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Calculation of CIDNP field dependences in biradicals in the photolysis of large-ring cycloalkanones
Chemical Physics 252 Ž2000. 83–95 www.elsevier.nlrlocaterchemphys
Calculation of CIDNP field dependences in biradicals in the photolysis of large-ring cycloalkanones A.V. Popov a , P.A. Purtov b
b,c,)
, A.V. Yurkovskaya
d
a KemeroÕo Technological Institute, 650067, KemeroÕo 67, Russian Federation Institute of Chemical Kinetics and Combustion, 630090, NoÕosibirsk 90, Russian Federation c NoÕosibirsk State UniÕersity, 630090, NoÕosibirsk 90, Russian Federation d International Tomography Center, 630090, NoÕosibirsk 90, Russian Federation
Received 11 March 1999; in final form 3 September 1999
Abstract Calculations of CIDNP effects in recombination of acyl–alkyl biradicals arising in the photolysis of cycloalkanones have been performed. Singlet–triplet transitions occurring in the terms intersection zone and essential for the effect are taken into account in the balance approximation. The calculated CIDNP field dependences have been compared with experimental data obtained for biradicals with different lifetimes. Good agreement has been achieved. Interpretation of CIDNP field dependences is given. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Over the last two decades spin polarization effects arising in the course of singlet–triplet evolution and spin-selective recombination of radical pairs ŽRP. have been the subject of extensive investigation Žfor reviews see Refs. w1,2x.. It opens up the possibility of obtaining unique information about the kinetics and mechanism of radical reactions. Lately, particular attention has been given to the studies of magneto-spin effects in RP with restricted molecular volume Žflexible biradicals, RP in micelles, plastic crystals. w3–10x. Exactly in the latter spin polarization effects and external magnetic field influence on the course of chemical reaction are particularly pronounced. In a certain approximation such RP may be treated as model ones for complicated biological substances. The motion of radical centers in restricted volume provides rather long lifetime for them and effective singlet–triplet evolution determined both by hyperfine interaction of electrons with magnetic nuclei and by the difference in electron Zeeman frequency between radicals. Not moving far apart, the partners reside in the exchange interaction zone for a long time. The value of the exchange interaction depends both on the
)
Corresponding author.
0301-0104r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 9 . 0 0 2 9 3 - 1
84
A.V. PopoÕ et al.r Chemical Physics 252 (2000) 83–95
distance between radical centers and on their relative position, and is modulated by molecular mobility of radicals. The exchange interaction splits singlet and triplet RP terms, while application of external magnetic field removes degeneration of triplet terms ŽTq, T0 and Ty .. The exchange interaction in long-lived RP produces particularly strong effect on spin dynamics in such magnetic fields where its value turns equal to the Zeeman energy, and two terms, S and Ty, become resonant. In this case the Ty–S transition where the flip-flop of electron and nuclear spins takes place becomes the main channel of spin conversion. This gives rise to integral electron and nuclear polarization in RP and reaction products. However, the increase in the distance between partners or the change in their relative orientation leads to a very rapid decrease in the exchange interaction. Thus the residence time of radicals in S and Ty terms intersection zone may be very short as compared to the time spent in space regions where the exchange interaction is almost equal to zero. At zero exchange interaction effective S–T0 transitions take place in RP the frequency of which depends on nuclear spin configuration m4 . No spin reversals occur in such transitions, and nuclear polarization in reaction products is formed owing to the competition between the transition into singlet reactive state and other processes, for example, the escape from the cage or the acceptance reaction. For this reason CIDNP S–T0 mechanism efficiency depends essentially on geminate radical pair lifetime, and is maximum when lifetimes are comparable to singlet–triplet evolution lifetime. At long RP lifetimes S–T0-transitions are considerably averaged and make no contribution into CIDNP effect. In weak magnetic fields S–T " transitions determined by hfi nonsecular part play an important role w1,11–13x. CIDNP field dependence is determined by variations in energy spectrum of RP spins. For example, polarization in S–T " transitions is brought about by the difference between energy barriers for electron spin flip-flops of radicals with different nuclear spin orientations. Polarization in S–T0 transitions results from the difference in dephasing rate Ždistinction between precession frequency differences. between electron spins of radicals with different nuclear spin orientations. Of course, in weak magnetic fields transitions of both types affect one another, however, such a separation is justified in the case of qualitative interpretation of CIDNP field dependence behaviour. If RP contains several magnetic nuclei, then CIDNP field dependence usually changes sign. Sometimes, the sign is changed several times. Recently we have experimentally studied CIDNP field dependences in acyl–alkyl biradicals produced in the photolysis of cycloalkanones Žcyclodecanone and cycloundecanone. in the presence of effective acceptor of radicals w14x. In these biradicals radical centers are connected by a flexible chain consisting of 11 and 10 methylene links. As is seen from experiments conducted by CIDNP method with time resolution w15x, the main decay channel of biradicals investigated is their recombination resulting in the production of initial ketone molecule. Comparison of CIDNP field dependences for different protons in the initial ketone shows that reduction of biradical lifetime due to the addition of acceptors does lead to noticeable decrease in polarization contribution by Ty–S channel of intercombination transitions occurring in the terms intersection zone. Paper w14x presents a solely qualitative interpretation of the results obtained, and the attempts to calculate CIDNP field dependences by single-nucleus model of de Kanter and co-authors proposed in papers w5,16x do not reproduce the regularities for CIDNP field dependences in biradicals discovered experimentally. To calculate magneto-spin effects, we have developed a kinematic approach w17–20x. The approach is based on the formalism of the Green functions specifying the character of a relative motion of radical centers in a pair. Singlet–triplet transitions taking place in the terms intersection zone may be taken into account in different ways. The simplest way is a balance approximation wherein the S–Ty transition matrix element and the shape of the terms intersection zone are given w19x. This paper presents consistent theoretical treatment of singlet–triplet evolution in radical pairs with restricted molecular volume in arbitrary magnetic fields based on the Green’s function formalism. Calculations of chemically induced nuclear polarization effects depending on external magnetic field induction for biradicals with different lifetimes have been compared with experimental findings given in w14x. The paper seeks to verify the adequacy of the theoretical model proposed, and to obtain some quantitative information about molecularkinetic and magneto-resonance parameters of biradicals.
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2. Theory The CIDNP effect of the k-th nuclear spin is defined by the mean value of its operator Iˆk z in recombination product w1x ² Iˆk z : s Tr1,2 Iˆk z wˆ Ž t . ' Ý m k wm Ž t . ,
ž
Ž 1.
/
m4
where Tr1,2 is the trace operation over all spin states of RP, index m defines nuclear spin configuration in spin states of the pair. The Laplace transform of the desired quantity ˆ˜Ž s . has been calculated in w18x, and is of the form s P w˜ˆ Ž s . s Kˆ r gˆ Ž s . 1 q Kˆ r gˆ Ž s .
y1
r0 ,
Ž 2.
where Kˆ r is the constant matrix of recombination from one or another spin state, r 0 is coordinate-independent initial matrix of RP Uˆ Ž q . s Kˆ r cu Ž q . , r 0 Ž q . s cu Ž q . w Ž q . .
Ž 3.
The function cuŽ q . specifies the reaction zone shape Žand the exchange interaction zone shape., and wŽ q . is the stationary normalized distribution in pairs of radical centers. It is expressed in terms of wŽ q,qX ,t . – the conditional probability density of realizing the coordinates q at the instant of time t if they were qX at the initial moment, and the configuration space volume accessible for relative motion of reactants V s Hd q
w Ž q . s V lim w Ž q,qX ;t . . t
™`
Ž 4.
The Green function wŽ q,qX ,t . satisfies the equation
E Et
y Lˆ Ž q . P w Ž q,qX ;t . s d Ž q y qX . d Ž t . , X
Ž 5.
X
w Ž q,q ;0 . s d Ž q y q . . The matrix gˆ Ž s . appearing in Eq. Ž2. defines spin and molecular dynamics, and calculated in the balance approximation is of the form w19,20x gˆ Ž s . s Lˆ Ž s . 1 q iKˆ J Lˆ Ž s .
y1
,
Lˆ Ž s . s gˆ 0 Ž s . y gˆ 0 u c Ž s . 1 q Kˆ c gˆ 0 cc Ž s .
y1
Ž 6. Kˆ c gˆ 0 u c Ž s . .
Here Kˆ J is the exchange interaction constant matrix JˆŽ q . s Kˆ J c J Ž q . , Kˆ J s JˆŽ q . P w Ž q . d q .
H
Ž 7.
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86
For brevity we use the designations gˆ 0 Ž s . s ² gˆ 0 u Ž q, s . :u '
HHc Ž q . Gˆ Ž q ,q ; s . c Ž q . w Ž q . d qd q , X
gˆ 0 u c Ž s . s ² gˆ 0 u Ž q, s . :c ' gˆ 0 cc Ž s . s ² gˆ 0 c Ž q, s . :c '
X
u
X
0
u
HHc Ž q . Gˆ Ž q ,q ; s . c Ž q . w Ž q . d qd q , X
u
X
0
X
c
Ž 8.
HHc Ž q . Gˆ Ž q ,q ; s . c Ž q . w Ž q . d qd q . X
c
X
0
X
c
The function cc Ž q . defines the shape of the zone of level crossing. For simplicity it is assumed to be the same for all terms. ² . . . :u is the average over the reaction zone defined as a scalar product in the Hilbert space; ² . . . :c means the average over the terms intersection zone. The matrix Gˆ 0 Ž q,qX ; s . describes RP spin evolution outside the exchange interaction zone, and is the Laplace transform of the matrix wˆ 0 Ž q,qX ;t .. This matrix satisfies the equation
E Et
y Lˆ Ž q . y i Vˆ P wˆ 0 Ž q,qX ;t . s d Ž q y qX . d Ž t . , X
Ž 9.
X
wˆ 0 Ž q,q ;0 . s d Ž q y q . , where Vˆ is the coordinate-independent component of the Liouvillian Lˆ Ž q .. The shape of the terms intersection zone is the characteristic of the interaction. It may be defined by the relations w21x X
X
HG Ž q ,q ; s . dq ' Cc Ž q . , c
Ž 10 . Cs
X
X
HHG Ž q ,q ; s . w Ž q . d q d q ,
where GŽ q,qX ; s . is the Laplace transform of the function w Ž q,qX ;t .. This function satisfies the equation
E Et
y Lˆ Ž q . y i Ž J Ž q . y E . P w Ž q,qX ;t . s d Ž q y qX . d Ž t . , X
Ž 11 .
X
w Ž q,q ;0 . s d Ž q y q . , where J Ž q . is the exchange integral, E is the splitting of the intersecting levels outside the exchange interaction zone. Finally, Kˆ c involved in Eq. Ž6. is the constant matrix of singlet–triplet transitions in the terms intersection zone. If these transitions are determined by hfi, then matrix elements of Kˆ c are proportional to hfi constant squares. Thus calculation of the CIDNP effect is reduced to the calculation of the pair spin dynamics outside the exchange interaction zone Žin the intervals between recontacts., determination Žor prescription. of the relative motion kinematics, and calculation Žor prescription. of the exchange integral as a function of a relative position of radical centers. Now we shall apply the kinematic approach to the calculation of CIDNP field dependence of acyl–alkyl biradicals. Biradicals came into existence in the triplet state in the photolysis of cycloalkanones in the presence of the effective acceptor of radicals ŽCBrCl 3 .. In biradicals under study only a-CH 2 and b-CH 2 protons of alkyl ring have essential hfi constants Ž A a s –2.2 and Ab s 2.8 mT for structurally similar monoradicals w22x..
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However, even in the case where only four nuclei are taken into account the problem is still rather complicated, since the experiment has been conducted in weak Žand moderate. magnetic fields, while in practice spin evolution involves a large number of states. Represent the coordinate-independent spin-Hamiltonian of the system under study consisting of two pairs of equivalent magnetic nuclei as
ž ™ˆ ™ˆ / ™ˆ
ž ™ˆ ™ˆ / ™ˆ
Hˆ s v 1 Sˆ1 z q v 2 Sˆ2 z q Aa I1 q I2 S1 q Ab I3 q I4 S1 ,
Ž 12 .
where v 1Ž2. is the precession frequency of electron spin 1Ž2. in constant magnetic field B directed along the z axis. Matrix elements of the Liouvillianˆ and Hamiltonian Hˆ are related as
Ž Vˆ . i f , k l s di k Hˆl j y d jl Hˆi k .
Ž 13 .
™ˆ ™ˆ During evolution the values of the following operators remain unchanged: Ž I1 q I2 . 2 s La Ž La q 1. and ™ ™ ˆ ˆ Ž I q I . 2 s L Ž L q 1.. Besides, the projection S of the total spin of electrons and nuclei on the direction of 3
4
b
b
external magnetic field is preserved. Obviously, the quantity La Žjust as Lb . assumes values 0 and 1, i.e., defines the multiplicity of the state of a Ž b . nuclei. Spin wave functions will be denoted as follows
Ž 14 .
where C s S, T0 , Tq, Ty are the states of electron spin pair, Ma Ž Mb . is the total projection of a Ž b . nuclei spins on the z axis. Indices S , La and Lb will be omitted where possible. Let us write out all subensembles necessary for the calculation of a-nuclei polarization. Here La / 0, and for different values of S , Lb , we have Lb s 1, S s 2: < S;1,1: , < T0 ;1,1: , < Tq ;1,0: , < Tq ;0,1: ; Lb s 1, S s y2: < S;y 1,y 1: , < T0 ;y 1,y 1: , < Ty ;y 1,0: , < Ty ;0,y 1: ; Lb s 1, S s 1: < S;1,0: , < S;0,1: , < T0 ;1,0: , < T0 ;0,1: , < Ty ;1,1: , < Tq ;0,0: , < Tq ;1,y 1: , < Tq ;y 1,1: ; Lbs1, Ss y 1: < S;y 1,0: , < S;0,y 1: , < T0 ;y 1,0: , < T0 ;0,y 1: , < Ty ;0,0: , < Tq ;y 1,y 1: , < Ty ;1,y 1: , < Ty ;y 1,1: ; Lbs1, Ss0: < S;0,0: , < S;1,y 1: , < S;y 1,1: , < T0 ;0,0: , < T0 ,1,y 1: , < T0 ;y 1,1: , < Tq ;0,y 1: , < Tq ;y 1,0: , < Ty ;1,0: , < Ty ;0,1: ; Lb s 0, S s 1: < S;1,0: , < T0 ;1,0: , < Tq ;0,0: ; Lb s 0, S s y1: < S;y 1,0: , < T0 ;y 1,0: , < Ty ;0,0: ; Lb s 0, S s 0: < S;0,0: , < T0 ;0,0: , < Tq ;y 1,0: , < Ty ;1,0: .
Ž 15 .
For example, consider the subensemble with Lb s 0, S s y1. In the Liouville space, the basis states of this subensemble may be written as <1: s < S;y 1,0;S;y 1,0: , <2: s < S;y 1,0;T0 ;y 1,0: , <3: s < T0 ;y 1,0;S;y 1,0: , <4: s < T0 ;y 1,0;T0 ;y 1,0: , <5: s < S;y 1,0;Ty ;0,0: , <6: s < Ty ;0,0;S;y 1,0: , <7: s < Ty ;0,0;Ty ;0,0: , <8: s < T0 ;y 1,0;Ty ;0,0: , <9: s < Ty ;0,0;T0 ;y 1,0: .
Ž 16 .
Most commonly, when the reactants get into the reaction zone, a strong exchange interaction is switched on that changes the phase between singlet and triplet states. In this case the phase change due to the reaction may
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be ignored. Besides, recombination proceeds from the singlet state, therefore, only one matrix element in recombination constant matrix differs from zero. Denoting it by K we obtain for recombination probability of such a subensemble from Eq. Ž2. s P wˆ˜ Ž s . s
Kgˆ Ž s . 1,1 1 q Kgˆ Ž s . 1,1
.
Ž 17 .
The CIDNP effect is easily obtained from the above expression. Thus the desired result is expressed in terms of the only element of the matrix gˆ Ž s . defined by Eq. Ž6.. In basis Ž16. the Liouvillian Kˆ J has six elements different from zero
ž Kˆ /
J 2,2 s
ž Kˆ /
J 5,5 s
ž Kˆ /
J 8,8 s y
ž Kˆ /
J 3,3 s y
ž Kˆ /
J 6,6 s y
ž Kˆ /
J 9,9 s K J
.
Ž 18 .
As for S–Ty transitions, in this subensemble four matrix elements in the matrix Kˆ c differ from zero
ž Kˆ /
c 1,1 s
ž Kˆ /
c 7 ,7 s y
ž Kˆ /
c 1,7 s y
ž Kˆ /
c 7,1 s K c
.
Ž 19 .
The matrix elements of the operator Gˆ 0 are calculated by the formula
ž Gˆ /
0 i k ,l m s
Ý ² i < cp :² cp < l :² m < cq :² cq < k : G Ž q,qX ; s q i Ž ´ p y ´ q . . ,
Ž 20 .
p, q
where < cp : are the eigenfunctions, and ´ p – the eigenvalues of spin-Hamiltonian Ž12.. The function GŽ q,qX ; s . is the Laplace transform of the function wŽ q,qX ;t .. Consider another subensemble with Lb s 1, S s 1. With this aim we introduce the following designations in the Liouville space for this subensemble: <1: s
s P w˜ˆ 2 Ž s . s K
gˆ Ž s . 1,1 q gˆ Ž s . 1,2 q K Ž gˆ Ž s . 1,1 gˆ Ž s . 2,2 y gˆ Ž s . 1,2 gˆ Ž s . 2,1 . 1 q K Ž gˆ Ž s . 1,1 q gˆ Ž s . 2,2 . q K 2 Ž gˆ Ž s . 1,1 gˆ Ž s . 2,2 y gˆ Ž s . 1,2 gˆ Ž s . 2,1 . gˆ Ž s . 2,2 q gˆ Ž s . 2,1 q K Ž gˆ Ž s . 1,1 gˆ Ž s . 2,2 y gˆ Ž s . 1,2 gˆ Ž s . 2,1 . 1 q K Ž gˆ Ž s . 1,1 q gˆ Ž s . 2,2 . q K 2 Ž gˆ Ž s . 1,1 gˆ Ž s . 2,2 y gˆ Ž s . 1,2 gˆ Ž s . 2,1 .
,
.
Ž 21 .
The matrix elements gˆ Ž s .1,2 and gˆ Ž s . 2,1 define the efficiency of transitions from state <1: into state <2:. For example, gˆ Ž0.1,2 has the meaning of the average residence time of RPs in a unit of the reaction zone size in state <1: provided that at the initial instant of time RPs were in the reaction zone in state <2: w18x. Since such transitions are accompanied by the flip-flops of both the first and the second nuclear spins, their efficiency is almost always low and will not be taken into account. Thus RP subensemble with two singlet states may be transformed into two subensembles each involving only one singlet state
Ž 22 .
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Then Eq. Ž17. that is more simple than Eq. Ž21. can be employed to find recombination probabilities. Transforming all subensembles Eq. Ž15. in a similar way, one can find the recombination probabilities in the case of both singlet and triplet RP precursors for each of the subensembles. Recall that RP recombination probability for singlet and triplet precursors are related by w1x TW s
1 3
Ž Ktp ys w Ž1 q Ktp . . ,
Ž 23 .
where tp s Õg Ž0. s Õg ² g uŽ q,0.:u ' ÕHHcuŽ qX .GŽ qX ,q;0. cuŽ q .wŽ q . dqdqX – is the full residence time of radicals in the reaction zone; Õ s Ž² cuŽ q .:u .y1 ' Ž Hcu Ž q . cuŽ q .wŽ q . dq .y1 is the effective volume of the reaction zone. Note that in the experiment under discussion biradicals were generated in the triplet state. Obviously, at the given magneto-resonance parameters in spin-Hamiltonian Ž12. the CIDNP effect also depends on the character of the relative motion of RP, i.e., on particular form of the functions wŽ q ., gˆ 0 Ž s ., gˆ 0 u c Ž s . and gˆ 0 cc Ž s .. As is shown in w19,20x, in the case of complete mixing in the configuration space we have the following equality for long-lived systems gˆ 0 Ž s . s gˆ 0 u c Ž s . s gˆ 0 cc Ž s . .
Ž 24 .
Eq. Ž24. is valid at sy1 ; t 4 t L , where t L is a characteristic time of the divergence of radical ends of the biradicals Žfor example, for diffusion t L ; L2rD, L is the length of biradical, D is the coefficient of relative diffusion.. However, taking into account that the reaction zone is close in space to the terms intersection zone we put gˆ 0 Ž s . f gˆ 0 u c Ž s . f gˆ 0 cc Ž s .. This simplifying assumption brings Eq. Ž6. into the form gˆ Ž s . s gˆ 0 Ž s . 1 q Kˆ c q iKˆ J gˆ 0 Ž s .
ž ž
/
y1
/
.
Ž 25 .
Note that the elements of the matrix gˆ 0 Ž s ., and, therefore, of the matrix gˆ Ž s . are expressed in terms of the combination of functions g Ž s . s ² g uŽ q, s .:u ' HHcuŽ qX .GŽ qX ,q; s . cu Ž q .wŽ q .d qd qX .
3. Computer simulation, comparison with experiment, and discussion of results To perform particular calculations, we have chosen the simplest two-positional model of RP w23x. In the framework of this model two possible states are introduced. The first state is the immediate contact Žthe pair is in the reaction zone.. The escape from this state is described as a monomolecular decay with the characteristic time t 1. The second state is the cage. This state is defined by two parameters: the time t 2 it takes the pair to return from the cage into the reaction zone, and the irreversible decay time tc . In this model g Ž s . is of the form g Ž s. s
t 1 Ž t 2 Ž s q 1rtc . q 1 .
Ž s q 1rtc . Ž t 1 q t 2 Ž t 1 Ž s q 1rtc . q 1 . . Õ
.
Ž 26 .
When doing particular calculations, we have taken the parameters t 1 and t 2 equal to 0.67 P 10y1 0 and 0.67 P 10y8 s, respectively. The two positional exponential models represent the processes in a cage qualitatively adequately, thus we believe it suitable for the description of CIDNP effects in biradicals. Of course, different, more realistic models of RP motion may be employed, for example, the well-known model of diffusion motion in restricted molecular volume w24x. In the framework of this model two spherical particles are considered that are linked by a flexible
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chain which prevents them from moving apart. The particles execute diffusion motion in such a way that their centers cannot be separated by the distance shorter than R s R A q R B and longer than L Ž R A and R B are the radii of spherical particles.. For this case the function g Ž s . is of the form w25x g Ž s. s
kR s
k L y ey2 b Ž LyR.
1
4p DR Ž 1 q b R . k L y k R ey2 b Ž LyR. ,
1ybR 1qbR
, kL s
1ybL 1qbL
, bs
(
s q 1rtc D
,
Ž 27 .
Žhere D is the relative diffusion coefficient.. Parameters of the given model may be chosen in such a way that Eq. Ž27. will give the same results as Eq. Ž26. for a wide range of the Laplace variable s. This only requires that in the expansion of functions Ž26. and Ž27. in the parameter z s Ž s q 1rtc . that determines the system decay rate the coefficients at zy1 and z 0 coincide. At L )) R and t 2 )) t 1 we have Õ 4 p DR
st1 ;
4 p Ž L3 y L3 . r3 Õ
s
t1 qt2 t1
.
Ž 28 .
˚ and L s 15 A. ˚ This approximately corresponds to actual distance distribution For example, let R s 5 A between biradical ends w5x. Finding Õ and D from Eq. Ž28. and using them in Eq. Ž26. and Ž27., one can compare Green’s functions g Ž s . for different types of motion ŽFig. 1.. The given model takes no account of the effective interaction potential of the reaction centers resulting from conformation energies of a biradical chain. However, now it is evident that for the function g Ž s . this is not of crucial importance, since one can always use a simple two positional model with appropriate parameters. Note that there are other functions g Ž s . available in the literature that correspond to more complicated models of molecular mobility organization, for example, to a hopping motion in a restricted molecular volume w26x or to taking account of the reactivity anisotropy w26–28x. However, we think there is no necessity in using them now. Taking that the singlet recombination is a fast process, its average rate U0 in a contact zone has been varied from 10 10 to 10 12 sy1 . The constant K has been found from the relation K s U0 Õ. The quantity K c is still to be defined. In the general case it depends both on hfi constants, and on the character of the relative motion of radical centers. However, approximate calculation of K c may be done in the
Fig. 1. Green’s functions g Ž s . for different types of motion: 1. For the two positional exponential model Ž26.; 2. for the model of diffusion motion in restricted molecular volume Ž27..
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91
following simple way. Since transitions take place in a narrow region of the configuration space where the energies of the intersecting terms become equal, we put w19x K c cc Ž q . s 2 px 2d Ž ES Ž q . y E Ty Ž q . . , where x s
A
'8
(I Ž I q 1. y m Ž m q 1. , and
Ž 29. ESŽT _.Ž q . specifies the SŽTy . term energy dependence on the
configuration space coordinates. Consider the case of the isotropic exchange interaction where ES Ž q . s yJ Ž q . s yJ Ž r . ,
ETy Ž q . s J Ž q . y E s J Ž r . y E .
Ž 30 .
Here J Ž r . is the exchange integral; E s Ž g 1 q g 2 . b B0r2 q A P mr2 is the Zeeman splitting of the terms S and Ty. Using the typical dependence J Ž r . s J0 expŽ r–R .rD . and integrating Eq. Ž29. in spherical coordinates, we obtain K c s 8p 2x 2 rc2D P w Ž rc . rE ,
Ž 31 .
where rc is the coordinate of the terms intersection point, J Ž rc . s Er2. However, actually, the exchange integral depends both on the distance between radical centers, and on their relative orientation. In a simple case the angular dependence of the exchange integral may be related solely to the distance r between radical centers: J Ž r, V . s J Ž r, V Ž r ... Thus the exchange integral is modulated by the function that relates the angular dependence to the distance between the partners. For definiteness, assume that
ž
J Ž r . s J0 exp y
ryR
D
/
P u Ž b y r . P u Ž r y a. ,
Ž 32 .
where u Ž x . is the Heavyside function, and a and b are variable parameters. Then 2
Kcs
8 Ž p rc x . D E
w Ž rc . u Ž b y rc . u Ž rc y a . .
Ž 33 .
It is seen that S–Ty transitions take place only at a - rc - b. The appearance of extra multiplier in Eq. Ž33. as compared to Eq. Ž31. may be understood as the change in the distribution wŽ r ., and this affects the CIDNP field dependences. Actually the effective function variation may be essentially more complicated, because this function, just as the exchange integral, can involve both the distance between radical centers and a combination of angular coordinates as arguments. We have chosen the form of the function wŽ r . from phenomenological standpoint only, just to provide the best agreement between calculations and experiment, and then analyzed the results obtained. The stationary distribution function wŽ r . has been modelled using piecewise linear function. The function wŽ r . for biradical C 11 H 20 O is shown in Fig. 2. The requirement for the agreement between experimental and theoretical results leads to the fact that its specific form is sensitive to parameters J0 and D. However, if the ˚ for better agreement values of these parameters are chosen properly Žwe used J0 s 10 12 sy1 and D s 0.93 A with experiment., the above function is much narrower than the distribution function used in calculations in w5,16x. Therefore, the exchange interaction anisotropy appears to be essential in quantitative calculations of the CIDNP effect. It is taken that the decay on acceptors is a pseudomonomolecular reaction, and the RP decay rate ty1 on the acceptors is found from the equation ty1 s k s C s where k s s Ž2.3 " 0.4. P 10 9 My1 sy1 w14x, and the parameter 1rtc is calculated by the formula 1rtc s u 0 q k s C s .
Ž 34 .
Here u 0 – the decay rate of biradicals on the solvent molecules – is taken equal to 10 6 My1 cy1 w14x. Concentration of acceptors in the experiment varied from 0 to 0.016 M.
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Fig. 2. The function w Ž r . used in calculations of CIDNP in biradicals: Ž1. In the present paper; Ž2. in w5,16x.
Actually the decay on acceptors is a bimolecular process, and its rate may be determined from Eq. Ž34. only in the case where the concentration C s of reactants is a constant value. We shall show that we discuss just this case. In the experiment CIDNP is formed as a result of three stages: 1. The originally triplet pair goes into a singlet state; the matrix element of such a transition is of the order of A ; 20 G, thus the rate of the process is approximately 10 8 sy1 ; 2. to form a product, radical centers must come in contact, the rate of this process is 8 ty1 sy1 , since it is related to the return time of the pair from the cage to the reaction zone; 3. 2 ; 10 recombination time of the pair is K ; 10 10 y 10 12 sy1 . Obviously, the slowest process is the liming stage, i.e., CIDNP is formed approximately during 10y8 s. The initial concentration of ketones is C ketone ; 10y2 M. Hence, the decay rate of acceptors is k s C ketone ; 10 7 sy1 , and the number is not essentially reduced during the CIDNP formation. Figs. 3 and 4 present the results of numerical CIDNP calculations of a- and b-protons of biradical C 11 H 20 O. It is seen that even in the region of weak Žless than 200 G. fields theoretical calculation shows rather good agreement with experimental evidence. In the absence of acceptors of radicals when the pair lifetime is fairly long polarization results from S–Ty transitions occurring in the terms intersection zone. And only in very weak fields polarization changes its sign. Here CIDNP is formed outside the exchange interaction zone. The energy barrier for the flip-flops of electron spins with positive projections of nuclear spins exceeds the energy barrier for the flip-flops of electron spins with negative projections of nuclear spins w13x. Consequently, outside the exchange interaction zone S–T " transitions will be more effective for biradicals with negative projections of magnetic nuclei spins. Since we consider the case where the biradical is generated in the triplet state, and recombines from the singlet one, this means that S–T " transitions form positive nuclear polarization Žfor a-protons with negative hfi constants. in the reaction products. We believe that some discrepancy between
A.V. PopoÕ et al.r Chemical Physics 252 (2000) 83–95
93
Fig. 3. Polarization of a-protons calculated in the balance approximation Žlines., and experimental data Žsymbols. at different concentrations of C s acceptor.
theoretical calculations and experimental behaviour of CIDNP field dependence at C s s 0 is determined exactly by use of the balance approximation. At fairly high concentrations of acceptors the efficiency of S–Ty transitions in the terms intersection zone decreases sharply due to the decrease in the lifetime of biradicals. Here decisive contribution into the CIDNP effect is made by singlet–triplet ŽS–T " and S–T0 . transitions taking place outside the exchange interaction zone. For a-protons the theory even predicts double change of CIDNP sign. Here additional dip in the negative region is determined by the intersection of stationary energy levels Žeigenvalues of spin-Hamiltonian Ž12.. of biradicals and negative projections of nuclear spins, thus singlet–triplet transitions are retarded. However, actually the intersection zone of stationary energy levels appears to be rather wide, and double change of sign is not observed. Finally, we have checked which of the transitions outside the exchange interaction zone Žwe mean S–T " and S–T0 transitions. play a predominant role in polarization formation. As it turned out, calculation results do not change if we assume that the g-factors of unpaired electrons of biradicals are the same, g 1 s g 2 . Thus, a change in polarization sign in fields 500 G - B0 - 800 G is caused not by S–T0 Žhigh field. CIDNP mechanism. This is quite clear. For the biradicals used, g 1 s 2.0026 and g 2 s 2.0008, i.e., D g ; 10y3 . This means that the matrix elements of transitions between the terms S and T0 in different subensembles differ by D gb B0r" ; 10 7 sy1 ; this is considerably less than the characteristic rate of S–T " transitions estimated for A ; 20 G as 10 8 sy1 .
Fig. 4. Polarization of b-protons calculated in the balance approximation Žlines., and experimental data Žsymbols. at different concentrations of C s acceptor.
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4. Conclusion The Green function formalism developed by the authors earlier is used to calculate the CIDNP effect in recombination of acyl–alkyl biradicals produced in the photolysis of cycloalkanones. Singlet–triplet transitions occurring in the terms intersection zones and essential for the effect are taken into consideration in the balance approximation by specifying transition matrix element and the terms intersection zone shape. The CIDNP effects calculated for biradicals with different lifetimes depending on external magnetic field induction have been compared with experimental data. Rather good agreement has been achieved. Single-proton model of biradical is not sufficient to compare theory with experiment, so four magnetic nuclei Žtwo a-protons and two b-protons. have been taken into account. When the number of acceptors is rather large, polarization is formed outside the exchange interaction zone with allowance for both S–T " and S–T0 transitions, and S–T0 transitions Žhigh field CIDNP mechanism. do not play a decisive role here. Positive polarization of a-protons in very weak fields is determined by the difference between energy barriers for electron spin flip-flops of a radical center with different orientations of nuclear spin. As one would expect, at C s s 0 S–Ty transitions taking place in the terms intersection zone form the main polarization mechanism.
Acknowledgements This work was supported by the Russian Foundation for Basic Research ŽProject 96-03-32 956. and High School Department of Russian Federation Žgrant No. 97-0-14.0-90..
References w1x K.M. Salikhov, Yu.N. Molin, R.Z. Sagdeev, A.L. Buchachenko, in: Y.N. Molin ŽEd.., Spin Polarization and Magnetic Effects in Radical Reactions, Elsevier, Amsterdam, 1984. w2x U.E. Steiner, H.-J. Wolff, in: J.F. Rabek, G.W. Scott ŽEds.., Photochemistry and Photophysics, Vol. 4, CRC Press, Boca Raton, 1991, p. 1. w3x G.L. Closs, C.E. Doubleday, J. Am. Chem. Soc. 94 Ž1972. 9248. w4x G.L. Closs, C.E. Doubleday, J. Am. Chem. Soc. 95 Ž1973. 2735. w5x F.J.J. de Kanter, R. Kaptein, J. Am. Chem. Soc. 104 Ž1982. 4759. w6x G.L. Closs, O.D. Redwine, J. Am. Chem. Soc. 107 Ž1985. 6131. w7x I.R. Gould, M.B. Zimmt, N.J. Turro, B.H. Barets, G.F. Lehr, J. Am. Chem. Soc. 107 Ž1985. 4607. w8x E.G. Bagryanskaya, V.F. Tarasov, N.I. Avdievich, I.A. Shkrob, Chem. Phys. 162 Ž1992. 213. w9x E.G. Bagryanskaya, R.Z. Sagdeev, Prog. React. Kinet. 18 Ž1993. 63. w10x A.A. Obynochny, P.A. Purtov, A.G. Maryasov, Yu.N. Molin, K.M. Salikhov, E.D. Skakovsky, Appl. Magn. Reson. 9 Ž1995. 355. w11x S.A. Sukhenko, P.A. Purtov, K.M. Salikhov, Khim. Fiz. 1 Ž1983. 21. w12x N.E. Polyakov, P.A. Purtov, T.V. Leshina, M.B. Taraban, R.Z. Sagdeev, K.M. Salikhov, Chem. Phys. Lett. 129 Ž1986. 357. w13x A.M. Osintsev, P.A. Purtov, K.M. Salikhov, Khim. Fiz. 11 Ž1992. 1192. w14x A.V. Yurkovskaya, O.B. Morozova, R.Z. Sagdeev, S.V. Dvinskih, G. Buntkovsky, H.-M. Vieth, Chem. Phys. 197 Ž1995. 157. w15x Yu.P. Tsentalovich, A.V. Yurkovskaya, R.Z. Sagdeev, A.A. Obynochny, P.A. Purtov, A.A. Shargorodsky, Chem. Phys. 139 Ž1989. 307. w16x F.J.J. Kanter, J.A. den Hollander, A.H. Huiser, R. Kaptein, Mol. Phys. 34 Ž1977. 857. w17x A.B. Doktorov, S.A. Mikhailov, P.A. Purtov, Chem. Phys. 160 Ž1992. 223. w18x P.A. Purtov, A.B. Doktorov, Chem. Phys. 178 Ž1993. 47. w19x A.V. Popov, P.A. Purtov, Chem. Phys. Reports 16 Ž1997. 27. w20x A.P. Parnachev, P.A. Purtov, E.G. Bagryanskaya, R.Z. Sagdeev, J. Chem. Phys. 107 Ž1997. 9942. w21x A.V. Popov, Calculation of magneto-spin effects in molecularly organized systems, Ph.D. Thesis, Kemerovo Technological Institute, Russia, 1997. w22x H. Fischer ŽEd.., Magnetic Properties of Free Radicals, Landolt-Bornstein New Series, V. II 9b, Springer, Berlin, 1980.
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K.M. Salikhov, S.A. Mikhailov, Teor. Eksp. Khim. 19 Ž1983. 550. V.F. Tarasov, N.D. Ghatlia, A.L. Buchachenko, N.J. Turro, J. Phys. Chem. 95 Ž1991. 10220. P.A. Purtov, A.B. Doktorov, A.V. Popov, Chem. Phys. 182 Ž1994. 149. A.M. Osintsev, P.A. Purtov, A.B. Doktorov, Chem. Phys. 185 Ž1994. 149. A.B. Doktorov, N.N. Lukzen, Khim. Fiz. 4 Ž1985. 616. G.S. Ananchenko, P.A. Purtov, E.G. Bagryanskaya, R.Z. Sagdeev, J. Phys. Chem. A 103 Ž1999. 3430.
95
Calculation of CIDNP field dependences in biradicals in the photolysis of large-ring cycloalkanones A.V. Popov a , P.A. Purtov b
b,c,)
, A.V. Yurkovskaya
d
a KemeroÕo Technological Institute, 650067, KemeroÕo 67, Russian Federation Institute of Chemical Kinetics and Combustion, 630090, NoÕosibirsk 90, Russian Federation c NoÕosibirsk State UniÕersity, 630090, NoÕosibirsk 90, Russian Federation d International Tomography Center, 630090, NoÕosibirsk 90, Russian Federation
Received 11 March 1999; in final form 3 September 1999
Abstract Calculations of CIDNP effects in recombination of acyl–alkyl biradicals arising in the photolysis of cycloalkanones have been performed. Singlet–triplet transitions occurring in the terms intersection zone and essential for the effect are taken into account in the balance approximation. The calculated CIDNP field dependences have been compared with experimental data obtained for biradicals with different lifetimes. Good agreement has been achieved. Interpretation of CIDNP field dependences is given. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Over the last two decades spin polarization effects arising in the course of singlet–triplet evolution and spin-selective recombination of radical pairs ŽRP. have been the subject of extensive investigation Žfor reviews see Refs. w1,2x.. It opens up the possibility of obtaining unique information about the kinetics and mechanism of radical reactions. Lately, particular attention has been given to the studies of magneto-spin effects in RP with restricted molecular volume Žflexible biradicals, RP in micelles, plastic crystals. w3–10x. Exactly in the latter spin polarization effects and external magnetic field influence on the course of chemical reaction are particularly pronounced. In a certain approximation such RP may be treated as model ones for complicated biological substances. The motion of radical centers in restricted volume provides rather long lifetime for them and effective singlet–triplet evolution determined both by hyperfine interaction of electrons with magnetic nuclei and by the difference in electron Zeeman frequency between radicals. Not moving far apart, the partners reside in the exchange interaction zone for a long time. The value of the exchange interaction depends both on the
)
Corresponding author.
0301-0104r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 9 . 0 0 2 9 3 - 1
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A.V. PopoÕ et al.r Chemical Physics 252 (2000) 83–95
distance between radical centers and on their relative position, and is modulated by molecular mobility of radicals. The exchange interaction splits singlet and triplet RP terms, while application of external magnetic field removes degeneration of triplet terms ŽTq, T0 and Ty .. The exchange interaction in long-lived RP produces particularly strong effect on spin dynamics in such magnetic fields where its value turns equal to the Zeeman energy, and two terms, S and Ty, become resonant. In this case the Ty–S transition where the flip-flop of electron and nuclear spins takes place becomes the main channel of spin conversion. This gives rise to integral electron and nuclear polarization in RP and reaction products. However, the increase in the distance between partners or the change in their relative orientation leads to a very rapid decrease in the exchange interaction. Thus the residence time of radicals in S and Ty terms intersection zone may be very short as compared to the time spent in space regions where the exchange interaction is almost equal to zero. At zero exchange interaction effective S–T0 transitions take place in RP the frequency of which depends on nuclear spin configuration m4 . No spin reversals occur in such transitions, and nuclear polarization in reaction products is formed owing to the competition between the transition into singlet reactive state and other processes, for example, the escape from the cage or the acceptance reaction. For this reason CIDNP S–T0 mechanism efficiency depends essentially on geminate radical pair lifetime, and is maximum when lifetimes are comparable to singlet–triplet evolution lifetime. At long RP lifetimes S–T0-transitions are considerably averaged and make no contribution into CIDNP effect. In weak magnetic fields S–T " transitions determined by hfi nonsecular part play an important role w1,11–13x. CIDNP field dependence is determined by variations in energy spectrum of RP spins. For example, polarization in S–T " transitions is brought about by the difference between energy barriers for electron spin flip-flops of radicals with different nuclear spin orientations. Polarization in S–T0 transitions results from the difference in dephasing rate Ždistinction between precession frequency differences. between electron spins of radicals with different nuclear spin orientations. Of course, in weak magnetic fields transitions of both types affect one another, however, such a separation is justified in the case of qualitative interpretation of CIDNP field dependence behaviour. If RP contains several magnetic nuclei, then CIDNP field dependence usually changes sign. Sometimes, the sign is changed several times. Recently we have experimentally studied CIDNP field dependences in acyl–alkyl biradicals produced in the photolysis of cycloalkanones Žcyclodecanone and cycloundecanone. in the presence of effective acceptor of radicals w14x. In these biradicals radical centers are connected by a flexible chain consisting of 11 and 10 methylene links. As is seen from experiments conducted by CIDNP method with time resolution w15x, the main decay channel of biradicals investigated is their recombination resulting in the production of initial ketone molecule. Comparison of CIDNP field dependences for different protons in the initial ketone shows that reduction of biradical lifetime due to the addition of acceptors does lead to noticeable decrease in polarization contribution by Ty–S channel of intercombination transitions occurring in the terms intersection zone. Paper w14x presents a solely qualitative interpretation of the results obtained, and the attempts to calculate CIDNP field dependences by single-nucleus model of de Kanter and co-authors proposed in papers w5,16x do not reproduce the regularities for CIDNP field dependences in biradicals discovered experimentally. To calculate magneto-spin effects, we have developed a kinematic approach w17–20x. The approach is based on the formalism of the Green functions specifying the character of a relative motion of radical centers in a pair. Singlet–triplet transitions taking place in the terms intersection zone may be taken into account in different ways. The simplest way is a balance approximation wherein the S–Ty transition matrix element and the shape of the terms intersection zone are given w19x. This paper presents consistent theoretical treatment of singlet–triplet evolution in radical pairs with restricted molecular volume in arbitrary magnetic fields based on the Green’s function formalism. Calculations of chemically induced nuclear polarization effects depending on external magnetic field induction for biradicals with different lifetimes have been compared with experimental findings given in w14x. The paper seeks to verify the adequacy of the theoretical model proposed, and to obtain some quantitative information about molecularkinetic and magneto-resonance parameters of biradicals.
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2. Theory The CIDNP effect of the k-th nuclear spin is defined by the mean value of its operator Iˆk z in recombination product w1x ² Iˆk z : s Tr1,2 Iˆk z wˆ Ž t . ' Ý m k wm Ž t . ,
ž
Ž 1.
/
m4
where Tr1,2 is the trace operation over all spin states of RP, index m defines nuclear spin configuration in spin states of the pair. The Laplace transform of the desired quantity ˆ˜Ž s . has been calculated in w18x, and is of the form s P w˜ˆ Ž s . s Kˆ r gˆ Ž s . 1 q Kˆ r gˆ Ž s .
y1
r0 ,
Ž 2.
where Kˆ r is the constant matrix of recombination from one or another spin state, r 0 is coordinate-independent initial matrix of RP Uˆ Ž q . s Kˆ r cu Ž q . , r 0 Ž q . s cu Ž q . w Ž q . .
Ž 3.
The function cuŽ q . specifies the reaction zone shape Žand the exchange interaction zone shape., and wŽ q . is the stationary normalized distribution in pairs of radical centers. It is expressed in terms of wŽ q,qX ,t . – the conditional probability density of realizing the coordinates q at the instant of time t if they were qX at the initial moment, and the configuration space volume accessible for relative motion of reactants V s Hd q
w Ž q . s V lim w Ž q,qX ;t . . t
™`
Ž 4.
The Green function wŽ q,qX ,t . satisfies the equation
E Et
y Lˆ Ž q . P w Ž q,qX ;t . s d Ž q y qX . d Ž t . , X
Ž 5.
X
w Ž q,q ;0 . s d Ž q y q . . The matrix gˆ Ž s . appearing in Eq. Ž2. defines spin and molecular dynamics, and calculated in the balance approximation is of the form w19,20x gˆ Ž s . s Lˆ Ž s . 1 q iKˆ J Lˆ Ž s .
y1
,
Lˆ Ž s . s gˆ 0 Ž s . y gˆ 0 u c Ž s . 1 q Kˆ c gˆ 0 cc Ž s .
y1
Ž 6. Kˆ c gˆ 0 u c Ž s . .
Here Kˆ J is the exchange interaction constant matrix JˆŽ q . s Kˆ J c J Ž q . , Kˆ J s JˆŽ q . P w Ž q . d q .
H
Ž 7.
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86
For brevity we use the designations gˆ 0 Ž s . s ² gˆ 0 u Ž q, s . :u '
HHc Ž q . Gˆ Ž q ,q ; s . c Ž q . w Ž q . d qd q , X
gˆ 0 u c Ž s . s ² gˆ 0 u Ž q, s . :c ' gˆ 0 cc Ž s . s ² gˆ 0 c Ž q, s . :c '
X
u
X
0
u
HHc Ž q . Gˆ Ž q ,q ; s . c Ž q . w Ž q . d qd q , X
u
X
0
X
c
Ž 8.
HHc Ž q . Gˆ Ž q ,q ; s . c Ž q . w Ž q . d qd q . X
c
X
0
X
c
The function cc Ž q . defines the shape of the zone of level crossing. For simplicity it is assumed to be the same for all terms. ² . . . :u is the average over the reaction zone defined as a scalar product in the Hilbert space; ² . . . :c means the average over the terms intersection zone. The matrix Gˆ 0 Ž q,qX ; s . describes RP spin evolution outside the exchange interaction zone, and is the Laplace transform of the matrix wˆ 0 Ž q,qX ;t .. This matrix satisfies the equation
E Et
y Lˆ Ž q . y i Vˆ P wˆ 0 Ž q,qX ;t . s d Ž q y qX . d Ž t . , X
Ž 9.
X
wˆ 0 Ž q,q ;0 . s d Ž q y q . , where Vˆ is the coordinate-independent component of the Liouvillian Lˆ Ž q .. The shape of the terms intersection zone is the characteristic of the interaction. It may be defined by the relations w21x X
X
HG Ž q ,q ; s . dq ' Cc Ž q . , c
Ž 10 . Cs
X
X
HHG Ž q ,q ; s . w Ž q . d q d q ,
where GŽ q,qX ; s . is the Laplace transform of the function w Ž q,qX ;t .. This function satisfies the equation
E Et
y Lˆ Ž q . y i Ž J Ž q . y E . P w Ž q,qX ;t . s d Ž q y qX . d Ž t . , X
Ž 11 .
X
w Ž q,q ;0 . s d Ž q y q . , where J Ž q . is the exchange integral, E is the splitting of the intersecting levels outside the exchange interaction zone. Finally, Kˆ c involved in Eq. Ž6. is the constant matrix of singlet–triplet transitions in the terms intersection zone. If these transitions are determined by hfi, then matrix elements of Kˆ c are proportional to hfi constant squares. Thus calculation of the CIDNP effect is reduced to the calculation of the pair spin dynamics outside the exchange interaction zone Žin the intervals between recontacts., determination Žor prescription. of the relative motion kinematics, and calculation Žor prescription. of the exchange integral as a function of a relative position of radical centers. Now we shall apply the kinematic approach to the calculation of CIDNP field dependence of acyl–alkyl biradicals. Biradicals came into existence in the triplet state in the photolysis of cycloalkanones in the presence of the effective acceptor of radicals ŽCBrCl 3 .. In biradicals under study only a-CH 2 and b-CH 2 protons of alkyl ring have essential hfi constants Ž A a s –2.2 and Ab s 2.8 mT for structurally similar monoradicals w22x..
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However, even in the case where only four nuclei are taken into account the problem is still rather complicated, since the experiment has been conducted in weak Žand moderate. magnetic fields, while in practice spin evolution involves a large number of states. Represent the coordinate-independent spin-Hamiltonian of the system under study consisting of two pairs of equivalent magnetic nuclei as
ž ™ˆ ™ˆ / ™ˆ
ž ™ˆ ™ˆ / ™ˆ
Hˆ s v 1 Sˆ1 z q v 2 Sˆ2 z q Aa I1 q I2 S1 q Ab I3 q I4 S1 ,
Ž 12 .
where v 1Ž2. is the precession frequency of electron spin 1Ž2. in constant magnetic field B directed along the z axis. Matrix elements of the Liouvillianˆ and Hamiltonian Hˆ are related as
Ž Vˆ . i f , k l s di k Hˆl j y d jl Hˆi k .
Ž 13 .
™ˆ ™ˆ During evolution the values of the following operators remain unchanged: Ž I1 q I2 . 2 s La Ž La q 1. and ™ ™ ˆ ˆ Ž I q I . 2 s L Ž L q 1.. Besides, the projection S of the total spin of electrons and nuclei on the direction of 3
4
b
b
external magnetic field is preserved. Obviously, the quantity La Žjust as Lb . assumes values 0 and 1, i.e., defines the multiplicity of the state of a Ž b . nuclei. Spin wave functions will be denoted as follows
Ž 14 .
where C s S, T0 , Tq, Ty are the states of electron spin pair, Ma Ž Mb . is the total projection of a Ž b . nuclei spins on the z axis. Indices S , La and Lb will be omitted where possible. Let us write out all subensembles necessary for the calculation of a-nuclei polarization. Here La / 0, and for different values of S , Lb , we have Lb s 1, S s 2: < S;1,1: , < T0 ;1,1: , < Tq ;1,0: , < Tq ;0,1: ; Lb s 1, S s y2: < S;y 1,y 1: , < T0 ;y 1,y 1: , < Ty ;y 1,0: , < Ty ;0,y 1: ; Lb s 1, S s 1: < S;1,0: , < S;0,1: , < T0 ;1,0: , < T0 ;0,1: , < Ty ;1,1: , < Tq ;0,0: , < Tq ;1,y 1: , < Tq ;y 1,1: ; Lbs1, Ss y 1: < S;y 1,0: , < S;0,y 1: , < T0 ;y 1,0: , < T0 ;0,y 1: , < Ty ;0,0: , < Tq ;y 1,y 1: , < Ty ;1,y 1: , < Ty ;y 1,1: ; Lbs1, Ss0: < S;0,0: , < S;1,y 1: , < S;y 1,1: , < T0 ;0,0: , < T0 ,1,y 1: , < T0 ;y 1,1: , < Tq ;0,y 1: , < Tq ;y 1,0: , < Ty ;1,0: , < Ty ;0,1: ; Lb s 0, S s 1: < S;1,0: , < T0 ;1,0: , < Tq ;0,0: ; Lb s 0, S s y1: < S;y 1,0: , < T0 ;y 1,0: , < Ty ;0,0: ; Lb s 0, S s 0: < S;0,0: , < T0 ;0,0: , < Tq ;y 1,0: , < Ty ;1,0: .
Ž 15 .
For example, consider the subensemble with Lb s 0, S s y1. In the Liouville space, the basis states of this subensemble may be written as <1: s < S;y 1,0;S;y 1,0: , <2: s < S;y 1,0;T0 ;y 1,0: , <3: s < T0 ;y 1,0;S;y 1,0: , <4: s < T0 ;y 1,0;T0 ;y 1,0: , <5: s < S;y 1,0;Ty ;0,0: , <6: s < Ty ;0,0;S;y 1,0: , <7: s < Ty ;0,0;Ty ;0,0: , <8: s < T0 ;y 1,0;Ty ;0,0: , <9: s < Ty ;0,0;T0 ;y 1,0: .
Ž 16 .
Most commonly, when the reactants get into the reaction zone, a strong exchange interaction is switched on that changes the phase between singlet and triplet states. In this case the phase change due to the reaction may
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be ignored. Besides, recombination proceeds from the singlet state, therefore, only one matrix element in recombination constant matrix differs from zero. Denoting it by K we obtain for recombination probability of such a subensemble from Eq. Ž2. s P wˆ˜ Ž s . s
Kgˆ Ž s . 1,1 1 q Kgˆ Ž s . 1,1
.
Ž 17 .
The CIDNP effect is easily obtained from the above expression. Thus the desired result is expressed in terms of the only element of the matrix gˆ Ž s . defined by Eq. Ž6.. In basis Ž16. the Liouvillian Kˆ J has six elements different from zero
ž Kˆ /
J 2,2 s
ž Kˆ /
J 5,5 s
ž Kˆ /
J 8,8 s y
ž Kˆ /
J 3,3 s y
ž Kˆ /
J 6,6 s y
ž Kˆ /
J 9,9 s K J
.
Ž 18 .
As for S–Ty transitions, in this subensemble four matrix elements in the matrix Kˆ c differ from zero
ž Kˆ /
c 1,1 s
ž Kˆ /
c 7 ,7 s y
ž Kˆ /
c 1,7 s y
ž Kˆ /
c 7,1 s K c
.
Ž 19 .
The matrix elements of the operator Gˆ 0 are calculated by the formula
ž Gˆ /
0 i k ,l m s
Ý ² i < cp :² cp < l :² m < cq :² cq < k : G Ž q,qX ; s q i Ž ´ p y ´ q . . ,
Ž 20 .
p, q
where < cp : are the eigenfunctions, and ´ p – the eigenvalues of spin-Hamiltonian Ž12.. The function GŽ q,qX ; s . is the Laplace transform of the function wŽ q,qX ;t .. Consider another subensemble with Lb s 1, S s 1. With this aim we introduce the following designations in the Liouville space for this subensemble: <1: s
s P w˜ˆ 2 Ž s . s K
gˆ Ž s . 1,1 q gˆ Ž s . 1,2 q K Ž gˆ Ž s . 1,1 gˆ Ž s . 2,2 y gˆ Ž s . 1,2 gˆ Ž s . 2,1 . 1 q K Ž gˆ Ž s . 1,1 q gˆ Ž s . 2,2 . q K 2 Ž gˆ Ž s . 1,1 gˆ Ž s . 2,2 y gˆ Ž s . 1,2 gˆ Ž s . 2,1 . gˆ Ž s . 2,2 q gˆ Ž s . 2,1 q K Ž gˆ Ž s . 1,1 gˆ Ž s . 2,2 y gˆ Ž s . 1,2 gˆ Ž s . 2,1 . 1 q K Ž gˆ Ž s . 1,1 q gˆ Ž s . 2,2 . q K 2 Ž gˆ Ž s . 1,1 gˆ Ž s . 2,2 y gˆ Ž s . 1,2 gˆ Ž s . 2,1 .
,
.
Ž 21 .
The matrix elements gˆ Ž s .1,2 and gˆ Ž s . 2,1 define the efficiency of transitions from state <1: into state <2:. For example, gˆ Ž0.1,2 has the meaning of the average residence time of RPs in a unit of the reaction zone size in state <1: provided that at the initial instant of time RPs were in the reaction zone in state <2: w18x. Since such transitions are accompanied by the flip-flops of both the first and the second nuclear spins, their efficiency is almost always low and will not be taken into account. Thus RP subensemble with two singlet states may be transformed into two subensembles each involving only one singlet state
Ž 22 .
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Then Eq. Ž17. that is more simple than Eq. Ž21. can be employed to find recombination probabilities. Transforming all subensembles Eq. Ž15. in a similar way, one can find the recombination probabilities in the case of both singlet and triplet RP precursors for each of the subensembles. Recall that RP recombination probability for singlet and triplet precursors are related by w1x TW s
1 3
Ž Ktp ys w Ž1 q Ktp . . ,
Ž 23 .
where tp s Õg Ž0. s Õg ² g uŽ q,0.:u ' ÕHHcuŽ qX .GŽ qX ,q;0. cuŽ q .wŽ q . dqdqX – is the full residence time of radicals in the reaction zone; Õ s Ž² cuŽ q .:u .y1 ' Ž Hcu Ž q . cuŽ q .wŽ q . dq .y1 is the effective volume of the reaction zone. Note that in the experiment under discussion biradicals were generated in the triplet state. Obviously, at the given magneto-resonance parameters in spin-Hamiltonian Ž12. the CIDNP effect also depends on the character of the relative motion of RP, i.e., on particular form of the functions wŽ q ., gˆ 0 Ž s ., gˆ 0 u c Ž s . and gˆ 0 cc Ž s .. As is shown in w19,20x, in the case of complete mixing in the configuration space we have the following equality for long-lived systems gˆ 0 Ž s . s gˆ 0 u c Ž s . s gˆ 0 cc Ž s . .
Ž 24 .
Eq. Ž24. is valid at sy1 ; t 4 t L , where t L is a characteristic time of the divergence of radical ends of the biradicals Žfor example, for diffusion t L ; L2rD, L is the length of biradical, D is the coefficient of relative diffusion.. However, taking into account that the reaction zone is close in space to the terms intersection zone we put gˆ 0 Ž s . f gˆ 0 u c Ž s . f gˆ 0 cc Ž s .. This simplifying assumption brings Eq. Ž6. into the form gˆ Ž s . s gˆ 0 Ž s . 1 q Kˆ c q iKˆ J gˆ 0 Ž s .
ž ž
/
y1
/
.
Ž 25 .
Note that the elements of the matrix gˆ 0 Ž s ., and, therefore, of the matrix gˆ Ž s . are expressed in terms of the combination of functions g Ž s . s ² g uŽ q, s .:u ' HHcuŽ qX .GŽ qX ,q; s . cu Ž q .wŽ q .d qd qX .
3. Computer simulation, comparison with experiment, and discussion of results To perform particular calculations, we have chosen the simplest two-positional model of RP w23x. In the framework of this model two possible states are introduced. The first state is the immediate contact Žthe pair is in the reaction zone.. The escape from this state is described as a monomolecular decay with the characteristic time t 1. The second state is the cage. This state is defined by two parameters: the time t 2 it takes the pair to return from the cage into the reaction zone, and the irreversible decay time tc . In this model g Ž s . is of the form g Ž s. s
t 1 Ž t 2 Ž s q 1rtc . q 1 .
Ž s q 1rtc . Ž t 1 q t 2 Ž t 1 Ž s q 1rtc . q 1 . . Õ
.
Ž 26 .
When doing particular calculations, we have taken the parameters t 1 and t 2 equal to 0.67 P 10y1 0 and 0.67 P 10y8 s, respectively. The two positional exponential models represent the processes in a cage qualitatively adequately, thus we believe it suitable for the description of CIDNP effects in biradicals. Of course, different, more realistic models of RP motion may be employed, for example, the well-known model of diffusion motion in restricted molecular volume w24x. In the framework of this model two spherical particles are considered that are linked by a flexible
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90
chain which prevents them from moving apart. The particles execute diffusion motion in such a way that their centers cannot be separated by the distance shorter than R s R A q R B and longer than L Ž R A and R B are the radii of spherical particles.. For this case the function g Ž s . is of the form w25x g Ž s. s
kR s
k L y ey2 b Ž LyR.
1
4p DR Ž 1 q b R . k L y k R ey2 b Ž LyR. ,
1ybR 1qbR
, kL s
1ybL 1qbL
, bs
(
s q 1rtc D
,
Ž 27 .
Žhere D is the relative diffusion coefficient.. Parameters of the given model may be chosen in such a way that Eq. Ž27. will give the same results as Eq. Ž26. for a wide range of the Laplace variable s. This only requires that in the expansion of functions Ž26. and Ž27. in the parameter z s Ž s q 1rtc . that determines the system decay rate the coefficients at zy1 and z 0 coincide. At L )) R and t 2 )) t 1 we have Õ 4 p DR
st1 ;
4 p Ž L3 y L3 . r3 Õ
s
t1 qt2 t1
.
Ž 28 .
˚ and L s 15 A. ˚ This approximately corresponds to actual distance distribution For example, let R s 5 A between biradical ends w5x. Finding Õ and D from Eq. Ž28. and using them in Eq. Ž26. and Ž27., one can compare Green’s functions g Ž s . for different types of motion ŽFig. 1.. The given model takes no account of the effective interaction potential of the reaction centers resulting from conformation energies of a biradical chain. However, now it is evident that for the function g Ž s . this is not of crucial importance, since one can always use a simple two positional model with appropriate parameters. Note that there are other functions g Ž s . available in the literature that correspond to more complicated models of molecular mobility organization, for example, to a hopping motion in a restricted molecular volume w26x or to taking account of the reactivity anisotropy w26–28x. However, we think there is no necessity in using them now. Taking that the singlet recombination is a fast process, its average rate U0 in a contact zone has been varied from 10 10 to 10 12 sy1 . The constant K has been found from the relation K s U0 Õ. The quantity K c is still to be defined. In the general case it depends both on hfi constants, and on the character of the relative motion of radical centers. However, approximate calculation of K c may be done in the
Fig. 1. Green’s functions g Ž s . for different types of motion: 1. For the two positional exponential model Ž26.; 2. for the model of diffusion motion in restricted molecular volume Ž27..
A.V. PopoÕ et al.r Chemical Physics 252 (2000) 83–95
91
following simple way. Since transitions take place in a narrow region of the configuration space where the energies of the intersecting terms become equal, we put w19x K c cc Ž q . s 2 px 2d Ž ES Ž q . y E Ty Ž q . . , where x s
A
'8
(I Ž I q 1. y m Ž m q 1. , and
Ž 29. ESŽT _.Ž q . specifies the SŽTy . term energy dependence on the
configuration space coordinates. Consider the case of the isotropic exchange interaction where ES Ž q . s yJ Ž q . s yJ Ž r . ,
ETy Ž q . s J Ž q . y E s J Ž r . y E .
Ž 30 .
Here J Ž r . is the exchange integral; E s Ž g 1 q g 2 . b B0r2 q A P mr2 is the Zeeman splitting of the terms S and Ty. Using the typical dependence J Ž r . s J0 expŽ r–R .rD . and integrating Eq. Ž29. in spherical coordinates, we obtain K c s 8p 2x 2 rc2D P w Ž rc . rE ,
Ž 31 .
where rc is the coordinate of the terms intersection point, J Ž rc . s Er2. However, actually, the exchange integral depends both on the distance between radical centers, and on their relative orientation. In a simple case the angular dependence of the exchange integral may be related solely to the distance r between radical centers: J Ž r, V . s J Ž r, V Ž r ... Thus the exchange integral is modulated by the function that relates the angular dependence to the distance between the partners. For definiteness, assume that
ž
J Ž r . s J0 exp y
ryR
D
/
P u Ž b y r . P u Ž r y a. ,
Ž 32 .
where u Ž x . is the Heavyside function, and a and b are variable parameters. Then 2
Kcs
8 Ž p rc x . D E
w Ž rc . u Ž b y rc . u Ž rc y a . .
Ž 33 .
It is seen that S–Ty transitions take place only at a - rc - b. The appearance of extra multiplier in Eq. Ž33. as compared to Eq. Ž31. may be understood as the change in the distribution wŽ r ., and this affects the CIDNP field dependences. Actually the effective function variation may be essentially more complicated, because this function, just as the exchange integral, can involve both the distance between radical centers and a combination of angular coordinates as arguments. We have chosen the form of the function wŽ r . from phenomenological standpoint only, just to provide the best agreement between calculations and experiment, and then analyzed the results obtained. The stationary distribution function wŽ r . has been modelled using piecewise linear function. The function wŽ r . for biradical C 11 H 20 O is shown in Fig. 2. The requirement for the agreement between experimental and theoretical results leads to the fact that its specific form is sensitive to parameters J0 and D. However, if the ˚ for better agreement values of these parameters are chosen properly Žwe used J0 s 10 12 sy1 and D s 0.93 A with experiment., the above function is much narrower than the distribution function used in calculations in w5,16x. Therefore, the exchange interaction anisotropy appears to be essential in quantitative calculations of the CIDNP effect. It is taken that the decay on acceptors is a pseudomonomolecular reaction, and the RP decay rate ty1 on the acceptors is found from the equation ty1 s k s C s where k s s Ž2.3 " 0.4. P 10 9 My1 sy1 w14x, and the parameter 1rtc is calculated by the formula 1rtc s u 0 q k s C s .
Ž 34 .
Here u 0 – the decay rate of biradicals on the solvent molecules – is taken equal to 10 6 My1 cy1 w14x. Concentration of acceptors in the experiment varied from 0 to 0.016 M.
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A.V. PopoÕ et al.r Chemical Physics 252 (2000) 83–95
Fig. 2. The function w Ž r . used in calculations of CIDNP in biradicals: Ž1. In the present paper; Ž2. in w5,16x.
Actually the decay on acceptors is a bimolecular process, and its rate may be determined from Eq. Ž34. only in the case where the concentration C s of reactants is a constant value. We shall show that we discuss just this case. In the experiment CIDNP is formed as a result of three stages: 1. The originally triplet pair goes into a singlet state; the matrix element of such a transition is of the order of A ; 20 G, thus the rate of the process is approximately 10 8 sy1 ; 2. to form a product, radical centers must come in contact, the rate of this process is 8 ty1 sy1 , since it is related to the return time of the pair from the cage to the reaction zone; 3. 2 ; 10 recombination time of the pair is K ; 10 10 y 10 12 sy1 . Obviously, the slowest process is the liming stage, i.e., CIDNP is formed approximately during 10y8 s. The initial concentration of ketones is C ketone ; 10y2 M. Hence, the decay rate of acceptors is k s C ketone ; 10 7 sy1 , and the number is not essentially reduced during the CIDNP formation. Figs. 3 and 4 present the results of numerical CIDNP calculations of a- and b-protons of biradical C 11 H 20 O. It is seen that even in the region of weak Žless than 200 G. fields theoretical calculation shows rather good agreement with experimental evidence. In the absence of acceptors of radicals when the pair lifetime is fairly long polarization results from S–Ty transitions occurring in the terms intersection zone. And only in very weak fields polarization changes its sign. Here CIDNP is formed outside the exchange interaction zone. The energy barrier for the flip-flops of electron spins with positive projections of nuclear spins exceeds the energy barrier for the flip-flops of electron spins with negative projections of nuclear spins w13x. Consequently, outside the exchange interaction zone S–T " transitions will be more effective for biradicals with negative projections of magnetic nuclei spins. Since we consider the case where the biradical is generated in the triplet state, and recombines from the singlet one, this means that S–T " transitions form positive nuclear polarization Žfor a-protons with negative hfi constants. in the reaction products. We believe that some discrepancy between
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93
Fig. 3. Polarization of a-protons calculated in the balance approximation Žlines., and experimental data Žsymbols. at different concentrations of C s acceptor.
theoretical calculations and experimental behaviour of CIDNP field dependence at C s s 0 is determined exactly by use of the balance approximation. At fairly high concentrations of acceptors the efficiency of S–Ty transitions in the terms intersection zone decreases sharply due to the decrease in the lifetime of biradicals. Here decisive contribution into the CIDNP effect is made by singlet–triplet ŽS–T " and S–T0 . transitions taking place outside the exchange interaction zone. For a-protons the theory even predicts double change of CIDNP sign. Here additional dip in the negative region is determined by the intersection of stationary energy levels Žeigenvalues of spin-Hamiltonian Ž12.. of biradicals and negative projections of nuclear spins, thus singlet–triplet transitions are retarded. However, actually the intersection zone of stationary energy levels appears to be rather wide, and double change of sign is not observed. Finally, we have checked which of the transitions outside the exchange interaction zone Žwe mean S–T " and S–T0 transitions. play a predominant role in polarization formation. As it turned out, calculation results do not change if we assume that the g-factors of unpaired electrons of biradicals are the same, g 1 s g 2 . Thus, a change in polarization sign in fields 500 G - B0 - 800 G is caused not by S–T0 Žhigh field. CIDNP mechanism. This is quite clear. For the biradicals used, g 1 s 2.0026 and g 2 s 2.0008, i.e., D g ; 10y3 . This means that the matrix elements of transitions between the terms S and T0 in different subensembles differ by D gb B0r" ; 10 7 sy1 ; this is considerably less than the characteristic rate of S–T " transitions estimated for A ; 20 G as 10 8 sy1 .
Fig. 4. Polarization of b-protons calculated in the balance approximation Žlines., and experimental data Žsymbols. at different concentrations of C s acceptor.
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A.V. PopoÕ et al.r Chemical Physics 252 (2000) 83–95
4. Conclusion The Green function formalism developed by the authors earlier is used to calculate the CIDNP effect in recombination of acyl–alkyl biradicals produced in the photolysis of cycloalkanones. Singlet–triplet transitions occurring in the terms intersection zones and essential for the effect are taken into consideration in the balance approximation by specifying transition matrix element and the terms intersection zone shape. The CIDNP effects calculated for biradicals with different lifetimes depending on external magnetic field induction have been compared with experimental data. Rather good agreement has been achieved. Single-proton model of biradical is not sufficient to compare theory with experiment, so four magnetic nuclei Žtwo a-protons and two b-protons. have been taken into account. When the number of acceptors is rather large, polarization is formed outside the exchange interaction zone with allowance for both S–T " and S–T0 transitions, and S–T0 transitions Žhigh field CIDNP mechanism. do not play a decisive role here. Positive polarization of a-protons in very weak fields is determined by the difference between energy barriers for electron spin flip-flops of a radical center with different orientations of nuclear spin. As one would expect, at C s s 0 S–Ty transitions taking place in the terms intersection zone form the main polarization mechanism.
Acknowledgements This work was supported by the Russian Foundation for Basic Research ŽProject 96-03-32 956. and High School Department of Russian Federation Žgrant No. 97-0-14.0-90..
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