Chirality Induced Spin Coherence in Electron Transfer Reactions
CChirality Induced Spin Coherence in ElectronTransfer Reactions
Thomas P. Fay ∗ Department of Chemistry, University of Oxford, Physical and Theoretical Chemistry Laboratory,South Parks Road, Oxford, OX1 3QZ, UK
E-mail: [email protected] a r X i v : . [ phy s i c s . c h e m - ph ] J a n bstract Recently there has been much interest in the chirality induced spin selectivity effect,whereby electron spin polarisation, which is dependent on the molecular chirality, is pro-duced in electrode-molecule electron transfer processes. Naturally, one might consider if asimilar effect can be observed in simple molecular charge transfer reactions, for example inlight-induced electron transfer from an electron donor to an electron acceptor. In this workwe explore the effect of electron transfer on spins in chiral single radicals and chiral radicalpairs using Nakajima-Zwanzig theory. In these cases chirality, in conjuction with spin-orbitcoupling, does not lead to spin polarisation, but instead the electron transfer generates quantumcoherence between spins states. In principle, this chirality induced spin coherence could man-ifest in a range of experiments, and in particular we demonstrate that the OOP-ESEEM pulseEPR experiment would be able to detect this effect in oriented radical pairs.
Graphical TOC Entry e - D–B*–A e - D–B*–A on and theoretical descrip-tions of the chirality induced spin selectivity (CISS) effect. Much of this has been motivatedby the potential for this effect to be exploited in spintronic devices, in which spins can be usedto transfer and store information, with the potential for use in quantum computation and quantuminformation transfer. The CISS effect involves the generation of electron spin polarisation bythe transmission of an electron through a chiral environment, and this effect is generally explainedby the presence of the spin-orbit interaction in these systems.
The majority of experimen-tal and theoretical studies involve the transmission of electrons between electrodes and chiralmolecules, however there has been relatively little investigation into the role of chirality on thespin in molecular electron transfers, and so this is precisely what we explore here for reactions ininvolving radicals and radical pair formation.It is instructive to first consider a charge transfer reaction in an 𝑆 = / [ D •− A ] 𝑘 f −−− ⇀↽ −−− 𝑘 b [ DA •− ] . A simple model for this is a system which can exist in one of two charge transfer (CT) states: | (cid:105) = | D •− A (cid:105) and | (cid:105) = | DA •− (cid:105) . Associated with each of these diabatic states are two spinstates | 𝑀 𝑆 = ± / (cid:105) , so overall there are four electronic states | 𝑗 , 𝑀 𝑆 (cid:105) . For this system we can writethe total Hamiltonian as ˆ 𝐻 = ˆ 𝐻 ˆ Π + ˆ 𝐻 ˆ Π + ˆ 𝑉 DC + ˆ 𝑉 SOC , (1)where ˆ Π 𝑗 = (cid:205) 𝑀 𝑆 = ± / | 𝑗 , 𝑀 𝑆 (cid:105)(cid:104) 𝑗 , 𝑀 𝑆 | is a projection operator onto charge transfer state 𝑗 , ˆ 𝐻 𝑗 is theHamiltonian for charge transfer state 𝑗 , ˆ 𝑉 DC is the diabatic coupling Hamiltonian and ˆ 𝑉 SOC is thespin-orbit coupling Hamiltonian. The Hamiltonian for each charge transfer state ˆ 𝐻 𝑗 can be dividedinto a nuclear part ˆ 𝐻 𝑗 n = ˆ 𝑇 n + 𝑉 𝑗 ( ˆ Q ) which is a sum of nuclear kinetic energy and potential energyterms, and a spin part ˆ 𝐻 𝑗 s , and for simplicity we can assume the spin part is independent of thenuclear coordinates Q . 𝑉 DC does not act on the spin of the system, instead it simply couplesthe two charge transfer states, and within the Condon approximation (coupling independent of Q )this is ˆ 𝑉 DC = Δ (| (cid:105)(cid:104) | + | (cid:105)(cid:104) |) (2)where Δ is a real-valued constant. The spin-orbit coupling also couples the charge transfer states,but it also acts on the spin as follows ˆ 𝑉 SOC = − 𝑖 (| (cid:105)(cid:104) | 𝚲 · ˆ S − 𝚲 · ˆ S | (cid:105)(cid:104) |) . (3)ˆ S here is the unitless electron spin operator and the vector coupling 𝚲 is real valued for the spin-orbitcoupling between bound electronic states (a detailed justification of this is given in the SI). Overallthis means we can write the total charge transfer state coupling Hamiltonian ˆ 𝑉 = ˆ 𝑉 DC + ˆ 𝑉 SOC asˆ 𝑉 = Γ (cid:16) ˆ 𝑈 † | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ˆ 𝑈 (cid:17) , (4)where ˆ 𝑈 is a unitary spin operator given by ˆ 𝑈 = ( Δ + 𝑖 𝚲 · ˆ S )/ Γ , and Γ = √︁ Δ + | 𝚲 | /
4. This unitaryoperator rotates the electron spin about the axis n = 𝚲 /| Λ | by an angle 2 𝜃 , with 𝜃 given by 𝜃 = atan (| 𝚲 |/( Δ )) . (5)Coherences between the spin states of each charge transfer state will be long-lived because thepotential energy surfaces for different spin states 𝑉 𝑗 𝑀 𝑆 ( Q ) = 𝑉 𝑗 ( Q ) are identical, but this is nottrue for coherences between the different charge transfer states, and as such we describe the systemwith two spin density operators, one for each charge transfer state, ˆ 𝜎 𝑗 s ( 𝑡 ) which are related to the4ull density operator ˆ 𝜌 ( 𝑡 ) by ˆ 𝜎 𝑗 s ( 𝑡 ) = Tr n [ (cid:104) 𝑗 | ˆ 𝜌 ( 𝑡 ) | 𝑗 (cid:105)] . (6)Our aim is now to find a quantum master equation for these spin density operators. This isdone by employing the Nakajima-Zwanzig equation following the same steps as outlined inthe derivation of radical pair spin master equations in Ref. 17, which we summarise in the SI.Assuming that the charge transfer coupling ˆ 𝑉 can be treated perturbatively, and that the populationtransfer is described accurately as an incoherent kinetic process, with rates independent of the spininteractions in ˆ 𝐻 𝑗 s , the resulting quantum master equations at second order in Γ for ˆ 𝜎 𝑗 s ( 𝑡 ) are,dd 𝑡 ˆ 𝜎 ( 𝑡 ) = − 𝑖 ℏ [ ˆ 𝐻 , ˆ 𝜎 ( 𝑡 )] − 𝑘 f ˆ 𝜎 ( 𝑡 ) + 𝑘 b ˆ 𝑈 † ˆ 𝜎 ˆ 𝑈 (7)dd 𝑡 ˆ 𝜎 ( 𝑡 ) = − 𝑖 ℏ [ ˆ 𝐻 , ˆ 𝜎 ( 𝑡 )] − 𝑘 b ˆ 𝜎 ( 𝑡 ) + 𝑘 f ˆ 𝑈 ˆ 𝜎 ˆ 𝑈 † , (8)where [· , ·] denotes the commutator. We see that as an electron is transferred, its spin is rotatedabout the vector 𝚲 , and as it is transferred back it is rotated about the vector − 𝚲 .Thus far we do not appear to have included the idea of chirality in any of the above discussion, sowe shall now discuss how this manifests in these equations. Chirality of the system determines therelative sign of Δ and 𝚲 . On spatial reflection, the sign of the diabatic coupling Δ in unchanged, butthe sign of the spin-orbit coupling vector 𝚲 is changed because the spin-orbit coupling operator is anaxial vector operator. This means that for an initially spin polarised state of | (cid:105) , ˆ 𝜎 𝑠 ( ) = | 𝛼 (cid:105)(cid:104) 𝛼 | , this polarisation will be transformed into coherence between spin states | 𝛼 (cid:105) and | 𝛽 (cid:105) in CT state | (cid:105) if 𝚲 has a component perpendicular to the 𝑧 axis. Interestingly this means that photo-initiatedelectron transfer in a spatially oriented chiral doublet system, initially at thermal equilibrium in a Here we use the standard notation for the electron spin states | 𝛼 (cid:105) = | 𝑀 𝑆 = + / (cid:105) and | 𝛽 (cid:105) = | 𝑀 𝑆 = − / (cid:105) . This rotation of the electron spin is reminiscent of the (cid:174) 𝐶 term appearing in the theory of Dalum & Hedegård inRef. 10, derived for spin transport through a molecular junction from the non-equilibrium Green’s function approach.A spin polarisation term analogous to the (cid:174) 𝐷 term in in Ref. 10 does not appear in our theory for spin transport betweenbound electronic states because the initial and final electronic states have no linear or angular momentum, so theirwavefunctions are real valued and the spin-orbit coupling vector is purely real, so no spin polarisation appears. 𝑥 / 𝑦 magnetisation, and therefore a free-induction decay signal, without a microwave pulse. For an achiral system there are equal contributions from configurations with ± 𝚲 , which means that spin polarisation perpendicular to the 𝚲 axis is simply lost in the chargetransfer process, and as such microwave pulse-free free-induction decay could not be observed.We are now in a position to consider the effect of spin-orbit coupling and chirality on the chargetransfer process between a singlet photoexcited precursor state DA ∗ and a charge separated radicalpair state D •+ A •− , [ DA ∗ ] 𝑘 f −−− ⇀↽ −−− 𝑘 b 𝑆 + [ D •+ A •− ] . Analogous to the above situation, a simple model for this includes the excited precursor diabaticstate | (cid:105) = | DA ∗ (cid:105) and a charge separated radical pair state | (cid:105) = | D •+ A •− (cid:105) . We assume that the state | (cid:105) can only exist in a singlet spin state, so only | , 𝑆 = , 𝑀 𝑆 = (cid:105) exists, but the radical pair state | (cid:105) can exist in both singlet and triplet states which lie very close in energy, therefore there are four(near-)degenerate | , 𝑆, 𝑀 𝑆 (cid:105) states which exist, with 𝑆 = 𝑆 =
1. The total Hamiltonian for thissystem has the same form as Eq. (1), ˆ 𝐻 = ˆ 𝐻 ˆ Π + ˆ 𝐻 ˆ Π + ˆ 𝑉 DC + ˆ 𝑉 SOC , where again ˆ 𝐻 𝑗 = ˆ 𝐻 𝑗 s + ˆ 𝐻 𝑗 n is the Hamiltonian for state 𝑗 and ˆ Π 𝑗 is a projection operator onto the 𝑗 states of the allowed spinmultiplicity for that charge transfer state, i.e. ˆ Π = | (cid:105)(cid:104) | ˆ 𝑃 S and ˆ Π = | (cid:105)(cid:104) | ( ˆ 𝑃 S + ˆ 𝑃 T ) , where ˆ 𝑃 S and ˆ 𝑃 T are singlet and triplet spin state projection operators. As beforeˆ 𝑉 DC = Δ ( ˆ 𝑃 S | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ˆ 𝑃 S ) (9)is the diabatic coupling Hamiltonian, which now comes with a singlet projection operator ˆ 𝑃 S because the diabatic coupling is spin conserving. The electron transfer can be regarded as a oneelectron transfer process, where only electron 1 transfers, and therefore the spin-orbit coupling termcan be written as ˆ 𝑉 SOC = − 𝑖 ( ˆ 𝑃 S | (cid:105)(cid:104) | 𝚲 · ˆ S − 𝚲 · ˆ S | (cid:105)(cid:104) | ˆ 𝑃 S ) . (10)6here ˆ S is the spin operator for electron 1 (this is justified in more detail in the SI). This meansthe total charge transfer Hamiltonian can be written asˆ 𝑉 = Γ ( ˆ 𝑃 S ˆ 𝑈 † | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ˆ 𝑈 ˆ 𝑃 S ) (11)where now ˆ 𝑈 = ( Δ + 𝑖 𝚲 · ˆ S )/ Γ only acts on electron 1.From this we can obtain the quantum master equations for the spin density operators of the twocharge transfer states, once again using the Nakajima-Zwanzig equation as in Ref. 17, which givesdd 𝑡 ˆ 𝜎 ( 𝑡 ) = − 𝑖 ℏ [ ˆ 𝐻 , ˆ 𝜎 ( 𝑡 )] − 𝑘 f ˆ 𝜎 ( 𝑡 ) + 𝑘 b ˆ 𝑃 S ˆ 𝑈 † ˆ 𝜎 ˆ 𝑈 ˆ 𝑃 S (12)dd 𝑡 ˆ 𝜎 ( 𝑡 ) = − 𝑖 ℏ [ ˆ 𝐻 , ˆ 𝜎 ( 𝑡 )] − (cid:26) 𝑘 b 𝑈 ˆ 𝑃 S ˆ 𝑈 † , ˆ 𝜎 ( 𝑡 ) (cid:27) − 𝑖 ℏ (cid:2) 𝛿𝐽 ˆ 𝑈 ˆ 𝑃 S ˆ 𝑈 † , ˆ 𝜎 ( 𝑡 ) (cid:3) + 𝑘 f ˆ 𝑈 ˆ 𝑃 S ˆ 𝜎 ( 𝑡 ) ˆ 𝑃 S ˆ 𝑈 † . (13)Here {· , ·} denotes the anti-commutator, and 2 𝛿𝐽 ˆ 𝑈 ˆ 𝑃 S ˆ 𝑈 † is an effective spin coupling term in theradical pair state (expressions for the master equation parameters 𝑘 f , 𝑘 b and 𝛿𝐽 are given in the SI).Now assuming the radical pair, state | (cid:105) , is formed very rapidly and irreversibly from an excitedsinglet precursor | (cid:105) , as is often the case for radical pairs formed by photoexcitation of a singletground state, then we can take 𝑘 b =
0, and the radical pair spin density operator as being formed inthe state ˆ 𝜎 ( ) = ˆ 𝜎 ns ˆ 𝑈 ˆ 𝑃 S ˆ 𝑈 † (14)where ˆ 𝜎 ns is the initial nuclear spin state of the precursor state. The initial electron spin state canbe written as ˆ 𝑈 ˆ 𝑃 S ˆ 𝑈 † = | 𝜓 (cid:105)(cid:104) 𝜓 | , where the initial spin state | 𝜓 (cid:105) = ˆ 𝑈 | S (cid:105) is | 𝜓 (cid:105) = cos 𝜃 | S (cid:105) + 𝑖 sin 𝜃 | T ( n )(cid:105) , (15)in which | S (cid:105) is an electron spin singlet state and | T ( n )(cid:105) is a T triplet state defined with respect to7 − B* − A D •+ − B* − A • − h ν Λ B || Λ h ν ( π /4) x ( π ) x τ τ FID (B)(A)
Figure 1: (A) A schematic representation radical pair formation in a magnetic field, with donor, D,and acceptor, A, covalently linked by a chiral bridge, B * . (B) A summary of the pulse sequence inthe OOP-ESEEM experiment with 𝑡 = n = 𝚲 /| 𝚲 | , and 𝜃 is given by Eq. (5). Once again for the opposite enantiomer the sign of 𝚲 changes. Denoting the different enantiomers + and − , this means that n + = − n − , or equivalentlywe can write the initial state as (cid:12)(cid:12) 𝜓 ± (cid:11) = cos 𝜃 + | S (cid:105) ± 𝑖 sin 𝜃 + | T ( n + )(cid:105) . (16)This means that the phase of the initial coherence between singlet and triplet states is changed bythe chirality in the radical pair. In an achiral system, there will be equal but opposite contributionsto charge transfer from configurations with opposite signs of n , and therefore there will be noinitial coherence between singlet and triplet electron spin states. It is well known that spin-orbitcoupled charge transport (SOCT) in radical pair reactions leads to the formation of triplet radicalpairs, but this shows that in chiral systems spin coherence between singlet and triplet statesis also generated by SOCT.We will now consider how chirality induced spin coherence could be detected experimentally byEPR. Because the spin coherence generated is orientation dependent, any experiment probing thiswould have to be performed on radical pairs somehow fixed in a specific spatial orientation. Here weconsider the OOP-ESEEM (out of phase electron spin echo envelope modulation) EPR experiment,8hich has been developed as a technique for probing zero-quantum coherences in photo-generatedradical pairs, with the applied field B parallel to 𝚲 for oriented radical pairs as illustrated in Fig. 1(A). The standard OOP-ESEEM experiment consists a laser flash ( ℎ𝜈 ), in which the radicalpairs are generated, followed by a waiting time of 𝑡 , followed by a non-selective ( 𝜋 / ) 𝑥 microwavepulse, and another ( 𝜋 ) 𝑥 pulse after a time 𝜏 , and after another waiting time 𝜏 the free-inductiondecay spin echo is recorded over times 𝑡 . This ℎ𝜈 − 𝑡 − ( 𝜋 / ) 𝑥 − 𝜏 − ( 𝜋 ) 𝑥 − 𝜏 − FID pulse sequenceis illustrated schematically in Fig. 1 (B). This pulse sequence gives access to information about thezero quantum coherences in the spin density operator, which evolve in the waiting times 𝑡 and 𝜏 . If we are interested in the initial state of the spin density operator, in particular the phases of theinitial zero-quantum coherences, then we should minimise the time for the coherences to evolve ontheir own, as well as any decoherence processes, and therefore here we only consider the case of 𝑡 =
0, as shown in Fig. 1 (B).Using the standard high-field radical pair spin Hamiltonian in the rotating frame, and invokingthe Schulten-Wolynes semiclassical approximation for the nuclear spins in the radical pair, we canobtain expressions for the 𝑥 -channel OOP-ESEEM signal 𝑓 𝑥 ( 𝑡 ) = Tr s [( ˆ 𝑆 𝑥 + ˆ 𝑆 𝑥 ) ˆ 𝜎 ( 𝑡 )] (details ofthis are given in the SI). In the weak coupling limit ( | Ω − Ω | (cid:29) | 𝐽 | , | 𝑑 | ) we find the OOP-ESEEMFID signal to be, 𝑓 𝑥 ( 𝑡 ) =
12 sin ( 𝜃 ) sin (cid:18) (cid:18) 𝐽 − 𝑑 (cid:19) ( 𝑡 + 𝜏 ) (cid:19) ∑︁ 𝑖 = , (cid:18) 𝑒 − 𝑡 /( 𝜏 𝑖 ) (cid:16) sin ( 𝜃 ) cos ( Ω 𝑖 𝑡 ) −√ ( 𝜃 ) sin ( Ω 𝑖 𝑡 ) (cid:17) (cid:19) . (17) Ω 𝑖 is the resonant frequency of electron spin 𝑖 (in the rotating frame), 𝐽 is the scalar coupling constantand 𝑑 is the effective dipolar coupling constant, and 𝜏 − 𝑖 = (cid:205) 𝑁 𝑖 𝑘 = 𝑎 𝑖,𝑘 𝐼 𝑖,𝑘 ( 𝐼 𝑖,𝑘 + ) , where 𝑎 𝑖,𝑘 and 𝐼 𝑖,𝑘 are the hyperfine coupling constants and spin quantum numbers for the hyperfine couplednuclei in radical 𝑖 . For simplicity we have ignored spin relaxation and radical pair recombinationprocesses. Importantly here we see that changing the sign of 𝜃 changes the sign of certain termsin the 𝑥 channel FID signal after the OOP-ESEEM pulse sequence. This means the signal would9igure 2: OOP-ESEEM signals for chiral radical pairs with Ω = − Ω = . 𝛾 e , 𝐽 − 𝑑 / = . 𝛾 e and 𝜏 = 𝜏 = . − 𝛾 − , typical parameters for a real organic radical ion pair( 𝛾 e = e 𝑔 e / 𝑚 e is the modulus of the gyromagnetic ratio of the free electron spin). Achiral signalsare an average of the 𝜃 = ± 𝜋 /
16 signals. Top panel: the FID echo for 𝜏 =
200 ns calculated usingEq. (17). Bottom panel: the integrated echo as a function of 𝜏 calculated using Eq. (S.26).be sensitive to the chirality of the molecule. As described in the SI, we can also calculate theOOP-ESEEM signal in the intermediate–strong coupling limit numerically. Often in OOP-ESEEMexperiments, the waiting time 𝜏 is varied, so we also consider the integrated FID echo signal as afunction of 𝜏 , 𝐹 𝑥 ( 𝜏 ) = ∫ ∞ 𝑓 𝑥 ( 𝑡 ) d 𝑡 .As an example of this we plot the 𝑡 dependent OOP-ESEEM signal of two enantiomers of achiral radical pair with 𝜃 = ± 𝜋 /
16 in Fig. 2. This corresponds to an initial triplet fraction of ∼
4% which is typical of organic radical pairs in solution.
We see that the two OOP-ESEEMsignals are very different even in the case of a modest SOCT contribution. The signals are notthe exact negative of each other because the spin Hamiltonian is chirality independent but chiralitydoes change the sign of the phase of coherences in the initial state, leading to a phase shift in theOOP-ESEEM signals. In Fig. 3 we show these signals for the same model in the intermediatecoupling regime, calculated by sampling 10 realisations of the hyperfine fields, where again we10igure 3: OOP-ESEEM signals for chiral radical pairs with Ω = − Ω = . 𝛾 e , 𝐽 − 𝑑 / = 𝛾 e and 𝜏 = 𝜏 = . − 𝛾 − . Achiral signals are an average of the 𝜃 = ± 𝜋 /
16 signals. Toppanel: the FID echo for 𝜏 =
200 ns calculated using Eq. (S.25). Bottom panel: the integrated echoas a function of 𝜏 calculated using Eq. (S.25).see that there is a phase shift between the 𝐹 𝑥 ( 𝜏 ) signals.In this work we have discussed how spin-orbit coupling can lead to chirality induced spincoherence in radicals and radical pair systems, an effect which has not before been explored inspin chemistry. In contrast to the case of electrode-molecule electron transfers, where chiralityinduces spin polarisation, chirality in intramolecular electron transfers generates specific coherentsuperpositions of spin states. This effect should manifest in certain EPR experiments and otherexperiments such as those that probe magnetic field effects (which we describe briefly in the SI).Chirality induced spin coherence could be important in a range of systems. For example,it has been suggested that radical pair reactions in cryptochrome proteins form the basis of themagnetic compass sense of migratory birds, and because chirality induced spin coherenceis orientation dependent, this effect could hypothetically play a role in avian magnetoreception.Radical-based systems have also been proposed as potential molecular qubits, and variousquantum information theoretic ideas have been explored using these molecules. Electron11ransport in chiral molecular systems could provide a way of further manipulating spin coherencesin these systems. Other chiral open-shell molecular systems, aside from radicals and radicalpairs, could potentially display chirality induced spin coherence effects. For example it has beendemonstrated that spin-orbit coupling can play a role in singlet fission and triplet-triplet up-conversion, so it stands to reason that in chiral systems spin-orbit coupling could lead to interestingeffects on these processes. Furthermore chirality induced spin coherence effects could manifestin the photophysics of open-shell chiral transition metal complexes with 𝑆 >
0. Overall we hopethat this work will lay foundations for exploring chirality induced spin effects in a wide variety ofmolecular systems, beyond what has already been studied in electrode-molecule interfaces.
Acknowledgements
I would like to thank Peter Hore and David Manolopoulos for their comments on this manuscript.I would also like to thank both Peter Hore and Jiate Luo for introducing me to the problem ofspin effects in electron transfer in chiral molecules. I would like to acknowledge financial supportfrom the Clarendon Scholarship from Oxford University, an E.A. Haigh Scholarship from CorpusChristi College, Oxford, the EPRSC Centre for Doctoral Training in Theory and Modelling inthe Chemical Sciences, EPSRC Grant No. EP/L015722/1, and the Air Force Office of ScientificResearch (Air Force Materiel Command, USAF award no. FA9550-14-1-0095).
Supporting Information
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