Studies on coherent and incoherent spin dynamics that control the magnetic field effect on photogenerated radical pairs

Since 1970s, magnetic field effects (MFEs) on photogenerated radical pairs have been the centre of focus in the field of spin chemistry. The MFE attributes to quantum mechanical interconversion between the singlet and triplet radical pair states and subsequent spin-selective recombination reactions. In this New View article, the author picks up two hot topics studied during the last two decades, which are (i) so-called low field effect (LFE) and (ii) 2J-resonance MFE on fixed distance donor–acceptor linked molecules. In both of the topics, quantum mechanical explanations are given referring to recent reports, and some novel calculations have been carried out for bridging theoretical and experimental data for long-lived radical pairs. For the first topic, time domain calculations of coherent state mixing have been carried out for elucidation of hyperfine (HF) structure dependence of the LFE. For the second topic, Monte Carlo simulations of the torsional motion of polyaromatic linker unit have been carried out for the demonstration of fast decoherence in such rigid molecules. From these considerations, future possibilities of MFE studies on photo-functional materials and biomolecules have been indicated. GRAPHICAL ABSTRACT


Introduction
Magnetic field effects (MFEs) on chemical reactions involving organic paramagnetic molecules had first been discovered in 1970s, and significant developments were made on both the experimental and theoretical studies in 1980s as described in the legendary review by Steiner et al. [1].Radical pairs are one of the most important paramagnetic intermediates seen in numbers of photo-, thermal-and radiation-induced chemical reactions.The MFE due to the radical pair mechanism (RPM) attributes to singlet (S)-triplet (T) quantum mechanical mixing and following spin-selective recombination reactions in the non-equilibrium situation [1,2].
CONTACT Tomoaki Miura t-miura@chem.sc.niigata-u.ac.jpThis article was originally published with errors, which have now been corrected in the online version.Please see Correction (https://doi.org/10.1080/00268976.2024.2307242)Supplemental data for this article can be accessed https://doi.org/10.1080/00268976.2019.1643510.
In this New View article, the author picks up two hot topics regarding the MFE on radical pairs studied during the last two decades, which are (i) a peculiar effect of very low magnetic fields (so-called low field effect, LFE [42]) and (ii) 2J-resonance MFE on rigid donor-acceptor linked molecules [43].The two phenomena respectively derive from quantum mechanical coherences of spin and electronic wavefunctions.However, discrepancy between theoretical and experimental studies exists especially for long-lived radical pairs, so that, in this article, particular emphasis has been put on theoretical understanding of 'realistic' chemical/physical phenomena of radical pairs.Before the main two topics, a basic theory for spin dynamics simulations is given.In the main topics, some new theoretical results will be presented for explaining previously reported experimental results.The article ends with conclusion and a future perspective.

Theory of MFE on radical pairs -Liouville equation
In this section, the explanation of a simple Liouville equation will be given for simulations of MFEs due to RPM.Here we consider a situation where the spin-orbit coupling (SOC) of the radical pair is so small that the radical pair exhibits distinct spin states.This situation is realised when the two radicals do not contain heavy atoms and are separated so much that the orbital overlap of the two radicals is negligibly small.
In the case of radical pairs in solutions, anisotropic terms in the spin Hamiltonian vanish as a result of rotational averaging, and the isotropic terms remain as where g i , μ B and B 0 are the g values, Bohr magnetron and magnetic field strength, respectively, a n and a m are hyperfine (HF) coupling constants, and 2J is the exchange coupling between the two electrons.In this article, energies are represented in angular frequency unit.Rotational modulations of the g tensor and the HF tensor are so fast in solutions that the anisotropies do not directly affect the MFE in solutions, but such rotational modulations induce incoherent spin relaxations for individual radicals, which potentially affect the MFE known as the relaxation mechanism [44].The Zeeman interaction (the first term in Equation 1) induces field-dependent energy splitting of between the S−T 0 mixed states and the T ±1 states.In this article, gyromagnetic ratio γ is defined either for a radical pair (see Equation ( 2)) or a radical ( = gμB 0 /h), which are used to convert couplings in angular frequency unit between those in magnetic field unit.Note that this splitting is very small as ∼ 1 cm −1 /T compared to the thermal energy at room temperature (k B T ∼ 200 cm −1 ).The exchange interaction (the fourth term in Equation 1) gives the ST energy splitting as The energy diagram in the absence of HF couplings is shown in Figure 1.S−T 0 mixing due to the difference in g values ( g mixing) can be ignored in the low magnetic fields below ∼ 1 T. Thus the S−T mixing is governed solely by the HF interactions (the second and third terms in Equation 1), which occurs conserving the total spin angular momentum F = S + I.
The effect of field-dependent S−T mixing on the reaction yield/rate is realised by the spin-selective chemical reactions.For long-lived radical pairs (τ > a), effects of the spin-selective reactions and spin relaxation on the spin dynamics should be calculated by the Liouville equation of a density matrix ρ as where Ĥ× , Ŵ and ˆ denote the commutator of the spin Hamiltonian, the super operator for chemical reactions and the relaxation operator, respectively.In the framework of the exponential model, recombination reactions from the singlet and triplet manifolds are described respectively by Haberkorn operators as where k recS and k recT are the recombination rate constants from the singlet and triplet manifolds, respectively [45,46].Here Ê denotes the identity operator.Dissociation of the radical pair to free radicals or escape from chemical cages such as micelles can simply be simulated by the escape operator as which dumps all the matrix elements with the non-spindependent escape (dissociation) rate constant of k esc .The dumped radical pair population can be stored on the computer memory as the free radical population.
In the framework of the exponential model, the mutual diffusion motion of the radicals, which modulates the inter-electron exchange and magnetic dipolar interactions, cannot directly be taken into account.Such an effect can be treated as the spin 'dephasing' terms, which will be explained in detail later in this article.The dephasing terms as well as the conventional relaxations of the individual radicals are involved in the relaxation super operator ˆ .
The initial density matrix ρ(0) for singlet-and tripletborn radical pairs are respectively described as and where χ (n) denotes nuclear spin states (α or β, Z = 2 for the single proton model).
The time evolution of the spin state is calculated by numerically solving Equation (4).The radical pair population tr ρ(t) or the populations of recombination/escape products can be calculated as functions of time and magnetic field for the simulation of magnetically affected reaction yield (MARY).

Advancement in researches on the LFE
In this section, the peculiar MFE at low magnetic fields, so-called LFE, will be treated.Here we consider radical pairs with a negligible exchange interaction (2J a).

Conventional HF mechanism versus LFE
MFE due to so-called HF mechanism is frequently observed in low magnetic fields ( < 1 T), which has been explained as follows.The Zeeman splitting among the triplet sublevels (see Equation 2) results in less efficient HF-induced S−T ±1 mixing at higher fields, which results in saturation of the mixing efficiency at ω 0 a.Thus the application of a magnetic field to a singlet/tripletborn radical pair that recombines from the singlet manifold results in monotonical acceleration/deceleration of recombination reaction.The half width at half maximum (B 1/2 ) of the MARY spectra agrees with an effective HF constant (a eff ) calculated by the Weller's equation as where where I n is the nuclear spin of nth nuclear [47].However, simulations based on the simplest Liouville equation do not reproduce such an effect in many cases.Figure 2(a) shows the simulated field-dependent kinetics for a long-lived singlet-born radical pair with a single proton (a/2π = 100 MHz) that recombines from the singlet manifold with a rate constant of k recS = 4 × 10 7 s −1 and escapes from a chemical cage with a rate constant of k esc = 1 × 10 6 s −1 .Here, all the spin relaxations were ignored, and the lifetime of the escaped free radical was assumed to be infinite.The recombination dynamics at 0.4 mT (ω 0 /2π = 11 MHz) exhibits slower decay than that at 0 mT.This trend is opposite to that expected from the conventional HF mechanism.The blue solid line in Figure 2(b) shows the magnetic field dependence of the observed decay rate k obs obtained by monoexponential fitting of the simulated kinetics.k obs steeply drops from 0 to 0.4 mT, then rises at higher fields, and saturates at high magnetic fields.
The opposite MFE at very low magnetic fields of ω 0 < a, although it had been theoretically known for several decades [48], recently drew attention as 'LFE' in the context of biophysics.The anisotropic LFE has been regarded as the most possible mechanism by which a migrating bird detect the orientation of small earth's magnetic fields (a few tens of microtesla) [49][50][51].

Quantum mechanical interpretation of LFE
Basic concepts of the LFE are overviewed in the review article by Timmel et al. [42].At zero magnetic field, only HF interactions are active, and some of the zero-field eigenstates degenerate.In the case of a radical pair with a single proton, for example, some of the F = +1/2 and −1/2 zero-field eigenstates energetically degenerate as where where ω m−n = ω m −ω n , and ω m and ω n denote eigenenergies for |m and |n .Lewis et al. [52] point out on the basis of this formalism that breakdown of the zerofield degeneracy causes enhancement of only S−T 0 mixing, and S−T ±1 mixing is unaffected by the unlocked degeneracies.
It had been empirically known that LFE tends to be larger for radical pairs with large HF constants for only one radical and small (or no) HF constants for the other one [53].This rule-of-thumb was experimentally demonstrated by Rodgers et al. [39,54] for the pyrene (Py) − dimethyl aniline (DMA) singlet-born short-lived (less than a few nanosecond) radical ion pair in polar solvents by fully deuterating one or both of the radicals.The relative LFE amplitude (LFE/high-field MFE) correlates well with the ratio a eff (DMA •+ )/a eff (Py •− ).They also point out from the theoretical calculation that radical pairs with any HF structure should show a significant LFE if their lifetime is longer enough than the mixing timescale.Their conclusion indicates that the 'rise time' of the LFE is shorter for radical pairs with more unbalanced HF structure.
In this article, the author proves this hypothesis for radical pairs with two proton spins I 1 and I 2 , each of which is coupling with the electron spins S 1 and S 2 by the coupling constants a 1 and a 2 , respectively (for details, see the supplementary file 1).It has been reported by Lewis et al. [52] that breakdown of the zero-field degeneracy occurs for four pairs among the 16 eigenstates.If we assume that the total coupling a = a 1 + a 2 is constant (a 1 > a 2 > 0) and r = a 2 /a 1 , the mixing angular frequencies for the four pairs of the eigenstates can be calculated by Equation (12) as where R = ω 0 /a.It is clear that all the mixing angular frequencies linearly increase with a at very low magnetic fields of ω 0 ≤ a (R ∼ 0).More importantly, differential calculations indicate these angular frequencies monotonically decrease with increasing r from 0 to 1 at any R, which confirms that the appearance time for LFE is longer for more balanced HF couplings for the two radicals.Numerical calculations have also been carried out for a singlet-born radical pairs with a 1 /2π = 60 MHz, a 2 /2π = 40 MHz (r = 0.67) and a 1 /2π = 90 MHz, a 2 /2π = 10 MHz (r = 0.11).P S (t) values were calculated without any chemical reactions or spin relaxations, and their time-averaged yields S (t) were evaluated as The S (t) value monitors the 'steady-state' population at each time, which is convenient for long-lived radical pairs where megahertz-order oscillations are less important.For both the HF structures, the amplitudes of LFE and high-field MFE are not very different at a very late time of 2 μs as shown in Figure 3(a).However, one can see a clear difference between the time dependences of S (t) for the two HF structures as shown in Figure 3(b).The subtracted S (t) signal is expressed as at B 0 = 0.6 mT ( ∼ peak field for LFE) rises much faster for the unbalanced HF structure (r = 0.11) than that for the balanced HF structure (r = 0.67).On the other hand, S rises quickly for the two HF structures at a high field of 19 mT.From these calculations, it is demonstrated that the rise time of LFE is slower than high-field MFE and particularly slower for similar HF couplings in the two radicals.

LFE on long-lived radical pairs -theories and experiments
Maeda et al. [29] studied a covalently linked donor (carotenoid, C)-chromophore (porphyrin, P)-acceptor (C 60 fullerene, F) triad in a low temperature solution (113−250 K) as a model system of the birds' magnetoreceptor.Photoexcitation of P gives a long-lived ( ∼ 190 ns) radical ion pair of C •+ (a eff /γ > 3 mT) and F •− (a eff /γ ∼ 0 mT), of which the transient population responds to a ultra-small magnetic field of 39 μT.Furthermore, anisotropic MFEs are demonstrated for the first time, which are due to the anisotropic HF interactions in C •+ .
In this context, magnetic response of cryptochrome, which is the most possible candidate for the magnetosensing protein in the birds' eyes, is focus of attention [49].Maeda et al. [55] reported MFE on the photogenerated radical pair in Arabidopsis thaliana (plant) cryptochrome.Photoexcitation of the flavin adenine dinucleotide (FAD) cofactor gives rise to electron transfer from the tryptophan (Trp) triad in the protein and generation of the long-lived (a few microseconds) radical ion pair of FAD •− (a eff /γ = 1.4 mT) and Trp •+ (a eff /γ = 1.94 mT).The MARY spectra exhibit a large MFE over 20% at above 20 mT and a not very large but clear LFE of a few% at ∼ 1 mT.
Numbers of previously reported radical pairs in micelles or similar amphiphilic cages exhibit very long radical pair lifetimes of 100 ns-a few microseconds and large high-field MFEs (sometimes over 100%) [6,22,[56][57][58], but the number of reports on the clear LFE is still small [31,59].The research group including the author of this article [31] reported UV-photocleavage of diphenyl(2,4,6-trimethylbenzoyl)phosphine oxide (TMDPO) in an SDS micelle, which gives a radical pair of the phosphorous centred diphenylphosphonyl radical (a eff /γ = 38.5 mT) and the trimethyl benzoyl radical (a eff /γ ∼ 0 mT).The reaction yield of the generated long-lived ( ∼ 120 ns) radical pair exhibits a clear LFE of 6% at peak magnetic field of 20 mT and time-dependent MFEs of 20% (80 ns)−60% (200 ns) at 500 mT.It is considered that the LFE in this system is observed at not-sosmall magnetic fields because of the large phosphorous HF coupling.
Considering the 100 MHz order HF couplings for regular organic radicals, the rise time for LFE, which is HF-structure dependent, should be on the order of a few hundreds of nanoseconds or shorter as has been demonstrated in Figure 3. On the other hand, lifetimes of most micellar radical pairs are on the same order as this 'maximum limit', so that those lifetimes seem to be long enough for observation of LFEs.This paradox has not frequently been mentioned in previous studies but indicates severe effect of spin relaxations.In the report on TMDPO in the SDS micelle [31], we tried a simulation based on the Liouville equation, but the LFE is overestimated even if the microsecond-order conventional spin relaxations (deriving from anisotropic HF interactions) [60] were taken into account.It turned out that the LFE is dramatically quenched by fast spin dephasing processes of 10 8 −10 9 s −1 , which damps coherences between the sin-glet-triplet and triplet-triplet states (ST and TT dephasings) as where k STD and k TTD are the rate constants for ST and TT dephasing, respectively [61,62].
It is considered that the ST and TT dephasing processes are induced by diffusional fluctuation of exchange and dipolar interactions, respectively [61,62].We have been extensively studying effects of these dephasing processes in low magnetic fields on the spin dynamics of long-lived radical pairs in micelles or similar supercages.Experimental studies using nanosecond magnetic field switching [31][32][33] and simulations including direct Monte Carlo simulations of the diffusive fluctuation of the exchange/dipolar interactions [37,38] have revealed that the fast dephasing of 10 8 −10 9 s −1 due to in-cage diffusion occurs in low magnetic fields comparable to HF couplings.Such a timescale is comparable to or shorter than the rise time for the LFE but longer than that for high-field MFE, which is the most likely the reason why the LFE is rarely observed for micellar radical pairs.The series of studies indicates that interplay of the fast dephasings and HF interactions results in incoherent S−T ±1 mixing of a few tens of nanosecond, which is a cause of broadening of MARY spectra at magnetic fields comparable to or slightly larger than a eff /γ [32,33,37].These features are clearly demonstrated in Figure 2(b) with the red trace where the calculation was carried out with dephasing terms of k STD = k TTD ( = k dep ) = 1 × 10 8 s −1 ; the LFE is quenched, and highfield MFE is broadened by the fast dephasing.
Thus it is important for observation of large LFEs to accelerate the rise time of the LFE or to slowdown the dephasing.The dephasing rate for the CPF triad in the low temperature solution [29] seems to be small because the diffusive reencounter of the radicals is entirely suppressed by the rigid molecular units connecting the chromophores.In the plant cryptochrome [55], it is expected that the reencounter of the FAD and Trp radicals is also suppressed by the rigid polypeptide framework.However, the LFE is not very large, and simulation of the MARY spectra requires ST dephasing of ∼ 10 7 s −1 .The authors attribute this dephasing to hole hopping among the three Trp residues, which would modulate the exchange coupling between the two radicals.

2J-resonance MFE and electron transfer mechanisms
In this section, radical pairs with a certain amplitude of the exchange interaction will be treated.So-called 2Jresonance mechanism is typical of such radical pairs [1].

Basic theory of MFE due to 2J-resonance
As shown in Figure 1, S state becomes energetically degenerated with T −1 or T +1 state for negative or positive 2J, respectively, at the magnetic field point At this magnetic field, the efficiency of HF-induced mixing is maximised (irrespective of the sign of 2J), which results in peak behaviour of MARY spectra.When 2|J| |a|, the spin dynamics of a single proton radical pair in the vicinity of the resonance point approximates to a simple two-state model (Sβ-T −1 α or Sα-T +1 β states for negative or positive 2J, respectively).The calculated singlet yield S (∞) as a function of ω 0 exhibits Lorentzian line shape with full width of half maximum (fwhm) of (see [63] for a full description).The sign of the MFE due to 2J-resonance is opposite to that for the conventional MFE due to the HF mechanism, so that the MARY spectra with phase inversion in the low magnetic field region, which are similar to those for the LFE, are frequently observed.If 2J is comparable to a eff , it is difficult to experimentally differentiate the 2J-resonance MFE from the LFE.

2J-resonance MFE on fixed distance rigid donor-bridge-acceptor molecules
In the context of artificial photosynthesis, donor− acceptor molecules that are covalently linked by rigid molecular 'bridges' such as para-phenylenes have been synthesised in the last few decades [64][65][66][67].In some of these systems, clear 2J-resonance MFEs are observed [18,19,43,63,[68][69][70]. Since such molecules have discrete centre-to-centre donor−acceptor distances (r DA ), distance dependence of the electron transfer rate and the exchange coupling can be studied by changing the repeating number of the bridge units [63,68,69,[71][72][73].Of much interest is that long-distance ( > 1 nm) exchange couplings mediated by aromatic bridge units are characterised by the through-bond coherent electronic coupling as where indicated matrix elements of V RP−l and V RP−m couple the radical pair state (ψ RP ) to singlet ( 1 ψ l ) and triplet ( 3 ψ m ) electronic states, E RP , 1 E l and 3 E m are energies of these states, respectively, and λ denotes the reorganisation energy for the charge recombination reaction [74,75].Validity of this theory has been ensured by discovery of the positive 2J by time-resolved EPR spectroscopy [74,76,77] and observations of long-range 2J coupling stated in the next paragraph, both of which cannot be explained by the conventional Heitler−London model [78].
Weiss et al. [68] reported distance dependence of the 2J coupling for phenothiazine (PTZ) donorpara-(n)phenylene -perylene diimide (PDI) acceptor molecules (PTZ−Ph n −PDI).In the case of PTZ−PDA linked molecules in toluene, charge separation occurs from the excited singlet state to give the singlet radical pair, and recombination occurs both from the triplet and singlet manifolds of the radical pair, so that 2J-resonance MARY spectra are observed both for the lifetime of the radical pair and the yield of the local excited triplet state ( 3 PDI*).They demonstrated that the 2J coupling exponentially decreases with r DA as with a decay factor β J = 0.37 Å −1 .This value is significantly smaller than those obtained from the simulation of MFEs on polymethylene-linked radical ion pairs (β J ∼ 2 Å −1 ), where the two ions are coupled by the short-range direct orbital overlap [79].Thus the MFE on such systems can be a powerful probe for the long-range electronic coupling.It is noteworthy in many fixed distance donor−bridge −acceptor systems with large 2J ( a eff ) that the linewidths of MARY spectra are much larger than those theoretically expected from the HF interaction (see Equation ( 18)) [68,69,72,73].In a recent report by Fay et al., the broad MARY spectra for the PTZ−Ph n −PDI molecules are simulated with extremely large k recT values (k recT = 2.8 × 10 10 s −1 k recS = 4.7 × 10 7 s −1 for n = 2).However, the broad linewidths are observed for linked molecules where k recT is smaller than k recS , so that the lifetime broadening by very fast triplet recombination cannot be a standard solution for this problem.A recent report by Steiner et al. [70] points out that broad MARY spectra for a fixed distance donor-acceptor molecule with a microsecond order lifetime can be simulated introducing relaxation times rigorously derived from anisotropic HF interactions.However, the broad linewidths seem to be ubiquitous to such fixed distance donor-acceptor molecules irrespective of their lifetimes (a few tens of nanosecond to a few microsecond) [63,68,69,73], which implies contribution of much faster relaxation.
The existence of the fast ST dephasing in such molecules seems to be inconsistent with the fact that diffusive reencounter of the two radicals is totally suppressed by the rigid molecular bridges.The most possible source of the ST dephasing is the modulation of the electronic coupling by fluctuation of dihedral angles between adjacent aromatic rings connected by σ bonds.An electronic coupling between the two aromatic rings (i and j) is described as where θ ij is the dihedral angle between the two rings [83][84][85][86].
In this article, the author demonstrates this hypothesis by a spin dynamics simulation directly taking into account the twisting motion of the aromatic rings without using phenomenological relaxation/dephasing terms (Monte Carlo method, see the supplemental file 2 for details).Adopting McConnell's tight binding model [87], Equations ( 19) and (21) give the 2J coupling between donor (D) and acceptor (A) through N equivalent bridge (B) sites as where for lth singlet and mth triplet tunnelling channels.E DB denotes the charge injection energy from the D site to the B site for the charge recombination reaction.2J 0 corresponds to a virtual maximum coupling for the totally flat conformation.The time evolution of the torsional twisting angle for a small time-step t is evaluated by the motion equation as where D rot and U tor (θ ) are the rotational diffusion constant and the potential energy for the torsional twisting, mean square value of θ diff is defined as and the sign of it is chosen randomly [85].Using reported torsional DFT-potentials U tor for D−B, B−B and B−A twisting of PTZ−FL n −PDI molecule [88], trajectories for dihedral angles are obtained, which are used to calculate 2J coupling at each time (see Figure S2-1 in the supplemental file 2).The time evolution of the spin state for 1000 torsional trajectories are statistically averaged to obtain the final result.Figure 4(b) shows the calculated MARY spectra for the normalised relative triplet yield of the PTZ−FL 2 −PDI molecule with 2J 0 /γ = +9.06T and D rot = 2.0 × 10 9 s −1 associated with the experimental result in room temperature toluene.The linewidth much larger than the HF constants is well-reproduced without using the phenomenological dephasing/relaxation terms.Deviation seen in the high-field region is considered to be due to the lack of conventional relaxations, which are induced by anisotropic HF interactions [60].Such conventional relaxations induce gradual decrease in the triplet yield in the high magnetic field region as previously studied by the exponential Liouville equation [63], which is consistent with so-called relaxation mechanism [44].
In order to extract parameters that can be used for the single-site simulation, which are k STD and a static 2J value (2J obs ), from the Monte Carlo simulations, time evolution of the initially given S−T +1 coherence (ρ ST+1 ) under the pure Hamiltonian of fluctuating 2J coupling was calculated (for details, see the supplementary file 2) [38].Since the Hamiltonian does not have the matrix elements between the S and T +1 states, the coherence oscillates with the frequency that corresponds to the effective energy difference between these two states (2J in this case).It turns out that the coherence calculated with D rot ∼ 10 9 −10 10 s −1 , which is a realistic range for organic molecules in solutions, at 2J 0 /γ = +9.06T can be analysed by the damped harmonic oscillation as as shown in Figure 4(c).The 2J obs values obtained with this method are in good agreement with those obtained by the peak field for MARY spectra calculated by the Monte Carlo method with the same D rot parameters (for example, 2J obs /γ = 29 mT at D rot = 2.0 × 10 9 s −1 obtained from Figure 4c agrees with the MARY peak field shown in Figure 4b).Furthermore, k STD obtained with this method at D rot = 2.0 × 10 9 s −1 is 1.1 × 10 9 s −1 , which is close to that previously obtained from the analysis of the MARY spectra with single-site Liouville equation (2 × 10 9 s −1 ) [63].These facts indicate that the spin dynamics for the present molecule can be approximated to the static 2J coupling and ST dephasing.Further important implication is that the spin dynamics can be a powerful probe for the important molecular motion that gates the D−A electronic coupling.The detailed studies including parameter dependences will be on future publications.

Conclusion
As has been reviewed in this New View article, researches on the LFE and 2J-resonance MFE during the last two decades have introduced a new paradigm into the spin chemistry.Importance of coherent spin dynamics on the reaction yield has been demonstrated mainly from the theoretical contributions, but it had seemed that discrepancy existed from the real situation for long-lived radical pairs.Detailed studies on coherent spin dynamics and decoherence mechanisms revealed important implications for both artificial molecules and biological systems that utilise coherent spin dynamics to their functionalities.In order to minimise the dephasing rate for the artificial donor-acceptor molecules, for example, dihedral twisting motion of poly-para-phenylenes can be inhibited by addition of covalent linkage at meta-positions [86].Ultimately in the future, long-lived radical pair molecules with negligible dephasing rate could be developed and utilised for application such as light and fieldcontrolled molecular devices for quantum informatics and quantum biology.Alternatively, there is a great possibility that the molecular dynamics can be quantitatively probed by observing the decoherence processes of radical pairs.Such an approach would help understanding complicated electron transfer mechanisms in biological and artificial systems.In the future, the mechanism of avian magnetoreception would be clarified by studying controlling mechanisms of the spin decoherence in avian cryptochrome with detailed analysis of experimental MFE by MD-assisted spin dynamics simulations.In the case of electron−hole pairs in organic semiconductor thin films, diffusion modelling based on the microscopic structures of the disordered organic solids and its combination with spin dynamics simulations would clarify the molecularlevel mechanisms of charge extinction (mainly recombination) from electrically and/or optically detected MFE.Such studies would offer important feedback to development and engineering of these future materials.

Figure 1 .
Figure 1.Magnetic field dependence of diagonal energies for the shown radical pair states in the frequency unit.HF interactions are not taken into account.2J/2π = 200 MHz.

Figure 2 .
Figure 2. Simulations for a singlet born radical pair with a single proton (a/2π = 100 MHz).k recS = 4 × 10 7 s −1 , k esc = 1 × 10 6 s −1 2J = 0.The sum of populations for the radical pair and the free radical is presented.(a) Kinetic traces calculated at shown magnetic fields without dephasing terms.(b) Magnetic field dependence of k obs obtained with (red dashed line) and without (blue solid line) dephasing terms.

Figure 3 .
Figure 3. Pure quantum mechanical simulations for a radical pair with two protons, each of which is coupling with one electron by HF couplings a 1 /2π = 90 MHz, a 2 /2π = 10 MHz (red) and a 1 /2π = 60 MHz, a 2 /2π = 40 MHz (blue).(a) Field dependence of time-averaged singlet yield S .(b) Kinetic traces for the subtracted singlet yield S = S (B 0 ) − S (0) at shown magnetic fields.

Figure 4 .
Figure 4. (a) Molecular structure of PTZ−FL 2 −PDI and dihedral angles taken into account for the Monte Carlo simulations.(b) MARY spectra for the yield of the excited triplet state generated from the singlet-born PTZ +• −FL 2 −PDI −• radical pair.Blue dashed line: Monte Carlo simulation with 2J 0 /γ = +9.06T and D rot = 2.0 × 10 9 s −1 .Red solid line: Experimental result.(c) Time evolution of ST +1 coherence under the Hamiltonian of fluctuating 2J coupling at shown rotational diffusion constants.The initially given coherence is 0.5.The black dashed lines indicate fitting by damped harmonic oscillation.
or β denotes the nuclear spin state.Note that |1 = |T +1 α and |8 = |T −1 β do not contribute to the state mixing at any magnetic fields.The application of a magnetic field smaller than a shifts the energy levels of eigenstates having a T +1 or T −1 character, which results in breakdown of the zero-field degeneracies (for |2 −|3 and |6 −|7 states in the single proton model).This causes unlocking of coherent oscillations of the S−T mixing that are disabled at zero magnetic field.This feature is understood by time evolution of a spin state populations for a singlet-born radical pair described by eigenstates of the Hamiltonian at a finite field (|n and |m ) P ψ (t) = 1 Z m,n n| PS |m m| Pψ |n e iω m−n t (ψ