Simulation of the Two-Dimensional Electronic Spectroscopy and Energy Transfer Dynamics of Light-Harvesting Complex II at Ambient Temperature

Simulation of the Pump-Probe Spectra and Excitation Energy Relaxation of the B850 Band of the LH2 Complex in Purple Bacteria

Ultrafast spectroscopic techniques have been vital in studying excitation energy transfer (EET) in photosynthetic light harvesting complexes. In this paper, we simulate the pump-probe spectra of the B850 band of the light harvesting complex 2 (LH2) of purple bacteria, by using the hierarchical equation of motion method and the optical response function approach. The ground state bleach, stimulated emission, and excited state absorption components of the pump-probe spectra are analyzed in detail. The laser pulse-induced population dynamics are also simulated to help understand the main features of the pump-probe spectra and the EET process. It is shown that the excitation energy relaxation is an ultrafast process with multiple time scales. The first 40 fs of the pump-probe spectra is dominated by the relaxation of the k = ±1 states to both the k = 0 and higher energy states. Dynamics on a longer time scale around 200 fs reflects the relaxation of higher energy states to the k = 0 state.

Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra

Modern 4-wave mixing spectroscopies are expensive to obtain experimentally and computationally. In certain cases, the unfavorable scaling of quantum dynamics problems can be improved using a generalized quantum master equation (GQME) approach. However, the inclusion of multiple (light-matter) interactions complicates the equation of motion and leads to seemingly unavoidable cubic scaling in time. In this paper, we present a formulation that greatly simplifies and reduces the computational cost of previous work that extended the GQME framework to treat arbitrary numbers of quantum measurements. Specifically, we remove the time derivatives of quantum correlation functions from the modified Mori-Nakajima-Zwanzig framework by switching to a discrete-convolution implementation inspired by the transfer tensor approach. We then demonstrate the method's capabilities by simulating 2D electronic spectra for the excitation-energy-transfer dimer model. In our method, the resolution of data can be arbitrarily coarsened, especially along the t2 axis, which mirrors how the data are obtained experimentally. Even in a modest case, this demands O(103) fewer data points. We are further able to decompose the spectra into one-, two-, and three-time correlations, showing how and when the system enters a Markovian regime where further measurements are unnecessary to predict future spectra and the scaling becomes quadratic. This offers the ability to generate long-time spectra using only short-time data, enabling access to timescales previously beyond the reach of standard methodologies.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

All-Atom Nonadiabatic Semiclassical Mapping Dynamics for Photoinduced Charge Transfer of Organic Photovoltaic Molecules in Explicit Solvents

Direct all-atom simulation of nonadiabatic dynamics in disordered condensed phases like liquid solutions and amorphous solids has been challenging. The first all-atom simulation of the photoinduced charge-transfer dynamics of a prototypical organic photovoltaic carotenoid-porphyrin-C₆₀ molecular triad in explicit tetrahydrofuran is presented. Based on the Meyer-Miller mapping Hamiltonian, various semiclassical and mixed quantum-classical dynamics are employed, including the linearized semiclassical, symmetrical quasiclassical, mean-field Ehrenfest, classical mapping model, and spin-mapping model approaches. The all-atom nonadiabatic dynamics were compared to multi-state harmonic models with a globally shared bath, and the models built using the ensemble averages on the initial electronic state could reproduce the all-atom results. The solvent effect was found to be critical for the photoinduced charge transfer, and the time-dependent solute-solvent radial distribution functions revealed that only the nonadiabatic dynamics started with the effective forces on the initial electronic state could capture the correct nuclear dynamics. The proposed strategy for modeling condensed-phase nonadiabatic dynamics with atomistic details is readily applied to complex condensed-phase systems.

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

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

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.

Time-dependent simulation of photocurrent-detected two-dimensional spectroscopy of open systems

A new time-domain simulation protocol of two-dimensional electronic spectra with photocurrent detection is presented. Time-dependent density functional theory for open systems at finite temperature is applied to evaluate the photocurrent response to four laser pulses, and a non-perturbative phase-matching approach is implemented to extract the fourth-order photocurrent signal with a desired phase-matching condition. Simulations for an open three-level model indicates that transition dipoles interact resonantly with the incident pulses and that different sample-electrode couplings may be identified by appearance of different peaks/valleys in photocurrent spectra from different electrodes. Moreover, qualitative reproduction of experimental spectra of a PbS quantum dot photocell [Karki et al., Nat. Commun. 5(1), 5869 (2014)] reveals the stimulated electron dynamics.

Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors

We develop new methods to efficiently propagate the hierarchical equations of motion (HEOM) by using the Tucker and hierarchical Tucker (HT) tensors to represent the reduced density operator and auxiliary density operators. We first show that by employing the split operator method, the specific structure of the HEOM allows a simple propagation scheme using the Tucker tensor. When the number of effective modes in the HEOM increases and the Tucker representation becomes intractable, the split operator method is extended to the binary tree structure of the HT representation. It is found that to update the binary tree nodes related to a specific effective mode, we only need to propagate a short matrix product state constructed from these nodes. Numerical results show that by further employing the mode combination technique commonly used in the multi-configuration time-dependent Hartree approaches, the binary tree representation can be applied to study excitation energy transfer dynamics in a fairly large system including over 10⁴ effective modes. The new methods may thus provide a promising tool in simulating quantum dynamics in condensed phases.

A quantum Langevin equation approach for two-dimensional electronic spectra of coupled vibrational and electronic dynamics

We present an efficient method to simulate two-dimensional (2D) electronic spectra of condensed-phase systems with an emphasis on treating quantum nuclear wave packet dynamics explicitly. To this end, we combine a quantum Langevin equation (QLE) approach for dissipation and a perturbative scheme to calculate three-pulse photon-echo polarizations based on wave packet dynamics under the influence of external fields. The proposed dynamical approach provides a consistent description of nuclear quantum dynamics, pulse-overlap effects, and vibrational relaxation, enabling simulations of 2D electronic spectra with explicit and non-perturbative treatment of coupled electronic-nuclear dynamics. We apply the method to simulate 2D electronic spectra of a displaced-oscillator model in the condensed phase and discuss the spectral and temporal evolutions of 2D signals. Our results show that the proposed QLE approach is capable of describing vibrational relaxation, decoherence, and vibrational coherence transfer, as well as their manifestations in spectroscopic signals. Furthermore, vibrational quantum beats specific for excited-state vs ground-state nuclear wave packet dynamics can also be identified. We anticipate that this method will provide a useful tool to conduct theoretical studies of 2D spectroscopy for strong vibronically coupled systems and to elucidate intricate vibronic couplings in complex molecular systems.

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.

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.

Insights into the mechanisms and dynamics of energy transfer in plant light-harvesting complexes from two-dimensional electronic spectroscopy

During the past two decades, two-dimensional electronic spectroscopy (2DES) and related techniques have emerged as a potent experimental toolset to study the ultrafast elementary steps of photosynthesis. Apart from the highly engaging albeit controversial analysis of the role of quantum coherences in the photosynthetic processes, 2DES has been applied to resolve the dynamics and pathways of energy and electron transport in various light-harvesting antenna systems and reaction centres, providing unsurpassed level of detail. In this paper we discuss the main technical approaches and their applicability for solving specific problems in photosynthesis. We then recount applications of 2DES to study the exciton dynamics in plant and photosynthetic light-harvesting complexes, especially light-harvesting complex II (LHCII) and the fucoxanthin-chlorophyll proteins of diatoms, with emphasis on the types of unique information about such systems that 2DES is capable to deliver. This article is part of a Special Issue entitled Light harvesting, edited by Dr. Roberta Croce.

Correction of spectral distortion in two-dimensional electronic spectroscopy arising from the wedge-based delay line

Unlike the probe wavelength, which is spectrally resolved by monochromator, the excitation wavelength in two-dimensional electronic spectroscopy is retrieved by means of Fourier transform of the interference signal introduced by the coherence delay time between the first and second excitation laser pulses. Hence, the calibration of delay lines would determine its accuracy. In this work, we showed that an inaccurate calibration factor of wedge-based delay line would result in a global peak shift and asymmetric spectral twists along the excitation axis. Both theoretical analysis and experiments have shown that such spectral distortions can be corrected by an accurately predetermined calibration factor. The relative accuracy of calibration factor reaches 3 × 10^-5 in our setup. The dispersion effect of wedges also has been considered for the broadband excitation.

Coherent and dissipative quantum process tensor reconstructions in two-dimensional electronic spectroscopy

A major goal of time-resolved spectroscopy is to resolve the dynamical processes that follow photoexcitation. This amounts to identifying all the quantum states involved and the rates of population transfer between them. Unfortunately, such quantum state and kinetic reconstructions are ambiguous using one-dimensional methods such as transient absorption even when all the states of the system are fully resolved. Higher-dimensionality methods like two-dimensional spectroscopy lift some of the ambiguity, but unless the spectral features are well-separated, current inversion methods generally fail. Here, we show that, using both coherence and population signals of the nonlinear response, it is indeed possible to accurately extract both static and dynamic information from the 2D spectrum even when features are highly congested. Coherences report on the positions of the vibronic states of the system, providing a useful constraint for extracting the full kinetic scheme. We model time-resolved 2D photon echo spectra using a sum-over-states approach and show in which regimes the Hamiltonian and kinetic schemes may be recovered. Furthermore, we discuss how such algorithms may be applied to experimental data and where some of the underlying assumptions may fail. The ability to systematically extract the maximal information content of multidimensional spectroscopic data is an important step toward utilizing the full power of these techniques and elucidating the structure and dynamics of increasingly complex molecular systems.