All-Mode Quantum–Classical Path Integral Simulation of Bacteriochlorophyll Dimer Exciton-Vibration Dynamics

Variational Quantum Algorithm for Non-Markovian Quantum Dynamics Using an Ensemble of Ehrenfest Trajectories

The simulation of non-Markovian quantum dynamics plays an important role in the understanding of charge and exciton dynamics in the condensed phase environment, yet such a simulation remains computationally expensive on classical computers. In this work, we develop a variational quantum algorithm that is capable of simulating non-Markovian quantum dynamics on quantum computers. The algorithm captures the non-Markovian effect by employing the Ehrenfest trajectories and Monte Carlo sampling of their thermal distribution. We test the algorithm with the spin-boson model on the quantum simulator, and the results match quantitatively with the exact ones. The algorithm naturally fits into the parallel computing platform of the NISQ devices and can be extended to anharmonic system-bath interactions and multistate systems.

Simulating Non-Markovian Dynamics in Multidimensional Electronic Spectroscopy via Quantum Algorithm

Including the effect of the molecular environment in the numerical modeling of time-resolved electronic spectroscopy remains an important challenge in computational spectroscopy. In this contribution, we present a general approach for the simulation of the optical response of multichromophore systems in a structured environment and its implementation as a quantum algorithm. A key step of the procedure is the pseudomode embedding of the system-environment problem resulting in a finite set of quantum states evolving according to a Markovian quantum master equation. This formulation is then solved by a collision model integrated into a quantum algorithm designed to simulate linear and nonlinear response functions. The workflow is validated by simulating spectra for the prototypical excitonic dimer interacting with fast (memoryless) and finite-memory environments. The results demonstrate, on the one hand, the potential of the pseudomode embedding for simulating the dynamical features of nonlinear spectroscopy, including lineshape, spectral diffusion, and relaxations along delay times. On the other hand, the explicit synthesis of quantum circuits provides a fully quantum simulation protocol of nonlinear spectroscopy harnessing the efficient quantum simulation of many-body dynamics promised by the future generation of fault-tolerant quantum computers.

Understanding the Photoprocesses in Biological Systems: Need for Accurate Multireference Treatment

Light-matter interaction is crucial to life itself and revolves around many of the central processes in biology. The need for understanding these photochemical and photophysical processes cannot be overemphasized. Interaction of light with biological systems starts with the absorption of light and subsequent phenomena that occur in the excited states of the system. However, excited states are typically difficult to understand within the mean field approximation of quantum chemical methods. Therefore, suitable multireference methods and methodologies have been developed to understand these phenomena. In this Perspective, we will describe a few methods and methodologies suitable for these descriptions and discuss some persisting difficulties.

PigmentHunter: A point-and-click application for automated chlorophyll-protein simulations

Chlorophyll proteins (CPs) are the workhorses of biological photosynthesis, working together to absorb solar energy, transfer it to chemically active reaction centers, and control the charge-separation process that drives its storage as chemical energy. Yet predicting CP optical and electronic properties remains a serious challenge, driven by the computational difficulty of treating large, electronically coupled molecular pigments embedded in a dynamically structured protein environment. To address this challenge, we introduce here an analysis tool called PigmentHunter, which automates the process of preparing CP structures for molecular dynamics (MD), running short MD simulations on the nanoHUB.org science gateway, and then using electrostatic and steric analysis routines to predict optical absorption, fluorescence, and circular dichroism spectra within a Frenkel exciton model. Inter-pigment couplings are evaluated using point-dipole or transition-charge coupling models, while site energies can be estimated using both electrostatic and ring-deformation approaches. The package is built in a Jupyter Notebook environment, with a point-and-click interface that can be used either to manually prepare individual structures or to batch-process many structures at once. We illustrate PigmentHunter's capabilities with example simulations on spectral line shapes in the light harvesting 2 complex, site energies in the Fenna-Matthews-Olson protein, and ring deformation in photosystems I and II.

Empirical asymmetric phonon sideband due to phonons in the protein matrix present in photosynthetic complexes: Time-domain response theory

The phonon spectral density plays a key role in probing the dynamical and spectral behavior of molecular aggregates. One may utilize the intimate connection between the one-phonon profile and the phonon spectral density to extract a plausible form of the spectral density of media with rich structure using advanced optical spectroscopy. The excitonic transition is normally accompanied by a broad, asymmetric phonon-side band due to the coupling to the phonons in the surrounding protein matrix present in photosynthetic complexes. The asymmetry in the one-phonon profile of a homogeneous absorption spectrum and other experiments performed on photosynthetic bacterial reaction centers (BRCs) led the Small group to employ a half-Gaussian distribution function on the red side and half-Lorentzian distribution function on the blue side of the absorption lineshape to account for the one-phonon profile asymmetrical shape and relaxation effects contributing to spectroscopy and dynamics of BRCs at hand. Different research groups successfully employed the theory of Small to simulate their photosynthetic spectral data so they could calculate the homogeneous absorption and hole-burned spectra of photosynthetic complexes. Although this report does not directly use the formulae of homogeneous absorption, hole-burning, and fluorescence line-narrowed spectra of BRCs, and photosynthetic complexes, developed by Hayes-Small, it builds on their idea of the phonon sideband asymmetric shape in deriving an accurate and computationally efficient linear electronic transition dipole moment time correlation function. Besides the compelling tractability and efficiency of this correlation function, it accounts for excitonic coupling and eliminates all the inconsistencies arising in the Hayes-Small theory.

Path Integral over Equivalence Classes for Quantum Dynamics with Static Disorder

An efficient, fully quantum mechanical, real-time path integral method for including the effects of static disorder in the dynamics of systems coupled to common or local harmonic baths is presented. Rather than performing a large number of demanding calculations for different realizations of the system Hamiltonian, the influence of the bath is captured through a single evaluation of the path sum by grouping the system paths into equivalence classes of fixed system amplitudes. The method is illustrated with several analytical and numerical examples that show a variety of nontrivial effects arising from the interplay among coherence, dissipation, thermal fluctuations, and geometric phases.

Electron-Binding Dynamics of the Dipole-Bound State: Correlation Effect on the Autodetachment Dynamics

The nature of the electron-binding forces in the dipole-bound states (DBS) of anions is interrogated through experimental and theoretical means by investigating the autodetachment dynamics from DBS Feshbach resonances of ortho-, meta-, and para-bromophenoxide (BrPhO-). Though the charge-dipole electrostatic potential has been widely regarded to be mainly responsible for the electron binding in DBS, the effect of nonclassical electron correlation has been conceived to be quite significant in terms of its static and/or dynamic contributions toward the binding of the excess electron to the neutral core. State-specific real-time autodetachment dynamics observed by picosecond time-resolved photoelectron velocity-map imaging spectroscopy reveal that the autodetachment processes from the DBS Feshbach resonances of BrPhO- anions cannot indeed be rationalized by the conventional charge-dipole potential. Specifically, the autodetachment lifetime is drastically lengthened depending on differently positioned Br-substitution, and this rate change cannot be explained within the framework of Fermi's golden rule based on the charge-dipole assumption. High-level ab initio quantum chemical calculations with EOM-EA-CCSD, which intrinsically takes into account electron correlations, generate more reasonable predictions on the binding energies than density functional theory (DFT) calculations, and semiclassical quantum dynamics simulations based on the EOM-EA-CCSD data excellently predict the trend in the autodetachment rates. These findings illustrate that static and dynamic properties of the excess electron in the DBS are strongly influenced by correlation interactions among electrons in the nonvalence orbital of the dipole-bound electron and highly polarizable valence orbitals of the bromine atom, which, in turn, dictate the interesting chemical fate of exotic anion species.

Impact of Spatial Inhomogeneity on Excitation Energy Transport in the Fenna-Matthews-Olson Complex

The dynamics of the excitation energy transfer (EET) in photosynthetic complexes is an interesting question both from the perspective of fundamental understanding and the research in artificial photosynthesis. Over the past decade, very accurate spectral densities have been developed to capture spatial inhomogeneities in the Fenna-Matthews-Olson (FMO) complex. However, challenges persist in numerically simulating these systems, both in terms of parameterizing them and following their dynamics over long periods of time because of long non-Markovian memories. We investigate the dynamics of FMO with the exact treatment of various theoretical spectral densities using the new tensor network path integral-based methods, which are uniquely capable of addressing the difficulty of long memory length and incoherent Förster theory. It is also important to be able to analyze the pathway of EET flow, which can be difficult to identify given the non-trivial structure of connections between bacteriochlorophyll molecules in FMO. We use the recently introduced ideas of relating coherence to population derivatives to analyze the transport process and reveal some new routes of transport. The combination of exact and approximate methods sheds light on the role of coherences in affecting the fine details of the transport and promises to be a powerful toolbox for future exploration of other open systems with quantum transport.

Coherence Maps and Flow of Excitation Energy in the Bacterial Light Harvesting Complex 2

We present and analyze coherence maps [ J. Phys. Chem. B 2022, 126, 9361-9375] to investigate the quantum coherences that are created, sustained, and damped by vibrational modes during the transfer of excitation energy from the B800 (outer) to the B850 (inner) ring of the light harvesting complex 2 (LH2) of purple bacteria with a variety of initial conditions. The reduced density matrix of the 24-pigment complex, where the ground and excited electronic states of each bacteriochlorophyll are explicitly coupled to 50 intramolecular vibrations at room temperature, is obtained from fully quantum-mechanical small matrix path integral (SMatPI) calculations. The coherence maps show a very rapid localization within the outer ring, accompanied by the formation of inter-ring quantum superpositions that evolve to a partial quantum delocalization at equilibrium, and quantify in state-to-state detail the flow of energy within the complex.

Topological aspects of system-bath Hamiltonians and a vector model for multisite systems coupled to local, correlated, or common baths

Some topological features of multisite Hamiltonians consisting of harmonic potential surfaces with constant site-to-site couplings are discussed. Even in the absence of Duschinsky rotation, such a Hamiltonian assumes the system-bath form only if severe constraints exist. The simplest case of a common bath that couples to all sites is realized when the potential minima are collinear. The bath reorganization energy increases quadratically with site distance in this case. Another frequently encountered situation involves exciton-vibration coupling in molecular aggregates, where the intramolecular normal modes of the monomers give rise to local harmonic potentials. In this case, the reorganization energy accompanying excitation transfer is independent of site-to-site separation, thus this situation cannot be described by the usual system-bath Hamiltonian. A vector system-bath representation is introduced, which brings the exciton-vibration Hamiltonian in system-bath form. In this, the system vectors specify the locations of the potential minima, which in the case of identical monomers lie on the vertices of a regular polyhedron. By properly choosing the system vectors, it is possible to couple each bath to one or more sites and to specify the desired initial density. With a collinear choice of system vectors, the coupling reverts to the simple form of a common bath. The compact form of the vector system-bath coupling generalizes the dissipative tight-binding model to account for local, correlated, and common baths. The influence functional for the vector system-bath Hamiltonian is obtained in a compact and simple form.

B800-to-B850 relaxation of excitation energy in bacterial light harvesting: All-state, all-mode path integral simulations

We report fully quantum mechanical simulations of excitation energy transfer within the peripheral light harvesting complex (LH2) of Rhodopseudomonas molischianum at room temperature. The exciton–vibration Hamiltonian comprises the 16 singly excited bacteriochlorophyll states of the B850 (inner) ring and the 8 states of the B800 (outer) ring with all available electronic couplings. The electronic states of each chromophore couple to 50 intramolecular vibrational modes with spectroscopically determined Huang–Rhys factors and to a weakly dissipative bath that models the biomolecular environment. Simulations of the excitation energy transfer following photoexcitation of various electronic eigenstates are performed using the numerically exact small matrix decomposition of the quasiadiabatic propagator path integral. We find that the energy relaxation process in the 24-state system is highly nontrivial. When the photoexcited state comprises primarily B800 pigments, a rapid intra-band redistribution of the energy sharply transitions to a significantly slower relaxation component that transfers 90% of the excitation energy to the B850 ring. The mixed character B850* state lacks the slow component and equilibrates very rapidly, providing an alternative energy transfer channel. This (and also another partially mixed) state has an anomalously large equilibrium population, suggesting a shift to lower energy by virtue of exciton–vibration coupling. The spread of the vibrationally dressed states is smaller than that of the eigenstates of the bare electronic Hamiltonian. The total population of the B800 band is found to decay exponentially with a 1/ e time of 0.5 ps, which is in good agreement with experimental results.

Tensor Network Path Integral Study of Dynamics in B850 LH2 Ring with Atomistically Derived Vibrations

The recently introduced multisite tensor network path integral (MS-TNPI) allows simulation of extended quantum systems coupled to dissipative media. We use MS-TNPI to simulate the exciton transport and the absorption spectrum of a B850 bacteriochlorophyll (BChl) ring. The MS-TNPI network is extended to account for the ring topology of the B850 system. Accurate molecular-dynamics-based description of the molecular vibrations and the protein scaffold is incorporated through the framework of Feynman-Vernon influence functional. To relate the present work with the excitonic picture, an exploration of the absorption spectrum is done by simulating it using approximate and topologically consistent transition dipole moment vectors. Comparison of these numerically exact MS-TNPI absorption spectra are shown with second-order cumulant approximations. The effect of temperature on both the exact and the approximate spectra is also explored.

Intramolecular Vibrations in Excitation Energy Transfer: Insights from Real-Time Path Integral Calculations

Excitation energy transfer (EET) is fundamental to many processes in chemical and biological systems and carries significant implications for the design of materials suitable for efficient solar energy harvest and transport. This review discusses the role of intramolecular vibrations on the dynamics of EET in nonbonded molecular aggregates of bacteriochlorophyll, a perylene bisimide, and a model system, based on insights obtained from fully quantum mechanical real-time path integral results for a Frenkel exciton Hamiltonian that includes all vibrational modes of each molecular unit at finite temperature. Generic trends, as well as features specific to the vibrational characteristics of the molecules, are identified. Weak exciton-vibration (EV) interaction leads to compact, near-Gaussian densities on each electronic state, whose peak follows primarily a classical trajectory on a torus, while noncompact densities and nonlinear peak evolution are observed with strong EV coupling. Interaction with many intramolecular modes and increasing aggregate size smear, shift, and damp these dynamical features. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Computational elucidations on the role of vibrations in energy transfer processes of photosynthetic complexes

Coupling between pigment excitations and nuclear movements in photosynthetic complexes is known to modulate the excitation energy transfer (EET) efficiencies. Toward providing microscopic information, researchers often apply simulation techniques and investigate how vibrations are involved in EET processes. Here, reports on such roles of nuclear movements are discussed from a theory perspective. While vibrations naturally present random thermal fluctuations that can affect energy transferring characteristics, they can also be intertwined with exciton structures and create more specific non-adiabatic energy transfer pathways. For reliable simulations, a bath model that accurately mimics a given molecular system is required. Methods for obtaining such a model in combination with quantum chemical electronic structure calculations and molecular dynamics trajectory simulations are discussed. Various quantum dynamics simulation tools that can handle pigment-to-pigment energy transfers together with their vibrational characters are also touched on. Behaviors of molecular vibrations often deviate from ideality, especially when all-atom details are included, which practically forces us to treat them classically. We conclude this perspective by considering some recent reports that suggest that classical descriptions of bath effects with all-atom details may still produce valuable information for analyzing sophisticated contributions by vibrations to EET processes.

Cooperation between Excitation Energy Transfer and Antisynchronously Coupled Vibrations

The effects of the environment in energy transfer systems have been continuously studied for decades. Here, we investigate how the energy transfer and the emergence of vibrational correlations cooperate with each other based on simulations with a few numerically approximate mixed quantum classical (MQC) methods. By adopting a two-state system with locally coupled underdamped vibrations that are resonant with the electronic energy gap, we observe prominent energy dissipations from the electronic system to the vibrations, rehighlighting the role of underdamped vibrations as a temporal electronic energy buffer. More importantly, this energy dissipation generates specific phase relations between the two vibrations. Namely, the vibrations become anticorrelated right after the initiation of the energy transfer but then synchronized as the transfer completes. These phase relations are interpreted as a selective activation of an anticorrelated motion of the vibrations and a subsequent deactivation by thermal energy redistribution. Furthermore, we show that a single vibration simultaneously coupled to the two electronic states with opposite phases induces a completely equivalent energy transfer dynamics as the two localized vibrations. Finally, we discuss how the vibrational energy dissipation dynamics is affected by the adopted MQC approaches and warn about the increased subtlety toward properly treating dissipation effects over having reliable population dynamics.

Density matrix and purity evolution in dissipative two-level systems: I. Theory and path integral results for tunneling dynamics

The time evolution of the purity (the trace of the square of the reduced density matrix) and von Neumann entropy in a symmetric two-level system coupled to a dissipative harmonic bath is investigated through analytical arguments and accurate path integral calculations on simple models and the singly excited bacteriochlorophyll dimer. A simple theoretical analysis establishes bounds and limiting behaviors. The contributions to purity from a purely incoherent term obtained from the diagonal elements of the reduced density matrix, a term associated with the difference of the two eigenstate populations, and a third term related to the square of the time derivative of a site population, are discussed in various regimes. In the case of tunneling dynamics from a localized initial condition, the complex interplay among these contributions leads to the recovery of purity under low-temperature, weakly dissipative conditions. Memory effects from the bath are found to play a critical role to the dynamics of purity. It is shown that the strictly quantum mechanical decoherence process associated with spontaneous phonon emission is responsible for the long-time recovery of purity. These analytical and numerical results show clearly that the loss of quantum coherence during the evolution toward equilibrium does not necessarily imply the decay of purity, and that the time scales relevant to these two processes may be entirely different.

Quantum-Classical Path Integral Simulation of Excess Proton Dynamics in a Water Dimer Embedded in the Gramicidin Channel

We use the quantum-classical path integral (QCPI) methodology to investigate the relaxation dynamics of an excess proton that has been inserted in a water dimer embedded in the gramicidin A channel at room temperature. We obtain one-dimensional potential slices for the quantum degree of freedom through a proper transformation to internal coordinates. Our results indicate that the proton transfer is driven by the oscillation of the oxygen pair, and that the transfer occurs primarily at single-well or nearby low-barrier configurations. Yet, we find that tunneling and zero-point energy lead to a significant acceleration of the proton transfer dynamics.

Electronic-vibrational density evolution in a perylene bisimide dimer: mechanistic insights into excitation energy transfer

The process of excitation energy transfer (EET) in molecular aggregates is etched with the signatures of a multitude of electronic and vibrational time scales that often are extremely difficult to resolve. The effect of the motion associated with one molecular vibration on that of another is fundamental to the dynamics of EET. In this paper we present simple theoretical ideas along with fully quantum mechanical calculations to develop a comprehensive mechanistic picture of EET in terms of the time evolution of electronic-vibrational densities (EVD) in a perylene bisimide (PBI) dimer, where 28 intramolecular normal modes couple to the ground and excited electronic states of each molecule. The EVD motion exhibits a plethora of dynamical features, which impart physical justification for the composite effects observed in the EET dynamics. Weakly coupled vibrations lead to classical-like motion of the EVD center on each electronic state, while highly nontrivial EVD characteristics develop under moderate or strong exciton-vibration interaction, leading to the formation of split or crescent-shaped densities, as well as density retention that slows down energy transfer and creates new peaks in the electronic populations. Pronounced correlation effects are observed in two-mode projections of the EVD, as a consequence of indirect vibrational coupling between uncoupled normal modes induced by the electronic coupling. Such indirect coupling depends on the strength of exciton-vibration interactions as well as the frequency mismatch between the two modes and leaves nontrivial signatures in the electronic population dynamics. The collective effects of many vibrational modes cause a partial smearing of these features through dephasing.

Spectroscopic response theory with classical mapping Hamiltonians

Exact quantum dynamics with a time-independent Hamiltonian in a discrete state space can be computed using classical mechanics through the classical Meyer-Miller-Stock-Thoss mapping Hamiltonian. In order to compute quantum response functions from classical dynamics, we extend this mapping to a quantum Hamiltonian with time-dependence arising from a classical field. This generalization requires attention to time-ordering in quantum and classical propagators. Quantum response theory with the original quantum Hamiltonian is equivalent to classical response theory with the classical mapping Hamiltonian. We elucidate the structure of classical response theory with the mapping Hamiltonian, thereby generating classical versions of the two-sided quantum density operator diagrams conventionally used to describe spectroscopic processes. This formal development can provide a foundation for new semiclassical approximations to spectroscopic observables for models in which classical nuclear degrees of freedom are introduced into a mapping Hamiltonian describing electronic states. Calculations of the temperature-dependence of two-dimensional electronic spectra for an exciton dimer using two semiclassical approaches are compared with benchmark calculations using the hierarchical equations of motion method.

Real-Time Path Integral Simulation of Exciton-Vibration Dynamics in Light-Harvesting Bacteriochlorophyll Aggregates

The mechanism of excitation energy transfer in photoexcited bacteriochlorophyll (BChl) aggregates poses intriguing questions, which have important implications for the observed efficiency of photosynthesis. We investigate this process through fully quantum mechanical calculations of exciton-vibration dynamics in chains and rings of BChl a molecules, with parameters characterizing the B850 ring of the LH2 complex of photosynthetic bacteria. The calculations are performed using the modular path integral methodology, which allows the exact treatment of 50 intramolecular vibrations on each pigment using parameters obtained from spectroscopic Huang-Rhys factors with computational effort that scales linearly with aggregate length. Our results indicate that the interplay between electronic and vibrational time scales leads to the rapid suppression but not the overdamping of electronic coherence, which facilitates the spreading of excitation energy throughout the aggregate.