Recombination and separation of photochemically created radical–ion pairs subjected to incoherent spin-conversion

Chemical Physics 323 (2006) 341–350 www.elsevier.com/locate/chemphys

Recombination and separation of photochemically created radical–ion pairs subjected to incoherent spin-conversion A.I. Burshtein

*

Weizmann Institute of Science, Rehovot 76100, Israel Received 10 May 2005; accepted 10 October 2005 Available online 7 November 2005

Abstract The recombination/separation of the photochemically created singlet radical pair to the ground state and excited triplet products is studied. The spin-conversion in a pair is considered as a stochastic (incoherent) process, assuming that the recombination of both singlet and triplet radical pairs is contact. The quantum yields of recombination products are calculated for any initial separation of radicals in a pair. A special attention is paid to the dissipation of contact born pairs from both singlet and triplet states or via one of them. It is shown that the viscosity dependence of the product yields of the double-channel recombination differs qualitatively from the single-channel one whatever channel is open for the recombination. The validity range of incoherent description of spin-conversion is clarified by a straightforward comparison of the recombination efficiencies with 1those obtained within the Hamiltonian description of spin-conversion. In addition the role of the radical pair distribution over initial (starting) distance is discussed as well as contact approximation for recombination. Ó 2005 Elsevier B.V. All rights reserved.

1. Introduction In a number of our and other works [1–7] the incoherent model of spin-conversion in radical pairs was used as a reasonable approximation. It implies that the spin transitions between all spin sub-levels of a pair occur with the rate constants which originated from the Dg or hyperfine interaction when the spin relaxation is relatively fast. In the simplest model of incoherent spin-conversion in the photoinduced radical ion pair (RIP), the transition rates from and to the singlet state differ three times [1,2]. An acceptor of electron (A) excited at t = 0 meets with electron donor (D) diffusing in a liquid solution and the electron transfer (ionization) during their encounter produces RIPs subjected to spin-conversion, recombination and separation. After spin-conversion the recombination proceeds to either the ground state (from singlet RIP) or to the excited triplet state 3A* (from triplet RIP) (see Scheme 1). *

Tel.: +972 8934 3708; fax: +972 8934 4123. E-mail address: [email protected].

0301-0104/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.10.008

If there is no recombination as well as ion separation, the spin-conversion in a zero magnetic field results in equipartition of the spin population between all spin sub-levels. Since the triplet consists of three of them (T-, T0, T+) its equilibrium population, pT, is three times larger than that of the singlet, pS. This is a stationary solution of kinetic equations for these quantities: p_ S ¼ 3k s pS þ k s pT ; p_ T ¼ 3k s pS  k s pT .

ð1:2aÞ ð1:2bÞ

They constituted a formal basis to account for the incoherent spin-conversion in either Unified or Integral Encounter Theories: UT [1] and IET [2]. As a matter of fact Eqs. (1.2) represent only a rough model of what happens in reality. Even in a limit when the transverse spin relaxation times, T2, is fast enough to make conversion incoherent the true set of 4 equations for populations of singlet level, qS  pS, and three sub-levels of triplet state, q, q0 and q+ should be considered instead. These are Eqs. (8.8) in [1] obtained from the relaxation operator proposed earlier [8]:

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A.I. Burshtein / Chemical Physics 323 (2006) 341–350

Then from the four equations (1.3), remain only the first two: q_ S ¼ k s ðq0  qS Þ; q_ 0 ¼ k s ðqS  q0 Þ; where   1 ks ¼ k0 þ . T2

Scheme 1.

 q_ S ¼

   1 1 1 1 q þ q k0 þ  ; q0  k 0 þ þ qS þ þ T 2 2T 1 T 2 2T 1 2T 1

 q_ 0 ¼



k0 þ





ð1:3aÞ

1 1 1 1 q þ q  ; qS  k 0 þ þ q0 þ þ T 2 2T 1 T 2 2T 1 2T 1 ð1:3bÞ

q_ þ ¼

qS þ q0 qþ  2T 1 T1

and

q_  ¼

qS þ q0 q  ; 2T 1 T1

ð1:3cÞ

2

k0 ¼ X T 2 is the rate of incoherent spin-conversion. If it is carried out by Dg-mechanism the conversion frequency xA  xB ; where x ¼ gbH = h ð1:4Þ X¼ 2 is a frequency of magnetic resonance in field H, which is different for two radicals in a pair, A and B, provided they have different g-factors (Dg = gA  gB 5 0) [9]. In principle, the transversal and longitudinal relaxation times are also different, T2 6 T1, and Eqs. (1.3) hold when T2 is rather short k 0 T 2  1 that is XT 2  1.

ð1:5Þ

This is the necessary condition for the spin-conversion to be incoherent when radicals in a pair are immobile and stable which they are not. In liquid solutions the encounter time of the reactants sd is finite and during this time RIP is subjected to recombination. It will be shown that under these conditions the incoherent description of spin-conversion makes the theory semi-quantitative even in a weak magnetic field obeying the inequality (1.5). The incoherent approximation is better if sd  T2 and worse if recombination is too fast (diffusional). 2. Two level system If the longitudinal relaxation is too slow it can not be ~ completed during the short encounter time sd ¼ r2 =D, ~ where r is contact distance between the reactants and D is the coefficient of the encounter diffusion in the RIPs. Moreover, one can completely neglect this relaxation provided T 1  T 2.

ð2:1Þ

ð2:3Þ

Unlike equations Eqs. (1.2) representing the rough model the set (2.2) is not a model at all but the regular ‘‘golden rule’’ approximation providing the rate description of spin-conversion accompanied by much faster transversal relaxation. Such a description is a bit different if the relaxation times T1 and T2 are comparable or even equal. This situation considered in Appendix A is described by the set of three rate equations instead of two equations (2.2). The total population of the interacting levels, q = q0 + qS, is conserved in time, so that Eqs. (2.2) may be rewritten as follows: q_ 0 ¼ 2k s q0 þ k s q; q_ ¼ 0.

where

ð2:2aÞ ð2:2bÞ

ð2:4aÞ ð2:4bÞ

The reduction of the problem to only two interacting states is also useful for treating other mechanisms of the spin-conversion. For instance, such a reduction is common for the conversion induced by the hyperfine interaction (HFI) in radical pairs when the large splitting of triplet in high magnetic field turns T± out of the game [9–11]. For the RIP with only a single proton the conversion frequency X = a/2, where a is the hyperfine splitting in organic radical. However, the spin-conversion carried out by the HFI mechanism can hardly be considered as incoherent, proceeding with the 2 rate k 0 ¼ X2 T 2 ¼ a4 T 2 . The spin-conversion frequency in radicals X  a is usually about 107–109 s1, while the relaxation times are relatively long: T1,T2  105–106 s. That is XT2  100–1000 is very large, that contradicts with inequality (1.5) making the incoherent model of spin-conversion inapplicable. The coherent (Hamiltonian) description used instead leads to the set of three equations for diagonal and off-diagonal elements of the density matrix. The reduction of these equations and their solutions to the incoherent analogs of them studied here is possible under definite conditions discussed in Appendix B. Adding to Eqs. (2.2) the recombination of the singlet and triplet RIPs assisted by encounter diffusion, we obtain the complete kinetic equations: q_ S ¼ k s q0  k s qS þ LqS  W S ðrÞqS ; q_ 0 ¼ k s q0 þ k s qS þ Lq0  W T ðrÞq0 ;

ð2:5aÞ ð2:5bÞ

where ~ L¼D

1 o 2 rc =r o rc =r re e r2 or or

ð2:6Þ

is an operator of the encounter diffusion of counter-ions in the Coulomb well with Onsager radius rc = e2/kBT (at

A.I. Burshtein / Chemical Physics 323 (2006) 341–350

temperature T and dielectric constant ). As usual, we presume that r is much greater than the discretion length of a medium and the encounter time sd is longer than the dielectric relaxation time of a medium. The set of Eqs. (2.5) should be solved with reflecting boundary conditions at contact: jqS jr¼r ¼ 0;

jq0 jr¼r ¼ 0;

3. Contact recombination In most of the previous works the problem was greatly simplified by using the contact approximation for recombination. In such an approximation the space dependence of recombination rates is ignored and instead of them the contact rate constants Z Z 3 T S k c ¼ W T ðrÞ d r and k c ¼ W S ðrÞ d3 r; ð3:1Þ are used in the boundary conditions presented below. This allows us to neglect the recombination in Eqs. (2.5) and obtain the following set of equations for RIPs starting from distance r0 where they were born in the singlet state: q_ S ¼ k s q0  k s qS þ LqS ; qS ð0Þ ¼ dðr  r0 Þ=4pr2 ; q_ 0 ¼ k s q0 þ k s qS þ Lq0 ; q0 ð0Þ ¼ 0.

ð3:2aÞ ð3:2bÞ

The contact RIPs recombination is accounted for by the boundary conditions for these equations: jqS jr¼r ¼ k Sc qS ðr; r0 ; tÞ;

jq0 jr¼r ¼ k Tc q0 ðr; r0 ; tÞ.

ð3:3Þ

Alternatively we can present the same set of kinetic equations using other variables, q0 and q, as in Eqs. (2.4). Making the Laplace transformation of them we have ~0 þ k s q ~; j~ ~0 ðr; r0 ; sÞ; s~ q0 ¼ Lq0  2k s q q0 ¼ k Tc q 2

q¼ s~ q ¼ L~ q þ dðr  r0 Þ=4pr ; j~

G_ ¼ LG; Gðr; r0 ; 0Þ ¼ dðr  r0 Þ=4pr2 ; .jGjr¼r ¼ k Tc Gðr; r0 ; tÞ.

ð3:4aÞ

~ðr; r0 ; sÞ þ j~ k Sc q q0 ðr; r0 ; sÞ;

c

þ j~ q0 ðr; r0 ; sÞ;

~0 ðr; r0 ; 0Þ; ut ðr0 Þ ¼ k Tc q ð3:5aÞ Z ~ðr; r0 ; 0Þ  j~ qðsÞ d3 r ¼ 1  k Sc q q0 ðr; r0 ; 0Þ. uðr0 Þ ¼ lim s~ s!0

ð3:5bÞ Here the Eq. (3.4b) integrated over space has been used to obtain the last result, Eq. (3.5b). The general solution of Eq. (3.4a), Z ~ r0 ; s þ 2k s Þ~ ~0 ðr; r0 ; sÞ ¼ k s Gðr; q qðr0 ; r0 ; sÞ d3 r0 ; ð3:6Þ is expressed here through the Green function which obeys the following equation:

ð3:8Þ

where G0 obeys the free diffusion equation with different initial and reflecting boundary conditions: G_ 0 ¼ LG0 ; G0 ðr; r0 ; 0Þ ¼ dðr  rÞ. ð3:9Þ Integrating Eq. (3.8) over r we return to the already obtained result (3.5b). ~ r0 ; sÞ and G~0 ðr; r0 ; sÞ are related The Green functions Gðr; to each other as in [13,14] ~ r0 ; sÞ ¼ G~0 ðr; r0 ; sÞ  k T G ~ r0 ; sÞ. ~ 0 ðr; r; sÞGðr; Gðr; c

At r = r it follows from this relationship that ~ 0 ðr; r0 ; sÞ G ~ r0 ; sÞ ¼ . Gðr; ~ 0 ðr; r; sÞ 1 þ kTG c

Setting now r = r in Eq. (3.8) we can resolve it for ~ðr; r0 ; sÞ and find the following expression for the density q of ion pairs at the contact: ~ 0 ðr; r; sÞ~ ~ 0 ðr; r0 ; sÞ  jG q0 ðr; r0 ; sÞ G ~ðr; r0 ; sÞ ¼ . ð3:10Þ q S~ 1 þ k G0 ðr; r; sÞ c

After substituting this result backward, into Eq. (3.8), we obtain ~ 0 ðr; r0 ; sÞ  G ~ 0 ðr; r; sÞ ~ðr; r0 ; sÞ ¼ G q 

~ 0 ðr; r0 ; sÞ þ j~ k Sc G q0 ðr; r0 ; sÞ . ~ 0 ðr; r; sÞ 1 þ k Sc G

ð3:11Þ

Using the last result in the right-hand side of Eq. (3.6) the closed equation for triplet pair density may be found: ~0 ðr; r0 ; sÞ ¼ Jðr; r0 ; sÞ  Jðr; r; sÞ q

ð3:4bÞ where j ¼ k Tc  k Sc . The yields of triplet production and charge separation from the starting distance r0, ut(r0) and u(r0), were shown to be [12]

ð3:7Þ

The solution of Eq. (3.4b) in the contact approximation is similar to that obtained in [13]: ~ 0 ðr; r0 ; sÞ  G ~ 0 ðr; r; sÞ½k S q ~ðr; r0 ; sÞ ~ðr; r0 ; sÞ ¼ G q

ð2:7Þ

~ o erc =r is an operator of flux at distance where jðrÞ ¼ 4pr2 D or r.

343



~ 0 ðr; r0 ; sÞ þ j~ k Sc G q0 ðr; r0 ; sÞ ; S~ 1 þ k c G0 ðr; r; sÞ

ð3:12Þ

where Jðr; r0 ; sÞ ¼ k s J 0 ðr; sÞ and Jðr; r; sÞ ¼ k s J r ðr; sÞ in the notations of [14]. As was shown there, Z ~ r0 ; s þ 2k s ÞG ~ 0 ðr0 ; r0 ; sÞ d3 r0 ; Jðr; r0 ; sÞ ¼ k s Gðr; has the contact value (see Eq. (4.11) in [14]): ~ 0 ðr; r0 ; s þ 2k s Þ ~ 0 ðr; r0 ; sÞ  G G   . Jðr; r0 ; sÞ ¼ T~ 2 1 þ k G0 ðr; r; s þ 2k s Þ

ð3:13Þ

c

At r = r Eq. (3.12) may be resolved for triplet density on the contact: ~0 ðr; r0 ; sÞ q

  ~ 0 ðr; r0 ; sÞ ~ 0 ðr; r; sÞ  k S Jðr; r; sÞG Jðr; r0 ; sÞ 1 þ k Sc G c   ¼ . T S ~ 1 þ k Jðr; r; sÞ þ k G0 ðr; r; sÞ  Jðr; r; sÞ c

c

ð3:14Þ

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A.I. Burshtein / Chemical Physics 323 (2006) 341–350

Eqs. (3.14) and (3.10) solve the contact problem in principle. Being substituted into Eqs. (3.5), they determine the yields, ~0 ðr;r0 ;0Þ ut ðr0 Þ ¼ k Tc q





~ 0 ðr;r;0Þ  k S Jðr;r;0ÞG ~ 0 ðr;r0 ;0Þ Jðr;r0 ;0Þ 1 þ k Sc G c   ¼ k Tc ; T S ~ 1 þ k c Jðr;r;0Þ þ k c G0 ðr;r;0Þ  Jðr;r;0Þ ð3:15aÞ ~ 0 ðr;r0 ;0Þ þ j~ kS G q0 ðr;r0 ;0Þ ; uðr0 Þ ¼ 1  c S~ 1 þ k c G0 ðr;r;0Þ

while the yield of the us(r0) = 1  u(r0)  ut(r0).

ground

ð3:15bÞ

state

product

is

4. Highly polar solvents As a matter of fact all the quantum yields are expressed above via the Green function of the free diffusion (diffusion without reactions), which obeys Eq. (3.9). In a highly polar solvent one can set rc = 0 to simplify the diffusional operator (2.6). In this limit called in [18] the‘‘free Brownian motion’’ (without Coulomb and any other inter-particle interactions), we have the well known analytic expression for the Green functions [14,19]: pffiffiffiffiffiffi ~ ðr0 rÞ s=D 1 e ~ 0 ðr; r0 ; sÞ ¼ qffiffiffiffiffiffiffiffi . G ð4:1Þ ~ 4pr0 D ~ 1 þ r s=D

1 ~ 0 ðr; r; 0Þ ¼ 1 ; ; G ð4:2Þ ~ kD 4pr0 D ~ is the usual diffusional rate constant of where k D ¼ 4prD the contact reaction. Taking into account the relationships (4.2), the general expressions (3.15) reduce to the following:

~ 0 ðr; r0 ; 0Þ ¼ G

~0 ðr; r0 ; 0Þ ut ðr0 Þ ¼ k Tc q ¼

    ~ Jðr; r0 ; 0Þ 1 þ k Sc =k D  Jðr; r; 0Þ k Sc =4pr0 D   ; 1 þ k Sc =k D þ k Tc  k Sc Jðr; r; 0Þ ð4:3aÞ

uðr0 Þ ¼ 1 

~ þ ðk T  k S Þ~ k Sc =4pr0 D c c q0 ðr; r 0 ; 0Þ . 1 þ k Sc =k D

ð4:3bÞ

Substituting the same relationships into Eq. (3.13) we get r0 r

1 1 þ a  ea r ; Jðr; r0 ; 0Þ ¼  ~ 2 1 þ a þ k Tc =k D 4pr0 D where qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ~ ¼ 2k s sd a ¼ r 2k s =D

For this particular case, Eqs. (4.3) reduce to the following set

kc 1 eaðr0 rÞ=r ut ðr0 Þ ¼  ; ð4:6aÞ ~ 1 þ k c =k D 1 þ a þ k c =k D 24pr0 D kc 1 . ð4:6bÞ uðr0 Þ ¼ 1  ~ 1 þ k 4pr0 D c =k D It is seen that the total yield of separated ions, u(r0), does not depend on the spin-conversion because in this case the transitions between singlet and triplet RIPs do not modulate the rate of their recombination. Therefore, their separation yield remains the same at any a, and exactly coincides with that obtained by the spin-less theory: this is for instance Eq. (3.15) in [1] at rc = 0 first obtained by Hong and Noolandi [20]. Different result follow from Eqs. (4.6) for the yield of triplet products of radical recombination, ut(r0). It turns to zero at a = ks = 0 as should be expected in the absence of spin-conversion provided the RIP was born in the singlet state. In the opposite limit a = ks = 1 the instant mixing of different spin states equilibrates their populations which both become 1/2. Therefore in this limit the total number of recombined radicals, 1  u(r0),is divided equally between the triplet and singlet products of the geminate reaction: kc 1 ¼ us ðr0 Þ ~ 1 þ k c =k D 2 4pr0 D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  uðr0 Þ ~ ! 1. at a ¼ 2k s r2 =D ¼ ð4:7Þ 2 ~ ! 0. Scanning D ~ at This result holds at either ks ! 1 or D ~ finite ks in the interval 0 6 D 6 1 one can actually trace the diffusional dependence of the triplet and free ions quantum yields between the limits of a = 0 and a = 1. The quantum yields are usually presented in the very convenient form [1,2,21]: ut ðr0 Þ ¼

When s = 0 we have

k Tc

k Tc ¼ k Sc ¼ k c .

ut ðr0 Þ ¼

~ Z T =D ; ~ 1 þ Z=D

uðr0 Þ ¼

1 ; ~ 1 þ Z=D

ð4:8Þ

where Z and ZT are the efficiencies of total recombination and production of triplets, respectively, Using in these formulae Eqs. (4.6) one can easily find how these efficiencies ~ and ks at any given r0 P 0. The results are depend on D shown in Fig. 1.

ð4:4Þ 5. Contact start ð4:5Þ

is the only parameter which is a measure of the spinconversion during a single encounter. Let us turn to the simplest hypothetical case first considered by Schulten and Schulten [18], when the total recombination is separable from the spin-conversion dynamics because

In principle, all the quantum yields should be averaged over the initial space distribution of RIPs, f(r0): Z  t ¼ ut ðr0 Þf ðr0 Þ d3 r0 ; ð5:1aÞ u Z  ¼ uðr0 Þf ðr0 Þ d3 r0 . ð5:1bÞ u

A.I. Burshtein / Chemical Physics 323 (2006) 341–350

345

contact starts as well as the spin-conversion [1,2]. In the ~ ¼ 0 and presence of the spin-conversion ZT = 0 at D ~ D ¼ 1, passing through the maximum in between: 8 qffiffiffiffiffi > ~ ¼ r ks D~ at D ~ ! 0; < 1 aD 2 2 ð5:7Þ ZT ¼ qffiffiffiffi > : 1 az ¼ zr k~s at D ~ ! 1. 2 2D The same results may be represented in another form more suitable for interpretation: 8 T <1a at kkDc  a  1; 2 ~ Z T =D ¼ : 1 a kTc at kTc < a  1. 2 kD kD Fig. 1. The diffusional dependence of the total recombination efficiency, ˚, r=7A ˚ and kc = 104 A ˚ 3/ns (upper solid curve) and Z(r0), at r0 = 7.5 A the family of triplet ones, ZT(r0), for a few spin-conversion rates ks: 1, 2 and 10 ns1(dashed curves from bottom to top). In insert: the same, but for narrow region of slow diffusion.

However, for the qualitative investigation of the phenomenon it is easier and instructive to analyze the recombination of RIPs started only from contact: f ðr0 Þ ¼ dðr0  rÞ=4pr20 .

ð5:2Þ

 t  ut ðrÞ; u   uðrÞ and one can get In this particular case u the final result setting r0 = r in Eqs. (4.3). After that any J-factor appearing in these equations is a . J ¼ Jðr; r; 0Þ ¼  ð5:3Þ 2 k D ð1 þ aÞ þ k Tc Using it we obtain from there: ~0 ðr; r; 0Þ ¼ q

a . 2½k D ð1 þ aÞ þ k Tc ð1 þ k Sc =k D Þ þ a½k Tc  k Sc  ð5:4Þ

Using the result (5.4) in formulae (4.3) under condition r = r, we can present all the yields in the same way as in ~ Eq. (4.8), with recombination efficiencies depending on D: T a ~ ¼ kc Z T =D ; k D 2ð1 þ a þ k Tc =k D Þ " # S T S k aðk  k Þ c c ~ ¼ c 1þ . Z=D kD 2k Sc ð1 þ a þ k Tc =k D Þ

ð5:5Þ

They both as well as parameter a differ by only numerical factors from their analogs obtained for the four level model represented by Eqs. (1.2) and successfully used in [21] for fitting the real experimental data. At k Sc ¼ k Tc ¼ k c , it follows from Eqs. (5.5) that Z¼z¼

kc ; 4pr

ZT ¼

az ; ~ 2ð1 þ a þ z=DÞ

ZS ¼ Z  ZT. ð5:6Þ

The result Z = z = const is the only one obtained with the so called ‘‘exponential model’’ which ignores the non-

The upper line corresponds to the limit of very fast recombination to triplet at contact sphere which becomes ‘‘black’’ in extreme case k Tc ¼ 1 (diffusional recombination). Therefore the triplet yield is limited by only spinconversion to highly reactive RIP state which remains empty all the encounter time. In the lower line recombination is not fast as well as spin-conversion and them both control the process. However, if the spin-conversion is faster than everything (ks = a = 1) then ZT = z/2 = Z/2 as follows from the general formula (5.6). In such a case the recombination to triplet product is under kinetic control but it efficiency is half of the total one because the RIP population is equally divided between singlet and triplet levels by infinitely fast conversion. The similar results were obtained earlier in [21] but in the frame of the model (1.2). However, there and here the incoherent spin-conversion was employed while in the pioneering work of Schulten and Schulten [18], the alternative (purely coherent) mechanism of spin-conversion was considered, setting T2 = T1 = 1. Therefore the dynamics of the process in the high fields was given there by the expression W T ðtÞ ¼ 12 sin2 Xt instead of that following from Eqs. (2.4): W T ðtÞ  q0 ðtÞ ¼ 12 ½1  e2ks t  provided q0(0) = 0. However, the difference between the two approaches is not exhausted by this principle distinction. The equality of the singlet and triplet recombination rates was considered by the authors as a sufficient condition to separate the spin-conversion from the inter-particle dynamics. They obtained the triplet quantum yield integrating WT(t) over time with the weight kcq(t). The time dependence of the latter was obtained from the equation equivalent to (3.4b), while another equation like (3.4a) was ignored. Such an approach is actually based on the decoupling of the relationship (3.6) in space and time ~ its dynamic analog, WT. Fortuand the usage instead of G nately, all these inaccuracies do not affect qualitatively the results obtained in [18]. For this particular case ðk Sc ¼ k Tc Þ our own results can be represented in the same way as in Fig. 5 of ShultenÕs work. Using as an argument u, changing with diffusion from 0 to 1, we specified uT and uS as the functions of u taking into account that the sum of all the yields remains unchanged: u þ uT þ uS ¼ 1.

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A.I. Burshtein / Chemical Physics 323 (2006) 341–350

As shown in Fig. 2 only uT has the bell shape. The other yields change monotonously with diffusion: uS decreases from 1 to 0 while u increases from 0 to 1. At k Tc ¼ 0 there is also a single reaction parameter zs ¼ k Sc =4pr and it follows from Eq. (5.5) that Z T ¼ 0; but Z ¼ zs

1 þ a=2 ¼ ZS. 1þa

ð5:8Þ

Speeding up the spin-conversion we reduce ZS (not more than two times) because the conversion of the singlet radical pairs into the non-reactive triplet state blocks their recombination. This happens in the slow diffusion (high ~ ! 0, when viscosity) limit, D pthe ffiffiffiffi encounter time is large enpffiffiffiffiffi ~ ! 1. In this limit Z = z/2 ough so that a / sd / 1= D while in the opposite one we have lim Z ! zs , as is shown ~ D!1 by the upper curve in Fig. 3. For the particular case of fast relaxation ZS is given by the expression (5.8) but with sffiffiffiffiffiffiffi 2sd a¼ ð5:9Þ because k 0 ¼ X2 T 2 ¼ 0. T2 This is actually the stochastic spin-conversion but in a zero magnetic field limit. Assuming X = DgbH/⁄ = 0 one looses all the magnetic field effects that can be thoroughly studied in the first order approximation with respect to X2 T 22 [14,15]. Such a theory of incoherent spin-conversion was successfully fitted to experimental data on Ruthenium complexes quenched by methylviologen [16,17]. The quantum yield uS(r), calculated by Mints and Pukhov [22] for arbitrary spin-conversion rate X, can be compared with ours at T1 = 1. In this particular case, their   zs 1 1þ ZS ¼ ð5:10Þ Q 2 should reduce to that in Eq. (5.8), if the stochastic description of the spin-conversion is possible. For such a reduction it is enough to require the following equality: Q ¼ 1 þ a.

ð5:11Þ

Fig. 3. The diffusional dependence of the single-channel recombination efficiencies from the initial singlet state at ks = 10 ns1: the upper curve is for Z = ZS (singlet recombination channel), the lower curve is forZ = ZT (triplet recombination channel). The rest of the parameters are the same as in Fig. 1.

The relationship between a calculated with our incoherent (stochastic) and their general (Hamiltonian) theory of spinconversion is discussed in Appendix B. At k Sc ¼ 0, the recombination through the singlet channel is switched off and it proceeds through only the triplet one with reaction parameter zt ¼ k Tc =4pr. In such a situation azt Z S ¼ 0 but Z ¼ ¼ ZT. ð5:12Þ ~ 2ð1 þ a þ zt =DÞ The diffusional dependence of Z in this case differs qualitatively from the previous one but coincides with that given for ZT in Eqs. (5.6) and (5.7): initially Z increases with diffusion but then passing through the maximum tends to 0 (lower curve in Fig. 3). ~ follows from the results obtained Similar behavior ZðDÞ in [22] for recombination of radicals born in the T0 state through the singlet channel, provided the two state model holds (T1 = 1). It can be obtained from Eq. (5.12) by just a substitution zs for zt: ZS ¼

azs ¼ Z at qS ð0Þ ¼ 0; ~ 2ð1 þ a þ zs =DÞ

q0 ð0Þ ¼

dðr  rÞ . 4pr2 ð5:13Þ

Fig. 2. The partition of all the products of spin-independent recombination into three components: uT, uS and u.

An equivalent result was also obtained later [11] for the conversion carried out by the hyperfine interaction (HFI) in a high magnetic field limit. In this limit the transfer also occurs between only two spin sub-levels, reactive S and non-reactive T0, because the other two, T±, are out of resonance and do not participate in the process. However, considering the non-polarized initial state of the triplet, the authors took all triplet sub-levels equally populated, leaving the singlet one empty. Therefore, the recombination quantum yield RT calculated in this work should be normalized before comparing it with the previous result:

A.I. Burshtein / Chemical Physics 323 (2006) 341–350

347

~ 1 Z S =D ¼ 3RT ¼ ~ 3 1 þ Z S =D 1 dðr  rÞ . at qS ð0Þ ¼ 0; q0 ð0Þ ¼ 3 4pr2

uS ¼ RT :

Using RT obtained for the diffusion controlled singlet state recombination (see Eq. (93) in [11]) we get from there: zs ab ZS ¼ ~ 3 2ð1 þ ab þ zs =3DÞ  ) ab D=2 ðblack sphere limitÞ. zs !1

ð5:14Þ

In general, Eq. (5.14) differs from the result (5.13) by the substitution zs/3 for zs because only one third of triplet population is redistributed with the singlet state from where the recombination is occurring. As to ab, this is an exact value of a for the ‘‘black sphere’’ limit (diffusional recombination): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sd ab ¼ lim a ¼ 1 þ 1 þ X2 T 22 T2 k Sc !1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   sd 1 2 2 ! 2 1 þ X T 2 at XT 2  1. ð5:15Þ 4 T2 It is considered and discussed in Appendix B. 6. Concluding remarks Using the stochastic (rate) description of spin-conversion in a two level system,we solved the problem of geminate recombination/separation of singlet born radical pairs via the singlet and triplet channels of contact recombination at any initial radical separation in a pair. In the case of starting from contact, the general expressions for recombination and separation efficiencies, (5.5), were derived and confirmed by comparison with the results obtained in a few particular cases by means of the coherent (Hamiltonian) approach. The difference between the stochastic and Hamiltonian approaches is reduced to the specification of a single parameter a representing the spin-conversion in the general formulae. This difference disappears utterly only in the single point X = 0 as follows from Fig. 4.In the case of the Dg-mechanism of spin-conversion this is possible in the zero magnetic field when X / H = 0 and therefore k0 = 0 so that the spin-conversion rate ks = 1/T2. Moreover, the stochastic approach is approximately valid also in the low field region, though in rather rigid limits (1.5): 0 6 XðH Þ 

1 . T2

There the exact result and its lowest order expansion in X2 T 22 (dotted line) coincide, while in the stochastic approach the term proportional to such a parameter is four times larger. However, this is true for only the black sphere model; for the grey sphere model (B.6) the difference is three times less.

Fig. 4. Parameter aas a function of spin-conversion frequency X for the black sphere model (solid lines) and for the incoherent spin-conversion (dashed lines) at rather long T2 = sd = 1 ns (lower thick curves) and for 10 times shorter T2 (upper thin curves). The low frequency approximation to ab in the latter case shown by the dotted line is obtained from Eq. (5.15) at XT2  1.

For transition metal complexes with strong spin–orbital coupling, [7] (X 6 109  1/T2  1011) the stochastic approach is not only exact in the zero field but can also describe semi-quantitatively the field dependence of the effect at reasonable H. This justifies a posteriori the appliance of this approach to the study of the magnetic field effects in a number of earlier works [14–17,1,2]. The situation is quite the opposite in the usual radical ion reactions, like that following the quenching of the excited pyrene (Py) by dimethylaniline (DMA) and a few other electron donors [10,25], or the quenching of perylene (Per) by DMA or o-DMT (N,N-dimethyl-o-toluidine) [26,27]. Fixed by the particular hyperfine interaction in given system, a, the spin-conversion frequency X = a/2 does not change. Usually X  107 s1 is much larger than the typical values of the spin–lattice relaxation rate in radicals: 1/T2 105–106 s1. As a result their X lie outside the applicability interval for the incoherent approximation. Hence, the absolute value of a in these systems can be properly estimated with only general Hamiltonian approach. ~ as well However, the dependencies of all the yields on D as on r0 remain the same, provided ks is considered as a phenomenological (fitting) parameter. As a matter of fact, the contact model employs the initial distribution (5.2) which is not always admissible. The most favorable for the contact start model is the kinetic ionization: the initial distributions generated in this limit are the closest to contact. If the encounter diffusion of neutral ion precur~ is really fast then the normalsors (with coefficient D 6¼ D) ized initial distribution of RIPs coincides in shape with the space dependent ionization rate, WI(r) [1,2]: W I ðr0 Þ ; where the kinetic constant k 0 k0 Z ¼ W I ðrÞ d3 r  k D ¼ 4prD.

f ðr0 Þ ¼

ð6:1Þ

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A.I. Burshtein / Chemical Physics 323 (2006) 341–350

The electron transfer rate is most commonly assumed to be exponential: W I ðrÞ ¼ W c e2ðrrÞ=l ;

ð6:2Þ

where l is the effective tunnelling length. The contact approximation to this exponent, Eq. (5.2), is reasonably good provided l  r. However, the exponential model (6.2) is also just an admissible approximation to the real WI(r) when ionization occurs in the normal Marcus region [1,23]. Only then the transfer rate monotonously decreases with reactant separation, as shown in Fig. 5 A. This is the case for the quasiresonant electron transfer which really proceeds in the normal Marcus region, while the highly exergonic transfer lays in the inverted Marcus region. There WI(r) has a bell shape with the maximum shifted out of contact (Fig. 5B) [1,24]. The corresponding distribution which duplicates this shape when ionization is kinetic, differs qualitatively from the exponential one and even more from the contact model. The exaggerating form of this distribution, f(r) = d(r  r0)/ 4pr2, was the basis of our preceding consideration, implying that all RIPs started from the same initial separation r0. A more realistic model is represented by the bell-shaped distribution arising from (6.1) after substitution the rate of remote ionization used in [28] Wm W I ðrÞ ¼ 2 rr0  . ð6:3Þ ch D At finite D  l this is an analog of the exponential one but for the inverted Marcus region.

Fig. 5. The initial RIPs distributions in the normal (a) and inverted (b) regions borrowing from [23] and [24], respectively.

If the diffusion of neutral reactants slows down, the ionization falls under diffusional control and the resulting initial distribution of RIPs becomes bell-shaped and distant in either the inverted or normal regions. The maximum of such a distribution shifts out to the effective ionization radius RQ > r. This radius increases with viscosity making the contact approximation inapplicable. Instead one should use the real one, specified by the generally valid expression [1,2,12]: Z 1 f ðr0 Þ ¼ cW I ðr0 Þ nðr0 ; t0 ÞN ðt0 Þ dt0 ; ð6:4Þ 0

where c is the density of quenchers (c = [D] in our particular case). The quenching kinetics is given by Rt 0 0 c k ðt Þ dt N ðtÞ ¼ e 0 I ; where the time dependent rate constant of ionization (forward electron transfer) is Z k I ðtÞ ¼ W I ðrÞnðr; tÞ d3 r: The pair distribution function n(r,t) obeys the following equation:  on n_ ¼ W I ðrÞn þ DDn nðr; 0Þ ¼ 1; ¼ 0. or r¼r The kinetic limit is achieved at the fastest diffusion when n = 1 and N ¼ eck0 t . Substituting these results into Eq. (6.4) we return back to Eq. (6.1). Otherwise (at slower diffusion) one has to find the real distribution from Eq. (6.4) and use it in Eqs. (5.1) to average the quantum yields. As to the recombination of the RIPs it can also proceed in either the normal or inverted Marcus regions depending on the reaction channel. The low exergonic recombination to the excited triplet state occurs very probably in the normal region while highly exergonic recombination to the ground singlet state goes on in the inverted one. In the latter case the recombination layer is shifted out of the contact but can be considered as pseudo-contact if the starting distance is even larger: r0  RQ > r. This is possible under diffusional control of the ionization, while under kinetic control in the normal region the start is from the contact and the situation is quite different: r0 = r  RQ. Starting from contact ions should cross the remote recombination layer and the faster they do this the more of them separate. This is the reason for the non-monotonous diffusional dependence of ZS = Z, when the recombination proceeds in the inverted region while the ionization is scanning from the diffusional to kinetic limit with decreasing the solvent viscosity [29,30]. The classification of all possible situation regarding the extent of recombination and the remoteness of starting distance is given in Table 1. Our general results (Section 4) are actually applicable to only the lower line of the table, where the recombination is approximately contact. If the starts are distant the yields should be averaged over the distribution (6.4), while the

A.I. Burshtein / Chemical Physics 323 (2006) 341–350

349

Table 1 Division of all the situations arising in recombination following ionization

2 y ¼ pT . 3

Recombination

Diffusional ionization

Kinetic ionization

In inverted region

Pseudo-contact recomb./distant start Contact recomb./ distant start

Non-contact recomb./contact start Contact recomb. /contact start

Such an assumption reduces three Eq. (A.2), to the approximate set of Eqs. (1.2) though with another conversion constant

In normal region

results for contact starts can be taken like they are given in Section 5. With the upper line of the table things become more complex. The contact model of recombination is acceptable as a rough approximation, when r0 is in average much larger than the distance to the maximum of WT,S(r). However, the contact model of recombination is not appropriate when ionization is kinetic so that radicals start from inside the remote recombination layer [27,29,30].

k0 1 þ . 3 2T 0

The relationship between the Eq. (A.2) and their simplified (model) version (1.2) has been inspected separately [31]. Appendix B In the case of the two level system the coherent (Hamiltonian) approach to the problem used in [22] leads to the set of three equations instead of two Eqs. (3.2): 1 jqjr ¼ k Sc ½qðrÞ þ q0 ðrÞ; ðB:1aÞ 2 2 1 q_ 0 ¼ Lq0  q0  4Xq ; jq0 jr ¼ k Sc ½qðrÞ þ q0 ðrÞ; T2 2 ðB:1bÞ 2 1 q_  ¼ Lq  q þ Xq0 ; jq jr ¼ k Sc q ðrÞ. ðB:1cÞ T2 2

q_ ¼ Lq;

Acknowledgement The author is grateful to V.S. Gladkikh M. Sc. for help in illustration of the article. Appendix A In the extreme case when there is no difference between the relaxation times in the radical ions, T1 = T2 = T0, one can change the variables to x ¼ qS þ q0 and y ¼ qþ þ q and obtain from the set (1.3):     1 1 y q_ S ¼ 2 k 0 þ ; qS þ k 0 þ xþ T0 2T 0 2T 0 x y x_ ¼  þ ; T0 T0 x y  ; y_ ¼ T0 T0

ks ¼

ð6:5Þ

ðA:1aÞ

Here q = qS + q0 and q 0 = qS  q0 are the sum and difference of the diagonal elements of the density matrix while q* is the imaginary part of its off-diagonal element. These variables obey the initial conditions corresponding to the singlet born ion pair: qð0Þ ¼ q0 ð0Þ ¼ dðr  r0 Þ=4pr2 ;

q ð0Þ ¼ 0.

ðA:1bÞ

The reduction to the incoherent description of RIP recombination is very simple, provided T2 is so small that not only the inequality (1.5) holds but also

ðA:1cÞ

~ T 2  sd ¼ r2 =D.

ðB:2Þ

p ¼ x þ y ¼ qS þ q þ þ q 0 þ q 

Then the diffusion during the encounter time may be neglected in Eq. (B.1c) and solving it quasi-stationary we get

obeys the conservation low: p_ ¼ 0. Using this variable and the total population of the triplet,

q ¼

where

pT ¼ qþ þ q0 þ q ; one can rewrite Eqs. (A.1) in the following form:     1 3 p_ T ¼ 2 k 0 þ p þ k0y þ k0 þ p; T0 T 2T 0 2 p y_ ¼  y þ ; T0 T0 p_ ¼ 0.

XT 2 0 q. 2

Substituting this result into Eq. (B.1b) we obtain the set of two equations ðA:2aÞ ðA:2bÞ ðA:2cÞ

At T0 = 1 these equations reduce to Eqs. (2.4) with pT ! q0 and p ! q and y = 0. Instead of solving the exact set of Eq. (A.2) we assumed rather arbitrarily in our previous articles that the triplet sub-levels are equipopulated and therefore

q_ 0 ¼ Lq0  2k s q0 ;

q_ ¼ Lq;

ðB:3Þ

which are identical (at L = 0) to Eqs. (2.4) written in other variables. The solution of the set of Eqs. (B.1) obtained in [22] can be expressed as usual with ZS from (5.10), which actually coincides with that obtained for the incoherent conversion, provided Eq. (5.11) is an identity. To verify this point we have to compare our value of rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sd a ¼ 2 ð1 þ X2 T 22 Þ ðB:4Þ T2

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A.I. Burshtein / Chemical Physics 323 (2006) 341–350

with the expression that can be deduced from the results obtained by Mints and Pukhov: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sd 1 þ 1 þ X2 T 22 a¼ T2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ X2 T 22  1 sd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ . ðB:5Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T2 2 2 sd S 1 þ k c =2k D þ T 2 1 þ 1 þ X T 2 Of course, one should not expect an exact identity of these aÕs but under conditions of incoherent spin-conversion the latter should reduce to the former. Due to condition (B.2) the square root in the denominator of (B.5) is much larger than other terms provided k Sc =k D K 1. In particular, this is the case of a ‘‘grey sphere’’ ðk Sc  k D Þ when recombination is kinetic controlled. Neglecting small terms we can represent Eq. (B.5) in this particular case as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 Tsd2 ð1 þ X2 T 22 Þ ffi ag ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 þ 1 þ X T2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   sd 3 2 2

2 1 þ X T2 4 T2

rffiffiffiffiffi sd ;1  . at kD T2 k Sc

ðB:6Þ

In the opposite case of diffusional recombination, especially in the case of a ‘‘black sphere’’ ðk Sc ¼ 1Þ the whole last term in Eq. (B.5) is negligible and the corresponding a is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sd 2 2 ab ¼ 1 þ 1 þ X T2 T2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   rffiffiffiffiffi sd 1 2 2 k Sc sd at 1 þ X T2  1;

2 . ðB:7Þ 4 T2 kD T2 The incoherent description of spin-conversion is indifferent to the boundary conditions for contact recombination and therefore can not discriminate between the grey and black spheres, Eqs. (B.6) and (B.7). Fortunately the difference between them and the distinction between both of them and the incoherent approximation, Eq. (B.4), concerns only the numerical coefficient before the term X2 T 22 . Since this term is always small under the validity condition (1.5) of

the incoherent approximation, the latter is semi-quantitatively acceptable though in very narrow limits. References [1] [2] [3] [4] [5] [6] [7] [8]

[9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

A.I. Burshtein, Adv. Chem. Phys. 114 (2000) 419. A.I. Burshtein, Adv. Chem. Phys. 129 (2004) 105. M. Tomkiewicz, M. Cocivera, Chem. Phys. Lett. 8 (1971) 595. W. Bube, R. Haberkorn, M.E. Michel-Beyerle, J. Am. Chem. Soc. 100 (1978) 5993. L. Sterna, D. Ronis, S. Wolfe, A. Pines, J. Chem. Phys. 73 (1980) 5493. H. Hayashi, S. Nagakura, Bull. Chem. Soc. Jpn. 57 (1984) 322. U.E. Steiner, Th. Ulrich, Chem. Rev. 89 (1989) 51. D. Bu¨rßner, H.-J. Wolff, U. Steiner, Z. Phys. Chem. Bd. 182 (1993) 297; Angew. Chem. Int. Ed. Engl. 33 (1994) 1772. K.M. Salikhov, Yu.N. Molin, R.Z. Sagdeev, A.L. Buchachenko, in: Yu.N. Molin (Ed.), Spin Polarization and Magnetic Effects in Radical Reactions, Elsevier, 1984. K. Schulten, H. Staerk, A. Weller, H.-J. Werner, B. Nickel, Z. Phys. Chem. NF101 (1976) 371. M.J. Hansen, A.A. Neufeld, J.B. Pedersen, Chem. Phys. 260 (2000) 125. A.I. Burshtein, Chem. Phys. 247 (1999) 275. A.I. Burshtein, A.A. Zharikov, N.V. Shokhirev, O.B. Spirina, E.B. Krissinel, J. Chem. Phys. 95 (1991) 8013. A.I. Burshtein, E.B. Krissinel, J. Phys. Chem. A 102 (1998) 7541. A.I. Burshtein, E. Krissinel, J. Phys. Chem. A 102 (1998) 816. E.B. Krissinel, A.I. Burshtein, N.N. Lukzen, U.E. Steiner, Mol. Phys. 96 (1999) 1083. A.I. Burshtein, E. Krissinel, U.E. Steiner, Phys. Chem. Chem. Phys. 3 (2001) 198. Z. Schulten, K. Schulten, J. Chem. Phys. 66 (1977) 4616. A.A. Zharikov, A.I. Burshtein, J. Chem. Phys. 93 (1990) 5573. K.M. Hong, J. Noolandi, J. Chem. Phys. 68 (1978) 5163. V.S. Gladkikh, A.I. Burshtein, G. Angulo, G. Grampp, Phys. Chem. Chem. Phys. 5 (2003) 2581. R.G. Mints, A.A. Pukhov, Chem. Phys. 87 (1984) 467. V.S. Gladkikh, A.I. Burshtein, I. Rips, J. Phys. Chem. A 109 (2005) 4983. V.S. Gladkikh, A.I. Burshtein, J. Chem. Phys. (submitted). A. Weller, H. Staerk, R. Treichel, Faraday Discuss. Chem. Soc. 78 (1984) 217. N. Mataga, T. Asahi, J. Kanda, T. Okada, T. Kakitani, Chem. Phys. 127 (1988) 249. G. Angulo, G. Grampp, A. Neufeld, A.I. Burshtein, J. Phys. Chem. A 107 (2003) 6913. A.I. Burshtein, P.A. Frantsuzov, Chem. Phys. 212 (1996) 137. A.I. Burshtein, A.A. Neufeld, J. Phys. Chem. B 105 (2001) 12364. A.A. Neufeld, A.I. Burshtein, G. Angulo, G. Grampp, J. Chem. Phys. 116 (2002) 2472. V.S. Gladkikh, A.I. Burshtein, Chem. Phys., in press.