Consistent Treatment of Spin-Selective Recombination of a Radical

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J. Phys. Chem. A 2010, 114, 9447–9455

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Consistent Treatment of Spin-Selective Recombination of a Radical Pair Confirms the Haberkorn Approach Konstantin L. Ivanov,*,† Marina V. Petrova,†,‡ Nikita N. Lukzen,† and Kiminori Maeda*,§ International Tomography Center SB RAS, NoVosibirsk, 630090, Russia, NoVosibirsk State UniVersity, NoVosibirsk, 630090, Russia, and Centre for AdVanced Electron Spin Resonance, UniVersity of Oxford, Oxford, OX1 3QR, U.K. ReceiVed: May 26, 2010; ReVised Manuscript ReceiVed: July 26, 2010

In the present work, we have shown that consistent derivation of the kinetic equations describing the electron spin-selective recombination of radical pairs confirms the conventional Haberkorn approach. The derivation has been based on considering the interaction of the reactive system (radical pair and product state) with the thermal bath. The consistency of this approach has also been substantiated by numerical simulations performed for the purely quantum mechanical model of the recombining radical pair. Finally, we have shown that the quantum Zeno effect on radical pair recombination is not an exclusive feature of the approach recently proposed by Kominis, as it should be present at any rate of the singlet-triplet dephasing in the radical pair, which always accompanies the recombination process. I. Introduction 1

For many years the Haberkorn approach has been used to describe recombination of radical pairs (RP). In this approach, the spin-selective recombination of the RP is described by an anticommutator term in the kinetic equation for the density matrix, F, of the RP, which is as follows:

k k dF ˆ , F] - S {Q ˆ S, F} - T {Q ˆ , F} ) -i[H dt 2 2 T

(1)

ˆ is the spin Hamiltonian of the RP; kS and kT are the Here H reaction rates for the singlet and triplet recombination channels, ˆ T are the projection operators for the ˆ S and Q respectively; Q singlet and triplet electronic states of the RP, respectively; {Aˆ,Bˆ} stands for the anticommutator of Aˆ and Bˆ. The Haberkorn equation follows from a very simple physical consideration. Let us imagine that the RP is described by the wave function, |ψ〉, which is given by a superposition of the singlet and triplet states:

|ψ(t)〉 ) cS(t)|S〉 + cT(t)|T〉

c˙T ) -

kT c 2 T

kS + kT FST, 2 kS + kT )FTS, F˙ TT ) -kTFST (4) 2

F˙ ST ) F˙ TS

These expressions are formally equivalent to the Haberkorn equation (eq 1), which is written in matrix form. If the triplet recombination rate is zero, the singlet state population decays at a rate kS, whereas the coherences, FST and FTS, decay two times slower at a rate kS/2. This is the classical result, which has been used for many years to model spin-selective RP recombination. In general, the rate of (kS + kT)/2 represents the lower limit of coherence decay rate. Let us demonstrate this for a twolevel system (singlet level and only one triplet level). Let us imagine that at t ) 0 the density matrix is positively defined, which is the general property of the density operator. This means that the following relation between the matrix elements of F is fulfilled:2

(2) FSS(t ) 0) × FTT(t ) 0) g |FST(t ) 0)| 2

Recombination of the RP is responsible for the time evolution of the coefficients cS and cT, which can be described by equations

kS c˙S ) - cS, 2

F˙ SS ) -kSFSS,

(3)

Taking the definition of the matrix elements of F, which is Fij ) cicj*, one immediately obtains * Corresponding authors. E-mail: [email protected] (K.L.I.), [email protected] (K.M.). † International Tomography Center SB RAS. ‡ Novosibirsk State University. § University of Oxford.

(5)

When there is no dynamic or stochastic mixing of the singlet and triplet states but only chemical reactions going on, the populations of the states decay at rates kS and kT. Let us assume that the coherences, FST and FTS, decay at a rate k′. Since the density matrix has to be positively defined at any instant of time, we immediately obtain

FSS(t ) 0) × FTT(t ) 0)e-(kS+kT)t g |FST(t ) 0)| 2e-2k′t

(6) Since this inequality has to be fulfilled also at any starting conditions, which do not violate eq 5, we obtain that

10.1021/jp1048265  2010 American Chemical Society Published on Web 08/12/2010

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k′ g

kS + kT 2

Ivanov et al.

(7)

Thus, the rate (kS + kT)/2 gives the lower limit for the decoherence rate. In general, basic physical principles allow a higher rate of the coherence decay.3 Although the conventional Haberkorn equation is physically transparent and widely accepted by the spin chemistry community, recently some doubts concerning this approach have appeared. In the work of Kominis4 it has been stated that the conventional approach does not reproduce the so-called quantum Zeno effect, which is responsible for slowing down the radical pair recombination upon frequent “measurement” of its state by chemical reaction. Moreover, in this work there was given a derivation of the equation for the RP density matrix, which considerably differs from eq 1. However, both the statement concerning the Zeno effect and the derivation of the equation seem dubious to us. In earlier works5,6 the Zeno effect has been obtained even with the conventional form of the recombination operator. This fact has been confirmed recently by Jones and Hore.7 In addition we believe that the derivation proposed by Kominis is erroneous (see below). We should also emphasize that, recently, Jones and Hore7 have obtained the recombination operator under an assumption that the reaction and measurement events are completely nonseparable. They obtained that the measurement efficiency (decoherence rate) is k′ ) kS + kT in this approach, which is 2 times higher than in the Haberkorn approach. Such a relation is generally appropriate, since it does not contradict the basic properties of the density matrix and can be met for special cases of chemical reactions. Nonetheless, the physical situation, which corresponds to nonseparable recombination and measurement events, remains unclear. The idea of the Haberkorn theory, which is explained by eqs 2 and 3, is physically clear in contrast to that of the “quantum measurement” based approaches. In fact, the concept of “quantum measurement” of the RP electronic spin state seems to be rather artificial in the case of chemical reaction. In the conventional quantum measurement theory, interaction of quantum system with a classical object (measuring device) destroys the coherence between the quantum states not resulting in the decay of the total population. Chemical reaction represents a different case, since not only decoherence is present but also the population decay. In fact, every chemical reaction would result in the decoherence proceeding at a rate k′ g (kS + kT)/2, i.e., every chemical reaction “measures” the RP spin state in this sense. However, the ratio between the decoherence rate and the chemical decay rate is generally not defined unless a derivation of the reaction operator is given. This means that the claim “chemical reaction measures the spin state of the RP” is an empty saying unless the ratio between the recombination and dephasing rates is not specified explicitly. Theoretical calculations of spin effects on chemical reactions performed in ref 7 for three types of recombination operators proposed by Haberkorn and Jones-Hore have shown that the difference between these two approaches was minor. At the same time, the results obtained for the Kominis recombination operator are expected to be strongly different. To make a comparison of the results predicted by these three approaches, we performed calculations for the chemically induced dynamic electron polarization (CIDEP)8 by taking account of the electronic exchange interaction and recombination in the contact approximation.9,10 Here we will not discuss the details of calculation, which are exhausted in refs 11, 12. In calculations, a radical pair without magnetic nuclei was

Figure 1. cw-CIDEP spectra as calculated by means of approaches proposed by Haberkorn (1), Jones-Hore (2), and Kominis (3). Parameters of calculation: R ) 8 Å, kSR∆/D ) kTR∆/D ) 10, ∆gB ) 3.9 mT, J ) -90 ns-1, RP lifetime due to scavenging is τRP ) 200 ns. Cases a and b correspond to the diffusion coefficient D of 10-8 and 10-5 cm2 s-1, respectively. Maximal line intensities in the CIDEP spectra are related as follows: IKom:IH:IJH ) 2.1:1:1 (case a) and IKom: IH:IJH ) 8.5:1:0.4 (case b).

considered with different g factors, g1 and g2, of the two radical centers. We assumed that the radicals move by means of free diffusion in three dimensions; reflecting boundary condition at their closes approach distance, R, was used. In our model, the exchange integral, J, as well as the singlet-state and tripletstate recombination probabilities, kS and kT, are nonzero only within a thin zone of the width, δ, which is much smaller than the contact radius: δ , R. The results of the calculation for the cw-CIDEP spectra are shown in Figure 1. At normal viscosity (larger diffusion coefficient D equal to 10-5 cm2/s), the shape of the CIDEP spectrum is the same for all three recombination operators used, although the CIDEP line intensities are different in all three approaches. However, for higher viscosity only the Haberkorn and Jones-Hore approaches predict similar results (identical spectrum shape though different line intensity), in contrast to the Kominis approach. Since the calculated magnetic field and spin effects on chemical reactions are sensitive to the form of the recombination operator used (as clearly seen when CIDEP is taken as an example), the problem of choosing its proper form becomes vital for spin chemistry. Once the commonly used Haberkorn approach turns out to be wrong, a large body of the results in the field of spin chemistry8 has to be at least reconsidered if not refuted. The aim of the present work is deriving the correct form of the recombination operator. The key step in this derivation will be the interaction of the reactive system with the medium. When the RP recombines in solids or liquids, for instance, by means of electron or hydrogen atom transfer, a large amount of energy is released (or absorbed). To fulfill the energy conservation law, this energy has to be absorbed (or released) by the medium, which thus plays a very important role for the condensed-phase reactions. We will model the interaction of the reactive system with the medium by assuming that the latter represents a bath, which is stabilized at a certain temperature and cannot be perturbed by the chemical reactions. As will be seen later, our treatment will support the conventional theory of Haberkorn. It is important to emphasize that recently Il’ichov and Anishchik13 have independently derived the Haberkorn equation under similar assumptions. The validity of the Haberkorn approach will also be tested by performing purely quantum mechanical calculations for the model of chemical reaction, in which the RP is dynamically coupled to the numerous product states. In addition, here we will discuss the so-called quantum Zeno effect in different approaches.

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II. Results and Discussion A. Reactive System Interacting with Thermal Bath. For the sake of clarity, in our derivation we will consider only two energy levels of the RP, namely, singlet level and T0 triplet level (hereafter denoted simply as the T level). This assumption is not of principal importance and is introduced only for simplicity of the derivation. When more levels are taken into account, the derivation remains essentially the same. We will assume that only the singlet state of the RP is reactive; extension to the case of reactive triplet RPs can be done in the same fashion. The reactive system will be considered as a three-level system (with states defined as S, T, S0) with the two RP states, S and T, and the singlet reaction product state S0. The density matrix of the reactive system is 3 × 3 matrix and its trace is preserved. Including the product state completely eliminates an artificial problem of nonpreserved trace of the RP density matrix, which has been stated by Kominis.4 The Hamiltonian of the reactive system is the following matrix (in units of p):

ˆs ) H

(

ω0 /2 0 0 ω0 /2 0 0 -ω0 /2 0 0

)

(8)

in the basis of states {S, T, S0}. Here ω0 stands for the energy difference between the RP and product state. In general, chemical reaction is a sophisticated process, during which the reactive system travels along the potential energy surface (PES) from the valley of reactants to the valley of reaction products through an appropriate saddle point. The shape of the PES is defined not only by the reactant properties but also by the chemically inert medium. The PES is different for the singlet and triplet RPs. During the elementary reaction event, the reactive system interacts with the medium degrees of freedom, which induces both chemical reaction (transitions between the reactant and the product states) and decay of the coherences between the reactant states. Here we will model chemical reaction in a rather simple but realistic way. We will consider reactive system interacting with thermal bath, which is kept at thermal equilibrium and is characterized by temperature T. All perturbations of the bath by chemical reaction will be neglected. Thus, the bath can absorb any amount of the energy released during the RP recombination. The bath will be considered as an harmonic oscillator (system with equally spaced energy levels) with frequency ω. Therefore, the bath can be described by the following Hamiltonian:

ˆ b ) ωaˆ+aˆ H

(9)

with rising, aˆ+, and lowering, aˆ, operators. Possible deviations from the harmonicity will affect the rates of forward reaction (RP recombination) and backward reaction (production of RPs from the S0 state). However, in the situation of irreversible recombination (which is the case when pω0/kT . 1), such deviations are unimportant. Extension of the consideration of the bath to the set of harmonic oscillators characterized by frequencies ωk is obvious (vide infra). Spin-selective recombination of the singlet ground state will be described by means of the following Hamiltonian:

Vˆ ) gaˆ+sˆ- + g*aˆsˆ+

(10)

which contains the rising and lowering operators of the reactive system and the bath. Here rising and lowering are assumed to be “spin-selective” for the reactive system and are described by the following matrices in the {S, T, S0} basis set:

( ) ( )

0 0 0 sˆ- ) 0 0 0 , 1 0 0

0 0 1 sˆ+ ) 0 0 0 0 0 0

(11)

Thus, sˆ- produces the reaction product S0 selectively from the singlet RPs, whereas sˆ+ results in the formation of the singlet RP from S0 state. Thus, the triplet RPs are not affected directly by the chemical reaction. However, the singlet-triplet evolution in the RP depends on the reaction, since it induces the decay of the FST matrix element, as will be shown below. The physical meaning of the operator defined by eq 10 is very simple. Each lowering (or rising) event for the reactive system is accompanied by rising (or lowering) of the bath state. As a result, the total energy is preserved during each elementary reaction event. The theoretical approach used for the derivation follows closely the one described in ref 14. It represents an extension of the Redfield relaxation theory15 to the systems with arbitrary temperature. Reaction dynamics can be described by the Liouville equation for the full density matrix, Ff, which is as follows:

dFf(t) ˆ b + Vˆ, Ff(t)] ˆs + H ) -i[H dt

(12)

In the following treatment, we will simplify the problem by considering it in the interaction representation. In the interaction picture, the density matrix and the interaction between the reactive system and the bath should be modified as follows:

ˆ 0t)Ff(t) exp(-iH ˆ 0t), FI(t) ) exp(iH ˆ 0t)Vˆ exp(-iH ˆ 0t) (13) VˆI(t) ) exp(iH ˆs + H ˆ b. The equation for the density matrix FI ˆ0 ) H Here H can be written in the usual way:

dFI(t) ) -i[VˆI(t), FI(t)] dt

(14)

Its formal solution can be written as follows:

FI(t) ) FI(0) - i

∫0 t[VˆI(t′), FI(t′)] dt′

(15)

An equivalent form of this equation can be obtained by replacing FI(t′) in the right-hand side by its explicit form:

dFI(t) ) -i[VˆI(t), FI(0)] dt

∫0 t[VˆI(t), [VˆI(t′), FI(t′)]] dt′ (16)

To go on further, several steps should be performed. First, the Hamiltonian Vˆ, which describes the chemical reaction, should be calculated in the interaction representation, which means that all the rising and lowering operators should be recalculated. Let us demonstrate how it can be done, for instance, for aˆ+

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(aˆ+)I ) exp(iωtaˆ+aˆ)aˆ+ exp(-iωtaˆ+aˆ) ) f (x)| x)iωt

(17) where f(x) ) exp (xaˆ+aˆ)aˆ+ exp (-xaˆ+aˆ). One can obtain the following differential equation for this quantity

df (x) ) exp(xaˆ+aˆ)(aˆ+aˆaˆ+ - aˆ+aˆ+aˆ) exp(-xaˆ+aˆ) ) f (x) dx (18) since aˆaˆ+ - aˆ+aˆ is the unity operator. As a result, f(x) ) exaˆ+ the bath degrees of freedom in eq, which allows us to obtain +

x +

iωt +

(aˆ )I ) f (iωt) ) e aˆ | x)iωt ) e aˆ

(sˆ+)I ) eiω0tsˆ+,

Fb )

1 ˆ b /kT) ) 1 exp(-βH ˆ b), exp(-H Z Z + 1 (Fb)mn ) δmn (e-βωaˆ aˆ)nn (24) Z

(19)

Here Z is the statistical sum and β ) 1/kT. To obtain the equation for the density matrix F, it is necessary to trace out the bath degrees of freedom in eq 22. After substituting the explicit form of the matrix Fb we obtain:

Similarly,

(aˆ)I ) e-iωtaˆ,

elements. This is an artificial complication, which comes from the erroneous factorization of the density matrix F. It also results in the incorrect trace-preserving form of the kinetic equation for the RP density matrix. It is clear that once the transitions (chemical reaction) are going on between the two subspaces of a quantum system, only the trace of density matrix of the entire system stays constant, but not that for any individual subsystem. The density matrix of the bath, which is characterized by the ˆ b and stabilized at temperature T, is given by the Hamiltonian H following expression (Gibbs distribution):

(sˆ-)I ) e-iω0tsˆ-

(20) As a result, the Hamiltonian in the interaction representation takes the form:

VˆI(t) ) gei(ω-ω0)taˆ+sˆ- + g*e-i(ω-ω0)taˆsˆ+

(21)

|g| 2 dF(t) )dt Z

g*e-i(ω-ω0)t[aˆsˆ+, FI(0)]) -

∫0 tdt′[(gei(ω-ω )taˆ+sˆ- + 0

g*e-i(ω-ω0)taˆsˆ+), gei(ω-ω0)t′[aˆ+sˆ-, FI(t′)] + g*e-i(ω-ω0)t′[aˆsˆ+, FI(t′)]] (22) This equation is still fairly complicated. Moreover, it is the equation for the full density matrix, whereas the equation for the density matrix of the reactive system is needed. Thus, additional simplifications are required for obtaining the kinetic equation, which describes the RP evolution caused by its recombination. Such simplifications are based on the assumption that the thermal bath is kept at equilibrium at any instant of time. In addition, there are no quantum coherences between the states of the bath and the reactive system. This means that the full density matrix can be written as the direct product of the density matrices of the two subsystems:

FI(t) ) F(t) X Fb

(23)

Here F is the density matrix of the reactive system. It cannot be factorized further into the direct product of the RP and S0 density operators (as has been done erroneously by Kominis4). The reason for this is that the chemical reaction continuously causes transitions between the two subspaces (RP and product). It is exactly because of this approximation that in the approach proposed by Kominis4 one should first solve the equation for the RP density matrix with the preserved trace and only later introduce the recombination rates via the FSS and FTT matrix

0

n

t

+

dt′ (〈n|[aˆ+sˆ-, [aˆsˆ+, F(t′)e-βωaˆ aˆ]]|n〉F+ + +

〈n|[aˆsˆ+, [aˆ+sˆ-, F(t′)e-βωaˆ aˆ]]|n〉F-)

(25)

Here we introduced the quantities

F( ) exp((i(ω - ω0)(t - t′))

Substitution of this expression into eq 16 leads to the following equation for the density matrix FI:

dFI(t) ) -i(gei(ω-ω0)t[aˆ+sˆ-, FI(0)] + dt

∑∫

(26)

The kinetic equation obtained can be simplified further. To do this, we first take the density matrix F(t′) out of the integral sign, because it evolves much slower than the oscillating exponents. Then we take only the real part of the F( terms, since the imaginary part gives only a slight shift of the energy levels and can be neglected. At long times t we can simplify the integral terms:

{ ∫ exp((i(ω - ω )t′)} ≈ πδ(ω - ω )

Re

t

0

0

0

(27)

To get rid of the delta-functions we assume that the bath is composed not by a single oscillator with frequency ω, but by a set of oscillators with distribution G(ω) over all possible frequencies. To perform the averaging over the oscillators, we should integrate over the frequencies, which gives the following result:

{ ∫ dt′ ∫ dωG(ω)F } ≈ πG(ω )

Re

t

(

0

(28)

0

Consequently, eq 25 can be modified as follows: π|g| 2G(ω0) dF(t) )dt Z

∑ (〈n|[aˆ sˆ , [aˆsˆ , F(t)e +

-

+

-βω0aˆ+aˆ

]]|n〉 +

n

+

〈n|[aˆsˆ+, [aˆ+sˆ-, F(t)e-βω0aˆ aˆ]]|n〉)

(29)

To get the final form of the kinetic equation, one more step is required, which is calculating the traces over the bath degrees of freedom, i.e., the explicit form of the operators of the type 〈n|f (aˆ+;aˆ)|n〉. They are as follows:

Spin-Selective Recombination of a Radical Pair

1 Z 1 Z

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∑ 〈n|aˆ+aˆe-βω aˆ aˆ|n〉 ) Z1 ∑ 〈n|aˆe-βω aˆ aˆaˆ+|n〉 ) 0

+

0

n

+

n

∑ n exp(-nβω0) ) exp(βω10) - 1 ) 〈n〉 n

(30) 1 Z

∑ 〈n|aˆaˆ+e-βω aˆ aˆ|n〉 ) Z1 ∑ 〈n|aˆ+e-βω aˆ aˆaˆ|n〉 ) 0

+

0

n

n

1 Z 1 Z

+

∑ 〈n|(1 + aˆ

+

+

aˆ)e-βω0aˆ aˆ |n〉 )

n

exp(βω )

∑ (n + 1) exp(-nβω0) ) exp(βω0) 0- 1 ) 〈n〉 + 1 n

Here 〈n〉 is the average population number for the bath states. Finally, we obtain

dF ) -k0〈n〉(sˆ-sˆ+F - 2sˆ+Fsˆ- + Fsˆ-sˆ+) dt k0(〈n〉 + 1)(sˆ+sˆ-F - 2sˆ-Fsˆ+ + Fsˆ+sˆ-) (31) where k0 ) π|g|2G(ω0) gives the effective reaction rates. The quantities k0(〈n〉 + 1) and k0〈n〉 are naturally associated with the rates of the forward reaction (RP recombination) and the backward reaction (RP production from S0), respectively. These rates are in full accordance with the detailed balance principle:

k0〈n〉 ) exp(-βω0) k0(〈n〉 + 1)

(32)

If we assume that the energy spacing between the RP and the product state is much greater than kT, 〈n〉 ) 0 and k0〈n〉 f 0. This situation corresponds to the irreversible RP recombination. Additionally, in this case all deviations from harmonicity of the bath degrees of freedom become irrelevant, since only the ground state of the bath is occupied and the average population index, 〈n〉, is equal to zero. As a consequence, in the limit βω0 . 1 we obtain the sought equation:

dF ) -k0(sˆ+sˆ-F - 2sˆ-Fsˆ+ + Fsˆ+sˆ-) dt

(33)

After substituting the explicit form of the operators sˆ+ and sˆ- from eq 11 and returning back from the interaction representation, we obtain the following system:

{

dFTT dFSS ) -2k0FSS ; )0 dt dt dFTS dFST ) -k0FST ; ) -k0FTS dt dt dFS0S0 ) 2k0FSS dt

(34)

These equations are in full agreement with the Haberkorn equation (eqs 1 and 4) when we set kS ) 2k0. It is obvious that consideration of the irreversible triplet RP recombination channel will result in the generalized Haberkorn equation (eq 1). In the recent work by Il’ichov and Anishchik,13 this form of the equations has been confirmed by a rigorous derivation under

similar assumptions. Finally, we have to emphasize that when a different type of interaction of the system with the thermal bath is put into the model, the ratio k′/kS can become greater than 1/2. This would mean that the chemical process “measures” the RP spin state more efficiently than in our model. However, we believe that unless solid evidence for the faster dephasing is given, one should employ the conventional eq 1.It is important to emphasize that such additional decoherence is also absent in the alternative model proposed by Schulten et al.,16,17 which relies neither on the quantum measurement ideology nor on the Haberkorn approach. In this model, the recombination rates for the singlet and triplet RP channels, kS and kT, are taken the same and equal to kR and the density matrix is obviously factorized into the following product:

F(t) ) e-kRtF′(t)

(35)

Here the new density matrix F′(t) is not affected by the recombination processes and evolves only due to the magnetic interactions and spin relaxation in the RP. Thus, the contributions of the chemical reaction and spin dynamics can be factorized. The rates of producing the singlet and triplet reaction products are e-kRtF′SS(t) and e-kRtF′TT(t), respectively. This physically obvious result can be reproduced only by the Haberkorn approach, but not by other theories, which predict additional dephasing. The result of this dephasing is the presence of extra terms in the equation for F′(t) that come from the quantum measurement, i.e., separation of the chemical and spin dynamics becomes incomplete. However, when chemical reaction is nonselective with respect to spin (kS ) kT ) kR) the “measurements” do not resolve the RP spin state and therefore cannot produce the singlet-triplet dephasing rate higher than kR. Though being obvious, this result is reproduced only by the Haberkorn approach, which does not overestimate the singlettriplet dephasing rate. B. Numerical Simulations of the Quantum Mechanical Electron Transfer. The form of the reaction operator obtained can be substantiated by another approach. To confirm the Haberkorn equation, we will consider a radical pair in which recombination (e.g., back electron transfer) is purely quantum mechanically driven and the dynamics of the entire reactive system (radical pair and all product states) can be described by a single Hamiltonian. This model is applicable to rigidly linked donor-acceptor systems such as the photochemical reaction center or the CPF triad system discussed as the model of the magneto reception for bird navigation.18 Description of the reactive system by the Hamiltonian can, nevertheless, provide exponential decay of the RPs. It is well-known that the overlap of multiple quantum beats with the Lorentzian distribution of frequencies causes effectively monoexponential decay kinetics. In other words, the exponential decay does not directly mean the incoherent dynamics. It is important to note that even in the gas phase exponential decay kinetics has been observed in the intramolecular electron transfer reactions.19,20 In this subsection, we test three effects by purely quantum mechanical calculations based on the solution of the Schro¨dinger equation for the reactive system. First, we examine whether the interference of many parallel coherent processes provides apparently incoherent recombination and check whether the decoherence is more efficient in this approach as compared to the Haberkorn approach (eq 1).1 Second, we check whether there is interference between the electron spin mixing and the quantum mechanically treated RP recombination process. Third and finally, we consider whether it is possible to treat the quantum

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|ψ(t)〉 )

∑ cpn(t)|Pn〉 + cS(t)|S〉 + cT (t)|T0〉 0

n

Figure 2. Scheme and energy diagram of the model system used for the numerical calculation of the geminate recombination process of the RP.

Zeno effect by introducing an artificial observation operator that “measures” the product formation. We perform direct numerical calculation of the purely quantum mechanical dynamics of the RP coupled to the product states; the electron spin mixing process also is taken into account in this calculation. Let us consider the geminate recombination of the RP as an intramolecular radiationless transition process,21,22 which allows us to use the model proposed by Bixon and Jortner (BJ model).23 This model can treat both the intramolecular radiationless transitions, which proceed from the singlet radical pair to the vibronic states of the ground state S0 and the spin dynamics of the RP. The structure of the energy levels of the reactive system is shown in Figure 2. The entire system is composed of two spaces, P and R. The P space is formed by the vibronic states |Pn〉 of the reaction product and R is formed by the two electronic spin states, |S〉 and |T0〉, of the RP. For simplicity, we ignore the vibronic states of the RP. Although it is generally possible that the vibronic motion of the radical pair causes the electron spin relaxation of the RP15,24 we neglect this effect. Consideration of spin relaxation inside of the RP (R space) can be performed by means of the Redfield theory15 and is out of the scope of the present work as is has no direct relationship to the recombination process (transformation from R to P). In contrast to the previous subsection, the multiple degrees of freedom belong to the reactive system but not to the bath. As a consequence, in the notation of the space chosen, the entire space of the basis states is formed by the direct sum, RxP, and not by the direct product RXP. The total Hamiltonian of the reactive system is written as follows:

(39)

with the initial condition |ψ(t ) 0)〉 ) |S〉 corresponding to the singlet-born RP. First of all, we tested the BJ model calculation taking the parameter V ) 0. The other parameters were g ) 2.0 × 106 rad s-1, ∆ ) 1.0 × 106 rad s-1, and δ ) 0.3 × 106 rad s-1. As shown in Figure 3a, the time evolution of the singlet-state population in the RP, FSS(t) ) |CS(t)|2, converges to the monoexponential decay when N > 700. The decay kinetic rate constant can be obtained from a monoexponential fit, giving kS ) 2.5 × 107 s-1. This value is well-approximated by the Fermi golden rule:

kS )

2πg2 ∆

(40)

The calculation result with V ) 1.0 × 108 rad s-1 is shown in Figure 3b. The results are very similar to the solution of the modified Liouville equation with the Haberkorn recombination operator (eq 1) when kS ) 2.5 × 107 s-1. This clearly indicates that the quantum mechanical approach to the electron transfer

N/2

ˆ )H ˆP + H

∑

ˆ RP g(|Pn〉〈S| + |S〉〈Pn |) + H

(36)

n)-N/2

where N/2

ˆP ) H

∑

|Pn〉(n∆ + δ)〈Pn |

(37)

n)-N/2

ˆ RP ) |S〉V〈T0 | + |T0〉V〈S| H

(38)

Here V describes the coherent spin mixing in the RP, g is responsible for the transitions between the S state and the product states, ∆ is the splitting between the vibronic states. In the calculation, we numerically solved the time-dependent Schro¨dinger equation and obtained the time evolution of the wave function

Figure 3. Calculation results of the time evolution of FSS(t) ) |CS(t)|2 in the singlet-born RP. Subplot a shows dependence on the number, N, of the product states. Parameters of calculation: V ) 0, g ) 2.0 × 106 rad s-1, ∆ ) 1.0 × 106 rad s-1, and δ ) 0.3 × 106 rad s-1. Subplot b shows a comparison of the calculation (N ) 700) with the Haberkorn model (kS ) 2.5 × 107 s-1). For both cases, two curves for V ) 0 (decaying) and 1.0 × 108 rad s-1 (damped oscillations) are plotted. Subplot c shows the case V ) 1.0 × 109 rad s-1.

Spin-Selective Recombination of a Radical Pair

J. Phys. Chem. A, Vol. 114, No. 35, 2010 9453

reaction does not introduce any additional dephasing as compared to the Haberkorn operator (eq 1). In other words, the quantum mechanical approach based on the solution of the Schro¨dinger equation does not reproduce additional dephasing due to the quantum measurement, as proposed in the previous papers.4,7 This result is physically reasonable, because our calculation is performed on a pure state. Let us introduce the normalized density matrix of the RP space, FNR , in the following way

FR )

[

]

cScS* cScT0* , cT0cS* cT0cT0*

FRN )

FR trFR

)

FRN

]

(43)

This superoperator does not destroy the coherences among the basis sets |Pn〉 because the measurement discussed here resolves neither the vibronic product states nor the RP states. Thus, the “measurement” only monitors the formation of the reaction product. The time evolution of the density matrix in the interval from one measurement to another is described by the following expression

(41)

At any instant of time FRN satisfies the condition of the pure state:

(FRN)2

[

F ˆˆ F ) P 0 O 0 FR

ˆˆ (U ˆ (∆t)F(t)U ˆ +(∆t)) F(t + ∆t) ) O

(44)

ˆ (∆t) ) exp(-iH ˆ ∆t) U

(45)

where

(42)

It is only in the Haberkorn model that the cS and cT coefficients in eq 2 satisfy the relations given by eqs 41 and 42. This is in contrast to the other models, which rely on the quantum measurement ideology. The result of the calculation with higher rate of S-T mixing, V ) 1.0 × 109 rad s-1, is shown in Figure 3c. When kS < V, the results of the Haberkorn model with kS calculated by the Fermi golden rule and the numerical calculation are considerably different. This discrepancy can be easily explained by the quantum mechanical interference between the spin mixing in the RP, which is given by V, and the coupling to the product states, which is described by g. One should note that this difference is not important because the parameter kS is usually determined phenomenologically. Therefore, one can hardly recognize this effect, even when the spin mixing is due to a large hyperfine coupling constant or large external magnetic field (∆g effect). The calculation presented above has not taken the concept of quantum measurement into account. Moreover, the present calculation demonstrates that the simple quantum mechanical calculation reasonably explains the reaction dynamics of the RP and supports the Haberkorn model. This fact strengthens suspicion concerning the concept that the RP recombination reactions are quantum measurement-like processes. Seeking a model, which would substantiate the concept of quantum measurement used by Kominis4 or Jones and Hore,7 we attempted to modify our approach in the following two ways. One of the modifications is regarding vibronic relaxation of the product states. Model calculation using the effective Hamiltoˆ RP ) H ˆ P - iγE allows us to take this effect into account. nian25 H The calculation with modified Hamiltonian changes the effective recombination rate constant, kS, providing the irreversible recombination process without recurrence in the period of 1/∆. However, the time evolution of FR is nothing else but the solution of the Haberkorn model with the appropriate ks value substituted. The other modification is based on the concept of continuous artificial quantum measurements. If the measurement of the product formation is introduced, the collapse of the wave function into the space P or R will occur. The same concept was employed earlier by Sugawara for describing the quantum measurements in the laser spectroscopy.26 By analogy with this work, we used the following measurement superoperator:

In the calculation, we set ∆t ) 1 ns, and the other parameters were g ) 4 × 106 rad s-1, ∆ ) 5 × 105 rad s-1, δ ) 3 × 105 rad s-1, and N ) 2000. The results are shown in Figure 4. Dramatic deceleration of the decay of the singlet RP from 2 × 108 to 5.8 × 107 s-1 is seen at V ) 0. This deceleration can be regarded as the true “quantum Zeno effect” originating from the decoherence between the P and R spaces. However, this is the “Zeno effect” on the recombination kinetics but not on the spin mixing process. When V ) 1 × 108 rad s-1, the quantum beats seen in the decay kinetics of the |S〉 and |T0〉 states decay slowly and the total dynamics of FSS(t) can be very well approximated by the Haberkorn model with the parameter kS ) 5.8 × 107 s-1 obtained at V ) 0. Thus, the purely quantum mechanical calculations presented in this subsection completely confirm the Haberkorn model (eq 1) and do not support any of the approaches that are based on the quantum measurement concept. C. Zeno Effect. In the work of Kominis,4 it has also been claimed that the Haberkorn equations cannot reproduce the Zeno effect on the RP recombination. However, this statement is in obvious contradication with the earlier works5,6 and a recent one by Jones and Hore.7 Here we would like to stress once again that the conventional approach does reproduce the Zeno effect, moreover, it should be present at any nonzero dephasing rate. Let us imagine an RP reacting at a rate kS from its singlet state, whereas the dephasing rate is kST. Let the coupling matrix element between the singlet and the triplet states be V, and the

ˆˆ on FSS(t). Figure 4. The effect of the measurement superoperator O Details are discussed in the text. Parameters used: g ) 4 × 106 rad s-1, ∆ ) 5 × 105 rad s-1, δ ) 3 × 105 rad s-1, and N ) 2000.

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Ivanov et al.

splitting between the S and T levels be ∆. The evolution of the RP can be described by the following system of equations:

{

dFSS dt dFTT dt dFST dt dFTS dt

) iV(FST - FTS) - kSFSS ) -iV(FST - FTS)

(46) ) iV(FSS - FTT) - i∆FST - kSTFST ) -iV(FSS - FTT) + i∆FTS - kSTFTS

In the general case (meaning the arbitrary relation between kS and kST),it is rather difficult to solve this system analytically, however, we can easily estimate the apparent RP lifetime, 〈τ〉, in the following way:

〈τ〉 )

∫0

∞

(FSS(t) + FTT(t)) dt

(47)

This is equivalent to calculating the Laplace transform of (FSS(t) + FTT(t)) at the zero value of the Laplace variable. If the initial RP state is the triplet state, the apparent lifetime is equal to

〈τ〉 )

∆2 + kST2 2 + kS 2V2kST

(48)

or to a simpler expression

〈τ〉 )

(

kST kSkST 2 1 + ) 2+ 2 kS kS 2V 2V2

)

(49)

at ∆ ) 0. When the dephasing is fast, kST g 1/2kS . V, the RP lifetime becomes much longer than 1/kS, which is the essence of the Zeno effect. It arises from the fact that fast dephasing slows down the dynamic singlet-triplet transitions, even making them stochastic at high dephasing rates. Since the S-T conversion is the bottleneck of the recombination process for the triplet-born RP, fast dephasing results in prolongation of the RP lifetime. At the same time, the ratio between the recombination and dephasing rates is not of a primary importance, as the effect should be present for any sufficiently high dephasing rate. The effect can be explained in a rather simple way, as was done in ref 5. If the radical pair is triplet born and rapidly reacts from its singlet state, the recombination rate will be directly proportional to the rate of the singlet-triplet transitions. According to the Fermi golden rule, this rate is proportional to the square of the coupling matrix element, V, introduced above and the density of the final singlet state. As the singlet state is reactive, it becomes broadened by a value of kS. As a consequence, the density of states is equal to 1/kS and the rate of singlet-triplet transitions and the RP recombination rate becomes proportional to V2/kS. As a result, the RP lifetime becomes proportional to

〈τ〉 ∝

kS V2

(50)

which is consistent with eq 49 and can be much longer than 1/kS.

Figure 5. Manifestation of the quantum Zeno effect on the tripletborn RP; time evolution of FT0T0(t) ) |CT0(t)|2 is shown. Parameters used: g ) 1.5 × 108 rad s-1, ∆ ) 2.5 × 107 rad s-1, V ) 1.0 × 108 rad s-1, and N ) 3000. The inset shows the decay kinetics of the singletborn RP, FSS(t) ) |CS(t)|2, at V ) 0. The results of the Haberkorn and Jones-Hore models are obtained for ks ) 5.7 × 109 s-1.

The quantum Zeno effect on the RP recombination can also be demonstrated numerically by the BJ model. In the calculation, we have taken the following set of parameters: g ) 1.5 × 108 rad s-1, ∆ ) 2.5 × 107 rad s-1, N ) 3000. The calculation for the singlet-born RP with V ) 0 is shown in the inset of Figure 5 and kS ∼ 5.7 × 109 rad s-1. The comparison of the calculation results performed for the triplet-born RP with V ) 1.0 × 108 rad s-1 is shown in Figure 5. The result of the numerical calculation fits the Haberkorn model and the spin mixing from |T0〉 to |S〉 is about twice as fast as the Jones and Hore model and is similar to Figure 2 of ref 7. III. Conclusions In the present work, we have considered the problem of calculating the reaction operator for the RP, that undergoes spinselective recombination. The relevance of this work is conditioned by the paper from Kominis,4 who has claimed that the whole body of theory of spin chemistry might be wrong because of the improper form of the conventional recombination operator used. In our work, we have derived a theoretical approach to the spin-selective recombination of the RP, which is based on considering interactions of the reactive system with thermal bath. Our results agree with those of Anishchik and Il’ichov13 and fully confirm the Haberkorn approach. Moreover, our treatment gives evidence for having strong doubts in the validity of the master equation proposed by Kominis. The derivation of the equation, which describes the RP recombination, has been supported by numerical calculations done for the purely quantum mechanical model. On the basis of the Schro¨dinger equation, we have shown that the dynamic interaction of the RP with the numerous product quantum states results in the effectively monoexponential decay of the singlet state population accompanied by the decoherence of the singlet and triplet RP states. The ratio of the singlet decay rate and decoherence rate is in agreement with the Haberkorn model. Moreover, the reaction rate constant in the Haberkorn model can be obtained in a simple way by using the Fermi golden rule. Introducing the artificial quantum measurement does not change the form of the reaction operator, since only the effective reaction rate changes due the “quantum measurements”. Our treatment gives

Spin-Selective Recombination of a Radical Pair strong evidence that the Haberkorn equation is generally valid, as it has been confirmed by two independent approaches. Thus, rigorous averaging over the bath degrees of freedom or reaction product states yields the Haberkorn reaction operator instead of those from the quantum measurement theory. The only difference from the Haberkorn approach, which we can expect, is faster dephasing of the singlet-triplet coherence. This effect is generally possible and may arise from the peculiarities of the reaction event. However, as there are no examples of such reactions known in literature, we suppose that the Haberkorn reaction operator should be used unless there is a solid evidence for faster dephasing produced by chemical reaction. In this case, extra terms should be added to the Haberkorn recombination operator.3,7,13,27,28 We have also examined whether the quantum Zeno effect5,6 can arise from the fast “measurement” of the RP spin state caused by the chemical reaction. In contrast to the statement of Kominis,4 it turned out that the quantum Zeno effect can be reproduced not exclusively by his theory, but also by the Haberkorn theory1 and the Jones-Hore approach;7 moreover, it persists at any nonzero rate of the singlet-triplet dephasing induced by the RP recombiation. Acknowledgment. This work was supported by RFBR (project No. 09-03-91006-FWF), Program of the Division of Chemistry and Material Science RAS (Project No. 5.1.1), FASI (contract 02.740.11.0262), EPSRC, and the EMF Biological Research Trust. The authors are thankful to the organizers of the seminar “Quantum Measurement and Chemical Spin Dynamics” (Lorentz Center, Leiden, 15-19 March, 2010) for the opportunity to meet with colleagues and discuss this work. We also acknowledge Prof. P. J. Hore and Dr. C. R. Timmel (University of Oxford) for inspiring discussions. References and Notes (1) Haberkorn, R. Mol. Phys. 1976, 32, 1491.

J. Phys. Chem. A, Vol. 114, No. 35, 2010 9455 (2) Blum, K. Density Matrix Theory and Applications (Physics of Atoms and Molecules); Springer: New York, 1995. (3) Doktorov, A. B.; Neufeld, A. A.; Pedersen, J. D. J. Chem. Phys. 1999, 110, 8869. (4) Kominis, I. Phys. ReV. E 2010, 80, 56115. (5) Koptyug, I. V.; Lukzen, N. N.; Bagryanskaya, E. G.; Doktorov, A. B. Chem. Phys. Lett. 1990, 175, 467. (6) Berdinskii, V. L.; Yakunin, I. N. Dokl. Phys. Chem. 2008, 421, 163. (7) Jones, J. Hore, P. Chem. Phys. Lett. 2010, 488, 90. (8) Salikhov, K. M., Molin, Y. N., Sagdeev, R. Z., Buchachenko, A. L. Spin Polarization and Magnetic Effects in Chemical Reaction; Amsterdam: Elsevier, 1984. (9) Ivanov, K. L.; Lukzen, N. N.; Vieth, H.-M.; Grosse, S.; Yurkovskaya, A. V.; Sagdeev, R. Z. Mol. Phys. 2002, 100, 1197. (10) Doktorov, A. B.; Purtov, P. A. SoV. Chem. Phys. 1987, 6, 484. (11) Neufeld, A. A.; Purtov, P. A.; Doktorov, A. B. Chem. Phys. Lett. 1997, 273, 311. (12) Lukzen, N. N.; Ivanov, K. L.; Morozov, V. A.; Kattnig, D. R.; Grampp, G. Chem. Phys. 2006, 328, 75. (13) Il’ichov, L. V.; Anishchik, S. V. arXiV 2010, 1003.1793v1. (14) Zubarev, D.; Morozov, V.; Repke, G. Statistical Mechanics of Nonequilibrium Processes; Akademie-Verlag: Berlin, 1996. (15) Redfield, A. G. In AdVances in Magnetic Resonance; Waugh, J. S., Ed.; Academic Press: New York: 1965; Vol. 1, p 1. (16) Schulten, K.; Wolynes, P. G. J. Chem. Phys. 1978, 68, 3292. (17) Knapp, E. W.; Schulten, K. J. Chem. Phys. 1979, 71, 1978. (18) Maeda, K.; Henbest, K.; Cintolesi, F.; Kuprov, I.; Rodgers, C.; Liddell, P.; Gust, D.; Timmel, C.; Hore, P. J. Nature 2008, 387. (19) Jortner, J.; Bixon, M.; Wegewijs, B.; Verhoeven, J.; Rettshnick, R. Chem. Phys. Lett. 1993, 205, 451. (20) Syage, J. A.; Felker, P. M.; Zewail, A. H. J. Chem. Phys. 1984, 81, 2233. (21) Bixon, M.; Jortner, J. J. Phys. Chem. 1993, 97, 13061. (22) Jortner, J.; Bixon, M. Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 1993, 234, 29. (23) Bixon, M.; Jortner, J. J. Chem. Phys. 1968, 48, 715. (24) Hayashi, H.; Nagakura, S. Bull. Chem. Soc. Jpn. 1984, 57, 322. (25) Ziv, A. J. Chem. Phys. 1978, 68, 152. (26) Sugawara, M. J. Chem. Phys. 2005, 123, 204115. (27) Shushin, A. Chem. Phys. Lett. 1991, 181, 274. (28) Fukuju, T.; Yashiro, H.; Maeda, K.; Murai, H. Chem. Phys. Lett. 1999, 304, 173.

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