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General relations for the recombination yield of spin 12 particles
Chemical Physics Letters 423 (2006) 208–211 www.elsevier.com/locate/cplett
General relations for the recombination yield of spin 12 particles A.B. Doktorov a b
a,b
, J. Boiden Pedersen
a,*
Physics Department, University of Southern Denmark, DK-5230 Odense M, Denmark Institute of Chemical Kinetics and Combustion, SB RAS 630090, Novosibirsk, Russia Received 27 December 2005; in final form 23 March 2006 Available online 4 April 2006
Abstract This Letter summarizes the general relationships for the recombination yields through different reactive channels (singlet or triplet) of radical pairs with different precursor states. It is sufficient to calculate one of these combinations, e.g., the singlet recombination from a triplet precursor; the others follow by simple transformations. For strongly dephasing systems, the dependence on the singlet reactivity can be factored out and the recombination yield through any channel and for any precursor state can be expressed in terms of a quantity F , which is independent of the singlet reactivity. 2006 Elsevier B.V. All rights reserved.
1. Introduction The recombination of two particles with spin depends on the initial spin state of the pair (precursor state), on external magnetic fields, on the relative motion of the particles, on interaction between the particles, and on the magnetic properties of the individual particles. For example, a geminate pair of spin 12 particles (radicals) may be created photolytically in a triplet spin state but recombination occurs predominantly from the singlet spin state. The particles are typically created at contact but separate by diffusion. The initial coherent spin state then develops coherently due to the magnetic interactions (hyperfine and Zeemann interaction) and incoherently due to spin relaxation. In order to recombine the particles must meet again, and when they do, be in the right spin state. The recombination probability thus depends on the combined effect of the stochastic (diffusive) motion of the pair and of the quantum evolution of the spin state of the pair. A calculation of the recombination probability of particles with spin is complicated and contains many parameters. At present, only numerical methods can be used for *
Corresponding author. E-mail addresses: [email protected] (A.B. Doktorov), [email protected] (J. Boiden Pedersen). 0009-2614/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.03.078
all but the simplest systems. The exploration of the parameter space is time consuming and the analysis of the results are complicated due to the large number of parameters. The most important parameters are the decay rates through the reactive channels, the precursor state of the particles, the characteristics of the relative motion of the particles, relaxation times, and intra- and inter-particle interactions. There have been only a few previous attempts to rationalize the different dependences. A rather complete set of dependences was found by Pedersen and Freed [1,2] by analyzing their numerical high field results. Since these authors did neither include triplet recombination nor relaxation it remained unclear whether the relations were valid also for these cases. Recently, we proved analytically [3] that these relations are valid also in the presence of relaxation, and that they can be generalized to account for triplet recombination. However, this proof was also based on the high field case. One of the relations, giving a relationship between the singlet recombination yields of singlet and triplet precursors has previously been derived for arbitrary field strength [4], but only for singlet reactivity, and more recently for a general system [5]. But, it was not expected that the remaining high field relations could be generalized to arbitrary fields and to radical pairs (RPs) with many nuclei.
A.B. Doktorov, J. Boiden Pedersen / Chemical Physics Letters 423 (2006) 208–211
In the present Letter we report a complete generalization of the high field relations to arbitrary field strengths. They relate the recombination yields for different precursor states as well as recombination yields through different channels but for the same precursor. Moreover we have, rather surprisingly, shown that the dependence on the singlet channel decay rate can be given analytically for systems with a strong singlet–triplet dephasing, e.g., caused by a strong exchange interaction or a fast decay through one of the reactive channels. This result is a generalization of the high field expression but, as might be expected, is more complex although the structure is the same. All these relations are valid for any relative motion of the partners, in the presence of relaxation, and for double channel recombination (i.e., when both singlet and triplet pairs are reactive). The formulas imply that it is sufficient to calculate the singlet diffusion controlled recombination through the singlet channel for a triplet precursor. The recombination yield for the other parameter combinations can be derived analytically from this quantity. The complete derivation of the relations, based on a Green’s function technique, will be given separately [6]. It is assumed that the recombination can be described as a contact recombination, which for many applications is a useful approximation. It is, at present, not clear whether the relations are also valid for long range transfer reactions.
209
and a superscript indicating the precursor spin state. Notice that although RS indicates the recombination yield through the singlet channel it does not exclude that the triplet channel is also reactive. The relation connecting the singlet and triplet recombination yields for the same, arbitrary precursor state, is RS RT þ ¼ 1. KS KT
ð2Þ
This relation was recently derived for high fields [3] and later for arbitrary fields [5], in a slightly different form. It implies that the triplet recombination yield RT and the escape yield (Y ¼ 1 RS RT ) can be calculated directly from the singlet recombination yield and thus only the singlet recombination yield is needed. The relation connecting the singlet recombination yield for singlet and triplet precursors is ðRSS KS Þð1 KT Þ ¼ 3RTS ð1 KS Þ
ð3Þ
Notice, that the relation is valid even when the triplet state is also reactive. This, rather deep relationship was recently derived in its most general form for high fields [3] and arbitrary fields [6]. It generalizes the previous results for unreactive triplets derived for high field [1,7] and low field [4], respectively. 3. Relations for strong dephasing
2. General relations There exist two general relations, that are valid for any spin dynamics and any model for the relative motion of the radicals. The first, and simplest relation connects the recombination yields through the two reactive channels and has only recently been discovered [3,5], probably because two channel recombination was not previously considered or observed. The second relation relates the singlet recombination yield for different precursor spin states and generalizes the previous high field [1,7] and low field [4] results for unreactive triplets. It is convenient to use the dimensionless recombination radii for the singlet and triplet channel, KS and KT, to describe the reactivity rather than giving the decay rates through the reactive channels. For contact like recombination, they can be expressed as Kj ¼
k j sj 1 þ k j sj
j ¼ S; T
ð1Þ
where kj is the decay rate through channel j and sj is the total time spend in the reaction zone. In general neither the decay constants nor the residence times in the reaction zones are known and thus the recombination radii may be considered experimental parameters. These are purely classical quantities, and e.g., KS can be interpreted as the recombination yield of a singlet radical pair, formed at the contact distance, in the absence of quantum interactions. The spin dependent recombination yield is denoted R with a subscript indicating the recombination channel
Perhaps the most important and surprising result is that the dependence on the singlet reactivity KS can be given explicitly for strong dephasing, e.g., due to a strong exchange interaction. These relations are generalizations of the high field relations [3] which, for unreactive triplets, were first discovered by an analysis of numerical data [2]. In high magnetic fields, the nuclear spin states are conserved and thus only appear as a parameter. However, this is no longer true at low magnetic fields, where it is necessary to consider the nuclear spin configurations in details. For ease in presentation, we will first consider the high field case and then introduces the modifications needed for the general case with multi-nuclei radical pairs. 3.1. The high field case 3.1.1. Singlet recombination yield The singlet recombination yield for a triplet precursor can be written as [3] RTS ¼
1 KS ð1 KT ÞF 3 1 þ F ð1 KS Þ
ð4Þ
where the fundamental quantity F , defined as F ¼ lim
KS!1
3RTS 1 KT
ð5Þ
is independent of the singlet reactivity KS but may depend on the triplet reactivity KT. All information on the spin dynamics and the diffusional motion is contained in this
210
A.B. Doktorov, J. Boiden Pedersen / Chemical Physics Letters 423 (2006) 208–211
fundamental quantity. For free diffusion, in the absence of relaxation, and for unreactive triplets this is given by the pffiffiffi pffiffiffi well know expression q=ð2 þ qÞ where the dimensionless quantity q = Qd2/D and Q is half the difference between the resonance frequencies of the two radicals in the specified nuclear spin state. From Eq. (4) and the general relations (3) and (2) one can easily derive expressions for all quantities of interest. The expression for the singlet yield from a singlet precursor is thus RSS ¼
KS 1 þ F ð1 KS Þ
ð6Þ
or in an alternative, more symmetric, form KS RSS ¼
KS ð1 KS ÞF 1 þ F ð1 KS Þ
ð7Þ
3.1.2. Triplet recombination yield Explicit expressions for the triplet recombination yields for the two precursor spin states are similarly derived by use of the above relations. One obtains 1 KT ð1 KT ÞF 3 1 þ F ð1 KS Þ KT ð1 KS ÞF RST ¼ 1 þ F ð1 KS Þ
RTT ¼ KT
ð8Þ ð9Þ
The first expression can be written in a more symmetric form, similar to Eq. (7), as KT RTT ¼
1 KT ð1 KT ÞF 3 1 þ F ð1 KS Þ
ð10Þ
3.1.3. Total recombination yield The total recombination yield, i.e., the sum of the singlet and triplet recombination yields for the same precursor state can be written as 1 ðKS KT Þð1 KT ÞF 3 1 þ F ð1 KS Þ ðKT KS Þð1 KS ÞF RS ¼ K S þ 1 þ F ð1 KS Þ RT ¼ K T þ
ð11Þ ð12Þ
These expressions show, as expected, that there is no magnetic field dependence if the singlet and triplet states are equally reactive (KS = KT). 3.1.4. Escape yield In some experiments, the escape yield Y rather than the recombination yield is observed. Expressions for this quantity is easily obtained from the definition Y ¼ 1 R ¼ 1 RT RS and the above relations. 3.2. Single nucleus radical pair in low fields In high fields the Hamiltonian is diagonal in the nuclear spin states and the Eqs. (4) and (5) are therefore valid for
any nuclear spin state consisting of a direct product of the individual nuclear spin states, and the nuclear spin state only enters as a parameter indicating the hyperfine energy of the state. For a single magnetic nuclei 12 in low fields, the situation is slightly more complicated. Neglecting relaxation, the Hamiltonian is block diagonal with the blocks denoted by the z-component of the total spin, electrons plus nuclei. There are 4 blocks, of which two do not contribute to singlet recombination as they contain only aaa and bbb, respectively. The notation used for the direct product spin state of the radical pair is (ve1ve2vn). The remaining two blocks have singlet character, and as basis for these blocks we can use {aba, baa, aab} and {abb, bab, bba}, where the order is electron spin 1, electron spin 2, and the nuclear spin. For each of these blocks, Eqs. (4) and (5) are valid. At first, it is rather surprising that the relations for a one nuclei RP in an arbitrary field are identical to the high field relations. The following discussion of the multi-nuclei case will give some indication as to why this is the case. Actually, there is one more simplification for this case. The two blocks corresponds to opposite directions of the magnetic field, and it is therefore sufficient to calculate F as a function of the magnetic field B for one of the blocks; the result for the other block is obtained by simply reversing the sign of B. It is clear that the above relations tremendously simplify the calculations of the parameter dependences of the recombination probability. It is sufficient to calculate the scalar quantity F , that only depends on the magnetic interaction and the triplet reactivity KT. In a recent work [8], an approximate, but accurate expression for the diffusion controlled singlet reactivity of a triplet precursor was derived; triplet pairs were assumed unreactive. This quantity is precisely F and by using the above relations, this expression can now be easily generalized to arbitrary singlet reactivity and other precursor states. 3.3. Multi-nuclei radical pairs in low fields The generalizations of the above relations, Eqs. (6) and (4), for singlet recombination yield to multi-nuclei RPs can be written as ^^ 1 0 ^S RSS ¼ KS Tr½ð1 þ ð1 KS ÞF Þ q
ð13Þ
and 1 ^^ 1 ^^ 0 ^S RTS ¼ KS ð1 KT ÞTr½ð1 þ ð1 KS ÞF Þ Fq ð14Þ 3 ^^ is a matrix of dimension N2 where N is the numwhere F ber of nuclear spin configurations that are connected with ^0S is the initial spin denthe singlet electron spin state, and q sity matrix in the singlet electron spin subspace, i.e., it has ^^ dimension N2. The quantity F is independent of KS, similar to the high field and single nuclei case. If the RP is only exposed to a static magnetic field of arbitrary strength, i.e., no rf-field, then the Hamiltonian
A.B. Doktorov, J. Boiden Pedersen / Chemical Physics Letters 423 (2006) 208–211
is block diagonal, where each block is characterized by the z-component of the total spin (including all electrons and nuclei). The total recombination probability (yield) can then be written as a sum of contributions from the individual blocks, i.e., X RSS ¼ RSiS ð15Þ i
where the sum is over the blocks denoted i. The crucial ^ ^ is now smalpoint is that the dimensions of the matrix F ler since only the nuclear spin configurations in the block that are associated with the singlet electron spin state are included. Let this number be n. For a single nuclear spin one half, the block with m ¼ 12 contains only one nuclear spin state, the a state, cf. the previous section and thus F becomes a scalar quantity. For radical pairs with two inequivalent nuclei, the blocks with singlet character are specified by m = ±1, 0. The m = ±1 blocks contain only one nuclear spin configuration aa and bb, respectively and thus F is a scalar for these two blocks. The m = 0 block contains two nuclear spin configu^ ^ is therefore a 4 · 4 matrix. rations ab and ba and F In the presence of a rf-field, the Hamiltonian is no longer block diagonal and all 4 states has to be included. ^ ^ becomes a 16 · 16 matrix. However, even for this Thus F case the calculational resources (cpu time) is dramatically reduced since one can vary KS essentially without additional computer time. The diffusion controlled singlet recombination is particularly simple since Eq. (14) reduces to 1 ^ ^q ^0S RTS ¼ ð1 KT ÞTr½F 3
ð16Þ
211
^ ^ are which implies that only the diagonal elements of F needed. 4. Conclusion In this Letter we have presented the complete set of general relations for spin dependent recombination problems that are known today and we have discussed the computational simplifications that result from their use. The derivation of some of the relations are complicated and will be presented elsewhere together with a discussion of possible ^^ procedures for calculations of the quantity F . Acknowledgement This work has been partially supported by a research grant from the Danish Council for Strategic Research under the program: Non-ionizing radiation. References [1] [2] [3] [4]
[5] [6] [7] [8]
J.B. Pedersen, J.H. Freed, J. Chem. Phys. 61 (1974) 1517. J.H. Freed, J.B. Pedersen, Adv. Mag. Res. 8 (1976) 1. A.B. Doktorov, J.B. Pedersen, Chem. Phys. 322 (2006) 433. K.M Salikhov, Y.N Molin, R.Z Sagdeev, A.L Buchachenko, Spin Polarization and Magnetic Field Effects in Radical Reactions, Elsevier, Amsterdam, 1984. N.N. Lukzen, J.B. Pedersen, A.I. Burshtein, J. Phys. Chem. A 109 (2005) 11914. A.B. Doktorov, J.B. Pedersen, Chem. Phys., in preparation. M.J. Hansen, A.A. Neufeld, J.B. Pedersen, Chem. Phys. 260 (2000) 125. M.J. Hansen, J.B. Pedersen, Chem. Phys. Lett. 361 (2002) 219.
General relations for the recombination yield of spin 12 particles A.B. Doktorov a b
a,b
, J. Boiden Pedersen
a,*
Physics Department, University of Southern Denmark, DK-5230 Odense M, Denmark Institute of Chemical Kinetics and Combustion, SB RAS 630090, Novosibirsk, Russia Received 27 December 2005; in final form 23 March 2006 Available online 4 April 2006
Abstract This Letter summarizes the general relationships for the recombination yields through different reactive channels (singlet or triplet) of radical pairs with different precursor states. It is sufficient to calculate one of these combinations, e.g., the singlet recombination from a triplet precursor; the others follow by simple transformations. For strongly dephasing systems, the dependence on the singlet reactivity can be factored out and the recombination yield through any channel and for any precursor state can be expressed in terms of a quantity F , which is independent of the singlet reactivity. 2006 Elsevier B.V. All rights reserved.
1. Introduction The recombination of two particles with spin depends on the initial spin state of the pair (precursor state), on external magnetic fields, on the relative motion of the particles, on interaction between the particles, and on the magnetic properties of the individual particles. For example, a geminate pair of spin 12 particles (radicals) may be created photolytically in a triplet spin state but recombination occurs predominantly from the singlet spin state. The particles are typically created at contact but separate by diffusion. The initial coherent spin state then develops coherently due to the magnetic interactions (hyperfine and Zeemann interaction) and incoherently due to spin relaxation. In order to recombine the particles must meet again, and when they do, be in the right spin state. The recombination probability thus depends on the combined effect of the stochastic (diffusive) motion of the pair and of the quantum evolution of the spin state of the pair. A calculation of the recombination probability of particles with spin is complicated and contains many parameters. At present, only numerical methods can be used for *
Corresponding author. E-mail addresses: [email protected] (A.B. Doktorov), [email protected] (J. Boiden Pedersen). 0009-2614/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.03.078
all but the simplest systems. The exploration of the parameter space is time consuming and the analysis of the results are complicated due to the large number of parameters. The most important parameters are the decay rates through the reactive channels, the precursor state of the particles, the characteristics of the relative motion of the particles, relaxation times, and intra- and inter-particle interactions. There have been only a few previous attempts to rationalize the different dependences. A rather complete set of dependences was found by Pedersen and Freed [1,2] by analyzing their numerical high field results. Since these authors did neither include triplet recombination nor relaxation it remained unclear whether the relations were valid also for these cases. Recently, we proved analytically [3] that these relations are valid also in the presence of relaxation, and that they can be generalized to account for triplet recombination. However, this proof was also based on the high field case. One of the relations, giving a relationship between the singlet recombination yields of singlet and triplet precursors has previously been derived for arbitrary field strength [4], but only for singlet reactivity, and more recently for a general system [5]. But, it was not expected that the remaining high field relations could be generalized to arbitrary fields and to radical pairs (RPs) with many nuclei.
A.B. Doktorov, J. Boiden Pedersen / Chemical Physics Letters 423 (2006) 208–211
In the present Letter we report a complete generalization of the high field relations to arbitrary field strengths. They relate the recombination yields for different precursor states as well as recombination yields through different channels but for the same precursor. Moreover we have, rather surprisingly, shown that the dependence on the singlet channel decay rate can be given analytically for systems with a strong singlet–triplet dephasing, e.g., caused by a strong exchange interaction or a fast decay through one of the reactive channels. This result is a generalization of the high field expression but, as might be expected, is more complex although the structure is the same. All these relations are valid for any relative motion of the partners, in the presence of relaxation, and for double channel recombination (i.e., when both singlet and triplet pairs are reactive). The formulas imply that it is sufficient to calculate the singlet diffusion controlled recombination through the singlet channel for a triplet precursor. The recombination yield for the other parameter combinations can be derived analytically from this quantity. The complete derivation of the relations, based on a Green’s function technique, will be given separately [6]. It is assumed that the recombination can be described as a contact recombination, which for many applications is a useful approximation. It is, at present, not clear whether the relations are also valid for long range transfer reactions.
209
and a superscript indicating the precursor spin state. Notice that although RS indicates the recombination yield through the singlet channel it does not exclude that the triplet channel is also reactive. The relation connecting the singlet and triplet recombination yields for the same, arbitrary precursor state, is RS RT þ ¼ 1. KS KT
ð2Þ
This relation was recently derived for high fields [3] and later for arbitrary fields [5], in a slightly different form. It implies that the triplet recombination yield RT and the escape yield (Y ¼ 1 RS RT ) can be calculated directly from the singlet recombination yield and thus only the singlet recombination yield is needed. The relation connecting the singlet recombination yield for singlet and triplet precursors is ðRSS KS Þð1 KT Þ ¼ 3RTS ð1 KS Þ
ð3Þ
Notice, that the relation is valid even when the triplet state is also reactive. This, rather deep relationship was recently derived in its most general form for high fields [3] and arbitrary fields [6]. It generalizes the previous results for unreactive triplets derived for high field [1,7] and low field [4], respectively. 3. Relations for strong dephasing
2. General relations There exist two general relations, that are valid for any spin dynamics and any model for the relative motion of the radicals. The first, and simplest relation connects the recombination yields through the two reactive channels and has only recently been discovered [3,5], probably because two channel recombination was not previously considered or observed. The second relation relates the singlet recombination yield for different precursor spin states and generalizes the previous high field [1,7] and low field [4] results for unreactive triplets. It is convenient to use the dimensionless recombination radii for the singlet and triplet channel, KS and KT, to describe the reactivity rather than giving the decay rates through the reactive channels. For contact like recombination, they can be expressed as Kj ¼
k j sj 1 þ k j sj
j ¼ S; T
ð1Þ
where kj is the decay rate through channel j and sj is the total time spend in the reaction zone. In general neither the decay constants nor the residence times in the reaction zones are known and thus the recombination radii may be considered experimental parameters. These are purely classical quantities, and e.g., KS can be interpreted as the recombination yield of a singlet radical pair, formed at the contact distance, in the absence of quantum interactions. The spin dependent recombination yield is denoted R with a subscript indicating the recombination channel
Perhaps the most important and surprising result is that the dependence on the singlet reactivity KS can be given explicitly for strong dephasing, e.g., due to a strong exchange interaction. These relations are generalizations of the high field relations [3] which, for unreactive triplets, were first discovered by an analysis of numerical data [2]. In high magnetic fields, the nuclear spin states are conserved and thus only appear as a parameter. However, this is no longer true at low magnetic fields, where it is necessary to consider the nuclear spin configurations in details. For ease in presentation, we will first consider the high field case and then introduces the modifications needed for the general case with multi-nuclei radical pairs. 3.1. The high field case 3.1.1. Singlet recombination yield The singlet recombination yield for a triplet precursor can be written as [3] RTS ¼
1 KS ð1 KT ÞF 3 1 þ F ð1 KS Þ
ð4Þ
where the fundamental quantity F , defined as F ¼ lim
KS!1
3RTS 1 KT
ð5Þ
is independent of the singlet reactivity KS but may depend on the triplet reactivity KT. All information on the spin dynamics and the diffusional motion is contained in this
210
A.B. Doktorov, J. Boiden Pedersen / Chemical Physics Letters 423 (2006) 208–211
fundamental quantity. For free diffusion, in the absence of relaxation, and for unreactive triplets this is given by the pffiffiffi pffiffiffi well know expression q=ð2 þ qÞ where the dimensionless quantity q = Qd2/D and Q is half the difference between the resonance frequencies of the two radicals in the specified nuclear spin state. From Eq. (4) and the general relations (3) and (2) one can easily derive expressions for all quantities of interest. The expression for the singlet yield from a singlet precursor is thus RSS ¼
KS 1 þ F ð1 KS Þ
ð6Þ
or in an alternative, more symmetric, form KS RSS ¼
KS ð1 KS ÞF 1 þ F ð1 KS Þ
ð7Þ
3.1.2. Triplet recombination yield Explicit expressions for the triplet recombination yields for the two precursor spin states are similarly derived by use of the above relations. One obtains 1 KT ð1 KT ÞF 3 1 þ F ð1 KS Þ KT ð1 KS ÞF RST ¼ 1 þ F ð1 KS Þ
RTT ¼ KT
ð8Þ ð9Þ
The first expression can be written in a more symmetric form, similar to Eq. (7), as KT RTT ¼
1 KT ð1 KT ÞF 3 1 þ F ð1 KS Þ
ð10Þ
3.1.3. Total recombination yield The total recombination yield, i.e., the sum of the singlet and triplet recombination yields for the same precursor state can be written as 1 ðKS KT Þð1 KT ÞF 3 1 þ F ð1 KS Þ ðKT KS Þð1 KS ÞF RS ¼ K S þ 1 þ F ð1 KS Þ RT ¼ K T þ
ð11Þ ð12Þ
These expressions show, as expected, that there is no magnetic field dependence if the singlet and triplet states are equally reactive (KS = KT). 3.1.4. Escape yield In some experiments, the escape yield Y rather than the recombination yield is observed. Expressions for this quantity is easily obtained from the definition Y ¼ 1 R ¼ 1 RT RS and the above relations. 3.2. Single nucleus radical pair in low fields In high fields the Hamiltonian is diagonal in the nuclear spin states and the Eqs. (4) and (5) are therefore valid for
any nuclear spin state consisting of a direct product of the individual nuclear spin states, and the nuclear spin state only enters as a parameter indicating the hyperfine energy of the state. For a single magnetic nuclei 12 in low fields, the situation is slightly more complicated. Neglecting relaxation, the Hamiltonian is block diagonal with the blocks denoted by the z-component of the total spin, electrons plus nuclei. There are 4 blocks, of which two do not contribute to singlet recombination as they contain only aaa and bbb, respectively. The notation used for the direct product spin state of the radical pair is (ve1ve2vn). The remaining two blocks have singlet character, and as basis for these blocks we can use {aba, baa, aab} and {abb, bab, bba}, where the order is electron spin 1, electron spin 2, and the nuclear spin. For each of these blocks, Eqs. (4) and (5) are valid. At first, it is rather surprising that the relations for a one nuclei RP in an arbitrary field are identical to the high field relations. The following discussion of the multi-nuclei case will give some indication as to why this is the case. Actually, there is one more simplification for this case. The two blocks corresponds to opposite directions of the magnetic field, and it is therefore sufficient to calculate F as a function of the magnetic field B for one of the blocks; the result for the other block is obtained by simply reversing the sign of B. It is clear that the above relations tremendously simplify the calculations of the parameter dependences of the recombination probability. It is sufficient to calculate the scalar quantity F , that only depends on the magnetic interaction and the triplet reactivity KT. In a recent work [8], an approximate, but accurate expression for the diffusion controlled singlet reactivity of a triplet precursor was derived; triplet pairs were assumed unreactive. This quantity is precisely F and by using the above relations, this expression can now be easily generalized to arbitrary singlet reactivity and other precursor states. 3.3. Multi-nuclei radical pairs in low fields The generalizations of the above relations, Eqs. (6) and (4), for singlet recombination yield to multi-nuclei RPs can be written as ^^ 1 0 ^S RSS ¼ KS Tr½ð1 þ ð1 KS ÞF Þ q
ð13Þ
and 1 ^^ 1 ^^ 0 ^S RTS ¼ KS ð1 KT ÞTr½ð1 þ ð1 KS ÞF Þ Fq ð14Þ 3 ^^ is a matrix of dimension N2 where N is the numwhere F ber of nuclear spin configurations that are connected with ^0S is the initial spin denthe singlet electron spin state, and q sity matrix in the singlet electron spin subspace, i.e., it has ^^ dimension N2. The quantity F is independent of KS, similar to the high field and single nuclei case. If the RP is only exposed to a static magnetic field of arbitrary strength, i.e., no rf-field, then the Hamiltonian
A.B. Doktorov, J. Boiden Pedersen / Chemical Physics Letters 423 (2006) 208–211
is block diagonal, where each block is characterized by the z-component of the total spin (including all electrons and nuclei). The total recombination probability (yield) can then be written as a sum of contributions from the individual blocks, i.e., X RSS ¼ RSiS ð15Þ i
where the sum is over the blocks denoted i. The crucial ^ ^ is now smalpoint is that the dimensions of the matrix F ler since only the nuclear spin configurations in the block that are associated with the singlet electron spin state are included. Let this number be n. For a single nuclear spin one half, the block with m ¼ 12 contains only one nuclear spin state, the a state, cf. the previous section and thus F becomes a scalar quantity. For radical pairs with two inequivalent nuclei, the blocks with singlet character are specified by m = ±1, 0. The m = ±1 blocks contain only one nuclear spin configuration aa and bb, respectively and thus F is a scalar for these two blocks. The m = 0 block contains two nuclear spin configu^ ^ is therefore a 4 · 4 matrix. rations ab and ba and F In the presence of a rf-field, the Hamiltonian is no longer block diagonal and all 4 states has to be included. ^ ^ becomes a 16 · 16 matrix. However, even for this Thus F case the calculational resources (cpu time) is dramatically reduced since one can vary KS essentially without additional computer time. The diffusion controlled singlet recombination is particularly simple since Eq. (14) reduces to 1 ^ ^q ^0S RTS ¼ ð1 KT ÞTr½F 3
ð16Þ
211
^ ^ are which implies that only the diagonal elements of F needed. 4. Conclusion In this Letter we have presented the complete set of general relations for spin dependent recombination problems that are known today and we have discussed the computational simplifications that result from their use. The derivation of some of the relations are complicated and will be presented elsewhere together with a discussion of possible ^^ procedures for calculations of the quantity F . Acknowledgement This work has been partially supported by a research grant from the Danish Council for Strategic Research under the program: Non-ionizing radiation. References [1] [2] [3] [4]
[5] [6] [7] [8]
J.B. Pedersen, J.H. Freed, J. Chem. Phys. 61 (1974) 1517. J.H. Freed, J.B. Pedersen, Adv. Mag. Res. 8 (1976) 1. A.B. Doktorov, J.B. Pedersen, Chem. Phys. 322 (2006) 433. K.M Salikhov, Y.N Molin, R.Z Sagdeev, A.L Buchachenko, Spin Polarization and Magnetic Field Effects in Radical Reactions, Elsevier, Amsterdam, 1984. N.N. Lukzen, J.B. Pedersen, A.I. Burshtein, J. Phys. Chem. A 109 (2005) 11914. A.B. Doktorov, J.B. Pedersen, Chem. Phys., in preparation. M.J. Hansen, A.A. Neufeld, J.B. Pedersen, Chem. Phys. 260 (2000) 125. M.J. Hansen, J.B. Pedersen, Chem. Phys. Lett. 361 (2002) 219.