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Theoretical treatment of the recombination probability of radical pairs with consideration of singlet-triplet transitions induced by paramagnetic relaxation
113
Chemical Physics 117 (1987) 113-131 North-Holland, Amsterdam
THEORETICAL TREATMENT OF THE RECOMBINATION PROBABILITY CJF RADICAL PAIRS WITH CONSIDERATION OF SINGLET-TRIPLET TRANSITIONS INDUCED BY PARAMAGNETIC RELAXATION K. LijDERS Sektion Physik der Karl-Marx-Universitiit,
7010 Leipzig, GDR
and K.M. SALIKHOV Institute of Chemical Kinetics and Combustion, Novosibirsk
630090, USSR
Received 29 January 1987; in final form 11 June 1987
The recombination probability was calculated for several models of radical pairs (RPs). The singlet-triplet transitions induced by the relaxation mechanism and by the difference of Larmor frequencies of the unpaired electrons of the two radicals of the pair were taken into account. The dependence of the recombination probability on the external magnetic field and on the lifetime of the RP was analyzed. It was pointed out that the different mechanisms of S-T mixing make non-additive contributions. An important part is played by the interferences of the contributions of longitudinal and transverse relaxations. In systems with sufficiently long lifetimes of RPs the recombination probability changes strongly in relatively low magnetic fields, whereas in the same systems the values of paramagnetic relaxation rate change strongly in higher magnetic fields.
1. Introduction
In the recombination of RPs appear several mechanisms of singlet-triplet (S-T) transitions: (a) isotropic hyperfine interaction (ihf mechanism), (b) difference of the g factors of the two partners (Ag mechanism), and (c) relaxation mechanism (see, e.g., ref. [l]). In reactions of organic radicals in most cases the first two mechanisms are regarded. In non-viscous liquids the lifetime of pairs of neutral radicals is in the range of nanoseconds, but typical values of the times of paramagnetic relaxation of unpaired electrons lie in the microsecond range. This being so, paramagnetic relaxation hardly appears in the recombination process of RPs. This situation changes essentially if the reaction occurs in micelles [2-51, in viscous systems [6-81 or if the recombination of ion-RPs is considered [9-121. In all these systems the lifetimes of RPs may be equivalent to, or even longer than, the paramagnetic relaxation times. The influence of S-T transitions on the recombination probability was first regarded based on the relaxation mechanism [13]. Soon afterwards the part of Ag and ihf mechanisms was elucidated and particularly these mechanisms of S-T transitions were investigated in detail. The theory of the relaxation mechanism remained less investigated. Several attempts at a theoretical analysis of possible occurrences of paramagnetic relaxation in RP recombination were made in refs. [4,14-161. In the present work the recombination probability was calculated for several models of RPs with consideration of the paramagnetic relaxation process. Besides the relaxation transitions, S-T, transitions resulting from the Ag mechanism are considered. The results may also be used for calculating the 0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
114
K. Liiders, KM. Salikhov / Recombination probability
of
radicalpairs
recombination probability of RPs in the presence of the ihf mechanism, if the external magnetic field is high in comparison with the local hf field and the isotropic hf interaction actually mixes only the S and T, states.
2. Models of radical pairs There exist several models of RPs, in which the diffusion escaping process of the .RPs are considered in a different determined by the concrete properties of the physical system by convenience of mathematical treatment. In this work we three models. 2.1. Exponential
one-positional
of the partners, their recombination and the manner. The choice of the actual model is considered. Sometimes this choice is dictated calculated the recombination probability for
model of RPs
This is the simplest phenomenological model of RPs. Within this model the following processes are taken into account: decomposition of the RPs into independent radicals, recombination and S-T transitions. The kinetic equation for the spin density matrix (p) of the RP is [1,17] +/at
= &/3t)sp.&IL-
JG( Fp + PF)/2
- p/r,
0)
where 7 is the mean lifetime of the RPs, K, the recombination rate constant of the RP in the reactive spin state and fi the projection operator into this state, and sp. dyn. is the abbreviation of spin dynamics. Usually the reactive state is the singlet state. On this account $ is always the projection operator into the singlet state below: @ = 1S) (S I. The first term in the right-hand part of (1) describes the change of p caused by the motion of the RP spins. Usually the model under discussion is simply called the exponential model. In our work we call it exponential onepositional model, in order to distinguish it from another model which is considered below. In the exponential one-positional model the radicals of the RPs are permanently in the reaction area. In principle they can always recombine, as soon as there is the corresponding multiplicity. This model poorly reflects the character of molecular motion of radicals in dense agents, especially in liquids. In liquids, during random thermal motion the partners undergo consecutive approaches (re-encounters) [18]. At the moment of contact the wavefunctions of the valence electrons overlap, the radicals are in the reaction area and may recombine. In the reaction area between the radicals exists a considerably strong exchange interaction, the S and T states get out of resonance. Besides this the lifetime of the RP in the reaction area lies in the picosecond range. Therefore the abovementioned mechanisms of S-T transitions are ineffective in the reaction area. In the intervals between re-encounters the radicals are incapable of recombination. But at these moments the S-T mixing in the RP gets effective. The decribed processes are reflected in the following phenomenological model, which was proposed in refs. [19,20]. 2.2. Exponential
two-positional
model of RPs
We suppose that the RPs may be in two states: (1) within the reaction area and (2) outside the reaction area. Within the reaction area the RPs may recombine and leave the area with the rate constant l/~i_ Within the second area the RPs do not recombine but S-T evolution takes place. With the rate constant 1/+r2RPs may enter into the reaction area and with the rate constant l/7 RPs leave the “cage”; they die. The death of RPs is caused by the decomposition of the RPs into independent radicals or by reactions of the radicals with acceptors, or by irreversible isomerization of the radicals.
K. L.iiders, K.M. Salikhov / Recombination probability
of radical pairs
115
We consider the probability density of re-encounters for this model. The probability that the first re-encounter occurs in the time interval (t, t -I-dt) is expressed by f(t) dt. It is equal to the product of two probabilities. The first is the probability that within the time t the RP is not decomposed and does not enter into the reaction area exp( - t/T2) exp( - t/T). The second factor is the probability of contact in the interval dt which is equal to df/r2. The following probability density of re-encounters results: r(t) The
= (I/r2)
exp[ - (I/r2 + I/+].
(2)
probability of a re-encounter is p=imf(t)
(3)
dt=7/(T+T2)
and the total number of re-encounters (see, e.g., ref. [l]) n=l/(l-p)=(7+7&7*.
(4)
The mean time between re-encounters is given by 7c = /omtf(t)
dt(iwf(t)
4.
dt)-1=~zp=vz/(7+
(5)
For the density matrices pi and pz of the RPs in the areas 1 and 2 we may write the following kinetic equations: aPl/at
= -Jq@P,
+ P2)/2
aP2m
= (~P*/~t)s,.dyn.+
- PI/? P*/?
+ P2h
- P2/72 -
P2/7
+ (~Pl/wex.de.ph> (6)
deph. stands for exchange dephasing). At the moment of closest approach the S and T states undergo a fast dephasing caused by exchange interaction. Therefore in the kinetic equation for p1 we may add the additional term
(ex.
(aPl/at)ex.deph. = -ift-‘[
ILhangeY p1] .
The following calculations were carried out with consideration of this supplementary dephasing as well as without it. The given model phenomenologically describes the most important peculiarity of the molecular dynamics of RPs: consecutive contacts. 2.3. The diffusion model of RPs The diffusion model of RPs without consideration of S-T transitions was elaborated by Noyes [18]. In this model the statistics of re-encounters is given by the distribution function f(t)
=mtC312
exp( --am2/p2t),
m = (27/81~)~‘~(1
-p)‘(b/XD)“$“,
(7) (8)
where b denotes the recombination radius, X, is the length of a diffusion step and 7s is the mean time of one elementary diffusion event. The statistics of re-encounters is different for the last two models. It is well known that in the Noyes model the mean lifetime of an RP is not defined because jOmtf(t) dt + CO. This can be explained by the fact that the partners may undergo a re-encounter after a very long diffusion path. In the exponential
116
K. Liiders, K. M. Salikhou / Recombination probability of radical pairs
fH1
116%‘I QZ--L
0
~2
Q3OrC
06081
T, 2
3
456
a
810
30405060 trlams 1
Fig. 1. Distribution function f(t) of re-encounters for the exponential two-positional model (curves 1, 2) and the model of continuous diffusion of RPs (curve 3). Parameters: (1) r= 5 x 10-p s, 72 = 5 x 10-1s s; (2) 7 = 10-S s, *a = 10-s s; (3) D = lo-” m’/s, b = 0.3 nm. The probability of a re-encounter p = ]Ff (t) dt is p = T/(T + TV) for the exponential two-positional model. The time r,, after which the differential probability of a re-encounter for the diffusion model reaches its maximum depends on the length of one elementary diffusion step A,. With decreasing A, the maximum increases and shifts to the left-hand side.
two-positional model the probability of a re-encounter is exponentially distributions (2) and (7) differ strongly also at small times (see fig. 1).
decreased for t > +rC.Moreover
3. The spin dynamics of radical pairs We regard the dynamics of S-T transitions caused by paramagnetic relaxation of the radicals A and B of the pair. The longitudinal and transverse relaxation times of the unpaired electron spins of the radicals A and B are called TIA (T& and TzA (TZB), respectively. The Larmor precession frequencies of the electron spins A and B are w, and or,. First we regard the spin dynamics of single isolated radicals. The kinetic equation for the density matrix, e.g., for the partner A, may be written in the form of Bloch equations +&at
= - ( P:~ - P;~)/zT,,,
aP&/at
= -aPk/k
aP$/at
= -it+P$
- P$/L,
(9)
where (Y, /3 are eigenfunctions of the operator of the projection of the electron spin on the direction of the external magnetic field. Rewriting these equations in the operator form we get
ap*/at = QApA.
00)
The spin dynamics of the two radicals is. described by (aP/%.dyn.=
&A
x
&P,
(11)
where &, X & is the outer product of the operators &A and $,. After transformation into the basis of S and T states we get the following kinetic equations for the spin dynamic part of the density matrix (only those equations are written which contain the populations of S and T states) (aPSS/at)sp.dp.=
-
wT,‘+
~/~T;)Pss
-(~-wd ( ap=,&iat)sp,dP,
= ( - l/457 +h-
+ (-VT:
+ l/2T;)prOr0
+ (PT+T++
P~_&~T;
Im psTo3 + l/W +A
1 PSS - (1/4T;
Im psr,,
+ VW’)
PT~T~ + ( PT+T+ + P,_T_)/~?
K. Liiders, K.M. Salikhov / Recombination probability ( aPT+-&$+,,,
(apr_da~),,,.= ( ap,/at
PT~~,)/~&’
= (PSS +
(pss + pToTo)/4?’
of radical pairs
- PT+T+/%’
- Re PST,/W”,
- PT_~/T’
+ Re P~T,,/W”~
117
)sp.dyn. =~(~,-WB)(PT~T~-PSS)/~-(PT+T+-PT_T_)/~T~'-(~/~T;+~/~T~')P~T~ + (- 1/4T;
+ 1/2T;)P,,,,,,
(apToS/a~)s&,,,.=-i(wA-wB)( + (-1/4T’+
PT,T,-PSS)/2-
(PT+T_-PT_T_)/~T;
~/~T;)PST~ - (V4T’
+ 1/2T;)P~,,s,
02)
where l/T,’ = l/TiA + l/T,,,
l/T;
= l/T,,
+ l/T,,,
l/T;
= l/TIA - l/T,,.
(13)
Only non-adiabatic interaction terms contribute to the spin-lattice relaxation process but the main contribution to the transverse relaxation is often made by adiabatic interactions, the adiabatic diffusion of the Larmor precession frequencies of the spins. Therefore T2 G 2Tr. Equality is obtained only if the relaxation is induced entirely by non-adiabatic interactions (spin flips) [21]. At the instant an RP originates its state is given by the population of S and T states, the off-diagonal elements of the psT type are equal to zero. In the evolution process appears psr # 0. But at the inStaM of contact psr becomes zero due to exchange interaction. In order to estimate the change of populations of S and T states of the RPs we solve the following problem. At the instant of origin we should have pST = 0 but the populations are given by the vector ( pss(0), p T~T,,CO>,PT+T+N + PT_T_WL Sol~%W with the% initial conditions we get
(14) [pT+T+Z!T_(j
=@(t)[pT+T+Z:_T_(o,)*
The spin evolution operator has the form
g(t)=
$+e+c
++e-c
a-e
++e-c
++e+c
4-e
1 2(+-e)
2(+-e)
2(++e)
,
(15)
I
where e= $ exp(-t/T,‘),
c=i
exp(-t/T,‘)
cos(Awt),
Aw=w*-wg.
(16)
For example, if the initial state of the RP is the triplet state (T precursor), subsequently it may be observed in the singlet state with the probability (see also fig. 2) PT+&)=$--
$ exp( - t/T;)
- $ cos( Awt) exp( - r/T,‘)
=&(l-exp(-t/T,‘))+i[l-cos(Awt)exp(-r/T;)]. The oscillations are dumped by the transverse relaxation becomes the well-known result [l] P-r+&)
= f sin’(AW2),
(17) of the spins. If there is no relaxation
(17)
(18)
which describes the oscillations of the RP between the S and T states caused by the difference of Larmor
118
K L.i.Ars, KM. Salikhov / Recombination probability of radical pairs
Fig. 2. Population of the singlet state of RP for a triplet precursor. Parameters: (1) Ag = 10w3, B, = 0.1 T, T;, T; + co; (2) Ag=O, T,‘=2xlO-’ s, T;=lO-’ s; (3) A.g=10-3, B,,=O.lT, T;=2xlO-‘s, T;=lO-‘s.Theinsetgivesthe area near the origin on an enlarged scale.
frequencies of the spins. Of great importance for the efficiency of S-T mixing for short times between re-encounters (see below) is the value of the time derivative @p/at), _ ,, which is non-zero if relaxation is taken into account as seen from eq. ‘(17) and fig. 2. Paramagnetic relaxation entirely mixes all the states and ~r+s(t)
+ l/4
if
t/T,‘, t/T*‘+
00
(19)
in correspondence with the statistical weight of the singlet state. The difference of the Larmor frequencies may be caused by Ag as well as by the isotropic hf interaction. In the latter case Aw depends on the nuclear spin configuration and on the hf coupling constants [l] AU = (g, - g,)/3fi-‘&
+ c aimi - c akmkY i(A) k(B)
(20)
where /I is the Bohr magneton. B, the magnetic flux density. The population of the singlet term for the whole ensemble of RPs is obtained by averaging (17) over all subensembles with different nuclear spin orientations in the radicals A and B of the RPs. The distribution of local hf fields in the case of a sufficiently large number of coupling nuclei may be described by a Gauss function. In this case, after averaging (17), we get PT&)
= + - & exp(-VT,‘)
where aA,eff radical A
aA,eff
and
=
(
f
aB,eff
c
i(A)
Ui’Ii(Ii
are
+
- gn)Ph-%
effective hf coupling constants
the
1)
- i cos(g,
1
i/2 ,
exp[ --t/T;
-
i(&.rr + &,,)t2],
(21)
of the radicals of the pair, e.g., for the
(24
where Ii is the spin of the magnetic nucleus i of the radical A. The transverse (phase) relaxation of the spins as well as the difference of the Larmor frequencies mixes the singlet state only with one of the triplet states, namely with T,,. The longitudinal (spin-lattice) relaxation mixes the S state also with the T, and T_ states. Therefore in the dynamics of S-T transitions in the RPs the longitudinal relaxation can play an important role, even if the Ag and ihf mechanisms warrant an effective S-T,, mixing.
K. Ciders, KM
Salikhov / Recombination probabiliv
of radicalpairs
119
In ref. [Vi] an expression was estimated similar to eq. (17) for the time dependence of singlet state population for a singlet precursor, ps _ s (2). As a proof of the correctness of the calculated populations ps ~ s [15] and p T _ s (17) we can use the cormection 3Pr-s(t)
+Ps+sW
(23)
= 1,
which is fulfilled. The recombination probability was calculated in ref. [15] for the model of continuous diffusion where only the first contact was taken into account. As we will see below for the relaxation mechanism of RF recombination just consecutive contacts play an important role.
4. Paramagnetic relaxation times For free organic radicals in liquids paramagnetic relaxation is caused mainly by random modulation by rotation of the g tensor anisotropy (gta), the anisotropic hf interaction (ahfi) and the spin-rotational interaction (sri) [1,21,22]. In the present work numerical calculations were carried out for illustration of the effect of this relaxation mechanisms in connection with the Ag and ihf mechanisms of S-T,, mixing. If the relaxation is caused by the anisotropy of the g tensor the longitudinal and transverse relaxation rates are l/T, = &-+32A-2B&,/(l l/T, = &&?2ti-2B&[4 where @
+ 3/(1+
(24) Y:@;)],
(25)
characterizes the scale of the gta and is expressed through its principal values
&?=g:+gzz+g32-3g2, ye
+ y,zB,#),
iT=(g,+g,+gd/%
(26)
is the gyromagnetic ratio of the unpaired electron, TV the rotational relaxation time, which is equal to 7, = 4qb3/3kT,
(27)
with b the radius of the radical and 7) the viscosity. With the Stokes-Einstein can be expressed by the diffusion coefficient D r. = 2b2/9D. For organic radicals @= zero to the value (l/T,),
equation D = kT/6Tbv,
7.
(28) 10-2-10-3.
-+ 5 x 10-2L\g2/~o.
From (24) it follows that l/T,
increases with increasing B, from
(29)
In non-viscous liquids, if r. = lo- l1 s, (l/T,),
--, 5 x 10-9bgz
s-l.
(30)
If, e.g., @= 10m2, TI lim = 2 x low8 s. With increasing viscosity this time limit T,,, is prolonged proportional to the viscosity. The transverse relaxation rate increases with increasing B, corresponding to (25). For very high B. formula (25) leaves the area of applicability of time-dependent perturbation theory ((@)“‘ptt-iB o~o< 1). In practice T2 cannot be less than ro. On that account in the numerical calculations we arbitrarily put T, = 7. in cases where the values T2 calculated by (25) became less than ro. Such a situation obtains if
K. Eiakrs, KM. Salikhov / Recombination probabiliv of radical pairs
120
If B, is measured in tesla and 7s in seconds, (31) yields I&, > 10-11,'~o(~)1'2 T.
(32)
In non-viscous liquids condition (32) is fulfilled at magnetic flux densities of dozens of tesla. With increasing viscosity condition (32) is fulfilled at lower values of B,. The relaxation due to the dipole-dipole anisotropic interaction of an electron and a spin I nucleus, in the point-dipole approximation, is described by the relations l/T, = 2w/(1+ l/T,
= W[l
+
$E,27,2),
(1+ Y:%‘~)-~]
(33) 7
(34)
where W= (2/3)1(1+ l)y&?A*r-%a, r is the distance between the unpaired electron and the nucleus, yr is the nuclear gyromagnetic ratio. In our model calculations we have considered only the interaction of the unpaired electron with a single proton. The S-T transition rate is seen from (33) and (34) to decrease with increasing field in this case. The spin-rotational relaxation caused by the random modulation of the rotational angular momentum and the sri tensor is given by [22] l/T,
= l/T,
= (8g; + 26g:)
kT/12nb3v,
where 6g = g - 2.0023 1. With the Stokes-Einstein
(35) equation we get
l/T, = l/T, = v/97,,
(36)
Usually 6g2> hg2, therefore the contribution of the sri relaxation to the where 6gz= 6gi + 2gg:. longitudinal relaxation rate is higher than that due to the gta relaxation. Because the sri relaxation is independent of the magnetic field it will reduce the dependence of RP recombination on the magnetic field. In general, all mechanisms described above contribute to the relaxation rates l/T, and l/T,. In our model calculations we will consider the ahfi relaxation separately from the other mechanisms. If sri relaxation is taken into account in all cases we set v= @. For illustration, in fig. 3 the typical field dependences of the relaxation rates are given.
5. Calculation of the recombination
probability of radical pairs
The recombination probability was calculated by two methods: by immediately solving the kinetic equations (1) and (6) or by addition of the contributions of all consecutive contacts [l]. For the exponential two-positional model with complete dephasing of S and T states at the instant of contact caused by exchange ‘interaction the calculations were performed by both methods. The results were identical as could be expected. The recombination probability depends on the multiplicity of the RP precursor. But there exists a connection between the recombination probabilities of RP for the S-precursor, “p and for the T-precursor, ‘P 1119 “P =sPcl - 3(I -sPO)TP,
(37)
where “p, is the recombination probability of the RP if the S-T mixing were “switched off’. In view of (37) we shall give explicit formulae only for ‘p. In all our calculations we assume that initially the radicals of the pairs are in the reaction area.
121
K. Liiders, KM Salikhov / Recombination probability of radicalpairs
+I*-, IO8
10'
IO6
Fig. 3. Longitudinal and transverse relaxation rates of a radical as a function of B,,. Parameters: D = lo-” m’/s, b = 0.3 nm, ) g-tensor ani1 - @= 10-3, 2 - dg2= 10-2. ( sotropy, (- - -) g-tensor anisotropy and spin-rotational relaxation (the calculations were performed for the case &L&+10-3), (.-.-.) anisotropic hyperfime interaction relaxation (point-dipole approximation; electron-proton distance r = 0.15 mu).
lo5
IO'
5.1. Exponential one-positional model The recombination
probability is
(38)
PI= K, / ~ss(t) dtSolving (1) we get
=pl = K,7[ (K,7/2
+ r/T; + l)( 7*/4T;T;
+ r/6T; + 7/12T;)
+ Aw*r*( 7/T,’ + 2/3)/41/A,, (39)
A, = (K,7/2
+ T/T,’ + l)[ K17( 7*/4T;T,’ + 3r/4T,’ + 7/2T; + 1) + (1 + r/T,‘)(l
+ 7/T;)],
+A~*r~[K~7/2+1+(ZQ/4+1)7/T~],
Sp()= &T/(1
+ K*r).
5.2. Exponential two-positional model For this model the recombination =pz = T$%ss(t)
dt-
probability is (4W
It was calculated using two approximations. In one case it was assumed that at the instant of contact the dephasing of S and T states is caused only by the reaction and drifting apart after contact. In the other case we assumed that at the instant of contact the exchange interaction causes a strong dephasing, i.e.
K. Liiders, K.M. Salikhov / Recombination probability
122
of radical pairs
z+ Ii. This is equivalent to plsr = 0. Corresponding with these two approximations following results: - Without consideration of exchange dephasing of S and T states:
+L&Ulge
‘pz
=
(K7,/A,)
{ ( T~/~T,‘T; + 7/6T; + 7/12T,‘) [ (K7J2
+ Ao~~~( K7t/2 + l)( T/T; A,=
[(I&J2
+ l)(l
+7/T,‘)
x [ ( KT~ + l)( 7/T;
+ l)(l
7/T;)
+
we get the
+ K717/272] r/r2
+ 2/3) ~/47~ } ,
(41)
+Ky~/27~]
+ l)( 7/T,’ + 1) + KT,( T~/~T;T;
+A~~7~(K7~/2+1){K7~(7/27~+1)+1+
+ 3~/4T;
+ r/2T;
+ 1)7/~~]
[K71(7/472+1)+1]r/T;}.
- With complete dephasing of S and T states during contact due to strong exchange interaction instant of contact: ‘p2 = (K7,/A2)[(r2/4T;T; +Aw27’(7/T; A,=
+ 7/6T,‘+
7/12T;)(l+~/T;
at the
+ ~/7~)7/7~
+ 2/3)~/4~~],
(42)
(1 +~/T;+‘T/~~)[(K~~+~)(~/T;+~)(~/T;+~) +I$(
T~/~T,‘T;
+ 37/4T;
+ r/2T;
+A~~7’{&~(7/27~+1)+1+
+ 1)~/~~2]
[K71(7/472+l)+l]7/T;}.
With complete exchange dephasing the recombination probability was calculated solving (6) and also by use of the method of adding the contributions of all re-encounters. Using the latter method the recombination probability is given by [l] p = Tr[ s(l
-@d)-’
p(O)],
(43.)
where # = /cm$(t) f(t)
d2 is the operator of spin state evolution, averaged over the RP lifetime distribution between re-encounters, ~(0) = (PSSVO, PT,,T,(O), PT+T+(O) + PT_ T_(O)) is the vector of the initial state of the RP but the operators P and & in the basis of S and T states are given by the matrices (see ref. [l]) P=(i
i
i),
e=(lih
;
Ej,
where A = K7J(l + KT~) is the recombination probability of the S-state RP during a single contact. With the aid of these relations and using (15), and (16) we get ‘p = K7, [ pn (1 - IF) + 2pn (1 - z) + 3pszy1
- Z)(l - z)] /3( K7,A + B)
(44)
where A = 4 + 2pn(l-
c”) + 3pn(l
B = 4n2(1 -pZ)(l F= f /,“exp( p=
,z /
t ()
+p2n2(1
dt,
E= $imcos(Aot)
- E),
-PC?),
-t/T,‘) dt,
- ;)(l
-Z)
f(t)
n=l/(l-p),
rr = 71”.
exp( -t/T;)
f(t)
dt, (45)
K. Liiah,
KM
We note that without consideration spa = Kqn/(l
n -
of S-T
123
of radical pairs
transitions the recombination
probability becomes
+ Kqn).
In the limit of full S-T case from (44) =p = KT,(
Salikhov / Recombination probability
(46)
mixing between two re-encounters
(Z, ? --* 0) we get an interesting result. In this
l)/[ K7,(3 + n) + 4n].
(47)
With increasing reactivity of the radicals ‘p-qn-1)/(3+n).
(48)
Only if n =*- 1, ‘p + 1 ! If S is mixed only with the T,, state (corresponding
to Z + 1, Z + 0 in (44)) we get
‘p-0Cr&2-1)/3[K+2+1)+2n], ‘p --) (n -
1)/3(n
+ 1)
(49) if
KT, z+ 1.
(50)
The above result (44) is valid for any statis$cs of re-encounters, which is defined by the type of motion of the reagents in the “cage”. Averaging F(t) using (2) we get the recombination probability for the exponential two-positional model =p2 =
K*,n ( T/T;
+ 27/T;
+ 3r2/T,‘T:)
rc/r23( KqnA2
+ B2),
(51)
where A, = 4(1+
q/T;)(l
B, = 4(1+
7/T;)(1+7/T:),
l/TC = l/7 + 1/T2,
+ q/T?)
+ 3(1+
7C/r,E)?.7C/72T; + 2(1 + ~,/T;)T~&T~~
l/7’ = l/7, + l/T;,
By algebraic transformations
l/Tz” = l/T;
+ T~T~/$T;T~~,
+ hti2~‘.
it can be shown that (51) and (42) are identical.
5.3. Model of continuous diffusion For the model of continuous diffusion we perform fully analogous calculations, only (2) is replaced by the distribution function (7), (8). The averaged spin state evolution operator is defined by f?= exp( - /mm/p), E=exp(
-/mm/p)
cos(+~(l/T~~-z/T;)
m/p),
(52)
where 1/T2D = (l/T; + d-)/2. Putting (52) into (44) and going to the Iimit of continuous diffusion (X ,,, 7s + 0; x’,/rs + 60, where D is the relative diffusion coefficient) we get =pD = Kq( /z/3
+ 2/m/3
-t /m)
~[K~+(4+2/~+3\/7,/T,‘+/~)+4(1+/~)(1+/~)]-~. where 7r is the total time the RP stays in the reaction area, 7r = qn, 7u = b2/D the RP.
(53) is the effective lifetime of
K. L.iiders,K. M. Salikhov / Recombination probability of radical pairs
124
The RP recombination probability for the model of continuous diffusion was also calculated in ref. [16] solving the stochastic Liouville equation. In the treatment of ref. [16] the exchange interaction was not taken into account. As mentioned above the exchange interaction causes a strong dephasing of S and T states during contact. This may be considered in the corresponding boundary condition for the off-diagonal matrix elements psr. Without consideration of exchange interaction there remains a dephasing between S and T, states resulting from the RP recombination in the S state. The dephasing rate constant is K’ = K/2, where K is a recombination rate constant. The exchange interaction causes a supplementary strong S-T dephasing, therefore one expects K’r z+-K7/2, in the limit K’r --* do. The kinetic equations for PST, ad PST_ are not coupled with the system of kinetic equations for the S- and T-state populations. Therefore it is sufficient in the expressions for the recombination probability in ref. [16] to change the rate constant K’ for the off-diagonal matrix elements psr, K’r + co in order to consider the effect of strong exchange dephasing on the recombination probability. By algebraic transformations it may be shown that the result corresponds to eq. (53) which we have calculated by the method of consecutive contacts.
6. Discussion of results 6.1. Comparison of the exponential
models
First we consider the situation if the S-T transitions are caused only by spin relaxation. In this case the kinetic equations for the S- and T-state populations of the RPs are not coupled with the off-diagonal elements psr of the density matrix. Therefore eqs. (41) and (42) (which differ only in the magnitude of dephasing of S and T states at the instant of contact) are equivalent if only the relaxation mechanism of S-T transitions works and give =pz = Kr,( r2/47’;Z”
+ r/6T;
xr/r2 [ (1 + K7,)(1+
+ 7/12T;)
7/T/)(1
+ 7/T;)
+ Kq(
T~/~T,‘T,’ + 37/4T;
+ 7/2T;
+ 1) T/T,]. (54)
In the case of the triplet precursor the RPs are unable to recombine at the instant of contact, contributions to the reaction product are made only by the subsequent re-encounters. Therefore (54) contains the factor r/r2 = n - 1, the number of re-encounters without consideration of the first contact. The quantity 7i7/r2 denotes the total time during which in principle the RPs may recombine. For the pure relaxation mechanism of S-T transitions a one-to-one relation can be established between the recombination probabilities calculated by the exponential one- and two-positional models. To this end the phenomenological parameters K and K, must be connected with the equation KI = K~/T~ (1 + K-r,),
(55)
which has an obvious interpretation: In the two-positional model l/r2 is the rate constant of RPs entering the reaction area and Kri/(l + Kq) is the recombination probability during a single contact. If the reactivity of the radicals increases, the parameter K in the two-positional model also increases while in the one-positional model K, at first increases and after that goes to the limit K, + l/r*. This means that using the one-positional model the phenomenological parameter K, must by no means be increased to such a degree as the reactivity of the radicals is increased. If S-T transitions are induced not only by relaxation but also by the difference of Larmor frequencies it is impossible to accurately relate the result of the two-positional model to the result of the one-positional model redefining the parameters corresponding to (55). Nevertheless the numerical calculations show that even if Aw # 0 the one-positional model (39) and the two-positional models (41), (42) give similar results if the parameters of the models obey (55) (see fig. 4).
K. Liiders, KM. Salikhov / Recombination probabiliy of radicalpairs
125
Fig. 4. Dependence of recombination probability on magnetic flux density for the exponential models: curves 1 and 2 pertain
to the onepositional model, 3 and 4 to the two-positional one. Curve 4 was calculated considering the spin dephasing at the instant of contact caused by exchange interaction. In all cases was assumed K = 10” s-l. The curves 1 and 2 were calculated from formula (39), but the parameter KI in the case of the curve 1 was chosen to be KI = 10” s-l (the renormalization formula (55) was not used). For curve 2 the renormahzed value of the recombination rate constant was chosen. The remaining parameters are: q = lOWi2 s, r2 = 10m9 s, r = lo-’ s, Ag = 2 x10e3, @+a=lO-‘, D=lO-lo m2/s, b=0.3 nm. From the figure it follows, that the results from the one-positional model using the renormalization (55) are in good agreement with the results from the more consistent two-positional model. The consideration of a suplementary dephasing caused by exchange interaction somewhat decreases the recombination probability as may be expected.
It is interesting to note that corresponding to (39) in the one-positional model the efficiency of S-T mixing caused by Au-at large Kl decreases with increasing Kl. Actually, if K1r + do and r/T;, r/T; -+ 0 we get =pl =
Aw2r/3K, + 0.
(56)
This decrease of S-T mixing can be explained by the “smearing out” of the singlet term caused by very fast recombination of the RP. This peculiarity of the one-positional model has been mentioned in the literature [19,20] and is a serious insufficiency of this model. In the models with m-encounters under consideration, the efficiency of S-T mixing does not decrease, the recombination probability monotonically increases and approaches a limit with increasing K. Assuming T,, T, + cc, 7’ = +rcand Kqn --, 00, from (51) we get a limit for the recombination probability different from (56) =pz +
Aw2rC2 ( n - 1)/3 [2 + Aw=r,=( n + l)] ,
(57)
which does not depend on K. 6.2. Effective rates of S-T, transitions From (51) and (53) follows a tremendously important conclusion. In the models with re-encounters under consideration the S-T,, mixing can be characterized by some effective rates 1/T2E, 1/T2D, if the relaxation, Ag and ihf mechanisms work simultaneously. This effective parameter for the exponential two-positional model is 1/T2E = l/T; where l/7
+ Ao=r’,
(58)
= l/7 + l/7= + l/T=‘, and, for the model of continuous diffusion,
1/T2D = (l/T;
+ /+))/2.
(59)
From these equations it follows that in the general case the dephasing of S and T, states caused by Aw and by transverse spin relaxation is not additive. Only in situations where T; is sufficiently large, T; > r, r2; TiAo B 1, the two parts of S-T, mixing can be added. In this case we get from (58) and (59), respectively
l/GE = l/T; + Aw=r,,
(60)
126
K. L.iiders,KM. Salikhov / Recombination probability of radical pairs
Fig. 5. Phase relaxation rate l/T; (curves 1) and effective phase relaxation rates l/TaD (curves 2), l/TrE (curves 3, 4) as a function of Ba. With increasing time between reencounters the efficiency of the Ag mechanism increases in comparison with the relaxation mechanism of S-T,, mixing. In this sense the long diffusional trajectories in the diffusional model (curves 2) play an important role. For short times between re-encounters (curves 3) the relaxation rate l/TaE differs only slightly form the relaxation rate l/T;. Parameters: (3) r = lo-* s, 72 = 10-s s; (4) r = lo-’ s, ra = 10-s s. The other parameters are the same as in fig. 4. ( ) Ag mechanism, g-tensor anisotropy relaxation; (- - -) Ag mechanism, g-tensor anisotropy and spin-rotational relaxation; (. -. -. ) anisotropic hyperfine interaction relaxation, ihf mechanism (r = 0.15 mn, a=lmT).
and l/T,D
= (l/T;
(61)
+ Aw)/2.
Obviously, in this limiting case the two mechanisms are additive. Even in the regarded situation this is an astonishing result because the character of spin dynamics caused by the difference of Larmor frequencies differs greatly from that caused by relaxation. What the two mechanisms have in common is only that they cause a dephasing of S and T, states. For illustration the dependence of the relaxation rate l/T; and the effective relaxation rates l/TzE and l/TzD on the magnetic flux density is shown in fig. 5. It is very interesting that for typical values of the parameters of the radicals the relaxation mechanism may well compete with and even predominate over the Ag mechanism of S-T,, mixing. 6.3. The recombination probability
of radical pairs
In ref. [4] calculations were made of the recombination probability of RF’ considering ihf, Ag and relaxation mechanisms. But in ref. [4] it was assumed that these mechanisms make additive contributions. We separate the contribution of S-T+ and S-T_ transitions to the recombination probability. We assume that the transverse relaxation, Ag and ihf mechanisms mix the S and To states extremely effectively, that is, r/TzE, rD/Tp B 1. Then we get from (51) and (53), respectively, =p2 = Kq( n - 1)(2/3
x [4n(l+ TpD = h(2/3
7/T,‘)
+ T/T;) + &,(4(1+7,/T,‘)
+ &777)/[
h(2
+ &Z)
+ 3rr,/~~T; + 4(1+
+ 2(1+ /X)].
T,/T;)T/T~
+ T~T,/~~T;)] -l,
(62) (63)
K. L.&&q K. M. Solikhov / Recombination probability of radical pairs
From this we can find the maximum contribution which can be made by the spin-lattice (S-T+, S-T_ transitions) to the recombination probability of RPs. For this we should find A=p==p(l/T,‘--)
127
relaxation
co) -=p(l/~,‘-+o),
where Kr, + 00. From (62) and (63) we get for the exponential two-positional
model
ATpz--, 2n(n - 1)/3(n* + 4n + 3)
(64)
and for the model of continuous diffusion
(65)
=po-+2/3. Usually radical recombination relation
in liquids is a process which is controlled by diffusion. This means that the
KT,>~
(66)
should be fulfilled. Under this condition, from (51) and (53) we get, assuming Kr, --, cc, ‘p2 + (q/T;
+ ~T,/T*~ + 3mc/T,‘T2E)(
n - 1)/3A,,
Above we have mentioned that the Ag and ihf mechanisms and the transverse relaxation contribute non-additively to the recombination probability of RPs. It is very important to note that also the transverse and the longitudinal spin relaxation appear non-additively in recombination probability. For example, corresponding to (67), with increasing relaxation rate ‘pD -+ l/9 if l/Tp ---,0 but l/T,’ + co; in the opposite situation l/T; --, 0, l/T2D --) cc we have ‘pD + l/3. On the other hand, if the transverse and the longitudinal relaxation are simultaneously considered if l/T,‘, l/Tp --, cc we get ‘pD + 1. A comparison of the results from the exponential two-positional model and the model of continuous diffusion (see, e.g., (67)) shows that the molecular dynamics of the radicals essentially influences the behaviour of recombination probability. Very important is the kinematics of the motion of the radicals in the intervals between m-encounters. For the two models considered, which differ from each other only in the statistics of radical reencounters we get very different dependences of recombination probability on paramagnetic relaxation rates, the g-value difference and lifetime of the RPs. For example, corresponding to (67) in the range of slow S-T transitions, if r/T:, r/T/ -K 1, we get for the exponential two-positional model =p2 = (7,/T;
+ 2rJqE)(n
- 1)/12
(68)
and for the model of continuous diffusion
The possible limiting values of the recombination probability of the RP also depend on the kinematics of the radicals of the pair. The maximum possible values of recombination probability, if only S-T,-, mixing works, are given by (50). If there is an effective mixing of the S state with all the triplet states eq. (48) is obtained. From these equations it can be seen that the maximum possible values of recombination probability reach the expected values l/3 and 1, respectively, only in cases where the number of re-encounters increases without limitation. Just such a situation is realized in the model of continuous diffusion.
128
K. Liiders, K. M. Salikhov / Recombination probabiliv
of radical pairs
6.4. Long-lived radical pairs We discuss the field dependence of the recombination probability of RPs in the situation where S-T transitions are caused by the Ag mechanism and by paramagnetic relaxation which is caused by g-tensor anisotropy. The parameters l/T; and l/TzD, l/TzE which characterize the S-T transition rate increase with increasing magnetic flux density BO. Simultaneously the recombination probability of RPs also increases. But the experimentally observable field dependence of ‘p may not simply reflect the field dependence of l/T,’ and l/T, E, l/TzD. Differences are especially great for long-lived RPs. Really, for long-lived RPs (large r, r2, To in comparison with T; and T2E, TzD) ‘p approaches the limiting value (47) and ceases to depend on B,, although l/T,’ and l/T, E, l/TzD may further increase with increasing magnitude of the B, field. Corresponding to (24) l/T; reaches half its limiting value (29) at 1’2B0,
which means that in the range Be = ‘/2B0 the maximum change of l/T{ takes place. Therefore one could think that for the relaxation mechanism of S-T transitions the characteristic range of maximum changes of =p should also lie in the range B, = 1/280. However, for long-lived RPs it is not necessarily so. In this case the transition range Tp(BO) should lie in fields B,, < 1’2B0. Such a situation may occur, for excample, in micelles, in which the lifetime of RPs may reach several lo-’ s [2,5]. Therefore customary values T,‘, T2E, Tp = 10e6 s will mix the S and T states of RPs very effectively and further shortening of the relaxation times with increasing Be is irrelevant to ‘p: there appears a peculiar saturation. For long-lived RPs one should expect the following behaviour ‘p( B,): with increasing lifetime of the RP the value of B, decreases for which =I, approaches the limiting value (47). If the relaxation process is caused mainly by the ahfi we expect another behaviour of ‘p(B,). As seen from (33), (34) and fig. 3 both relaxation rates l/T; and l/T; decrease with increasing B0 field, for l/T; eq. (70) is valid. If the ihf mechanism of S-T, mixing works simultaneously 1/T2E and 1/T2D only slowly depend on the & field. Relative small coupling constants l-2 mT warrant an effective S-T, mixing in long-living RPs. If we follow the abovementioned idea the characteristic range of maximum changes of ‘p should also lie in the range B,, = 1/2B0. Due to the opposite dependence l/T,( B,) for ahfi relaxation in comparison with the gta relaxation we expect the opposite behaviour of ‘p( B,): with increasing lifetime of the RP that value of B, increases in which the maximum change of ‘p (B,) takes place. In general, beside the Ag and ihf mechanisms of S-T, mixing all the three abovementioned relaxation mechanisms work simultaneously. In real cases some of these mechanisms can preponderate. If, e.g., the recombination probability monotonically increases with increasing B, field as measured in refs. [8,23] the Ag and ihf mechanisms together with gta relaxation may be supposed. In the opposite case, if the recombination probability decreases with increasing Be field the main contribution to the magnetic field dependence of RP recombination probability results from the ihf mechanism and the ahfi relaxation [4,14,24]. 6.5. Results of numerical calculations For an illustration of the above expressions and qualitative discussion we performed several numerical calculations. The results are given in figs. 6-9. These curves corroborate the abovementioned basic regularities of the change of ‘p dependent on the molecular kinetic and magnetic resonance parameters of RP. We have made a detailed study of the recombination probability considering the relaxation and Ag mechanisms of S-T transitions. In order to cover systems in which the isotropic hyperfine interaction also
K. Liiders, KM.
ci.1
E!JTI
d5
Salikhov / Recombination probability
129
ojradicalpairs
i
Fig. 6. Recombination probability of triplet-born RPs; exponential two-positional model with consideration of dephasing during contact caused by exchange interaction. The contributions of the Ag and relaxation mechanisms of S-T mixing are compared in the regarded case of relatively short times between consecutive re-encounters. The dashed curves contain a contribution from spin-rotational interaction (@v). Parameters: (1) Ag- 2 X 10e3, a= a= 0; (2) Ag = 0, m+a=10-3; (3) Ag=2x10-3, a+a=10-3. The other parameters are the same as in fig. 4.
Fig. 7. Influence of the kinematics of radical motion on the character of field dependence Tp. Parameters: Ag = 0.02, a + a= 0.01, (1) model of continuous diffusion, (2) exponential two-positional model. The other parameters are the same as in fig. 4, symbols see fig. 5 (a = 2 mT).
a4 a3
.-.-._.
a2 0.1 .-.-.-.
Fig. 8. Influence of RP lifetime on field dependence ‘pz. Parameters:(1)7=10-*s,(2)7=10-7s,(3)7=10-6s.The other parameters are the same as in fig. 4. Curve 4 shows the field dependence of the effective spin dephasing rate l/7”. For the actual values r and 7s the l/Z’,” curves differ insignificantly from each other. The dashed curves contain a contribution from spin-rotational interaction (Q= v).
Fig. 9. Field dependence ‘pz of RPs with large anisotropy of the g tensor, large difference of the isotropic g values and long lifetimes in comparison with the corresponding curves resulting from ihf mechanism and anisotropic hyperfine interaction relaxation. Parameters: (1) 7 = lo-’ s, (2) 7 = lo-’ s, (3) r= 10-6 s; Ag = 0.02, @ + e= 0.01. The other parameters are the same as in fig. 4, symbols see fig. 5 (a = 2
130
K. L.iArs,
K.M. Salikhov / Recombination probabiliv
contributes to the S-T transitions ‘p(Aw) has been calculated using the abovegiven formulae. The total probability is
of
radicalpairs
for each configuration
of nuclear spins
(71) where N = & (21, + 1) is the total number of nuclear configurations of RP, Au(m) for different nuclear spin configurations is given by eq. (20). In our model calculations we have considered only coupling with one proton in the RP. It is necessary, of course, to remember that (71) is valid only for high magnetic fields (in comparison with the isotropic hyperfine interaction), where the non-secular terms of the isotropic hyperfine interaction can be neglected. For organic free radicals this is fulfilled in fields B, 2 0.1 T. In fig. 6 the recombination probability is shown as calculated by the exponential two-positional model. The Ag and relaxation mechanisms are considered separately as well as coupled. In the case under consideration the lifetime and the time between m-encounters are relatively small and the S-T mixing is caused mainly by phase relaxation (see figs. 2 and 3). On that account the consideration of spin-rotational relaxation only slightly changes ‘p(B,,). It is seen that the contributions of the Ag and relaxation mechanisms are not additive. In fig. 7 it is shown that ‘pD varies less than ‘pz with changing magnetic flux density. This is connected with the influence of the kinematics of radical motion, the statistics of re-encounters. As expected, the area in which the value of ‘p(B,-,) changes strongly is shifted to smaller values of B, (in the case of Ag mechanism and gta relaxation, we discuss now) as the lifetime of RP increases (see fig. 8). In the low-field region ‘pz depends predominantly on r and. the spin-rotational relaxation times T,, T,. From (36) and (68) we get =pz =
7/4T,.
(72)
In the given examples in most cases ‘p does not exceed l/3. This is connected with the fact that for the values of parameters chosen the spin-lattice relaxation influences the shape of the curves only slightly. It becomes apparent at large lifetimes of RP and large anisotropies, e.g., @= 0.01 (see fig. 9).
Acknowledgement The authors wish to thank S.A. Mikhailov and S.N. Smirnov for assistance in the accomplishment numerical calculations.
of
References [l] K.M. Salikhov, Yu.N. Mohn, RZ. Sagdeev and A.L. Buchachenko, Spin polarization and magnetic effects in radical reactions (Akademiai Kiad6, Budapest, 1984). [2] N.J. Turro and B. Kraeutler, J. Am. Chem. Sot. 100 (1978) 7432. [3] V.F. Tarasov, A.D. Pershin and A.L. Buchachenko, Izv. Akad. Nauk SSSR Ser. Khim. (1980) 1927. [4] H. Hayashi and S. Nagakura, Bull. Chem. Sot. Japan 57 (1984) 322. [5] K. Lnders and K.M. Salikhov, Chem. Phys. 98 (1985) 259. [6] L. Sterna and A. Pines, Annual Report of Materials and Molecular Research Division of Lawrence Berkeley Laboratory, University of California, CA, USA, LBL-7355, HC-13, TID-4500-R66 (1977). [7] L. Sterna, D. Ronis, S. Wolfe and A. Pines, J. Chem. Phys. 73 (1980) 5493. [8] L.A. Marguhs, I.V. Khudyakov and V.A. Kuzmin, Chem. Phys. Letters 119 (1985) 244. [9] F.P. Sargent, B. Brocklehurst, R.S. Dixon, E.M. Gordy, V.J. Lopata and A. Singh, J. Phys. Chem. 80 (1977) 815. [lo] K. Schuhen, H. Staerk, A. Weller, H.-J. Werner and B. Nickel, Z. Physik. Chem. NF 101 (1976) 371.
K. Liiders, K. M. Salikhov / Recombination probability
of radical pairs
131
[ll] M.E. Michel-Beyerle, R. Haberkom, W. Bube, E. Steffens, H. Schroder, H.J. Neusser, E.W. Schlag and H. Seidhtz, Chem. Phys. 17 (1976) 139. [12] Yu.N. Mohn, O.A. Anisimov, V.M. Grigoryants, V.K. Molchanov and K.M. Sahkhov, J. Phys. Chem. 84 (1980) 1853. [13] B. Brocklehurst, Nature 221 (1969) 921. [14] T. Uhich and U. Steiner, Chem. Phys. Letters 112 (1984) 365. [15] S.I. Kubarev, E.A. Pshenicnov and AS. Shustov, Teor. Exp. Khim. 15 (1979) 16. [16] R.G. Mints and A.A. Pukhov, Chem. Phys. 87 (1984) 467. [17] M. Tomkiewicz, A. Groen and M. Cocivera, J. Chem. Phys. 56 (1972) 5850. [18] R.M. Noyes, J. Am. Chem. Sot. 77 (1954) 2042. [19] K.M. Sahkhov and F.S. Sarvarov, Tear. Exp. Khim. 18 (1982) 146. [20] K.M. Sahkhov and S.A. MikhaiIov, Teor. Exp. Khim. 19 (1983) 550. [21] A. Carrington and A.D. McLachlan, Introduction to magnetic resonance (Harper and Row, London, 1967). [22] P.W. Atkins and D. Kivelson, J. Chem. Phys. 44 (1966) 169. [23] L.A. Marguhs, I.V. Khudyakov and V.A. Kuzmin, Chem. Phys. Letters 124 (1986) 483. [24] T. Uhich, U.E. Steiner and W. Schlenker, Tetrahedron 42 (1986) 6131.
Chemical Physics 117 (1987) 113-131 North-Holland, Amsterdam
THEORETICAL TREATMENT OF THE RECOMBINATION PROBABILITY CJF RADICAL PAIRS WITH CONSIDERATION OF SINGLET-TRIPLET TRANSITIONS INDUCED BY PARAMAGNETIC RELAXATION K. LijDERS Sektion Physik der Karl-Marx-Universitiit,
7010 Leipzig, GDR
and K.M. SALIKHOV Institute of Chemical Kinetics and Combustion, Novosibirsk
630090, USSR
Received 29 January 1987; in final form 11 June 1987
The recombination probability was calculated for several models of radical pairs (RPs). The singlet-triplet transitions induced by the relaxation mechanism and by the difference of Larmor frequencies of the unpaired electrons of the two radicals of the pair were taken into account. The dependence of the recombination probability on the external magnetic field and on the lifetime of the RP was analyzed. It was pointed out that the different mechanisms of S-T mixing make non-additive contributions. An important part is played by the interferences of the contributions of longitudinal and transverse relaxations. In systems with sufficiently long lifetimes of RPs the recombination probability changes strongly in relatively low magnetic fields, whereas in the same systems the values of paramagnetic relaxation rate change strongly in higher magnetic fields.
1. Introduction
In the recombination of RPs appear several mechanisms of singlet-triplet (S-T) transitions: (a) isotropic hyperfine interaction (ihf mechanism), (b) difference of the g factors of the two partners (Ag mechanism), and (c) relaxation mechanism (see, e.g., ref. [l]). In reactions of organic radicals in most cases the first two mechanisms are regarded. In non-viscous liquids the lifetime of pairs of neutral radicals is in the range of nanoseconds, but typical values of the times of paramagnetic relaxation of unpaired electrons lie in the microsecond range. This being so, paramagnetic relaxation hardly appears in the recombination process of RPs. This situation changes essentially if the reaction occurs in micelles [2-51, in viscous systems [6-81 or if the recombination of ion-RPs is considered [9-121. In all these systems the lifetimes of RPs may be equivalent to, or even longer than, the paramagnetic relaxation times. The influence of S-T transitions on the recombination probability was first regarded based on the relaxation mechanism [13]. Soon afterwards the part of Ag and ihf mechanisms was elucidated and particularly these mechanisms of S-T transitions were investigated in detail. The theory of the relaxation mechanism remained less investigated. Several attempts at a theoretical analysis of possible occurrences of paramagnetic relaxation in RP recombination were made in refs. [4,14-161. In the present work the recombination probability was calculated for several models of RPs with consideration of the paramagnetic relaxation process. Besides the relaxation transitions, S-T, transitions resulting from the Ag mechanism are considered. The results may also be used for calculating the 0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
114
K. Liiders, KM. Salikhov / Recombination probability
of
radicalpairs
recombination probability of RPs in the presence of the ihf mechanism, if the external magnetic field is high in comparison with the local hf field and the isotropic hf interaction actually mixes only the S and T, states.
2. Models of radical pairs There exist several models of RPs, in which the diffusion escaping process of the .RPs are considered in a different determined by the concrete properties of the physical system by convenience of mathematical treatment. In this work we three models. 2.1. Exponential
one-positional
of the partners, their recombination and the manner. The choice of the actual model is considered. Sometimes this choice is dictated calculated the recombination probability for
model of RPs
This is the simplest phenomenological model of RPs. Within this model the following processes are taken into account: decomposition of the RPs into independent radicals, recombination and S-T transitions. The kinetic equation for the spin density matrix (p) of the RP is [1,17] +/at
= &/3t)sp.&IL-
JG( Fp + PF)/2
- p/r,
0)
where 7 is the mean lifetime of the RPs, K, the recombination rate constant of the RP in the reactive spin state and fi the projection operator into this state, and sp. dyn. is the abbreviation of spin dynamics. Usually the reactive state is the singlet state. On this account $ is always the projection operator into the singlet state below: @ = 1S) (S I. The first term in the right-hand part of (1) describes the change of p caused by the motion of the RP spins. Usually the model under discussion is simply called the exponential model. In our work we call it exponential onepositional model, in order to distinguish it from another model which is considered below. In the exponential one-positional model the radicals of the RPs are permanently in the reaction area. In principle they can always recombine, as soon as there is the corresponding multiplicity. This model poorly reflects the character of molecular motion of radicals in dense agents, especially in liquids. In liquids, during random thermal motion the partners undergo consecutive approaches (re-encounters) [18]. At the moment of contact the wavefunctions of the valence electrons overlap, the radicals are in the reaction area and may recombine. In the reaction area between the radicals exists a considerably strong exchange interaction, the S and T states get out of resonance. Besides this the lifetime of the RP in the reaction area lies in the picosecond range. Therefore the abovementioned mechanisms of S-T transitions are ineffective in the reaction area. In the intervals between re-encounters the radicals are incapable of recombination. But at these moments the S-T mixing in the RP gets effective. The decribed processes are reflected in the following phenomenological model, which was proposed in refs. [19,20]. 2.2. Exponential
two-positional
model of RPs
We suppose that the RPs may be in two states: (1) within the reaction area and (2) outside the reaction area. Within the reaction area the RPs may recombine and leave the area with the rate constant l/~i_ Within the second area the RPs do not recombine but S-T evolution takes place. With the rate constant 1/+r2RPs may enter into the reaction area and with the rate constant l/7 RPs leave the “cage”; they die. The death of RPs is caused by the decomposition of the RPs into independent radicals or by reactions of the radicals with acceptors, or by irreversible isomerization of the radicals.
K. L.iiders, K.M. Salikhov / Recombination probability
of radical pairs
115
We consider the probability density of re-encounters for this model. The probability that the first re-encounter occurs in the time interval (t, t -I-dt) is expressed by f(t) dt. It is equal to the product of two probabilities. The first is the probability that within the time t the RP is not decomposed and does not enter into the reaction area exp( - t/T2) exp( - t/T). The second factor is the probability of contact in the interval dt which is equal to df/r2. The following probability density of re-encounters results: r(t) The
= (I/r2)
exp[ - (I/r2 + I/+].
(2)
probability of a re-encounter is p=imf(t)
(3)
dt=7/(T+T2)
and the total number of re-encounters (see, e.g., ref. [l]) n=l/(l-p)=(7+7&7*.
(4)
The mean time between re-encounters is given by 7c = /omtf(t)
dt(iwf(t)
4.
dt)-1=~zp=vz/(7+
(5)
For the density matrices pi and pz of the RPs in the areas 1 and 2 we may write the following kinetic equations: aPl/at
= -Jq@P,
+ P2)/2
aP2m
= (~P*/~t)s,.dyn.+
- PI/? P*/?
+ P2h
- P2/72 -
P2/7
+ (~Pl/wex.de.ph> (6)
deph. stands for exchange dephasing). At the moment of closest approach the S and T states undergo a fast dephasing caused by exchange interaction. Therefore in the kinetic equation for p1 we may add the additional term
(ex.
(aPl/at)ex.deph. = -ift-‘[
ILhangeY p1] .
The following calculations were carried out with consideration of this supplementary dephasing as well as without it. The given model phenomenologically describes the most important peculiarity of the molecular dynamics of RPs: consecutive contacts. 2.3. The diffusion model of RPs The diffusion model of RPs without consideration of S-T transitions was elaborated by Noyes [18]. In this model the statistics of re-encounters is given by the distribution function f(t)
=mtC312
exp( --am2/p2t),
m = (27/81~)~‘~(1
-p)‘(b/XD)“$“,
(7) (8)
where b denotes the recombination radius, X, is the length of a diffusion step and 7s is the mean time of one elementary diffusion event. The statistics of re-encounters is different for the last two models. It is well known that in the Noyes model the mean lifetime of an RP is not defined because jOmtf(t) dt + CO. This can be explained by the fact that the partners may undergo a re-encounter after a very long diffusion path. In the exponential
116
K. Liiders, K. M. Salikhou / Recombination probability of radical pairs
fH1
116%‘I QZ--L
0
~2
Q3OrC
06081
T, 2
3
456
a
810
30405060 trlams 1
Fig. 1. Distribution function f(t) of re-encounters for the exponential two-positional model (curves 1, 2) and the model of continuous diffusion of RPs (curve 3). Parameters: (1) r= 5 x 10-p s, 72 = 5 x 10-1s s; (2) 7 = 10-S s, *a = 10-s s; (3) D = lo-” m’/s, b = 0.3 nm. The probability of a re-encounter p = ]Ff (t) dt is p = T/(T + TV) for the exponential two-positional model. The time r,, after which the differential probability of a re-encounter for the diffusion model reaches its maximum depends on the length of one elementary diffusion step A,. With decreasing A, the maximum increases and shifts to the left-hand side.
two-positional model the probability of a re-encounter is exponentially distributions (2) and (7) differ strongly also at small times (see fig. 1).
decreased for t > +rC.Moreover
3. The spin dynamics of radical pairs We regard the dynamics of S-T transitions caused by paramagnetic relaxation of the radicals A and B of the pair. The longitudinal and transverse relaxation times of the unpaired electron spins of the radicals A and B are called TIA (T& and TzA (TZB), respectively. The Larmor precession frequencies of the electron spins A and B are w, and or,. First we regard the spin dynamics of single isolated radicals. The kinetic equation for the density matrix, e.g., for the partner A, may be written in the form of Bloch equations +&at
= - ( P:~ - P;~)/zT,,,
aP&/at
= -aPk/k
aP$/at
= -it+P$
- P$/L,
(9)
where (Y, /3 are eigenfunctions of the operator of the projection of the electron spin on the direction of the external magnetic field. Rewriting these equations in the operator form we get
ap*/at = QApA.
00)
The spin dynamics of the two radicals is. described by (aP/%.dyn.=
&A
x
&P,
(11)
where &, X & is the outer product of the operators &A and $,. After transformation into the basis of S and T states we get the following kinetic equations for the spin dynamic part of the density matrix (only those equations are written which contain the populations of S and T states) (aPSS/at)sp.dp.=
-
wT,‘+
~/~T;)Pss
-(~-wd ( ap=,&iat)sp,dP,
= ( - l/457 +h-
+ (-VT:
+ l/2T;)prOr0
+ (PT+T++
P~_&~T;
Im psTo3 + l/W +A
1 PSS - (1/4T;
Im psr,,
+ VW’)
PT~T~ + ( PT+T+ + P,_T_)/~?
K. Liiders, K.M. Salikhov / Recombination probability ( aPT+-&$+,,,
(apr_da~),,,.= ( ap,/at
PT~~,)/~&’
= (PSS +
(pss + pToTo)/4?’
of radical pairs
- PT+T+/%’
- Re PST,/W”,
- PT_~/T’
+ Re P~T,,/W”~
117
)sp.dyn. =~(~,-WB)(PT~T~-PSS)/~-(PT+T+-PT_T_)/~T~'-(~/~T;+~/~T~')P~T~ + (- 1/4T;
+ 1/2T;)P,,,,,,
(apToS/a~)s&,,,.=-i(wA-wB)( + (-1/4T’+
PT,T,-PSS)/2-
(PT+T_-PT_T_)/~T;
~/~T;)PST~ - (V4T’
+ 1/2T;)P~,,s,
02)
where l/T,’ = l/TiA + l/T,,,
l/T;
= l/T,,
+ l/T,,,
l/T;
= l/TIA - l/T,,.
(13)
Only non-adiabatic interaction terms contribute to the spin-lattice relaxation process but the main contribution to the transverse relaxation is often made by adiabatic interactions, the adiabatic diffusion of the Larmor precession frequencies of the spins. Therefore T2 G 2Tr. Equality is obtained only if the relaxation is induced entirely by non-adiabatic interactions (spin flips) [21]. At the instant an RP originates its state is given by the population of S and T states, the off-diagonal elements of the psT type are equal to zero. In the evolution process appears psr # 0. But at the inStaM of contact psr becomes zero due to exchange interaction. In order to estimate the change of populations of S and T states of the RPs we solve the following problem. At the instant of origin we should have pST = 0 but the populations are given by the vector ( pss(0), p T~T,,CO>,PT+T+N + PT_T_WL Sol~%W with the% initial conditions we get
(14) [pT+T+Z!T_(j
=@(t)[pT+T+Z:_T_(o,)*
The spin evolution operator has the form
g(t)=
$+e+c
++e-c
a-e
++e-c
++e+c
4-e
1 2(+-e)
2(+-e)
2(++e)
,
(15)
I
where e= $ exp(-t/T,‘),
c=i
exp(-t/T,‘)
cos(Awt),
Aw=w*-wg.
(16)
For example, if the initial state of the RP is the triplet state (T precursor), subsequently it may be observed in the singlet state with the probability (see also fig. 2) PT+&)=$--
$ exp( - t/T;)
- $ cos( Awt) exp( - r/T,‘)
=&(l-exp(-t/T,‘))+i[l-cos(Awt)exp(-r/T;)]. The oscillations are dumped by the transverse relaxation becomes the well-known result [l] P-r+&)
= f sin’(AW2),
(17) of the spins. If there is no relaxation
(17)
(18)
which describes the oscillations of the RP between the S and T states caused by the difference of Larmor
118
K L.i.Ars, KM. Salikhov / Recombination probability of radical pairs
Fig. 2. Population of the singlet state of RP for a triplet precursor. Parameters: (1) Ag = 10w3, B, = 0.1 T, T;, T; + co; (2) Ag=O, T,‘=2xlO-’ s, T;=lO-’ s; (3) A.g=10-3, B,,=O.lT, T;=2xlO-‘s, T;=lO-‘s.Theinsetgivesthe area near the origin on an enlarged scale.
frequencies of the spins. Of great importance for the efficiency of S-T mixing for short times between re-encounters (see below) is the value of the time derivative @p/at), _ ,, which is non-zero if relaxation is taken into account as seen from eq. ‘(17) and fig. 2. Paramagnetic relaxation entirely mixes all the states and ~r+s(t)
+ l/4
if
t/T,‘, t/T*‘+
00
(19)
in correspondence with the statistical weight of the singlet state. The difference of the Larmor frequencies may be caused by Ag as well as by the isotropic hf interaction. In the latter case Aw depends on the nuclear spin configuration and on the hf coupling constants [l] AU = (g, - g,)/3fi-‘&
+ c aimi - c akmkY i(A) k(B)
(20)
where /I is the Bohr magneton. B, the magnetic flux density. The population of the singlet term for the whole ensemble of RPs is obtained by averaging (17) over all subensembles with different nuclear spin orientations in the radicals A and B of the RPs. The distribution of local hf fields in the case of a sufficiently large number of coupling nuclei may be described by a Gauss function. In this case, after averaging (17), we get PT&)
= + - & exp(-VT,‘)
where aA,eff radical A
aA,eff
and
=
(
f
aB,eff
c
i(A)
Ui’Ii(Ii
are
+
- gn)Ph-%
effective hf coupling constants
the
1)
- i cos(g,
1
i/2 ,
exp[ --t/T;
-
i(&.rr + &,,)t2],
(21)
of the radicals of the pair, e.g., for the
(24
where Ii is the spin of the magnetic nucleus i of the radical A. The transverse (phase) relaxation of the spins as well as the difference of the Larmor frequencies mixes the singlet state only with one of the triplet states, namely with T,,. The longitudinal (spin-lattice) relaxation mixes the S state also with the T, and T_ states. Therefore in the dynamics of S-T transitions in the RPs the longitudinal relaxation can play an important role, even if the Ag and ihf mechanisms warrant an effective S-T,, mixing.
K. Ciders, KM
Salikhov / Recombination probabiliv
of radicalpairs
119
In ref. [Vi] an expression was estimated similar to eq. (17) for the time dependence of singlet state population for a singlet precursor, ps _ s (2). As a proof of the correctness of the calculated populations ps ~ s [15] and p T _ s (17) we can use the cormection 3Pr-s(t)
+Ps+sW
(23)
= 1,
which is fulfilled. The recombination probability was calculated in ref. [15] for the model of continuous diffusion where only the first contact was taken into account. As we will see below for the relaxation mechanism of RF recombination just consecutive contacts play an important role.
4. Paramagnetic relaxation times For free organic radicals in liquids paramagnetic relaxation is caused mainly by random modulation by rotation of the g tensor anisotropy (gta), the anisotropic hf interaction (ahfi) and the spin-rotational interaction (sri) [1,21,22]. In the present work numerical calculations were carried out for illustration of the effect of this relaxation mechanisms in connection with the Ag and ihf mechanisms of S-T,, mixing. If the relaxation is caused by the anisotropy of the g tensor the longitudinal and transverse relaxation rates are l/T, = &-+32A-2B&,/(l l/T, = &&?2ti-2B&[4 where @
+ 3/(1+
(24) Y:@;)],
(25)
characterizes the scale of the gta and is expressed through its principal values
&?=g:+gzz+g32-3g2, ye
+ y,zB,#),
iT=(g,+g,+gd/%
(26)
is the gyromagnetic ratio of the unpaired electron, TV the rotational relaxation time, which is equal to 7, = 4qb3/3kT,
(27)
with b the radius of the radical and 7) the viscosity. With the Stokes-Einstein can be expressed by the diffusion coefficient D r. = 2b2/9D. For organic radicals @= zero to the value (l/T,),
equation D = kT/6Tbv,
7.
(28) 10-2-10-3.
-+ 5 x 10-2L\g2/~o.
From (24) it follows that l/T,
increases with increasing B, from
(29)
In non-viscous liquids, if r. = lo- l1 s, (l/T,),
--, 5 x 10-9bgz
s-l.
(30)
If, e.g., @= 10m2, TI lim = 2 x low8 s. With increasing viscosity this time limit T,,, is prolonged proportional to the viscosity. The transverse relaxation rate increases with increasing B, corresponding to (25). For very high B. formula (25) leaves the area of applicability of time-dependent perturbation theory ((@)“‘ptt-iB o~o< 1). In practice T2 cannot be less than ro. On that account in the numerical calculations we arbitrarily put T, = 7. in cases where the values T2 calculated by (25) became less than ro. Such a situation obtains if
K. Eiakrs, KM. Salikhov / Recombination probabiliv of radical pairs
120
If B, is measured in tesla and 7s in seconds, (31) yields I&, > 10-11,'~o(~)1'2 T.
(32)
In non-viscous liquids condition (32) is fulfilled at magnetic flux densities of dozens of tesla. With increasing viscosity condition (32) is fulfilled at lower values of B,. The relaxation due to the dipole-dipole anisotropic interaction of an electron and a spin I nucleus, in the point-dipole approximation, is described by the relations l/T, = 2w/(1+ l/T,
= W[l
+
$E,27,2),
(1+ Y:%‘~)-~]
(33) 7
(34)
where W= (2/3)1(1+ l)y&?A*r-%a, r is the distance between the unpaired electron and the nucleus, yr is the nuclear gyromagnetic ratio. In our model calculations we have considered only the interaction of the unpaired electron with a single proton. The S-T transition rate is seen from (33) and (34) to decrease with increasing field in this case. The spin-rotational relaxation caused by the random modulation of the rotational angular momentum and the sri tensor is given by [22] l/T,
= l/T,
= (8g; + 26g:)
kT/12nb3v,
where 6g = g - 2.0023 1. With the Stokes-Einstein
(35) equation we get
l/T, = l/T, = v/97,,
(36)
Usually 6g2> hg2, therefore the contribution of the sri relaxation to the where 6gz= 6gi + 2gg:. longitudinal relaxation rate is higher than that due to the gta relaxation. Because the sri relaxation is independent of the magnetic field it will reduce the dependence of RP recombination on the magnetic field. In general, all mechanisms described above contribute to the relaxation rates l/T, and l/T,. In our model calculations we will consider the ahfi relaxation separately from the other mechanisms. If sri relaxation is taken into account in all cases we set v= @. For illustration, in fig. 3 the typical field dependences of the relaxation rates are given.
5. Calculation of the recombination
probability of radical pairs
The recombination probability was calculated by two methods: by immediately solving the kinetic equations (1) and (6) or by addition of the contributions of all consecutive contacts [l]. For the exponential two-positional model with complete dephasing of S and T states at the instant of contact caused by exchange ‘interaction the calculations were performed by both methods. The results were identical as could be expected. The recombination probability depends on the multiplicity of the RP precursor. But there exists a connection between the recombination probabilities of RP for the S-precursor, “p and for the T-precursor, ‘P 1119 “P =sPcl - 3(I -sPO)TP,
(37)
where “p, is the recombination probability of the RP if the S-T mixing were “switched off’. In view of (37) we shall give explicit formulae only for ‘p. In all our calculations we assume that initially the radicals of the pairs are in the reaction area.
121
K. Liiders, KM Salikhov / Recombination probability of radicalpairs
+I*-, IO8
10'
IO6
Fig. 3. Longitudinal and transverse relaxation rates of a radical as a function of B,,. Parameters: D = lo-” m’/s, b = 0.3 nm, ) g-tensor ani1 - @= 10-3, 2 - dg2= 10-2. ( sotropy, (- - -) g-tensor anisotropy and spin-rotational relaxation (the calculations were performed for the case &L&+10-3), (.-.-.) anisotropic hyperfime interaction relaxation (point-dipole approximation; electron-proton distance r = 0.15 mu).
lo5
IO'
5.1. Exponential one-positional model The recombination
probability is
(38)
PI= K, / ~ss(t) dtSolving (1) we get
=pl = K,7[ (K,7/2
+ r/T; + l)( 7*/4T;T;
+ r/6T; + 7/12T;)
+ Aw*r*( 7/T,’ + 2/3)/41/A,, (39)
A, = (K,7/2
+ T/T,’ + l)[ K17( 7*/4T;T,’ + 3r/4T,’ + 7/2T; + 1) + (1 + r/T,‘)(l
+ 7/T;)],
+A~*r~[K~7/2+1+(ZQ/4+1)7/T~],
Sp()= &T/(1
+ K*r).
5.2. Exponential two-positional model For this model the recombination =pz = T$%ss(t)
dt-
probability is (4W
It was calculated using two approximations. In one case it was assumed that at the instant of contact the dephasing of S and T states is caused only by the reaction and drifting apart after contact. In the other case we assumed that at the instant of contact the exchange interaction causes a strong dephasing, i.e.
K. Liiders, K.M. Salikhov / Recombination probability
122
of radical pairs
z+ Ii. This is equivalent to plsr = 0. Corresponding with these two approximations following results: - Without consideration of exchange dephasing of S and T states:
+L&Ulge
‘pz
=
(K7,/A,)
{ ( T~/~T,‘T; + 7/6T; + 7/12T,‘) [ (K7J2
+ Ao~~~( K7t/2 + l)( T/T; A,=
[(I&J2
+ l)(l
+7/T,‘)
x [ ( KT~ + l)( 7/T;
+ l)(l
7/T;)
+
we get the
+ K717/272] r/r2
+ 2/3) ~/47~ } ,
(41)
+Ky~/27~]
+ l)( 7/T,’ + 1) + KT,( T~/~T;T;
+A~~7~(K7~/2+1){K7~(7/27~+1)+1+
+ 3~/4T;
+ r/2T;
+ 1)7/~~]
[K71(7/472+1)+1]r/T;}.
- With complete dephasing of S and T states during contact due to strong exchange interaction instant of contact: ‘p2 = (K7,/A2)[(r2/4T;T; +Aw27’(7/T; A,=
+ 7/6T,‘+
7/12T;)(l+~/T;
at the
+ ~/7~)7/7~
+ 2/3)~/4~~],
(42)
(1 +~/T;+‘T/~~)[(K~~+~)(~/T;+~)(~/T;+~) +I$(
T~/~T,‘T;
+ 37/4T;
+ r/2T;
+A~~7’{&~(7/27~+1)+1+
+ 1)~/~~2]
[K71(7/472+l)+l]7/T;}.
With complete exchange dephasing the recombination probability was calculated solving (6) and also by use of the method of adding the contributions of all re-encounters. Using the latter method the recombination probability is given by [l] p = Tr[ s(l
-@d)-’
p(O)],
(43.)
where # = /cm$(t) f(t)
d2 is the operator of spin state evolution, averaged over the RP lifetime distribution between re-encounters, ~(0) = (PSSVO, PT,,T,(O), PT+T+(O) + PT_ T_(O)) is the vector of the initial state of the RP but the operators P and & in the basis of S and T states are given by the matrices (see ref. [l]) P=(i
i
i),
e=(lih
;
Ej,
where A = K7J(l + KT~) is the recombination probability of the S-state RP during a single contact. With the aid of these relations and using (15), and (16) we get ‘p = K7, [ pn (1 - IF) + 2pn (1 - z) + 3pszy1
- Z)(l - z)] /3( K7,A + B)
(44)
where A = 4 + 2pn(l-
c”) + 3pn(l
B = 4n2(1 -pZ)(l F= f /,“exp( p=
,z /
t ()
+p2n2(1
dt,
E= $imcos(Aot)
- E),
-PC?),
-t/T,‘) dt,
- ;)(l
-Z)
f(t)
n=l/(l-p),
rr = 71”.
exp( -t/T;)
f(t)
dt, (45)
K. Liiah,
KM
We note that without consideration spa = Kqn/(l
n -
of S-T
123
of radical pairs
transitions the recombination
probability becomes
+ Kqn).
In the limit of full S-T case from (44) =p = KT,(
Salikhov / Recombination probability
(46)
mixing between two re-encounters
(Z, ? --* 0) we get an interesting result. In this
l)/[ K7,(3 + n) + 4n].
(47)
With increasing reactivity of the radicals ‘p-qn-1)/(3+n).
(48)
Only if n =*- 1, ‘p + 1 ! If S is mixed only with the T,, state (corresponding
to Z + 1, Z + 0 in (44)) we get
‘p-0Cr&2-1)/3[K+2+1)+2n], ‘p --) (n -
1)/3(n
+ 1)
(49) if
KT, z+ 1.
(50)
The above result (44) is valid for any statis$cs of re-encounters, which is defined by the type of motion of the reagents in the “cage”. Averaging F(t) using (2) we get the recombination probability for the exponential two-positional model =p2 =
K*,n ( T/T;
+ 27/T;
+ 3r2/T,‘T:)
rc/r23( KqnA2
+ B2),
(51)
where A, = 4(1+
q/T;)(l
B, = 4(1+
7/T;)(1+7/T:),
l/TC = l/7 + 1/T2,
+ q/T?)
+ 3(1+
7C/r,E)?.7C/72T; + 2(1 + ~,/T;)T~&T~~
l/7’ = l/7, + l/T;,
By algebraic transformations
l/Tz” = l/T;
+ T~T~/$T;T~~,
+ hti2~‘.
it can be shown that (51) and (42) are identical.
5.3. Model of continuous diffusion For the model of continuous diffusion we perform fully analogous calculations, only (2) is replaced by the distribution function (7), (8). The averaged spin state evolution operator is defined by f?= exp( - /mm/p), E=exp(
-/mm/p)
cos(+~(l/T~~-z/T;)
m/p),
(52)
where 1/T2D = (l/T; + d-)/2. Putting (52) into (44) and going to the Iimit of continuous diffusion (X ,,, 7s + 0; x’,/rs + 60, where D is the relative diffusion coefficient) we get =pD = Kq( /z/3
+ 2/m/3
-t /m)
~[K~+(4+2/~+3\/7,/T,‘+/~)+4(1+/~)(1+/~)]-~. where 7r is the total time the RP stays in the reaction area, 7r = qn, 7u = b2/D the RP.
(53) is the effective lifetime of
K. L.iiders,K. M. Salikhov / Recombination probability of radical pairs
124
The RP recombination probability for the model of continuous diffusion was also calculated in ref. [16] solving the stochastic Liouville equation. In the treatment of ref. [16] the exchange interaction was not taken into account. As mentioned above the exchange interaction causes a strong dephasing of S and T states during contact. This may be considered in the corresponding boundary condition for the off-diagonal matrix elements psr. Without consideration of exchange interaction there remains a dephasing between S and T, states resulting from the RP recombination in the S state. The dephasing rate constant is K’ = K/2, where K is a recombination rate constant. The exchange interaction causes a supplementary strong S-T dephasing, therefore one expects K’r z+-K7/2, in the limit K’r --* do. The kinetic equations for PST, ad PST_ are not coupled with the system of kinetic equations for the S- and T-state populations. Therefore it is sufficient in the expressions for the recombination probability in ref. [16] to change the rate constant K’ for the off-diagonal matrix elements psr, K’r + co in order to consider the effect of strong exchange dephasing on the recombination probability. By algebraic transformations it may be shown that the result corresponds to eq. (53) which we have calculated by the method of consecutive contacts.
6. Discussion of results 6.1. Comparison of the exponential
models
First we consider the situation if the S-T transitions are caused only by spin relaxation. In this case the kinetic equations for the S- and T-state populations of the RPs are not coupled with the off-diagonal elements psr of the density matrix. Therefore eqs. (41) and (42) (which differ only in the magnitude of dephasing of S and T states at the instant of contact) are equivalent if only the relaxation mechanism of S-T transitions works and give =pz = Kr,( r2/47’;Z”
+ r/6T;
xr/r2 [ (1 + K7,)(1+
+ 7/12T;)
7/T/)(1
+ 7/T;)
+ Kq(
T~/~T,‘T,’ + 37/4T;
+ 7/2T;
+ 1) T/T,]. (54)
In the case of the triplet precursor the RPs are unable to recombine at the instant of contact, contributions to the reaction product are made only by the subsequent re-encounters. Therefore (54) contains the factor r/r2 = n - 1, the number of re-encounters without consideration of the first contact. The quantity 7i7/r2 denotes the total time during which in principle the RPs may recombine. For the pure relaxation mechanism of S-T transitions a one-to-one relation can be established between the recombination probabilities calculated by the exponential one- and two-positional models. To this end the phenomenological parameters K and K, must be connected with the equation KI = K~/T~ (1 + K-r,),
(55)
which has an obvious interpretation: In the two-positional model l/r2 is the rate constant of RPs entering the reaction area and Kri/(l + Kq) is the recombination probability during a single contact. If the reactivity of the radicals increases, the parameter K in the two-positional model also increases while in the one-positional model K, at first increases and after that goes to the limit K, + l/r*. This means that using the one-positional model the phenomenological parameter K, must by no means be increased to such a degree as the reactivity of the radicals is increased. If S-T transitions are induced not only by relaxation but also by the difference of Larmor frequencies it is impossible to accurately relate the result of the two-positional model to the result of the one-positional model redefining the parameters corresponding to (55). Nevertheless the numerical calculations show that even if Aw # 0 the one-positional model (39) and the two-positional models (41), (42) give similar results if the parameters of the models obey (55) (see fig. 4).
K. Liiders, KM. Salikhov / Recombination probabiliy of radicalpairs
125
Fig. 4. Dependence of recombination probability on magnetic flux density for the exponential models: curves 1 and 2 pertain
to the onepositional model, 3 and 4 to the two-positional one. Curve 4 was calculated considering the spin dephasing at the instant of contact caused by exchange interaction. In all cases was assumed K = 10” s-l. The curves 1 and 2 were calculated from formula (39), but the parameter KI in the case of the curve 1 was chosen to be KI = 10” s-l (the renormalization formula (55) was not used). For curve 2 the renormahzed value of the recombination rate constant was chosen. The remaining parameters are: q = lOWi2 s, r2 = 10m9 s, r = lo-’ s, Ag = 2 x10e3, @+a=lO-‘, D=lO-lo m2/s, b=0.3 nm. From the figure it follows, that the results from the one-positional model using the renormalization (55) are in good agreement with the results from the more consistent two-positional model. The consideration of a suplementary dephasing caused by exchange interaction somewhat decreases the recombination probability as may be expected.
It is interesting to note that corresponding to (39) in the one-positional model the efficiency of S-T mixing caused by Au-at large Kl decreases with increasing Kl. Actually, if K1r + do and r/T;, r/T; -+ 0 we get =pl =
Aw2r/3K, + 0.
(56)
This decrease of S-T mixing can be explained by the “smearing out” of the singlet term caused by very fast recombination of the RP. This peculiarity of the one-positional model has been mentioned in the literature [19,20] and is a serious insufficiency of this model. In the models with m-encounters under consideration, the efficiency of S-T mixing does not decrease, the recombination probability monotonically increases and approaches a limit with increasing K. Assuming T,, T, + cc, 7’ = +rcand Kqn --, 00, from (51) we get a limit for the recombination probability different from (56) =pz +
Aw2rC2 ( n - 1)/3 [2 + Aw=r,=( n + l)] ,
(57)
which does not depend on K. 6.2. Effective rates of S-T, transitions From (51) and (53) follows a tremendously important conclusion. In the models with re-encounters under consideration the S-T,, mixing can be characterized by some effective rates 1/T2E, 1/T2D, if the relaxation, Ag and ihf mechanisms work simultaneously. This effective parameter for the exponential two-positional model is 1/T2E = l/T; where l/7
+ Ao=r’,
(58)
= l/7 + l/7= + l/T=‘, and, for the model of continuous diffusion,
1/T2D = (l/T;
+ /+))/2.
(59)
From these equations it follows that in the general case the dephasing of S and T, states caused by Aw and by transverse spin relaxation is not additive. Only in situations where T; is sufficiently large, T; > r, r2; TiAo B 1, the two parts of S-T, mixing can be added. In this case we get from (58) and (59), respectively
l/GE = l/T; + Aw=r,,
(60)
126
K. L.iiders,KM. Salikhov / Recombination probability of radical pairs
Fig. 5. Phase relaxation rate l/T; (curves 1) and effective phase relaxation rates l/TaD (curves 2), l/TrE (curves 3, 4) as a function of Ba. With increasing time between reencounters the efficiency of the Ag mechanism increases in comparison with the relaxation mechanism of S-T,, mixing. In this sense the long diffusional trajectories in the diffusional model (curves 2) play an important role. For short times between re-encounters (curves 3) the relaxation rate l/TaE differs only slightly form the relaxation rate l/T;. Parameters: (3) r = lo-* s, 72 = 10-s s; (4) r = lo-’ s, ra = 10-s s. The other parameters are the same as in fig. 4. ( ) Ag mechanism, g-tensor anisotropy relaxation; (- - -) Ag mechanism, g-tensor anisotropy and spin-rotational relaxation; (. -. -. ) anisotropic hyperfine interaction relaxation, ihf mechanism (r = 0.15 mn, a=lmT).
and l/T,D
= (l/T;
(61)
+ Aw)/2.
Obviously, in this limiting case the two mechanisms are additive. Even in the regarded situation this is an astonishing result because the character of spin dynamics caused by the difference of Larmor frequencies differs greatly from that caused by relaxation. What the two mechanisms have in common is only that they cause a dephasing of S and T, states. For illustration the dependence of the relaxation rate l/T; and the effective relaxation rates l/TzE and l/TzD on the magnetic flux density is shown in fig. 5. It is very interesting that for typical values of the parameters of the radicals the relaxation mechanism may well compete with and even predominate over the Ag mechanism of S-T,, mixing. 6.3. The recombination probability
of radical pairs
In ref. [4] calculations were made of the recombination probability of RF’ considering ihf, Ag and relaxation mechanisms. But in ref. [4] it was assumed that these mechanisms make additive contributions. We separate the contribution of S-T+ and S-T_ transitions to the recombination probability. We assume that the transverse relaxation, Ag and ihf mechanisms mix the S and To states extremely effectively, that is, r/TzE, rD/Tp B 1. Then we get from (51) and (53), respectively, =p2 = Kq( n - 1)(2/3
x [4n(l+ TpD = h(2/3
7/T,‘)
+ T/T;) + &,(4(1+7,/T,‘)
+ &777)/[
h(2
+ &Z)
+ 3rr,/~~T; + 4(1+
+ 2(1+ /X)].
T,/T;)T/T~
+ T~T,/~~T;)] -l,
(62) (63)
K. L.&&q K. M. Solikhov / Recombination probability of radical pairs
From this we can find the maximum contribution which can be made by the spin-lattice (S-T+, S-T_ transitions) to the recombination probability of RPs. For this we should find A=p==p(l/T,‘--)
127
relaxation
co) -=p(l/~,‘-+o),
where Kr, + 00. From (62) and (63) we get for the exponential two-positional
model
ATpz--, 2n(n - 1)/3(n* + 4n + 3)
(64)
and for the model of continuous diffusion
(65)
=po-+2/3. Usually radical recombination relation
in liquids is a process which is controlled by diffusion. This means that the
KT,>~
(66)
should be fulfilled. Under this condition, from (51) and (53) we get, assuming Kr, --, cc, ‘p2 + (q/T;
+ ~T,/T*~ + 3mc/T,‘T2E)(
n - 1)/3A,,
Above we have mentioned that the Ag and ihf mechanisms and the transverse relaxation contribute non-additively to the recombination probability of RPs. It is very important to note that also the transverse and the longitudinal spin relaxation appear non-additively in recombination probability. For example, corresponding to (67), with increasing relaxation rate ‘pD -+ l/9 if l/Tp ---,0 but l/T,’ + co; in the opposite situation l/T; --, 0, l/T2D --) cc we have ‘pD + l/3. On the other hand, if the transverse and the longitudinal relaxation are simultaneously considered if l/T,‘, l/Tp --, cc we get ‘pD + 1. A comparison of the results from the exponential two-positional model and the model of continuous diffusion (see, e.g., (67)) shows that the molecular dynamics of the radicals essentially influences the behaviour of recombination probability. Very important is the kinematics of the motion of the radicals in the intervals between m-encounters. For the two models considered, which differ from each other only in the statistics of radical reencounters we get very different dependences of recombination probability on paramagnetic relaxation rates, the g-value difference and lifetime of the RPs. For example, corresponding to (67) in the range of slow S-T transitions, if r/T:, r/T/ -K 1, we get for the exponential two-positional model =p2 = (7,/T;
+ 2rJqE)(n
- 1)/12
(68)
and for the model of continuous diffusion
The possible limiting values of the recombination probability of the RP also depend on the kinematics of the radicals of the pair. The maximum possible values of recombination probability, if only S-T,-, mixing works, are given by (50). If there is an effective mixing of the S state with all the triplet states eq. (48) is obtained. From these equations it can be seen that the maximum possible values of recombination probability reach the expected values l/3 and 1, respectively, only in cases where the number of re-encounters increases without limitation. Just such a situation is realized in the model of continuous diffusion.
128
K. Liiders, K. M. Salikhov / Recombination probabiliv
of radical pairs
6.4. Long-lived radical pairs We discuss the field dependence of the recombination probability of RPs in the situation where S-T transitions are caused by the Ag mechanism and by paramagnetic relaxation which is caused by g-tensor anisotropy. The parameters l/T; and l/TzD, l/TzE which characterize the S-T transition rate increase with increasing magnetic flux density BO. Simultaneously the recombination probability of RPs also increases. But the experimentally observable field dependence of ‘p may not simply reflect the field dependence of l/T,’ and l/T, E, l/TzD. Differences are especially great for long-lived RPs. Really, for long-lived RPs (large r, r2, To in comparison with T; and T2E, TzD) ‘p approaches the limiting value (47) and ceases to depend on B,, although l/T,’ and l/T, E, l/TzD may further increase with increasing magnitude of the B, field. Corresponding to (24) l/T; reaches half its limiting value (29) at 1’2B0,
which means that in the range Be = ‘/2B0 the maximum change of l/T{ takes place. Therefore one could think that for the relaxation mechanism of S-T transitions the characteristic range of maximum changes of =p should also lie in the range B, = 1/280. However, for long-lived RPs it is not necessarily so. In this case the transition range Tp(BO) should lie in fields B,, < 1’2B0. Such a situation may occur, for excample, in micelles, in which the lifetime of RPs may reach several lo-’ s [2,5]. Therefore customary values T,‘, T2E, Tp = 10e6 s will mix the S and T states of RPs very effectively and further shortening of the relaxation times with increasing Be is irrelevant to ‘p: there appears a peculiar saturation. For long-lived RPs one should expect the following behaviour ‘p( B,): with increasing lifetime of the RP the value of B, decreases for which =I, approaches the limiting value (47). If the relaxation process is caused mainly by the ahfi we expect another behaviour of ‘p(B,). As seen from (33), (34) and fig. 3 both relaxation rates l/T; and l/T; decrease with increasing B0 field, for l/T; eq. (70) is valid. If the ihf mechanism of S-T, mixing works simultaneously 1/T2E and 1/T2D only slowly depend on the & field. Relative small coupling constants l-2 mT warrant an effective S-T, mixing in long-living RPs. If we follow the abovementioned idea the characteristic range of maximum changes of ‘p should also lie in the range B,, = 1/2B0. Due to the opposite dependence l/T,( B,) for ahfi relaxation in comparison with the gta relaxation we expect the opposite behaviour of ‘p( B,): with increasing lifetime of the RP that value of B, increases in which the maximum change of ‘p (B,) takes place. In general, beside the Ag and ihf mechanisms of S-T, mixing all the three abovementioned relaxation mechanisms work simultaneously. In real cases some of these mechanisms can preponderate. If, e.g., the recombination probability monotonically increases with increasing B, field as measured in refs. [8,23] the Ag and ihf mechanisms together with gta relaxation may be supposed. In the opposite case, if the recombination probability decreases with increasing Be field the main contribution to the magnetic field dependence of RP recombination probability results from the ihf mechanism and the ahfi relaxation [4,14,24]. 6.5. Results of numerical calculations For an illustration of the above expressions and qualitative discussion we performed several numerical calculations. The results are given in figs. 6-9. These curves corroborate the abovementioned basic regularities of the change of ‘p dependent on the molecular kinetic and magnetic resonance parameters of RP. We have made a detailed study of the recombination probability considering the relaxation and Ag mechanisms of S-T transitions. In order to cover systems in which the isotropic hyperfine interaction also
K. Liiders, KM.
ci.1
E!JTI
d5
Salikhov / Recombination probability
129
ojradicalpairs
i
Fig. 6. Recombination probability of triplet-born RPs; exponential two-positional model with consideration of dephasing during contact caused by exchange interaction. The contributions of the Ag and relaxation mechanisms of S-T mixing are compared in the regarded case of relatively short times between consecutive re-encounters. The dashed curves contain a contribution from spin-rotational interaction (@v). Parameters: (1) Ag- 2 X 10e3, a= a= 0; (2) Ag = 0, m+a=10-3; (3) Ag=2x10-3, a+a=10-3. The other parameters are the same as in fig. 4.
Fig. 7. Influence of the kinematics of radical motion on the character of field dependence Tp. Parameters: Ag = 0.02, a + a= 0.01, (1) model of continuous diffusion, (2) exponential two-positional model. The other parameters are the same as in fig. 4, symbols see fig. 5 (a = 2 mT).
a4 a3
.-.-._.
a2 0.1 .-.-.-.
Fig. 8. Influence of RP lifetime on field dependence ‘pz. Parameters:(1)7=10-*s,(2)7=10-7s,(3)7=10-6s.The other parameters are the same as in fig. 4. Curve 4 shows the field dependence of the effective spin dephasing rate l/7”. For the actual values r and 7s the l/Z’,” curves differ insignificantly from each other. The dashed curves contain a contribution from spin-rotational interaction (Q= v).
Fig. 9. Field dependence ‘pz of RPs with large anisotropy of the g tensor, large difference of the isotropic g values and long lifetimes in comparison with the corresponding curves resulting from ihf mechanism and anisotropic hyperfine interaction relaxation. Parameters: (1) 7 = lo-’ s, (2) 7 = lo-’ s, (3) r= 10-6 s; Ag = 0.02, @ + e= 0.01. The other parameters are the same as in fig. 4, symbols see fig. 5 (a = 2
130
K. L.iArs,
K.M. Salikhov / Recombination probabiliv
contributes to the S-T transitions ‘p(Aw) has been calculated using the abovegiven formulae. The total probability is
of
radicalpairs
for each configuration
of nuclear spins
(71) where N = & (21, + 1) is the total number of nuclear configurations of RP, Au(m) for different nuclear spin configurations is given by eq. (20). In our model calculations we have considered only coupling with one proton in the RP. It is necessary, of course, to remember that (71) is valid only for high magnetic fields (in comparison with the isotropic hyperfine interaction), where the non-secular terms of the isotropic hyperfine interaction can be neglected. For organic free radicals this is fulfilled in fields B, 2 0.1 T. In fig. 6 the recombination probability is shown as calculated by the exponential two-positional model. The Ag and relaxation mechanisms are considered separately as well as coupled. In the case under consideration the lifetime and the time between m-encounters are relatively small and the S-T mixing is caused mainly by phase relaxation (see figs. 2 and 3). On that account the consideration of spin-rotational relaxation only slightly changes ‘p(B,,). It is seen that the contributions of the Ag and relaxation mechanisms are not additive. In fig. 7 it is shown that ‘pD varies less than ‘pz with changing magnetic flux density. This is connected with the influence of the kinematics of radical motion, the statistics of re-encounters. As expected, the area in which the value of ‘p(B,-,) changes strongly is shifted to smaller values of B, (in the case of Ag mechanism and gta relaxation, we discuss now) as the lifetime of RP increases (see fig. 8). In the low-field region ‘pz depends predominantly on r and. the spin-rotational relaxation times T,, T,. From (36) and (68) we get =pz =
7/4T,.
(72)
In the given examples in most cases ‘p does not exceed l/3. This is connected with the fact that for the values of parameters chosen the spin-lattice relaxation influences the shape of the curves only slightly. It becomes apparent at large lifetimes of RP and large anisotropies, e.g., @= 0.01 (see fig. 9).
Acknowledgement The authors wish to thank S.A. Mikhailov and S.N. Smirnov for assistance in the accomplishment numerical calculations.
of
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