm are first generated and propagated using the HEOM. It is found that the failure of factorization is accompanied by a loss of fidelity and mutual information. An important challenge in using the HEOM to simulate quantum computers in a dissipative environment is how to efficiently treat systems with many qubits. We propose a two-dimensional tensor network scheme for this problem and demonstrate its capability by simulating a one-dimensional random circuit model with 21 qubits.

Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra

Modern 4-wave mixing spectroscopies are expensive to obtain experimentally and computationally. In certain cases, the unfavorable scaling of quantum dynamics problems can be improved using a generalized quantum master equation (GQME) approach. However, the inclusion of multiple (light-matter) interactions complicates the equation of motion and leads to seemingly unavoidable cubic scaling in time. In this paper, we present a formulation that greatly simplifies and reduces the computational cost of previous work that extended the GQME framework to treat arbitrary numbers of quantum measurements. Specifically, we remove the time derivatives of quantum correlation functions from the modified Mori-Nakajima-Zwanzig framework by switching to a discrete-convolution implementation inspired by the transfer tensor approach. We then demonstrate the method's capabilities by simulating 2D electronic spectra for the excitation-energy-transfer dimer model. In our method, the resolution of data can be arbitrarily coarsened, especially along the t2 axis, which mirrors how the data are obtained experimentally. Even in a modest case, this demands O(103) fewer data points. We are further able to decompose the spectra into one-, two-, and three-time correlations, showing how and when the system enters a Markovian regime where further measurements are unnecessary to predict future spectra and the scaling becomes quadratic. This offers the ability to generate long-time spectra using only short-time data, enabling access to timescales previously beyond the reach of standard methodologies.

First-principles simulation of excitation energy transfer and transient absorption spectroscopy in the CP29 light-harvesting complex

The dynamics of delocalized excitons in light-harvesting complexes (LHCs) can be investigated using different experimental techniques, and transient absorption (TA) spectroscopy is one of the most valuable methods for this purpose. A careful interpretation of TA spectra is essential for the clarification of excitation energy transfer (EET) processes occurring during light-harvesting. However, even in the simplest LHCs, a physical model is needed to interpret transient spectra as the number of EET processes occurring at the same time is very large to be disentangled from measurements alone. Physical EET models are commonly built by fittings of the microscopic exciton Hamiltonians and exciton-vibrational parameters, an approach that can lead to biases. Here, we present a first-principles strategy to simulate EET and transient absorption spectra in LHCs, combining molecular dynamics and accurate multiscale quantum chemical calculations to obtain an independent estimate of the excitonic structure of the complex. The microscopic parameters thus obtained are then used in EET simulations to obtain the population dynamics and the related spectroscopic signature. We apply this approach to the CP29 minor antenna complex of plants for which we follow the EET dynamics and transient spectra after excitation in the chlorophyll b region. Our calculations reproduce all the main features observed in the transient absorption spectra and provide independent insight on the excited-state dynamics of CP29. The approach presented here lays the groundwork for the accurate simulation of EET and unbiased interpretation of transient spectra in multichromophoric systems.

What is Missing in the Mean Field Description of Spatial Distribution of Population? Important Role of Auxiliary Wave Packets in Trajectory Branching

When the traditional Ehrenfest mean field approach is employed to simulate nonadiabatic dynamics, an effective wave packet (WP) on the average potential energy surface (PES) is utilized to describe the nuclear motion. In the fully quantum picture, however, the WP components on different adiabatic PESs gradually separate in space because they evolve under different velocities and forces. Due to trajectory branching of the WP components, proper decoherence needs to be taken into account, and the spatial distribution of population cannot be described by a single effective WP. Here, we propose an auxiliary branching corrected mean field (A-BCMF) method, where trajectories of auxiliary WPs on adiabatic PESs are introduced. As benchmarked in the three standard Tully models, A-BCMF not only gives correct channel populations but also captures an accurate time-dependent spatial distribution of population. Thereby, we reveal the important role of auxiliary WPs in solving intrinsic problems of the widely used mean field description of nonadiabatic dynamics.

Learning Decoherence Time Formulas for Surface Hopping from Quantum Dynamics

Surface hopping simulations have achieved great success in many different fields, but their reliability has long been limited by the overcoherence problem. We here present a general machine learning assisted approach to identify optimal decoherence time formulas for surface hopping using exact quantum dynamics as references. In order to avoid computationally expensive force calculations, we use the nuclear kinetic energy and the adiabatic energy difference to iteratively generate the descriptor space. Through multilayer screening of the candidate descriptors and discrete optimization of the relevant parameters, we obtain new energy-based decoherence time formulas. As benchmarked in thousands of diverse multilevel systems and six standard scattering models, surface hopping with our new decoherence time formulas nearly reproduces the exact quantum dynamics while maintaining high efficiency. Thereby, our approach provides a promising avenue for systematically improving the accuracy of surface hopping simulations in complex systems from quantum dynamics data.

Two-dimensional fluorescence excitation spectroscopy: A novel technique for monitoring excited-state photophysics of molecular species with high time and frequency resolution

We propose a novel UV/Vis femtosecond spectroscopic technique, two-dimensional fluorescence-excitation (2D-FLEX) spectroscopy, which combines spectral resolution during the excitation process with exclusive monitoring of the excited-state system dynamics at high time and frequency resolution. We discuss the experimental feasibility and realizability of 2D-FLEX, develop the necessary theoretical framework, and demonstrate the high information content of this technique by simulating the 2D-FLEX spectra of a model four-level system and the Fenna-Matthews-Olson antenna complex. We show that the evolution of 2D-FLEX spectra with population time directly monitors energy transfer dynamics and can thus yield direct qualitative insight into the investigated system. This makes 2D-FLEX a highly efficient instrument for real-time monitoring of photophysical processes in polyatomic molecules and molecular aggregates.

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.

A Comprehensive Approach to Exciton Delocalization and Energy Transfer

Electrostatic intermolecular interactions lie at the heart of both the Förster model for resonance energy transfer (RET) and the exciton model for energy delocalization. In the Förster theory of RET, the excitation energy incoherently flows from the energy donor to a weakly coupled energy acceptor. The exciton model describes instead the energy delocalization in aggregates of identical (or nearly so) molecules. Here, we introduce a model that brings together molecular aggregates and RET. We will consider a couple of molecules, each described in terms of two diabatic electronic states, coupled to an effective molecular vibration. Electrostatic intermolecular interactions drive energy fluxes between the molecules, that, depending on model parameters, can be described as RET or energy delocalization. At variance with the standard Förster model for RET and of the exciton model for aggregates, our approach applies both in the weak and in the strong coupling regimes and fully accounts for the quantum nature of molecular vibrations in a nonadiabatic approach. Coupling the system to a thermal bath, we follow RET and energy delocalization in real time and simulate time-resolved emission spectra.

Trajectory Propagation of Symmetrical Quasi-classical Dynamics with Meyer-Miller Mapping Hamiltonian Using Machine Learning

The long short-term memory recurrent neural network (LSTM-RNN) approach is applied to realize the trajectory-based nonadiabatic dynamics within the framework of the symmetrical quasi-classical dynamics method based on the Meyer-Miller mapping Hamiltonian (MM-SQC). After construction, the LSTM-RNN model allows us to propagate the entire trajectory evolutions of all involved degrees of freedoms (DOFs) from initial conditions. The proposed idea is proven to be reliable and accurate in the simulations of the dynamics of several site-exciton electron-phonon coupling models and three Tully's scattering models. It indicates that the LSTM-RNN model perfectly captures the dynamical information on the trajectory evolution in the MM-SQC dynamics. Our work proposes a novel machine learning approach in the simulation of trajectory-based nonadiabatic dynamic of complex systems with a large number of DOFs.

Coherence and Efficient Energy Transfer in Molecular Wires: Insights from Surface Hopping Simulations

Understanding the dynamics of electronic energy transfer through a molecular wire is essential to understand the working of natural processes like photosynthesis. We investigate simpler 2 and 3-site model Hamiltonians in this work to understand the importance of coherence to efficient energy transfer. We compare the results of surface hopping simulation with that of numerically exact results and rate theories. Different parameters are analyzed, motivated by a photosynthetic molecular wire - the FMO complex. A comparison of results from different theories shows that coherence can play an important role towards efficient energy transfer for certain parameters. When these coherences are important, even small couplings (of the order of 5 cm^-1 ) in the Hamiltonian can significantly affect rates. Surface hopping simulations capture all the results correctly qualitatively. Rate theories, on the other hand, can differ significantly from numerically exact results when coherences become important. The results of this work should provide design guidelines for efficient energy transfer in molecular wires.

Substituent Effect on Vibrationally Resolved Absorption Spectra and Exciton Dynamics of Dipyrrolonaphthyridinedione Aggregates

Dipyrrolonaphthyridinedione (DPND) thin films exhibit interesting photophysical properties and singlet fission (SF) processes. A recent experimental work found that the alkyl substitution in the DPND skeleton has the remarkable influence on the characteristics of electronic absorption spectra and SF rates. Here, we theoretically elucidate the microscopic mechanism of the substituent effect on the optical properties and exciton dynamics of materials by combining the electronic structure calculations and the quantum dynamics simulations. The results show that the alkyl substituent has a minor effect on the single molecular properties but dramatically changes those of DPND aggregates via varying the intermolecular interactions. The aggregates of DPND with and without alkyl side chains exhibit the more likely characters of H-type aggregations. In the former (DPND6), the weak degree of mixing of intramolecular localized excited (LE) states and intermolecular charge transfer (CT) states makes the low-energy absorption band possess the predominant optical absorption, while in the latter (DPND), the CT and LE states are close in energy, together with their strong interaction, resulting in the substantial state-mixing, so that its two low-energy absorption bands have nearly equal oscillator strengths and a wide energy spacing of more than 0.5 eV. The simulation of exciton dynamics elucidates that the photoinitiated states in both aggregates cannot generate the free charge carrier because of the lack of enough driving forces. However, the population exchanges between LE and CT states in DPND aggregates are much faster than in DPND6 aggregates, indicating the different SF behaviors, consistent with the experimental observation.

Coherent excitation energy transfer in model photosynthetic reaction center: Effects of non-Markovian quantum environment

Excitation energy transfer (EET) and electron transfer (ET) are crucially involved in photosynthetic processes. In reality, the photosynthetic reaction center constitutes an open quantum system of EET and ET, which manifests an interplay of pigments, solar light and phonon baths. So far theoretical studies have been mainly based on master equation approaches in the Markovian condition. The non-Markovian environmental effect, which may play a crucial role, has not been sufficiently considered. In this work, we propose a mixed dynamic approach to investigate this open system. The influence of phonon bath is treated via the exact dissipaton equation of motion (DEOM) while that of photon bath is via the Lindblad master equation. Specifically, we explore the effect of non-Markovian quantum phonon bath on the coherent transfer dynamics and its manipulation on the current--voltage behavior. Distinguished from the results of completely Markovian Lindblad equation and those adopting classical environment description, the mixed DEOM--Lindblad simulations exhibittransfer coherence up to a few hundreds femtosecondsand the related environmental manipulation effect on current.These non-Markovian quantum coherent effects may be extended tomore complex and realistic systems and be helpful to thedesign of organic photovoltaic devices.

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.

The influence of a Hamiltonian vibration vs a bath vibration on the 2D electronic spectra of a homodimer

We elucidate the influence of the system–bath boundary placement within an open quantum system, with emphasis on the two-dimensional electronic spectra, through the application of the hierarchical equations of motion formalism for an exciton system. We apply two different models, the Hamiltonian vibration model (HVM) and bath vibration model (BVM), to a monomer and a homodimer. In the HVM, we specifically include the vibronic states in the Hamiltonian capturing vibronic quenching, whereas in the BVM, all vibrational details are contained within the bath and described by an underdamped spectral density. The resultant spectra are analyzed in terms of energetic peak position and thermodynamic broadening precision in order to evaluate the efficacy of the two models. The HVM produces 2D spectra with accurate peak positional information, while the BVM is well suited to modeling dynamic peak broadening. For the monomer, both models produce equivalent spectra in the limit where additional damping associated with the underdamped vibration in the BVM approaches zero. This is supported by analytical results. However, for the homodimer, the BVM spectra are redshifted with respect to the HVM due to an absence of vibronic quenching in the BVM. The computational efficiency of the two models is also discussed in order to inform us of the most appropriate use of each method.

Ultrafast Internal Conversion Dynamics through the on-the-Fly Simulation of Transient Absorption Pump-Probe Spectra with Different Electronic Structure Methods

An on-the-fly surface-hopping simulation protocol is developed for the evaluation of transient absorption (TA) pump-probe (PP) signals of molecular systems exhibiting internal conversion to the electronic ground state. We study the nonadiabatic dynamics of azomethane and the associating TA PP spectra at three levels of the electronic-structure theory, OM2/MRCI, SA-CASSCF, and XMS-CASPT2. The impact of these methods on the population dynamics and time-resolved TA PP signals is substantially different. This difference is attributed to the strong non-Condon effects that must be taken into account for the proper understanding and interpretation of time-resolved TA PP signals of nonadiabatic polyatomic systems. This shows that the combination of the dynamical and spectral simulations definitely provides more accurate and detailed information on the microscopic mechanisms of photophysical and photochemical processes. Hence the simulation of time-resolved spectroscopic signals provides another important dimension to examine the accuracy of quantum chemistry methods.

Simulation of Open Quantum Dynamics with Bootstrap-Based Long Short-Term Memory Recurrent Neural Network

The recurrent neural network with the long short-term memory cell (LSTM-NN) is employed to simulate the long-time dynamics of open quantum systems. The bootstrap method is applied in the LSTM-NN construction and prediction, which provides a Monte Carlo estimation of a forecasting confidence interval. Within this approach, a large number of LSTM-NNs are constructed by resampling time-series sequences that were obtained from the early stage quantum evolution given by numerically exact multilayer multiconfigurational time-dependent Hartree method. The built LSTM-NN ensemble is used for the reliable propagation of the long-time quantum dynamics, and the simulated result is highly consistent with the exact evolution. The forecasting uncertainty that partially reflects the reliability of the LSTM-NN prediction is also given. This demonstrates the bootstrap-based LSTM-NN approach is a practical and powerful tool to propagate the long-time quantum dynamics of open systems with high accuracy and low computational cost.

A quantum Langevin equation approach for two-dimensional electronic spectra of coupled vibrational and electronic dynamics

We present an efficient method to simulate two-dimensional (2D) electronic spectra of condensed-phase systems with an emphasis on treating quantum nuclear wave packet dynamics explicitly. To this end, we combine a quantum Langevin equation (QLE) approach for dissipation and a perturbative scheme to calculate three-pulse photon-echo polarizations based on wave packet dynamics under the influence of external fields. The proposed dynamical approach provides a consistent description of nuclear quantum dynamics, pulse-overlap effects, and vibrational relaxation, enabling simulations of 2D electronic spectra with explicit and non-perturbative treatment of coupled electronic-nuclear dynamics. We apply the method to simulate 2D electronic spectra of a displaced-oscillator model in the condensed phase and discuss the spectral and temporal evolutions of 2D signals. Our results show that the proposed QLE approach is capable of describing vibrational relaxation, decoherence, and vibrational coherence transfer, as well as their manifestations in spectroscopic signals. Furthermore, vibrational quantum beats specific for excited-state vs ground-state nuclear wave packet dynamics can also be identified. We anticipate that this method will provide a useful tool to conduct theoretical studies of 2D spectroscopy for strong vibronically coupled systems and to elucidate intricate vibronic couplings in complex molecular systems.