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Recombination yield of geminate radical pairs in low magnetic fields – A Green’s function method
Chemical Physics 328 (2006) 333–337 www.elsevier.com/locate/chemphys
Recombination yield of geminate radical pairs in low magnetic fields – A Green’s function method A.B. Doktorov b
a,b
, M.J. Hansen b, J. Boiden Pedersen
b,*
a Institute of Chemical Kinetics and Combustion, SB RAS, 530090 Novosibirsk, Russia Department of Physics and Chemistry, University of Southern Denmark, DK-5230 Odense M, Denmark
Received 2 June 2006; accepted 17 July 2006 Available online 29 July 2006
Abstract An analytic expression for the recombination yield of a geminate radical pair with a single spin one half nuclei is derived. The expression is valid for any field strength of the static magnetic field. It is assumed that the spin mixing is caused solely by the hyperfine interaction of the nuclear spin and the difference in Zeeman energies of the two radical partners, that the recombination occurs at the distance of closest approach, and that there is a locally strong dephasing at contact. This is a special result of a new general approach where a Green’s function technique is used to recast the stochastic Liouville equation into a low dimensional matrix equation that is particularly convenient for locally strong dephasing systems. The equation is expressed in terms of special values (determined by the magnetic parameters) of the Green’s function for the relative motion of the radicals and it is therefore valid for any motional model, e.g. diffusion, one and two site models. The applicability of the strong dephasing approximation is illustrated by comparison with numerical exact results. 2006 Elsevier B.V. All rights reserved. Keywords: Radical pair; Recombination; Magnetic field effect; Coherences; Low fields
1. Introduction The recombination yield of a geminate radical pair depends on the strength of an external magnetic field. The effect arises from the time evolution of the coherent state of the radical pair spin state and the spin selectivity of the recombination. Typically, a radical pair (RP) is created in a triplet electron spin state and can only recombine in the singlet state. In order to recombine, the RP must simultaneously encounter (collide) and be in a singlet state. The problem of calculating the average recombination probability therefore involves a complicated, combined description of the evolution of the spin system and the spatial separation of the radicals, where the former depends on the latter. The problem is particularly complicated when the radicals performs a random motion with multiple reencounters. Accurate numerical investigations for radical *
Corresponding author. Tel.: +45 6550 3516; fax: +45 66 158760. E-mail address: [email protected] (J.B. Pedersen).
0301-0104/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.07.038
pairs with up to five different nuclei [1] are possible, but a general analytic solution of the problem is complicated and has not yet been obtained. In order to reduce the complexity of the problem for diffusive systems, it is common to approximate the diffusion process by a one or two site model with exponential decay kinetics, see e.g. [2,3]. This may be a reasonable approximation for micelles, see for example the discussion in Ref. [4], but its applicability to homogeneous liquids is questionable and the model parameters have no direct physical interpretation. A major problem with calculations in low magnetic fields is that the dimensions of the spin matrices grow as the square of the number of coupled spin states. The Johnson–Merrifield approximation (JMA) [5,6] reduces the problem by excluding all off-diagonal elements of the spin density matrix in the basis that diagonalizes the free Hamiltonian. Consequently it cannot describe coherent effects, such as the usual singlet triplet (Q) mixing and its applicability appears to be restricted to systems with exponential
334
A.B. Doktorov et al. / Chemical Physics 328 (2006) 333–337
time dependences, such as micelles, and to incoherent mixing of states (relaxation). In the present work we use a Green’s function technique to recast the stochastic Liouville equation into a low dimensional matrix equation. The approach is a generalization of a steady state approach previously introduced [7]. The derived matrix equation is particularly convenient for locally strong dephasing systems, where the S-T offdiagonal elements are zero at contact. The simplest version of the equation is obtained by additionally assuming, as suggested in [7], that the T-T off-diagonal elements are also zero at contact. The derived analytic expressions for the recombination yield are expressed in terms of special values, determined by the magnetic interactions, of the a general Green’s function for the relative motion. They can therefore be applied to any motional model for which the Green’s function can be evaluated. The accuracies of the approximations are determined by comparison with numerical results.
derivation will be performed for an arbitrary operator and all results will be expressed in terms of contact values of its Green’s function, defined as GðsÞ Gðd; d; sÞ;
where ½s Lr Gðr; r0 ; sÞ ¼
dðr r0 Þ : 4prr0 ð4Þ
3. Green’s function method Let us introduce the ‘‘free Green’s operator’’ [9] by 0 b b ðr; r0 ; sÞ ¼ dðr r Þ ^^1: ½s þ iH Lr G ð5Þ 4prr0 Eq. (1) can then be rewritten as Z 1 b b b b b ðr; r0 ; sÞq0 ðr0 Þ4pr02 dr0 G bq ~ðd; sÞ: ~ðr; sÞ ¼ ðr; d; sÞ K G q 0
ð6Þ 2. Model description Our treatment is based on the Laplace transformed Stochastic Liouville Equation [8] b b ðrÞ~ ½s þ iH Lr þ K qðr; sÞ ¼ q0 ðrÞ;
ð1Þ
~ðr; sÞ is the Laplace transformed spin density matrix where q and the initial spin density matrix is denoted q0(r). The three (super)operators on the left hand side describes respectively the coherent spin evolution, the relative motion of the radicals, and the recombination and dephasing processes. The coherent evolution of the spin system is determined by the usual commutator generated superoperator H of the spin Hamiltonian H ¼ bBðgA S ZA þ gB S ZB Þ þ aIA SA ;
ð2Þ
where it has been assumed that anisotropic effects can be neglected and that the effect of the exchange interaction can be included via an additional dephasing process. Notice that we have h = 1, and arbitrarily chosen the single set nuclear spin I ¼ 12 to be situated on radical A. The Liouville representation of the decay of the RPs through the singlet and triplet channels, at contact, is given by the superoperator 1 dðr dÞ b ^ ^ ^ ^ ^ ^ b ^ ^ ; K q ¼ K S P S qP S þ K T P T qP T þ K d ðP S qP T þ P T qP S Þ 2 4prd ð3Þ where P^ S and P^ T are projection operators onto the singlet state and triplet manifold, respectively, and the dephasing constant K d P 12 ðK S þ K T Þ; the inequality sign accounts for additional contact dephasing processes, e.g. due to the exchange interaction. The relative (stochastic) motion of the radicals is modeled by the spin independent operator Lr . The following
At the contact distance, r = d, this yields Z 1 b b 1 0 0 02 0 b b ~ðd; sÞ G ðd; d; sÞ q G ðd; r ; sÞq0 ðr Þ4pr dr 0
b bq ~ðr; sÞ: ¼ K
ð7Þ
This is the main equation of the paper. For a contact precursor, i.e. dðr dÞ q0 ðrÞ ¼ q0 ; ð8Þ 4prd the stationary limit (s = 0) of Eq. (7) is b b b 1 ðd; d; 0Þ~ bq ~ðd; 0Þ: G qðd; 0Þ ¼ q0 K ð9Þ The reasons for separating the two terms with the un~ðd; 0Þ will be clear in the following. It should be known q noted that the inverse Green’s operator is no more difficult to calculate than the Green’s operator, cf. Appendix A. The total recombination yield is given by b bq ~ðd; 0Þ: R ¼ Tr½ K
ð10Þ
In the words of the previous work [7], one may consider the first term on the right hand side as the steady state flux into the system at the contact distance and the last term of Eq. (9) to be the stationary flux through the different channels at the contact boundary. Eq. (9) is therefore the Green’s function generalisation of the ‘‘steady state’’ approach previously used for free diffusion [7]. We will now solve Eq. (9) for a RP with a single nuclear spin 12. The rotational symmetry of the Hamiltonian about the z-axis implies conservation of the z-component of the total spin (m = mA + mB + mI). The Hamiltonian therefore splits up into four sub-matrices according to the four values of m (±3/2, ±1/2). Only recombination from the blocks with m = ±1/2 has a magnetic field dependence. These two blocks can be obtained from each another by reversing the sign of the magnetic field B. Each block is spanned by three
A.B. Doktorov et al. / Chemical Physics 328 (2006) 333–337
states, e.g. {a1b2a, b1a2a, a1a2b} for m ¼ 12. The Hamiltonian for this block is therefore a 3 · 3 matrix, and the corresponding density matrix q has nine elements. Eq. (9) is therefore a complicated 9 · 9 matrix equation. Eq. (9) is considerably simplified if some of the matrix elements of q are identical zero at contact, e.g. for strong contact S-T dephasing for which qST i ðd; sÞ ¼ qT i S ðd; sÞ ¼ 0:
ð11Þ
Strong contact dephasing can be caused by a strong exchange interaction, but diffusion controlled recombination through either the singlet or the triplet channel will also give strong S-T dephasing and in addition cause the corresponding diagonal element to be zero. In the following we will assume that the strong dephasing condition, Eq. (11), is applicable. In the singlet–triplet basis, the m = 1/2 block is spanned by {Sa, T0a, T+b}, and four of the matrix elements of ~ðd; 0Þ are automatically zero, cf. Eq. (11). Consequently q ~ðd; 0Þ, but the corwe have only five unknown elements of q responding flux terms (Kq) for the vanishing elements are not zero, and thus we are still left with nine unknowns. However, a closer look at the matrix equation shows that it decouples into two matrix equations. We only have to solve a set of five closed equations for the non-vanishing matrix elements, followed by insertion of the results into the remaining four decoupled equations for the flux of the vanishing elements. For diffusion controlled recombination through the sin~ðd; sÞ that have singlet glet state (KS ! 1), all elements of q character are zero thus reducing the dimension of the rele~SS ðd; sÞ is vant block of Eq. (9) from 9 · 9 to 4 · 4. Since q zero we cannot calculate the recombination yield through the singlet channel directly by Eq. (10). Instead, it follows from Eq. (9) that the singlet recombination yield for a triplet precursor can be calculated as X b b b 1 ðd; d; 0Þ~ bq ~ðd; 0Þ RTS ¼ K ¼ qðd; 0ÞT i T j ; G SS;Ti Tj SS
i;j¼0;
ð12Þ where the summation is over the indicated elements that are present in the block. If we furthermore assume, as in Ref. [7], that the off~ðd; sÞ between the triplet states are diagonal elements of q also zero (strong T-T contact dephasing) then we are left with a 2 · 2 block. Then only two of the elements of ~ðd; 0Þ are non-zero and they are determined by solving q two coupled algebraic equations.
335
results are the recombination yield through the individual reactive channels. The recombination yield through the singlet channel for a singlet precursor is conveniently written as RSS ¼
KS ð1 KT Þ KS ¼ ; 1 KT þ 3ð1 KS ÞRTS 1 þ ð1 K Þ 3RTS S 1KT
ð13Þ
where the decay rates through the reactive channels are expressed in terms of the dimensionless recombination radii, defined as KT ¼
K TC0 ; 1 þ K T C0
KS ¼
K SC0 ; 1 þ K SC0
ð14Þ
where C0 = G(0) is the steady state value of the contact Green’s function, cf. Eq. (4). The quantum parameters are contained in the fundamental quantity RTS ¼
1 ð1 BÞð1 4AÞ þ A þ 2ðA BÞD 3 ð1 þ BÞð1 2AÞ A þ ð1 A þ BÞD
ð15Þ
which is independent of the singlet reactivity. This quantity is equal to the diffusion controlled singlet recombination yield of an unpolarised triplet precursor, which follows from a comparison of Eq. (13) with the recently derived set of general relationships, that relate the yields through the different channels and for different precursor states [10]. These relations were expressed in terms of F ¼ 3RTS = ð1 KT Þ, and the combination of these relations and the above results therefore cover all possible cases. The quantities appearing in Eq. (15) are defined as A ¼ ð1 KT ÞðA0 Dce þ Dcc Þ;
ð16Þ
B ¼ KT þ ð1 KT ÞðB0 Dce Dcc Þ; D ¼ ð1 KT ÞðDcc Dee Þ:
ð17Þ ð18Þ
It is interesting to note that all D-terms are zero in the limit of strong contact T-T dephasing. Thus these terms account for the effects of the off-diagonal T-T elements. In this limit, and for unreactive triplet states, Eq. (15) simplifies to the previous result [7] with A ¼ A0 , B ¼ B0 , and D = 0, where cC 0 1 1 A0 ¼ Re ; ð19Þ 2 C0 G ð1 þ cÞ C0 1 c Re B0 ¼ þ ð20Þ Gþ G 1þc and we have introduced an abbreviated notation for the four needed values of the contact Green’s function G(s). C 0 ¼ Gð0Þ;
G ¼ Gðiðkþ k ÞÞ;
G ¼ Gðiðk k0 ÞÞ: ð21Þ
4. Results Eq. (9) can be solved analytically for strong dephasing, defined by Eq. (11), for one of the two blocks corresponding to m ¼ 12. The solution for the other block is obtained by reversing the sign of the magnetic field B. The total yield is the weighted sum of the contributions from the individual blocks, the weights being equal to projection of the initial spin density matrix onto the block. The most informative
The parameter c and the eigenvalues are 2 kþ þ k0 ; a=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g bB a 1 2 ðgA bBÞ þ a2 ; k ¼ B 2 4 2 ðg gB ÞbB a k0 ¼ A þ : 2 4 c¼
ð22Þ ð23Þ ð24Þ
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The D-terms are more complicated. They can be written as Dab ¼ 2
Re½aððf þ C 0 K T Þb gb Þ 2
jf þ C 0 K T j g2
ð25Þ
;
where a, b 2 {c, e}, and pffiffiffi c 1c 1 C0 c 1 1 1 1 1 c ¼ pffiffiffi þ þ ; ð26Þ 2 2 1 þ c 1 þ c C 0 1 þ c G 1 þ c G Gþ G pffiffiffi c C0 1c 1 c 1 1 1 1 1 e ¼ pffiffiffi þ ; þ 1 þ c C 0 1 þ c G 1 þ c G Gþ G 2 2 1þc
2
ð27Þ
C0 1 2c 1 c 1 1 1 1 1 þ þ þ c þ ; ð28Þ 2 1 þ c 1 þ c C 0 1 þ c G 1 þ c G Gþ G 1 1 c g ¼ C0 Re : ð29Þ C0 G ð1 þ cÞ2 f¼
5. Discussion The strong S-T dephasing is exact both for a diffusion controlled reaction and for a strong exchange interaction. However, it is not a priori clear when the strong T-T dephasing approximation is accurate. In order to test the accuracy of this approximation we evaluated the general expressions for free diffusion and compare the results. The Green’s function for free diffusion is well known and the needed contact values are sffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi C0 C xd 2 0 ¼ 1 þ sd 2 =D or ¼ 1 þ ð1 þ iÞ : GðsÞ GðixÞ 2D
Fig. 1. Illustration of the relative percentage error of the recombination yield introduced by the strong T-T dephasing approximation for a diffusion controlled singlet reaction. The full lines show the errors on the recombination yield from the individual blocks of the Hamiltonian, where positive (negative) field values correspond to m ¼ þ 12 12 , respectively. The stippled line shows the error on the total recombination yield from the two blocks. The parameter values are shown on the figure.
qffiffiffiffiffiffiffiffiffiffiffiffiffi The plateau error is approximately 0:5% ad 2 =D. This order of deviation is found in the region gbB 2 [a/4,a]. The error decays approximately algebraic as (gbB)1.2 for gbB 2a.
ð30Þ The resulting expressions are evaluated for a diffusion controlled singlet reaction with un-reactive triplet states. The strong S-T dephasing is exact for this situation and the results are compared with those obtained by additionally applying the strong T-T dephasing approximation. Fig. 1 illustrates the percentage error introduced by the T-T approximation for a range of magnetic fields corresponding to a Zeeman interaction both smaller and larger than the hyperfine energy. The results are given both for the total recombination yield and for the contribution from the two blocks of the Hamiltonian; where positive (negative) field values correspond to m ¼ þ 12 m ¼ 12 , respectively. The error on the total recombination yield, shown by the broken curve, is smaller than that of the individual blocks. Figures calculated with other values of the dimensionless hyperfine constant ad2/D are almost identical to the one displayed in Fig.q 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi if the x-axis is scaled with ad2/D and the y-axis with ad 2 =D. The observed dependence of the percentage error, introduced by using the strong T-T dephasing approximation, can therefore be summaqffiffiffiffiffiffiffiffiffiffiffiffiffi rized as: The maximum error is approximately 10% ad 2 =D and is only realised in a very narrow region around B = 0, approximately gbB 6 a/10. It decays exponentially fast to an intermediate plateau value.
6. Conclusion In this work we have presented a Green’s function method for solving the stochastic Liouville equation for local recombination and exchange interactions. The method generalises the steady state approach and the associated approximation introduced in a previous work [7]. The derived matrix equation (9) becomes particularly simple when supplemented with the assumption of a locally strong dephasing mechanism or applied to diffusion controlled recombination. In both cases the equation decouples into two smaller matrix equations. An even larger reduction of the problem is obtained by assuming locally strong T-T dephasing. The latter approximation is not so physically sound as the S-T dephasing, but the simplification is achieved without significantly affecting the accuracy of the results. An analytic expression was derived for the recombination yield of a geminate radical pair with a single magnetic nuclei with spin one half where both singlet and triplet states may be reactive. This relatively simple and accurate expression appears to be the most general expression that covers both low and high magnetic fields. The previous analytic low field works have included some restrictions, e.g. un-reactive triplet states [11] or zero field [12].
A.B. Doktorov et al. / Chemical Physics 328 (2006) 333–337
Appendix A. The inverse Green’s operator b b and H are diagonal in the The Green’s operator G b b are same basis. In this basis the diagonal elements of G Z 1 b b G eikp t eþikq t uðd; d;tÞest dt ¼ Gðiðkp kq ÞÞ; pq;pq ðd; d;sÞ ¼ 0
ðA:1Þ where u(d, d; t) is the conditional probability that a RP will be at contact at time t when it was at contact at time 0. This quantity is related to the diffusional Green’s function, introduced in Eq. (4), by Z 1 GðsÞ ¼ uðd; d; tÞest dt: ðA:2Þ 0
In the diagonal basis, the diagonal elements of the inverse Green’s operator is just the reciprocal of the Green’s function values of Eq. (A.1). In any other basis, e.g. the S-T basis, the matrix elements will be linear combinations of these characteristic values of the diffusional Green’s function. However, these linear combinations are identical for the Green’s operator and its inverse. Consequently, the inverse Green’s operator can be obtained from
337
the Green’s operator by replacing all the Green’s function values by their inverse and keeping all other quantities unchanged. References [1] J.B. Pedersen, J.S. Jørgensen, A.I. Shushin, IV International Symposium on Magnetic Field and Spin Effects in Chemistry and Related Phenomena, Novosibirsk, Russia, 1996. [2] C.R. Timmel, U. Till, B. Brocklehurst, K.A. McLauchlan, P.J. Hore, Mol. Phys. 95 (1998) 71. [3] K.M. Salikhov, Chem. Phys. 82 (1983) 145. [4] A.A. Neufeld, M.J. Hansen, J.B. Pedersen, Chem. Phys. 278 (2002) 129. [5] R.C. Johnson, R.E. Merrifield, Phys. Rev. B1 (1970) 896. [6] J.B. Pedersen, A.I. Shushin, J.S. Jørgensen, Chem. Phys. 189 (1994) 479. [7] M.J. Hansen, J.B. Pedersen, Chem. Phys. Lett. 361 (2002) 219. [8] J.B. Pedersen, J.H. Freed, J. Chem. Phys. 61 (1974) 1517. [9] A.B. Doktorov, A.A. Neufeld, J.B. Pedersen, J. Chem. Phys. 110 (1999) 8869. [10] A.B. Doktorov, J.B. Pedersen, Chem. Phys. Lett. 423 (2006) 208. [11] P.A. Purtov, K.M. Salikhov, Teor. Eksp. Khim. 16 (1980) 737. [12] N.N. Lukzen, J.B. Pedersen, A.I. Burshtein, J. Phys. Chem. 109 (2005) 11914.
Recombination yield of geminate radical pairs in low magnetic fields – A Green’s function method A.B. Doktorov b
a,b
, M.J. Hansen b, J. Boiden Pedersen
b,*
a Institute of Chemical Kinetics and Combustion, SB RAS, 530090 Novosibirsk, Russia Department of Physics and Chemistry, University of Southern Denmark, DK-5230 Odense M, Denmark
Received 2 June 2006; accepted 17 July 2006 Available online 29 July 2006
Abstract An analytic expression for the recombination yield of a geminate radical pair with a single spin one half nuclei is derived. The expression is valid for any field strength of the static magnetic field. It is assumed that the spin mixing is caused solely by the hyperfine interaction of the nuclear spin and the difference in Zeeman energies of the two radical partners, that the recombination occurs at the distance of closest approach, and that there is a locally strong dephasing at contact. This is a special result of a new general approach where a Green’s function technique is used to recast the stochastic Liouville equation into a low dimensional matrix equation that is particularly convenient for locally strong dephasing systems. The equation is expressed in terms of special values (determined by the magnetic parameters) of the Green’s function for the relative motion of the radicals and it is therefore valid for any motional model, e.g. diffusion, one and two site models. The applicability of the strong dephasing approximation is illustrated by comparison with numerical exact results. 2006 Elsevier B.V. All rights reserved. Keywords: Radical pair; Recombination; Magnetic field effect; Coherences; Low fields
1. Introduction The recombination yield of a geminate radical pair depends on the strength of an external magnetic field. The effect arises from the time evolution of the coherent state of the radical pair spin state and the spin selectivity of the recombination. Typically, a radical pair (RP) is created in a triplet electron spin state and can only recombine in the singlet state. In order to recombine, the RP must simultaneously encounter (collide) and be in a singlet state. The problem of calculating the average recombination probability therefore involves a complicated, combined description of the evolution of the spin system and the spatial separation of the radicals, where the former depends on the latter. The problem is particularly complicated when the radicals performs a random motion with multiple reencounters. Accurate numerical investigations for radical *
Corresponding author. Tel.: +45 6550 3516; fax: +45 66 158760. E-mail address: [email protected] (J.B. Pedersen).
0301-0104/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.07.038
pairs with up to five different nuclei [1] are possible, but a general analytic solution of the problem is complicated and has not yet been obtained. In order to reduce the complexity of the problem for diffusive systems, it is common to approximate the diffusion process by a one or two site model with exponential decay kinetics, see e.g. [2,3]. This may be a reasonable approximation for micelles, see for example the discussion in Ref. [4], but its applicability to homogeneous liquids is questionable and the model parameters have no direct physical interpretation. A major problem with calculations in low magnetic fields is that the dimensions of the spin matrices grow as the square of the number of coupled spin states. The Johnson–Merrifield approximation (JMA) [5,6] reduces the problem by excluding all off-diagonal elements of the spin density matrix in the basis that diagonalizes the free Hamiltonian. Consequently it cannot describe coherent effects, such as the usual singlet triplet (Q) mixing and its applicability appears to be restricted to systems with exponential
334
A.B. Doktorov et al. / Chemical Physics 328 (2006) 333–337
time dependences, such as micelles, and to incoherent mixing of states (relaxation). In the present work we use a Green’s function technique to recast the stochastic Liouville equation into a low dimensional matrix equation. The approach is a generalization of a steady state approach previously introduced [7]. The derived matrix equation is particularly convenient for locally strong dephasing systems, where the S-T offdiagonal elements are zero at contact. The simplest version of the equation is obtained by additionally assuming, as suggested in [7], that the T-T off-diagonal elements are also zero at contact. The derived analytic expressions for the recombination yield are expressed in terms of special values, determined by the magnetic interactions, of the a general Green’s function for the relative motion. They can therefore be applied to any motional model for which the Green’s function can be evaluated. The accuracies of the approximations are determined by comparison with numerical results.
derivation will be performed for an arbitrary operator and all results will be expressed in terms of contact values of its Green’s function, defined as GðsÞ Gðd; d; sÞ;
where ½s Lr Gðr; r0 ; sÞ ¼
dðr r0 Þ : 4prr0 ð4Þ
3. Green’s function method Let us introduce the ‘‘free Green’s operator’’ [9] by 0 b b ðr; r0 ; sÞ ¼ dðr r Þ ^^1: ½s þ iH Lr G ð5Þ 4prr0 Eq. (1) can then be rewritten as Z 1 b b b b b ðr; r0 ; sÞq0 ðr0 Þ4pr02 dr0 G bq ~ðd; sÞ: ~ðr; sÞ ¼ ðr; d; sÞ K G q 0
ð6Þ 2. Model description Our treatment is based on the Laplace transformed Stochastic Liouville Equation [8] b b ðrÞ~ ½s þ iH Lr þ K qðr; sÞ ¼ q0 ðrÞ;
ð1Þ
~ðr; sÞ is the Laplace transformed spin density matrix where q and the initial spin density matrix is denoted q0(r). The three (super)operators on the left hand side describes respectively the coherent spin evolution, the relative motion of the radicals, and the recombination and dephasing processes. The coherent evolution of the spin system is determined by the usual commutator generated superoperator H of the spin Hamiltonian H ¼ bBðgA S ZA þ gB S ZB Þ þ aIA SA ;
ð2Þ
where it has been assumed that anisotropic effects can be neglected and that the effect of the exchange interaction can be included via an additional dephasing process. Notice that we have h = 1, and arbitrarily chosen the single set nuclear spin I ¼ 12 to be situated on radical A. The Liouville representation of the decay of the RPs through the singlet and triplet channels, at contact, is given by the superoperator 1 dðr dÞ b ^ ^ ^ ^ ^ ^ b ^ ^ ; K q ¼ K S P S qP S þ K T P T qP T þ K d ðP S qP T þ P T qP S Þ 2 4prd ð3Þ where P^ S and P^ T are projection operators onto the singlet state and triplet manifold, respectively, and the dephasing constant K d P 12 ðK S þ K T Þ; the inequality sign accounts for additional contact dephasing processes, e.g. due to the exchange interaction. The relative (stochastic) motion of the radicals is modeled by the spin independent operator Lr . The following
At the contact distance, r = d, this yields Z 1 b b 1 0 0 02 0 b b ~ðd; sÞ G ðd; d; sÞ q G ðd; r ; sÞq0 ðr Þ4pr dr 0
b bq ~ðr; sÞ: ¼ K
ð7Þ
This is the main equation of the paper. For a contact precursor, i.e. dðr dÞ q0 ðrÞ ¼ q0 ; ð8Þ 4prd the stationary limit (s = 0) of Eq. (7) is b b b 1 ðd; d; 0Þ~ bq ~ðd; 0Þ: G qðd; 0Þ ¼ q0 K ð9Þ The reasons for separating the two terms with the un~ðd; 0Þ will be clear in the following. It should be known q noted that the inverse Green’s operator is no more difficult to calculate than the Green’s operator, cf. Appendix A. The total recombination yield is given by b bq ~ðd; 0Þ: R ¼ Tr½ K
ð10Þ
In the words of the previous work [7], one may consider the first term on the right hand side as the steady state flux into the system at the contact distance and the last term of Eq. (9) to be the stationary flux through the different channels at the contact boundary. Eq. (9) is therefore the Green’s function generalisation of the ‘‘steady state’’ approach previously used for free diffusion [7]. We will now solve Eq. (9) for a RP with a single nuclear spin 12. The rotational symmetry of the Hamiltonian about the z-axis implies conservation of the z-component of the total spin (m = mA + mB + mI). The Hamiltonian therefore splits up into four sub-matrices according to the four values of m (±3/2, ±1/2). Only recombination from the blocks with m = ±1/2 has a magnetic field dependence. These two blocks can be obtained from each another by reversing the sign of the magnetic field B. Each block is spanned by three
A.B. Doktorov et al. / Chemical Physics 328 (2006) 333–337
states, e.g. {a1b2a, b1a2a, a1a2b} for m ¼ 12. The Hamiltonian for this block is therefore a 3 · 3 matrix, and the corresponding density matrix q has nine elements. Eq. (9) is therefore a complicated 9 · 9 matrix equation. Eq. (9) is considerably simplified if some of the matrix elements of q are identical zero at contact, e.g. for strong contact S-T dephasing for which qST i ðd; sÞ ¼ qT i S ðd; sÞ ¼ 0:
ð11Þ
Strong contact dephasing can be caused by a strong exchange interaction, but diffusion controlled recombination through either the singlet or the triplet channel will also give strong S-T dephasing and in addition cause the corresponding diagonal element to be zero. In the following we will assume that the strong dephasing condition, Eq. (11), is applicable. In the singlet–triplet basis, the m = 1/2 block is spanned by {Sa, T0a, T+b}, and four of the matrix elements of ~ðd; 0Þ are automatically zero, cf. Eq. (11). Consequently q ~ðd; 0Þ, but the corwe have only five unknown elements of q responding flux terms (Kq) for the vanishing elements are not zero, and thus we are still left with nine unknowns. However, a closer look at the matrix equation shows that it decouples into two matrix equations. We only have to solve a set of five closed equations for the non-vanishing matrix elements, followed by insertion of the results into the remaining four decoupled equations for the flux of the vanishing elements. For diffusion controlled recombination through the sin~ðd; sÞ that have singlet glet state (KS ! 1), all elements of q character are zero thus reducing the dimension of the rele~SS ðd; sÞ is vant block of Eq. (9) from 9 · 9 to 4 · 4. Since q zero we cannot calculate the recombination yield through the singlet channel directly by Eq. (10). Instead, it follows from Eq. (9) that the singlet recombination yield for a triplet precursor can be calculated as X b b b 1 ðd; d; 0Þ~ bq ~ðd; 0Þ RTS ¼ K ¼ qðd; 0ÞT i T j ; G SS;Ti Tj SS
i;j¼0;
ð12Þ where the summation is over the indicated elements that are present in the block. If we furthermore assume, as in Ref. [7], that the off~ðd; sÞ between the triplet states are diagonal elements of q also zero (strong T-T contact dephasing) then we are left with a 2 · 2 block. Then only two of the elements of ~ðd; 0Þ are non-zero and they are determined by solving q two coupled algebraic equations.
335
results are the recombination yield through the individual reactive channels. The recombination yield through the singlet channel for a singlet precursor is conveniently written as RSS ¼
KS ð1 KT Þ KS ¼ ; 1 KT þ 3ð1 KS ÞRTS 1 þ ð1 K Þ 3RTS S 1KT
ð13Þ
where the decay rates through the reactive channels are expressed in terms of the dimensionless recombination radii, defined as KT ¼
K TC0 ; 1 þ K T C0
KS ¼
K SC0 ; 1 þ K SC0
ð14Þ
where C0 = G(0) is the steady state value of the contact Green’s function, cf. Eq. (4). The quantum parameters are contained in the fundamental quantity RTS ¼
1 ð1 BÞð1 4AÞ þ A þ 2ðA BÞD 3 ð1 þ BÞð1 2AÞ A þ ð1 A þ BÞD
ð15Þ
which is independent of the singlet reactivity. This quantity is equal to the diffusion controlled singlet recombination yield of an unpolarised triplet precursor, which follows from a comparison of Eq. (13) with the recently derived set of general relationships, that relate the yields through the different channels and for different precursor states [10]. These relations were expressed in terms of F ¼ 3RTS = ð1 KT Þ, and the combination of these relations and the above results therefore cover all possible cases. The quantities appearing in Eq. (15) are defined as A ¼ ð1 KT ÞðA0 Dce þ Dcc Þ;
ð16Þ
B ¼ KT þ ð1 KT ÞðB0 Dce Dcc Þ; D ¼ ð1 KT ÞðDcc Dee Þ:
ð17Þ ð18Þ
It is interesting to note that all D-terms are zero in the limit of strong contact T-T dephasing. Thus these terms account for the effects of the off-diagonal T-T elements. In this limit, and for unreactive triplet states, Eq. (15) simplifies to the previous result [7] with A ¼ A0 , B ¼ B0 , and D = 0, where cC 0 1 1 A0 ¼ Re ; ð19Þ 2 C0 G ð1 þ cÞ C0 1 c Re B0 ¼ þ ð20Þ Gþ G 1þc and we have introduced an abbreviated notation for the four needed values of the contact Green’s function G(s). C 0 ¼ Gð0Þ;
G ¼ Gðiðkþ k ÞÞ;
G ¼ Gðiðk k0 ÞÞ: ð21Þ
4. Results Eq. (9) can be solved analytically for strong dephasing, defined by Eq. (11), for one of the two blocks corresponding to m ¼ 12. The solution for the other block is obtained by reversing the sign of the magnetic field B. The total yield is the weighted sum of the contributions from the individual blocks, the weights being equal to projection of the initial spin density matrix onto the block. The most informative
The parameter c and the eigenvalues are 2 kþ þ k0 ; a=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g bB a 1 2 ðgA bBÞ þ a2 ; k ¼ B 2 4 2 ðg gB ÞbB a k0 ¼ A þ : 2 4 c¼
ð22Þ ð23Þ ð24Þ
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The D-terms are more complicated. They can be written as Dab ¼ 2
Re½aððf þ C 0 K T Þb gb Þ 2
jf þ C 0 K T j g2
ð25Þ
;
where a, b 2 {c, e}, and pffiffiffi c 1c 1 C0 c 1 1 1 1 1 c ¼ pffiffiffi þ þ ; ð26Þ 2 2 1 þ c 1 þ c C 0 1 þ c G 1 þ c G Gþ G pffiffiffi c C0 1c 1 c 1 1 1 1 1 e ¼ pffiffiffi þ ; þ 1 þ c C 0 1 þ c G 1 þ c G Gþ G 2 2 1þc
2
ð27Þ
C0 1 2c 1 c 1 1 1 1 1 þ þ þ c þ ; ð28Þ 2 1 þ c 1 þ c C 0 1 þ c G 1 þ c G Gþ G 1 1 c g ¼ C0 Re : ð29Þ C0 G ð1 þ cÞ2 f¼
5. Discussion The strong S-T dephasing is exact both for a diffusion controlled reaction and for a strong exchange interaction. However, it is not a priori clear when the strong T-T dephasing approximation is accurate. In order to test the accuracy of this approximation we evaluated the general expressions for free diffusion and compare the results. The Green’s function for free diffusion is well known and the needed contact values are sffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi C0 C xd 2 0 ¼ 1 þ sd 2 =D or ¼ 1 þ ð1 þ iÞ : GðsÞ GðixÞ 2D
Fig. 1. Illustration of the relative percentage error of the recombination yield introduced by the strong T-T dephasing approximation for a diffusion controlled singlet reaction. The full lines show the errors on the recombination yield from the individual blocks of the Hamiltonian, where positive (negative) field values correspond to m ¼ þ 12 12 , respectively. The stippled line shows the error on the total recombination yield from the two blocks. The parameter values are shown on the figure.
qffiffiffiffiffiffiffiffiffiffiffiffiffi The plateau error is approximately 0:5% ad 2 =D. This order of deviation is found in the region gbB 2 [a/4,a]. The error decays approximately algebraic as (gbB)1.2 for gbB 2a.
ð30Þ The resulting expressions are evaluated for a diffusion controlled singlet reaction with un-reactive triplet states. The strong S-T dephasing is exact for this situation and the results are compared with those obtained by additionally applying the strong T-T dephasing approximation. Fig. 1 illustrates the percentage error introduced by the T-T approximation for a range of magnetic fields corresponding to a Zeeman interaction both smaller and larger than the hyperfine energy. The results are given both for the total recombination yield and for the contribution from the two blocks of the Hamiltonian; where positive (negative) field values correspond to m ¼ þ 12 m ¼ 12 , respectively. The error on the total recombination yield, shown by the broken curve, is smaller than that of the individual blocks. Figures calculated with other values of the dimensionless hyperfine constant ad2/D are almost identical to the one displayed in Fig.q 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi if the x-axis is scaled with ad2/D and the y-axis with ad 2 =D. The observed dependence of the percentage error, introduced by using the strong T-T dephasing approximation, can therefore be summaqffiffiffiffiffiffiffiffiffiffiffiffiffi rized as: The maximum error is approximately 10% ad 2 =D and is only realised in a very narrow region around B = 0, approximately gbB 6 a/10. It decays exponentially fast to an intermediate plateau value.
6. Conclusion In this work we have presented a Green’s function method for solving the stochastic Liouville equation for local recombination and exchange interactions. The method generalises the steady state approach and the associated approximation introduced in a previous work [7]. The derived matrix equation (9) becomes particularly simple when supplemented with the assumption of a locally strong dephasing mechanism or applied to diffusion controlled recombination. In both cases the equation decouples into two smaller matrix equations. An even larger reduction of the problem is obtained by assuming locally strong T-T dephasing. The latter approximation is not so physically sound as the S-T dephasing, but the simplification is achieved without significantly affecting the accuracy of the results. An analytic expression was derived for the recombination yield of a geminate radical pair with a single magnetic nuclei with spin one half where both singlet and triplet states may be reactive. This relatively simple and accurate expression appears to be the most general expression that covers both low and high magnetic fields. The previous analytic low field works have included some restrictions, e.g. un-reactive triplet states [11] or zero field [12].
A.B. Doktorov et al. / Chemical Physics 328 (2006) 333–337
Appendix A. The inverse Green’s operator b b and H are diagonal in the The Green’s operator G b b are same basis. In this basis the diagonal elements of G Z 1 b b G eikp t eþikq t uðd; d;tÞest dt ¼ Gðiðkp kq ÞÞ; pq;pq ðd; d;sÞ ¼ 0
ðA:1Þ where u(d, d; t) is the conditional probability that a RP will be at contact at time t when it was at contact at time 0. This quantity is related to the diffusional Green’s function, introduced in Eq. (4), by Z 1 GðsÞ ¼ uðd; d; tÞest dt: ðA:2Þ 0
In the diagonal basis, the diagonal elements of the inverse Green’s operator is just the reciprocal of the Green’s function values of Eq. (A.1). In any other basis, e.g. the S-T basis, the matrix elements will be linear combinations of these characteristic values of the diffusional Green’s function. However, these linear combinations are identical for the Green’s operator and its inverse. Consequently, the inverse Green’s operator can be obtained from
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the Green’s operator by replacing all the Green’s function values by their inverse and keeping all other quantities unchanged. References [1] J.B. Pedersen, J.S. Jørgensen, A.I. Shushin, IV International Symposium on Magnetic Field and Spin Effects in Chemistry and Related Phenomena, Novosibirsk, Russia, 1996. [2] C.R. Timmel, U. Till, B. Brocklehurst, K.A. McLauchlan, P.J. Hore, Mol. Phys. 95 (1998) 71. [3] K.M. Salikhov, Chem. Phys. 82 (1983) 145. [4] A.A. Neufeld, M.J. Hansen, J.B. Pedersen, Chem. Phys. 278 (2002) 129. [5] R.C. Johnson, R.E. Merrifield, Phys. Rev. B1 (1970) 896. [6] J.B. Pedersen, A.I. Shushin, J.S. Jørgensen, Chem. Phys. 189 (1994) 479. [7] M.J. Hansen, J.B. Pedersen, Chem. Phys. Lett. 361 (2002) 219. [8] J.B. Pedersen, J.H. Freed, J. Chem. Phys. 61 (1974) 1517. [9] A.B. Doktorov, A.A. Neufeld, J.B. Pedersen, J. Chem. Phys. 110 (1999) 8869. [10] A.B. Doktorov, J.B. Pedersen, Chem. Phys. Lett. 423 (2006) 208. [11] P.A. Purtov, K.M. Salikhov, Teor. Eksp. Khim. 16 (1980) 737. [12] N.N. Lukzen, J.B. Pedersen, A.I. Burshtein, J. Phys. Chem. 109 (2005) 11914.