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

Many-Body Models for Chirality-Induced Spin Selectivity in Electron Transfer

We present the first microscopic model for the chirality-induced spin selectivity effect in electron-transfer, in which the internal degrees of freedom of the chiral bridge are explicitly included. By exactly solving this model on short chiral chains we demonstrate that a sizable spin polarization on the acceptor arises from the interplay of coherent and incoherent dynamics, with strong electron-electron correlations yielding many-body states on the bridge as crucial ingredients. Moreover, we include the coherent and incoherent dynamics induced by interactions with vibrational modes and show that they can play an important role in determining the long-time polarized state probed in experiments.

The Effect of Spin Relaxation on Magnetic Compass Sensitivity in ErCry4a

This study explores the impact of thermal motion on the magnetic compass mechanism in migratory birds, focusing on the radical pair mechanism within cryptochrome photoreceptors. The coherence of radical pairs, crucial for magnetic field inference, is curbed by spin relaxation induced by intra-protein motion. Molecular dynamics simulations, density-functional-theory-based calculations, and spin dynamics calculations were employed, utilizing Bloch-Redfield-Wangsness (BRW) relaxation theory, to investigate compass sensitivity. Previous research hypothesized that European robin's cryptochrome 4a (ErCry4a) optimized intra-protein motion to minimize spin relaxation, enhancing magnetic sensing compared to the plant Arabidopsis thaliana's cryptochrome 1 (AtCry1). Different correlation times of the nuclear hyperfine coupling constants in AtCry1 and ErCry4a were indeed found, leading to distinct radical pair recombination yields in the two species, with ErCry4a showing optimized sensitivity. However, this optimization is likely negligible in realistic spin systems with numerous nuclear spins. Beyond insights in magnetic sensing, the study presents a detailed method employing molecular dynamics simulations to assess for spin relaxation effects on chemical reactions with realistically modelled protein motion, relevant for studying radical pair reactions at finite temperature.

Evolution of spin coherence of radical pairs due to spin-selective recombination: Comparison of three models

This study looked for a way to evaluate the validity of previously suggested models for describing the spin-selective recombination of radical pairs. As an example, for analysis, we used the conventional model, the model by Jones and Hore [Chem. Phys. Lett. 488, 90 (2010)], and the model by Salikhov [Am. J. Phys. Chem. 11, 67 (2022)]. To do this, analytical solutions to the evolution of the radical pair density matrix due to a radical pair's spin-selective recombination and singlet-triplet transitions in a strong magnetic field were obtained for the conventional model and the Jones and Hore model. Spin interactions included in the Hamiltonian were time-independent exchange interactions as well as Zeeman and hyperfine interactions. The most striking difference between the models' predictions appeared when considering the fraction of singlet pairs among all currently existing ones. In the Jones and Hore model, this ratio has the form of damped oscillations for any values of the spin-hamiltonian parameters. The conventional model and the Salikhov model both predicted that this ratio had the form of undamped oscillations in the absence of exchange interaction and at a sufficiently low recombination rate. Besides, the conventional model predicts the possibility of a resonance-like behavior of this ratio when singlet-triplet transitions in a part of the radical pair ensemble are completely suppressed by tuning the magnetic field strength. Possible experimental conditions in which distinguishing between the models seems to be most straightforward were suggested.

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.

Modeling spin relaxation in complex radical systems using MolSpin

Spin relaxation is an important aspect of the spin dynamics of free radicals and can have a significant impact on the outcome of their spin-selective reactions. Examples range from the use of radicals as spin qubits in quantum information processing to the radical pair reactions in proteins that may allow migratory birds to sense the direction of the Earth's magnetic field. Accurate modeling of spin relaxation, however, is non-trivial. Bloch-Redfield-Wangsness theory derives a quantum mechanical master equation from system-bath interactions in the Markovian limit that provides a comprehensive framework for describing spin relaxation. Unfortunately, the construction of the master equation is system-specific and often resource-heavy. To address this challenge, we introduce a generalized and efficient implementation of BRW theory as a new feature of the spin dynamics toolkit MolSpin which offers an easy-to-use approach for studying systems of reacting radicals of varying complexity.

A spin statistical factor in electron transfer to oxygen molecules

The oxygen molecule in its ground triplet state (³O₂) is a strong electron acceptor. Electron transfer to ³O₂ to form a superoxide anion is an important elementary step in many chemical and biological processes. If this transfer occurs from a spin 1/2 paramagnetic particle where the total spin of the reactants is equal to 3/2, the reaction is spin-forbidden. In liquids, the significant dipole-dipole electron spin interaction in ³O₂ is supposed to mix the non-reactive quartet and reactive doublet states at a time scale of ∼10 ps, thus avoiding the barrier. To elucidate the role of spin effects in the electron transfer to ³O₂, we studied this reaction over a range of more than three orders of magnitude of the relative diffusion coefficient (D) of the reactants. It was found that spin effects during electron transfer to ³O₂ become insignificant when D < 10^-9 m² s^-1. In the range of intermediate D values (10^-9 m² s^-1 < D < 10^-8 m² s^-1) - which corresponds to some reactions of oxygen with small radicals in aqueous solutions - the effective spin factor decreases with increasing D value. If D > 10^-8 m² s^-1, the electron transfer is spin-selective with the spin factor of 1/3 as determined by the spin statistics. At such D values, the reaction encounter time may exceed the expected quartet-doublet mixing time by almost an order of magnitude. The reduced rate of quartet-doublet transitions within the encounter complex in the reaction with ³O₂ has been explained by the spin-exchange interaction and chemical Zeno effect.

Radical triads, not pairs, may explain effects of hypomagnetic fields on neurogenesis

Adult hippocampal neurogenesis and hippocampus-dependent cognition in mice have been found to be adversely affected by hypomagnetic field exposure. The effect concurred with a reduction of reactive oxygen species in the absence of the geomagnetic field. A recent theoretical study suggests a mechanistic interpretation of this phenomenon in the framework of the Radical Pair Mechanism. According to this model, a flavin-superoxide radical pair, born in the singlet spin configuration, undergoes magnetic field-dependent spin dynamics such that the pair's recombination is enhanced as the applied magnetic field is reduced. This model has two ostensible weaknesses: a) the assumption of a singlet initial state is irreconcilable with known reaction pathways generating such radical pairs, and b) the model neglects the swift spin relaxation of free superoxide, which abolishes any magnetic sensitivity in geomagnetic/hypomagnetic fields. We here suggest that a model based on a radical triad and the assumption of a secondary radical scavenging reaction can, in principle, explain the phenomenon without unnatural assumptions, thus providing a coherent explanation of hypomagnetic field effects in biology.

The role of Heisenberg spin exchange and the quantum Zeno effect in the spin-selective reaction between spin-1/2 and spin-1 particles

The kinetics of spin-selective reactions involving triplet molecules, such as triplet-triplet annihilation or electron transfer to dioxygen molecules in the ground triplet spin state, are strongly dependent on the dipole-dipole interaction (DDI) of electron spins in spin-1 particles. The effect of this interaction on the intersystem crossing in the reaction encounter complex of the paramagnetic particles was previously considered for some particular cases using oversimplified approaches. In this study, we consider a rigorous kinetic model of the irreversible reaction between the spin-1/2 and spin-1 particles in an encounter complex with the reactive doublet state. This model explicitly includes both isotropic exchange coupling of the reactants and spin dependence of the reaction rate in the form of the Haberkorn reaction term. For the time-independent DDI, an analytical expression for the reaction kinetics was derived. The effect of DDI fluctuations was analyzed using numerical simulations. It was found that increasing both the exchange coupling and the reaction rate constants can significantly slow down the quartet-doublet spin transitions and, as a consequence, the observed spin-selective reaction rate. Additionally, the presence of the irreversible reaction in the doublet states affects a coherent evolution in the non-reactive quartet subsystem.

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 the 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-Metthews-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.

Observations about utilitarian coherence in the avian compass

It is hypothesised that the avian compass relies on spin dynamics in a recombining radical pair. Quantum coherence has been suggested as a resource to this process that nature may utilise to achieve increased compass sensitivity. To date, the true functional role of coherence in these natural systems has remained speculative, lacking insights from sufficiently complex models. Here, we investigate realistically large radical pair models with up to 21 nuclear spins, inspired by the putative magnetosensory protein cryptochrome. By varying relative radical orientations, we reveal correlations of several coherence measures with compass fidelity. Whilst electronic coherence is found to be an ineffective predictor of compass sensitivity, a robust correlation of compass sensitivity and a global coherence measure is established. The results demonstrate the importance of realistic models, and appropriate choice of coherence measure, in elucidating the quantum nature of the avian compass.

Anisotropic magnetic field effects in the re-oxidation of cryptochrome in the presence of scavenger radicals

The avian compass and many other of nature's magnetoreceptive traits are widely ascribed to the protein cryptochrome. There, magnetosensitivity is thought to emerge as the spin dynamics of radicals in the applied magnetic field enters in competition with their recombination. The first and dominant model makes use of a radical pair. However, recent studies have suggested that magnetosensitivity could be markedly enhanced for a radical triad, the primary radical pair of which undergoes a spin-selective recombination reaction with a third radical. Here, we test the practicality of this supposition for the reoxidation reaction of the reduced FAD cofactor in cryptochrome, which has been implicated with light-independent magnetoreception but appears irreconcilable with the classical radical pair mechanism (RPM). Based on the available realistic cryptochrome structures, we predict the magnetosensitivity of radical triad systems comprising the flavin semiquinone, the superoxide, and a tyrosine or ascorbyl scavenger radical. We consider many hyperfine-coupled nuclear spins, the relative orientation and placement of the radicals, their coupling by the electron-electron dipolar interaction, and spin relaxation in the superoxide radical in the limit of instantaneous decoherence, which have not been comprehensively considered before. We demonstrate that these systems can provide superior magnetosensitivity under realistic conditions, with implications for dark-state cryptochrome magnetoreception and other biological magneto- and isotope-sensitive radical recombination reactions.

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.

Influence of Spin Decoherence on the Yield of Photodriven Quantum Teleportation in Molecular Triads

The evolution of spin coherences due to spin-selective recombination in the system with three unpaired electrons is discussed. It is shown that in the case of bidirectional kinetics, the decoherence processes can significantly change the quantum yield of the products. This enables one to discriminate between approaches that model spin-selective recombination but predict different decoherence rates. The rigid donor-acceptor-radical molecular triad is suggested to study the decay rate of singlet-triplet coherence. A modification of the photodriven quantum teleportation protocol is proposed to measure the quantum yields of the monoradical products.

F-cluster: Reaction-induced spin correlation in multi-radical systems

We provide a theoretical analysis of spin-selective recombination processes in clusters of n ≥ 3 radicals. Specifically, we discuss how spin correlation can ensue from random encounters of n radicals, i.e., "F-clusters" as a generalization of radical F-pairs, acting as precursors of spin-driven magnetic field effects. Survival probabilities and the spin correlation of the surviving radical population, as well as transients, are evaluated by expanding the spin density operator in an operator basis that is closed under application of the Haberkorn recombination operator and singlet-triplet dephasing. For the primary spin cluster, the steady-state density operator is found to be independent of the details of the recombination network, provided that it is irreducible; pairs of surviving radicals are triplet-polarized independent of whether they are actually reacting with each other. The steady state is independent of the singlet-triplet dephasing, but the kinetics and the population of sister clusters of smaller size can depend on the degree of dephasing. We also analyze reaction-induced singlet-triplet interconversion in radical pairs due to radical scavenging by initially uncorrelated radicals ("chemical Zeno effect"). We generalize previous treatments for radical triads by discussing the effect of spin-selective recombination in the original pair and extending the analysis to four radicals, i.e., radical pairs interacting with two radical scavengers.

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.

Chirality-Induced Spin Coherence in Electron Transfer Reactions

Recently, there has been much interest in the chirality-induced spin selectivity effect, whereby electron spin polarization, which is dependent on molecular chirality, is produced in electrode-molecule electron transfer processes. Naturally, one might consider if a similar effect can be observed in simple molecular charge transfer reactions, for example, in light-induced electron transfer from an electron donor to an electron acceptor. In this work, I explore the effect of electron transfer on spins in chiral single radicals and chiral radical pairs using Nakajima-Zwanzig theory. In these cases, chirality, in conjuction with spin-orbit coupling, does not lead to spin polarization, but instead, the electron transfer generates quantum coherence between spins states. In principle, this chirality-induced spin coherence could manifest in a range of experiments, and in particular, I demonstrate that the out of phase electron spin echo envelope modulation pulse electron paramagnetic resonance experiment would be able to detect this effect in oriented radical pairs.

Electronic spin separation induced by nuclear motion near conical intersections

Though the concept of Berry force was proposed thirty years ago, little is known about the practical consequences of this force as far as chemical dynamics are concerned. Here, we report that when molecular dynamics pass near a conical intersection, a massive Berry force can appear as a result of even a small amount of spin-orbit coupling (<10⁻³ eV), and this Berry force can in turn dramatically change pathway selection. In particular, for a simple radical reaction with two outgoing reaction channels, an exact quantum scattering solution in two dimensions shows that the presence of a significant Berry force can sometimes lead to spin selectivity as large as 100%. Thus, this article opens the door for organic chemists to start designing spintronic devices that use nuclear motion and conical intersections (combined with standard spin-orbit coupling) in order to achieve spin selection. Vice versa, for physical chemists, this article also emphasizes that future semiclassical simulations of intersystem crossing (which have heretofore ignored Berry force) should be corrected to account for the spin polarization that inevitably arises when dynamics pass near conical intersections.

Spin polarization is at the basis of quantum information and underlies some natural processes, but many aspects still need to be explored. Here, the authors, by quantum mechanical computations, show that even a weak spin-orbit coupling near a conical intersection can induce large spin selection, with consequences for spin manipulation in photochemical or electrochemical reactions.

Spin-chemical effects on intramolecular photoinduced charge transfer reactions in bisphenanthroline copper(i)-viologen dyad assemblies

Two covalently linked donor-acceptor copper phenanthroline complexes (C-A dyads) of interest for solar energy conversion/storage schemes, [Cu(i)(Rphen(OMV)2 4+)2]9+ = RC+A4 8+ with RC+ = [Cu(i)Rphen2]+ involving 2,9-methyl (R = Me) or 2,9-phenyl (R = Ph)-phenanthroline ligands that are 5,6-disubstituted by 4-(n-butoxy) linked methylviologen electron acceptor groups (A2+ = OMV2+), have been synthesized and investigated via quantum chemical calculations and nanosecond laser flash spectroscopy in 1,2-difluorobenzene/methanol (dfb/MeOH) mixtures. Upon photoexcitation, charge transfer (CT) states RC2+A+A3 6+ are formed in less than one ns and decay by charge recombination on a time scale of 6-45 ns. The CT lifetime of RC2+A+A3 6+ has a strong dependence on MeOH solvent fraction when R = Me, but is unaffected if R = Ph. This solvent effect is due to coordination of MeOH solvent in MeC+A4 8+ (i.e. exciplex formation) allowed by conformational flattening of the ligand sphere, which cannot occur in PhC+A4 8+ having bulkier Phphen ligand framework. Interestingly, the decay time of the CT state increases for both species at low magnetic fields with a maximum increase of ca. 30% at ca. 150 mT, then decreases as the field is increased up to 1500 mT, the highest field investigated. This magnetic field effect (MFE) is due to magnetic modulation of the spin dynamics interconverting 3CT and 1CT states. A quantitative modeling according to the radical pair mechanism involving ab initio multireference calculations of the complexes revealed that the spin process is dominated by the effect of Cu hyperfine coupling. The external magnetic field suppresses the hyperfine coupling induced spin state mixing thereby lengthening the CT decay time. This effect is counteracted by the field dependent processes of T0-S mixing through the Δg-mechanism and by a local mode spin-orbit mechanism. Further, the maximum MFE is limited by a finite rate of direct recombination of 3CT states and the spin-rotational mechanism of spin relaxation. This study provides a first comprehensive characterization of Cu(ii)-complex spin chemistry and highlights how spin chemistry can be used to manipulate solar energy harvesting and storage materials.

How quantum is radical pair magnetoreception?

Semiclassical methods cannot accurately simulate magnetic field effects relevant to avian magnetoreception, which may therefore deserve the label “quantum biology”.

On the magnetosensitivity of lipid peroxidation: two- versus three-radical dynamics

We present a theoretical analysis of the putative magnetosensitivity of lipid peroxidation. We focus on the widely accepted radical pair mechanism (RPM) and a recently suggested idea based on spin dynamics induced in three-radical systems by the mutual electron-electron dipolar coupling (D3M). We show that, contrary to claims in the literature, lipid peroxides, the dominant chain carriers of the autoxidation process, have associated non-zero hyperfine coupling interactions. This suggests that their recombination could, in principle, be magnetosensitive due to the RPM. While the RPM indeed goes a long way to explaining magnetosensitivity in these systems, we show that the simultaneous interaction of three peroxyl radicals via the D3M can achieve larger magnetic field effects (MFE), even if the third radical is remote from the recombining radical pair. For randomly oriented three-radical systems, the D3M induces a low-field effect comparable to that of the RPM. The mechanism furthermore immunizes the spin dynamics to the presence of large exchange coupling interactions in the recombining radical pair, thereby permitting much larger MFE at magnetic field intensities comparable to the geomagnetic field than would be expected for the RPM. Based on these characteristics, we suggest that the D3M could be particularly relevant for MFE at low fields, provided that the local radical concentration is sufficient to allow for three-spin radical correlations. Eventually, our observations suggest that MFEs could intricately depend on radical concentration and larger effects could ensue under conditions of oxidative stress.