Influence of Force Fields and Quantum Chemistry Approach on Spectral Densities of BChl a in Solution and in FMO Proteins

Axial H-Bonding Solvent Controls Inhomogeneous Spectral Broadening, While Peripheral H-Bonding Solvent Controls Vibronic Broadening: Cresyl Violet in Methanol

The dynamics of the nuclei of both a chromophore and its condensed-phase environment control many spectral features, including the vibronic and inhomogeneous broadening present in spectral line shapes. For the cresyl violet chromophore in methanol, we here analyze and isolate the effect of specific chromophore-solvent interactions on simulated spectral densities, reorganization energies, and linear absorption spectra. Employing both chromophore and its condensed-phase environment control many spectral features, including the vibronic and inhomogeneous broadening present in spectral line shapes. For the cresyl violet chromophore in methanol, we here analyze and isolate the effect of specific chromophore-solvent interactions on simulated spectral densities, reorganization energies, and linear absorption spectra. Employing both force field and ab initio molecular dynamics trajectories along with the inclusion of only certain solvent molecules in the excited-state calculations, we determine that the methanol molecules axial to the chromophore are responsible for the majority of inhomogeneous broadening, with a single methanol molecule that forms an axial hydrogen bond dominating the response. The strong peripheral hydrogen bonds do not contribute to spectral broadening, as they are very stable throughout the dynamics and do not lead to increased energy-gap fluctuations. We also find that treating the strong peripheral hydrogen bonds as molecular mechanical point charges during the molecular dynamics simulation underestimates the vibronic coupling. Including these peripheral hydrogen bonding methanol molecules in the quantum-mechanical region in a geometry optimization increases the vibronic coupling, suggesting that a more advanced treatment of these strongly interacting solvent molecules during the molecular dynamics trajectory may be necessary to capture the full vibronic spectral broadening.

The synergy between the PscC subunits for electron transfer to the P840 special pair in Chlorobaculum tepidum

The primary photochemical reaction of photosynthesis in green sulfur bacteria occurs in the homodimer PscA core proteins by a special chlorophyll pair. The light induced excited state of the special pair producing P840+ is rapidly reduced by electron transfer from one of the two PscC subunits. Molecular dynamics (MD) simulations are combined with bioinformatic tools herein to provide structural and dynamic insight into the complex between the two PscA core proteins and the two PscC subunits. The microscopic dynamic model involves extensive sampling at atomic resolution and at a cumulative time-scale of 22µs and reveals well defined protein-protein interactions. The membrane complex is composed of the two PscA and the two PscC subunits and macroscopic connections are revealed within a putative electron transfer pathway from the PscC subunit to the special pair P840 located within the PscA subunits. Our results provide a structural basis for understanding the electron transport to the homodimer RC of the green sulfur bacteria. The MD based approach can provide the basis to further probe the PscA-PscC complex dynamics and observe electron transfer therein at the quantum level. Furthermore, the transmembrane helices of the different PscC subunits exert distinct dynamics in the complex.

Protein Effects on the Excitation Energies and Exciton Dynamics of the CP24 Antenna Complex

In this study, the site energy fluctuations, energy transfer dynamics, and some spectroscopic properties of the minor light-harvesting complex CP24 in a membrane environment were determined. For this purpose, a 3 μs-long classical molecular dynamics simulation was performed for the CP24 complex. Furthermore, using the density functional tight binding/molecular mechanics molecular dynamics (DFTB/MM MD) approach, we performed excited state calculations for the chlorophyll a and chlorophyll b molecules in the complex starting from five different positions of the MD trajectory. During the extended simulations, we observed variations in the site energies of the different sets as a result of the fluctuating protein environment. In particular, a water coordination to Chl-b 608 occurred only after about 1 μs in the simulations, demonstrating dynamic changes in the environment of this pigment. From the classical and the DFTB/MM MD simulations, spectral densities and the (time-dependent) Hamiltonian of the complex were determined. Based on these results, three independent strongly coupled chlorophyll clusters were revealed within the complex. In addition, absorption and fluorescence spectra were determined together with the exciton relaxation dynamics, which reasonably well agrees with experimental time scales.

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

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

Electronic Excited States from Physically Constrained Machine Learning

Data-driven techniques are increasingly used to replace electronic-structure calculations of matter. In this context, a relevant question is whether machine learning (ML) should be applied directly to predict the desired properties or combined explicitly with physically grounded operations. We present an example of an integrated modeling approach in which a symmetry-adapted ML model of an effective Hamiltonian is trained to reproduce electronic excitations from a quantum-mechanical calculation. The resulting model can make predictions for molecules that are much larger and more complex than those on which it is trained and allows for dramatic computational savings by indirectly targeting the outputs of well-converged calculations while using a parametrization corresponding to a minimal atom-centered basis. These results emphasize the merits of intertwining data-driven techniques with physical approximations, improving the transferability and interpretability of ML models without affecting their accuracy and computational efficiency and providing a blueprint for developing ML-augmented electronic-structure methods.

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.

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.

Predicting Mutation-Induced Changes in the Electronic Properties of Photosynthetic Proteins from First Principles: The Fenna-Matthews-Olson Complex Example

Multiscale molecular modeling is utilized to predict optical absorption and circular dichroism spectra of two single-point mutants of the Fenna-Matthews-Olson photosynthetic pigment-protein complex. The modeling approach combines classical molecular dynamics simulations with structural refinement of photosynthetic pigments and calculations of their excited states in a polarizable protein environment. The only experimental input to the modeling protocol is the X-ray structure of the wild-type protein. The first-principles modeling reproduces changes in the experimental optical spectra of the considered mutants, Y16F and Q198V. Interestingly, the Q198V mutation has a negligible effect on the electronic properties of the targeted bacteriochlorophyll a pigment. Instead, the electronic properties of several other pigments respond to this mutation. The molecular modeling demonstrates that a single-point mutation can induce long-range effects on the protein structure, while extensive structural changes near a pigment do not necessarily lead to significant changes in the electronic properties of that pigment.

Spectral densities and absorption spectra of the core antenna complex CP43 from photosystem II

Besides absorbing light, the core antenna complex CP43 of photosystem II is of great importance in transferring excitation energy from the antenna complexes to the reaction center. Excitation energies, spectral densities, and linear absorption spectra of the complex have been evaluated by a multiscale approach. In this scheme, quantum mechanics/molecular mechanics molecular dynamics simulations are performed employing the parameterized density functional tight binding (DFTB) while the time-dependent long-range-corrected DFTB scheme is applied for the excited state calculations. The obtained average spectral density of the CP43 complex shows a very good agreement with experimental results. Moreover, the excitonic Hamiltonian of the system along with the computed site-dependent spectral densities was used to determine the linear absorption. While a Redfield-like approximation has severe shortcomings in dealing with the CP43 complex due to quasi-degenerate states, the non-Markovian full second-order cumulant expansion formalism is able to overcome the drawbacks. Linear absorption spectra were obtained, which show a good agreement with the experimental counterparts at different temperatures. This study once more emphasizes that by combining diverse techniques from the areas of molecular dynamics simulations, quantum chemistry, and open quantum systems, it is possible to obtain first-principle results for photosynthetic complexes, which are in accord with experimental findings.

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.

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.

The Influence of Electronic Polarization on Nonlinear Optical Spectroscopy

The environment surrounding a chromophore can dramatically affect the energy absorption and relaxation process, as manifested in optical spectra. Simulations of nonlinear optical spectroscopy, such as two-dimensional electronic spectroscopy (2DES) and transient absorption (TA), will be influenced by the computational model of the environment. We here compare a fixed point charge molecular mechanics model and a quantum mechanical (QM) model of the environment in computed 2DES and TA spectra of Nile red in water and the chromophore of photoactive yellow protein (PYP) in water and protein environments. In addition to simulating these nonlinear optical spectra, we directly juxtapose the computed excitation energy correlation function to the dynamic Stokes shift function often used to analyze environment dynamics. Overall, we find that for the three systems studied here the mutual electronic polarization provided by the QM environment manifests in broader 2DES signals, as well as a larger reorganization energy and a larger static Stokes shift due to stronger coupling between the chromophore and the environment.

Computational spectroscopy of complex systems

Numerous linear and non-linear spectroscopic techniques have been developed to elucidate structural and functional information of complex systems ranging from natural systems, such as proteins and light-harvesting systems, to synthetic systems, such as solar cell materials and light-emitting diodes. The obtained experimental data can be challenging to interpret due to the complexity and potential overlapping spectral signatures. Therefore, computational spectroscopy plays a crucial role in the interpretation and understanding of spectral observables of complex systems. Computational modeling of various spectroscopic techniques has seen significant developments in the past decade, when it comes to the systems that can be addressed, the size and complexity of the sample types, the accuracy of the methods, and the spectroscopic techniques that can be addressed. In this Perspective, I will review the computational spectroscopy methods that have been developed and applied for infrared and visible spectroscopies in the condensed phase. I will discuss some of the questions that this has allowed answering. Finally, I will discuss current and future challenges and how these may be addressed.

Accurate prediction of mutation-induced frequency shifts in chlorophyll proteins with a simple electrostatic model

Photosynthetic pigment-protein complexes control local chlorophyll (Chl) transition frequencies through a variety of electrostatic and steric forces. Site-directed mutations can modify this local spectroscopic tuning, providing critical insight into native photosynthetic functions and offering the tantalizing prospect of creating rationally designed Chl proteins with customized optical properties. Unfortunately, at present, no proven methods exist for reliably predicting mutation-induced frequency shifts in advance, limiting the method's utility for quantitative applications. Here, we address this challenge by constructing a series of point mutants in the water-soluble chlorophyll protein of Lepidium virginicum and using them to test the reliability of a simple computational protocol for mutation-induced site energy shifts. The protocol uses molecular dynamics to prepare mutant protein structures and the charge density coupling model of Adolphs et al. [Photosynth. Res. 95, 197-209 (2008)] for site energy prediction; a graphical interface that implements the protocol automatically is published online at http://nanohub.org/tools/pigmenthunter. With the exception of a single outlier (presumably due to unexpected structural changes), we find that the calculated frequency shifts match the experiment remarkably well, with an average error of 1.6 nm over a 9 nm spread in wavelengths. We anticipate that the accuracy of the method can be improved in the future with more advanced sampling of mutant protein structures.

Influence of non-adiabatic effects on linear absorption spectra in the condensed phase: Methylene blue

Modeling linear absorption spectra of solvated chromophores is highly challenging as contributions are present both from coupling of the electronic states to nuclear vibrations and from solute-solvent interactions. In systems where excited states intersect in the Condon region, significant non-adiabatic contributions to absorption line shapes can also be observed. Here, we introduce a robust approach to model linear absorption spectra accounting for both environmental and non-adiabatic effects from first principles. This model parameterizes a linear vibronic coupling (LVC) Hamiltonian directly from energy gap fluctuations calculated along molecular dynamics (MD) trajectories of the chromophore in solution, accounting for both anharmonicity in the potential and direct solute-solvent interactions. The resulting system dynamics described by the LVC Hamiltonian are solved exactly using the thermalized time-evolving density operator with orthogonal polynomials algorithm (T-TEDOPA). The approach is applied to the linear absorption spectrum of methylene blue in water. We show that the strong shoulder in the experimental spectrum is caused by vibrationally driven population transfer between the bright S₁ and the dark S₂ states. The treatment of the solvent environment is one of many factors that strongly influence the population transfer and line shape; accurate modeling can only be achieved through the use of explicit quantum mechanical solvation. The efficiency of T-TEDOPA, combined with LVC Hamiltonian parameterizations from MD, leads to an attractive method for describing a large variety of systems in complex environments from first principles.

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.

Vibronic and Environmental Effects in Simulations of Optical Spectroscopy

Including both environmental and vibronic effects is important for accurate simulation of optical spectra, but combining these effects remains computationally challenging. We outline two approaches that consider both the explicit atomistic environment and the vibronic transitions. Both phenomena are responsible for spectral shapes in linear spectroscopy and the electronic evolution measured in nonlinear spectroscopy. The first approach utilizes snapshots of chromophore-environment configurations for which chromophore normal modes are determined. We outline various approximations for this static approach that assumes harmonic potentials and ignores dynamic system-environment coupling. The second approach obtains excitation energies for a series of time-correlated snapshots. This dynamic approach relies on the accurate truncation of the cumulant expansion but treats the dynamics of the chromophore and the environment on equal footing. Both approaches show significant potential for making strides toward more accurate optical spectroscopy simulations of complex condensed phase systems.

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.

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.

Normal mode analysis of spectral density of FMO trimers: Intra- and intermonomer energy transfer

The intermolecular contribution to the spectral density of the exciton-vibrational coupling of the homotrimeric Fenna-Matthews-Olson (FMO) light-harvesting protein of green sulfur bacteria P. aestuarii is analyzed by combining a normal mode analysis of the protein with the charge density coupling method for the calculation of local transition energies of the pigments. Correlations in site energy fluctuations across the whole FMO trimer are found at low vibrational frequencies. Including, additionally, the high-frequency intrapigment part of the spectral density, extracted from line-narrowing spectra, we study intra- and intermonomer exciton transfer. Whereas the intrapigment part of the spectral density is important for fast intramonomer exciton relaxation, the intermolecular contributions (due to pigment-environment coupling) determine the intermonomer exciton transfer. Neither the variations of the local Huang-Rhys factors nor the correlations in site energy fluctuations have a critical influence on energy transfer. At room temperature, the intermonomer transfer in the FMO protein occurs on a 10 ps time scale, whereas intramonomer exciton equilibration is roughly two orders of magnitude faster. At cryogenic temperatures, intermonomer transfer limits the lifetimes of the lowest exciton band. The lifetimes are found to increase between 20 ps in the center of this band up to 100 ps toward lower energies, which is in very good agreement with the estimates from hole burning data. Interestingly, exciton delocalization in the FMO monomers is found to slow down intermonomer energy transfer, at both physiological and cryogenic temperatures.

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.

Analysis of Photosynthetic Systems and Their Applications with Mathematical and Computational Models

In biological and life science applications, photosynthesis is an important process that involves the absorption and transformation of sunlight into chemical energy. During the photosynthesis process, the light photons are captured by the green chlorophyll pigments in their photosynthetic antennae and further funneled to the reaction center. One of the most important light harvesting complexes that are highly important in the study of photosynthesis is the membrane-attached Fenna–Matthews–Olson (FMO) complex found in the green sulfur bacteria. In this review, we discuss the mathematical formulations and computational modeling of some of the light harvesting complexes including FMO. The most recent research developments in the photosynthetic light harvesting complexes are thoroughly discussed. The theoretical background related to the spectral density, quantum coherence and density functional theory has been elaborated. Furthermore, details about the transfer and excitation of energy in different sites of the FMO complex along with other vital photosynthetic light harvesting complexes have also been provided. Finally, we conclude this review by providing the current and potential applications in environmental science, energy, health and medicine, where such mathematical and computational studies of the photosynthesis and the light harvesting complexes can be readily integrated.

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.

Quantum biology revisited

Photosynthesis is a highly optimized process from which valuable lessons can be learned about the operating principles in nature. Its primary steps involve energy transport operating near theoretical quantum limits in efficiency. Recently, extensive research was motivated by the hypothesis that nature used quantum coherences to direct energy transfer. This body of work, a cornerstone for the field of quantum biology, rests on the interpretation of small-amplitude oscillations in two-dimensional electronic spectra of photosynthetic complexes. This Review discusses recent work reexamining these claims and demonstrates that interexciton coherences are too short lived to have any functional significance in photosynthetic energy transfer. Instead, the observed long-lived coherences originate from impulsively excited vibrations, generally observed in femtosecond spectroscopy. These efforts, collectively, lead to a more detailed understanding of the quantum aspects of dissipation. Nature, rather than trying to avoid dissipation, exploits it via engineering of exciton-bath interaction to create efficient energy flow.

Theoretical Study on the Effect of Environment on Excitation Energy Transfer in Photosynthetic Light-Harvesting Systems

In recent years, two-dimensional (2D) electronic spectroscopy experiments prove that the excitation energy transfer (EET) in photosynthetic light-harvesting systems presents long-lived electronic quantum beating signals. After being discovered in the light-harvesting system, the quantum coherence effect has aroused widespread discussion. To illustrate the EET process in the Fenna-Matthews-Olson (FMO) and phycocyanin 645 (PC645) complex, the local protein environment is often thought to be the same; however, this is ambivalent to the practical structural analysis of the light-harvesting complex. By adopting the dissipaton equation of motion theory, we present the effect of a heterogeneous protein environment on the energy transfer process with accurate numerical results. We demonstrate that the energy transfer process relies on the local heterogeneous environment for the FMO complex. A similar good agreement is found for the PC645 complex. Furthermore, we discuss the optimal value of different chromophores in the excitation energy transfer process by controlling the environmental characteristics.

Predictive First-Principles Modeling of a Photosynthetic Antenna Protein: The Fenna-Matthews-Olson Complex

High efficiency of light harvesting in photosynthetic pigment-protein complexes is governed by evolutionary-perfected protein-assisted tuning of individual pigment properties and interpigment interactions. Due to the large number of spectrally overlapping pigments in a typical photosynthetic complex, experimental methods often fail to unambiguously identify individual chromophore properties. Here, we report a first-principles-based modeling protocol capable of predicting properties of pigments in protein environment to a high precision. The technique was applied to successfully uncover electronic properties of the Fenna-Matthews-Olson (FMO) pigment-protein complex. Each of the three subunits of the FMO complex contains eight strongly coupled bacteriochlorophyll a (BChl a) pigments. The excitonic structure of FMO can be described by an electronic Hamiltonian containing excitation (site) energies of BChl a pigments and electronic couplings between them. Several such Hamiltonians have been developed in the past based on the information from various spectroscopic measurements of FMO; however, fine details of the excitonic structure and energy transfer in FMO, especially assignments of short-lived high-energy sites, remain elusive. Utilizing polarizable embedding quantum mechanics/molecular mechanics with the effective fragment potentials, we computed the electronic Hamiltonian of FMO that is in general agreement with previously reported empirical Hamiltonians and quantitatively reproduces experimental absorption and circular dichroism spectra of the FMO protein. The developed computational protocol is sufficiently simple and can be utilized for predictive modeling of other wild-type and mutated photosynthetic pigment-protein complexes.

FMOxFMO: Elucidating Excitonic Interactions in the Fenna-Matthews-Olson Complex with the Fragment Molecular Orbital Method

In order to study Förster resonance energy transfer (FRET), the fragment molecular orbital (FMO) method is extended to compute electronic couplings between local excitations via the excited state transition density model, enabling efficient calculations of nonlocal excitations in a large molecular system and overcoming the previous limitation of being able to compute only local excitations. The results of these simple but accurate models are validated against full quantum calculations without fragmentation. The developed method is applied to a very important photosynthetic pigment-protein complex, the Fenna-Matthews-Olson complex (FMOc), that is responsible for the energy transfer from a chlorosome to the reaction center in the green sulfur bacteria. Absorption and circular dichroism spectra of FMOc are simulated, and the role of the molecular environment on the excitations is revealed.

Benchmark and performance of long-range corrected time-dependent density functional tight binding (LC-TD-DFTB) on rhodopsins and light-harvesting complexes

The chromophores of rhodopsins (Rh) and light-harvesting (LH) complexes still represent a major challenge for a quantum chemical description due to their size and complex electronic structure. Since gradient corrected and hybrid density functional approaches have been shown to fail for these systems, only range-separated functionals seem to be a promising alternative to the more time consuming post-Hartree-Fock approaches. For extended sampling of optical properties, however, even more approximate approaches are required. Recently, a long-range corrected (LC) functional has been implemented into the efficient density functional tight binding (DFTB) method, allowing to sample the excited states properties of chromophores embedded into proteins using quantum mechanical/molecular mechanical (QM/MM) with the time-dependent (TD) DFTB approach. In the present study, we assess the accuracy of LC-TD-DFT and LC-TD-DFTB for rhodopsins (bacteriorhodopsin (bR) and pharaonis phoborhodopsin (ppR)) and LH complexes (light-harvesting complex II (LH2) and Fenna-Matthews-Olson (FMO) complex). This benchmark study shows the improved description of the color tuning parameters compared to standard DFT functionals. In general, LC-TD-DFTB can exhibit a similar performance as the corresponding LC functionals, allowing a reliable description of excited states properties at significantly reduced cost. The two chromophores investigated here pose complementary challenges: while huge sensitivity to external field perturbation (color tuning) and charge transfer excitations are characteristic for the retinal chromophore, the multi-chromophoric character of the LH complexes emphasizes a correct description of inter-chromophore couplings, giving less importance to color tuning. None of the investigated functionals masters both systems simultaneously with satisfactory accuracy. LC-TD-DFTB, at the current stage, although showing a systematic improvement compared to TD-DFTB cannot be recommended for studying color tuning in retinal proteins, similar to some of the LC-DFT functionals, because the response to external fields is still too weak. For sampling of LH-spectra, however, LC-TD-DFTB is a viable tool, allowing to efficiently sample absorption energies, as shown for three different LH complexes. As the calculations indicate, geometry optimization may overestimate the importance of local minima, which may be averaged over when using trajectories. Fast quantum chemical approaches therefore may allow for a direct sampling of spectra in the near future.

Adiabatic-Molecular Dynamics Generalized Vertical Hessian Approach: A Mixed Quantum Classical Method To Compute Electronic Spectra of Flexible Molecules in the Condensed Phase

We present a general mixed quantum classical method that couples classical molecular dynamics (MD) and vibronic models to compute the shape of electronic spectra of flexible molecules in the condensed phase without, in principle, any phenomenological broadening. It is based on a partition of the nuclear motions of the solute + solvent system in "soft" and "stiff" vibrational modes and an adiabatic hypothesis that assumes that stiff modes are much faster than soft ones. In this framework, the spectrum is rigorously expressed as a conformational integral of quantum vibronic spectra along the stiff coordinates only. Soft modes enter at the classical level through the conformational distribution that is sampled with classical MD runs. In each configuration, reduced-dimensionality quadratic Hamiltonians are built in the space of the stiff coordinates only, thanks to a generalization of the Vertical Hessian harmonic model and an iterative application of projectors in internal coordinates to remove soft modes. Quantum vibronic spectra, specific for each sampled configuration of the soft coordinates, are then computed at the desired temperature with efficient time-dependent techniques, and the global spectrum simply arises from their average. For consistency of the whole procedure, classical MD runs are performed with quantum-mechanically derived force fields, parameterized at the same level of theory selected for generating the quadratic Hamiltonians along the stiff coordinates. Application to N-methyl-6-oxyquinolinium betaine in water, dithiophene in ethanol, and cyanidine in water is presented to show the performance of the method.

Does electronic coherence enhance anticorrelated pigment vibrations under realistic conditions?

The light-harvesting efficiency of a photoactive molecular complex is largely determined by the properties of its electronic quantum states. Those, in turn, are influenced by molecular vibrational states of the nuclear degrees of freedom. Here, we reexamine two recently formulated concepts that a coherent vibronic coupling between molecular states would either extend the electronic coherence lifetime or enhance the amplitude of the anticorrelated vibrational mode at longer times. For this, we study a vibronically coupled dimer and calculate the nonlinear two-dimensional (2D) electronic spectra that directly reveal electronic coherence. The time scale of electronic coherence is initially extracted by measuring the antidiagonal bandwidth of the central peak in the 2D spectrum at zero waiting time. Based on the residual analysis, we identify small-amplitude long-lived oscillations in the cross-peaks, which, however, are solely due to groundstate vibrational coherence, regardless of having resonant or off-resonant conditions. Our studies neither show an enhancement of the electronic quantum coherence nor an enhancement of the anticorrelated vibrational mode by the vibronic coupling under ambient conditions.

Efficient Simulation of Finite-Temperature Open Quantum Systems

Chain-mapping techniques in combination with the time-dependent density matrix renormalization group are a powerful tool for the simulation of open-system quantum dynamics. For finite-temperature environments, however, this approach suffers from an unfavorable algorithmic scaling with increasing temperature. We prove that the system dynamics under thermal environments can be nonperturbatively described by temperature-dependent system-environmental couplings with the initial environment state being in its pure vacuum state, instead of a mixed thermal state. As a consequence, as long as the initial system state is pure, the global system-environment state remains pure at all times. The resulting speed-up and relaxed memory requirements of this approach enable the efficient simulation of open quantum systems interacting with highly structured environments in any temperature range, with applications extending from quantum thermodynamics to quantum effects in mesoscopic systems.

Optical spectra in the condensed phase: Capturing anharmonic and vibronic features using dynamic and static approaches

Simulating optical spectra in the condensed phase remains a challenge for theory due to the need to capture spectral signatures arising from anharmonicity and dynamical effects, such as vibronic progressions and asymmetry. As such, numerous simulation methods have been developed that invoke different approximations and vary in their ability to capture different physical regimes. Here, we use several models of chromophores in the condensed phase and ab initio molecular dynamics simulations to rigorously assess the applicability of methods to simulate optical absorption spectra. Specifically, we focus on the ensemble scheme, which can address anharmonic potential energy surfaces but relies on the applicability of extreme nuclear-electronic time scale separation; the Franck-Condon method, which includes dynamical effects but generally only at the harmonic level; and the recently introduced ensemble zero-temperature Franck-Condon approach, which straddles these limits. We also devote particular attention to the performance of methods derived from a cumulant expansion of the energy gap fluctuations and test the ability to approximate the requisite time correlation functions using classical dynamics with quantum correction factors. These results provide insights as to when these methods are applicable and able to capture the features of condensed phase spectra qualitatively and, in some cases, quantitatively across a range of regimes.

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.

Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures

The time evolution in open quantum systems, such as a molecular aggregate in contact with a thermal bath, still poses a complex and challenging problem. The influence of the thermal noise can be treated using a plethora of schemes, several of which decompose the corresponding correlation functions in terms of weighted sums of exponential functions. One such scheme is based on the hierarchical equations of motion (HEOM), which is built using only certain forms of bath correlation functions. In the case where the environment is described by a complex spectral density or is at a very low temperature, approaches utilizing the exponential decomposition become very inefficient. Here, we utilize an alternative decomposition scheme for the bath correlation function based on Chebyshev polynomials and Bessel functions to derive a HEOM approach up to an arbitrary order in the environmental coupling. These hierarchical equations are similar in structure to the popular exponential HEOM scheme, but are formulated using the derivatives of the Bessel functions. The proposed scheme is tested up to the fourth order in perturbation theory for a two-level system and compared to benchmark calculations for the case of zero-temperature quantum Ohmic and super-Ohmic noise. Furthermore, the benefits and shortcomings of the present Chebyshev-based hierarchical equations are discussed.

Structure of supramolecular astaxanthin aggregates revealed by molecular dynamics and electronic circular dichroism spectroscopy

Biomolecular aggregation is omnipresent in nature and important for metabolic processes or in medical treatment; however, the phenomenon is rather difficult to predict or understand on the basis of computational models. Recently, we found that electronic circular dichroism (ECD) spectroscopy and closely related resonance Raman optical activity (RROA) are extremely sensitive to the aggregation mechanism and structure of the astaxanthin dye. In the present study, molecular dynamics (MD) and quantum chemical (QC) computations (ZIndo/S, TDDFT) are used to link the aggregate structure with ECD spectral shapes. Realistic absorption and ECD intensities were obtained and the simulations reproduced many trends observed experimentally, such as the prevalent sign pattern and dependence of the aggregate structure on the solvent type. The computationally cheaper ZIndo/S method provided results very similar to those obtained by TDDFT. In the future, the accuracy of the combined MD/QC methodology of spectra interpretation should be improved to provide more detailed information on astaxanthin aggregates and similar macromolecular systems.

Nonadiabatic Dynamics via the Symmetrical Quasi-Classical Method in the Presence of Anharmonicity

The symmetrical quasi-classical (SQC) method recently proposed by Miller and Cotton allows one to simulate nonadiabatic dynamics based on an algorithm with classical-like scaling with respect to system size. This is made possible by casting the electronic degrees of freedom in terms of mapping variables that can be propagated in a classical-like manner. While SQC was shown to be rather accurate when applied to benchmark models with harmonic electronic potential energy surfaces, it was also found to become inaccurate and to suffer numerical instabilities when applied to anharmonic systems. In this paper, we propose an extended SQC (E-SQC) methodology for overcoming those discrepancies by describing the anharmonic nuclear modes, which are coupled to the electronic degrees of freedom, in terms of classical-like mapping variables. The accuracy of E-SQC relative to standard SQC is demonstrated on benchmark models with quartic and Morse potential energy surfaces.

The modeling of the absorption lineshape for embedded molecules through a polarizable QM/MM approach

We present a polarizable QM/MM strategy to simulate the absorption line shape of chromophores embedded in complex matrices, including both homogeneous and inhomogeneous broadening.