Local protein solvation drives direct down-conversion in phycobiliprotein PC645 via incoherent vibronic transport

Energetic Landscape and Terminal Emitters of Phycobilisome Cores from Quantum Chemical Modeling

Phycobilisomes (PBs) are giant antenna supercomplexes of cyanobacteria that use phycobilin pigments to capture sunlight and transfer the collected energy to membrane-bound photosystems. In the PB core, phycobilins are bound to particular allophycocyanin (APC) proteins. Some phycobilins are thought to be terminal emitters (TEs) with red-shifted fluorescence. However, the precise identification of TEs is still under debate. In this work, we employ multiscale quantum-mechanical calculations to disentangle the excitation energy landscape of PB cores. Using the recent atomistic PB structures from Synechoccoccus PCC 7002 and Synechocystis PCC 6803, we compute the spectral properties of different APC trimers and assign the low-energy pigments. We show that the excitation energy of APC phycobilins is determined by geometric and electrostatic factors and is tuned by the specific protein-protein interactions within the core. Our findings challenge the simple picture of a few red-shifted bilins in the PB core and instead suggest that the red-shifts are established by the entire TE-containing APC trimers. Our work provides a theoretical microscopic basis for the interpretation of energy migration and time-resolved spectroscopy in phycobilisomes.

TD-DMRG Study of Exciton Dynamics with both Thermal and Static Disorders for Fenna-Matthews-Olson Complex

Photosynthesis is a fundamental process that converts solar energy into chemical energy. Understanding the microscopic mechanisms of energy transfer in photosynthetic systems is crucial for the development of novel optoelectronic materials. Simulating these processes poses significant challenges due to the intricate interactions between electrons and phonons, compounded by static disorder. In this work, we present a numerically nearly exact study using the time-dependent density matrix renormalization group (TD-DMRG) method to simulate the quantum dynamics of the Fenna-Matthews-Olson (FMO) complex considering an eight-site model with both thermal and static disorders. We employ the thermo-field dynamics formalism for temperature effects. We merge all electronic interactions into one large matrix product state (MPS) site, boosting accuracy efficiently without increasing complexity. Previous combined experimental and computational studies indicated that the static disorders range from 30 to 90 cm-1 for different FMO sites. We employ a Gaussian distribution and the auxiliary bosonic operator approach to consider the static disorder in our TD-DMRG algorithm. We investigate the impact of different initial excitation sites, temperatures, and degrees of static disorder on the exciton dynamics and temporal coherence. It is found that under the influence of the experimentally determined static disorder strength, the exciton population evolution shows a non-negligible difference at zero temperature, while it is hardly affected at room temperature.

General framework for quantifying dissipation pathways in open quantum systems. II. Numerical validation and the role of non-Markovianity

In the previous paper [C. W. Kim and I. Franco, J. Chem. Phys. 160, 214111-1-214111-13 (2024)], we developed a theory called MQME-D, which allows us to decompose the overall energy dissipation process in open quantum system dynamics into contributions by individual components of the bath when the subsystem dynamics is governed by a Markovian quantum master equation (MQME). Here, we contrast the predictions of MQME-D against the numerically exact results obtained by combining hierarchical equations of motion (HEOM) with a recently reported protocol for monitoring the statistics of the bath. Overall, MQME-D accurately captures the contributions of specific bath components to the overall dissipation while greatly reducing the computational cost compared to exact computations using HEOM. The computations show that MQME-D exhibits errors originating from its inherent Markov approximation. We demonstrate that its accuracy can be significantly increased by incorporating non-Markovianity by exploiting time scale separations (TSS) in different components of the bath. Our work demonstrates that MQME-D combined with TSS can be reliably used to understand how energy is dissipated in realistic open quantum system dynamics.

Systematic Coarse Graining of Environments for the Nonperturbative Simulation of Open Quantum Systems

Conducting precise electronic-vibrational dynamics simulations of molecular systems poses significant challenges when dealing with realistic environments composed of numerous vibrational modes. Here, we introduce a technique for the construction of effective phonon spectral densities that capture accurately open-system dynamics over a finite time interval of interest. When combined with existing nonperturbative simulation tools, our approach can reduce significantly the computational costs associated with many-body open-system dynamics.

Taming the third order cumulant approximation to linear optical spectroscopy

The second order cumulant method offers a promising pathway to predicting optical properties in condensed phase systems. It allows for the computation of linear absorption spectra from excitation energy fluctuations sampled along molecular dynamics (MD) trajectories, fully accounting for vibronic effects, direct solute-solvent interactions, and environmental polarization effects. However, the second order cumulant approximation only guarantees accurate line shapes for energy gap fluctuations obeying Gaussian statistics. A third order correction has recently been derived but often yields unphysical spectra or divergent line shapes for moderately non-Gaussian fluctuations due to the neglect of higher order terms in the cumulant expansion. In this work, we develop a corrected cumulant approach, where the collective effect of neglected higher order contributions is approximately accounted for through a dampening factor applied to the third order cumulant term. We show that this dampening factor can be expressed as a function of the skewness and kurtosis of energy gap fluctuations and can be parameterized from a large set of randomly sampled model Hamiltonians for which exact spectral line shapes are known. This approach is shown to systematically remove unphysical contributions in the form of negative absorbances from cumulant spectra in both model Hamiltonians and condensed phase systems sampled from MD and dramatically improves over the second order cumulant method in describing systems exhibiting Duschinsky mode mixing effects. We successfully apply the approach to the coumarin-153 dye in toluene, obtaining excellent agreement with experiment.

How can Phycobilisome, the unique light harvesting system in certain algae working highly efficiently: The connection in between structures and functions

Algae, which are ubiquitous in ecosystems, have evolved a variety of light-harvesting complexes to better adapt to diverse habitats. Phycobilisomes/phycobiliproteins, unique to cyanobacteria, red algae, and certain cryptomonads, compensate for the lack of chlorophyll absorption, allowing algae to capture and efficiently transfer light energy in aquatic environments. With the advancement of microscopy and spectroscopy, the structure and energy transfer processes of increasingly complex phycobilisomes have been elucidated, providing us with a vivid portrait of the dynamic adaptation of their structures to the light environment in which algae thrive: 1) Cyanobacteria living on the surface of the water use short, small phycobilisomes to absorb red-orange light and reduce the damage from blue-violet light via multiple methods; 2) Large red algae inhabiting the depths of the ocean have evolved long and dense phycobilisomes containing phycoerythrin to capture the feeble blue-green light; 3) In far-red light environments such as caves, algae use special allophycocyanin cores to optimally utilize the far-red light; 4) When the environment shifts, algae can adjust the length, composition and density of their rods to better adapt; 5) By carefully designing the position of the pigments, phycobilisomes can transfer light energy to the reaction center with nearly 100% efficiency via three energy transfer processes.

Reversible disassembly-reassembly of C-phycocyanin in pressurization-depressurization cycles of high hydrostatic pressure

Hydrostatic pressure can reversibly modulate protein-protein and protein-chromophore interactions of C-phycocyanin (C-PC) from Spirulina platensis. Small-angle X-ray scattering combined with UV-Vis spectrophotometry and protein modeling was used to explore the color and structural changes of C-PC under high pressure conditions at different pH levels. It was revealed that pressures up to 350 MPa were enough to fully disassemble C-PC from trimers to monomers at pH 7.0, or from monomers to detached subunits at pH 9.0. These disassemblies were accompanied by protein unfolding that caused these high-pressure induced structures to be more extended. These changes were reversible following depressurization. The trimer-to-monomer transition proceeded through a collection of previously unrecognized, L-shaped intermediates resembling C-PC dimers. Additionally, pressurized C-PC showed decayed Q-band absorption and fortified Soret-band absorption. This was evidence that the folded tetrapyrroles, which had folded at ambient pressure, formed semicyclic unfolded conformations at a high pressure. Upon depressurization, the peak intensity and shift all recovered stepwise, showing pressure can precisely manipulate C-PC's structure as well as its color. Overall, a protein-chromophore regulatory theory of C-PC was unveiled. The pressure-tunability could be harnessed to modify and stabilize C-PC's structure and photochemical properties for designing new delivery and optical materials.

Mapping electronic decoherence pathways in molecules

Establishing the fundamental chemical principles that govern molecular electronic quantum decoherence has remained an outstanding challenge. Fundamental questions such as how solvent and intramolecular vibrations or chemical functionalization contribute to the decoherence remain unanswered and are beyond the reach of state-of-the-art theoretical and experimental approaches. Here we address this challenge by developing a strategy to isolate electronic decoherence pathways for molecular chromophores immersed in condensed phase environments that enables elucidating how electronic quantum coherence is lost. For this, we first identify resonance Raman spectroscopy as a general experimental method to reconstruct molecular spectral densities with full chemical complexity at room temperature, in solvent, and for fluorescent and non-fluorescent molecules. We then show how to quantitatively capture the decoherence dynamics from the spectral density and identify decoherence pathways by decomposing the overall coherence loss into contributions due to individual molecular vibrations and solvent modes. We illustrate the utility of the strategy by analyzing the electronic decoherence pathways of the DNA base thymine in water. Its electronic coherences decay in [Formula: see text]30 fs. The early-time decoherence is determined by intramolecular vibrations while the overall decay by solvent. Chemical substitution of thymine modulates the decoherence with hydrogen-bond interactions of the thymine ring with water leading to the fastest decoherence. Increasing temperature leads to faster decoherence as it enhances the importance of solvent contributions but leaves the early-time decoherence dynamics intact. The developed strategy opens key opportunities to establish the connection between molecular structure and quantum decoherence as needed to develop chemical strategies to rationally modulate it.

Predicting rate kernels via dynamic mode decomposition

Simulating dynamics of open quantum systems is sometimes a significant challenge, despite the availability of various exact or approximate methods. Particularly when dealing with complex systems, the huge computational cost will largely limit the applicability of these methods. In this work, we investigate the usage of dynamic mode decomposition (DMD) to evaluate the rate kernels in quantum rate processes. DMD is a data-driven model reduction technique that characterizes the rate kernels using snapshots collected from a small time window, allowing us to predict the long-term behaviors with only a limited number of samples. Our investigations show that whether the external field is involved or not, the DMD can give accurate prediction of the result compared with the traditional propagations, and simultaneously reduce the required computational cost.

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.

Unraveling the role of thermal fluctuations on the exciton structure of the cryptophyte PC612 and PC645 photosynthetic antenna complexes

Protein scaffolds play a crucial role in tuning the light harvesting properties of photosynthetic pigment-protein complexes, influencing pigment-protein and pigment-pigment excitonic interactions. Here, we investigate the influence of thermal dynamic effects on the protein tuning mechanisms of phycocyanin PC645 and PC612 antenna complexes of cryptophyte algae, featuring closed or open quaternary structures. We employ a dual molecular dynamics (MD) strategy that combines extensive classical MD simulations with multiple short Born-Oppenheimer quantum/molecular mechanical (QM/MM) simulations to accurately account for both static and dynamic disorder effects. Additionally, we compare the results with an alternative protocol based on multiple QM/MM geometry optimizations of the pigments. Subsequently, we employ polarizable QM/MM calculations using time-dependent density functional theory (TD-DFT) to compute the excited states, and we adopt the full cumulant expansion (FCE) formalism to describe the absorption and circular dichroism spectra. Our findings indicate that thermal effects have only minor impacts on the energy ladder in PC612, despite its remarkable flexibility owing to an open quaternary structure. In striking contrast, thermal effects significantly influence the properties of PC645 due to the absence of a hydrogen bond controlling the twist of ring D in PCB β82 bilins, as well as the larger impact of fluctuations on the excited states of MBV pigments, which possess a higher conjugation length compared to other bilin types. Overall, the dual MD protocol combined with the FCE formalism yields excellent spectral properties for PC612 and PC645, and the resultant excitonic Hamiltonians pave the way for future investigations concerning the implications of open and closed quaternary structures on phycocyanin light harvesting properties.

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

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

Infrared Signatures of Phycobilins within the Phycocyanin 645 Complex

Aquatic photosynthetic organisms evolved to use a variety of light frequencies to perform photosynthesis. Phycobiliprotein phycocyanin 645 (PC645) is a light-harvesting complex in cryptophyte algae able to transfer the absorbed green solar light to other antennas with over 99% efficiency. The infrared signatures of the phycobilin pigments embedded in PC645 are difficult to access and could provide useful information to understand the mechanism behind the high efficiency of energy transfer in PC645. We use visible-pump IR-probe and two-dimensional electronic vibrational spectroscopy to study the dynamical evolution and assign the fingerprint mid-infrared signatures to each pigment in PC645. Here, we report the pigment-specific vibrational markers that enable us to track the spatial flow of excitation energy between the phycobilin pigment pairs. We speculate that two high-frequency modes (1588 and 1596 cm^-1) are involved in the vibronic coupling leading to fast (<ps) and direct energy transfer from the highest to lowest exciton, bypassing the intermediate excitons.

Understanding Primary Charge Separation in the Heliobacterial Reaction Center

The homodimeric reaction center of heliobacteria retains features of the ancestral reaction center and can thus provide insights into the evolution of photosynthesis. Primary charge separation is expected to proceed in a two-step mechanism along either of the two reaction center branches. We reveal the first charge-separation step from first-principles calculations based on time-dependent density functional theory with an optimally tuned range-separated hybrid and ab initio Born-Oppenheimer molecular dynamics: the electron is most likely localized on the electron transfer cofactor 3 (EC3, OH-chlorophyll a), and the hole on the adjacent EC2. Including substantial parts of the surrounding protein environment into the calculations shows that a distinct structural mechanism is decisive for the relative energetic positioning of the electronic excitations: specific charged amino acids in the vicinity of EC3 lower the energy of charge-transfer excitations and thus facilitate efficient charge separation. These results are discussed considering recent experimental insights.

Recent progress in atomistic modeling of light-harvesting complexes: a mini review

In this mini review, we focus on recent advances in the atomistic modeling of biological light-harvesting (LH) complexes. Because of their size and sophisticated electronic structures, multiscale methods are required to investigate the dynamical and spectroscopic properties of such complexes. The excitation energies, in this context also known as site energies, excitonic couplings, and spectral densities are key quantities which usually need to be extracted to be able to determine the exciton dynamics and spectroscopic properties. The recently developed multiscale approach based on the numerically efficient density functional tight-binding framework followed by excited state calculations has been shown to be superior to the scheme based on pure classical molecular dynamics simulations. The enhanced approach, which improves the description of the internal vibrational dynamics of the pigment molecules, yields spectral densities in good agreement with the experimental counterparts for various bacterial and plant LH systems. Here, we provide a brief overview of those results and described the theoretical foundation of the multiscale protocol.

An efficient protocol for excited states of large biochromophores

Efficient energy transport in photosynthetic antenna is a long-standing source of inspiration for artificial light harvesting materials. However, characterizing the excited states of the constituent chromophores poses a considerable challenge to mainstream quantum chemical and semiempirical excited state methods due to their size and complexity and the accuracy required to describe small but functionally important changes in their properties. In this paper, we explore an alternative approach to calculating the excited states of large biochromophores, exemplified by a specific method for calculating the Q_y transition of bacteriochlorophyll a, which we name Chl-xTB. Using a diagonally dominant approximation to the Casida equation and a bespoke parameterization scheme, Chl-xTB can match time-dependent density functional theory's accuracy and semiempirical speed for calculating the potential energy surfaces and absorption spectra of chlorophylls. We demonstrate that Chl-xTB (and other prospective realizations of our protocol) can be integrated into multiscale models, including concurrent excitonic and point-charge embedding frameworks, enabling the analysis of biochromophore networks in a native environment. We exploit this capability to probe the low-frequency spectral densities of excitonic energies and interchromophore interactions in the light harvesting antenna protein LH2 (light harvesting complex 2). The impact of low-frequency protein motion on interchromophore coupling and exciton transport has routinely been ignored due to the prohibitive costs of including it in simulations. Our results provide a more rigorous basis for continued use of this approximation by demonstrating that exciton transition energies are unaffected by low-frequency vibrational coupling to exciton interaction energies.

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.

The atomistic modeling of light-harvesting complexes from the physical models to the computational protocol

The function of light-harvesting complexes is determined by a complex network of dynamic interactions among all the different components: the aggregate of pigments, the protein, and the surrounding environment. Complete and reliable predictions on these types of composite systems can be only achieved with an atomistic description. In the last few decades, there have been important advances in the atomistic modeling of light-harvesting complexes. These advances have involved both the completeness of the physical models and the accuracy and effectiveness of the computational protocols. In this Perspective, we present an overview of the main theoretical and computational breakthroughs attained so far in the field, with particular focus on the important role played by the protein and its dynamics. We then discuss the open problems in their accurate modeling that still need to be addressed. To illustrate an effective computational workflow for the modeling of light harvesting complexes, we take as an example the plant antenna complex CP29 and its H111N mutant.

Photosynthesis tunes quantum-mechanical mixing of electronic and vibrational states to steer exciton energy transfer

Photosynthetic species evolved to protect their light-harvesting apparatus from photoxidative damage driven by intracellular redox conditions or environmental conditions. The Fenna-Matthews-Olson (FMO) pigment-protein complex from green sulfur bacteria exhibits redox-dependent quenching behavior partially due to two internal cysteine residues. Here, we show evidence that a photosynthetic complex exploits the quantum mechanics of vibronic mixing to activate an oxidative photoprotective mechanism. We use two-dimensional electronic spectroscopy (2DES) to capture energy transfer dynamics in wild-type and cysteine-deficient FMO mutant proteins under both reducing and oxidizing conditions. Under reducing conditions, we find equal energy transfer through the exciton 4-1 and 4-2-1 pathways because the exciton 4-1 energy gap is vibronically coupled with a bacteriochlorophyll-a vibrational mode. Under oxidizing conditions, however, the resonance of the exciton 4-1 energy gap is detuned from the vibrational mode, causing excitons to preferentially steer through the indirect 4-2-1 pathway to increase the likelihood of exciton quenching. We use a Redfield model to show that the complex achieves this effect by tuning the site III energy via the redox state of its internal cysteine residues. This result shows how pigment-protein complexes exploit the quantum mechanics of vibronic coupling to steer energy transfer.

Time-dependent atomistic simulations of the CP29 light-harvesting complex

Light harvesting as the first step in photosynthesis is of prime importance for life on earth. For a theoretical description of photochemical processes during light harvesting, spectral densities are key quantities. They serve as input functions for modeling the excitation energy transfer dynamics and spectroscopic properties. Herein, a recently developed procedure is applied to determine the spectral densities of the pigments in the minor antenna complex CP29 of photosystem II, which has recently gained attention because of its active role in non-photochemical quenching processes in higher plants. To this end, the density functional-based tight binding (DFTB) method has been employed to enable simulation of the ground state dynamics in a quantum-mechanics/molecular mechanics (QM/MM) scheme for each chlorophyll pigment. Subsequently, the time-dependent extension of the long-range corrected DFTB approach has been used to obtain the excitation energy fluctuations along the ground-state trajectories also in a QM/MM setting. From these results, the spectral densities have been determined and compared for different force fields and to spectral densities from other light-harvesting complexes. In addition, time-dependent and time-independent excitonic Hamiltonians of the system have been constructed and applied to the determination of absorption spectra as well as exciton dynamics.

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.

Semiclassical Modified Redfield and Generalized Förster Theories of Exciton Relaxation/Transfer in Light-Harvesting Complexes: The Quest for the Principle of Detailed Balance

A conceptual problem of transfer theories that use a semiclassical description of the electron-vibrational coupling is the neglect of the correlation between momenta and coordinates of nuclei. In the Redfield theory of exciton relaxation, this neglect leads to a violation of the principle of detailed balance; equal “uphill” and “downhill” transfer rate constants are obtained. Here, we investigate how this result depends on nuclear reorganization effects, neglected in Redfield but taken into account in the modified Redfield theory. These reorganization effects, resulting from a partial localization of excited states, are found to promote a preferential “downhill” relaxation of excitation energy. However, for realistic spectral densities of light-harvesting antennae in photosynthesis, the reorganization effects are too small to compensate for the missing coordinate–momentum uncertainty. For weaker excitonic couplings as they occur between domains of strongly coupled pigments, we find the principle of detailed balance to be fulfilled in a semiclassical variant of the generalized Förster theory. A qualitatively correct description of the transfer is obtained with this theory at a significantly lower computational cost as with the quantum generalized Förster theory. Larger deviations between the two theories are expected for large energy gaps as they occur in complexes with chemically different pigments.

Machine-learned potentials for next-generation matter simulations

The choice of simulation methods in computational materials science is driven by a fundamental trade-off: bridging large time- and length-scales with highly accurate simulations at an affordable computational cost. Venturing the investigation of complex phenomena on large scales requires fast yet accurate computational methods. We review the emerging field of machine-learned potentials, which promises to reach the accuracy of quantum mechanical computations at a substantially reduced computational cost. This Review will summarize the basic principles of the underlying machine learning methods, the data acquisition process and active learning procedures. We highlight multiple recent applications of machine-learned potentials in various fields, ranging from organic chemistry and biomolecules to inorganic crystal structure predictions and surface science. We furthermore discuss the developments required to promote a broader use of ML potentials, and the possibility of using them to help solve open questions in materials science and facilitate fully computational materials design.

Multiscale QM/MM molecular dynamics simulations of the trimeric major light-harvesting complex II

Photosynthetic processes are driven by sunlight. Too little of it and the photosynthetic machinery cannot produce the reductive power to drive the anabolic pathways. Too much sunlight and the machinery can get damaged. In higher plants, the major Light-Harvesting Complex (LHCII) efficiently absorbs the light energy, but can also dissipate it when in excess (quenching). In order to study the dynamics related to the quenching process but also the exciton dynamics in general, one needs to accurately determine the so-called spectral density which describes the coupling between the relevant pigment modes and the environmental degrees of freedom. To this end, Born-Oppenheimer molecular dynamics simulations in a quantum mechanics/molecular mechanics (QM/MM) fashion utilizing the density functional based tight binding (DFTB) method have been performed for the ground state dynamics. Subsequently, the time-dependent extension of the long-range-corrected DFTB scheme has been employed for the excited state calculations of the individual chlorophyll-a molecules in the LHCII complex. The analysis of this data resulted in spectral densities showing an astonishing agreement with the experimental counterpart in this rather large system. This consistency with an experimental observable also supports the accuracy, robustness, and reliability of the present multi-scale scheme. To the best of our knowledge, this is the first theoretical attempt on this large complex system is ever made to accurately simulate the spectral density. In addition, the resulting spectral densities and site energies were used to determine the exciton transfer rate within a special pigment pair consisting of a chlorophyll-a and a carotenoid molecule which is assumed to play a role in the balance between the light harvesting and quenching modes.

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.

Coherent Speedup of Excitation Energy Transfer in PC645

The fundamental principle governing the robust excitation energy transfer (EET) in the light-harvesting complexes remained elusive. Recent spectroscopic observations of the EET attract attention toward the role of the quantum effects in the biological systems. In this study, along with the time evolution, the distance between the quantum states of the pigments is also considered for a detailed illustration of the effect of the quantum coherence on the speed of the excitation transfer. The comparative analysis of the coherently delocalized EET with the incoherent discrete hopping reveals that the coherent superposition speedup the excitation transfer process, for instance, with the initially localized excitation, coherence provides a ∼4-fold enhancement to the excitation transfer rate.

Static Disorder in Excitation Energies of the Fenna-Matthews-Olson Protein: Structure-Based Theory Meets Experiment

Inhomogeneous broadening of optical lines of the Fenna-Matthews-Olson (FMO) light-harvesting protein is investigated by combining a Monte Carlo sampling of low-energy conformational substates of the protein with a quantum chemical/electrostatic calculation of local transition energies (site energies) of the pigments. The good agreement between the optical spectra calculated for the inhomogeneous ensemble and the experimental data demonstrates that electrostatics is the dominant contributor to static disorder in site energies. Rotamers of polar amino acid side chains are found to cause bimodal distribution functions of site energy shifts, which can be probed by hole burning and single-molecule spectroscopy. When summing over the large number of contributions, the resulting distribution functions of the site energies become Gaussians, and the correlations in site energy fluctuations at different sites practically average to zero. These results demonstrate that static disorder in the FMO protein is in the realm of the central limit theorem of statistics.

Frenkel excitons in heat-stressed supramolecular nanocomposites enabled by tunable cage-like scaffolding

Delocalized Frenkel excitons-coherently shared excitations among chromophores-are responsible for the remarkable efficiency of supramolecular light-harvesting assemblies within photosynthetic organisms. The translation of nature's design principles to applications in optoelectronic devices has been limited by the fragility of the supramolecular structures used and the delicate nature of Frenkel excitons, particularly under mildly changing solvent conditions and elevated temperatures and upon deposition onto solid substrates. Here, we overcome those functionalization barriers through composition of stable supramolecular light-harvesting nanotubes enabled by tunable (~4.3-4.9 nm), uniform (±0.3 nm) cage-like scaffolds. High-resolution cryogenic electron microscopy, combined with scanning electron microscopy, broadband femtosecond transient absorption spectroscopy and near-field scanning optical microscopy revealed that excitons within the cage-like scaffolds are robust, even under extreme heat stress, and control over nanocomposite dimensions is maintained on solid substrates. Our bio-inspired nanocomposites provide a general framework for the development of next-generation organic devices made from stable supramolecular materials.

Herzberg-Teller Effect on the Vibrationally Resolved Absorption Spectra of Single-Crystalline Pentacene at Finite Temperatures

The line shape of an electronic spectrum conveys the coupling between electronic and vibrational degrees of freedom. In the present study, the light absorption spectra of single-crystalline pentacene were measured by polarized UV-vis microscopy at 77, 185, and 293 K. The vibronic coupling encoded in each spectrum was resolved by the Herzberg-Teller theory that considers the contributions from the Franck-Condon (FC) factor, Franck-Condo/Herzberg-Teller (FC/HT) interference, and Herzberg-Teller (HT) coupling. Specifically, excitation energies, electronic transition dipole moments, and their nuclear gradients were evaluated by the GW method to ensure numerical accuracy, while the computationally efficient density function theory was employed to determine the optimized structures and vibrational normal modes. For every pair of electronic transition and normal mode that gives rise to a strong vibronic transition intensity, we examined their spatial characteristics by projecting them onto the three crystal axes. It was found that all normal modes strongly coupled to the lowest-lying a-polarized electronic transitions oscillate along axis a, whereas none of their counterparts for the lowest-lying b-polarized electronic transitions is predominantly along axis b. This notable difference on the alignment between the electronic transition and molecular vibration could help the directional control of charge dissociation and/or spin separation. Moreover, a significant variance of the destructive FC/HT interference was discovered with increasing temperatures that can well explain the a-polarized fading tableland near 650 nm. Finally, the importance of HT coupling was corroborated by comparing its intensity with those of FC factor and FC/HT interference. Taken all together, the vibrational dependence of the electronic wave function is critical to resolve the light absorption spectra of single-crystalline pentacene and its temperature effects, facilitating the systematic design of functional optical materials based on pentacene and its derivatives.

DFTB/MM Molecular Dynamics Simulations of the FMO Light-Harvesting Complex

Because of the size of light-harvesting complexes and the involvement of electronic degrees of freedom, computationally these systems need to be treated with a combined quantum-classical description. To this end, Born-Oppenheimer molecular dynamics simulations have been employed in a quantum mechanics/molecular mechanics (QM/MM) fashion for the ground state followed by excitation energy calculations again in a QM/MM scheme for the Fenna-Matthews-Olson (FMO) complex. The self-consistent-charge density functional tight-binding (DFTB) method electrostatically coupled to a classical description of the environment was applied to perform the ground-state dynamics. Subsequently, long-range-corrected time-dependent DFTB calculations were performed to determine the excitation energy fluctuations of the individual bacteriochlorophyll a molecules. The spectral densities obtained using this approach show an excellent agreement with experimental findings. In addition, the fluctuating site energies and couplings were used to estimate the exciton transfer dynamics.

Exploiting Machine Learning to Efficiently Predict Multidimensional Optical Spectra in Complex Environments

The excited-state dynamics of chromophores in complex environments determine a range of vital biological and energy capture processes. Time-resolved, multidimensional optical spectroscopies provide a key tool to investigate these processes. Although theory has the potential to decode these spectra in terms of the electronic and atomistic dynamics, the need for large numbers of excited-state electronic structure calculations severely limits first-principles predictions of multidimensional optical spectra for chromophores in the condensed phase. Here, we leverage the locality of chromophore excitations to develop machine learning models to predict the excited-state energy gap of chromophores in complex environments for efficiently constructing linear and multidimensional optical spectra. By analyzing the performance of these models, which span a hierarchy of physical approximations, across a range of chromophore-environment interaction strengths, we provide strategies for the construction of machine learning models that greatly accelerate the calculation of multidimensional optical spectra from first principles.

Designing and understanding light-harvesting devices with machine learning

Understanding the fundamental processes of light-harvesting is crucial to the development of clean energy materials and devices. Biological organisms have evolved complex metabolic mechanisms to efficiently convert sunlight into chemical energy. Unraveling the secrets of this conversion has inspired the design of clean energy technologies, including solar cells and photocatalytic water splitting. Describing the emergence of macroscopic properties from microscopic processes poses the challenge to bridge length and time scales of several orders of magnitude. Machine learning experiences increased popularity as a tool to bridge the gap between multi-level theoretical models and Edisonian trial-and-error approaches. Machine learning offers opportunities to gain detailed scientific insights into the underlying principles governing light-harvesting phenomena and can accelerate the fabrication of light-harvesting devices.

Nonlinear spectroscopy in the condensed phase: The role of Duschinsky rotations and third order cumulant contributions

First-principles modeling of nonlinear optical spectra in the condensed phase is highly challenging because both environment and vibronic interactions can play a large role in determining spectral shapes and excited state dynamics. Here, we compute two dimensional electronic spectroscopy (2DES) signals based on a cumulant expansion of the energy gap fluctuation operator, with specific focus on analyzing mode mixing effects introduced by the Duschinsky rotation and the role of the third order term in the cumulant expansion for both model and realistic condensed phase systems. We show that for a harmonic model system, the third order cumulant correction captures effects introduced by a mismatch in curvatures of ground and excited state potential energy surfaces, as well as effects of mode mixing. We also demonstrate that 2DES signals can be accurately reconstructed from purely classical correlation functions using quantum correction factors. We then compute nonlinear optical spectra for the Nile red and methylene blue chromophores in solution, assessing the third order cumulant contribution for realistic systems. We show that the third order cumulant correction is strongly dependent on the treatment of the solvent environment, revealing the interplay between environmental polarization and the electronic-vibrational coupling.

Toward monitoring the dissipative vibrational energy flows in open quantum systems by mixed quantum-classical simulations

In open quantum system dynamics, rich information about the major energy relaxation channels and corresponding relaxation rates can be elucidated by monitoring the vibrational energy flow among individual bath modes. However, such calculations often become tremendously difficult as the complexity of the subsystem-bath coupling increases. In this paper, we attempt to make this task feasible by using a mixed quantum-classical method, the Poisson-bracket mapping equation with non-Hamiltonian modification (PBME-nH) [H. W. Kim and Y. M. Rhee, J. Chem. Phys. 140, 184106 (2014)]. For a quantum subsystem bilinearly coupled to harmonic bath modes, we derive an expression for the mode energy in terms of the classical positions and momenta of the nuclei, while keeping consistency with the energy of the quantum subsystem. The accuracy of the resulting expression is then benchmarked against a numerically exact method by using relatively simple models. Although our expression predicts a qualitatively correct dissipation rate for a range of situations, cases involving a strong vibronic resonance are quite challenging. This is attributed to the inherent lack of quantum back reaction in PBME-nH, which becomes significant when the subsystem strongly interacts with a small number of bath modes. A rigorous treatment of such an effect will be crucial for developing quantitative simulation methods that can handle generic subsystem-bath coupling.

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.

On the arrangement of chromophores in light harvesting complexes: chance versus design

We used a homogeneous computational approach to derive the excitonic Hamiltonian for five light harvesting complexes containing only one type of chromophore and compare them in terms of statistical descriptors. We then studied the approximate exciton dynamics for the five complexes introducing a measure, the (averaged and time-dependent) inverse participation ratio, that enables the comparison between very diverse complexes on the same ground. We find that the global dynamics are very similar across the set of systems despite the variety of geometric structures of the complexes. In particular, the dynamics of four out of five light harvesting complexes are barely distinguishable with a small variation from the norm seen only for the Fenna-Matthews-Olson complex. We use the information from the realistic Hamiltonians to build a reduced model system that shows how the global dynamics are ultimately dominated by a single parameter, the degree of localization of the excitonic Hamiltonian eigenstates. Considering the physically plausible range of system parameters, the reduced model explains why the dynamics are so similar across most light harvesting complexes containing a single type of chromophore regardless of the detailed pattern of the inter-chromophore excitonic coupling.

Multicolor Quantum Control for Suppressing Ground State Coherences in Two-Dimensional Electronic Spectroscopy

The measured multidimensional spectral response of different light harvesting complexes exhibits oscillatory features which suggest an underlying coherent energy transfer. However, making this inference rigorous is challenging due to the difficulty of isolating excited state coherences in highly congested spectra. In this work, we provide a coherent control scheme that suppresses ground state coherences, thus making rephasing spectra dominated by excited state coherences. We provide a benchmark for the scheme using a model dimeric system and numerically exact methods to analyze the spectral response. We argue that combining temporal and spectral control methods can facilitate a second generation of experiments that are tailored to extract desired information and thus significantly advance our understanding of complex open many-body structure and dynamics.

Quantum dynamics of vibration-assisted excitation energy transfer in phycobiliprotein light-harvesting complex

Phycobiliprotein is a light-harvesting complex containing linear tetrapyrrole bilin pigments that are responsible for absorption and funneling the sun's energy in cryptophytes algae. In particular, the protein structure determines relative positions and orientations of the pigments and thus controls energy transfer pathways. The present research reveals the impact of molecular vibrations (in the 850-2700 cm^-1 region) on excitation energy transfer in phycobiliprotein. The analysis of the excitation energy transfer pathways indicates a possibility of the coherent mechanism of energy transfer (delocalization) in central dihydrobiliverdin pigments and incoherent vibration-assisted energy transfer to peripheral phycocyanobilin pigments at a sub-picosecond time scale. A computational approach that enables modeling the dynamics of the excitation energy transfer with the quantum master equation formalism employing Huang-Rhys factors to describe electronic-vibrational coupling has been developed. The computational methodology has been implemented in PyFREC software.

Dissipation-Assisted Matrix Product Factorization

Charge and energy transfer in biological and synthetic organic materials are strongly influenced by the coupling of electronic states to a highly structured dissipative environment. Nonperturbative simulations of these systems require a substantial computational effort, and current methods can only be applied to large systems if environmental structures are severely coarse grained. Time evolution methods based on tensor networks are fundamentally limited by the times that can be reached due to the buildup of entanglement in time, which quickly increases the size of the tensor representation, i.e., the bond dimension. In this Letter, we introduce a dissipation-assisted matrix product factorization (DAMPF) method that combines a tensor network representation of the vibronic state within a pseudomode description of the environment where a continuous bosonic environment is mapped into a few harmonic oscillators under Lindblad damping. This framework is particularly suitable for a tensor network representation, since damping suppresses the entanglement growth among oscillators and significantly reduces the bond dimension required to achieve a desired accuracy. We show that dissipation removes the "time-wall" limitation of existing methods, enabling the long-time simulation of large vibronic systems consisting of 10-50 sites coupled to 100-1000 underdamped modes in total and for a wide range of parameter regimes. For these reasons, we believe that our formalism will facilitate the investigation of spatially extended systems with applications to quantum biology, organic photovoltaics, and quantum thermodynamics.

Successes & challenges in the atomistic modeling of light-harvesting and its photoregulation

Light-harvesting is a crucial step of photosynthesis. Its mechanisms and related energetics have been revealed by a combination of experimental investigations and theoretical modeling. The success of theoretical modeling is largely due to the application of atomistic descriptions combining quantum chemistry, classical models and molecular dynamics techniques. Besides the important achievements obtained so far, a complete and quantitative understanding of how the many different light-harvesting complexes exploit their structural specificity is still missing. Moreover, many questions remain unanswered regarding the mechanisms through which light-harvesting is regulated in response to variable light conditions. Here we show that, in both fields, a major role will be played once more by atomistic descriptions, possibly generalized to tackle the numerous time and space scales on which the regulation takes place: going from the ultrafast electronic excitation of the multichromophoric aggregate, through the subsequent conformational changes in the embedding protein, up to the interaction between proteins.

The amazing phycobilisome

Cyanobacteria and red-algae share a common light-harvesting complex which is different than all other complexes that serve as photosynthetic antennas - the Phycobilisome (PBS). The PBS is found attached to the stromal side of thylakoid membranes, filling up most of the gap between individual thylakoids. The PBS self assembles from similar homologous protein units that are soluble and contain conserved cysteine residues that covalently bind the light absorbing chromophores, linear tetra-pyrroles. Using similar construction principles, the PBS can be as large as 16.8 MDa (68×45×39nm), as small as 1.2 MDa (24 × 11.5 × 11.5 nm), and in some unique cases smaller still. The PBS can absorb light between 450 nm to 650 nm and in some cases beyond 700 nm, depending on the species, its composition and assembly. In this review, we will present new observations and structures that expand our understanding of the distinctive properties that make the PBS an amazing light harvesting system. At the end we will suggest why the PBS, for all of its excellent properties, was discarded by photosynthetic organisms that arose later in evolution such as green algae and higher plants.

Effect of an underdamped vibration with both diagonal and off-diagonal exciton-phonon interactions on excitation energy transfer

A numerically exact approach, named as the hierarchical stochastic Schrödinger equation, is employed to investigate the resonant vibration-assisted excitation energy transfer in a dimer system, where an underdamped vibration with both diagonal and off-diagonal exciton-phonon interactions is incorporated. From a large parameter space over the site-energy difference, excitonic coupling, and reorganization energy, it is found that the promotion effect of the underdamped vibration is significant only when the excitonic coupling is smaller than the site-energy difference. Under the circumstance, there is an optimal strength ratio between diagonal and off-diagonal exciton-phonon interactions for the resonant vibration-assisted excitation energy transfer as the site-energy difference is greater than the reorganization energy, whereas in the opposite situation the most efficient energy transfer occurs as the exciton-phonon interaction is totally off-diagonal. © 2018 Wiley Periodicals, Inc.