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Theory of magnetic effects in radical reactions at zero field
Chemical
Physics
North-Holland
82 (1983)
Publishing
THEORY
145-162 Company
OF MAGNETIC
EFFECTS
IN RADlCAL
REACTIONS
AT ZERO
FIELD
1. Introduction
The hyperfine interaction (hfi) of unpaired electrons with magnetic nuclei induces singlet-triplet (S-T) transitions in radical pairs (RPs) thereby affectin, 0 their recombination probability [l-4]. This influence of nuclei upon radical reactions can be manifested in various experiments. For instance_ the hfi \vith magnetic nuclei can affect the quantum yield of molecular photodecomposition which fact must be taken into account when estimating the cage effect in liquid-phase reactions. Magnetic isotopic effects are an important manifestation of the hfi. These effects consist in a change of the RP recombination probability due to the substitution of either magnetic isotopes for non-magnetic ones (e.g.. “C for “C). or one magnetic isotope for another (e.g,, D for H). Theory of RP recombination has been developed in a number of publications [l- 141. The conclusions of this theory. as applied to the zero-field case. can be outlined as follows. (Note th;lt the small e;1rth‘s magnetic field cannot show up within a RP lifetime and thus can be assumed zero. Therefore. the results obtained for zero field can be applied to the earth’s field.) One of the most important results concerns short-lived RP recombination. For such a pair. irrespective of the type and the number of magnetic nuck. the singlet-triplet mixing efficiency induced by the radical hfi is determined by a single parameter: thr effective constant a,,,. given by
1 I/’
fCu;z,(z,
JGrr =
[
+ 1)
k
.
(1)
where U~. is the hfi constant. I, the nuclear spin [5-71. In the case of the simplest. one-nuclear. RP \vith Z = l/2, analytic expressions for the RP recombination probability were obtained in terms of the exponential [S] and diffusion [9] models. For more complex systems (a one-nuclear RP with arbitrary spin. a RP \vith several mugnctically non-equivalent nuclei) exact kinetic equations for the RP density matrix and the RP recombination probability were calculated only numerically (see. e.g. refs. [ 10-131). As the number of magnetic nuclei increases, precise calculations prove to be practically unrealizable even with a computer. Therefore. it was proposed to investigate systems with a great number of magnetic nuclei in terms of a semiclassical
0301-0104/83/0000-0000/$03.00
8 1983 North-Holland
description
fO\\r-fdil
of tl\r: RI’
spin rlyn;\niics
[ 141.In tlris c:\sc. the recolrrhinatii~i~
w:\s prcsc\~lcd :\s
prcrh:\bilily
i\IWgri\lS [ IS).
NOIL’. finally. ;\ppr\>xi\ll;\tc: Csti\lli\ICs OT rl\r: RI’ r~ccl\\~lli\~;\tit~\~ prob;\hili\y \:\ki\lp i\r\o i\~c~~\\\lI 111~ I\lLi\iil\\ccd S-l‘ mixing. The p\\l~licolions on lhis poinl c;\n he tlividcd into IWO groups. 0nc CI~~I~IXCCS prccisr: Ci\lClllillillllS Of IllC RI spin dy\lilllliCS. will1 IIlL’ lll\dCC\llN dyixrmics of ri\JiCi\l pairs considcrcd in the framework of i\ simplified IMXI~I (c.g.. tl\c K:\pteiu-C’llrss-~l~sterl\~~r~ nwrlel. or 111~Knptoin-r~drinii 11111dCl [ 16.171. Lee illSO rcfs. [l-4)). l3xi\lllplCS elf Sllcll CS~illlillCS ClI\l h! fc~lllld in r&s. (6.18-X]. Thr‘ olhw group (see. c.g. rcfs. [21-Z]) dcscrilws Ill12 llll~lcX\llilrll~lli\llliCS in tcrnu 0r :\ fiiirly Well dcvcli~pcclnioild. howcvcr. tilr: spin J~l~illlliCSis CillC\\ti\~Cd \\ndcr \\ns\\l~sta\~li\\leii i\SS\llllplil~\~S.For CSillll~~lC~ ir is ilSS\llllCll Illill in lligll lklils Ihe S-l‘ \nixi\lg cfficicncy \ll\\slIW oiic-llrird Of llli\l in mu lklti. This ~\ss\\mption holds only for shorl-lived RI’s (this prohleni IlilS hail discussed in rrfs. [1.6.24]). The prcsrnt pilpcr rcporls priiclicol formiiliis for lhc rccoiiilriii;\lioii prohilhilil~ (\l\lcllilrg~d) radiciils ill zero IlXlgllClic field.
The RP recombination
is considered
~Sec\\ltf i\ ra\ldo\ll llllXllli\l niolion.
l
life~iillr
I.iistribulioIl
instance. this distribution
/(I)
=
JJIf-3/2
hc~wwil
rtxIlco\IIlws
has
found
been
where p is the re-encounter
;\I
11wdtA (SW
rds.
nci\tr;\I
Il.241). The RI’ partners rccomhinc. I’llC hy lllr: function j(I). For
llic rtZ\clill\l
rildilIS
is ilclliWd
rndia\ls: (2)
probability,
111is rxprrssed
vii\ molecular-kinetic
p;\r;\meters,
1 -~)$7/X,,)‘7~/‘.
exp( -7rn,nz/pzr
d
10 ClVllCinto CO\llilCl illld tIllIS Cilll
by Noyes 1251for neutrnl
(3)
b is the reaction radius. X,. T/. are the mean the presence of radical acceptors [ 11, /(f)=JJlI-“’
prow
pilirS
-Tn,J12//J21).
L’Sp(
)I?= (27/8n)“3
in terms of ~hc following
At limes lllrg
0r
length
and the me;\n time of
a11
clemcntnry
diffusion
- K,f ).
step. In
(4)
where EC, is the rate constant for decay of a’RP by reaction with the acceptors. If at the initial moment the radicals of the pair are not in a direct contact, 011e needs also the RP time distribution function/,,(r) before the first encounter. For instance, in the continuous diffusion model [LX]: f,,( 1) = r,1,r-3”
exp( -vimrn~/ptf
- K,r ).
p. = h/r,,.
I?Z[,= b( r, - h)/2r,,(
7rDy.
where r0 is the initial interradical distance, p. is the first-contact probability. At an encounter. singlet RPs recombine with a constant KS, and triplet RPs with a constan\ exchange interaction is assumed to be negligible between re-encounters. and sufficiently encounters to completely dephase the S and T states. Within this model the total contribution contacts to the recombination probability is [ 1.41: p
=
i’(
TT
1 - FQ)-\&p(O).
(5) K-,-. The strong at from all
(6)
where pp is the recombination proiability at a single encounter. 0 describes the change in the density matrix induced:by one encounter, F is the spin evolution operator averaged over time intervals between re-encounters, F, is the average spin evolution operator before the first encounter. The forms of 8’ and 0 are known (see refs. [ 1.41). Therefore, to calculate p one must only find the S-T transition dynamics.
where A, = KS7/( 1 + KS7 ) and X r = K:-l-T/( I + K-,-T ) ;irE: the raxm~hination proh:ihililirs of singIt aid triplet RPs at an encounter. T is the Illtill duration of an r‘ncounwr. Averaging i-‘c f ) hy j-t I ) a11d I;,( I ) and the rtxon~binntion prob:tbilitv. Prior 1~1 substituting the result obtained into eq. (6). One deterniines Lrt ‘p(J’) be thr‘ RP rcxombination adducing the final results. we introduce so~lle useful notation. the notation in par~nthrtsrs d~nows the probability. The pre-superscript shows the initial RP multiplicity. multiplicity of that RP state of which the rccon~binntion product is undtx study. Fm instance. ‘p(S)
denotes the RP recombination probability when the initial state is a triplet and the recombination of singlet pairs is considered. With eqs. (11). (12) and (6) we obtain +K+)(l
sp(S)=p,KsTn[(l
-‘s;,/J?o)+~i;rl]/A.
-‘> (S) = p. Ksm [ ( 1 + K,m
=pOKTm[(l
“p(T)
(14)
G]/A.
+ K,-TII)+~~,,K,-TJz[(~
‘~p(T)=
(13)
) ,i’o/Po + ii%] /3?1,
+ K,n’)i?;,/P,+
product
(15) + K?;TJz)(~ - ~T,,/p,,)+~i;rr]/A.
(16)
where A = (1 -I- K,Tu)(
,T;=
I + K,.T~I)
=w( r)f( r)dr. /0
+ iFn(2 + KSrn + Kl.~n).
Is;,,=
-I(.( I)_&( r)dr. I0
(17)
and II = l/(1 - p) is the overall number of encauncers of a RP. p,, is the probability of the first encounter at the recombination radius. TIJ the total residence time of a RP in the reaction zone. RelaCons 1\3j-{ 13) xe u&d TOT xrg statistics of re-encounters. Unfortun;lte\s. the re-encountrr stat isk5 +CI~&5x1 sQ&& in *xX W% Cm Y&WA{ ix&i~ S ‘[25]. FGY FLY& ~fiXZ555~C+K CkSfCSISC~I+YL~~~~~ between RP n-encounters and before the first contact are defined by eqs. (2)-(S). They can be used to calculaIe G% and &&. Subsdtufing eqs. (4) and (1 I j inio eq.
[pc/2(1
+c)][esp(-g,)-cos(g,).exp(-gJ)J.
g, = pK,‘/‘. p = Z,?,m’/2/p.
(18)
~~=I.LI[-K~+(K~+I~~)“~],Z~“~_ u2=a’[I(Z+
1)+
.5~=PC[K,+(KZ+211)‘/2],2)“2_
l/4].
The overall number of encounters is [l -il/(r)dr]-‘=
II=
[1 -esp(-g,)]-‘-
(1%
With eqs. (18) and (19) one can calculate the RP recombination the limit of A,, TV --, 0 (A’D/6~L - D) we obtain the following (tramktiton-ro -2ri~‘imirc’ts ti~scusseh. es..‘m t-&_:c?~~:
Ttl
-
Tr =
‘G/P, =
Orb/+ + (&To)"']> [c/31 +c)l[exp(-g,,)--os(g,,)
St0= (&,#‘,
probability for various A,, and T/.values. In results for the continuous diffusion model
exp(-th,)].
g,, = ([-K,+(K,‘+u’)1’2]T~,,2)“2,
&,,=([K;i+(K~+U')"2]To,2)"2. Here 7r is the total residence lime of a RP in the reaction zone. a, and b are the width and the radius of the reacTi‘JI1-5sxk. itCiijX32&&~,7 D-k'~+fth~-ztzz UT WViht, TD
=
b’/D,
(23)
and q, is the mean time before the first encounter. r,, = (r,, - b)‘/D.
(14)
Substituting eqs. (20)-(24) into eqs. (13)-( 16). we calculate the RP recombination prohahilitv. are especially simple in the absence of radical acceptors (K, = 0) and if the radicals recombination radius at the initial moment (r,, = h. p,, = 1. I?,,= 0). In this c;1se_ Sp(S)
= KgTr( 1 + K+,
+ q)/A.
‘p(T)
= K-p&A.
rp(T)
The results arc at the
‘p (S) = K,T$//3A. = 2K+J3(
1 + K,.T~) t- A-,-7,( 1 + Ksrr + y)/U_
(25)
il=(l+KsTr)(l+K,.T,)+~(~+iisT,+h’,-T,). q = { I(
1 +
I)/[
41( 1 +
1 +
~)]3’4)(~t~~Tr>)“‘-
At the moment of a RP generation the partners can be out of contact. For instance. ;I RP c:~n result from the decomposition of a molecule into three fragments: two radicals and a diamagnsric molecule. in this case. e.g.. in the absence of radical acceptors: ‘T;,,/P,,=q0={21(1+ & = { [ 1 + 41( 1 + Eq. (14) affords,
i)/[i
+41(1+
ij])[i
-COS(,Q)
r~p(-g,,)]_
l)]“‘lU17,}‘i2/2.
e.g.. for a T precursor of a RP:
TP(S) = (b/r,,)&&?+
4o(t +
(‘7)
K~Tr)]/3A-
The S-T mixing before the first encounter (parameter q,,) oscillat:s dcprndin, distance_ If q, is sufficiently large and ]a] To > 1. then. according to eq. (26). qo=21(1+
(26)
1)/[1+41(1+
l)]_
0 on the initial
intcrpartncr (2s)
The S-T transition oscillations can also be manifested in the recombination probability. They are disguised and not observed experimentally if there is a distribution of RPs over initial inrsrparrncr distances. Thus, we have derived analytic expressions for the recombination probability of a RP \virh one ma%nsric nucleus with an arbitrary spin in zero field. These expressions can be readily used IO calculate conrriburions from magnetic nuclei and hfi to the RP recombination_ to estimate the~nuclear magnetic effect on the quantum yield of molecular photodecomposition. to determine the change in the RP recombination probability due to the substitution of magnetic isotopes for non-magnetic ones. SIC.
4. RPs with magnetically
equivalent
nuclei
The above results allow one to calculate the recombination probability of a RP v.ith one radical having any number of magnetically equivalent nuclei. Such pairs can be subdivided into subensembles. each having on!y one magnetic nucleus with the spin equal to one of all possible values of the total spin of the equivalent nuclei. For instance, let one radical of a pair have three equivalent protons. The total nuclear spin equals l/2 or 3/2. Taking into account the statistical weights of RPs \vith equal tom1 nuclear spins. the RP recombination probability is p = [P(V.
%)
(3)
+p(3/2.41/2.
where p( 1, a) is the recombination
probability
for a one-nuclear
RP havin,0 _
consider a deuterated molecule instead of a protonated compound. the RP recombination probability 10/X7. 9/27 and varies. The total spin of D nuclei becomes 3. 2. 1 or 0 with statistical weights of 7/27. l/27. respectively. The recombination probability of a RP with a deuteratcd radical is calculated from the formula p = [7p(3.
ut,) + IOp(2,
a,,)
+@(I.
o,,)
-tP(O.
N,,)]/27.
(30)
The difference in p calculated from eqs. (29) and (30) determines the magnetic isotope effect. The magnetic isotope effect for other RPs with magnetically equivalent nuclei can be calculated in a similar manner using the rule of summation of moments and the results of the above section.
5. RPs with magnetically non-equivalent nuclei scheme remains the same as that for a In the case of these complex systems. the general calculation one-nuclear RP. In practice. however. such calculations can be realized only with a computer_ As noted in section 1, the semiclassical approach to RP spin motion [14] can be used in the case of RPs with many magnetically non-equivalent nuclei. In terms of this approach [ 141. the hfi is represented us fi = gp( H,;S,
+ H,&
= h( w,,.$
+ o,$).
(31)
where H, is the local magnetic field created by nuclei at the site of an unpaired the hfi constants and the nuclear spins: o, = $A‘H,.
The whole RP ensemble is divided into subensembles. local field distribution obeys the gaussian function v(r,+)do,=
(2/5i)1”(w7/a~,,)
the local radical
exp( -&2a;,r)d+
electron
and expressed
fields in each having
via
set values. The
(33)
approach the where aerf is the effective hfi constant given by eq. (1). In the framework of the semiclassical recombination probability calculations include two types of averaging: over all re-encounters of a RP, and over all possible values and orientations of the local radical hfi fields. The correct sequence of the averagings is as follows. First of all it is necessary to sum up all re-encounter contributions to the with set w, and o,,, thereafter the result recombination (to find p( a,,_ a,?)) for each RP subensemble should be averaged over all 0,: P = 0&J,,,
a,,))
=J!P(w,,.
0,2)(P(W,,)(P(W,I)d0,PW,~-
(34)
This calculation procedure has already been realized [ 151. the final results being represented as integrals of the type (34) which need numerical calculations. There is another, however less consistent, averaging procedure [ 121: the S-T transition dynamics is first calculated; then the S-T efficiency is averaged over all local field realizations; then the RP recombination is calculated by numerical means taking into account all re-encounters. In the present paper we make use of the averaging technique proposed in ref. [12]. It allows analytic expressions for the RP recombination probability in zero field. The final results will be compared with those obtained in ref. [15] on the basis of the correct sequence of averaging over the realizations of radical re-encounters and local hfi fields. Use the hamiltonian (31) and find the operator k(r) which describes the variation in the RP density
matrix between laxv of variation
x-encounters. After averaging thr operator of the singlet and triplet populations:
‘Pd’) PlJ,,(
\
il-3F,
fI
P7.J.W
=
F,
F,
I-3F,
F, F,
Fl
F,
1 -2F,-L;
where F, is the probability
of S-T,.
S-T.,
F, =~(+s~+s~+s~).
und S-T_
over all If,, ad
Ii,,.
ax
nht3in the follo\ving
1 P,,(O)
F, 6
P l..,,.(O)
I_i
P ,-_ , _(0)
transitions. I-1:is th:it of T _ -T
.
trxnsition?;:
F;=6(2.~~+2_~~-.s~-s1?_).
sk = (sin( w,,f/2)).
Ii = 1, 2.
(35)
(36)
.s-+=
Her2 ( ) denotes averaging of the type (34). By avzragin g 2q. (36) o\+zr th2 tim2 distributions hrt\vcen re-encounters (2) and (4) and before the first contact (5). and substituting the rcsu1t.sinto ~‘y. (6). \ve obtain the following recombination probabilitizs: %(S)
=PoK&
-i- K,?)(l
.‘iP(T)=p”K-r?[3(l
-r~(S)=p,&TJ(l
- 3f;,) +f]/z.
+ Ksq)f,
‘/J(T)
+ 3j]l/Z-
=J@-+-,[(I
+K&)./;,+-f]/z. i h’,~:)( 1 -1,)
+ 3_/-],‘Z_
(37)
where 7, = 711.
z = (1 + KsTr)( 1 + KTT,) +f(4
and fO, f are average respectively:
efficiencies
of S-T
+ /\lsrr + 3K., 7,).
mixin, 0 before
thz first 2ncount2r
and bctt\ve:i‘n t\vo w2ncount2rs.
where ( ) denotes averaging over the local firld distribution. With 2qs. (-I)_ (5). (34) and (36) on2 can determine f, and f_ Below are the final results for the c~sc xvhen the radical motion fits the continuous diffusion moment.
model. there are no radical acceptors. In this case.
7r = a,b/D.
f. = 0.
f = (l/lSfi)[{
L&I;(‘) + (GJ;;2) + ((w,,
p”=
and the radicals of a pair are in contact at the generation
1. + w,J’~2)
+ ((ILL./,- +,l)’
I)]
T;, :_
In the case of a RP with only one radical having magnetic nuclei. f is expressed through the effective constant (1) in line with the equation:
(40, hfi
where r is the gamma function. Substitutin, 0 sqs. (40) and (41) into sqs. (34)-(37). \ve calculate the recombination probability from various RP multiplet states and for vxious RP precursors. For instance. if a triplet-born RP can recombine only from its singlet state. then. according to eqs. (37). (40) and (41). ~‘5
152
l-able I Values of J al various ratios of cffrcCvc hfi constams
~‘Lcrl/~xcrf
J+
J-
J
0 0.16
0.16 0.14
0.16 0.17
0.32 0.3 1
0.32 0.5 1
0.12 0.10
0.18 0.19
0.30 0.29
0.73
0.09
0.19
0.2s
I.00
0.09
0.19
0.2s
have (K., = 0):
.‘p(S) = 0.145Ks7,q/[l
+ KSrr + 0_145q(4
+ K,T,)].
(42)
where q = (u,_~~~T~,)‘/‘_ Now let us wnsider ;1 nlore general case. when both radicals have many magnetically nuclei. In this case. j is expressed via the effective hfi constants of both radicals: f = 0.048[((r,.,r‘r,,)“2 Here J denotes
+ (u2.erg,,)“2]
the sum of two integrals
J,= / z’zcos2~sin’v[[Icos( 0
+0.30[(&,,+
J = J + + J _.
q i_ a)l]“‘dg7,
non-equivalent
U~.err)“2To]“~J(u,.~r~/u~.~~~).
(43)
where (44)
tg a = ul.crr/u2.crr-
Table 1 lists numerical values of J,. J_ and J. It is seen that J weakly depends on the ratio of the effective hfi constants. Therefore. in lieu of eq. (43) one can use the approximate relation (45) Substituting we obtain
eq.
(45)
into
“p(S)
= @,(l
“p(T)
= 3&~,f/Z.
eq. (37) and assumin g the radicals
+ KTT~ +f)/Z, Tp(T)
In deriving eqs. (37). we violated
?(S)
of a pair to be in contact
at the initial moment.
= @J/Z.
= KT~r( 1 + KS~r -I-3f )/Z_ the sequence
of averaging
(46) over all re-encounters
and all local hfi fields.
Table 2 Comparison of Ul.CffTD
Tp valuescalculated =‘Z.errrD
by eq. (47) and according to ref. 5%
[ 151 TP
IW
TP(47)
0.29
0.5
5
0.090
0.092
0.18 0.22
0.09 0.01
1 10
0.032 0.069
0.032 0.060
0.22
3.97
10
0.205
0.201
5.8
5
5
0.211
0.234
5 0.1
0.256 7.39x
10-S
0.282 7.43x
10-J
9.54x
lo-’
9.74x
10-Z
11.6 0.17
10 0.44
0.25
0.44
10
Thrrefore. the results obtained must be compared with a more consistent theory [ 151. As an esample. we consider a triplet-born RP recombination under the assumption that the reaction can occur only in thr singlet
term.
Tp(s)
Eq. (46) affords: = K&f/[
I + I&T, +f(4
+ I&T, ,] _
(47)
obtained from this formula are compared in table 2 with numerical calculations rcalizcd lvithin the consistent theory [ 151. Relation (47) is seen to fairly \vell agree Lvith the data obtained in ref. [ 151. This means that the change in the sequence of the above averagings affects negligibly the RP recombination probability and thus is quite justified. We now consider a more general case. when the RP partners are out of contact at the generation moment, Q,Z- h. The recombination probability obeys the general relations (37). They become practically applicable. however. only after calculating the integral (3s). Let us find it for a comparativcl_v simple situation when there are no radical acceptors. Substituting eq. (36) into eq. (3s) and using cq_ (5). xvc have
The data
fo=(l/lS)(J,+J,+J++J_).
(4s) exp[ -x1
JI = 1 - (4/fi)~x_x-2 I-
J+= Q),(P) where
B,
(164 =/u;;“dq{
cos( B,_~)“2ds.
=05 exp(-p’)Q,.(n)do. co+
and y+ are expressed
first encounter;
- (B,.\-~“‘1
sir+
cos(ly,lp)“’
exp[ - (lu,lp)““]}_
via the effective
radical
hfi constants
and
rhe time of approach
-
B, = akm,,,7,/2”‘.
k = 1, 2.
Table 3 Values of _r( s) calculated by numerical
yl=
integration
( “1_‘To/2’q
and by approximate
cos( v A- n).
formula
(53)
1’(S) (48)
_r(1)(53)
0.126 0.154 0.16s 0.229 0.27s
0.121 0.15s 0.193 0.237 0.2SY
0.242
0.337 0.405
0.352 0.126
0.362 0.542 O.S16 1.22 1.83
0.484 0.573 0.669 0.769 0.864
0.511 0.60-I 0.70-I 0.79Y
2.76
0.947
O.QAS
4.12 6.20 9.2s 13.9 21.0
1.01 1.04 1.05 1.03 1.o’,
0.950 0.996 0.999 0.999 1.00
31.4 47.0
1.01 1.00
1.00 1.00
s 0.0212 0.0318 0.0476 0.07 16 0.107 0.161
OS.;
befors
the
Ul.Clf/(‘Z.Cff
5,+-J-(48)
l_r(u,.zq,)
0.0707
0.15s
0.450
0.47 I
0.0707
0.324
0.436
0.47 I
0.0707
0.509
0.41s
0.47 1
0.0707
0.726
0.404
0.47 1
0.0707 0.0707
1.00 0.1%
0.395 0.450
0.47 I 0.47 I
0.353
0.1%
0.946
1.01
I.77
0.158
1xi9
1.75
t-i.84 44.2
0.1% 0.158
1.m xl0
2.00
a I.231
LOO
where u,.? = (~:,rr + o~.,.rr)““. and cy has already been defined. The integrals J&_ J,+J_ were calculated numerically (see tables 3 and 4). Analysis has shown that ./. + .I_ depends wrakly on the ratio of the effectilre hfi constants and is determined. within an accuracy of some 10%. solely by u,~T,,. We introduce the function J(S) which defines the dependence of JI. upon CJJ,.Cff%: J1, =.,.(u,-.,,,T”).
(49)
This function was determined by numerical approximate the parameter A, as follows: fo=
(1/18)[_+
,.&I,)
In the range s < OS. y(s)
+?‘(%.,rr%)
integration
+
of eq. (48) and listed in table 3. It can be used to
~r.(h~“)l-
(50)
a ~‘1’~ and
f,, = O-145( (77U)‘/2. where d is defined I0 -
(51)
by eq. (45). As &-, increases.&
aquires
the limit:
7/9-
(52)
Generally speaking. as c7;,, g rows. the parameter f0 oscillates (see table 3). however. these oscillations are slightly manifested and thus hardly deserve attention in the case of radicals with many magnetically non-equivalent nuclei. J(S ) can be approximated by the function _Y(x) = 1 - exp( -0.87s’/
- 0.53s).
With eqs. (SO) and (53) one can calculate the recombination probability the recombination radius at the initial time moment. ‘0 > h.
6. A RP with a particular
nucleus
(53) of a RP whose partners
are not at
with I = l/2
One
of the most interesting manifestations of nuclear effects in radical recombination is magnetic effects. The magnetic 13C enrichment observed at dibenzylketone photolysis [27.28] can serve here as a well-studied illustration. The decomposition occurs from the triplet state and gives: isotope
PhCH,“CO
-. - PhCH,,
RP I;
PhCH,“CO
. - - PhCH?.
RP 11.
(54)
An appreciable “C enrichment in the initial ketone has been measured csperimcntall?_. The effect is associated with a high hfi with ‘-‘C in RP II at cl‘- = 125 G [29]. The hfi constants \vith the protons in ii benzyl radical are an order of magnitude smaller than II<- [29]_ To theoretically intcrprrte thr esprrimcntal data. one must calculate the recombination probabilities of RP I and RP 11 (54). Hwvcvsr. strictI> speaking. the theory of the previous section is inapplicable to calculations of the RP I1 recombination probability_ The theory of recombination of RPs \vith many magnetically non-equivalent nuclei cn~plo~s the gaussian distribution of local hfi fields in radicals. For the radical PhCH,‘-’ CO this supposition doe3 not hold since here the hfi with H nuclei and with “C differs by several orders of magnitude ((I,, < 0.4 G [29]). As a result of this ratio of the hfi constants. the total local radical field disohrys the applicability requirement of the central limit theorem of probability theory and thus the local field distribution c;mno~ be treated as gaussian (33) in the radical considered. Hence. it is necessary to verify the possibility of applying the theory of the above section to the experimental situation of isotopic enrichment \vhcn the RI’ has a magnetic nucleus with a hfi constant greatly differin, 0 from those of the other nuclei in the pair. having man\ To this end we investigated the recombination of a model pair. lvith one radical magnetically non-equivalent nuclei and the other only one nucleus with an I = l/7 spin. We descrihc rhc spin dynamics of the latter radical in the proper quantum-mechanical way. and that of the former radical in the semiclassical approximation [ 141. As a result. the spin-hamiltoiiian takes the form ri = ho,,-& The
following
+ Ii&L
RP states
II) = IT, ,.
a>.
16) = IS. P>.
are defined:
12)= IT_ 13 P>-
13) = IT,, . m) _
17) =IT_,.
W=lT-,.P)-
cr).
14) = IS. a)_
15) = IT,,. /I>_
(‘6,
Here S and T are singlet and triplet states of electron spins. (Y and j3 arc the eigenstatss of i, conforming to the eigenvalues l/2 and - l/2. The change in the state populations (56) induced hy the hamiltonion (55) obeys
the equation
Phh(t)=CFh~.,,,,(f)~,*,l(0)-
(57)
,I
Averaging F(I) over the time distribution between re-encounters and over 1hr2 locc~l field distribution the radical with many magnetic nuclei. we obtain in the limit of continuous diffusion:
F II.33
Fu..u = f&s = F,,.ss =f,. F2z.m= 5~. = J;;;.,, = F-4 7, =J-;F33.14 = I;;%, =A. . I/2
fl= 0-145(uI.,ff7~,) . f3= -f,/Zt-f,/2 + J/24.
J,, =/oldzz‘
proportionally
.,J
=
=
Feb.::
=
i.. -
Fz2.x = 2 <:3_55= 2
_t-?=f,, - fI/2
exp( -z’)[(j_x
5 lists J,, values
increases
--.
=.i, _
+ J/24_
f,=f,,/2 +f,/3-J/24_
_& =j,/4
i J/16.
J = 1.903(aI.cfl~1,)‘i2Jo.
/o = (la17,,/2)“‘/6.
Table
F,,.,, = I=--.
=
for
for various to
ISI”’
+ zl)“’
+ (IS - :I)‘/‘].
s = L’,/z’51,.ccr_
radical hfi constant ratios. AS seen. J,, %triss ncgligibi~ at Is] 5 1. and at 1.1-l> 1. Quantitative calculations cm be made in the follo\ving
0.1
0.917
1.2
0.929
2.0
1.18
0.5 0.8
0x9 I 0.8S2
1.4 1.6
0.9so 1.04
2.5 3.0
1.35 1so
R.0
2.50
10 I5
2.8 3.43
approximation J,, = 0.9.
.I = 1.7( “,.crrT*,)“~.
(59)
when 1.~15 1. and
(mlsly/z.
J,, =
J = 1_42(pJ~7,,)?
(60)
when the hfi constant of the particular nucleus exceeds that of the magnetically non-equivalent nuclei. j_\j > 1. Substitute k kto eq. (6) and calculate the RP recombination probability. Let us consider the case when RPs can recombine only from the sin&t state. The recombination probability is represented as follows (K,=
A-)):
“p = KT,(
uj
+
u,),
Here 114and 14~are solutions /
G, 0
1;;
(
0
-A
0
-13
0
-f4
0
0
with constant
of the system
-f,
-f../2
-/4/z --A 0
+AI~~~(II~+z+,)]_
of algebraic 0
-f,
-f, G, -_fz G, --A -A
0
-(I
1P=(~7r/3)[~
-f2 -f G.3 -f4/2 -f,/2 -f3 0
(61)
equations 0
0
0
\
-Lx --J-i 0 --A -f,/Z --f/2 -f3 0 -f,/2 -f,/l -h 0 G, -fs -f2 -f, -f5 G4 -fz -f, G, 0 -f, -f2 0 -f, -f, G, I
I
0' 0 0 1 0
-
(62)
0 0 i
01
coefficients
Gz=l +2f2+2f-x+f,. G4 = G, + Kr,. G,= 1+f,+fi+f~+h+L-
G,=
1 i-21,.
In principle. eqs. (62) can be solved analytically_ Unfortunately, the formulas are too cumbersome. We solved eqs. (62) numerically and calculated the RP recombination probability (61). table 6 listing some results. Though eqs. (62) can be readily solved by a computer for any particular system. the absence of a simple analytic solution may be an obstacle to a routine application of the present results to an interpretation of experimental data on magnetic isotopic enrichment. In this connection it would be interesting to see whether the theory of the above section can be applied to the analysis of magnetic isotopic effects in a RP with a particular nucleus, as is the case in experiments on magnetic isotope enrichment.
Table 6
Values of'p ohtaincd from at?;. (61) and(62)and eq.(47)(thrsecond column ~~hulat~sculuulati~~~~?; hy q(h3). q.
(4s)
at a2.cT( = u/2).
RP pxametcrs:
a = 2.2 X 10’ rad/s.
u,.<~~ = ‘.2x
to” r;ld/s.
K I\‘, _-
‘p (61.62)
‘I, (47.63)
‘p (45.47)
2.13x 2.09x
2.15x to-’ 2.11 x lo->
2.07X 2.03x
IO-” to-”
0.5 0.5
IO” 10’
‘.5X10-”
I.&OX 10-z
1.82X
1.74x
lo-’
0.5
10”’
0.25
7.46x10-' 0.109 0.114
7.54x10-' 0.110 0.115
7.20x10-~ 0.105 0.110
0.5 0.5 0.5
IO" 10'2 10"
3.76x10-" 3.36x10-' 0.162
3.85 x lo-”
3.62 x10-'
3.44x10-'
3.22x10-'
5.0 5.0
10" 10"
0.262
0.167 0.273
0.153 0.245
5.0 5.0
10”’ IO”
0.279 0.281
0.29 I 0.293
0.26 1 0.263
5.0 5.0
10’: IO”
4.70x 0.263
1o-J lo->
lo-’
0.487
4.80X 0.273 0.512
lo-’
lo-:
4.67X 0.260 0.4so
IO-’
IO%,, (St
rhe third one those hy
7r = s,,,“O A-,(s-‘)
50 50
10” 10’
50
IO’O
0.533
0.562
0.525
50
10”
0.538 0.538
0.567 0.568
0.530 0.530
50 50
10’z to’J
2.5 X 10. J
2.5 25 250 ~.jXlW~ 0.25 2.5 ‘5 250 2.5.x to’ 0.15 7.5 25 250 2.5 f 10’ 2.5 :i loa
According to the theory given in the previous sectron. the hfi effect in the recombination is determined by the effective hfi constants of the radicals (see eq. (45)). By the definition (1). LI,{~= (I/) at I = I,.?. When applying eqs. (45) and (47) to our case of a RP with a particular nuc1eus. the question arises xvhsthcr we should substitute the hfi constant CIof the radical with the particular nucleus for CJ~~,~,. in sq. (15). or set ~z.err = a/2 in line with eq. (1). To answer this question. \ve used both ways (see table 6). The data of table 6 demonstrate that calculations by eq. (47) are in fair agreement with those b_v sqs. (61) and (62). If akff = a/2 in eq. (47). Tp becomes underestimated (by some 105). If in eq. (17) a,.,ii = t~._f-= O.l-Z(Lir,, 1’ ‘_
eq. (47) overestimates the recombination probability approximately by lo!%_ Thus. the theory of the above section. based on a semi-classical approach to RP spin dynamics. affords satisfactory values of the recombination probability for radicals with magnetically non-cquivslsnt nuclei even if the hfi constant of the particular nucleus differs greatly from those of the other nuclei. 7. Discussion Above we found the recombination probabilities for pairs of uncharged radicals \vith magnetically equivalent and non-equivalent nuclei. The recombination probability is determined by the multiplicity of the RP precursor and by that of the state from which the RP recombines. The p dependence on hfi. viscosity and radical reactivity is determined by three dimensionless parameters: lal~,-, (or cir,). KT,. and la1~c (or circ). The residence times of a RP in a “cage” (ro). in a reac:ion zone (TV). and the time of the first approach (me) all are proportional to the medium viscosity (q)_ Hence. the results obtained alloa analysis of the influence of viscosity and reactivity upon magnetic effects in RP recombination. A typical reaction showing magnetic isotopic effects is molecular decomposition from a triplet excited state under photolysis or radiolysis. We therefore analyze the case when a RP is triplet-born and the
reaction can occur only in the singlet state. For all systems considered (equivalent and non-equivalent magnetic nuclei) the theory predicts the same character of p dependence upon the medium viscosity and the and tends to the limit radical reactivity. As the latter increases (I\’ s grows). ‘rp(S) rises monotonically
-‘P(S)~(h/r,,)(y+4,,)/3(1 for :I one-nuclear -‘P(S)
(64)
+4)
RP (see eq. (27)).
= (~~/%)(/+I;,)/(1
and to the limit (65)
+f)
for a RP with many magnetically non-equivalent nuclei.(see eq. (37)). In the case of a RP with magnetically equivalent nuclei this limit is always smaller than l/3; for radicals with non-equivalent nuclei it can reach values as high as 1. A detailed discussion of the p upper-limit problem can be found in ref. [24]. The probability “b(S) increases with the medium viscosity. and rqs. (64) and (65) afford: ‘p(S) respectively.
= h/3r;.
.‘I,(S)
It is necessary
3 h/l;,.
(66)
to note that these extreme
~alurs
are directly
connected
with the conservation
of the total electron
and nuclear spin for the case when the isotropic hfi is the predonkmnt mechanism of singlet-triplet mixing. This assumption is quite valid in zero field provided the medium viscosity is not very high. In very viscous media with 9 L 10 P the in-cage RP lifetime becomes so long (TV, 2 IO-” s) that S-T transitions occur in all the RPs due to paramagnetic relaxation and thus ‘/, tends to unity. The quantum yield of molecular photodecomposition is expressed via the geminate recombination probability as: O=y(l
(67)
-PI.
where y is the probability of generating a RP from an excited molecule. If the quantum yield before and after an isotopic substitution is 0 and +*. respectively. then the ratio +*/+ determines the magnetic isotopic effect. The recombination probability of geminate pairs depends on the initial interpartner distance. If it is sufficiently long. the RP spin states become disordered before the first encounter at the recombination radius. and the singlet and triplet states get populated according to their statistical weights. Hence. if the time interval before the first encounter. 7”. is longer than the radical paramagnetic relaxation times.
where “p the recombination it is possible to find ‘Y(S)
= I\‘&[(
‘p(T)
= K-&3(
probability
1 + KTTr)/4 1 + K,T,)/~
of diffusion
(encountered
randomly)
pairs of radicals_ With eq. (37)
+f]/Z. i- 3f]/Z_
(68)
Let us analyze now the case when the partners are in contact at the initial time moment. For a molecule with many magnetically non-equivalent nuclei which decomposes from the triplet excited state we have, using eq. (47).
(6% As K, grows. T+ reduces from y to y/( 1 +/)_ of low viscosities (fe 1, KT~ -=E1)
As the viscosity
increases.
I-& falls from y to zero. In the range
(70)
while in highly viscous .‘& a y/fa
media
yC27j- ‘I’.
(71)
\vhere C, and C, arc viscous-independent
coefficients.
in our cast.
‘i+*/‘(s = A/B.
(72)
.1=(l+Als~,+4f*~[1+KsT,+/~4+~‘,~,~]. S=(1+Als7~+4f)[1+AIS7,+f*(4+~S~,)]. As KS increases.
‘i$*/-‘+ = When
the magnetic (1 +I)/(
the viscosity (“+“/‘i+),
isotopic
and reaches
an rstrsmr
vaiuc
1 i-f*)_
(733
rises. the magnetic =
effect reduces
isotopic
effect tends
to the limit
(d/&f’.
(74)
where Csand
fi* are the effective RP hfi constants before and after the isotopic substitution. rs~prctivel~. under a complete deuteration (all protons are substituted by deutsrium ittums) 11lr effecti;.e hfi constants reduce by fourfold_ It means that under deuteration (‘&*/ I+), = 7. To interprete data on magnetic isotopic enrichment. one introduces one additional p:tr:tmctcr [Xl: For instance.
a=(O-+*)/0=(p*-~))/(l In our case of a triplet-born give
(75)
-P).
RP. eqs. (45) and (47) for RP I and cqs. (47) and (63) for RP II (SW cq. (54))
7n increases monotonically with K, and the \*iscosity. -In - 1 - (2/d’)” ‘_ To compare the present theory with experiment quantitatively. one must have data on 111s \xscosirv dependence of 9. +*/+. or CL The RP recombination rate constant is a \ariablr in the thsory and thus th> latter can always be fitted to experiment by a proper choice of K provided the experimental data are taken at a definite viscosity of the medium. The magnetic isotopic enrichment seems IO he most adequatelv studied for dibenzylketone photodecomposition [_____L 77 7’ 77.X.30]. III benzene at 9 = 0.6 cP tx = 0.03 [30]. Ii1 a glycerine-alcohol mixture with 11 = 540 CP a grows up to 0.24 [El_ Dibenz_vlkstons photol_vsis in cyclohexanol (17 = 60 cP) gives a = 0.09 [23]. There are also data on dibsnzvlktons photol\-sis in micsllss [28]. In these systems the microviscosity is 17 = 30-40 cP. The measured values of n in various micrllar systems are 0.26-0.30. In micelles, radical pairs diffuse in a restricted volume and the in-cage RP must be longer than in the case of partners freely diffusin, 0 in a solution. Therefcwr. the data for are compatible with those for highly viscous glycerine-alcohol solutions. The experimental hfi constants for RP I and RP II obtained at dibsnzylkstone photol_vsis [29] used to determine effective hfi constants. ii. in RP I (by eq_ (45)) and RP I1 in\-olving ‘:C isotopes
lifetime micellrs can be (bv eq_
(63)): r7 = 0.22 x 10’ rad/s. The RP lifetimes ,rI, = b’/D,
rl* = 2.70 x 10’ rad/s.
in a cage and in a reaction
7r = orb/D.
z@ne are calculated
(77) from sqs;. (23) and (JO):
(7s)
Table 7 RP I and RP II parameters (see cq. (54))
On determining
the mutual
4 (cP)
T,, (ns)
f(45) RP I
/’ (63) RP II
0.6 60 540
0.15 15 132
0.026 0.26 0.78
0.092 0.92 2.73
diffusion
coefficient
by the Stokes
formula.
D = 2kT/3~inh. we obtain TV, = (3&‘/2kT)t~.
TI) = 024Sq
r,.= (3m,b’/XT)q_
ns.
-rr= 0.0124
ns.
(SO)
In eqs. (SO)viscosity must be in centipoise units. Substituting obtain the parameters f and f* necessary for the comparison
eqs. (77) and (SO) into eqs. (45) and (63). we of theory and experiment (see table 7). In a
glycerine-alcohol mixture ho is sufficiently high: the relaxation processes can reduce the isotopic enrichment. Therefore. the above theory might overestimate the magnetic isotope enrichment coefficients since it neglects slow relaxation processes. At given /and /* values a maximum isotopic enrichment is achieved for radicals with a high reactivity_ at KS7r B 1. The maximum value of Ta is (see eq. (73))
‘-a,,, = (f*-f Ml
+f*)-
(81)
Table 8 lists LY,,, estimated by this formula. A comparison with experiment demonstrates the extreme ratio (Sl) to give a values which exceed experimental data by a factor of 1.5-3. According to eq. (79). in cyclohexanol TVis 100-fold higher than in benzene and one-tenth of that in a glycerine-alcohol mixture. Taking this into account. we tried to find the K, at which the calculated Tcu coincides with CY According to eq. (SO), KATE = 1.2 X IO-” K,o. In benzene Ksrr = 0.7 X IO-” A’,. If . K, = 2.0 x 10” s-“F’ . 1-e. Kgr = 1.5, then eq. (75) gives Ta = 0.032 which is an attractive fit to experiment_ However. at this very K, for the other solvents studied, Ks~r Z+ 1, Ta ~~a,,,,, so that the calculated (Y exceeds the experimental one (cf. a, with as,,, in table 8). If we set K, = 3 X lo9 s- ‘, Ks7, = 1 and 10 in cyclohexanol and a glycerine-alcohol mixture, respectively, the theoretical a is coinciding with experiment_ For benzene. however. the theoretical a is much less than the experimental one (cf. Ta, and CYST,,in table 8).
T;tbls 8 Coctfficirms of magnetic “C isotope enrichment in dibenqlketone Solvent
(Kg,
values used IO calculate Ta (76) are shown in parmthsses)
=c.p 0.03 1191 - .
T%-
Ta*
Tag
Ta3
brnzsns
0.060
cyclohexanol
0.09 1301
0.34
glycsrine + alcohol
0.24 [26]
0.52
4.3 x 10-j (0.0 1) 0.066 (1) 0.31 (10)
0.032 (1.5) 0.34 (150) 0.52 (1500)
0.032 (1.5) 0.092 (1.5) 0.24 (6)
Within the theory in question, experiment can be quantitatively described only undrr the supposition that either 7,) and/or 7r should be calculated in some other way than eq. (80). or K, may depend on the solvent. in particular. on the solvent viscosity. Table 8 tabulates calculations with varyin, 0 Ksrr_ As seen. K,r, can be chosen so that the calculated a is in good agreement with experiment. To this end. holvcver. OIW has to assume that either the RP recombination constant at the moment of an encounter reduces in more viscous media. or -rr depends weakly on the medium viscosity. The assumption of a weak dependence of KsTr on the medium viscosity is undoubtedly a11 extremely promising way leading to important consequences. Providin, Q K,T, changes negligibly in various solvents or at various viscosities. the viscosity dependence of the rate constant K, in the kinetic limit lvould resemble the diffusion-controlled case, however. the value of K, would be lower than that in the diffusion limit: K,aDa
l/n.
K, < 4z;hD.
(S2)
However. the available experimental data are too meagre to give an adequate description of the Ki,i, dependence on the viscosity and properties of the solvent_ Further investigations are required. Indeed. only recently Turro and Kraeutler 1311 have put forward arguments that the exprrimentally observed Q = 0.03 for benzene cannot unambiguously be associated with a magnetic isotope effect. The fact is that 311 isotope mass effect can also result in a similar (Y value. If we ignore the data for benzene. the present rhror_v. as seen from table 8 (see T&). fairly well describes experimental results. Thus. the theory discussed needs also further development. First of all. paramagnetic relaxation processes must be taken into account \vhen analyzing
recombination
in viscous
media.
molecule may give pairs with different inevitably change the magnetic isotope
Furthermore_
kvhen decomposin,
a in different solvents. the same allo\ved for. the above factors lvill
interpartner separations. When effect predicted theoretically.
Acknowledgement The author is grateful to Z.V. Shapochanskaga and X.K_ -Salikhova for calculations, and to Dr. K.A. McLauchlan for dralving his attention to ref. [31].
assistance
in
numerical
References 111 A.L.
Buchachenko.
1978). in Russian. [2] Yu.N. Molin, R.Z. [3] R.Z. Sagdeev.
R_Z. Sagdeev
and K.hl.
Salikhov.
Xlugnctic
and spin effecls
in rhemicril
Sagdssv and K.M. Salikhov. Review of Soviet Authors. Chsnktn_
K.M. Salikhov
and Yu.N.
rcacrions
(Sx.&.a.
X’ovosihirsk.
Ssriss. Vol. 1 (1979) pp. I-70.
Molin. Uspekhi
Khim. 36 (1977) 569. in Russian. [4] K.M. Salikhov, A.L. Buchachenko. Yu.N. hlolin and R.Z. Sllgdeev.Spin polarization and magnetic (Akademiai Kiado. Budapest/Elsevier. Amsrsrdam. 1983). [5] K.M. Salikhov. Doct. Thesis. Kazan (1974). [6] F.S. Sarvarov and K.hl. Salikhov. Tear. Eksper. Khim. 11 (1975) 435. in Russian. [7] R. Haherkorn. Ph.D. Thesis. hlunich (1977). [S] F.S. Sarvarov. Cand. Thesis. Novosibirsk (1977). [9] P.A. Purtov and K.M. Salikhov. Teor. Eksper. Khim. 16 (19SO) 737. [IO] C.T. Evans and R.G. Lawler. hlol. Phys. 30 (1975) 1085. [ 111 H.-J. Werner. 2. Schulten and K. Schulten. J. Chcm. Phys. 67 (1977) 636. [12] 2. Schulten and K. Schulcen. J. Chem. Phvs. 66 (1977) 4616. [13] G.R. Zientara and J.H. Freed. J. Phys. Chem. 83 (1979) 3333. [ 141 K. Schulten and P.G. Wolynes. J. Chem. Phys. 6S (1975) 3292. [IS] P.A. Purlov and K.hl. Saiikhov. Tear. Eksprr. Khim. 16 (19SO) 579. in Russian. [16] R. Kaptrin and L.J. Oosterhoff. Chem. Phys. Letters 4 (1969) 195. ] 171 F.J. Adrian, J. Chem. Phys. 53 (1970) 3374.
sffec:s
in radical rwcrions
11x1 R.G. Lawler and G.T. Ewns. Ind. Chim. Belg. 36 (1971) 1087. [ IY] R.Z. Sagdeev. K.hl. Salikhov. T.V. Leshina. M.A. Kamtha. S.M. Shein and Yu.N. Molin. Pisma Zh. t&per. Tcor. Fiz. 16 (1972) 59Y. in Russian. 1201 O.A. Anisimrw. V.hl. Grigoryanls. S.V. Kiyunov, K.hl. Sslikhov. S.A. Sukhenko and Yu.N. Molin. Tear. Eksptx. Khim. IS ( I9ki2) 292. 1211 V.F. Tarasov. Zh. Fiz. Khim. 54 (1980) 2438, in Russian. 1221 A.L. Uuchachmko. V.F. Tarssov and V-1. hlnl~srv. Zh. Fiz. Khim. 55 (1981) 1649. in Russian. 123) L. Swna. D. Ronis. S. Wolfe and A. Pines. J. Chem. Phys. 73 (19SO) 5493. l2dj KM. Salikhov. Chcm. Phys. 82 (1983) 163. 1251 R.M. Noyes. J. Chem. Phys. 22 (1954) 1349. 1261 J.M. Dcuwh. J. Chem. Phys. 56 (1972) 6076. 1271 VI. Tamsov. A.D. Pershin and A.L. Buchachenko. Izv. AN SSSR Ser. Khim. 8 (1980) 1927. 1281 II. Kraeuricr and N. Turro. Chcm. J’hys. LCIIC~S 70 ( 19t;O) 270. 1291 II. Paul and 11. Fischer. Iirk. Chim. Acua 56 (1973) 1575. 1301 NJ. Turro and B. Kracudcr. J. Am. Chem. Sot. 100 (1978) 7432. 1311 N.J. Turm and B. Kraeuder. intersystem crossing in diradicals and radical pairs. ch. 6 in a book devwsd IO diradicals (preprint).
Physics
North-Holland
82 (1983)
Publishing
THEORY
145-162 Company
OF MAGNETIC
EFFECTS
IN RADlCAL
REACTIONS
AT ZERO
FIELD
1. Introduction
The hyperfine interaction (hfi) of unpaired electrons with magnetic nuclei induces singlet-triplet (S-T) transitions in radical pairs (RPs) thereby affectin, 0 their recombination probability [l-4]. This influence of nuclei upon radical reactions can be manifested in various experiments. For instance_ the hfi \vith magnetic nuclei can affect the quantum yield of molecular photodecomposition which fact must be taken into account when estimating the cage effect in liquid-phase reactions. Magnetic isotopic effects are an important manifestation of the hfi. These effects consist in a change of the RP recombination probability due to the substitution of either magnetic isotopes for non-magnetic ones (e.g.. “C for “C). or one magnetic isotope for another (e.g,, D for H). Theory of RP recombination has been developed in a number of publications [l- 141. The conclusions of this theory. as applied to the zero-field case. can be outlined as follows. (Note th;lt the small e;1rth‘s magnetic field cannot show up within a RP lifetime and thus can be assumed zero. Therefore. the results obtained for zero field can be applied to the earth’s field.) One of the most important results concerns short-lived RP recombination. For such a pair. irrespective of the type and the number of magnetic nuck. the singlet-triplet mixing efficiency induced by the radical hfi is determined by a single parameter: thr effective constant a,,,. given by
1 I/’
fCu;z,(z,
JGrr =
[
+ 1)
k
.
(1)
where U~. is the hfi constant. I, the nuclear spin [5-71. In the case of the simplest. one-nuclear. RP \vith Z = l/2, analytic expressions for the RP recombination probability were obtained in terms of the exponential [S] and diffusion [9] models. For more complex systems (a one-nuclear RP with arbitrary spin. a RP \vith several mugnctically non-equivalent nuclei) exact kinetic equations for the RP density matrix and the RP recombination probability were calculated only numerically (see. e.g. refs. [ 10-131). As the number of magnetic nuclei increases, precise calculations prove to be practically unrealizable even with a computer. Therefore. it was proposed to investigate systems with a great number of magnetic nuclei in terms of a semiclassical
0301-0104/83/0000-0000/$03.00
8 1983 North-Holland
description
fO\\r-fdil
of tl\r: RI’
spin rlyn;\niics
[ 141.In tlris c:\sc. the recolrrhinatii~i~
w:\s prcsc\~lcd :\s
prcrh:\bilily
i\IWgri\lS [ IS).
NOIL’. finally. ;\ppr\>xi\ll;\tc: Csti\lli\ICs OT rl\r: RI’ r~ccl\\~lli\~;\tit~\~ prob;\hili\y \:\ki\lp i\r\o i\~c~~\\\lI 111~ I\lLi\iil\\ccd S-l‘ mixing. The p\\l~licolions on lhis poinl c;\n he tlividcd into IWO groups. 0nc CI~~I~IXCCS prccisr: Ci\lClllillillllS Of IllC RI spin dy\lilllliCS. will1 IIlL’ lll\dCC\llN dyixrmics of ri\JiCi\l pairs considcrcd in the framework of i\ simplified IMXI~I (c.g.. tl\c K:\pteiu-C’llrss-~l~sterl\~~r~ nwrlel. or 111~Knptoin-r~drinii 11111dCl [ 16.171. Lee illSO rcfs. [l-4)). l3xi\lllplCS elf Sllcll CS~illlillCS ClI\l h! fc~lllld in r&s. (6.18-X]. Thr‘ olhw group (see. c.g. rcfs. [21-Z]) dcscrilws Ill12 llll~lcX\llilrll~lli\llliCS in tcrnu 0r :\ fiiirly Well dcvcli~pcclnioild. howcvcr. tilr: spin J~l~illlliCSis CillC\\ti\~Cd \\ndcr \\ns\\l~sta\~li\\leii i\SS\llllplil~\~S.For CSillll~~lC~ ir is ilSS\llllCll Illill in lligll lklils Ihe S-l‘ \nixi\lg cfficicncy \ll\\slIW oiic-llrird Of llli\l in mu lklti. This ~\ss\\mption holds only for shorl-lived RI’s (this prohleni IlilS hail discussed in rrfs. [1.6.24]). The prcsrnt pilpcr rcporls priiclicol formiiliis for lhc rccoiiilriii;\lioii prohilhilil~ (\l\lcllilrg~d) radiciils ill zero IlXlgllClic field.
The RP recombination
is considered
~Sec\\ltf i\ ra\ldo\ll llllXllli\l niolion.
l
life~iillr
I.iistribulioIl
instance. this distribution
/(I)
=
JJIf-3/2
hc~wwil
rtxIlco\IIlws
has
found
been
where p is the re-encounter
;\I
11wdtA (SW
rds.
nci\tr;\I
Il.241). The RI’ partners rccomhinc. I’llC hy lllr: function j(I). For
llic rtZ\clill\l
rildilIS
is ilclliWd
rndia\ls: (2)
probability,
111is rxprrssed
vii\ molecular-kinetic
p;\r;\meters,
1 -~)$7/X,,)‘7~/‘.
exp( -7rn,nz/pzr
d
10 ClVllCinto CO\llilCl illld tIllIS Cilll
by Noyes 1251for neutrnl
(3)
b is the reaction radius. X,. T/. are the mean the presence of radical acceptors [ 11, /(f)=JJlI-“’
prow
pilirS
-Tn,J12//J21).
L’Sp(
)I?= (27/8n)“3
in terms of ~hc following
At limes lllrg
0r
length
and the me;\n time of
a11
clemcntnry
diffusion
- K,f ).
step. In
(4)
where EC, is the rate constant for decay of a’RP by reaction with the acceptors. If at the initial moment the radicals of the pair are not in a direct contact, 011e needs also the RP time distribution function/,,(r) before the first encounter. For instance, in the continuous diffusion model [LX]: f,,( 1) = r,1,r-3”
exp( -vimrn~/ptf
- K,r ).
p. = h/r,,.
I?Z[,= b( r, - h)/2r,,(
7rDy.
where r0 is the initial interradical distance, p. is the first-contact probability. At an encounter. singlet RPs recombine with a constant KS, and triplet RPs with a constan\ exchange interaction is assumed to be negligible between re-encounters. and sufficiently encounters to completely dephase the S and T states. Within this model the total contribution contacts to the recombination probability is [ 1.41: p
=
i’(
TT
1 - FQ)-\&p(O).
(5) K-,-. The strong at from all
(6)
where pp is the recombination proiability at a single encounter. 0 describes the change in the density matrix induced:by one encounter, F is the spin evolution operator averaged over time intervals between re-encounters, F, is the average spin evolution operator before the first encounter. The forms of 8’ and 0 are known (see refs. [ 1.41). Therefore, to calculate p one must only find the S-T transition dynamics.
where A, = KS7/( 1 + KS7 ) and X r = K:-l-T/( I + K-,-T ) ;irE: the raxm~hination proh:ihililirs of singIt aid triplet RPs at an encounter. T is the Illtill duration of an r‘ncounwr. Averaging i-‘c f ) hy j-t I ) a11d I;,( I ) and the rtxon~binntion prob:tbilitv. Prior 1~1 substituting the result obtained into eq. (6). One deterniines Lrt ‘p(J’) be thr‘ RP rcxombination adducing the final results. we introduce so~lle useful notation. the notation in par~nthrtsrs d~nows the probability. The pre-superscript shows the initial RP multiplicity. multiplicity of that RP state of which the rccon~binntion product is undtx study. Fm instance. ‘p(S)
denotes the RP recombination probability when the initial state is a triplet and the recombination of singlet pairs is considered. With eqs. (11). (12) and (6) we obtain +K+)(l
sp(S)=p,KsTn[(l
-‘s;,/J?o)+~i;rl]/A.
-‘> (S) = p. Ksm [ ( 1 + K,m
=pOKTm[(l
“p(T)
(14)
G]/A.
+ K,-TII)+~~,,K,-TJz[(~
‘~p(T)=
(13)
) ,i’o/Po + ii%] /3?1,
+ K,n’)i?;,/P,+
product
(15) + K?;TJz)(~ - ~T,,/p,,)+~i;rr]/A.
(16)
where A = (1 -I- K,Tu)(
,T;=
I + K,.T~I)
=w( r)f( r)dr. /0
+ iFn(2 + KSrn + Kl.~n).
Is;,,=
-I(.( I)_&( r)dr. I0
(17)
and II = l/(1 - p) is the overall number of encauncers of a RP. p,, is the probability of the first encounter at the recombination radius. TIJ the total residence time of a RP in the reaction zone. RelaCons 1\3j-{ 13) xe u&d TOT xrg statistics of re-encounters. Unfortun;lte\s. the re-encountrr stat isk5 +CI~&5x1 sQ&& in *xX W% Cm Y&WA{ ix&i~ S ‘[25]. FGY FLY& ~fiXZ555~C+K CkSfCSISC~I+YL~~~~~ between RP n-encounters and before the first contact are defined by eqs. (2)-(S). They can be used to calculaIe G% and &&. Subsdtufing eqs. (4) and (1 I j inio eq.
[pc/2(1
+c)][esp(-g,)-cos(g,).exp(-gJ)J.
g, = pK,‘/‘. p = Z,?,m’/2/p.
(18)
~~=I.LI[-K~+(K~+I~~)“~],Z~“~_ u2=a’[I(Z+
1)+
.5~=PC[K,+(KZ+211)‘/2],2)“2_
l/4].
The overall number of encounters is [l -il/(r)dr]-‘=
II=
[1 -esp(-g,)]-‘-
(1%
With eqs. (18) and (19) one can calculate the RP recombination the limit of A,, TV --, 0 (A’D/6~L - D) we obtain the following (tramktiton-ro -2ri~‘imirc’ts ti~scusseh. es..‘m t-&_:c?~~:
Ttl
-
Tr =
‘G/P, =
Orb/+ + (&To)"']> [c/31 +c)l[exp(-g,,)--os(g,,)
St0= (&,#‘,
probability for various A,, and T/.values. In results for the continuous diffusion model
exp(-th,)].
g,, = ([-K,+(K,‘+u’)1’2]T~,,2)“2,
&,,=([K;i+(K~+U')"2]To,2)"2. Here 7r is the total residence lime of a RP in the reaction zone. a, and b are the width and the radius of the reacTi‘JI1-5sxk. itCiijX32&&~,7 D-k'~+fth~-ztzz UT WViht, TD
=
b’/D,
(23)
and q, is the mean time before the first encounter. r,, = (r,, - b)‘/D.
(14)
Substituting eqs. (20)-(24) into eqs. (13)-( 16). we calculate the RP recombination prohahilitv. are especially simple in the absence of radical acceptors (K, = 0) and if the radicals recombination radius at the initial moment (r,, = h. p,, = 1. I?,,= 0). In this c;1se_ Sp(S)
= KgTr( 1 + K+,
+ q)/A.
‘p(T)
= K-p&A.
rp(T)
The results arc at the
‘p (S) = K,T$//3A. = 2K+J3(
1 + K,.T~) t- A-,-7,( 1 + Ksrr + y)/U_
(25)
il=(l+KsTr)(l+K,.T,)+~(~+iisT,+h’,-T,). q = { I(
1 +
I)/[
41( 1 +
1 +
~)]3’4)(~t~~Tr>)“‘-
At the moment of a RP generation the partners can be out of contact. For instance. ;I RP c:~n result from the decomposition of a molecule into three fragments: two radicals and a diamagnsric molecule. in this case. e.g.. in the absence of radical acceptors: ‘T;,,/P,,=q0={21(1+ & = { [ 1 + 41( 1 + Eq. (14) affords,
i)/[i
+41(1+
ij])[i
-COS(,Q)
r~p(-g,,)]_
l)]“‘lU17,}‘i2/2.
e.g.. for a T precursor of a RP:
TP(S) = (b/r,,)&&?+
4o(t +
(‘7)
K~Tr)]/3A-
The S-T mixing before the first encounter (parameter q,,) oscillat:s dcprndin, distance_ If q, is sufficiently large and ]a] To > 1. then. according to eq. (26). qo=21(1+
(26)
1)/[1+41(1+
l)]_
0 on the initial
intcrpartncr (2s)
The S-T transition oscillations can also be manifested in the recombination probability. They are disguised and not observed experimentally if there is a distribution of RPs over initial inrsrparrncr distances. Thus, we have derived analytic expressions for the recombination probability of a RP \virh one ma%nsric nucleus with an arbitrary spin in zero field. These expressions can be readily used IO calculate conrriburions from magnetic nuclei and hfi to the RP recombination_ to estimate the~nuclear magnetic effect on the quantum yield of molecular photodecomposition. to determine the change in the RP recombination probability due to the substitution of magnetic isotopes for non-magnetic ones. SIC.
4. RPs with magnetically
equivalent
nuclei
The above results allow one to calculate the recombination probability of a RP v.ith one radical having any number of magnetically equivalent nuclei. Such pairs can be subdivided into subensembles. each having on!y one magnetic nucleus with the spin equal to one of all possible values of the total spin of the equivalent nuclei. For instance, let one radical of a pair have three equivalent protons. The total nuclear spin equals l/2 or 3/2. Taking into account the statistical weights of RPs \vith equal tom1 nuclear spins. the RP recombination probability is p = [P(V.
%)
(3)
+p(3/2.41/2.
where p( 1, a) is the recombination
probability
for a one-nuclear
RP havin,0 _
consider a deuterated molecule instead of a protonated compound. the RP recombination probability 10/X7. 9/27 and varies. The total spin of D nuclei becomes 3. 2. 1 or 0 with statistical weights of 7/27. l/27. respectively. The recombination probability of a RP with a deuteratcd radical is calculated from the formula p = [7p(3.
ut,) + IOp(2,
a,,)
+@(I.
o,,)
-tP(O.
N,,)]/27.
(30)
The difference in p calculated from eqs. (29) and (30) determines the magnetic isotope effect. The magnetic isotope effect for other RPs with magnetically equivalent nuclei can be calculated in a similar manner using the rule of summation of moments and the results of the above section.
5. RPs with magnetically non-equivalent nuclei scheme remains the same as that for a In the case of these complex systems. the general calculation one-nuclear RP. In practice. however. such calculations can be realized only with a computer_ As noted in section 1, the semiclassical approach to RP spin motion [14] can be used in the case of RPs with many magnetically non-equivalent nuclei. In terms of this approach [ 141. the hfi is represented us fi = gp( H,;S,
+ H,&
= h( w,,.$
+ o,$).
(31)
where H, is the local magnetic field created by nuclei at the site of an unpaired the hfi constants and the nuclear spins: o, = $A‘H,.
The whole RP ensemble is divided into subensembles. local field distribution obeys the gaussian function v(r,+)do,=
(2/5i)1”(w7/a~,,)
the local radical
exp( -&2a;,r)d+
electron
and expressed
fields in each having
via
set values. The
(33)
approach the where aerf is the effective hfi constant given by eq. (1). In the framework of the semiclassical recombination probability calculations include two types of averaging: over all re-encounters of a RP, and over all possible values and orientations of the local radical hfi fields. The correct sequence of the averagings is as follows. First of all it is necessary to sum up all re-encounter contributions to the with set w, and o,,, thereafter the result recombination (to find p( a,,_ a,?)) for each RP subensemble should be averaged over all 0,: P = 0&J,,,
a,,))
=J!P(w,,.
0,2)(P(W,,)(P(W,I)d0,PW,~-
(34)
This calculation procedure has already been realized [ 151. the final results being represented as integrals of the type (34) which need numerical calculations. There is another, however less consistent, averaging procedure [ 121: the S-T transition dynamics is first calculated; then the S-T efficiency is averaged over all local field realizations; then the RP recombination is calculated by numerical means taking into account all re-encounters. In the present paper we make use of the averaging technique proposed in ref. [12]. It allows analytic expressions for the RP recombination probability in zero field. The final results will be compared with those obtained in ref. [15] on the basis of the correct sequence of averaging over the realizations of radical re-encounters and local hfi fields. Use the hamiltonian (31) and find the operator k(r) which describes the variation in the RP density
matrix between laxv of variation
x-encounters. After averaging thr operator of the singlet and triplet populations:
‘Pd’) PlJ,,(
\
il-3F,
fI
P7.J.W
=
F,
F,
I-3F,
F, F,
Fl
F,
1 -2F,-L;
where F, is the probability
of S-T,.
S-T.,
F, =~(+s~+s~+s~).
und S-T_
over all If,, ad
Ii,,.
ax
nht3in the follo\ving
1 P,,(O)
F, 6
P l..,,.(O)
I_i
P ,-_ , _(0)
transitions. I-1:is th:it of T _ -T
.
trxnsition?;:
F;=6(2.~~+2_~~-.s~-s1?_).
sk = (sin( w,,f/2)).
Ii = 1, 2.
(35)
(36)
.s-+=
Her2 ( ) denotes averaging of the type (34). By avzragin g 2q. (36) o\+zr th2 tim2 distributions hrt\vcen re-encounters (2) and (4) and before the first contact (5). and substituting the rcsu1t.sinto ~‘y. (6). \ve obtain the following recombination probabilitizs: %(S)
=PoK&
-i- K,?)(l
.‘iP(T)=p”K-r?[3(l
-r~(S)=p,&TJ(l
- 3f;,) +f]/z.
+ Ksq)f,
‘/J(T)
+ 3j]l/Z-
=J@-+-,[(I
+K&)./;,+-f]/z. i h’,~:)( 1 -1,)
+ 3_/-],‘Z_
(37)
where 7, = 711.
z = (1 + KsTr)( 1 + KTT,) +f(4
and fO, f are average respectively:
efficiencies
of S-T
+ /\lsrr + 3K., 7,).
mixin, 0 before
thz first 2ncount2r
and bctt\ve:i‘n t\vo w2ncount2rs.
where ( ) denotes averaging over the local firld distribution. With 2qs. (-I)_ (5). (34) and (36) on2 can determine f, and f_ Below are the final results for the c~sc xvhen the radical motion fits the continuous diffusion moment.
model. there are no radical acceptors. In this case.
7r = a,b/D.
f. = 0.
f = (l/lSfi)[{
L&I;(‘) + (GJ;;2) + ((w,,
p”=
and the radicals of a pair are in contact at the generation
1. + w,J’~2)
+ ((ILL./,- +,l)’
I)]
T;, :_
In the case of a RP with only one radical having magnetic nuclei. f is expressed through the effective constant (1) in line with the equation:
(40, hfi
where r is the gamma function. Substitutin, 0 sqs. (40) and (41) into sqs. (34)-(37). \ve calculate the recombination probability from various RP multiplet states and for vxious RP precursors. For instance. if a triplet-born RP can recombine only from its singlet state. then. according to eqs. (37). (40) and (41). ~‘5
152
l-able I Values of J al various ratios of cffrcCvc hfi constams
~‘Lcrl/~xcrf
J+
J-
J
0 0.16
0.16 0.14
0.16 0.17
0.32 0.3 1
0.32 0.5 1
0.12 0.10
0.18 0.19
0.30 0.29
0.73
0.09
0.19
0.2s
I.00
0.09
0.19
0.2s
have (K., = 0):
.‘p(S) = 0.145Ks7,q/[l
+ KSrr + 0_145q(4
+ K,T,)].
(42)
where q = (u,_~~~T~,)‘/‘_ Now let us wnsider ;1 nlore general case. when both radicals have many magnetically nuclei. In this case. j is expressed via the effective hfi constants of both radicals: f = 0.048[((r,.,r‘r,,)“2 Here J denotes
+ (u2.erg,,)“2]
the sum of two integrals
J,= / z’zcos2~sin’v[[Icos( 0
+0.30[(&,,+
J = J + + J _.
q i_ a)l]“‘dg7,
non-equivalent
U~.err)“2To]“~J(u,.~r~/u~.~~~).
(43)
where (44)
tg a = ul.crr/u2.crr-
Table 1 lists numerical values of J,. J_ and J. It is seen that J weakly depends on the ratio of the effective hfi constants. Therefore. in lieu of eq. (43) one can use the approximate relation (45) Substituting we obtain
eq.
(45)
into
“p(S)
= @,(l
“p(T)
= 3&~,f/Z.
eq. (37) and assumin g the radicals
+ KTT~ +f)/Z, Tp(T)
In deriving eqs. (37). we violated
?(S)
of a pair to be in contact
at the initial moment.
= @J/Z.
= KT~r( 1 + KS~r -I-3f )/Z_ the sequence
of averaging
(46) over all re-encounters
and all local hfi fields.
Table 2 Comparison of Ul.CffTD
Tp valuescalculated =‘Z.errrD
by eq. (47) and according to ref. 5%
[ 151 TP
IW
TP(47)
0.29
0.5
5
0.090
0.092
0.18 0.22
0.09 0.01
1 10
0.032 0.069
0.032 0.060
0.22
3.97
10
0.205
0.201
5.8
5
5
0.211
0.234
5 0.1
0.256 7.39x
10-S
0.282 7.43x
10-J
9.54x
lo-’
9.74x
10-Z
11.6 0.17
10 0.44
0.25
0.44
10
Thrrefore. the results obtained must be compared with a more consistent theory [ 151. As an esample. we consider a triplet-born RP recombination under the assumption that the reaction can occur only in thr singlet
term.
Tp(s)
Eq. (46) affords: = K&f/[
I + I&T, +f(4
+ I&T, ,] _
(47)
obtained from this formula are compared in table 2 with numerical calculations rcalizcd lvithin the consistent theory [ 151. Relation (47) is seen to fairly \vell agree Lvith the data obtained in ref. [ 151. This means that the change in the sequence of the above averagings affects negligibly the RP recombination probability and thus is quite justified. We now consider a more general case. when the RP partners are out of contact at the generation moment, Q,Z- h. The recombination probability obeys the general relations (37). They become practically applicable. however. only after calculating the integral (3s). Let us find it for a comparativcl_v simple situation when there are no radical acceptors. Substituting eq. (36) into eq. (3s) and using cq_ (5). xvc have
The data
fo=(l/lS)(J,+J,+J++J_).
(4s) exp[ -x1
JI = 1 - (4/fi)~x_x-2 I-
J+= Q),(P) where
B,
(164 =/u;;“dq{
cos( B,_~)“2ds.
=05 exp(-p’)Q,.(n)do. co+
and y+ are expressed
first encounter;
- (B,.\-~“‘1
sir+
cos(ly,lp)“’
exp[ - (lu,lp)““]}_
via the effective
radical
hfi constants
and
rhe time of approach
-
B, = akm,,,7,/2”‘.
k = 1, 2.
Table 3 Values of _r( s) calculated by numerical
yl=
integration
( “1_‘To/2’q
and by approximate
cos( v A- n).
formula
(53)
1’(S) (48)
_r(1)(53)
0.126 0.154 0.16s 0.229 0.27s
0.121 0.15s 0.193 0.237 0.2SY
0.242
0.337 0.405
0.352 0.126
0.362 0.542 O.S16 1.22 1.83
0.484 0.573 0.669 0.769 0.864
0.511 0.60-I 0.70-I 0.79Y
2.76
0.947
O.QAS
4.12 6.20 9.2s 13.9 21.0
1.01 1.04 1.05 1.03 1.o’,
0.950 0.996 0.999 0.999 1.00
31.4 47.0
1.01 1.00
1.00 1.00
s 0.0212 0.0318 0.0476 0.07 16 0.107 0.161
OS.;
befors
the
Ul.Clf/(‘Z.Cff
5,+-J-(48)
l_r(u,.zq,)
0.0707
0.15s
0.450
0.47 I
0.0707
0.324
0.436
0.47 I
0.0707
0.509
0.41s
0.47 1
0.0707
0.726
0.404
0.47 1
0.0707 0.0707
1.00 0.1%
0.395 0.450
0.47 I 0.47 I
0.353
0.1%
0.946
1.01
I.77
0.158
1xi9
1.75
t-i.84 44.2
0.1% 0.158
1.m xl0
2.00
a I.231
LOO
where u,.? = (~:,rr + o~.,.rr)““. and cy has already been defined. The integrals J&_ J,+J_ were calculated numerically (see tables 3 and 4). Analysis has shown that ./. + .I_ depends wrakly on the ratio of the effectilre hfi constants and is determined. within an accuracy of some 10%. solely by u,~T,,. We introduce the function J(S) which defines the dependence of JI. upon CJJ,.Cff%: J1, =.,.(u,-.,,,T”).
(49)
This function was determined by numerical approximate the parameter A, as follows: fo=
(1/18)[_+
,.&I,)
In the range s < OS. y(s)
+?‘(%.,rr%)
integration
+
of eq. (48) and listed in table 3. It can be used to
~r.(h~“)l-
(50)
a ~‘1’~ and
f,, = O-145( (77U)‘/2. where d is defined I0 -
(51)
by eq. (45). As &-, increases.&
aquires
the limit:
7/9-
(52)
Generally speaking. as c7;,, g rows. the parameter f0 oscillates (see table 3). however. these oscillations are slightly manifested and thus hardly deserve attention in the case of radicals with many magnetically non-equivalent nuclei. J(S ) can be approximated by the function _Y(x) = 1 - exp( -0.87s’/
- 0.53s).
With eqs. (SO) and (53) one can calculate the recombination probability the recombination radius at the initial time moment. ‘0 > h.
6. A RP with a particular
nucleus
(53) of a RP whose partners
are not at
with I = l/2
One
of the most interesting manifestations of nuclear effects in radical recombination is magnetic effects. The magnetic 13C enrichment observed at dibenzylketone photolysis [27.28] can serve here as a well-studied illustration. The decomposition occurs from the triplet state and gives: isotope
PhCH,“CO
-. - PhCH,,
RP I;
PhCH,“CO
. - - PhCH?.
RP 11.
(54)
An appreciable “C enrichment in the initial ketone has been measured csperimcntall?_. The effect is associated with a high hfi with ‘-‘C in RP II at cl‘- = 125 G [29]. The hfi constants \vith the protons in ii benzyl radical are an order of magnitude smaller than II<- [29]_ To theoretically intcrprrte thr esprrimcntal data. one must calculate the recombination probabilities of RP I and RP 11 (54). Hwvcvsr. strictI> speaking. the theory of the previous section is inapplicable to calculations of the RP I1 recombination probability_ The theory of recombination of RPs \vith many magnetically non-equivalent nuclei cn~plo~s the gaussian distribution of local hfi fields in radicals. For the radical PhCH,‘-’ CO this supposition doe3 not hold since here the hfi with H nuclei and with “C differs by several orders of magnitude ((I,, < 0.4 G [29]). As a result of this ratio of the hfi constants. the total local radical field disohrys the applicability requirement of the central limit theorem of probability theory and thus the local field distribution c;mno~ be treated as gaussian (33) in the radical considered. Hence. it is necessary to verify the possibility of applying the theory of the above section to the experimental situation of isotopic enrichment \vhcn the RI’ has a magnetic nucleus with a hfi constant greatly differin, 0 from those of the other nuclei in the pair. having man\ To this end we investigated the recombination of a model pair. lvith one radical magnetically non-equivalent nuclei and the other only one nucleus with an I = l/7 spin. We descrihc rhc spin dynamics of the latter radical in the proper quantum-mechanical way. and that of the former radical in the semiclassical approximation [ 141. As a result. the spin-hamiltoiiian takes the form ri = ho,,-& The
following
+ Ii&L
RP states
II) = IT, ,.
a>.
16) = IS. P>.
are defined:
12)= IT_ 13 P>-
13) = IT,, . m) _
17) =IT_,.
W=lT-,.P)-
cr).
14) = IS. a)_
15) = IT,,. /I>_
(‘6,
Here S and T are singlet and triplet states of electron spins. (Y and j3 arc the eigenstatss of i, conforming to the eigenvalues l/2 and - l/2. The change in the state populations (56) induced hy the hamiltonion (55) obeys
the equation
Phh(t)=CFh~.,,,,(f)~,*,l(0)-
(57)
,I
Averaging F(I) over the time distribution between re-encounters and over 1hr2 locc~l field distribution the radical with many magnetic nuclei. we obtain in the limit of continuous diffusion:
F II.33
Fu..u = f&s = F,,.ss =f,. F2z.m= 5~. = J;;;.,, = F-4 7, =J-;F33.14 = I;;%, =A. . I/2
fl= 0-145(uI.,ff7~,) . f3= -f,/Zt-f,/2 + J/24.
J,, =/oldzz‘
proportionally
.,J
=
=
Feb.::
=
i.. -
Fz2.x = 2 <:3_55= 2
_t-?=f,, - fI/2
exp( -z’)[(j_x
5 lists J,, values
increases
--.
=.i, _
+ J/24_
f,=f,,/2 +f,/3-J/24_
_& =j,/4
i J/16.
J = 1.903(aI.cfl~1,)‘i2Jo.
/o = (la17,,/2)“‘/6.
Table
F,,.,, = I=--.
=
for
for various to
ISI”’
+ zl)“’
+ (IS - :I)‘/‘].
s = L’,/z’51,.ccr_
radical hfi constant ratios. AS seen. J,, %triss ncgligibi~ at Is] 5 1. and at 1.1-l> 1. Quantitative calculations cm be made in the follo\ving
0.1
0.917
1.2
0.929
2.0
1.18
0.5 0.8
0x9 I 0.8S2
1.4 1.6
0.9so 1.04
2.5 3.0
1.35 1so
R.0
2.50
10 I5
2.8 3.43
approximation J,, = 0.9.
.I = 1.7( “,.crrT*,)“~.
(59)
when 1.~15 1. and
(mlsly/z.
J,, =
J = 1_42(pJ~7,,)?
(60)
when the hfi constant of the particular nucleus exceeds that of the magnetically non-equivalent nuclei. j_\j > 1. Substitute k kto eq. (6) and calculate the RP recombination probability. Let us consider the case when RPs can recombine only from the sin&t state. The recombination probability is represented as follows (K,=
A-)):
“p = KT,(
uj
+
u,),
Here 114and 14~are solutions /
G, 0
1;;
(
0
-A
0
-13
0
-f4
0
0
with constant
of the system
-f,
-f../2
-/4/z --A 0
+AI~~~(II~+z+,)]_
of algebraic 0
-f,
-f, G, -_fz G, --A -A
0
-(I
1P=(~7r/3)[~
-f2 -f G.3 -f4/2 -f,/2 -f3 0
(61)
equations 0
0
0
\
-Lx --J-i 0 --A -f,/Z --f/2 -f3 0 -f,/2 -f,/l -h 0 G, -fs -f2 -f, -f5 G4 -fz -f, G, 0 -f, -f2 0 -f, -f, G, I
I
0' 0 0 1 0
-
(62)
0 0 i
01
coefficients
Gz=l +2f2+2f-x+f,. G4 = G, + Kr,. G,= 1+f,+fi+f~+h+L-
G,=
1 i-21,.
In principle. eqs. (62) can be solved analytically_ Unfortunately, the formulas are too cumbersome. We solved eqs. (62) numerically and calculated the RP recombination probability (61). table 6 listing some results. Though eqs. (62) can be readily solved by a computer for any particular system. the absence of a simple analytic solution may be an obstacle to a routine application of the present results to an interpretation of experimental data on magnetic isotopic enrichment. In this connection it would be interesting to see whether the theory of the above section can be applied to the analysis of magnetic isotopic effects in a RP with a particular nucleus, as is the case in experiments on magnetic isotope enrichment.
Table 6
Values of'p ohtaincd from at?;. (61) and(62)and eq.(47)(thrsecond column ~~hulat~sculuulati~~~~?; hy q(h3). q.
(4s)
at a2.cT( = u/2).
RP pxametcrs:
a = 2.2 X 10’ rad/s.
u,.<~~ = ‘.2x
to” r;ld/s.
K I\‘, _-
‘p (61.62)
‘I, (47.63)
‘p (45.47)
2.13x 2.09x
2.15x to-’ 2.11 x lo->
2.07X 2.03x
IO-” to-”
0.5 0.5
IO” 10’
‘.5X10-”
I.&OX 10-z
1.82X
1.74x
lo-’
0.5
10”’
0.25
7.46x10-' 0.109 0.114
7.54x10-' 0.110 0.115
7.20x10-~ 0.105 0.110
0.5 0.5 0.5
IO" 10'2 10"
3.76x10-" 3.36x10-' 0.162
3.85 x lo-”
3.62 x10-'
3.44x10-'
3.22x10-'
5.0 5.0
10" 10"
0.262
0.167 0.273
0.153 0.245
5.0 5.0
10”’ IO”
0.279 0.281
0.29 I 0.293
0.26 1 0.263
5.0 5.0
10’: IO”
4.70x 0.263
1o-J lo->
lo-’
0.487
4.80X 0.273 0.512
lo-’
lo-:
4.67X 0.260 0.4so
IO-’
IO%,, (St
rhe third one those hy
7r = s,,,“O A-,(s-‘)
50 50
10” 10’
50
IO’O
0.533
0.562
0.525
50
10”
0.538 0.538
0.567 0.568
0.530 0.530
50 50
10’z to’J
2.5 X 10. J
2.5 25 250 ~.jXlW~ 0.25 2.5 ‘5 250 2.5.x to’ 0.15 7.5 25 250 2.5 f 10’ 2.5 :i loa
According to the theory given in the previous sectron. the hfi effect in the recombination is determined by the effective hfi constants of the radicals (see eq. (45)). By the definition (1). LI,{~= (I/) at I = I,.?. When applying eqs. (45) and (47) to our case of a RP with a particular nuc1eus. the question arises xvhsthcr we should substitute the hfi constant CIof the radical with the particular nucleus for CJ~~,~,. in sq. (15). or set ~z.err = a/2 in line with eq. (1). To answer this question. \ve used both ways (see table 6). The data of table 6 demonstrate that calculations by eq. (47) are in fair agreement with those b_v sqs. (61) and (62). If akff = a/2 in eq. (47). Tp becomes underestimated (by some 105). If in eq. (17) a,.,ii = t~._f-= O.l-Z(Lir,, 1’ ‘_
eq. (47) overestimates the recombination probability approximately by lo!%_ Thus. the theory of the above section. based on a semi-classical approach to RP spin dynamics. affords satisfactory values of the recombination probability for radicals with magnetically non-cquivslsnt nuclei even if the hfi constant of the particular nucleus differs greatly from those of the other nuclei. 7. Discussion Above we found the recombination probabilities for pairs of uncharged radicals \vith magnetically equivalent and non-equivalent nuclei. The recombination probability is determined by the multiplicity of the RP precursor and by that of the state from which the RP recombines. The p dependence on hfi. viscosity and radical reactivity is determined by three dimensionless parameters: lal~,-, (or cir,). KT,. and la1~c (or circ). The residence times of a RP in a “cage” (ro). in a reac:ion zone (TV). and the time of the first approach (me) all are proportional to the medium viscosity (q)_ Hence. the results obtained alloa analysis of the influence of viscosity and reactivity upon magnetic effects in RP recombination. A typical reaction showing magnetic isotopic effects is molecular decomposition from a triplet excited state under photolysis or radiolysis. We therefore analyze the case when a RP is triplet-born and the
reaction can occur only in the singlet state. For all systems considered (equivalent and non-equivalent magnetic nuclei) the theory predicts the same character of p dependence upon the medium viscosity and the and tends to the limit radical reactivity. As the latter increases (I\’ s grows). ‘rp(S) rises monotonically
-‘P(S)~(h/r,,)(y+4,,)/3(1 for :I one-nuclear -‘P(S)
(64)
+4)
RP (see eq. (27)).
= (~~/%)(/+I;,)/(1
and to the limit (65)
+f)
for a RP with many magnetically non-equivalent nuclei.(see eq. (37)). In the case of a RP with magnetically equivalent nuclei this limit is always smaller than l/3; for radicals with non-equivalent nuclei it can reach values as high as 1. A detailed discussion of the p upper-limit problem can be found in ref. [24]. The probability “b(S) increases with the medium viscosity. and rqs. (64) and (65) afford: ‘p(S) respectively.
= h/3r;.
.‘I,(S)
It is necessary
3 h/l;,.
(66)
to note that these extreme
~alurs
are directly
connected
with the conservation
of the total electron
and nuclear spin for the case when the isotropic hfi is the predonkmnt mechanism of singlet-triplet mixing. This assumption is quite valid in zero field provided the medium viscosity is not very high. In very viscous media with 9 L 10 P the in-cage RP lifetime becomes so long (TV, 2 IO-” s) that S-T transitions occur in all the RPs due to paramagnetic relaxation and thus ‘/, tends to unity. The quantum yield of molecular photodecomposition is expressed via the geminate recombination probability as: O=y(l
(67)
-PI.
where y is the probability of generating a RP from an excited molecule. If the quantum yield before and after an isotopic substitution is 0 and +*. respectively. then the ratio +*/+ determines the magnetic isotopic effect. The recombination probability of geminate pairs depends on the initial interpartner distance. If it is sufficiently long. the RP spin states become disordered before the first encounter at the recombination radius. and the singlet and triplet states get populated according to their statistical weights. Hence. if the time interval before the first encounter. 7”. is longer than the radical paramagnetic relaxation times.
where “p the recombination it is possible to find ‘Y(S)
= I\‘&[(
‘p(T)
= K-&3(
probability
1 + KTTr)/4 1 + K,T,)/~
of diffusion
(encountered
randomly)
pairs of radicals_ With eq. (37)
+f]/Z. i- 3f]/Z_
(68)
Let us analyze now the case when the partners are in contact at the initial time moment. For a molecule with many magnetically non-equivalent nuclei which decomposes from the triplet excited state we have, using eq. (47).
(6% As K, grows. T+ reduces from y to y/( 1 +/)_ of low viscosities (fe 1, KT~ -=E1)
As the viscosity
increases.
I-& falls from y to zero. In the range
(70)
while in highly viscous .‘& a y/fa
media
yC27j- ‘I’.
(71)
\vhere C, and C, arc viscous-independent
coefficients.
in our cast.
‘i+*/‘(s = A/B.
(72)
.1=(l+Als~,+4f*~[1+KsT,+/~4+~‘,~,~]. S=(1+Als7~+4f)[1+AIS7,+f*(4+~S~,)]. As KS increases.
‘i$*/-‘+ = When
the magnetic (1 +I)/(
the viscosity (“+“/‘i+),
isotopic
and reaches
an rstrsmr
vaiuc
1 i-f*)_
(733
rises. the magnetic =
effect reduces
isotopic
effect tends
to the limit
(d/&f’.
(74)
where Csand
fi* are the effective RP hfi constants before and after the isotopic substitution. rs~prctivel~. under a complete deuteration (all protons are substituted by deutsrium ittums) 11lr effecti;.e hfi constants reduce by fourfold_ It means that under deuteration (‘&*/ I+), = 7. To interprete data on magnetic isotopic enrichment. one introduces one additional p:tr:tmctcr [Xl: For instance.
a=(O-+*)/0=(p*-~))/(l In our case of a triplet-born give
(75)
-P).
RP. eqs. (45) and (47) for RP I and cqs. (47) and (63) for RP II (SW cq. (54))
7n increases monotonically with K, and the \*iscosity. -In - 1 - (2/d’)” ‘_ To compare the present theory with experiment quantitatively. one must have data on 111s \xscosirv dependence of 9. +*/+. or CL The RP recombination rate constant is a \ariablr in the thsory and thus th> latter can always be fitted to experiment by a proper choice of K provided the experimental data are taken at a definite viscosity of the medium. The magnetic isotopic enrichment seems IO he most adequatelv studied for dibenzylketone photodecomposition [_____L 77 7’ 77.X.30]. III benzene at 9 = 0.6 cP tx = 0.03 [30]. Ii1 a glycerine-alcohol mixture with 11 = 540 CP a grows up to 0.24 [El_ Dibenz_vlkstons photol_vsis in cyclohexanol (17 = 60 cP) gives a = 0.09 [23]. There are also data on dibsnzvlktons photol\-sis in micsllss [28]. In these systems the microviscosity is 17 = 30-40 cP. The measured values of n in various micrllar systems are 0.26-0.30. In micelles, radical pairs diffuse in a restricted volume and the in-cage RP must be longer than in the case of partners freely diffusin, 0 in a solution. Therefcwr. the data for are compatible with those for highly viscous glycerine-alcohol solutions. The experimental hfi constants for RP I and RP II obtained at dibsnzylkstone photol_vsis [29] used to determine effective hfi constants. ii. in RP I (by eq_ (45)) and RP I1 in\-olving ‘:C isotopes
lifetime micellrs can be (bv eq_
(63)): r7 = 0.22 x 10’ rad/s. The RP lifetimes ,rI, = b’/D,
rl* = 2.70 x 10’ rad/s.
in a cage and in a reaction
7r = orb/D.
z@ne are calculated
(77) from sqs;. (23) and (JO):
(7s)
Table 7 RP I and RP II parameters (see cq. (54))
On determining
the mutual
4 (cP)
T,, (ns)
f(45) RP I
/’ (63) RP II
0.6 60 540
0.15 15 132
0.026 0.26 0.78
0.092 0.92 2.73
diffusion
coefficient
by the Stokes
formula.
D = 2kT/3~inh. we obtain TV, = (3&‘/2kT)t~.
TI) = 024Sq
r,.= (3m,b’/XT)q_
ns.
-rr= 0.0124
ns.
(SO)
In eqs. (SO)viscosity must be in centipoise units. Substituting obtain the parameters f and f* necessary for the comparison
eqs. (77) and (SO) into eqs. (45) and (63). we of theory and experiment (see table 7). In a
glycerine-alcohol mixture ho is sufficiently high: the relaxation processes can reduce the isotopic enrichment. Therefore. the above theory might overestimate the magnetic isotope enrichment coefficients since it neglects slow relaxation processes. At given /and /* values a maximum isotopic enrichment is achieved for radicals with a high reactivity_ at KS7r B 1. The maximum value of Ta is (see eq. (73))
‘-a,,, = (f*-f Ml
+f*)-
(81)
Table 8 lists LY,,, estimated by this formula. A comparison with experiment demonstrates the extreme ratio (Sl) to give a values which exceed experimental data by a factor of 1.5-3. According to eq. (79). in cyclohexanol TVis 100-fold higher than in benzene and one-tenth of that in a glycerine-alcohol mixture. Taking this into account. we tried to find the K, at which the calculated Tcu coincides with CY According to eq. (SO), KATE = 1.2 X IO-” K,o. In benzene Ksrr = 0.7 X IO-” A’,. If . K, = 2.0 x 10” s-“F’ . 1-e. Kgr = 1.5, then eq. (75) gives Ta = 0.032 which is an attractive fit to experiment_ However. at this very K, for the other solvents studied, Ks~r Z+ 1, Ta ~~a,,,,, so that the calculated (Y exceeds the experimental one (cf. a, with as,,, in table 8). If we set K, = 3 X lo9 s- ‘, Ks7, = 1 and 10 in cyclohexanol and a glycerine-alcohol mixture, respectively, the theoretical a is coinciding with experiment_ For benzene. however. the theoretical a is much less than the experimental one (cf. Ta, and CYST,,in table 8).
T;tbls 8 Coctfficirms of magnetic “C isotope enrichment in dibenqlketone Solvent
(Kg,
values used IO calculate Ta (76) are shown in parmthsses)
=c.p 0.03 1191 - .
T%-
Ta*
Tag
Ta3
brnzsns
0.060
cyclohexanol
0.09 1301
0.34
glycsrine + alcohol
0.24 [26]
0.52
4.3 x 10-j (0.0 1) 0.066 (1) 0.31 (10)
0.032 (1.5) 0.34 (150) 0.52 (1500)
0.032 (1.5) 0.092 (1.5) 0.24 (6)
Within the theory in question, experiment can be quantitatively described only undrr the supposition that either 7,) and/or 7r should be calculated in some other way than eq. (80). or K, may depend on the solvent. in particular. on the solvent viscosity. Table 8 tabulates calculations with varyin, 0 Ksrr_ As seen. K,r, can be chosen so that the calculated a is in good agreement with experiment. To this end. holvcver. OIW has to assume that either the RP recombination constant at the moment of an encounter reduces in more viscous media. or -rr depends weakly on the medium viscosity. The assumption of a weak dependence of KsTr on the medium viscosity is undoubtedly a11 extremely promising way leading to important consequences. Providin, Q K,T, changes negligibly in various solvents or at various viscosities. the viscosity dependence of the rate constant K, in the kinetic limit lvould resemble the diffusion-controlled case, however. the value of K, would be lower than that in the diffusion limit: K,aDa
l/n.
K, < 4z;hD.
(S2)
However. the available experimental data are too meagre to give an adequate description of the Ki,i, dependence on the viscosity and properties of the solvent_ Further investigations are required. Indeed. only recently Turro and Kraeutler 1311 have put forward arguments that the exprrimentally observed Q = 0.03 for benzene cannot unambiguously be associated with a magnetic isotope effect. The fact is that 311 isotope mass effect can also result in a similar (Y value. If we ignore the data for benzene. the present rhror_v. as seen from table 8 (see T&). fairly well describes experimental results. Thus. the theory discussed needs also further development. First of all. paramagnetic relaxation processes must be taken into account \vhen analyzing
recombination
in viscous
media.
molecule may give pairs with different inevitably change the magnetic isotope
Furthermore_
kvhen decomposin,
a in different solvents. the same allo\ved for. the above factors lvill
interpartner separations. When effect predicted theoretically.
Acknowledgement The author is grateful to Z.V. Shapochanskaga and X.K_ -Salikhova for calculations, and to Dr. K.A. McLauchlan for dralving his attention to ref. [31].
assistance
in
numerical
References 111 A.L.
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