Exciton dynamics from the mapping approach to surface hopping: comparison with Förster and Redfield theories.

We compare the recently introduced multi-state mapping approach to surface hopping (MASH) with the Förster and Redfield theories of excitation energy transfer. Whereas Förster theory relies on weak coupling between chromophores, and Redfield theory assumes the electronic excitations to be weakly coupled to fast chromophore vibrations, MASH is free from any perturbative or Markovian approximations. We illustrate this with an example application to the rate of energy transfer in a Frenkel-exciton dimer, showing that MASH interpolates correctly between the opposing regimes in which the Förster and Redfield results are reliable. We then compare the three methods for a realistic model of the Fenna-Matthews-Olson complex with a structured vibrational spectral density and static disorder in the excitation energies. In this case there are no exact results for comparison so we use MASH to assess the validity of Förster and Redfield theories. We find that Förster theory is the more accurate of the two on the picosecond timescale, as has been shown previously for a simpler model of this particular light-harvesting complex. We also explore various ways to sample the initial electronic state in MASH and find that they all give very similar results for exciton dynamics.

Extending non-adiabatic rate theory to strong electronic couplings in the Marcus inverted regime.

Electron transfer reactions play an essential role in many chemical and biological processes. Fermi's golden rule (GR), which assumes that the coupling between electronic states is small, has formed the foundation of electron transfer rate theory; however, in short range electron/energy transfer reactions, this coupling can become very large, and, therefore, Fermi's GR fails to make even qualitatively accurate rate predictions. In this paper, I present a simple modified GR theory to describe electron transfer in the Marcus inverted regime at arbitrarily large electronic coupling strengths. This theory is based on an optimal global rotation of the diabatic states, which makes it compatible with existing methods for calculating GR rates that can account for nuclear quantum effects with anharmonic potentials. Furthermore, the optimal GR (OGR) theory can also be combined with analytic theories for non-adiabatic rates, such as Marcus theory and Marcus-Levich-Jortner theory, offering clear physical insights into strong electronic coupling effects in non-adiabatic processes. OGR theory is also tested on a large set of spin-boson models and an anharmonic model against exact quantum dynamics calculations, where it performs well, correctly predicting rate turnover at large coupling strengths. Finally, an example application to a boron-dipyrromethane-anthracene photosensitizer reveals that strong coupling effects inhibit excited state charge recombination in this system, reducing the rate of this process by a factor of 4. Overall, OGR theory offers a new approach to calculating electron transfer rates at strong couplings, offering new physical insights into a range of non-adiabatic processes.

Coherent control from quantum commitment probabilities.

We introduce a general definition of a quantum committor in order to clarify reaction mechanisms and facilitate control in processes where coherent effects are important. With a quantum committor, we generalize the notion of a transition state to quantum superpositions and quantify the effect of interference on the progress of the reaction. The formalism is applicable to any linear quantum master equation supporting metastability for which absorbing boundary conditions designating the reactant and product states can be applied. We use this formalism to determine the dependence of the quantum transition state on coherences in a polaritonic system and optimize the initialization state of a conical intersection model to control reactive outcomes, achieving yields of the desired state approaching 100%. In addition to providing a practical tool, the quantum committor provides a conceptual framework for understanding reactions in cases when classical intuitions fail.

Spin selective charge recombination in chiral donor-bridge-acceptor triads.

In this paper, we outline a physically motivated framework for describing spin-selective recombination processes in chiral systems, from which we derive spin-selective reaction operators for recombination reactions of donor-bridge-acceptor molecules, where the electron transfer is mediated by chirality and spin-orbit coupling. In general, the recombination process is selective only for spin-coherence between singlet and triplet states, and it is not, in general, selective for spin polarization. We find that spin polarization selectivity only arises in hopping-mediated electron transfer. We describe how this effective spin-polarization selectivity is a consequence of spin-polarization generated transiently in the intermediate state. The recombination process also augments the coherent spin dynamics of the charge separated state, which is found to have a significant effect on the recombination dynamics and to destroy any long-lived spin polarization. Although we only consider a simple donor-bridge-acceptor system, the framework we present here can be straightforwardly extended to describe spin-selective recombination processes in more complex systems.

A simple improved low temperature correction for the hierarchical equations of motion.

The study of open system quantum dynamics has been transformed by the hierarchical equations of motion (HEOM) method, which gives the exact dynamics for a system coupled to a harmonic bath at arbitrary temperature and system-bath coupling strength. However, in its standard form, this method is only consistent with the weak-coupling quantum master equation at all temperatures when many auxiliary density operators are included in the hierarchy, even when low temperature corrections are included. Here, we propose a new low temperature correction scheme for the termination of the hierarchy based on Zwanzig projection, which alleviates this problem and restores consistency with the weak-coupling master equation with a minimal hierarchy. The utility of the new correction scheme is demonstrated on a range of model systems, including the Fenna-Matthews-Olson complex. The new closure is found to improve convergence of the HEOM even beyond the weak-coupling limit and is very straightforward to implement in existing HEOM codes.

Coupled charge and energy transfer dynamics in light harvesting complexes from a hybrid hierarchical equations of motion approach.

We describe a method for simulating exciton dynamics in protein-pigment complexes, including effects from charge transfer as well as fluorescence. The method combines the hierarchical equations of motion, which are used to describe quantum dynamics of excitons, and the Nakajima-Zwanzig quantum master equation, which is used to describe slower charge transfer processes. We study the charge transfer quenching in light harvesting complex II, a protein postulated to control non-photochemical quenching in many plant species. Using our hybrid approach, we find good agreement between our calculation and experimental measurements of the excitation lifetime. Furthermore, our calculations reveal that the exciton energy funnel plays an important role in determining quenching efficiency, a conclusion we expect to extend to other proteins that perform protective excitation quenching. This also highlights the need for simulation methods that properly account for the interplay of exciton dynamics and charge transfer processes.

Xanthophyll-cycle based model of the rapid photoprotection of Nannochloropsis in response to regular and irregular light/dark sequences.

We explore the photoprotection dynamics of Nannochloropsis oceanica using time-correlated single photon counting under regular and irregular actinic light sequences. The varying light sequences mimic natural conditions, allowing us to probe the real-time response of non-photochemical quenching (NPQ) pathways. Durations of fluctuating light exposure during a fixed total experimental time and prior light exposure of the algae are both found to have a profound effect on NPQ. These observations are rationalized with a quantitative model based on the xanthophyll cycle and the protonation of LHCX1. The model is able to accurately describe the dynamics of non-photochemical quenching across a variety of light sequences. The combined model and observations suggest that the accumulation of a quenching complex, likely zeaxanthin bound to a protonated LHCX1, is responsible for the gradual rise in NPQ. Additionally, the model makes specific predictions for the light sequence dependence of xanthophyll concentrations that are in reasonable agreement with independent chromatography measurements taken during a specific light/dark sequence.

Origin of Chirality Induced Spin Selectivity in Photoinduced Electron Transfer.

Here we propose a mechanism by which spin-polarization can be generated dynamically in chiral molecular systems undergoing photoinduced electron transfer. The proposed mechanism explains how spin-polarization emerges in systems where charge transport is dominated by incoherent hopping, mediated by spin-orbit and electronic exchange couplings through an intermediate charge transfer state. We derive a simple expression for the spin-polarization that predicts a nonmonotonic temperature dependence, consistent with recent experiments, and a maximum spin-polarization that is independent of the magnitude of the spin-orbit coupling. We validate this theory using approximate quantum master equations and the numerically exact hierarchical equations of motion. The proposed mechanism of chirality induced spin selectivity should apply to many chiral systems, and the ideas presented here have implications for the study of spin transport at temperatures relevant to biology and provide simple principles for the molecular control of spins in fluctuating environments.

Spin relaxation in radical pairs from the stochastic Schrödinger equation.

We show that the stochastic Schrödinger equation (SSE) provides an ideal way to simulate the quantum mechanical spin dynamics of radical pairs. Electron spin relaxation effects arising from fluctuations in the spin Hamiltonian are straightforward to include in this approach, and their treatment can be combined with a highly efficient stochastic evaluation of the trace over nuclear spin states that is required to compute experimental observables. These features are illustrated in example applications to a flavin-tryptophan radical pair of interest in avian magnetoreception and to a problem involving spin-selective radical pair recombination along a molecular wire. In the first of these examples, the SSE is shown to be both more efficient and more widely applicable than a recent stochastic implementation of the Lindblad equation, which only provides a valid treatment of relaxation in the extreme-narrowing limit. In the second, the exact SSE results are used to assess the accuracy of a recently proposed combination of Nakajima-Zwanzig theory for the spin relaxation and Schulten-Wolynes theory for the spin dynamics, which is applicable to radical pairs with many more nuclear spins. We also analyze the efficiency of trace sampling in some detail, highlighting the particular advantages of sampling with SU(N) coherent states.

Quantum mechanical spin dynamics of a molecular magnetoreceptor.

Radical pair recombination reactions are known to be sensitive to extremely weak magnetic fields and can therefore be said to function as molecular magnetoreceptors. The classic example is a carotenoid-porphyrin-fullerene (C•+PF•-) radical pair that has been shown to provide a "proof-of-principle" for the operation of a chemical compass [K. Maeda et al., Nature 453, 387 (2008)]. Previous simulations of this radical pair have employed semiclassical approximations, which are routinely applicable to its 47 coupled electronic and nuclear spins. However, calculating the exact quantum mechanical spin dynamics presents a significant challenge and has not been possible until now. Here, we use a recently developed method to perform numerically converged simulations of the C•+PF•- quantum mechanical spin dynamics, including all coupled spins. A comparison of these quantum mechanical simulations with various semiclassical approximations reveals that, while it is not perfect, the best semiclassical approximation does capture essentially all of the relevant physics in this problem.

How quantum is radical pair magnetoreception?

Currently the most likely mechanism of the magnetic compass sense in migratory songbirds relies on the coherent spin dynamics of pairs of photochemically formed radicals in the retina. Spin-conserving electron transfer reactions are thought to result in radical pairs whose near-degenerate electronic singlet and triplet states interconvert coherently as a result of hyperfine, exchange, and dipolar couplings and, crucially for a compass sensor, Zeeman interactions with the geomagnetic field. In this way, the yields of the reaction products can be influenced by magnetic interactions a million times smaller than kBT. The question we ask here is whether one can only account for the coherent spin dynamics using quantum mechanics. We find that semiclassical approximations to the spin dynamics of radical pairs only provide a satisfactory description of the anisotropic product yields when there is no electron spin-spin coupling, a situation unlikely to be consistent with a magnetic sensing function. Although these methods perform reasonably well for shorter-lived radical pairs with stronger electron-spin coupling, the accurate simulation of anisotropic magnetic field effects relevant to magnetoreception seems to require full quantum mechanical calculations.

Radical pair intersystem crossing: Quantum dynamics or incoherent kinetics?

Magnetic field effects on radical pair reactions arise due to the interplay of coherent electron spin dynamics and spin relaxation effects, a rigorous treatment of which requires the solution of the Liouville-von Neumann equation. However, it is often found that simple incoherent kinetic models of the radical pair singlet-triplet intersystem crossing provide an acceptable description of experimental measurements. In this paper, we outline the theoretical basis for this incoherent kinetic description, elucidating its connection to exact quantum mechanics. We show, in particular, how the finite lifetime of the radical pair spin states, as well as any additional spin-state dephasing, leads to incoherent intersystem crossing. We arrive at simple expressions for the radical pair spin state interconversion rates to which the functional form proposed recently by Steiner et al. [J. Phys. Chem. C 122, 11701 (2018)] can be regarded as an approximation. We also test the kinetic master equation against exact quantum dynamical simulations for a model radical pair and for a series of PTZ•+-Phn-PDI•- molecular wires.

Electron spin relaxation in radical pairs: Beyond the Redfield approximation.

Relaxation processes can have a large effect on the spin selective electron transfer reactions of radical pairs. These processes are often treated using phenomenological relaxation superoperators or with some model for the microscopic relaxation mechanism treated within Bloch-Redfield-Wangsness theory. Here, we demonstrate that an alternative perturbative relaxation theory, based on the Nakajima-Zwanzig equation, has certain advantages over Redfield theory. In particular, the Nakajima-Zwanzig equation does not suffer from the severe positivity problem of Redfield theory in the static disorder limit. Combining the Nakajima-Zwanzig approach consistently with the Schulten-Wolynes semiclassical method, we obtain an efficient method for modeling the spin dynamics of radical pairs containing many hyperfine-coupled nuclear spins. This is then used to investigate the spin-dependent electron transfer reactions and intersystem crossing of dimethyljulolidine-naphthalenediimide (DMJ-NDI) radical ion pairs. By comparing our simulations with experimental data, we find evidence for a field-independent contribution to the triplet quantum yields of these reactions which cannot be explained by electron spin relaxation alone.

Spin-selective electron transfer reactions of radical pairs: Beyond the Haberkorn master equation.

Radical pair recombination reactions are normally described using a quantum mechanical master equation for the electronic and nuclear spin density operator. The electron spin state selective (singlet and triplet) recombination processes are described with a Haberkorn reaction term in this master equation. Here we consider a general spin state selective electron transfer reaction of a radical pair and use Nakajima-Zwanzig theory to derive the master equation for the spin density operator, thereby elucidating the relationship between non-adiabatic reaction rate theory and the Haberkorn reaction term. A second order perturbation theory treatment of the diabatic coupling naturally results in the Haberkorn master equation with an additional reactive scalar electron spin coupling term. This term has been neglected in previous spin chemistry calculations, but we show that it will often be quite significant. We also show that beyond the second order in perturbation theory, i.e., beyond the Fermi golden rule limit, an additional reactive singlet-triplet dephasing term appears in the master equation. A closed form expression for the reactive scalar electron spin coupling in terms of the Marcus theory parameters that determine the singlet and triplet recombination rates is presented. By performing simulations of radical pair reactions with the exact hierarchical equations of motion method, we demonstrate that our master equations provide a very accurate description of radical pairs undergoing spin-selective non-adiabatic electron transfer reactions. The existence of a reactive electron spin coupling may well have implications for biologically relevant radical pair reactions such as those which have been suggested to play a role in avian magnetoreception.

On the low magnetic field effect in radical pair reactions.

Radical pair recombination reactions are known to be sensitive to the application of both low and high magnetic fields. The application of a weak magnetic field reduces the singlet yield of a singlet-born radical pair, whereas the application of a strong magnetic field increases the singlet yield. The high field effect arises from energy conservation: when the magnetic field is stronger than the sum of the hyperfine fields in the two radicals, S → T± transitions become energetically forbidden, thereby reducing the number of pathways for singlet to triplet interconversion. The low field effect arises from symmetry breaking: the application of a weak magnetic field lifts degeneracies among the zero field eigenstates and increases the number of pathways for singlet to triplet interconversion. However, the details of this effect are more subtle and have not previously been properly explained. Here we present a complete analysis of the low field effect in a radical pair containing a single proton and in a radical pair in which one of the radicals contains a large number of hyperfine-coupled nuclear spins. We find that the new transitions that occur when the field is switched on are between S and T0 in both cases, and not between S and T± as has previously been claimed. We then illustrate this result by using it in conjunction with semiclassical spin dynamics simulations to account for the observation of a biphasic-triphasic-biphasic transition with increasing magnetic field strength in the magnetic field effect on the time-dependent survival probability of a photoexcited carotenoid-porphyrin-fullerene radical pair.

Spin-dependent charge recombination along para-phenylene molecular wires.

We have used an efficient new quantum mechanical method for radical pair recombination reactions to study the spin-dependent charge recombination along PTZ•+-Phn-PDI•- molecular wires. By comparing our results with the experimental data of Weiss et al. [J. Am. Chem. Soc. 126, 5577 (2004)], we are able to extract the spin-dependent (singlet and triplet) charge recombination rate constants for wires with n = 2-5. These spin-dependent rate constants have not been extracted previously from the experimental data because they require fitting its magnetic field-dependence to the results of quantum spin dynamics simulations. We find that the triplet recombination rate constant decreases exponentially with the length of the wire, consistent with the superexchange mechanism of charge recombination. However, the singlet recombination rate constant is nearly independent of the length of the wire, suggesting that the singlet pathway is dominated by an incoherent hopping mechanism. A simple qualitative explanation for the different behaviours of the two spin-selective charge recombination pathways is provided in terms of Marcus theory. We also find evidence for a magnetic field-independent background contribution to the triplet yield of the charge recombination reaction and suggest several possible explanations for it. Since none of these explanations is especially compelling given the available experimental evidence, and since the result appears to apply more generally to other molecular wires, we hope that this aspect of our study will stimulate further experimental work.

An efficient quantum mechanical method for radical pair recombination reactions.

The standard quantum mechanical expressions for the singlet and triplet survival probabilities and product yields of a radical pair recombination reaction involve a trace over the states in a combined electronic and nuclear spin Hilbert space. If this trace is evaluated deterministically, by performing a separate time-dependent wavepacket calculation for each initial state in the Hilbert space, the computational effort scales as O(Z2log⁡Z), where Z is the total number of nuclear spin states. Here we show that the trace can also be evaluated stochastically, by exploiting the properties of spin coherent states. This results in a computational effort of O(MZlog⁡Z), where M is the number of Monte Carlo samples needed for convergence. Example calculations on a strongly coupled radical pair with Z>106 show that the singlet yield can be converged to graphical accuracy using just M=200 samples, resulting in a speed up by a factor of >5000 over a standard deterministic calculation. We expect that this factor will greatly facilitate future quantum mechanical simulations of a wide variety of radical pairs of interest in chemistry and biology.

Unraveling the mechanisms of triplet state formation in a heavy-atom free photosensitizer.

Triplet excited state generation plays a pivotal role in photosensitizers, however the reliance on transition metals and heavy atoms can limit the utility of these systems. In this study, we demonstrate that an interplay of competing quantum effects controls the high triplet quantum yield in a prototypical boron dipyrromethene-anthracene (BD-An) donor-acceptor dyad photosensitizer, which is only captured by an accurate treatment of both inner and outer sphere reorganization energies. Our ab initio-derived model provides excellent agreement with experimentally measured spectra, triplet yields and excited state kinetic data, including the triplet lifetime. We find that rapid triplet state formation occurs primarily via high-energy triplet states through both spin-orbit coupled charge transfer and El-Sayed's rule breaking intersystem crossing, rather than direct spin-orbit coupled charge transfer to the lowest lying triplet state. Our calculations also reveal that competing effects of nuclear tunneling, electronic state recrossing, and electronic polarizability dictate the rate of non-productive ground state recombination. This study sheds light on the quantum effects driving efficient triplet formation in the BD-An system, and offers a promising simulation methodology for diverse photochemical systems.

Spin Dynamics of Radical Pairs Using the Stochastic Schrödinger Equation in MolSpin.

The chemical reactivity of radical pairs is strongly influenced by the interactions of electronic and nuclear spins. A detailed understanding of these effects requires a quantum description of the spin dynamics that considers spin-dependent reaction rates, interactions with external magnetic fields, spin-spin interactions, and the loss of spin coherence caused by coupling to a fluctuating environment. Modeling real chemical and biochemical systems, which frequently involve radicals with multinuclear spin systems, poses a severe computational challenge. Here, we implement a method based on the stochastic Schrödinger equation in the software package MolSpin. Large electron-nuclear spin systems can be simulated efficiently, with asymmetric spin-selective recombination reactions, anisotropic hyperfine interactions, and nonzero exchange and dipolar couplings. Spin-relaxation can be modeled using the stochastic time-dependence of spin interactions determined by molecular dynamics and quantum chemical calculations or by allowing rate coefficients to be explicitly time-dependent. The flexibility afforded by this approach opens new avenues for exploring the effects of complex molecular motions on the spin dynamics of chemical transformations.

On the Mechanism of Polaritonic Rate Suppression from Quantum Transition Paths.

Polariton chemistry holds promise for facilitating mode-selective chemical reactions, but the underlying mechanism behind the rate modifications observed under strong vibrational coupling is not well understood. Using the recently developed quantum transition path theory, we have uncovered a mechanism of resonant suppression of a thermal reaction rate in a simple model polaritonic system consisting of a reactive mode in a bath confined to a lossless microcavity with a single photon mode. We observed the formation of a polariton during rate-limiting transitions on reactive pathways and identified the concomitant rate suppression as being due to hybridization between the reactive mode and the cavity mode, which inhibits bath-mediated tunneling. The transition probabilities that define the quantum master equation can be directly translated into a visualization of the corresponding polariton energy landscape. This landscape exhibits a double funnel structure with a large barrier between the initial and final states.