Origin of long-lived coherences in light-harvesting complexes

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.

The development and applications of multidimensional biomolecular spectroscopy illustrated by photosynthetic light harvesting

The parallel and synergistic developments of atomic resolution structural information, new spectroscopic methods, their underpinning formalism, and the application of sophisticated theoretical methods have led to a step function change in our understanding of photosynthetic light harvesting, the process by which photosynthetic organisms collect solar energy and supply it to their reaction centers to initiate the chemistry of photosynthesis. The new spectroscopic methods, in particular multidimensional spectroscopies, have enabled a transition from recording rates of processes to focusing on mechanism. We discuss two ultrafast spectroscopies - two-dimensional electronic spectroscopy and two-dimensional electronic-vibrational spectroscopy - and illustrate their development through the lens of photosynthetic light harvesting. Both spectroscopies provide enhanced spectral resolution and, in different ways, reveal pathways of energy flow and coherent oscillations which relate to the quantum mechanical mixing of, for example, electronic excitations (excitons) and nuclear motions. The new types of information present in these spectra provoked the application of sophisticated quantum dynamical theories to describe the temporal evolution of the spectra and provide new questions for experimental investigation. While multidimensional spectroscopies have applications in many other areas of science, we feel that the investigation of photosynthetic light harvesting has had the largest influence on the development of spectroscopic and theoretical methods for the study of quantum dynamics in biology, hence the focus of this review. We conclude with key questions for the next decade of this review.

Controlling Vibronic Coupling in Chlorophyll Proteins: The Effects of Excitonic Delocalization and Vibrational Localization

Vibrational-electronic (vibronic) coupling plays a critical role in excitation energy transfer in molecular aggregates and pigment-protein complexes (PPCs). But the interplay between excitonic delocalization and vibronic interactions is complex, often leaving even qualitative questions as to what conceptual framework (e.g., Redfield versus Förster theory) should be used to interpret experimental results. To shed light on this issue, we report here on the interplay between excitonic delocalization and vibronic coupling in site-directed mutants of the water-soluble chlorophyll protein (WSCP), as reflected in 77 K fluorescence spectra. Experimentally, we find that in PPCs where excitonic delocalization is disrupted (either by mutagenesis or heterodimer formation), the relative intensity of the vibrational sideband (VSB) in fluorescence spectra is suppressed by up to 37% compared to that of the native protein. Numerical simulations reveal that this effect results from the localization of high-frequency vibrations in the coupled system; while excitonic delocalization suppresses the purely electronic transition due to H-aggregate-like dipole-dipole interference, high-frequency vibrations are unaffected, leading to a relative enhancement of the VSB. By comparing VSB intensities of PPCs both in the presence and absence of excitonic delocalization, we extract a set of "local" Huang-Rhys (HR) factors for Chl a in WSCP. More generally, our results suggest a significant role for geometric effects in controlling energy-transfer rates (which depend sensitively on absorption/fluorescence line shapes) in molecular aggregates and PPCs.

Frequency-Dependent Vibronic Effects in Steady State Energy Transport

The interplay between electronic and intramolecular high-frequency vibrational degrees of freedom is ubiquitous in natural light-harvesting systems. Recent studies have indicated that an intramolecular vibrational donor-acceptor frequency difference can enhance energy transport. Here, we analyze the extent to which different intramolecular donor-acceptor vibrational frequencies affect excitation energy transport in the natural nonequilibrium steady state configuration. Comments are included on the less physical equilibrium case for comparison with the literature. It is found that for constant Huang-Rhys factors, whereas the acceptor population increases in the equilibrium case when the intramolecular vibrational frequency of the acceptor exceeds that of the donor, this increase is negligible for the nonequilibrium steady state. Therefore, these changes in acceptor population do not significantly enhance energy transport in the nonequilibrium steady state for the natural scenario of incoherent light excitation with biologically relevant parameters of typical photosynthetic complexes. Insight about a potential mechanism to optimize energy transfer in the nonequilibrium steady state based on increasing the harvesting time at the reaction center is analyzed.

Chirped Laser Pulse Control of Vibronic Wavepackets and Energy Transfer in Phycocyanin 645

Photosynthetic organisms use light-harvesting complexes to increase the spectrum of light that they absorb from solar photons. Recent ultrafast spectroscopic studies have revealed that efficient (sub-ps) energy transfer is mediated by vibronic coherence in the phycobiliprotein phycocyanin 645 (PC645). Here, we report studies that employ broadband pump-probe spectroscopy with linearly chirped excitation pulses to further investigate the relationship between vibronic state preparation and energy transfer dynamics in PC645. Negatively chirped pulse excitation is found to enhance wavepackets of a high-frequency mode (1580 cm-1) and increase the rate of downhill energy transfer, while on the other hand, positively chirped pulses suppress these oscillatory features and decrease this rate. Model calculations incorporating the influence of the chirped pump pulse are used to understand its effect on initial state preparation. These results provide mechanistic insight into how the overall nonequilibrium rate of energy transfer is influenced by initial state preparation.

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

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

Theoretical model of femtosecond coherence spectroscopy of vibronic excitons in molecular aggregates

When used as pump pulses in transient absorption spectroscopy measurements, femtosecond laser pulses can produce oscillatory signals known as quantum beats. The quantum beats arise from coherent superpositions of the states of the sample and are best studied in the Fourier domain using Femtosecond Coherence Spectroscopy (FCS), which consists of one-dimensional amplitude and phase plots of a specified oscillation frequency as a function of the detection frequency. Prior works have shown ubiquitous amplitude nodes and π phase shifts in FCS from excited-state vibrational wavepackets in monomer samples. However, the FCS arising from vibronic-exciton states in molecular aggregates have not been studied theoretically. Here, we use a model of vibronic-exciton states in molecular dimers based on displaced harmonic oscillators to simulate FCS for dimers in two important cases. Simulations reveal distinct spectral signatures of excited-state vibronic-exciton coherences in molecular dimers that may be used to distinguish them from monomer vibrational coherences. A salient result is that, for certain relative orientations of the transition dipoles, the key resonance condition between the electronic coupling and the frequency of the vibrational mode may yield strong enhancement of the quantum-beat amplitude and, perhaps, also cause a significant decrease of the oscillation frequency to a value far lower than the vibrational frequency. Future studies using these results will lead to new insights into the excited-state coherences generated in photosynthetic pigment-protein complexes.

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.

Electron transfer rate modulation with mid-IR in butadiyne-bridged donor-bridge-acceptor compounds

Controlling electron transfer (ET) processes in donor-bridge-acceptor (DBA) compounds by mid-IR excitation can enhance our understanding of the ET dynamics and may find practical applications in molecular sensing and molecular-scale electronics. Alkyne moieties are attractive to serve as ET bridges, as they offer the possibility of fast ET and present convenient vibrational modes to perturb the ET dynamics. Yet, these bridges introduce complexity because of the strong torsion angle dependence of the ET rates and transition dipoles among electronic states and a shallow torsion barrier. In this study, we implemented ultrafast 3-pulse laser spectroscopy to investigate how the ET from the dimethyl aniline (D) electron donor to the N-isopropyl-1,8-napthalimide (NAP) electron acceptor can be altered by exciting the CC stretching mode (νCC) of the butadiyne bridge linking the donor and acceptor. The electron transfer was initiated by electronically exciting the acceptor moiety at 400 nm, followed by vibrational excitation of the alkyne, νCC, and detecting the changes in the absorption spectrum in the visible spectral region. The experiments were performed at different delay times t1 and t2, which are the delays between UV-mid-IR and mid-IR-Vis pulses, respectively. Two sets of torsion-angle conformers were identified, one featuring a very fast mean ET time of 0.63 ps (group A) and another featuring a slower mean ET time of 4.3 ps (group B), in the absence of the mid-IR excitation. TD-DFT calculations were performed to determine key torsion angle dependent molecular parameters, including the electronic and vibrational transition dipoles, transition frequencies, and electronic couplings. To describe the 3-pulse data, we developed a kinetic model that includes a locally excited, acceptor-based S2 state, a charge separated S1 state, and their vibrationally excited counterparts, with either excited νCC (denoted as S1Atr, S1Btr, S2Atr, and S2Btr, where tr stands for the excited triplet bond, νCC) or excited daughter modes of the νCC relaxation (S1Ah, S1Bh, S2Ah, and S2Bh, where h stands for vibrationally hot species). The kinetic model was solved analytically, and the species-associated spectra (SAS) were determined numerically using a matrix approach, treating first the experiments with longer t1 delays and then using the already determined SAS for modeling the experiments with shorter t1 delays. Strong vibronic coupling of νCC and of vibrationally hot states makes the analysis complicated. Nevertheless, the SAS were identified and the ET rates of the vibrationally excited species, S2Atr, S2Btr and S2Bh, were determined. The results show that the ET rate for the S2A species is ca. 1.2-fold slower when the νCC mode is excited. The ET rate for species S2B is slower by ca. 1.3-fold if the compound is vibrationally hot and is essentially unchanged when the νCC mode is excited. The SAS determined for the tr and h species resemble the SAS for their respective precursor species in the 2-pulse transient absorption experiments, which validates the procedure used and the results.

Computational study of the photophysical properties and electronic structure of iridium(III) photosensitizer complexes with electron-withdrawing groups

A series of novel [Ir(tpy)(btp)Cl]+ complexes (Ir1-Ir4) have been reported to show excellent performance as photosensitizers. The introduction of electron-withdrawing groups increases visible light absorption and the lifetime of triplet states. To improve the photophysical properties, we theoretically design Ir5-Ir9 with electron-withdrawing groups (Cl, F, COOH, CN and NO2). Surprisingly, our findings indicate that the photosensitizer performance does not strictly increase with the electron-withdrawing ability of the substituents. In this work, the geometric and electronic structures, transition features, and photophysical properties of Ir1-Ir9 are investigated. The natural transition orbital (NTO) analysis indicates that the T1 and T2 states play a role in the photochemical pathways. Ultraviolet-visible (UV-vis) absorption spectra and charge-transfer spectra (CTS) have been investigated to show that the introduction of electron-withdrawing groups not only improves the visible light absorbing ability, but also changes the nature of electron excitation, providing a future molecular design strategy for similar series of photosensitizers. The rates of (reverse) intersystem crossing and the Huang-Rhys factors are evaluated to interpret the experimental results within the framework of Marcus theory. For complexes Ir1-Ir7, the introduction of electron-withdrawing groups leads to a lower efficiency of reverse intersystem crossing and a strong non-radiative process T2 → T1, resulting in a long triplet lifetime and excellent performance as a photosensitizer. Furthermore, some newly designed complexes (Ir7-Ir9) show great potential as thermally activated delayed fluorescence emitters, contrary to our initial expectations.

A quantum trajectory picture of single photon absorption and energy transport in photosystem II

We use quantum trajectory theory to study the dynamics of the first step in photosynthesis for a single photon interacting with photosystem II (PSII). By considering individual trajectories we are able to look beyond the ensemble average dynamics to compute the PSII system evolution conditioned upon individual photon counting measurements. Measurements of the transmitted photon beam strongly affects the system state, since detection of an outgoing photon confirms that the PSII must be in the electronic ground state, while a null measurement implies it is in an excited electronic state. We show that under ideal conditions, observing the null result transforms a state with a low excited state population to a state with nearly all population contained in the excited states. We study the PSII dynamics conditioned on such photon counting for both a pure excitonic model of PSII and a more realistic model with exciton-phonon coupling to a dissipative phononic environment. In the absence of such coupling, we show that the measured fluorescence rates show oscillations constituting a photon-counting witness of excitonic coherence. Excitonic coupling to the phonon environment has a strong effect on the observed rates of fluorescence, damping the oscillations. Addition of non-radiative decay and incoherent transitions to radical pair states in the reaction center to the phononic model allows extraction of a quantum efficiency of 92.5% from the long-time evolution, consistent with bulk experimental measurements.

Resonant vibrations produce quantum bridge over high-energy states in heterogeneous antenna

Photosynthetic light-harvesting complexes usually contain several pools of molecules with a big difference in transition energies, for example, chlorophylls a and b in plant antennas. Some pathways of the excitation energy transfer may include pigments from the low-energy pool separated by a site occupied by a high-energy molecule. We demonstrate that such pathways may be functional if high-frequency intramolecular vibrations fall in resonance with the energy gap between the neighboring molecules belonging to different pools. In this case, a vibration-assisted mixing of the excited states can produce delocalized vibronic states playing a role of 'quantum bridge' that facilitates a passage over high-energy barrier. We perform calculations of the excitation dynamics in the model three-state system with the parameters emerging from our previous studies of real antennas. Simulation of the dynamics in an explicit electron-vibrational basis demonstrates that the rate of transfer between the two chlorophylls a through the chlorophyll b intermediate is increased by a factor of 1.7-2 in the presence of resonant vibration. A possible influence of energetic disorder and other (non-resonant) vibrations on this effect is discussed.

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.

Bioexcitons by Design: How Do We Get There?

Biological pigment-protein complexes (PPCs) exhibit a remarkable ability to tune the optical properties of biological excitons (bioexcitons) through specific pigment-protein interactions. While such fine-tuning allows natural systems (e.g., photosynthetic proteins) to carry out their native functions with near-optimal performance, native function itself is often suboptimal for applications such as biofuel production or quantum technology development. This perspective offers a look at near-term prospects for the rational reoptimization of PPC bioexcitons for new functions using site-directed mutagenesis. The primary focus is on the "structure-spectrum" challenge of understanding the relationships between structural features and spectroscopic properties. While recent examples demonstrate that site-directed mutagenesis can be used to tune nearly all key bioexciton parameters (e.g., site energies, interpigment couplings, and electronic-vibrational interactions), critical challenges remain before we achieve truly rational design of bioexciton properties.

Electronic-Vibrational Resonance Does Not Significantly Alter Steady-State Transport in Natural Light-Harvesting Systems

Oscillations in time-dependent two-dimensional electronic spectra appear as evidence of quantum coherence in light-harvesting systems related to electronic-vibrational resonant interactions. Nature, however, takes place in a non-equilibrium steady-state; therefore, the relevance of these arguments to the natural process is unclear. Here, we examine the role of intramolecular vibrations in the non-equilibrium steady-state of photosynthetic dimers in the natural scenario of incoherent light excitation. Specifically, we analyze the PEB dimer in the cryptophyte algae PE545 antenna protein. It is found that vibrations resonant with the energy difference between exciton states only marginally increase the quantum yield and the imaginary part of the intersite coherence that is relevant for transport compared to non-resonant vibrations in the natural non-equilibrium steady-state. That is, the electronic-vibrational resonance interaction does not significantly enhance energy transport under natural incoherent light excitation conditions.

Dyads with tunable near-infrared donor-acceptor excited-state energy gaps: molecular design and Förster analysis for ultrafast energy transfer

Bacteriochlorophylls, nature's near-infrared absorbers, play an essential role in energy transfer in photosynthetic antennas and reaction centers. To probe energy-transfer processes akin to those in photosynthetic systems, nine synthetic bacteriochlorin-bacteriochlorin dyads have been prepared wherein the constituent pigments are joined at the meso-positions by a phenylethyne linker. The phenylethyne linker is an unsymmetric auxochrome, which differentially shifts the excited-state energies of the phenyl- or ethynyl-attached bacteriochlorin constituents in the dyad. Molecular designs utilized known effects of macrocycle substituents to engineer bacteriochlorins with S₀ → S₁ (Q_y) transitions spanning 725-788 nm. The design-predicted donor-acceptor excited-state energy gaps in the dyads agree well with those obtained from time dependent density functional theory calculations and with the measured range of 197-1089 cm^-1. Similar trends with donor-acceptor excited-state energy gaps are found for (1) the measured ultrafast energy-transfer rates of (0.3-1.7 ps)^-1, (2) the spectral overlap integral (J) in Förster energy-transfer theory, and (3) donor-acceptor electronic mixing manifested in the natural transition orbitals for the S₀ → S₁ transition. Subtle outcomes include the near orthogonal orientation of the π-planes of the bacteriochlorin macrocycles, and the substituent-induced shift in transition-dipole moment from the typical coincidence with the NH-NH axis; the two features together afforded the Förster orientation term κ² ranging from 0.55-1.53 across the nine dyads, a value supportive of efficient excited-state energy transfer. The molecular design and collective insights on the dyads are valuable for studies relevant to artificial photosynthesis and other processes requiring ultrafast energy transfer.

Effects of Heterogeneous Protein Environment on Excitation Energy Transfer Dynamics in the Fenna-Matthews-Olson Complex

The Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria has been serving as a prototypical light-harvesting protein for studying excitation energy transfer (EET) dynamics in photosynthesis. The most widely used Frenkel exciton model for FMO complex assumes that each excited bacteriochlorophyll site couples to an identical and isolated harmonic bath, which does not account for the heterogeneous local protein environment. To better describe the realistic environment, we propose to use the recently developed multistate harmonic (MSH) model, which contains a globally shared bath that couples to the different pigment sites according to the atomistic quantum mechanics/molecular mechanics simulations with explicit protein scaffold and solvent. In this work, the effects of heterogeneous protein environment on EET in FMO complexes from Prosthecochloris aestuarii and Chlorobium tepidum, specifically including realistic spectral density, site-dependent reorganization energies, and system-bath couplings are investigated. Semiclassical and mixed quantum-classical mapping dynamics were applied to obtain the nonadiabatic EET dynamics in several models ranging from the Frenkel exciton model to the MSH model and their variants. The MSH model with realistic spectral density and site-dependent system-bath couplings displays slower EET dynamics than the Frenkel exciton model. Our comparative study shows that larger average reorganization energy, heterogeneity in spectral densities, and low-frequency modes could facilitate energy dissipation, which is insensitive to the static disorder in reorganization energies. The effects of the spectral densities and system-bath couplings along with the MSH model can be used to optimize EET dynamics for artificial light-harvesting systems.

Characterizing coherences in chemical dynamics with attosecond time-resolved x-ray absorption spectroscopy

Coherence can drive wave-like motion of electrons and nuclei in photoexcited systems, which can yield fast and efficient ways to exert materials’ functionalities beyond the thermodynamic limit. The search for coherent phenomena has been a central topic in chemical physics although their direct characterization is often elusive. Here, we highlight recent advances in time-resolved x-ray absorption spectroscopy (tr-XAS) to investigate coherent phenomena, especially those that utilize the eminent light source of isolated attosecond pulses. The unparalleled time and state sensitivities of tr-XAS in tandem with the unique element specificity render the method suitable to study valence electronic dynamics in a wide variety of materials. The latest studies have demonstrated the capabilities of tr-XAS to characterize coupled electronic–structural coherence in small molecules and coherent light–matter interactions of core-excited excitons in solids. We address current opportunities and challenges in the exploration of coherent phenomena, with potential applications for energy- and bio-related systems, potential crossings, strongly driven solids, and quantum materials. With the ongoing developments in both theory and light sources, tr-XAS holds great promise for revealing the role of coherences in chemical dynamics.

From antenna to reaction center: Pathways of ultrafast energy and charge transfer in photosystem II

The photosystem II core complex (PSII-CC) is a photosynthetic complex that contains antenna proteins, which collect energy from sunlight, and a reaction center, which converts the collected energy to redox potential. Understanding the interplay between the antenna proteins and the reaction center will facilitate the development of more efficient solar energy conversion technologies. Here, we study the sub-100-ps dynamics of PSII-CC with two-dimensional electronic-vibrational spectroscopy, which connects energy flows with physical space, allowing a direct mapping of energy transfer pathways. Our results reveal a complex dynamical scheme which includes a specific pathway that connects CP43 to the reaction center. Resolving this pathway experimentally provides insights into the energy conversion processes in natural photosynthesis.

The photosystem II core complex (PSII-CC) is the smallest subunit of the oxygenic photosynthetic apparatus that contains core antennas and a reaction center, which together allow for rapid energy transfer and charge separation, ultimately leading to efficient solar energy conversion. However, there is a lack of consensus on the interplay between the energy transfer and charge separation dynamics of the core complex. Here, we report the application of two-dimensional electronic-vibrational (2DEV) spectroscopy to the spinach PSII-CC at 77 K. The simultaneous temporal and spectral resolution afforded by 2DEV spectroscopy facilitates the separation and direct assignment of coexisting dynamical processes. Our results show that the dominant dynamics of the PSII-CC are distinct in different excitation energy regions. By separating the excitation regions, we are able to distinguish the intraprotein dynamics and interprotein energy transfer. Additionally, with the improved resolution, we are able to identify the key pigments involved in the pathways, allowing for a direct connection between dynamical and structural information. Specifically, we show that C505 in CP43 and the peripheral chlorophyll Chlz_D1 in the reaction center are most likely responsible for energy transfer from CP43 to the reaction center.

Fingerprint and Universal Markovian Closure of Structured Bosonic Environments

We exploit the properties of chain mapping transformations of bosonic environments to identify a finite collection of modes able to capture the characteristic features, or fingerprint, of the environment. Moreover we show that the countable infinity of residual bath modes can be replaced by a universal Markovian closure, namely, a small collection of damped modes undergoing a Lindblad-type dynamics whose parametrization is independent of the spectral density under consideration. We show that the Markovian closure provides a quadratic speedup with respect to standard chain mapping techniques and makes the memory requirement independent of the simulation time, while preserving all the information on the fingerprint modes. We illustrate the application of the Markovian closure to the computation of linear spectra but also to nonlinear spectral response, a relevant experimentally accessible many body coherence witness for which efficient numerically exact calculations in realistic environments are currently lacking.

Probing the Effects of Electronic-Vibrational Resonance on the Rate of Excited-State Energy Transfer in Bacteriochlorin Dyads

The impact of vibrational-electronic resonances on the rate of excited-state energy transfer is examined in a set of bacteriochlorin dyads that employ the same phenylethyne linker. The donor/acceptor excited-state energy gap is tuned from ∼200 to ∼1100 cm^-1 using peripheral substituents on the donor and acceptor bacteriochlorin macrocycles. Ultrafast energy transfer is observed with rate constants of (0.3 ps)^-1 to (1.7 ps)^-1, which agree with those predicted by Förster theory to within a factor of 2. Furthermore, the measured rates follow a trend-line with only small deviations that do not correlate with the density of vibrations at the donor/acceptor excited-state energy gap. Thus, if vibrational-electronic resonances occur in any of these dyads, which seems likely, the impact on the rate of energy transfer is small.

Vibrational response functions for multidimensional electronic spectroscopy in the adiabatic regime: A coherent-state approach

Multi-dimensional spectroscopy represents a particularly insightful tool for investigating the interplay of nuclear and electronic dynamics, which plays an important role in a number of photophysical processes and photochemical reactions. Here, we present a coherent state representation of the vibronic dynamics and of the resulting response functions for the widely used linearly displaced harmonic oscillator model. Analytical expressions are initially derived for the case of third-order response functions in an N-level system, with ground state initialization of the oscillator (zero-temperature limit). The results are then generalized to the case of Mth order response functions, with arbitrary M. The formal derivation is translated into a simple recipe, whereby the explicit analytical expressions of the response functions can be derived directly from the Feynman diagrams. We further generalize to the whole set of initial coherent states, which form an overcomplete basis. This allows one, in principle, to derive the dependence of the response functions on arbitrary initial states of the vibrational modes and is here applied to the case of thermal states. Finally, a non-Hermitian Hamiltonian approach is used to include in the above expressions the effect of vibrational relaxation.

Coherent vibrational dynamics of Au144(SR)60 nanoclusters

The coherent vibrational dynamics of gold nanoclusters (NCs) provides important information on the coupling between vibrations and electrons as well as their mechanical properties, which is critical for understanding the evolution from a metallic state to a molecular state with diminishing size. Coherent vibrations have been widely explored in small-sized atomically precise gold NCs, while it remains a challenge to observe them in large-sized gold NCs. In this work, we report the coherent vibrational dynamics of atomically precise Au144(SR)60 NCs via temperature-dependent femtosecond transient absorption (TA) spectroscopy. The population dynamics of Au144(SR)60 consists of three relaxation processes: internal conversion, core-shell charge transfer and relaxation to the ground state. After removing the population dynamics from the TA kinetics, fast Fourier transform analysis on the residual oscillation reveals distinct vibrational modes at 1.5 THz (50 cm-1) and 2 THz (67 cm-1), which arise from the wavepacket motions along the ground-state and excited-state potential energy surfaces (PES), respectively. These results are helpful for understanding the physical properties of gold nanostructures with a threshold size that lies in between those of molecular-like NCs and metallic-state nanoparticles.

π-Stacking-Dependent Vibronic Couplings Drive Excited-State Dynamics in Perylenediimide Assemblies

Vibronic coupling, the interplay of electronic and nuclear vibrational motion, is considered a critical mechanism in photoinduced reactions such as energy transfer, charge transfer, and singlet fission. However, our understanding of how particular vibronic couplings impact excited-state dynamics is lacking due to the limited number of experimental studies of model molecular systems. Herein, we use two-dimensional electronic spectroscopy (2DES) to launch and interrogate a range of vibronic coherences in two distinct types of perylenediimide slip stacks─along the short and long molecular axes, which form either an excimer or a mixed state between the Frenkel exciton (FE) and charge transfer states. We explore the functionality of these vibronic coherences using quantum beatmaps, which display the Fourier amplitude signal oscillations as a function of pump and probe frequencies, along with knowledge of the characteristic signatures of the FE, ionic, and excimer species. We find that a low-frequency vibrational mode of the short-axis slip stack appears concomitantly with the formation of the excimer state, survives 2-fold longer than in the FE state in the reference monomer, and shows a phase shift compared to other modes. For the long-axis slip stacks, a pair of low-frequency modes coupled to a high-frequency coordinate of the FE state were found to play a critical role in mixed-state generation. Our findings thus experimentally reveal the complex and varying roles of vibronic couplings in tightly packed multimers undergoing a range of photoinduced processes.

Beating signals in CdSe quantum dots measured by low-temperature 2D spectroscopy

Advances in ultrafast spectroscopy can provide access to dynamics involving nontrivial quantum correlations and their evolutions. In coherent 2D spectroscopy, the oscillatory time dependence of a signal is a signature of such quantum dynamics. Here we study such beating signals in electronic coherent 2D spectroscopy of CdSe quantum dots (CdSe QDs) at 77 K. The beating signals are analyzed in terms of their positive and negative Fourier components. We conclude that the beatings originate from coherent LO-phonons of CdSe QDs. No evidence for the quantum dot size dependence of the LO-phonon frequency was identified.

Leveraging Dynamical Symmetries in Two-Dimensional Electronic Spectra to Extract Population Transfer Pathways

We present a method to deterministically isolate population transfer kinetics from two-dimensional electronic spectroscopic signals. Central to this analysis is the characterization of how all possible subensembles of excited state systems evolve through the population time. When these dynamics are diagrammatically mapped by using double-sided Feynman pathways where population time dynamics are included, a useful symmetry emerges between excited state absorption and ground state bleach recovery dynamics of diagonal and below diagonal cross-peak signals. This symmetry allows removal of pathways from the spectra to isolate signals that evolve according to energy transfer kinetics. We describe a regression procedure to fit to energy transfer time constants and characterize the accuracy of the method in a variety of complex excited state systems using simulated two-dimensional spectra. Our results show that the method is robust for extracting ultrafast energy transfer in multistate excitonic systems, systems containing dark states that affect the signal kinetics, and systems with interfering vibrational relaxation pathways. This procedure can be used to accurately extract energy transfer kinetics from a wide variety of condensed phase systems.

Exact simulation of pigment-protein complexes unveils vibronic renormalization of electronic parameters in ultrafast spectroscopy

The primary steps of photosynthesis rely on the generation, transport, and trapping of excitons in pigment-protein complexes (PPCs). Generically, PPCs possess highly structured vibrational spectra, combining many discrete intra-pigment modes and a quasi-continuous of protein modes, with vibrational and electronic couplings of comparable strength. The intricacy of the resulting vibronic dynamics poses significant challenges in establishing a quantitative connection between spectroscopic data and underlying microscopic models. Here we show how to address this challenge using numerically exact simulation methods by considering two model systems, namely the water-soluble chlorophyll-binding protein of cauliflower and the special pair of bacterial reaction centers. We demonstrate that the inclusion of the full multi-mode vibronic dynamics in numerical calculations of linear spectra leads to systematic and quantitatively significant corrections to electronic parameter estimation. These multi-mode vibronic effects are shown to be relevant in the longstanding discussion regarding the origin of long-lived oscillations in multidimensional nonlinear spectra.

Multimode vibronic mixing in model photosynthetic systems revealed by numerically exact simulations is shown to strongly modify linear and non-linear optical responses and facilitate the persistence of coherent dynamics.

Vibronic resonance along effective modes mediates selective energy transfer in excitonically coupled aggregates

We recently proposed effective normal modes for excitonically coupled aggregates that exactly transform the energy transfer Hamiltonian into a sum of one-dimensional Hamiltonians along the effective normal modes. Identifying physically meaningful vibrational motions that maximally promote vibronic mixing suggested an interesting possibility of leveraging vibrational-electronic resonance for mediating selective energy transfer. Here, we expand on the effective mode approach, elucidating its iterative nature for successively larger aggregates, and extend the idea of mediated energy transfer to larger aggregates. We show that energy transfer between electronically uncoupled but vibronically resonant donor-acceptor sites does not depend on the intermediate site energy or the number of intermediate sites. The intermediate sites simply mediate electronic coupling such that vibronic coupling along specific promoter modes leads to direct donor-acceptor energy transfer, bypassing any intermediate uphill energy transfer steps. We show that the interplay between the electronic Hamiltonian and the effective mode transformation partitions the linear vibronic coupling along specific promoter modes to dictate the selectivity of mediated energy transfer with a vital role of interference between vibronic couplings and multi-particle basis states. Our results suggest a general design principle for enhancing energy transfer through synergistic effects of vibronic resonance and weak mediated electronic coupling, where both effects individually do not promote efficient energy transfer. The effective mode approach proposed here paves a facile route toward four-wavemixing spectroscopy simulations of larger aggregates without severely approximating resonant vibronic coupling.

Accelerating symmetry-breaking charge separation in a perylenediimide trimer through a vibronically coherent dimer intermediate

Understanding the photophysics and photochemistry of molecular π-stacked chromophores is important for utilizing them as functional photonic materials. However, these investigations have been mostly limited to covalent molecular dimers, which can only approximate the electronic and vibronic interactions present in the higher oligomers typical of functional organic materials. Here we show that a comparison of the excited-state dynamics of a covalent slip-stacked perylenediimide dimer (2) and trimer (3) provides fundamental insights into electronic state mixing and symmetry-breaking charge separation (SB-CS) beyond the dimer limit. We find that coherent vibronic coupling to high-frequency modes facilitates ultrafast state mixing between the Frenkel exciton (FE) and charge-transfer (CT) states. Subsequently, solvent fluctuations and interchromophore low-frequency vibrations promote CT character in the coherent FE/CT mixed state. The coherent FE/CT mixed state persists in 2, but, in 3, low-frequency vibronic coupling collapses the coherence, resulting in ultrafast SB-CS between the distal perylenediimide units.

Vibrational and vibronic coherences in the energy transfer process of light-harvesting complex II revealed by two-dimensional electronic spectroscopy

The presence of quantum coherence in light-harvesting complex II (LHCII) as a mechanism to understand the efficiency of the light-harvesting function in natural photosynthetic systems is still debated due to its structural complexity and weak-amplitude coherent oscillations. Here, we revisit the coherent dynamics and clarify different types of coherences in the energy transfer processes of LHCII using a joint method of the high-S/N transient grating and two-dimensional electronic spectroscopy. We find that the electronic coherence decays completely within 50 fs at room temperature. The vibrational coherences of chlorophyll a dominate over oscillations within 1 ps, whereas a low-frequency mode of 340 cm−1 with a vibronic mixing character may participate in vibrationally assisted energy transfer between chlorophylls a. Our results may suggest that vibronic mixing is relevant for rapid energy transfer processes among chlorophylls in LHCII.

Dynamics of Excitons in Conjugated Molecules and Organic Semiconductor Systems

The exciton, an excited electron-hole pair bound by Coulomb attraction, plays a key role in photophysics of organic molecules and drives practically important phenomena such as photoinduced mechanical motions of a molecule, photochemical conversions, energy transfer, generation of free charge carriers, etc. Its behavior in extended π-conjugated molecules and disordered organic films is very different and very rich compared with exciton behavior in inorganic semiconductor crystals. Due to the high degree of variability of organic systems themselves, the exciton not only exerts changes on molecules that carry it but undergoes its own changes during all phases of its lifetime, that is, birth, conversion and transport, and decay. The goal of this review is to give a systematic and comprehensive view on exciton behavior in π-conjugated molecules and molecular assemblies at all phases of exciton evolution with emphasis on rates typical for this dynamic picture and various consequences of the above dynamics. To uncover the rich variety of exciton behavior, details of exciton formation, exciton transport, exciton energy conversion, direct and reverse intersystem crossing, and radiative and nonradiative decay are considered in different systems, where these processes lead to or are influenced by static and dynamic disorder, charge distribution symmetry breaking, photoinduced reactions, electron and proton transfer, structural rearrangements, exciton coupling with vibrations and intermediate particles, and exciton dissociation and annihilation as well.

Coherent Two-Dimensional and Broadband Electronic Spectroscopies

Over the past few decades, coherent broadband spectroscopy has been widely used to improve our understanding of ultrafast processes (e.g., photoinduced electron transfer, proton transfer, and proton-coupled electron transfer reactions) at femtosecond resolution. The advances in femtosecond laser technology along with the development of nonlinear multidimensional spectroscopy enabled further insights into ultrafast energy transfer and carrier relaxation processes in complex biological and material systems. New discoveries and interpretations have led to improved design principles for optimizing the photophysical properties of various artificial systems. In this review, we first provide a detailed theoretical framework of both coherent broadband and two-dimensional electronic spectroscopy (2DES). We then discuss a selection of experimental approaches and considerations of 2DES along with best practices for data processing and analysis. Finally, we review several examples where coherent broadband and 2DES were employed to reveal mechanisms of photoinitiated ultrafast processes in molecular, biological, and material systems. We end the review with a brief perspective on the future of the experimental techniques themselves and their potential to answer an even greater range of scientific questions.

Hidden vibronic and excitonic structure and vibronic coherence transfer in the bacterial reaction center

We report two-dimensional electronic spectroscopy (2DES) experiments on the bacterial reaction center (BRC) from purple bacteria, revealing hidden vibronic and excitonic structure. Through analysis of the coherent dynamics of the BRC, we identify multiple quasi-resonances between pigment vibrations and excitonic energy gaps, and vibronic coherence transfer processes that are typically neglected in standard models of photosynthetic energy transfer and charge separation. We support our assignment with control experiments on bacteriochlorophyll and simulations of the coherent dynamics using a reduced excitonic model of the BRC. We find that specific vibronic coherence processes can readily reveal weak exciton transitions. While the functional relevance of such processes is unclear, they provide a spectroscopic tool that uses vibrations as a window for observing excited state structure and dynamics elsewhere in the BRC via vibronic coupling. Vibronic coherence transfer reveals the upper exciton of the “special pair” that was weakly visible in previous 2DES experiments.

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.

Redox conditions correlated with vibronic coupling modulate quantum beats in photosynthetic pigment-protein complexes

Quantum coherences, observed as time-dependent beats in ultrafast spectroscopic experiments, arise when light-matter interactions prepare systems in superpositions of states with differing energy and fixed phase across the ensemble. Such coherences have been observed in photosynthetic systems following ultrafast laser excitation, but what these coherences imply about the underlying energy transfer dynamics remains subject to debate. Recent work showed that redox conditions tune vibronic coupling in the Fenna-Matthews-Olson (FMO) pigment-protein complex in green sulfur bacteria, raising the question of whether redox conditions may also affect the long-lived (>100 fs) quantum coherences observed in this complex. In this work, we perform ultrafast two-dimensional electronic spectroscopy measurements on the FMO complex under both oxidizing and reducing conditions. We observe that many excited-state coherences are exclusively present in reducing conditions and are absent or attenuated in oxidizing conditions. Reducing conditions mimic the natural conditions of the complex more closely. Further, the presence of these coherences correlates with the vibronic coupling that produces faster, more efficient energy transfer through the complex under reducing conditions. The growth of coherences across the waiting time and the number of beating frequencies across hundreds of wavenumbers in the power spectra suggest that the beats are excited-state coherences with a mostly vibrational character whose phase relationship is maintained through the energy transfer process. Our results suggest that excitonic energy transfer proceeds through a coherent mechanism in this complex and that the coherences may provide a tool to disentangle coherent relaxation from energy transfer driven by stochastic environmental fluctuations.

Computational Study of Photochemical Relaxation Pathways of Platinum(II) Complexes

A series of functional platinum(II) complexes (Pt1-Pt3), which present high activity in four-photon absorption, in vivo imaging, and precise cancer therapy, as previously reported by the experimental work of Zhang et al. (Inorg. Chem. 2021, 60, 2362-2371), are computationally investigated in the article. We find that after the complex goes through four-photon absorption to the S₁ state, it undergoes intersystem crossing to the T₂ state and eventually reaches the T₁ state through internal conversion. On the T₁ state, both radiative and nonradiative decay to S₀ exit. The radiative decay forms the basis for the phosphorescence imaging in tissues as reported in the original paper. In addition, the nonradiative decay can simultaneously generate cytotoxic singlet oxygen by the excited energy transfer process, also known as triplet oxygen's quenching of triplet states. We conclude that the phosphorescence property as well as the photosensitizer character jointly bring high activity of in vivo imaging and photodynamic therapy to these complexes.

Vibronic coupling in energy transfer dynamics and two-dimensional electronic-vibrational spectra

We introduce a heterodimer model in which multiple mechanisms of vibronic coupling and their impact on energy transfer can be explicitly studied. We consider vibronic coupling that arises through either Franck-Condon activity in which each site in the heterodimer has a local electron-phonon coupling or Herzberg-Teller activity in which the transition dipole moment coupling the sites has an explicit vibrational mode-dependence. We have computed two-dimensional electronic-vibrational (2DEV) spectra for this model while varying the magnitude of these two effects and find that 2DEV spectra contain static and dynamic signatures of both types of vibronic coupling. Franck-Condon activity emerges through a change in the observed excitonic structure, while Herzberg-Teller activity is evident in the appearance of significant side-band transitions that mimic the lower-energy excitonic structure. A comparison of quantum beating patterns obtained from analysis of the simulated 2DEV spectra shows that this technique can report on the mechanism of energy transfer, elucidating a means of experimentally determining the role of specific vibronic coupling mechanisms in such processes.

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

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

2D Electronic Spectroscopic Techniques for Quantum Technology Applications

2D electronic spectroscopy (2DES) techniques have gained particular interest given their capability of following ultrafast coherent and noncoherent processes in real-time. Although the fame of 2DES is still majorly linked to the investigation of energy and charge transport in biological light-harvesting complexes, 2DES is now starting to be recognized as a particularly valuable tool for studying transport processes in artificial nanomaterials and nanodevices. Particularly meaningful is the possibility of assessing coherent mechanisms active in the transport of excitation energy in these materials toward possible quantum technology applications. The diverse nature of these new target samples poses significant challenges and calls for a critical rethinking of the technique and its different realizations. With the confluence of promising new applications and rapidly developing technical capabilities, the enormous potential of 2DES techniques to impact the field of nanosystems, quantum technologies, and quantum devices is here delineated.

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.

Excitation relaxation in a molecular chain and energy transfer at steady state

We consider the reduced dynamics of a molecular chain weakly coupled to a phonon bath. With a small and constant inhomogeneity in the coupling, the excitation relaxation rates are obtained in closed form. They are dominated by transitions between exciton modes lying next to each other in the energy spectrum. The rates are quadratic in the number of sites in a long chain. Consequently, the evolution of site occupation numbers exhibits longer coherence lifetime for short chains only. When external source and sink are added, the rate equations of exciton occupation numbers are similar to those obtained earlier by Fröhlich to explain energy storage and energy transfer in biological systems. There is a clear separation of timescale into a faster one pertaining to internal influence of the chain and phonon bath, and a slower one determined by external influence, such as the pumping rate of the source, the absorption rate of the sink, and the rate of radiation loss. The energy transfer efficiency at steady state depends strongly on these external parameters and is robust against a change in the internal parameters, such as temperature and inhomogeneity. Excitations are predicted to concentrate to the lowest energy mode when the source power is sufficiently high. In the site basis, this implies that when sustained by a high power source, a sink positioned at the center of the chain is more efficient in trapping energy than a sink placed at its end. Analytic expressions of energy transfer efficiency are obtained in the high power and low-power source limit. Parameters of a photosynthetic system are used as examples to illustrate the results.

Effective normal modes identify vibrational motions which maximally promote vibronic mixing in excitonically coupled aggregates

Controlling energy transfer through vibronic resonance is an interesting possibility. Exact treatment of non-adiabatic vibronic coupling is necessary to fully capture its role in driving energy transfer. However, the exact treatment of vibrations in extended systems is expensive, sometimes requiring oversimplifying approximations to reduce vibrational dimensionality, and do not provide physical insights into which specific vibrational motions promote energy transfer. In this communication, we derive effective normal modes for understanding vibronically enhanced energy transfer in excitonically coupled aggregates. We show that the dynamics of the overall high-dimensional vibronic Hamiltonian can be better understood through one-dimensional Hamiltonians separable along these effective modes. We demonstrate this approach on a trimer toy model to analyze the role of an intermediate "trap" site in mediating energy transfer between electronically uncoupled sites. Bringing uncoupled sites into vibronic resonance converts the "trap" into a "shuttle" for energy transfer. By deconvolving the dynamics along the aggregate normal modes, our approach identifies the specific vibrational motions, which maximally promote energy transfer, against spectator modes, which do not participate in vibronic mixing.

Insights into Photosynthetic Energy Transfer Gained from Free-Energy Structure: Coherent Transport, Incoherent Hopping, and Vibrational Assistance Revisited

Giant strides in ultrashort laser pulse technology have enabled real-time observation of dynamical processes in complex molecular systems. Specifically, the discovery of oscillatory transients in the two-dimensional electronic spectra of photosynthetic systems stimulated a number of theoretical investigations exploring the possible physical mechanisms of the remarkable quantum efficiency of light harvesting processes. In this work, we revisit the elementary aspects of environment-induced fluctuations in the involved electronic energies and present a simple way to understand energy flow with the intuitive picture of relaxation in a funnel-type free-energy landscape. The presented free-energy description of energy transfer reveals that typical photosynthetic systems operate in an almost barrierless regime. The approach also provides insights into the distinction between coherent and incoherent energy transfer and the criteria by which the necessity of the vibrational assistance is considered.

Exciton Lifetime Distributions and Population Dynamics in the FMO Protein Complex from Prosthecochloris aestuarii

Significant protein rearrangement upon excitation and energy transfer in Fenna-Matthews-Olson protein of Prosthecochloris aestuarii results in a modified energy landscape, which induces more changes in pigment site energies than predicted by the "standard" hole-burning theory. The energy changes are elucidated by simulations while investigating the effects of site-dependent disorder, both static (site-energy distribution widths) and dynamic (spectral density shapes). The resulting optimized site energies and their fluctuations are consistent with relative differences observed in inhomogeneous widths calculated by recent molecular dynamic simulations. Two sets of different spectral densities reveal how their shapes affect the population dynamics and distribution of exciton lifetimes. Calculations revealed the wavelength-dependent distributions of exciton lifetimes (T ₁) in the femtosecond to picosecond time frame. We suggest that the calculated multimodal and asymmetric wavelength-dependent T ₁ distributions offer more insight into the interpretation of resonant hole-burned (HB) spectra, kinetic traces in two-dimensional (2D) electronic spectroscopy experiments, and widely used global analyses in fitting data from transient absorption experiments.

Automatic Feynman diagram generation for nonlinear optical spectroscopies and application to fifth-order spectroscopy with pulse overlaps

Perturbative nonlinear optical spectroscopies are powerful methods to understand the dynamics of excitonic and other condensed phase systems. Feynman diagrams have long provided the essential tool to understand and interpret experimental spectra and to organize the calculation of spectra for model systems. When optical pulses are strictly time ordered, only a small number of diagrams contribute, but in many experiments, pulse-overlap effects are important for interpreting results. When pulses overlap, the number of contributing diagrams can increase rapidly, especially with higher order spectroscopies, and human error is especially likely when attempting to write down all the diagrams. We present an automated Diagram Generator (DG) that generates all the Feynman diagrams needed to calculate any nth-order spectroscopic signal. We characterize all perturbative nonlinear spectroscopies by their associated phase-discrimination condition as well as the time intervals where pulse amplitudes are nonzero. Although the DG can be used to automate impulsive calculations, its greatest strength lies in automating finite pulse calculations where pulse overlaps are important. We consider third-order transient absorption spectroscopy and fifth-order exciton-exciton interaction 2D (EEI2D) spectroscopy, which are described by six or seven diagrams in the impulsive limit, respectively, but 16 or 240 diagrams, respectively, when pulses overlap. The DG allows users to automatically include all relevant diagrams at a relatively low computational cost, since the extra diagrams are only generated for the inter-pulse delays where they are relevant. For EEI2D spectroscopy, we show the important effects of including the overlap diagrams.

Delocalization Effects in Chlorophyll Fluorescence: Nonperturbative Line Shape Analysis of a Vibronically Coupled Dimer

Non-adiabatic vibrational/electronic (vibronic) interactions in photosynthetic pigment/protein complexes (PPCs) have recently attracted considerable interest as a potential source for long-lived dynamic coherence and optimized light harvesting. The analysis of such effects is limited, however, by the complexity of the vibrational spectrum of biological pigments such as chlorophyll (Chl) molecules, which often makes numerical calculations prohibitively expensive and complicates the interpretation of experimental spectroscopic data. This work contributes to both challenges by using numerically exact computational methods to systematically examine vibronic mixing effects in the low-temperature fluorescence spectra of a Chl dimer possessing a full complement of local vibrations, using parameters extracted from experimental data. The results highlight the varying roles local vibrations can play in energy-transfer dynamics, both enhancing delocalization through vibronic resonance and, conversely, inducing dynamic localization by acting as a "self-bath" for local electronic transitions. In the specific context of line-narrowed fluorescence, the results indicate that, while low-frequency features are strongly suppressed by delocalization, high-frequency modes are likely to be dynamically localized in the parameter regime relevant to most photosynthetic complexes.