Influence of phonons on exciton transfer dynamics: comparison of the Redfield, Förster, and modified Redfield equations

Electronic-vibrational resonance damping time-dependent photosynthetic energy transfer acceleration revealed by 2D electronic spectroscopy

The effects of damping time of electronic-vibrational resonance modes on energy transfer in photosynthetic light-harvesting systems are examined. Using the hierarchical equations of motion (HEOM) method, we simulate the linear absorption and two-dimensional electronic spectra (2DES) for a dimer model based on bottleneck sites in the light-harvesting complex of photosystem II. A site-dependent spectral density is incorporated, with only the low-energy site being coupled to the resonance mode. Similar patterns are observed in linear absorption spectra and early time 2DES for various damping times, owing to the weak coupling strength. However, notable differences emerge in the dynamics of the high-energy diagonal and cross-peaks in the 2DES. It is found that the coupling of electronic-vibrational resonance modes accelerates the energy transfer process, with rates being increased as the damping time is extended, but the impact becomes negligible when the damping time exceeds a certain threshold. To evaluate the reliability of the perturbation method, the modified Redfield (MR) method is employed to simulate 2DES under the same conditions. The results from the MR method are aligned with those obtained from the HEOM method, but the MR method predicts faster dynamics.

Key Chlorophyll a Molecules in the Uphill Energy Transfer from Chlorophyll f to P700 in Far-Red Light-Adapted Photosystem I

Multiple far-red light-adapted photosystem I (FR-PSI) reaction centers are recently found to work in oxygenic photosynthesis. They contain a small amount of a new type pigment chlorophyll f (Chl f) in addition to the major pigment chlorophyll a (Chl a). FR-PSI differs from the conventional PSIs in plants and cyanobacteria, which use only visible light absorbed by Chl a, although the mechanism of FR-PSI is not fully clear yet. We theoretically studied the light-harvesting mechanism of FR-PSI of Fischerella thermalis PCC 7521, in which a small amount of Chl f transfers the excitation energy of FR-light uphill to Chl a. We constructed two types of exciton models for FR-PSI using pigment arrangements based on the structural information. A model that assumes the same site energy value for all of the antenna Chl a molecules reproduced most of the experimentally obtained properties. The transient absorption spectra, excitation energy relaxation, and mean first passage time (MFPT) of the excitation energy transfer from Chls f and a to the special pair P700 (a pair of Chl a/Chl a') were numerically calculated. The model, however, could not reproduce the low but distinct absorption intensity between the Chl a- and Chl f-bands and predicted a rather slow energy transfer from Chl f to P700. Advanced "modified models" further tested the effect of modification of the site energy values at individual antenna Chl a molecules. The optical properties and MFPTs of FR-PSI were calculated for each model with modified site energy values to evaluate the uphill light-harvesting process. The analysis showed that Chl a-1131 and -1222 play key roles in the light-harvesting process from Chl f molecules to P700, regardless of the excitation wavelength. The locations and site energy values of these Chl a molecules were found to be essential to reproduce the unique uphill energy transfer function of FR-PSI.

Application of the Time-Domain Multichromophoric Fluorescence Resonant Energy Transfer Method in the NISE Programme

We present the implementation of the time-domain multichromophoric fluorescence resonant energy transfer (TC-MCFRET) approach in the numerical integration of the Schrödinger equation (NISE) program. This method enables the efficient simulation of incoherent energy transfer between distinct segments within large and complex molecular systems, such as photosynthetic complexes. Our approach incorporates a segmentation protocol to divide these systems into manageable components and a modified thermal correction to ensure detailed balance. The implementation allows us to calculate the energy transfer rate in the NISE program systematically and easily. To validate our method, we applied it to a range of test cases, including parallel linear aggregates and biologically relevant systems like the B850 rings from LH2 and the Fenna-Matthews-Olson complex. Our results show excellent agreement with previous studies, demonstrating the accuracy and efficiency of our TD-MCFRET method. We anticipate that this approach will be widely applicable to the calculation of energy transfer rates in other large molecular systems and will pave the way for future simulations of multidimensional electronic spectra.

Intramolecular charge transfer and the function of vibronic excitons in photosynthetic light harvesting

A widely discussed explanation for the prevalence of pairs or clusters of closely spaced electronic chromophores in photosynthetic light-harvesting proteins is the presence of ultrafast and highly directional excitation energy transfer pathways mediated by vibronic excitons, the delocalized optical excitations derived from mixing of the electronic and vibrational states of the chromophores. We discuss herein the hypothesis that internal conversion processes between exciton states on the <100 fs timescale are possible when the excitonic potential energy surfaces are controlled by the vibrational modes that induce charge transfer character in a strongly coupled system of chromophores. We discuss two examples, the peridinin-chlorophyll protein from marine dinoflagellates and the intact phycobilisome from cyanobacteria, in which the intramolecular charge-transfer (ICT) character arising from out-of-plane distortion of the conjugation of carotenoid or bilin chromophores also results in localization of the initially delocalized optical excitation on the vibrational timescale. Tuning of the ground state conformations of the chromophores to manipulate their ICT character provides a natural photoregulatory mechanism, which would control the overall quantum yield of excitation energy transfer by turning on and off the delocalized character of the optical excitations.

Resonant Vibrational Enhancement of Downhill Energy Transfer in the C-Phycocyanin Chromophore Dimer

Energy transfer between electronically coupled photosynthetic light-harvesting antenna pigments is frequently assisted by protein and chromophore nuclear motion. This energy transfer mechanism usually occurs in the weak or intermediate system-bath coupling regime. Redfield theory is frequently used to describe the energy transfer in this regime. Spectral densities describe vibronic coupling in visible transitions of the chromophores and govern energy transfer in the Redfield mechanism. In this work, we perform finely sampled broadband pump-probe spectroscopy on the phycobilisome antenna complex with sub-10-fs pump and probe pulses. The spectral density obtained by Fourier transforming the pump-probe time-domain signal is used to perform modified Redfield rate calculations to check for vibrational enhancement of energy transfer in a coupled chromophore dimer in the C-phycocyanin protein of the phycobilisome antenna. We find two low-frequency vibrations to be in near-resonance with the interexcitonic energy gap and a few-fold enhancement in the interexcitonic energy transfer rate due to these resonances at room temperature. Our observations and calculations explain the fast downhill energy transfer process in C-phycocyanin. We also observe high-frequency vibrations involving chromophore-protein residue interactions in the excited state of the phycocyanobilin chromophore. We suggest that these vibrations lock the chromophore nuclear configuration of the excited state and prevent the energetic relaxation that blocks energy transfer.

Vibrational Relaxation Completes the Excitation Energy Transfer and Localization of Vibronic Excitons in Allophycocyanin α84-β84

Phycobilisomes are light-harvesting complexes that play a key role in photosynthesis in cyanobacteria, which generate more than 40% of the world's oxygen. The near-unity excitation energy transfer efficiency from phycobilisomes to photosystems highlights its importance in understanding efficient energy transfer processes. Spectroscopic studies have shown that the 280 fs rapid excitonic downhill energy transfer within the α84-β84 chromophore dimer in allophycocyanin (APC), a subunit of phycobilisomes, is crucial to this efficiency. However, the role of strong chromophore-protein interactions and vibrational relaxation requires further exploration to fully explain this efficient downhill energy transfer. A theory is required that adequately describes exciton dynamics in an intermediate region while also incorporating vibrational relaxation mediated by protein bath modes. In this work, we incorporate vibrational relaxation into modified Redfield theory by introducing coupling fluctuation. We holistically simulate the rapid excitation energy transfer process of the α84-β84 chromophore dimer in APC and successfully model the recently observed rapid energy capture. We find that vibrational relaxation dictates capture of excitons by the localized state of the β84 chromophore. The calculated rate shows excellent agreement with previous ultrafast spectroscopic experiments. Our results show that the inclusion of vibrational relaxation is essential for systems that utilize vibronic coupling to enhance energy transfer and capture. Consequently, incorporating vibrational relaxation into Modified Redfield theory shows promise for accurately describing the excitation energy transfer process in other photosynthetic systems.

Reduced density matrix dynamics in multistate harmonic models via time-convolution and time-convolutionless quantum master equations with quantum-mechanical and semiclassical kernels

In this work, we explore the electronic reduced density matrix (RDM) dynamics using time-convolution (TC) and time-convolutionless (TCL) quantum master equations (QMEs) that are based on perturbative electronic couplings within the framework of multistate harmonic (MSH) models. The MSH model Hamiltonian consistently incorporates the electronic-vibrational correlations between all pairs of states by satisfying the pairwise reorganization energies directly obtained from all-atom simulations, representing the globally heterogeneous environments that couple to the multiple states differently. We derive the exact quantum-mechanical and a hierarchy of semiclassical approximate expressions for the kernels in TC and TCL QMEs that project the full RDM for general shifted harmonic systems, including the MSH model. These QMEs are applied to simulate RDM dynamics of photoinduced charge transfer (PICT) in organic photovoltaic carotenoid-porphyrin-fullerene triad solvated in tetrahydrofuran solution and the excitation energy transfer (EET) dynamics in photosynthetic Fenna-Matthews-Olson complexes from C. tepidum and P. aestuarii. Our results show that while both TC and TCL QMEs capture similar phenomena in PICT and EET processes, TC QME generally provides more accurate results than TCL QME, particularly in the initial oscillation of EET population dynamics. This study highlights the effectiveness of the TC and TCL QMEs in modeling RDM dynamics of nonadiabatic processes, offering insights for realistic condensed phase systems.

Coherence in Chemistry: Foundations and Frontiers

Coherence refers to correlations in waves. Because matter has a wave-particle nature, it is unsurprising that coherence has deep connections with the most contemporary issues in chemistry research (e.g., energy harvesting, femtosecond spectroscopy, molecular qubits and more). But what does the word "coherence" really mean in the context of molecules and other quantum systems? We provide a review of key concepts, definitions, and methodologies, surrounding coherence phenomena in chemistry, and we describe how the terms "coherence" and "quantum coherence" refer to many different phenomena in chemistry. Moreover, we show how these notions are related to the concept of an interference pattern. Coherence phenomena are indeed complex, and ambiguous definitions may spawn confusion. By describing the many definitions and contexts for coherence in the molecular sciences, we aim to enhance understanding and communication in this broad and active area of chemistry.

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.

Unraveling quantum coherences mediating primary charge transfer processes in photosystem II reaction center

Photosystem II (PSII) reaction center (RC) is a unique complex that is capable of efficiently separating electronic charges across the membrane. The primary energy- and charge-transfer (CT) processes occur on comparable ultrafast timescales, which makes it extremely challenging to understand the fundamental mechanism responsible for the near-unity quantum efficiency of the transfer. Here, we elucidate the role of quantum coherences in the ultrafast energy and CT in the PSII RC by performing two-dimensional (2D) electronic spectroscopy at the cryogenic temperature of 20 kelvin, which captures the distinct underlying quantum coherences. Specifically, we uncover the electronic and vibrational coherences along with their lifetimes during the primary ultrafast processes of energy and CT. We construct an excitonic model that provides evidence for coherent energy and CT at low temperature in the 2D electronic spectra. The principles could provide valuable guidelines for creating artificial photosystems with exploitation of system-bath coupling and control of coherences to optimize the photon conversion efficiency to specific functions.

Homogeneous Dephasing in Photosynthetic Bacterial Reaction Centers: Time Correlation Function Approach

A new tractable linear electronic transition dipole moment time correlation function (ETDMTCF) that accurately accounts for electronic dephasing, asymmetry, and width of 1-phonon profile, which the zero-phonon line (ZPL) contributes to it, in Rhodopseudomonas viridis bacterial reaction center is derived. This time correlation function proves to be superior to other frequency-domain expressions in case of strong electron-phonon coupling (which is often the case in bacterial RCs and pigment-protein complexes), many vibrational modes involved, and high temperature, whereby more vibronic and electronic (sequence) transitions would arise. The Fourier transform of this ETDMTCF leads to asymmetric multiphonon profiles composed of Lorentzian distribution and Gaussian distribution on the high- and low-energy sides, respectively, whereby the overtone widths fold themselves with that of the one-phonon profile. This ETDMTCF also features expedient computation in large systems using asymmetric phonon profiles to account correctly for dephasing and pigment-protein interaction (electron-phonon coupling). The derived ETDMTCF allows computing all nonlinear optical signals in both time and frequency domains, through the nonlinear dipole moment time correlation functions (as guided by nonlinear optical response theory) in line with the eight Liouville space pathways. The linear transition dipole moment time correlation function is of a central value as the nonlinear transition dipole moment time correlation function is expressed in terms of the linear transition dipole moment time correlation function, derived herein. One of the great advantages of presenting this ETDMTCF is its applicability to nonlinear transition dipole moment time correlation functions in line with the eight Liouville space pathways needed in computing nonlinear signals. As such, there is more to the utility and applicability of the presented ETDMTCF besides computational expediency and efficiency. Results show good agreement with the reported literature. The intimate connection between a one-phonon profile and the corresponding bath spectral density in photosynthetic complexes is discussed.

Formally Exact Simulations of Mesoscale Exciton Diffusion in a Light-Harvesting 2 Antenna Nanoarray

The photosynthetic apparatus of plants and bacteria combine atomically precise pigment-protein complexes with dynamic membrane architectures to control energy transfer on the 10-100 nm length scales. Recently, synthetic materials have integrated photosynthetic antenna proteins to enhance exciton transport, though the influence of artificial packing on the excited-state dynamics in these biohybrid materials is not fully understood. Here, we use the adaptive hierarchy of pure states (adHOPS) to perform a formally exact simulation of excitation energy transfer within artificial aggregates of light-harvesting complex 2 (LH2) with a range of packing densities. We find that LH2 aggregates support a remarkable exciton diffusion length ranging from 100 nm at a biological packing density to 300 nm at the densest packing previously suggested in an artificial aggregate. The unprecedented scale of these formally exact calculations also underscores the efficiency with which adHOPS simulates excited-state processes in molecular materials.

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 Schrodinger picture electronic coherences from them. A key experimental quantity that relies on the ability to obtain accurate Schrodinger picture electronic coherences is the absorption spectrum. In this paper, we propose using a recently introduced procedure for extracting Schrodinger picture electronic coherences from interaction picture inputs to calculate electronic absorption spectra from electronic dynamics generated by OD-QMEs. The accuracy of the absorption spectra obtained in 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 (HEOM) method.

Structure-based model of fucoxanthin-chlorophyll protein complex: Calculations of chlorophyll electronic couplings

Diatoms are a group of marine algae that are responsible for a significant part of global oxygen production. Adapted to life in an aqueous environment dominated by the blue-green light, their major light-harvesting antennae-fucoxanthin-chlorophyll protein complexes (FCPs)-exhibit different pigment compositions than of plants. Despite extensive experimental studies, until recently the theoretical description of excitation energy dynamics in these complexes was limited by the lack of high-resolution structural data. In this work, we use the recently resolved crystallographic information of the FCP complex from Phaeodactylum tricornutum diatom [Wang et al., Science 363, 6427 (2019)] and quantum chemistry-based calculations to evaluate the chlorophyll transition dipole moments, atomic transition charges from electrostatic potential, and the inter-chlorophyll couplings in this complex. The obtained structure-based excitonic couplings form the foundation for any modeling of stationary or time-resolved spectroscopic data. We also calculate the inter-pigment Förster energy transfer rates and identify two quickly equilibrating chlorophyll clusters.

Mechanism of Ir(ppy)3 Guest Exciton Formation with the Exciplex-Forming TCTA:TPBI Cohost within a Phosphorescent Organic Light-Emitting Diode Environment

Cohosts based on hole transporting and electron transporting materials often act as exciplexes in the form of intermolecular charge transfer complexes. Indeed, exciplex-forming cohosts have been widely developed as the host materials for efficient phosphorescent organic light-emitting diodes (OLEDs). In host-guest systems of OLEDs, the guest can be excited by two competing mechanisms, namely, excitation energy transfer (EET) and charge transfer (CT). Experimentally, it has been reported that the EET mechanism is dominant and the excitons are primarily formed in the host first and then transferred to the guest in phosphorescent OLEDs based on exciplex-forming cohosts. With this, exciplex-forming cohosts are widely employed for avoiding the formation of trapped charge carriers in the phosphorescent guest. However, theoretical studies are still lacking toward elucidating the relative importance between EET and CT processes in exciting the guest molecules in such systems. Here, we obtain the kinetics of guest excitation processes in a few trimer model systems consisting of an exciplex-forming cohost pair and a phosphorescent guest. We adopt the Förster resonance energy transfer (FRET) rate constants for the electronic transitions between excited states toward solving kinetic master equations. The input parameters for calculating the FRET rate constants are obtained from density functional theory (DFT) and time-dependent DFT. The results show that while the EET mechanism is important, the CT mechanism may still play a significant role in guest excitations. In fact, the relative importance of CT over EET depends strongly on the location of the guest molecule relative to the cohost pair. This is understandable as both the coupling for EET and the interaction energy for CT are strongly influenced by the geometric constraints. Understanding the energy transfer pathways from the exciplex state of cohost to the emissive state of guest may provide insights for improving exciplex-forming materials adopted in OLEDs.

Vibrational Modes Promoting Exciton Relaxation in the B850 Band of LH2

Exciton relaxation dynamics in multichromophore systems are often modeled using Redfield theory, where bath fluctuations mediate the relaxation among the exciton eigenstates. Identifying the vibrational or phonon modes that are implicated in exciton relaxation allows more detailed understanding of exciton dynamics. Here we focus on a well-studied light-harvesting II complex (LH2) isolated from the photosynthetic purple bacterium Rhodoblastus acidophilus strain 10050. Using two synchronized mode-locked lasers, we carried out a polarization-dependent two-dimensional electronic spectroscopy (2DES) study of an ultrafast exciton relaxation in the B850 band of LH2. 2DES data with different polarization configurations enable us to investigate the exciton relaxation between the k = ±1 exciton states. Then, we identify vibrational modes coupled to the exciton relaxation by analyzing the coherent wavepackets in the 2DES signals. Focusing on the coherent vibrational wavepackets, the data suggest that certain symmetry-breaking modes of monomeric units play a key role in exciton relaxation.

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.

On simulating the dynamics of electronic populations and coherences via quantum master equations based on treating off-diagonal electronic coupling terms as a small perturbation

Quantum master equations provide a general framework for describing the dynamics of electronic observables within a complex molecular system. One particular family of such equations is based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. In this paper, we show how different choices of projection operators, as well as whether one starts out with the time-convolution or the time-convolutionless forms of the generalized quantum master equation, give rise to four different types of such off-diagonal quantum master equations (OD-QMEs), namely, time-convolution and time-convolutionless versions of a Pauli-type OD-QME for only the electronic populations and an OD-QME for the full electronic density matrix (including both electronic populations and coherences). The fact that those OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. To address this, we also extend a procedure for extracting Schrödinger picture electronic coherences from interaction picture populations recently introduced by Trushechkin in the context of time-convolutionless Pauli-type OD-QME to the other three types of OD-QMEs. The performance of the aforementioned four types of OD-QMEs is explored in the context of the Garg-Onuchic-Ambegaokar benchmark model for charge transfer in the condensed phase across a relatively wide parameter range. The results show that time-convolution OD-QMEs can be significantly more accurate than their time-convolutionless counterparts, particularly in the case of Pauli-type OD-QMEs, and that rather accurate Schrödinger picture coherences can be obtained from interaction picture electronic inputs.

Electronic dephasing of polyatomic molecules interacting with mixed quantum-classical media

This paper offers an expedient, efficient, and unique treatment of multimode quantum subsystems (polyatomic molecules) interacting with a classical environment in which the time evolution of the coupling term is governed by the algebraic rules of statistical mechanics in mixed quantum-classical systems developed by Kapral and Nielsen [S. Nielsen, R. Kapral, and G. Ciccotti, J. Chem. Phys., 2001, 115, 5805]. This unique time evolution of the coupling term is neither quantal nor classical but rather something different that relies heavily on Wigner transform, thereby leading to non-Newtonian mechanics. As such, an argument is presented that the approach provided herein for treating polyatomic molecular systems in a mixed quantum-classical environment is new and different as opposed to the many other schemes of semiclassical dynamics that are normally employed to study such systems. The merits of expediency and efficiency of the herein mixed quantum-classical dynamics calculations emanate from avoiding using integrals for time evolutions, and, instead, employing matrix mechanics whereby LU decomposition and singular value decomposition (SVD) numerical techniques are utilized for diagonalization. An electronic 2-level subsystem interacting with a classical bath through the spin-boson model to render accurate pure electronic dephasing in multimode molecular systems by eliminating the unphysical asymmetry in the line shape of the zero-phonon line (ZPL) exhibited by other models is exploited. This work has a superior advantage over the single-mode spin-boson model, published previously, whereby a multitude of types of vibrational modes (slow, fast, or both) of the quantum subsystem may readily be handled using different spectral densities. The spin-boson model used here is a composite system made up of a quantum subsystem, i.e., a subsystem bilinearly coupled to a multidimensional harmonic oscillator (representing the intermediate quantum vibrational modes between the electronic subsystem and the bath), interacting with a classical bath, where the coupling term is governed by the mixed quantum-classical Liouville equation. A multidimensional coherent-state approach is employed to deal with the time evolution of the quantum subsystem. A closed-form expression of linear and nonlinear optical electronic transition dipole moment time correlation functions in mixed quantum-classical dissipative media is derived. Pure electronic dephasing is probed using the aforementioned approach. Linear absorption spectra and 4-wave mixing signals (e.g., photon echo and pump-probe) are calculated showing a reasonable thermal broadening, temporal decay, and accurate pure dephasing.

Excited state distribution function for probing Herzberg-Teller vibronic coupling using linear optical response theory: Application to glassy pheophytin a

The goal of the present work is to develop an excited-state distribution function that can be used to calculate electronic transition dipole moment time correlation functions at a considerably low computational cost. An additional merit of the distribution function is its capability to probe the Hertzberg-Teller vibronic coupling effect in terms of the previously reported Condon correlation functions in the literature without having to start from the equilibrium density operator to probe spectral non-Condon effects, thereby exploring Hertzberg-Teller vibronic coupling by building on the Condon regime. It is easily extendable to anharmonic systems. Model calculations are reported to show the high degree of accuracy and computational efficiency of the presented approach. Application to a photosynthetic system such as pheophytin a in triethylamine is provided.

Achieving two-dimensional optical spectroscopy with temporal and spectral resolution using quantum entangled three photons

Recent advances in techniques for generating quantum light have stimulated research on novel spectroscopic measurements using quantum entangled photons. One such spectroscopy technique utilizes non-classical correlations among entangled photons to enable measurements with enhanced sensitivity and selectivity. Here, we investigate the spectroscopic measurement utilizing entangled three photons. In this measurement, time-resolved entangled photon spectroscopy with monochromatic pumping [A. Ishizaki, J. Chem. Phys. 153, 051102 (2020)] is integrated with the frequency-dispersed two-photon counting technique, which suppresses undesired accidental photon counts in the detector and thus allows one to separate the weak desired signal. This time-resolved frequency-dispersed two-photon counting signal, which is a function of two frequencies, is shown to provide the same information as that of coherent two-dimensional optical spectra. The spectral distribution of the phase-matching function works as a frequency filter to selectively resolve a specific region of the two-dimensional spectra, whereas the excited-state dynamics under investigation are temporally resolved in the time region longer than the entanglement time. The signal is not subject to Fourier limitations on the joint temporal and spectral resolution, and therefore, it is expected to be useful for investigating complex molecular systems in which multiple electronic states are present within a narrow energy range.

Formally exact simulations of mesoscale exciton dynamics in molecular materials

Excited state carriers, such as excitons, can diffuse on the 100 nm to micron length scale in molecular materials but only delocalize over short length scales due to coupling between electronic and vibrational degrees-of-freedom. Here, we leverage the locality of excitons to adaptively solve the hierarchy of pure states equations (HOPS). We demonstrate that our adaptive HOPS (adHOPS) methodology provides a formally exact and size-invariant (i.e., ) scaling algorithm for simulating mesoscale quantum dynamics. Finally, we provide proof-of-principle calculations for exciton diffusion on linear chains containing up to 1000 molecules.

The adaptive hierarchy of pure states (adHOPS) algorithm leverages the locality of excitons in molecular materials to perform formally-exact simulations with size-invariant (i.e., ) scaling, enabling efficient simulations of mesoscale exciton dynamics.

Simulation of Time- and Frequency-Resolved Four-Wave-Mixing Signals at Finite Temperatures: A Thermo-Field Dynamics Approach

We propose a new approach to simulate four-wave-mixing signals of molecular systems at finite temperatures by combining the multiconfigurational Ehrenfest method with the thermo-field dynamics theory. In our approach, the four-time correlation functions at finite temperatures are mapped onto those at zero temperature in an enlarged Hilbert space with twice the vibrational degrees of freedom. As an illustration, we have simulated three multidimensional spectroscopic signals, time- and frequency-resolved fluorescence spectra, transient-absorption pump-probe spectra, and electronic two-dimensional (2D) spectra at finite temperatures, for a conical intersection-mediated singlet fission model of a rubrene crystal. It is shown that a detailed dynamical picture of the singlet fission process can be extracted from the three spectroscopic signals. An increasing temperature leads to lower intensities of the signals and broadened vibrational peaks, which can be attributed to faster singlet-triplet population transfer and stronger bath-induced electronic dephasing at higher temperatures.

Quantum and semiclassical dynamical studies of nonadiabatic processes in solution: achievements and perspectives

We concisely review the main methodological approaches to model nonadiabatic dynamics in isotropic solutions and their applications. Three general classes of models are identified as the most used to include solvent effects in the simulations. The first model describes the solvent as a set of harmonic collective modes coupled to the solute degrees of freedom, and the second as a continuum, while the third explicitly includes solvent molecules in the calculations. The issues related to the use of these models in semiclassical and quantum dynamical simulations are discussed, as well as the main limitations and perspectives of each approach.

Ultrafast Spectroscopy of Photoactive Molecular Systems from First Principles: Where We Stand Today and Where We Are Going

Computational spectroscopy is becoming a mandatory tool for the interpretation of the complex, and often congested, spectral maps delivered by modern non-linear multi-pulse techniques. The fields of Electronic Structure Methods, Non-Adiabatic Molecular Dynamics, and Theoretical Spectroscopy represent the three pillars of the virtual ultrafast optical spectrometer, able to deliver transient spectra in silico from first principles. A successful simulation strategy requires a synergistic approach that balances between the three fields, each one having its very own challenges and bottlenecks. The aim of this Perspective is to demonstrate that, despite these challenges, an impressive agreement between theory and experiment is achievable now regarding the modeling of ultrafast photoinduced processes in complex molecular architectures. Beyond that, some key recent developments in the three fields are presented that we believe will have major impacts on spectroscopic simulations in the very near future. Potential directions of development, pending challenges, and rising opportunities are illustrated.

Variety, the spice of life and essential for robustness in excitation energy transfer in light-harvesting complexes

For over a decade there has been some significant excitement and speculation that quantum effects may be important in the excitation energy transport process in the light harvesting complexes of certain bacteria and algae, in particular via the Fenna-Matthews-Olsen (FMO) complex. Whilst the excitement may have waned somewhat with the realisation that the observed long-lived oscillations in two-dimensional electronic spectra of FMO are probably due to vibronic coherences, it remains a question whether these coherences may play any important role. We review our recent work showing how important the site-to-site variation in coupling between chloroplasts in FMO and their protein scaffold environment is for energy transport in FMO and investigate the role of vibronic modes in this transport. Whilst the effects of vibronic excitations seem modest for FMO, we show that for bilin-based pigment-protein complexes of marine algae, in particular PC645, the site-dependent vibronic excitations seem essential for robust excitation energy transport, which may again open the door for important quantum effects to be important in these photosynthetic complexes.

How Well Does the Hole-Burning Action Spectrum Represent the Site-Distribution Function of the Lowest-Energy State in Photosynthetic Pigment-Protein Complexes?

For the first time, we combined Monte Carlo and nonphotochemical hole burning (NPHB) master equation approaches to allow for ultrahigh-resolution (<0.005 cm^-1, smaller than the typical homogeneous line widths at 5 K) simulations of the NPHB spectra of dimers and trimers of interacting pigments. These simulations reveal significant differences between the zero-phonon hole (ZPH) action spectrum and the site-distribution function (SDF) of the lowest-energy state. The NPHB of the lowest-energy pigment, following the excitation energy transfer (EET) from the higher-energy pigments which are excited directly, results in the shifts of all excited states. These shifts affect the ZPH action spectra and EET times derived from the widths of the spectral holes burned in the donor-dominated regions. The effect is present for a broad variety of realistic antihole functions, and it is maximal at relatively low values of interpigment coupling (V ≤ 5 cm^-1) where the use of the Förster approximation is justified. These findings need to be considered in interpreting various optical spectra of photosynthetic pigment-protein complexes for which SDFs (describing the inhomogeneous broadening) are often obtained directly from the ZPH action spectra. Water-soluble chlorophyll-binding protein (WSCP) was considered as an example.

Theory for polariton-assisted remote energy transfer

Strong-coupling between light and matter produces hybridized states (polaritons) whose delocalization and electromagnetic character allow for novel modifications in spectroscopy and chemical reactivity of molecular systems. Recent experiments have demonstrated remarkable distance-independent long-range energy transfer between molecules strongly coupled to optical microcavity modes. To shed light on the mechanism of this phenomenon, we present the first comprehensive theory of polariton-assisted remote energy transfer (PARET) based on strong-coupling of donor and/or acceptor chromophores to surface plasmons. Application of our theory demonstrates that PARET up to a micron is indeed possible. In particular, we report two regimes for PARET: in one case, strong-coupling to a single type of chromophore leads to transfer mediated largely by surface plasmons while in the other case, strong-coupling to both types of chromophores creates energy transfer pathways mediated by vibrational relaxation. Importantly, we highlight conditions under which coherence enhances or deteriorates these processes. For instance, while exclusive strong-coupling to donors can enhance transfer to acceptors, the reverse turns out not to be true. However, strong-coupling to acceptors can shift energy levels in a way that transfer from acceptors to donors can occur, thus yielding a chromophore role-reversal or "carnival effect". This theoretical study demonstrates the potential for confined electromagnetic fields to control and mediate PARET, thus opening doors to the design of remote mesoscale interactions between molecular systems.

Control of Excitation Energy Transfer in Condensed Phase Molecular Systems by Floquet Engineering

Excitation energy transfer (EET) is one of the most important processes in both natural and artificial chemical systems including, for example, photosynthetic complexes and organic solar cells. The EET rate, however, is strongly suppressed when there is a large difference in the excitation energy between the donor and acceptor molecules. Here, we demonstrate both analytically and numerically that the EET rate can be greatly enhanced by periodically modulating the excitation energy difference. The enhancement of EET by using this Floquet engineering, in which the system's Hamiltonian is made periodically time-dependent, turns out to be efficient even in the presence of strong fluctuations and dissipations induced by the coupling with a huge number of dynamic degrees of freedom in the surrounding molecular environments. As an effect of the environment on the Floquet engineering of EET, the optimal driving frequency is found to depend on the relative magnitudes of the system and environment's characteristic time scales with an observed frequency shift when moving from the limit of slow environmental fluctuations (inhomogeneous broadening limit) to that of fast fluctuations (homogeneous broadening limit).

Quantum corrections of the truncated Wigner approximation applied to an exciton transport model

We modify the path integral representation of exciton transport in open quantum systems such that an exact description of the quantum fluctuations around the classical evolution of the system is possible. As a consequence, the time evolution of the system observables is obtained by calculating the average of a stochastic difference equation which is weighted with a product of pseudoprobability density functions. From the exact equation of motion one can clearly identify the terms that are also present if we apply the truncated Wigner approximation. This description of the problem is used as a basis for the derivation of a new approximation, whose validity goes beyond the truncated Wigner approximation. To demonstrate this we apply the formalism to a donor-acceptor transport model.

Efficiency of energy funneling in the photosystem II supercomplex of higher plants

The investigation of energy transfer properties in photosynthetic multi-protein networks gives insight into their underlying design principles. Here, we discuss the excitonic energy transfer mechanisms of the photosystem II (PS-II) C2S2M2 supercomplex, which is the largest isolated functional unit of the photosynthetic apparatus of higher plants. Despite the lack of a definite energy gradient in C2S2M2, we show that the energy transfer is directed by relaxation to low energy states. C2S2M2 is not organized to form pathways with strict energetically downhill transfer, which has direct consequences for the transfer efficiency, transfer pathways and transfer limiting steps. The exciton dynamics is sensitive to small changes in the energetic layout which, for instance, are induced by the reorganization of vibrational coordinates. In order to incorporate the reorganization process in our numerical simulations, we go beyond rate equations and use the hierarchically coupled equation of motion approach (HEOM). While transfer from the peripheral antenna to the proteins in proximity to the reaction center occurs on a faster time scale, the final step of the energy transfer to the RC core is rather slow, and thus the limiting step in the transfer chain. Our findings suggest that the structure of the PS-II supercomplex guarantees photoprotection rather than optimized efficiency.

Energy transfer efficiency in the chromophore network strongly coupled to a vibrational mode

Using methods from condensed matter and statistical physics, we examine the transport of excitons through the photosynthetic complex from a receiving antenna to a reaction center. Writing the equations of motion for the exciton creation-annihilation operators, we are able to describe the exciton dynamics, even in the regime when the reorganization energy is of the order of the intrasystem couplings. We determine the exciton transfer efficiency in the presence of a quenching field and protein environment. While the majority of the protein vibrational modes are treated as a heat bath, we address the situation when specific modes are strongly coupled to excitons and examine the effects of these modes on the energy transfer efficiency in the steady-state regime. Using the structural parameters of the Fenna-Matthews-Olson complex, we find that, for vibrational frequencies below 16 meV, the exciton transfer is drastically suppressed. We attribute this effect to the formation of a "mixed exciton-vibrational mode" where the exciton is transferred back and forth between the two pigments with the absorption or emission of vibrational quanta, instead of proceeding to the reaction center. The same effect suppresses the quantum beating at the vibrational frequency of 25 meV. We also show that the efficiency of the energy transfer can be enhanced when the vibrational mode strongly couples to the third pigment only, instead of coupling to the entire system.

The lowest triplet states of bridged cis-2,2'-bithiophenes - theory vs. experiment

Theoretical methods that were previously used to give a good quantitative description of the 3(1)Bu state of trans-2,2'-bithiophene are applied to characterize the lowest triplet states of three bridged cis-2,2'-bithiophenes: 3,3'-cyclopentadithiophene (CPDT), 3,3'-dithienylpyrrole (DTP), and 3,3'-dithienylthiophene (DTT). By obtaining highly accurate reproductions of the phosphorescence spectra of all three compounds, we rationalize the observed vibronic activity, further explore the performance of the applied theoretical methods, and address the quality of the reported experimental spectra. Over the course of this study we have, first, characterized the changes in the electronic structures between the ground state and the lowest triplet state and, second, expressed the related geometrical differences in terms of the Huang-Rhys factors. The Huang-Rhys factors have then been used to generate theoretical emission spectra with vibronic resolution. The applied procedure has yielded quantitative reproductions of the previously reported experimental phosphorescence spectra of DTT and DTP. The experimental spectrum of CPDT, on the other hand, turned out to be considerably narrower and intensity-deficient in its low energy region when compared with the theoretical results. Our experimental reinvestigation of the CPDT phosphorescence has given a refined spectrum that is significantly wider than the previously reported one, and is in nearly quantitative agreement with the theoretical prediction. This enabled us to attribute the observed discrepancy to an experimental artifact associated with the sensitivity characteristics of the commonly used photomultipliers.

Resonant vibrational energy transfer in ice Ih

Fascinating anisotropy decay experiments have recently been performed on H2O ice Ih by Timmer and Bakker [R. L. A. Timmer, and H. J. Bakker, J. Phys. Chem. A 114, 4148 (2010)]. The very fast decay (on the order of 100 fs) is indicative of resonant energy transfer between OH stretches on different molecules. Isotope dilution experiments with deuterium show a dramatic dependence on the hydrogen mole fraction, which confirms the energy transfer picture. Timmer and Bakker have interpreted the experiments with a Förster incoherent hopping model, finding that energy transfer within the first solvation shell dominates the relaxation process. We have developed a microscopic theory of vibrational spectroscopy of water and ice, and herein we use this theory to calculate the anisotropy decay in ice as a function of hydrogen mole fraction. We obtain very good agreement with experiment. Interpretation of our results shows that four nearest-neighbor acceptors dominate the energy transfer, and that while the incoherent hopping picture is qualitatively correct, vibrational energy transport is partially coherent on the relevant timescale.

Microscopic theory of singlet exciton fission. III. Crystalline pentacene

We extend our previous work on singlet exciton fission in isolated dimers to the case of crystalline materials, focusing on pentacene as a canonical and concrete example. We discuss the proper interpretation of the character of low-lying excited states of relevance to singlet fission. In particular, we consider a variety of metrics for measuring charge-transfer character, conclusively demonstrating significant charge-transfer character in the low-lying excited states. The impact of this electronic structure on the subsequent singlet fission dynamics is assessed by performing real-time master-equation calculations involving hundreds of quantum states. We make direct comparisons with experimental absorption spectra and singlet fission rates, finding good quantitative agreement in both cases, and we discuss the mechanistic distinctions that exist between small isolated aggregates and bulk systems.