Recombination yield of geminate radical pairs in high magnetic fields: general results and application to free diffusion

Chemical Physics 260 (2000) 125±142

www.elsevier.nl/locate/chemphys

Recombination yield of geminate radical pairs in high magnetic ®elds: general results and application to free di€usion Martin J. Hansen, Anatole A. Neufeld 1, Jùrgen Boiden Pedersen * Fysisk Institut, SDU-Odense Universitet, Campusvej 55, DK-5230 Odense M, Denmark Received 22 May 2000

Abstract We have derived a general and exact expression for the quantum yield of geminate radical pairs (RPs) in high magnetic ®elds. We consider the dynamical S±T0 mixing caused by di€erences in Zeeman and hyper®ne interactions, intraradical (T1 and T2 ) relaxation, and a ®rst-order scavenging process. The recombination process and the exchange interaction are assumed to be isotropic and of the contact type. In order to solve the stochastic Liouville equation, we introduce e€ective methods to minimize the dimensionality of the problem and to calculate the matrix elements of the superoperators in an arbitrary coupled basis. Our general expression is expressed in terms of the classical Green's function for the relative motion of the radicals, and it is therefore valid for any kind of relative motion. Various limiting cases of the general expression, which are of particular interest, are derived. Explicit analytical formulas for freely di€using RPs are given, and the high accuracy of the expressions is illustrated by comparison with accurate numerical results. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction The fact that magnetic ®elds have an in¯uence on the quantum yield of radical reactions has been known for more than 30 years. The physical processes leading to the e€ect are well known, and a completely satisfactory theoretical description can be based on the stochastic Liouville equation (SLE) [1]. Accurate numerical methods for solving the SLE have been developed [2]. Numerical results have provided much information about the physical processes and various parameter depen*

Corresponding author. Fax: +45-66-15-87-60. E-mail address: [email protected] (J.B. Pedersen). 1 On leave from the Institute of Chemical Kinetics and Combustion, SB RAS, 630090 Novosibirsk, Russian Federation.

dences. From a theoretical point of view, the numerical results are useful for testing simpli®ed models and for investigating the accuracy of analytic results. However, it is clear that analytic solutions may provide more insight into the physical processes and also facilitate analysis of experimental data. However, there are relatively few analytic expressions, and those available are either approximate with a small region of applicability or are valid only for very special systems. Thus, at present, only numerical methods can be used for all but the simplest systems. The quantitative theoretical methods may be roughly divided into the following categories: reencounter methods [3,4], approximate solutions to the SLE [5±7], and exact solution to a special or approximate SLE [8]. In most of these methods, the exchange interaction is either neglected, or its

0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 2 5 0 - 0

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M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

spatial dependence is approximated by a delta function. Both the present and the previous works consider only high magnetic ®elds, where the nuclear spin state is conserved and thus appears in the calculation as a magnetic parameter. A Green's function formalism for calculation of recombination probabilities of geminate pairs was developed in Ref. [7]. Green's function in the presence of recombination was expressed in terms of quantities calculated using Green's function without recombination, and these quantities were evaluated by use of a decoupling procedure over di€erent shapes of the recombination and exchange zones. However, the accuracy of the decoupling procedure was not investigated and no formal justi®cation of the procedure was given. It has later been shown that this decoupling procedure does not work for CIDEP problems [9]. Moreover, the decoupling procedure gives rise to rather cumbersome calculations and an explicit expression was only given for the simplifying case of strong exchange. It is doubtful, however, whether it is applicable in the strong exchange limit. In principle, it might be possible to use the method to a more general case, but the calculations will be signi®cantly more dicult than those presented below using our variant. We are only considering isotropic interactions, and the contact approximation appears to be well justi®ed for a calculation of recombination probabilities when the exchange interaction is locally weak. We will assume that the exchange interaction and the recombination operator are both isotropic and that they can be described by a contact approximation, i.e. the spatial dependence is given by a d-function in the relative separation between the radicals. The strength of the exchange interaction is crucial for the development of electron spin polarization e€ects. But for the recombination process the exchange interaction only causes a dephasing and prevents singlet±triplet mixing, i.e. an excluded volume e€ect. The dephasing e€ect is well described by a contact approximation but the excluded volume e€ect is not. However, the excluded volume e€ect is only signi®cant for a strong exchange interaction and for intermediate values of the singlet±triplet mixing parameter. There are experimental indications that intermediate strengths

of the exchange interaction are common, and a contact approximation for the exchange interaction may thus be expected to describe the quantum yield quite satisfactorily even for large values of the mixing parameter. The present method is completely general with respect to the character of the relative motion of the radicals. It is valid for any di€usion model and may also be used for deterministic motion. The derived expressions for the quantum yield are completely expressed in terms of the Laplace transform of Green's function of the relative motion evaluated for the Laplace variable equal to some combinations of the characteristic parameter values (magnetic parameters, di€usion constant, etc). Green's function contains all information about the relative motion of the radical pair (RP). The determination of the appropriate Green's functions for a particular case is a separate problem, which has been solved for a few useful cases [10±13]. By substituting the relevant Green's functions into the general formula, we immediately obtain an analytical expression for the recombination probability. We apply the general formula to the free di€usion case. The Green's function for free di€usion is both well known and relatively simple. It is thus a perfect case for illustration of the structure of the formula. Moreover, it has signi®cant practical value as the radical motion in ordinary liquids is believed to be described quite accurately by the free di€usion model. The high accuracy and the large region of applicability of the derived equations are illustrated by comparison with numerical results.

2. Model description The most consistent way of calculating the recombination yield of RPs under in¯uence of magnetic ®elds and other spin e€ects is the density matrix approach. For the problem under consideration, it is convenient to solve the Laplace transformed SLE [1] that can be written as i h ^^ ^^ ÿ L qL …r; s† ‡R s ‡ ksc ‡ iH …r† ‡ K…r† r ˆ q0 …r†;

…1†

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

where qL …r; s† ˆ

Z

1

0

q…r; t† eÿst dt

…2†

is the Laplace transformed density matrix, q0 …r†, the initial density matrix of the RP, and r, the relative separation of the two radicals. The possible scavenging of the radicals by homogeneously distributed traps is described by a pseudo-®rstorder reaction with the rate constant ksc . H …r† is the standard commutator type superoperator that includes both the spin Hamiltonian of the free radicals and the exchange interactions between the ^^ ^ ^ describes the spin-selective radicals. K…r† and R recombination and the intraradical relaxation, respectively. The relative motion of the radicals is speci®ed by the linear functional operator Lr . For example, we can describe rectlinear motion with constant velocity ~ v by ~ v
r Lr ˆ ÿ~ while Lr ˆ

…3†

   1 d d 1 dU …r† 2 D…r†r ‡ r2 dr dr kT dr

…4†

describes the relative di€usion of the radicals in an isotropic potential, U …r†, which could include the Coulomb interaction or solvation e€ects. D…r† is the relative di€usion coecient, k, the Boltzmann constant, and T, the absolute temperature. All superoperators, that are also known as Liouville operators, in Eq. (1), are tensors of rank 4, and their operation on the density matrix is given by X ^^ Lik;lm qlm : …5† . ˆ Lq; .ik ˆ l;m

The product of two superoperators is de®ned as X ^^ ^^ ^^ ˆ A Aik;pq Bpq;lm : …6† L B; Lik;lm ˆ p;q

It is necessary to use the Liouville space to include the intraradical relaxation processes, which cannot be represented by a commutator or anticommutator superoperator generated by a Hilbert space operator and the density matrix.

127

The matrix elements of a commutator type superoperator are expressed in terms of the matrix elements of the corresponding Hamiltonian by the well-known formula [14] H ik;lm ˆ Hil dkm ÿ Hmk dil :

…7†

As noted before, we will exclusively consider the high ®eld limit where the free spin Hamiltonian, which includes the Zeeman e€ect and the hyper®ne interactions, can be written as H^0 ˆ xA S^ZA ‡ xB S^ZB ;

…8†

where xm ˆ gm bB ‡

X j2m

…m†

…m†

Aj m j

…m ˆ A; B†

…9†

are the ESR frequencies of the two radicals (A and B) for a nuclear spin con®guration given by the …m† quantum numbers fmj g; the corresponding hy…m† per®ne constants are denoted by fAj g. We use frequency units and have set h ˆ 1. Note that the spin Hamiltonian conserves the nuclear spin projection on the direction of the external magnetic ®eld. The recombination probability is thus calculated by averaging over the recombination probabilities of all nuclear spin con®gurations. The only interradical spin±spin interaction we consider in the present work is the exchange interaction that is described by the Hamiltonian   …10† J^…r† ˆ ÿJ …r† 12 ‡ 2S^A
S^B ; where J…r† is the exchange integral, and S^A and S^B are the spin operators for the unpaired electrons on radicals A and B. We assume that only RPs of singlet character can recombine, which implies that the recombination superoperator is given by K…r†ik;lm ˆ


KS …r† ÿ S Pmk dli ‡ PilS dkm ; 2

…11†

where dik is the Kronecker delta, KS …r† is the intrinsic decay rate through the singlet channel, and P^S ˆ jSihSj is the projection operator on the singlet state.

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M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

We assume that the individual electron spins relax independently, i.e. ^^ : ^ ^^ ‡ R ^ˆR R A B

…12†

Furthermore, we assume that the relaxation can be described phenomenologically by Bloch equations [15]. When the radicals are separated, i.e. noninteracting, the evolution of the electron spins are governed by the internal radical parameters and the external magnetic ®eld and hence o SZ …t† ˆ ÿ T1ÿ1 SZ …t†; ot ÿ
o S‡ …t† ˆ ix ÿ T2ÿ1 S‡ …t†; ot
ÿ o Sÿ …t† ˆ ÿ ix ÿ T2ÿ1 Sÿ …t†; ot

…13†

where T1 and T2 are the longitudinal and transversal relaxation times and x is the e€ective Larmor frequency of the radical in question, given by Eq. (9). It is convenient to separate the superoperators into two groups. The short range terms that are e€ective only when the radicals are close together (i.e. the recombination process and the exchange interaction), and the spatially independent terms that are independent of the distance between the radicals. We thus de®ne …14†

^^ ^^ Q ˆ ÿH 0 ‡ iR:

…15†

The Laplace transformed SLE can now be written in the form used for the calculations: i ^^ ^^ …r† ÿ L ^ r qL …r; s† ˆ q0 …r†: ÿ iW s ‡ ksc ÿ iQ

The rate of recombination is given by Z  ^ 3 _ ^ K…r†q…r; t† d r R…t† ˆ Tr V

and the total recombination probability is

1

Z

^^ K…r†q…r; t† dt d3 r 0 V Z  ^ 3 L ^ K…r†q …r; 0† d r : ˆ Tr



V

…18†

3. Green's function formulation In accordance with previous works [16,17] we de®ne the ``free Green's operator'' by   d…r ÿ r0 † ^^ ^^ ^^ 0 ^r G ÿL 1; …19† s ‡ ksc ÿ iQ 0 …r; r ; s† ˆ 4prr0 which describes the motion of the RP in the absence of recombination and exchange interaction. ^ Here ^1 is the unity superoperator in the Liouville space that has the matrix elements 1ik;lm ˆ dil dkm :

…20†

Using Eq. (19), we can transform SLE (16) into an equivalent integral equation Z ^^ 3 0 L 0 0 G q …r; s† ˆ 0 …r; r ; s†q0 …r †d r V Z ^^ ^^ 0 L 0 3 0 0 ‡ G …21† 0 …r; r ; s†iW …r †q …r ; s†d r : V

By using the identity h i X Jii;lm …r†qLlm …r; s† ˆ 0; Tr J^^…r†qL …r; s† ˆ

…22†

i;l;m

^^ …r† ˆ J^^…r† ‡ iK…r†; ^^ W

h

Z

R ˆ Tr

which simply says that the exchange interaction cannot give rise to a disappearance of radicals, and Eq. (18) can be rewritten as  Z   ^ ^ 3 L ^ ^ K…r† ÿ iJ …r† q …r; 0†d r R ˆ Tr   ZV ^^ …r†qL …r; 0†d3 r ˆ ÿTr‰r…0†Š; ˆ Tr ÿ iW V

…16†

…23† where we have de®ned

…17†

^^ …r†qL …r; s†; r…r; s† ˆ iW Z r…s† ˆ

V

r…r; 0†d3 r:

…24† …25†

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

^ ^ …r†, we obtain By multiplying Eq. (21) by iW Z ^^ ^^ …r†G 3 0 0 0 iW r…r; s† ˆ 0 …r; r ; s†q0 …r †d r V Z ^ ^^ …r†G ^ 0 …r; r0 ; s†r…r0 ; s†d3 r0 : ‡ iW …26† V

Eqs. (23)±(26) are the foundation on which the following calculations are based.

129

for the Green's superoperator evaluated at the distance of closest approach d. This is known as the ``contact''-value of Green's superoperator and is the only value needed for the calculations. The superoperator is represented by a 16  16 complex matrix since the density matrices are represented by column vectors with 42 ˆ 16 elements. Consequently, Eq. (30) represents 16 linear equations, but fortunately, they are not all coupled.

4. Contact approximation

4.1. Dimensionality of the problem

If both the exchange and the recombination interactions decrease fast with respect to the distance between the radicals, then the contact approximation is useful. The integral strength of ^^ …r† is W Z 1 ^^ …r†dr: ^ ^ 4pr2 W …27† W ˆ

We will now determine the number of linear equations that we need to solve in order to calculate the recombination probability. It is convenient to use the singlet±triplet basis. We have numbered the basis states in the following manner:

d

In the present work, we will only consider the contact approximation, and we approximate the spatial dependence of both KS …r† and J …r† by delta functions at the distance of closest approach d, i.e. ^^ …r†  W ^^ d…r ÿ d† : W 4prd

…28†

^ ^ is complex; the Note that, according to Eq. (14), W ^, and real part is due to the exchange interaction J^ the imaginary part is due to the recombination ^^ superoperator K. For simplicity, we will also assume that the radicals are created at the distance of closest approach, i.e. q0 …r† ˆ q0

d…r ÿ d† : 4prd

…29†

If we introduce the contact approximation (28) and the initial condition (29) into Eq. (26) and use Eq. (25), then we obtain a set of algebraic equations for r…s† ^^ ^ ^^ G ^ ^^ r…s† ˆ iW 0 …s†q0 ‡ iW G0 …s†r…s†:

…30†

Here we have used the abbreviated notation ^^ ^^ G 0 …s†  G0 …d; d; s†

…31†

j1i ˆ jSi; j3i ˆ jT‡ i;

j2i ˆj T0 i; j4i ˆj Tÿ i:

…32†

^^ is diagonal. Furthermore, the diIn this basis, W agonal elements Wlm;lm 6ˆ 0 if and only if l or m is equal to 1, cf. Eqs. (7), (10) and (11). This reduces the problem to a maximum of seven coupled linear equations. The recombination probability (23) has a particularly simple form in this basis, R ˆ ÿr11 …0†;

…33†

and therefore we only need the elements of r that are coupled with r11 . Eq. (30) shows that the couplings between di€erent elements of r are de^^ ^^ G termined by the block structure of W 0 …s†. Since ^ ^ is diagonal, we only need to ®nd the block W ^^ structure of G 0 …s†. This structure is determined by ^^ Q and can be found from Bloch equations (13) as indicated below. Neither H 0 nor the transverse relaxation can change the Z-component of the total electron spin (SZ ˆ SZA ‡ SZB ) of the RP. The longitudinal relaxation mixes the pure states jabi, jbai, jaai, and jbbi. This means that the only matrix elements of r ^^ that are coupled with r11 via G 0 are r12 , r21 , r22 , r33 , and r44 . However, r22 , r33 , and r44 are identically zero which follows from Eq. (30) since Eqs. ^^ (7), (10) and (11) give W jj;lm ˆ 0 for j 2 f2; 3; 4g.

130

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

Thus, we only need to solve three coupled linear equations in r11 , r12 , and r21 . Eq. (30) can be formally solved to ^^ G ^^ W ^^ 0 …0†q0 ; …34† r…s† ˆ Ai ^^ where A is de®ned by   ^^ ^^ ^^ ^^ ^^ G …35† A 1 ÿ iW 0 …0† ˆ 1: The recombination probability can then be written as X ÿA11;lm iWlm;lm G0 lm;rs …0†…q0 †rs : …36† Rˆ lmrs

If we assume that the initial state is a statistical mixture of S, T0 , T‡ and Tÿ states and use the information about the matrix elements discussed above, we can write X ÿA11;11 iW11;11 G0 11;rr …0†q0 rr Rˆ r

ÿ

X r

ÿ

X r

A11;12 iW12;12 G0 12;rr …0†q0 rr A11;21 iW21;21 G0 21;rr …0†q0 rr ;

…37†

^ ^ are given by where the matrix elements of W

pq

ˆ A11;lm ÿ A11;11 iW11;11 G0 11;lm …0† ÿ A11;12 iW12;12 G0 12;lm …0† ÿ A11;21 iW21;21 G0 21;lm …0†;

…40†

^^ where the block-diagonal form of A has been used. It is seen that lm 2 f11; 12; 21g give rise to three coupled linear equations in the elements A11;11 , A11;12 , and A11;21 . Thus, in order to determine the recombination ^^ probability, we need a total of 18 elements of G 0 because the above calculation of A included six ^^ elements of G 0 that did not appear in Eq. (37). ^^ 4.3. Calculation of G 0 The most ecient way to calculate the matrix elements of the free Green's superoperator is to use the shift operators

iW11;11 ˆ ÿKS ; iW12;12 ˆ ÿKS =2 ÿ 2iJ ; iW21;21 ˆ ÿKS =2 ‡ 2iJ ;

^^ ^^ G Eq. (35) that the block-diagonal form of iW 0 …0† ^^ implies that A is also block diagonal. Eq. (37) shows that we need to calculate the three matrix elements A11;11 , A11;12 , and A11;21 . For this purpose, it is convenient to use Eq. (35) to write X A11;pq iWpq;pq G0 pq;lm …0† 111;lm ˆ A11;lm ÿ

…38†

KS and J are the integrated values of the recombination rate and the exchange interaction, respectively, i.e. Z 1 4pr2 K…r†dr; KS ˆ d Z 1 …39† 2 4pr J …r†dr: Jˆ d

The remaining problem is to evaluate three ele^^ ^ ^ 0 …0†. ments of A and 12 elements of G ^^ 4.2. Calculation of A ^ ^ ^ 0 …0† is block diagonal ^G As discussed above, iW and the interesting block is the one that contains ^^ ^^ G the elements ‰iW 0 …0†Šjk;lm , where jk 2 f11; 12; 21g and lm 2 f11; 12; 21; 22; 33; 44g. It follows from

P^ik ˆ jiihkj;

…41†

which, for an orthonormal basis (as the singlet± triplet basis), has the property P^ik P^lm ˆ dkl P^im :

…42†

The Laplace transform of the free Green's operator of Eq. (19) may be written as Z 1 ^^ 0 u…r; tjr0 †T^^ 0 …t† eÿ…s‡ksc †t dt; …43† G 0 …r; r ; s† ˆ 0

where T^^ 0 …t† is the free propagator. The scavenging rate has been incorporated in the Laplace transform parameter for convenience, and this step is possible because ksc appears in the same way as s in Eq. (19) that de®nes the free Green's operator. u…r; tjr0 † is the conditional probability density for the RP to be separated by r at time t given that their separation was r0 at t ˆ 0. The latter quantity satis®es the equation

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142



 0 o ^ r u…r; tjr0 † ˆ d…t† d…r ÿ r † ÿL 0 ot 4prr

…44†

and its Laplace transform will be denoted as Z 1 0 u…r; tjr0 † exp…ÿst†dt: …45† g0 …r; r ; s† ˆ 0

Eq. (43) reduces the calculation of the matrix elements of the Green's operator to a calculation of the matrix elements of the free propagator. These can conveniently be derived from the spectral resolution X T0 ik;lm …t†P^il q…0†P^mk ; …46† q…t† ˆ T^^ 0 …t†q…0† ˆ ik;lm

ik;lm

ˆ

X lm

By use of these relations, we express the shift operators of the RP states in the singlet±triplet basis in terms of the single spin operators S^Z , S^‡ , and S^ÿ . The resulting expressions are given in Appendix A, which also contains the inverse relations where the single spin operators are written in terms of the shift operators in the singlet±triplet basis. To illustrate the procedure, we will show the derivation of P^11 in some detail P^11 ˆ jSihSj ˆ p12…jabi ÿ jbai†p12…habj ÿ hbaj† h ˆ 1 …1 ‡ S^ZA †…1 ÿ S^ZB † ÿ S^‡A S^ÿB 2

where q is the spin density matrix of the separated (free) RP. The validity of Eq. (46) may easily be veri®ed by taking matrix elements of the equation X T0 ik;lm …t†hnjiihljq…0†jmihkjji qnj …t† ˆ T0 nj;lm …t†qlm …0†:

…47†

The time-dependent expectation values of the shift operators can be written as h i P nj …t† ˆ Tr P^nj q…t† h i X T0 ik;lm …t†Tr P^nj P^il q…0†P^mk ˆ ik;lm

h i X T0 ik;lm …t†Tr P^mk P^nj P^il q…0† ˆ

P^ab ˆ S^‡

i

ˆ 14 ÿ 12‰S^‡A S^ÿB ‡ S^ÿA S^‡B Š ÿ S^ZA S^ZB :

…50†

In this calculation, it is convenient to assume that the initial density matrix is a direct product of the spin density matrices of the individual radicals. It is important to emphasize that the chosen type of initial state does not introduce any assumptions into T^^ 0 since this quantity obviously does not depend on the chosen initial state. The time dependence of the expectation values of the single spin operators are found by solving the Bloch equations (13). Inserting the results into Eq. (50) yields

ÿ 12‰e…iQÿk2 †t S‡A …0†SÿB …0†

…48†

Thus, if the initial conditions are given in terms of shift operators, the matrix element of the evolution operator T0 jn;lm …t† is the coecient to P ml …0† in the equation for P nj …t†. Our calculation of the matrix elements is based on the shift operators for a single electron spin. As usual, we designate the states jai ˆ `spin up' and jbi ˆ `spin down'. It is easily shown that the four shift operators for a single spin can be expressed in terms of the single spin operators as P^bb ˆ 12 ÿ S^Z

2

ÿ S^ÿA S^‡B ‡ …12 ÿ S^ZA †…12 ‡ S^ZB †

lm

P^aa ˆ 12 ‡ S^Z

2

P 11 …t† ˆ 14 ÿ eÿk1 t SZA …0†SZB …0†

ik;lm

X T0 jn;lm …t†P ml …0†: ˆ

131

P^ba ˆ S^ÿ : …49†

‡ e…ÿiQÿk2 †t SÿA …0†S‡B …0†Š;

…51†

where Q ˆ …xA ÿ xB †=2

k1;2 ˆ 1=T1;2A ‡ 1=T1;2B : …52†

Note that this calculation would have been more cumbersome had we not chosen the convenient initial state. Using the inverse relations (A.6) and expressing the right-hand side of Eq. (51) in terms of the initial values of the shift operator yields

132

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

 1

P 11 …t† ˆ 4 1 ‡ eÿk1 t ‡ e…iQÿk2 †t   ‡ e…ÿiQÿk2 †t P 11 …0† ‡ 14 1 ‡ eÿk1 t  ÿ e…iQÿk2 †t ÿ e…ÿiQÿk2 †t P 22 …0†   ÿ 14 e…iQÿk2 †t ÿ e…ÿiQÿk2 †t P 12 …0†    ÿ P 21 …0† ‡ 14 1 ÿ eÿk1 t P 33 …0†  ‡ P 44 …0† :

Rˆ

…53†

In order to write the subsequent expressions in a more compact form, we introduce ``e€ective'' relaxation rates including both intraradical relaxation and scavenging k~1;2 ˆ ksc ‡ k1;2 ;

…55† …56†

are the longitudinal relaxation rates for the individual radicals and the parameters k1;2 are given by Eq. (52). By using Eqs. (43), (45) and (54) we get h h ii G0 11;11 …0† ˆ 14 g0 …ksc † ‡ g0 …k~1 † ‡ 2 Re g0 …k~2 ÿ 2iQ† ;

The functions a0 and b0 are de®ned as h i a0 ˆ Re g0 …k~2 ÿ 2iQ† ; h i b0 ˆ Im g0 …k~2 ÿ 2iQ† ;

…60†

k~1;2 and Q are de®ned by Eqs. (52) and dimensionless parameter g is indepenand is given by   KS p 4J 2 1‡ 2 …61† gˆ1‡ 2 KS where the (55). The dent of Q

with

h i p ˆ 12 g0 …k~1A † ‡ g0 …k~1B † :

F ˆ

1 ; 1 ‡ … p ‡ a0 g†KS =2

U1 ˆ g0 …ksc † ‡ g0 …k~1 † ‡ 2a; U2 ˆ g0 …ksc † ‡ g0 …k~1 † ÿ 2a;

…62†

and

…58†

V1 ˆ ÿV2 ˆ KS b0 g; h i V3 ˆ ÿV4 ˆ J g0 …k~1A † ÿ g0 …k~1B † :

We only need to perform three calculations like the one in Eqs. (50)±(53) to ®nd all the needed ^^ elements of G 0 , i.e. G0 11;lm , G0 12;lm and, G0 21;lm . The elements involved in the calculations are listed in Appendix B.

5. General solutions The procedure outlined above gives the following general expression for the recombination yield:

…63†

…64†

U3 ˆ U4 ˆ g0 …ksc † ÿ g0 …k~1 †;

…57†

where g0 …s† ˆ g0 …d; d; s†:

n ‰Un ‡ b0 FVn Šq0 nn h i: g0 …ksc † ‡ g0 …k~1 † ‡ 2a0 ‡ KS b20 F g

The other quantities are

where k1A;B ˆ 1=T1A;B

1‡

KS 4

P

…59†

According to Eq. (48), this equation contains the time dependence of all matrix elements that can be written as T0 11;lm . For example, it is immediately seen that   T0 11;11 …t† ˆ 14 1 ‡ eÿk1 t ‡ e…iQÿk2 †t ‡ e…ÿiQÿk2 †t : …54†

k~1A;B ˆ ksc ‡ k1A;B ;

KS 4

…65†

This rather complicated looking expression simpli®es considerably when applied to the usual special cases. Note that all quantities are expressed in terms of the reduced free Green's function for the relative motion g0 …s†, cf. Eq. (58). For a singlet precursor, Eq. (59) reduces to RS ˆ

c1 g0 …ksc † ‡ g0 …k~1 † ‡ 2M…Q† ; 2 1 ‡ c1 M…Q†

…66†

and for an unpolarized triplet precursor Eq. (59) gives

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

RT ˆ

c1 3g0 …ksc † ÿ g0 …k~1 † ÿ 2M…Q† ; 6 1 ‡ c1 M…Q†

…67†

133

5.1. Fast relaxation (T2ÿ1  Q)

where only the function M…Q† is a€ected by the S±T0 mixing parameter Q. This function is ÿ
a0 ‡ Y a20 ‡ b20 ; …68† M…Q† ˆ 1 ‡ Ya0

In the limit of fast relaxation where T2ÿ1  Q, we are in e€ect looking at the limit where Q ˆ 0 (remember though that our results are only valid in the high ®eld limit). Note that the recombination yield become independent of the exchange interaction in this limit and that M…Q† simpli®es to

where

M…0† ˆ g0 …k~2 †:

c1 ˆ

KS =2 h i. ; 1 ‡ KS g0 …ksc † ‡ g0 …k~1 † 4

…69†

Y ˆ

KS g=2 ; 1 ‡ KS p=2

…70†

and the functions p and g are de®ned in Eqs. (61) and (62). In the case of no exchange interaction, Eq. (68) is considerably simpli®ed and gives ÿ
a0 ‡ KS a20 ‡ b20 =2 : …71† M…Q† ˆ 1 ‡ KS a0 =2 Eqs. (66) and (67) are not independent but are connected through the relation RS ˆ K…ksc † ÿ 3RT ‰1 ÿ K…ksc †Š;

…72†

K…ksc † ˆ

The recombination probability for an unpolarized triplet initial state (67) becomes equal to h i KS ~ ~ 3g …k † ÿ g … k † ÿ 2g … k † 0 sc 0 1 0 2 12 h i: …75† RT ˆ KS 1 ‡ 4 g0 …ksc † ‡ g0 …k~1 † ‡ 2g0 …k~2 † The recombination probability for a singlet precursor can be obtained from Eq. (72). 5.2. Di€usion controlled recombination For di€usion controlled recombination through the singlet channel (KS ! 1), Eq. (68) gives
1ÿ 2 a0 ‡ b20 …76† M…Q† ˆ a0 since Y ! 1 in this limit according to Eqs. (61) and (70). Eqs. (66) and (67) are thus simpli®ed to RSKS !1 ˆ 1;

where KS g0 …ksc † 1 ‡ KS g0 …ksc †

…73†

is the classical recombination probability in the presence of a scavenger. In our case, it corresponds to the recombination probability for a singlet RP initially together in the absence of any S±T mixing and moving in a medium with a homogeneous concentration of scavengers. In the absence of scavengers (ksc ˆ 0) this quantity is equal to the usual recombination probability K. In the following, we will mainly give results for a triplet precursor. The corresponding results for a singlet precursor can easily be derived from Eq. (72). In order to illustrate the implications of the general expressions (66) and (67), we will now consider several useful limiting cases.

…74†

and RTKS !1

…77†

ÿ
1 3g0 …ksc † ÿ g0 …k~1 † ÿ 2 a20 ‡ b20 =a0 ˆ : 3 g0 …ksc † ‡ g0 …k~1 † ‡ 2…a20 ‡ b20 †=a0 …78†

5.3. In®nitely strong exchange In the limit J =KS ! 1, M…Q† becomes identical to Eq. (76), and thus, Eq. (67) can be written as h ÿ
i KS 3g0 …ksc † ÿ g0 …k~1 † ÿ 2 a20 ‡ b20 =a0 12 h i RTJ !1 ˆ 1 ‡ K4S g0 …ksc † ‡ g0 …k~1 † ‡ 2…a20 ‡ b20 †=a0 …79† which was also found in Ref. [7].

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M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

5.4. No relaxation In the absence of intraradical relaxation (k1 ˆ k2 ˆ 0), Eq. (67) simpli®es to  ÿ
 c1 g0 …ksc †…1 ‡ Ya0 † ÿ a0 ‡ Y a20 ‡ b20 T ; R ˆ 3 1 ‡ Ya0 ‡ c1 ‰a0 ‡ Y …a20 ‡ b20 †Š …80† KS =2 ; 1 ‡ KS g0 …ksc †=2    KS =2 KS g0 …ksc † 4J 2 1‡ 1‡ 2 Y ˆ : 1 ‡ KS g0 …ksc †=2 2 KS

c1 ˆ

…81† 6. Freely di€using radical pair The free Green's function, de®ned in Eq. (45), is well known for freely di€using RPs [7]. For this case the ``contact'' value of the free Green's function as de®ned in Eq. (58) is 1 1 p ; KD 1 ‡ d s=D

KD ˆ 4pdD;

…82†

where KD is the di€usion controlled rate constant, d, the distance of the closest approach, and D, the relative di€usion coecient. Substituting this function into Eq. (60) and separating the real and imaginary parts give p 1 1 ‡ X‡ p ; …83† a0 ˆ KD 1 ‡ 2 X‡ ‡ 2X b0 ˆ

p 1 Xÿ p ; KD 1 ‡ 2 X‡ ‡ 2X

where the dimensionless parameters are q X ˆ X  k2 =2; X ˆ q2 ‡ k22 =4;

RT ˆ

1‡



 1‡

KS0



4

…84†

…85†

3  p ÿ ksc

1‡

1‡

1  p ‡ ksc

1  p ÿ 2M 0 …q† k1

1‡

1  p ‡ 2M 0 …q†

;

…87†

k1

where KS0 is the recombination rate constant in units of KD and M 0 …q† is the dimensionless value of M…q†, i.e. KS0 ˆ KS =KD ;

where, cf. Eqs. (61), (69) and (70),

g0 …s† ˆ

KS0 12

M 0 …q† ˆ KD M…q†

…88†

and ksc and k1 are ksc ˆ ksc d 2 =D;

k1 ˆ k~1 d 2 =D:

…89†

We will now consider some limiting cases. 6.1. No exchange interaction In the absence of exchange interaction, Eq. (71) transforms into p 1 ‡ KS0 =2 ‡ X‡ ÿ
  p p ; M 0 …q†J ˆ0 ˆ …1 ‡ KS0 =2† 1 ‡ X‡ ‡ X‡ ‡ 2X …90† where the parameters X and X‡ are de®ned in Eq. (85). Substituting this expression into Eq. (87) under the additional assumption of no scavenging ksc ˆ 0 we reproduce the result Mints and Pukhov [8] got for this limiting case. 6.2. Fast relaxation (T2ÿ1  Q) In this limiting case, we may use the approximation Q  0, but remember that our results are only valid in the high ®eld limit. Eqs. (74), (82), and (89) give the simple result M 0 …0† ˆ

1 p : 1 ‡ k2

…91†

Note that the recombination probability is independent of the exchange interaction in this case.

…86†

6.3. In®nitely strong exchange or di€usion controlled singlet state recombination

The recombination probability for an unpolarized triplet initial state from Eq. (67) can be written as

In®nitely strong exchange interaction is de®ned as J =KS  1. For both of these cases, M 0 …q† of Eq. (76) reduces to

2

q ˆ Qd =D;

k2 ˆ k~2 d =D: 2

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

M 0 …q† ˆ

1‡

1 p : X‡

…92†

135

and from Eqs. (61) and (62), we see that g can be written as
KS0 ÿ 1 ‡ 4J 2 =KS2 : 2

In absence of scavenging (ksc ˆ 0), we get the following simple expression for the recombination probability: p pp p 1 k1 ‡ 2 X‡ ‡ 3 k1 X‡ T p R ˆ p pp ; …93† 3 c ‡ 4 ‡ 3 k1 ‡ 2 X‡ ‡ k1 X‡

6.4.1. In®nitely strong exchange interaction If the exchange rate is much faster than the recombination rate such that

where

J 2 =KS2 ! 1;

p p 12…1 ‡ k1 †…1 ‡ X‡ † : cˆ KS0

…94†

Eq. (93) coincide with the results of Mints and Pukhov in Ref. [8] when KS ! 1 (c ˆ 0). The dephasing contact exchange interaction, which Mints and Pukhov did not consider, has no e€ect on the results in this limit because the dephasing e€ect of the recombination process is already in®nitely fast. Some of the simplest subcases are further speci®ed in Section 6.4. 6.4. Absence of scavenging and intraradical relaxation In absence of both scavenging and intraradical relaxation, we have p 1‡ q 1 ; a0 ˆ p KD 1 ‡ 2 q ‡ 2q …95† p q 1 : b0 ˆ p KD 1 ‡ 2 q ‡ 2q Substituting this equation into Eq. (80) and performing a few algebraic manipulations lead to    p ÿ p
KS0 KS0 q KS0 ‡ 1 ‡ 2 q 1 ‡ g 2 2 6  RT ˆ  ; p p p p A 1 ‡ q ‡ B q 1 ‡ 2 q ‡ g 2 ‡ q KS02 =4 …96†

where
KS0 ÿ 3 ‡ KS0 ‡ g ; 2 K 02 B ˆ 1 ‡ KS0 ‡ S 4 Aˆ1‡

…97†

gˆ1‡

then Eq. (96) simpli®es considerably to p q 1 K 0 =2 RT ˆ p h K 0 S p i : 1‡ q 3 2‡ q S ‡ 2‡pq 2

…98†

…99†

…100†

If furthermore the recombination is di€usion controlled (KS0 ! 1), then Eq. (100) simpli®es to p q 1 …101† RT ˆ p : 3 2‡ q

7. Comparison with numerical calculations Our numerical calculations are based on the model, described in Section 2, but the exchange interaction is assumed to have the usual exponential form J …r† ˆ J0 exp ‰ ÿ …r ÿ d†=`x Š;

…102†

where J0 is the ``contact''-value of the exchange integral. The spatially integrated value of J …r† is (cf. Eq. (39)) "  2 # `x `x 2 ; …103† J ˆ J0 4p`x d 1 ‡ 2 ‡ 2 d d where `x is the characteristic decay length of the exchange interaction. The numerical method is described in Ref. [2], and was extended to all three triplet states to be adequate for the comparison with the analytical expressions. The exchange interaction has no e€ect on the recombination yield when the dynamical S±T0 mixing parameter q is zero. Thus, we need a numerical test of the case where we have both exchange interaction and dynamical S±T0 mixing

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M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

since this is the only case where the obtained analytical formulas are not exact. There are two reasons for the deviations between our analytical formulae and the numerical results that take account for the realistic spatial decay of the exchange interaction of Eq. (102). The ®rst reason is that the character of the exchange interaction is essentially a non-contact phenomenon for a locally strong exchange de®ned by  2 `2 …104† j2c ˆ 2J0 x J 1; D which implies that the e€ective exchange radius `e is greater than d [9]. For small and intermediate values of q, there is no dynamical S±T0 mixing for interradical distances smaller than the e€ective exchange radius. The e€ective exchange radius grows logarithmically with the parameter jc introduced in Eq. (104). However, many experimental results [18] indicate that the exchange interaction is locally weak for solutions of moderate and low viscosity, i.e.  2 `2x 2 K 1: …105† jc ˆ 2J0 D The contact approximation seems to be a good approximation in this case. The second reason is that when the S±T0 mixing is very fast `2x J 1; …106† D ^^ 0 then G 0 …r; r ; s† decays spatially on a scale that is comparable with the spatial decay of the exchange interaction. In this case, it is not appropriate to use the contact approximation because the step from Eq. (26) to Eq. (30) becomes questionable. This problem can in principle be overcome by applying the modi®ed kinematic approximation (MKA) [17] that is valid for arbitrary values of q in the limit of locally weak exchange de®ned by Eq. (105). However, the expressions will be considerably more complicated. This complicating modi®cation is not justi®ed in our case since the recombination probability is close to its limiting value when we have fast dynamical S±T0 mixing

Eq. (106) and hence we could only get a minor improvement of the results. Figs. 1 and 2 show the q-dependence of the recombination probability for an unpolarized triplet precursor at di€erent values of the exchange interaction and in the absence of intraradical relaxation and scavenging. Fig. 1 was calculated for an intermediate value of the singlet state reactivity KS =KD ˆ 1. The di€usion controlled case, where

Fig. 1. The recombination probability for an unpolarized triplet initial state in absence of intraradical relaxation and scavenging. The recombination process is neither di€usion nor reaction controlled since KS =KD ˆ 1. The strength of the exchange interaction is characterized by the local strength (jc ˆ 2J0 `2x =D).

2Q

Fig. 2. Same as in Fig. 1 but the recombination is di€usion controlled (KS =KD ! 1).

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

KS =KD ! 1, is displayed in Fig. 2. The local strengths of the exchange interaction (jc ) is indicated in the ®gures since this parameter value determines the applicability of the analytical results. The ®gures demonstrate that the analytical formulas are very accurate for a locally weak exchange interaction (de®ned in Eq. (105)). With increasing local strengths of the exchange interaction considerable deviations between the analytical and the numerical results appear. For the value of jc used in Fig. 1, the deviations appear for q > 0:1. For very low values of q the recombination probability is independent of the exchange interaction. The actual value of q for which deviations appear depend on the local strength of the exchange interaction (jc ). The nature of the deviations may easily be explained qualitatively. For r < `e , the dynamical S± T0 mixing is suppressed and thus the RP needs to have longer trajectories between re-encounters than in the absence of exchange interaction in order to be converted from a triplet state to a singlet state. This, of course, leads to a decrease in the recombination probability. For small values of q, this e€ect is less important since the excluded spin-mixing area is small relative to the length of the needed trajectories. However, it has a signi®cant e€ect for intermediate q values since the trajectories necessary to mix S and T0 are shorter and thus the excluded area becomes important. For larger q values, the di€erence between the analytical and the numerical results for a locally strong exchange decreases and ®nally disappears. This e€ect is due to the fact, that for very large values of q, it is possible to have S±T0 mixing even for r < `e and the e€ect of the spin exchange becomes negligible, both for the realistic spatial decay of Eq. (102) and for the contact approximation. Figs. 3±5 illustrate the q-dependence of the recombination probability for an unpolarized triplet precursor for di€usion controlled recombination. The local strength of the exchange interaction has an intermediate value (jc ˆ 1; `x =d ˆ 0:1). Fig. 3 illustrates the e€ect of transverse relaxation. It is readily seen that the S±T0 mixing is dominated by the relaxation mechanism when q 6 k2 and thus the recombination probability is

137

Fig. 3. Illustration of the e€ect of transverse relaxation on the recombination probability for a di€usion controlled reaction. No longitudinal relaxation is included (T1ÿ1 ˆ 0). The values of ÿ1 ÿ1 the transverse relaxational rate k2 ˆ T2A ‡ T2B are indicated in the ®gure. Other non-zero parameter values are also indicated in the ®gure.

Fig. 4. Illustration of the combined e€ect of longitudinal and transverse relaxation on the recombination probability for a di€usion controlled reaction. The transverse (k1 ) and longitudinal (k2 ) relaxational rates are identical and the parameter values are indicated in the ®gure. Other parameter values are as in Fig. 3.

independent on q for q 6 k2 . The q mixing becomes increasingly important for increasing values of q and very little e€ect of the relaxation is observed for q > k2 . Fig. 4 shows the combined e€ect of transverse and longitudinal relaxation (T2 ˆ T1 ). Comparison

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M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

Fig. 5. Illustration of the e€ect of scavenging on the recombination probability for a di€usion controlled reaction in the absence of relaxation. Other parameter values are as in the previous ®gure.

with Fig. 4 shows that the longitudinal (T1 ) relaxation shifts the recombination probabilities upward for all q-values. This is because the longitudinal relaxation induces transitions between the T states and T0 and S. The e€ect on the recombination probability is largest for larger q values, since after recombination of the singlet pairs, a larger fraction of the so formed T0 pairs will have a chance to be converted to singlet state before re-encountering and thus have a chance to react. Note that if the q dependence is due to a variation of the static magnetic ®eld, then the magnetic ®eld dependence of the relaxation times must be taken into account. Fig. 5 shows the e€ect of di€erent scavenging rates. As expected, the recombination probability decreases for increasing values of the scavenging rate because the lifetime of the RP is reduced. The e€ect is diminished for larger q-values since the time scale on which the recombination takes place is shortened.

8. Comparison with other methods Only a few analytical expressions for the recombination probability of freely di€using RP can

be found in the literature [5,6,8]. Most of the previous works [6,8] use a contact approximation for the singlet state reactivity and either neglect the exchange interaction or approximates it by a delta function. The basic assumption in these works are thus similar to those used in the present work. However, the results of these approaches are different from our results; either because they invoke more assumptions or do not consider as general a system. Evans et al. [6] applied a delta function approximation for the exchange interaction and the singlet state reactivity. Intraradical transversal relaxation (T2 ) was included but longitudinal relaxation (T1 ) was excluded. The derived expressions are very di€erent from ours, simpli®ed to no longitudinal relaxation. The reason for the deviation is that the radicals in Ref. [6] were considered as point reactants (i.e. with a distance of the closest approach d ˆ 0) but with a delta±function exchange interaction and singlet state reactivity located at a radius R > 0, i.e. the radicals were allowed to di€use in the region r < R. This assumption was not clearly stated in the paper [6] and was not justi®ed. We have compared the analytical solutions for the simple limiting case of no exchange interaction (J ˆ 0), no intraradical relaxation (k1 ˆ k2 ˆ 0), no scavenging (ksc ˆ 0), and a vanishing singlet state reactivity (KS ! 0). This situation can be treated by perturbation theory in the reactivity and the resulting expressions have a very simple form. For an unpolarized triplet precursor, our approach gives the following general result: RT ˆ

KS KS G0 11;22 ˆ ‰g0 …0† ÿ a0 Š; 3 6

…107†

which upon insertion of Eq. (82) for the reduced free Green's function yields pÿ p
q 1‡2 q KS T ; …108† R ˆ p 6KD 1 ‡ 2 q ‡ 2q where KD ˆ 4pdD. In order to use our general analytical expression to make calculations for point reactants, we have to substitute another free Green's function

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

g0p …s† ˆ g0p …d; d; s† into Eq. (107). The required Green's function of freely di€using point radicals is readily obtained from the Green's function with re¯ective boundary condition at r ˆ d, which we use [7,9] "   p 1 p exp ÿ s=Djr ÿ r0 j g0 …r; r0 ; s† ˆ 8prr0 D s=D p 1 ÿ d s=D p ÿ 1 ‡ d s=D #   p 0  exp ÿ s=D…r ‡ r ÿ 2d† …109† by letting d ! 0. This immediately gives

The published expression for an unpolarized triplet precursor is [6] RT ˆ

KS A11 …Q1 ÿ Q3 † ÿ A12 Q2 : 3 A11 A22 ÿ A12 A21

g0p …0† ˆ ap0

ˆ

1 KD

1 ÿ eÿ2

…111† p ÿ q

p p
cos…2 q† ÿ sin…2 q† : p 4KD q

…112†

By substituting these expressions into Eq. (107), we obtain the following expression for the recombination probability of point reactants: RTp

p ÿ  p p
 1 ÿ eÿ2 q cos…2 q† ÿ sin…2 q† KS ˆ 1ÿ : p 6KD 4 q

…113†

…114†

In the considered low reactivity limit, we immediately obtain from Ref. [6] A11 ˆ 1;

A22 ˆ 2;

A12 ˆ A21 ˆ 0

…115†

and, assuming that the RP was created at the distance d (as used in our calculations) Q1  g0p …0† ˆ 1=KD ;

Q3  ap0 ;

…116†

where g0p and ap0 are given by Eqs. (111) and (112). Upon substitution of Eqs. (115) and (116) into Eq. (114), we get an expression for the recombination

 p   p  exp ÿ s=Djr ÿ r0 j ÿ exp ÿ s=D…r ‡ r0 † p : g0p …r; r; s† ˆ 8prr0 D s=D

Our method only requires the Green's function evaluated at the common point where the exchange and recombination are non-zero, i.e. at r ˆ r0 ˆ d. The resulting function was called g0 …s† and is given by Eq. (82); it is immediately obtained from Eq. (109) by setting r ˆ r0 ˆ d. The required values of the reduced Green's function of point radicals follows immediately from Eq. (110) and are:

139

…110†

probability which is identical to Eq. (113), derived by our method for point radicals. Eqs. (107) and (113) coincide only in the limit p 2 q  1, in which case, the unphysical S±T0 mixing in the region r < d, included in the point radical model, is negligible. In that case, both models give RT '

KS p q: 6KD

…117†

For larger, and thus realistic, values of the mixing parameter (q) signi®cant deviations occurs and Eq. (113) is no longer accurate. It should be mentioned that Evans later indicated that their results needed to be modi®ed [19]. The required modi®cation is essentially equivalent to replacing the incorrect Green's function with the correct one, but the expressions were not explicitly evaluated. Mints and Pukhov [8] obtained an exact solution of the SLE for free di€usion. They neglected the exchange interaction and used a contact approximation for the recombination. Scavenging was neglected but both longitudinal and transverse

140

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

intraradical relaxation were included. Their results are identical to our results for free di€usion, but do not include the e€ect of a contact exchange interaction and of a homogeneous scavenging process. A sudden perturbation approximation (SPA) was suggested by Shushin [5]. The method is valid for arbitrary values of the exchange interaction but was stated to have the following restriction on the mixing parameter: nˆ

p `x q  1; d

…118†

where `x is the characteristic decay length of the exchange interaction, cf. Eq. (102). When only singlet RPs are allowed to react, the recombination probability for an unpolarized triplet precursor is given in Ref. [5] as p LSS q=D 1 T ; …119† RSPA ˆ 6 1 ‡ D1 ‡ D2 ‡ 2D1 D2 where the quantities LSS and D1;2 were given by analytic expressions for the special cases: no exchange interaction and a locally strong exchange interaction [5]. Several special cases of this formula was numerically tested for a locally strong exchange interaction and found to have a high accuracy [20]. Analytic expressions for D1;2 for a locally weak exchange interaction were later derived in Ref. [9]. The simplest way to investigate the accuracy of the SPA expression (119) is to compare the expressions for no exchange interaction, no intraradical relaxation and no scavenging. For di€usion controlled recombination, p …120† LSS ˆ d; D1 ˆ q=2; D2 ˆ 0; and Eq. (119) gives the correct result (101). For low reactivity (KS ! 0) and no exchange interaction the quantities are [5] p D1 ˆ D2 ˆ q: …121† LSS ˆ dKS =KD ; By substituting these expressions into Eq. (119), we get RTSPA ˆ

p q KS ; p 6KD 1 ‡ 2 q ‡ 2q

…122†

which obviously di€ers from the exact expression p (108) by the lack of the multiplier 1 ‡ 2 q in the numerator. Thus, the validity range of the SPA for p vanishing reactivity is actually 2 q  1. This corresponds to the validity range of a perturbation expansion in the rate of S±T0 mixing and Eq. (122) should actually be reduced to Eq. (117). It is concluded that the SPA works best in the di€usion controlled limit and that, at least for locally weak exchange, its accuracy depends on the reactivity of the radicals in a non-predicted manner. Moreover, neither relaxation nor scavenging was included.

9. Conclusion The main result of the present work is the derivation of an exact and completely general expression for the recombination probability (or quantum yield) of a radical pair in a high magnetic ®eld. Both the e€ect of a homogeneous distributed scavenger and those of longitudinal and transversal intraradical relaxation are included. The approach is limited to short range exchange and recombination as it applies a contact approximation for these two interactions. It is noticeable that the expression is given solely in terms of a very small number of values of the classical Green's function for the relative motion of the radicals. In the present work, we have considered the free di€usion case in detail. Since the expressions are exact for the assumed model, only deviations for a ®nite range (non-contact) exchange interaction can be found. We have illustrated the high accuracy and the di€erent dependences by comparison with accurate numerical results. By evaluating Green's function for another model of the relative motion of the radicals one immediately obtains an exact expression for the recombination probability. We are currently working on the interesting micellar system. Appendix A. Relations between shift and spin operators The necessary relations between shift and single spin operators are

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

P^11 P^12 P^13 P^14

h i ˆ 14 ÿ 12 S^‡A S^ÿB ‡ S^ÿA S^‡B ÿ S^ZA S^ZB ; h i ˆ 12 S^ZA ÿ S^ZB ‡ S^‡A S^ÿB ÿ S^ÿA S^‡B ; hh i h ii ˆ p12 12 ‡ S^ZA S^ÿB ÿ S^ÿA 12 ‡ S^ZB ; h h i h i i ˆ p12 S^‡A 12 ÿ S^ZB ÿ 12 ÿ S^ZA S^‡B ;

h i P^21 ˆ 12 S^ZA ÿ S^ZB ÿ S^‡A S^ÿB ‡ S^ÿA S^‡B ; h i P^22 ˆ 14 ‡ 12 S^‡A S^ÿB ‡ S^ÿA S^‡B ÿ S^ZA S^ZB ; hh i h ii P^23 ˆ p12 12 ‡ S^ZA S^ÿB ‡ S^ÿA 12 ‡ S^ZB ; h h i h i i P^24 ˆ p12 S^‡A 12 ÿ S^ZB ‡ 12 ÿ S^ZA S^‡B ; i h ii ‡ S^ZA S^‡B ÿ S^‡A 12 ‡ S^ZB ; P ˆ hh i h ii P^32 ˆ p12 12 ‡ S^ZA S^‡B ‡ S^‡A 12 ‡ S^ZB ; h ih i P^33 ˆ 12 ‡ S^ZA 12 ‡ S^ZB ; ^31

hh

p1 2

…A:1†

…A:2†

Furthermore, completeness of the basis imply ^1 ˆ P^11 ‡ P^22 ‡ P^33 ‡ P^44 :

2

2

Appendix B. Matrix elements of the free Green's superoperator

…A:4†

P^43 ˆ S^ÿA S^ÿB ; h ih i P^44 ˆ 12 ÿ S^ZA 12 ÿ S^ZB :

The necessary matrix elements of Green's superoperator of Eq. (31) in the singlet±triplet basis are (s ˆ 0) i h …B:1† G0 11;11 ˆ 14 g0 …ksc † ‡ g0 …k~1 † ‡ 2a0 ; h i …B:2† G0 11;22 ˆ 14 g0 …ksc † ‡ g0 …k~1 † ÿ 2a0 ; G0 11;12 ˆ G0 12;11 ˆ G0 21;22 ˆ ÿG0 11;21

The inverse relations between the single spin and the shift operators are as follows: h i S^ZA ˆ 12 P^12 ‡ P^21 ‡ P^33 ÿ P^44 ; h i S^ZB ˆ 12 ÿ P^12 ÿ P^21 ‡ P^33 ÿ P^44 ; h i S^‡A ˆ p12 P^14 ‡ P^24 ÿ P^31 ‡ P^32 ; h i S^‡B ˆ p12 ÿ P^14 ‡ P^24 ‡ P^31 ‡ P^32 ; h i S^ÿA ˆ p12 P^41 ‡ P^42 ÿ P^13 ‡ P^23 ; h i S^ÿB ˆ p12 ÿ P^41 ‡ P^42 ‡ P^13 ‡ P^23 ;

…A:7†

…A:3†

P^34 ˆ S^‡A S^‡B ;

2

…A:6†

S^‡A S^‡B ˆ P^34 ; S^ÿA S^ÿB ˆ P^43 :

1 2

h h i h i i P^41 ˆ p12 S^ÿA 12 ÿ S^ZB ÿ 12 ÿ S^ZA S^ÿB ; h h i h i i P^42 ˆ p1 S^ÿA 1 ÿ S^ZB ‡ 1 ÿ S^ZA S^ÿB ;

i S^ZA S^ZB ˆ 14 ÿ P^11 ÿ P^22 ‡ P^33 ‡ P^44 ; h i S^‡A S^ZB ˆ 2p1 2 ÿ P^14 ÿ P^24 ÿ P^31 ‡ P^32 ; h i S^‡A S^ÿB ˆ 12 P^12 ÿ P^21 ÿ P^11 ‡ P^22 ; h i S^ÿA S^ZB ˆ 2p1 2 ÿ P^41 ÿ P^42 ÿ P^13 ‡ P^23 ; h i S^ÿA S^‡B ˆ 12 ÿ P^12 ‡ P^21 ÿ P^11 ‡ P^22 ; h i S^ZA S^‡B ˆ 2p1 2 P^14 ÿ P^24 ‡ P^31 ‡ P^32 ; h i S^ZA S^ÿB ˆ 2p1 2 P^41 ÿ P^42 ‡ P^13 ‡ P^23 ;

141

h

…A:5†

G0 11;33

ˆ ÿG0 21;11 ˆ ÿG0 12;22 ˆ ib0 =2; h i ˆ G0 11;44 ˆ 14 g0 …ksc † ÿ g0 …k~1 † ;

…B:3† …B:4†

G0 12;12 ˆ G0 21;21 h i ˆ 14 g0 …k~1A † ‡ g0 …k~1B † ‡ 2a0 ;

…B:5†

G0 12;21 ˆ G0 21;12 h i ˆ 14 g0 …k~1A † ‡ g0 …k~1B † ÿ 2a0 ;

…B:6†

G0 12;33 ˆ G0 21;33 ˆ ÿG0 12;44 ˆ ÿG0 21;44 h i ˆ 14 g0 …k~1A † ÿ g0 …k~1B † ;

…B:7†

142

M.J. Hansen et al. / Chemical Physics 260 (2000) 125±142

where the functions a0 and b0 are de®ned by Eq. (60), and the ``e€ective'' relaxation rates are de®ned in Eq. (55). References [1] [2] [3] [4] [5] [6]

J.B. Pedersen, J.H. Freed, J. Chem. Phys. 58 (1973) 2746. J.B. Pedersen, L.I. Lolle, Chem. Phys. 5 (1993) 89. J.B. Pedersen, Chem. Phys. Lett. 52 (1977) 333. K.M. Salikhov, Teor. Eksp. Khim. 13 (1977) 732. A.I. Shushin, Chem. Phys. 144 (1990) 201. G.T. Evans, P.D. Fleming, R.G. Lawler, J. Chem. Phys. 58 (1973) 2071. [7] P.A. Purtov, A.B. Doktorov, Chem. Phys. 178 (1993) 47. [8] R.G. Mints, A.A. Pukhov, Chem. Phys. 87 (1984) 467. [9] A.B. Doktorov, A.A. Neufeld, J. Boiden Pedersen, J. Chem. Phys. 110 (1999) 8881.

[10] A.B. Doktorov, S.A. Mikhailov, P.A. Purtov, Chem. Phys. 160 (1992) 223. [11] S.I. Temkin, B.I. Yakobson, Phys. Chem. 88 (1984) 2679. [12] A.I. Burshtein, A.B. Doktorov, V.A. Morozov, Chem. Phys. 104 (1986) 1. [13] A.A. Neufeld, A.B. Doktorov, J.Boiden Pedersen, J. Chem. Phys., submitted for publication. [14] U. Fano, Phys. Rev. 131 (1963) 259. [15] F. Bloch, Phys. Rev. 70 (1946) 460. [16] A.B. Doktorov, J. Boiden Pedersen, J. Chem. Phys. 108 (1998) 6868. [17] A.B. Doktorov, A.A. Neufeld, J. Boiden Pedersen, J. Chem. Phys. 110 (1999) 8869. [18] V.F. Tarasov, H. Yashiro, K. Maeda, T. Azumi, I.A. Shkrob, Chem. Phys. 212 (1996) 353. [19] G.T. Evans, Mol. Phys. 31 (1976) 777. [20] A.I. Shushin, J.B. Pedersen, L.I. Lolle, Chem. Phys. 177 (1993) 119.