Internal Conversion and Vibrational Energy Redistribution in Chlorophyll A

Generalized Semiclassical Ehrenfest Method: A Route to Wave Function-Free Photochemistry and Nonadiabatic Dynamics with Only Potential Energies and Gradients

We reconsider recent methods by which direct dynamics calculations of electronically nonadiabatic processes can be carried out while requiring only adiabatic potential energies and their gradients. We show that these methods can be understood in terms of a new generalization of the well-known semiclassical Ehrenfest method. This is convenient because it eliminates the need to evaluate electronic wave functions and their matrix elements along the mixed quantum-classical trajectories. The new approximations and procedures enabling this advance are the curvature-driven approximation to the time-derivative coupling, the generalized semiclassical Ehrenfest method, and a new gradient correction scheme called the time-derivative matrix (TDM) scheme. When spin-orbit coupling is present, one can carry out dynamics calculations in the fully adiabatic basis using potential energies and gradients calculated without spin-orbit coupling plus the spin-orbit coupling matrix elements. Even when spin-orbit coupling is neglected, the method is useful because it allows calculations by electronic structure methods for which nonadiabatic coupling vectors are unavailable. In order to place the new considerations in context, the article starts out with a review of background material on trajectory surface hopping, the semiclassical Ehrenfest scheme, and methods for incorporating decoherence. We consider both internal conversion and intersystem crossing. We also review several examples from our group of successful applications of the curvature-driven approximation.

Unraveling the excited-state vibrational cooling dynamics of chlorophyll-a using femtosecond broadband fluorescence spectroscopy

Understanding the dynamics of excited-state vibrational energy relaxation in photosynthetic pigments is crucial for elucidating the mechanisms underlying energy transfer processes in light-harvesting complexes. Utilizing advanced femtosecond broadband transient fluorescence (TF) spectroscopy, we explored the excited-state vibrational dynamics of Chlorophyll-a (Chl-a) both in solution and within the light-harvesting complex II (LHCII). We discovered a vibrational cooling (VC) process occurring over ∼6 ps in Chl-a in ethanol solution following Soret band excitation, marked by a notable ultrafast TF blueshift and spectral narrowing. This VC process, crucial for regulating the vibronic lifetimes, was further elucidated through the direct observation of the population dynamics of higher vibrational states within the Qy electronic state. Notably, Chl-a within LHCII demonstrated significantly faster VC dynamics, unfolding within a few hundred femtoseconds and aligning with the ultrafast energy transfer processes observed within the complex. Our findings shed light on the complex interaction between electronic and vibrational states in photosynthetic pigments, underscoring the pivotal role of vibrational dynamics in enabling efficient energy transfer within light-harvesting complexes.

Recommendations for Velocity Adjustment in Surface Hopping

This study investigates velocity adjustment directions after hopping in surface hopping dynamics. Using fulvene and a protonated Schiff base (PSB4) as case studies, we investigate the population decay and reaction yields of different sets of dynamics with the velocity adjusted in either the nonadiabatic coupling, gradient difference, or momentum directions. For the latter, in addition to the conventional algorithm, we investigated the performance of a reduced kinetic energy reservoir approach recently proposed. Our evaluation also considered velocity adjustment in the directions of approximate nonadiabatic coupling vectors. While results for fulvene are susceptible to the adjustment approach, PSB4 is not. We correlated this dependence to the topography near the conical intersections. When nonadiabatic coupling vectors are unavailable, the gradient difference direction is the best adjustment option. If the gradient difference is also unavailable, a semiempirical vector direction or the momentum direction with a reduced kinetic energy reservoir becomes an excellent option to prevent an artificial excess of back hoppings. The precise velocity adjustment direction is less crucial for describing the nonadiabatic dynamics than the kinetic energy reservoir's size.

NEXMD v2.0 Software Package for Nonadiabatic Excited State Molecular Dynamics Simulations

We present NEXMD version 2.0, the second release of the NEXMD (Nonadiabatic EXcited-state Molecular Dynamics) software package. Across a variety of new features, NEXMD v2.0 incorporates new implementations of two hybrid quantum-classical dynamics methods, namely, Ehrenfest dynamics (EHR) and the Ab-Initio Multiple Cloning sampling technique for Multiconfigurational Ehrenfest quantum dynamics (MCE-AIMC or simply AIMC), which are alternative options to the previously implemented trajectory surface hopping (TSH) method. To illustrate these methodologies, we outline a direct comparison of these three hybrid quantum-classical dynamics methods as implemented in the same NEXMD framework, discussing their weaknesses and strengths, using the modeled photodynamics of a polyphenylene ethylene dendrimer building block as a representative example. We also describe the expanded normal-mode analysis and constraints for both the ground and excited states, newly implemented in the NEXMD v2.0 framework, which allow for a deeper analysis of the main vibrational motions involved in vibronic dynamics. Overall, NEXMD v2.0 expands the range of applications of NEXMD to a larger variety of multichromophore organic molecules and photophysical processes involving quantum coherences and persistent couplings between electronic excited states and nuclear velocity.

New Gradient Correction Scheme for Electronically Nonadiabatic Dynamics Involving Multiple Spin States

It has been recommended that the best representation to use for trajectory surface hopping (TSH) calculations is the fully adiabatic basis in which the Hamiltonian is diagonal. Simulations of intersystem crossing processes with conventional TSH methods require an explicit computation of nonadiabatic coupling vectors (NACs) in the molecular-Coulomb-Hamiltonian (MCH) basis, also called the spin-orbit-free basis, in order to compute the gradient in the fully adiabatic basis (also called the diagonal representation). This explicit requirement destroys some of the advantages of the overlap-based algorithms and curvature-driven algorithms that can be used for the most efficient TSH calculations. Therefore, although these algorithms allow one to perform NAC-free simulations for internal conversion processes, one still requires NACs for intersystem crossing. Here, we show that how the NAC requirement is circumvented by a new computation scheme called the time-derivative-matrix scheme.

Surface hopping modeling of charge and energy transfer in active environments

An active environment is any atomic or molecular system changing a chromophore's nonadiabatic dynamics compared to the isolated molecule. The action of the environment on the chromophore occurs by changing the potential energy landscape and triggering new energy and charge flows unavailable in the vacuum. Surface hopping is a mixed quantum-classical approach whose extreme flexibility has made it the primary platform for implementing novel methodologies to investigate the nonadiabatic dynamics of a chromophore in active environments. This Perspective paper surveys the latest developments in the field, focusing on charge and energy transfer processes.

Direct Nonadiabatic Dynamics of Ammonia with Curvature-Driven Coherent Switching with Decay of Mixing and with Fewest Switches with Time Uncertainty: An Illustration of Population Leaking in Trajectory Surface Hopping Due to Frustrated Hops

Mixed quantum-classical nonadiabatic dynamics is a widely used approach to simulate molecular dynamics involving multiple electronic states. There are two main categories of mixed quantum-classical nonadiabatic dynamics algorithms, namely, trajectory surface hopping (TSH) in which the trajectory propagates on a single potential energy surface, interrupted by hops, and self-consistent-potential (SCP) methods, such as semiclassical Ehrenfest, in which propagation occurs on a mean-field surface without hops. In this work, we will illustrate an example of severe population leaking in TSH. We emphasize that such leaking is a combined effect of frustrated hops and long-time simulations that drive the final excited-state population toward zero as a function of time. We further show that such leaking can be alleviated-but not eliminated-by the fewest switches with time uncertainty TSH algorithm (here implemented in the SHARC program); the time uncertainty algorithm slows down the leaking process by a factor of 4.1. The population leaking is not present in coherent switching with decay of mixing (CSDM), which is an SCP method with non-Markovian decoherence included. Another result in this paper is that we find very similar results with the original CSDM algorithm, with time-derivative CSDM (tCSDM), and with curvature-driven CSDM (κCSDM). Not only do we find good agreement for electronically nonadiabatic transition probabilities but also we find good agreement of the norms of the effective nonadiabatic couplings (NACs) that are derived from the curvature-driven time-derivative couplings as implemented in κCSDM with the time-dependent norms of the nonadiabatic coupling vectors computed by state-averaged complete-active-space self-consistent field theory.

Exciton Dissociation in a Model Organic Interface: Excitonic State-Based Surface Hopping versus Multiconfigurational Time-Dependent Hartree

Quantum dynamical simulations are essential for a molecular-level understanding of light-induced processes in optoelectronic materials, but they tend to be computationally demanding. We introduce an efficient mixed quantum-classical nonadiabatic molecular dynamics method termed eXcitonic state-based Surface Hopping (X-SH), which propagates the electronic Schrödinger equation in the space of local excitonic and charge-transfer electronic states, coupled to the thermal motion of the nuclear degrees of freedom. The method is applied to exciton decay in a 1D model of a fullerene-oligothiophene junction, and the results are compared to the ones from a fully quantum dynamical treatment at the level of the Multilayer Multiconfigurational Time-Dependent Hartree (ML-MCTDH) approach. Both methods predict that charge-separated states are formed on the 10-100 fs time scale via multiple "hot-exciton dissociation" pathways. The results demonstrate that X-SH is a promising tool advancing the simulation of photoexcited processes from the molecular to the true nanomaterials scale.

Two-dimensional electronic spectroscopy of the Qx to Qy relaxation of chlorophylls a in photosystem II core complex

Using two-dimensional electronic spectroscopy, we measured the Qx to Qy transfer dynamics of the chlorophyll a (Chl a) manifold in the photosystem II (PSII) monomeric core complex from Arabidopsis thaliana. A PSII monomeric core consists of 35 Chls a and no Chl b, thus allowing for a clear window to study Chl a Qx dynamics in a large pigment-protein complex. Initial excitation in the Qx band results in a transfer to the Qy band in less than 60 fs. Upon the ultrafast transfer, regardless of the excitation frequency within the Qx band, the quasi-transient absorption spectra are very similar. This observation indicates that Chl a’s Qx to Qy transfer is not frequency selective. Using a simple model, we determined that this is not due to the lifetime broadening of the ultrafast transfer but predominantly due to a lack of correlation between the PSII core complex’s Chl a Qx and Qy bands. We suggest the origin to be the intrinsic loss of correlation during the Qx to Qy internal conversion as observed in previous studies of molecular Chl a dissolved in solvents.

Q-Band relaxation in chlorophyll: new insights from multireference quantum dynamics

The ultrafast relaxation within the Q-bands of chlorophyll plays a crucial role in photosynthetic light-harvesting. We investigate this process via nuclear and electronic quantum dynamics on multireference potential energy surfaces.

Back-and-Forth Energy Transfer during Electronic Relaxation in a Chlorin-Perylene Dyad

Donor-acceptor dyads represent a practical approach to tuning the photophysical properties of linear conjugated polymers in materials chemistry. Depending on the absorption wavelength, the acceptor and donor roles can be interchanged, and as such, the directionality of the energy transfer can be controlled. Herein, nonadiabatic excited state molecular dynamics simulations have been performed in an arylethylene-linked perylene-chlorin dyad. After an initial photoexcitation at the Soret band of chlorin, we observe an ultrafast sequential electronic relaxation to the lowest excited state. This process is accomplished through an efficient round-trip chlorin-to-perylene-to-chlorin energy transfer. It is characterized by successive intermittent localized and delocalized vibronic dynamics. Nonradiative relaxation takes place mainly through energy transfer events with perylene acting as a "heat sink" through which the nonradiative relaxation is efficiently funneled, and the excess energy is dispersed in a larger space of vibrational degrees of freedom. Thus, our findings suggest the use of donor-acceptor dyads as a useful strategy when one needs to deactivate an electronic excitation.

Cooperation between Excitation Energy Transfer and Antisynchronously Coupled Vibrations

The effects of the environment in energy transfer systems have been continuously studied for decades. Here, we investigate how the energy transfer and the emergence of vibrational correlations cooperate with each other based on simulations with a few numerically approximate mixed quantum classical (MQC) methods. By adopting a two-state system with locally coupled underdamped vibrations that are resonant with the electronic energy gap, we observe prominent energy dissipations from the electronic system to the vibrations, rehighlighting the role of underdamped vibrations as a temporal electronic energy buffer. More importantly, this energy dissipation generates specific phase relations between the two vibrations. Namely, the vibrations become anticorrelated right after the initiation of the energy transfer but then synchronized as the transfer completes. These phase relations are interpreted as a selective activation of an anticorrelated motion of the vibrations and a subsequent deactivation by thermal energy redistribution. Furthermore, we show that a single vibration simultaneously coupled to the two electronic states with opposite phases induces a completely equivalent energy transfer dynamics as the two localized vibrations. Finally, we discuss how the vibrational energy dissipation dynamics is affected by the adopted MQC approaches and warn about the increased subtlety toward properly treating dissipation effects over having reliable population dynamics.

Velocity Adjustment in Surface Hopping: Ethylene as a Case Study of the Maximum Error Caused by Direction Choice

The most common surface hopping dynamics algorithms require velocity adjustment after hopping to ensure total-energy conservation. Based on the semiclassical analysis, this adjustment must be made parallel to the nonadiabatic coupling vector's direction. Nevertheless, this direction is not always known, and the common practice has been to adjust the velocity in either the linear momentum or velocity directions. This paper benchmarks surface hopping dynamics of photoexcited ethylene with velocity adjustment in several directions, including those of the nonadiabatic coupling vector, the momentum, and the energy gradient difference. It is shown that differences in time constants and structural evolution fall within the statistical uncertainty of the method considering up to 500 trajectories in each dynamics set, rendering the three approaches statistically equivalent. For larger ensembles beyond 1000 trajectories, significant differences between the results arise, limiting the validity of adjustment in alternative directions. Other possible adjustment directions (velocity, single-state gradients, angular momentum) are evaluated as well. Given the small size of ethylene, the results reported in this paper should be considered an upper limit for the error caused by the choice of the velocity-adjustment direction on surface hopping dynamics.

Unraveling the internal conversion process within the Q-bands of a chlorophyll-like-system through surface-hopping molecular dynamics simulations

The non-radiative relaxation process within the Q-bands of chlorophylls represents a crucial preliminary step during the photosynthetic mechanism. Despite several experimental and theoretical efforts performed in order to clarify the complex dynamics characterizing this stage, a complete understanding of this mechanism is still far to be reached. In this study, non-adiabatic excited-state molecular dynamic simulations have been performed to model the non-radiative process within the Q-bands for a model system of chlorophylls. This system has been considered in the gas phase and then, to have a more representative picture of the environment, with implicit and mixed implicit-explicit solvation models. In the first part of this analysis, absorption spectra have been simulated for each model in order to guide the setup for the non-adiabatic excited-state molecular dynamic simulations. Then, non-adiabatic excited-state molecular dynamic simulations have been performed on a large set of independent trajectories and the population of the Q_x and Q_y states has been computed as the average of all the trajectories, estimating the rate constant for the process. Finally, with the aim of investigating the possible role played by the solvent in the Q_x-Q_y crossing mechanism, an essential dynamic analysis has been performed on the generated data, allowing one to find the most important motions during the simulated dynamics.

A general approach to study molecular fragmentation and energy redistribution after an ionizing event

We propose to combine quantum chemical calculations, statistical mechanical methods, and photoionization and particle collision experiments to unravel the redistribution of internal energy of the furan cation and its dissociation pathways. This approach successfully reproduces the relative intensity of the different fragments as a function of the internal energy of the system in photoelectron-photoion coincidence experiments and the different mass spectra obtained when ions ranging from Ar+ to Xe25+ or electrons are used in collision experiments. It provides deep insights into the redistribution of the internal energy in the ionized molecule and its influence on the dissociation pathways and resulting charged fragments. The present pilot study demonstrates the efficiency of a statistical exchange of excitation energy among various degrees of freedom of the molecule and proves that the proposed approach is mature to be extended to more complex systems.

Photoinduced Dynamics with Constrained Vibrational Motion: FrozeNM Algorithm

Ab initio molecular dynamics (AIMD) simulation, analyzed in terms of vibrational normal modes, is a widely used technique that facilitates understanding of complex structural motions and coupling between electronic and nuclear degrees of freedom. Usually, only a subset of vibrations is directly involved in the process of interest. The impact of these vibrations can be evaluated by performing AIMD simulations by selectively freezing certain motions. Herein, we present frozen normal mode (FrozeNM), a new algorithm to apply normal-mode constraints in AIMD simulations, as implemented in the nonadiabatic excited state molecular dynamics code. We further illustrate its capacity by analyzing the impact of normal-mode constraints on the photoinduced energy transfer between polyphenylene ethynylene dendrimer building blocks. Our results show that the electronic relaxation can be significantly slowed down by freezing a well-selected small subset of active normal modes characterized by their contributions in the direction of energy transfer. The application of these constraints reduces the nonadiabatic coupling between electronic excited states during the entire dynamical simulations. Furthermore, we validate reduced dimensionality models by freezing all the vibrations, except a few active modes. Altogether, we consider FrozeNM as a useful tool that can be broadly used to underpin the role of vibrational motion in a studied process and to formulate reduced models that describe essential physical phenomena.

Relaxation Dynamics of Chlorophyll b in the Sub-ps Ultrafast Timescale Measured by 2D Electronic Spectroscopy

A thorough characterization of the early time sub-100 fs relaxation dynamics of biologically relevant chromophores is of crucial importance for a complete understanding of the mechanisms regulating the ultrafast dynamics of the relaxation processes in more complex multichromophoric light-harvesting systems. While chlorophyll a has already been the object of several investigations, little has been reported on chlorophyll b, despite its pivotal role in many functionalities of photosynthetic proteins. Here the relaxation dynamics of chlorophyll b in the ultrafast regime have been characterized using 2D electronic spectroscopy. The comparison of experimental measurements performed at room temperature and 77 K allows the mechanisms and the dynamics of the sub-100 fs relaxation dynamics to be characterized, including spectral diffusion and fast internal conversion assisted by a specific set of vibrational modes.

Non-adiabatic Excited-State Molecular Dynamics: Theory and Applications for Modeling Photophysics in Extended Molecular Materials

Optically active molecular materials, such as organic conjugated polymers and biological systems, are characterized by strong coupling between electronic and vibrational degrees of freedom. Typically, simulations must go beyond the Born-Oppenheimer approximation to account for non-adiabatic coupling between excited states. Indeed, non-adiabatic dynamics is commonly associated with exciton dynamics and photophysics involving charge and energy transfer, as well as exciton dissociation and charge recombination. Understanding the photoinduced dynamics in such materials is vital to providing an accurate description of exciton formation, evolution, and decay. This interdisciplinary field has matured significantly over the past decades. Formulation of new theoretical frameworks, development of more efficient and accurate computational algorithms, and evolution of high-performance computer hardware has extended these simulations to very large molecular systems with hundreds of atoms, including numerous studies of organic semiconductors and biomolecules. In this Review, we will describe recent theoretical advances including treatment of electronic decoherence in surface-hopping methods, the role of solvent effects, trivial unavoided crossings, analysis of data based on transition densities, and efficient computational implementations of these numerical methods. We also emphasize newly developed semiclassical approaches, based on the Gaussian approximation, which retain phase and width information to account for significant decoherence and interference effects while maintaining the high efficiency of surface-hopping approaches. The above developments have been employed to successfully describe photophysics in a variety of molecular materials.

Symmetry controlled excited state dynamics

Symmetry effects in internal conversion are studied by means of two isomeric cyclic tertiary aliphatic amines in a velocity map imaging (VMI) experiment on the femtosecond timescale. We conclude that lessening the symmetry of the molecule leads to loss of coherence after internal conversion between Rydberg states.

Let Digons be Bygones: The Fate of Excitons in Curved π-Systems

We explore the diverse origins of unpolarized absorption and emission of molecular polygons consisting of π-conjugated oligomer chains held in a bent geometry by strain controlled at the vertex units. For this purpose, we make use of atomistic nonadiabatic excited-state molecular dynamics simulations of a bichromophore molecular polygon (digon) with bent chromophore chains. Both structural and photoexcited dynamics were found to affect polarization features. Bending strain induces exciton localization on individual chromophore units of the conjugated chains. The latter display different transition dipole moment orientations, a feature not present in the linear oligomer counterparts. In addition, bending makes exciton localization very sensitive to molecular distortions induced by thermal fluctuations. The excited-state dynamics reveals an ultrafast intramolecular energy redistribution that spreads the exciton equally among spatially separated chromophore fragments within the molecular system. As a result, digons become virtually unpolarized absorbers and emitters, in agreement with recent experimental studies on the single-molecule level.

Coherent exciton-vibrational dynamics and energy transfer in conjugated organics

Coherence, signifying concurrent electron-vibrational dynamics in complex natural and man-made systems, is currently a subject of intense study. Understanding this phenomenon is important when designing carrier transport in optoelectronic materials. Here, excited state dynamics simulations reveal a ubiquitous pattern in the evolution of photoexcitations for a broad range of molecular systems. Symmetries of the wavefunctions define a specific form of the non-adiabatic coupling that drives quantum transitions between excited states, leading to a collective asymmetric vibrational excitation coupled to the electronic system. This promotes periodic oscillatory evolution of the wavefunctions, preserving specific phase and amplitude relations across the ensemble of trajectories. The simple model proposed here explains the appearance of coherent exciton-vibrational dynamics due to non-adiabatic transitions, which is universal across multiple molecular systems. The observed relationships between electronic wavefunctions and the resulting functionalities allows us to understand, and potentially manipulate, excited state dynamics and energy transfer in molecular materials.

Relation between Vibrational Dephasing Time and Energy Gap Fluctuations

Dephasing processes are present in basically all applications in which quantum mechanics plays a role. These applications certainly include excitation energy and charge transfer in biological systems. In a previous study, we have analyzed the vibrational dephasing time as a function of energy gap fluctuation for a large set of molecular simulations. In that investigation, individual molecular subunits were the focus of the calculations. The set of studied molecules included bacteriochlorophylls in Fenna-Matthews-Olson and light-harvesting system 2 complexes as well as bilins in PE545 aggregates. The present work extends this study to entire complexes, including the respective intermolecular couplings. Again, it can be concluded that a universal and inverse proportionality exists between dephasing time and variance of the excitonic energy gap fluctuations, whereas the respective proportionality constants can be rationalized using the energy gap autocorrelation functions. Furthermore, these findings can be extended to the gaps between higher-lying neighboring excitonic states.

Electronic Delocalization, Vibrational Dynamics, and Energy Transfer in Organic Chromophores

The efficiency of materials developed for solar energy and technological applications depends on the interplay between molecular architecture and light-induced electronic energy redistribution. The spatial localization of electronic excitations is very sensitive to molecular distortions. Vibrational nuclear motions can couple to electronic dynamics driving changes in localization. The electronic energy transfer among multiple chromophores arises from several distinct mechanisms that can give rise to experimentally measured signals. Atomistic simulations of coupled electron-vibrational dynamics can help uncover the nuclear motions directing energy flow. Through careful analysis of excited state wave function evolution and a useful fragmenting of multichromophore systems, through-bond transport and exciton hopping (through-space) mechanisms can be distinguished. Such insights are crucial in the interpretation of fluorescence anisotropy measurements and can aid materials design. This Perspective highlights the interconnected vibrational and electronic motions at the foundation of nonadiabatic dynamics where nuclear motions, including torsional rotations and bond vibrations, drive electronic transitions.