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Analysis of the radical decay in a magnetic field by stochastic-Liouville equation
Volume 179, number I ,2
CHEMICAL PHYSICS LETTERS
12Aprill991
Analysis of the radical decay in a magnetic field by stochastic-Liouville equation Keisuke Tominaga ‘, Seigo Yamauchi * and Noboru Hirota Department of Chemistry,Faculty ofScrence, Kyoto University,Kyoto 606, Japan
Received 22 March 1990;in final form 7 January 1991.
A method for the simulation of the decay of the transient radical in a magnetic field is given by solving the stochastioLiouville equation numerically. The method is applied to analyze the decay of benzophenone ketyl radical in glycerol reported by Levin et al. It is suggested that the analysis of the decay by this method will provide information about the microscopic details of the geminate radical pairs.
1. Introduction The magnetic field effect (MFE) on the kinetics of chemical reactions is a subject of current interest [ 11. MFE has made a success in proving the existence of ,the spin-correlated geminate radical pairs (RP) and clarifying the details of reaction mechanisms. By the use of pulsed UV lasers it has now become possible to explore the details of MFE in the time domain with time resolution of nanoseconds [ 2,3 1. Such time-resolved studies have been made in micelles, biradicals, and homogeneous solutions [ I 1. MFE has its origin in the same mechanisms as those of the related phenomena of chemically induced nuclear and electron spin polarization (CIDNP and CIDEP) of which theories have been developed extensively during the last two decades [ 4131. The most general theoretical treatments are based on the stochastic-Liouville equation (SLE) which considers the effects of the diffusion, the spin Hamiltonian, the relaxation, and the reaction simultaneously [ 61. Since MFE takes place when the radicals in the pair are close enough to interact with each other, it is expected that the analysis of the time’ Present address: Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, USA.
’ Present address: Chemical Research Institute of Non-Aqueous Solutions, Tohoku University, Katahira 2-l-1, Sendai 980, Japan. Elsevier Science Publishers B.V. (North-Holland)
resolved MFE provides information about the microscopic details of RP, e.g. the interaction between the radicals, the diffusional and rotational motions, the spin relaxation, the geminate recombination, and so on. Such information should be obtained by solving time dependence of SLE. There have been a few studies to examine MFE from this point of view [ 7101, but they lack in generality. Furthermore, there is no study which includes the effect of the electron spin relaxation in the time dependence studies, though this is considered to be one of the main mechanisms of MFE [ 141. However, for a proper understanding of MFE a detailed analysis of time dependence of MFE based on SLE is desirable. Recently we have been concerned with CIDEP of the acetone ketyl radical in i-propanol to elucidate the microscopic properties of RP [ 15- 171. Wehave performed the simulation of the EPR spectra of RP and the temperature dependence of the spin polarization of the separated radicals (SR) by solving SLE numerically. It has been shown that the experimental results can be reproduced reasonably well in terms of a modified diffusion model which presuppose existence of the microscopic solvent structures [ 171. In this Letter we first present a method to solve SLE in the time domain numerically, and then demonstrate that this method can reproduce the result reported by Levin et al. [ 3 ] who studied MFE on the geminate recombination kinetics of the radical pairs 35
Volume 179,number 1,2
CHEMICALPHYSICS LETTERS
formed by the photolysis of benzophenone in glycerol.
12Aprill991
the homogeneous reactions, Kh. The concentration of the radical at time t, p(t), which is obtained from the experiment, is given by m
2. Theory and calculation
p(t)= rTr[p(r,i)]dr=
The spin motion, diffusion, spin relaxation, and reaction of radical pairs are described by the method of the density matrix p( r, t), which is the solution of SLE 161, ap(r,t)/at=-i[,n,p]+rp+Rp+~.
(1)
This equation implies p (r, t) to be spherically symmetric. The first term of eq. ( 1) describes the eleo tron spin motion of the radical pair which is governed by the spin Hamiltonian,
1 a2,Z2,S,-J(r)(2S1S2+f). m
(4)
We rewrite the stochastic-Liouville equation, eq. into the matrix form in the following fashion. and First, we choose AS, PsT~,~~s,~~,,~+~T+~~ h_,=_, as a basis set [ 111. All the terms in eq. (1) important in a high magnetic field are schematically illustrated in fig. 1 in this basis set except the diffusion operator. Eq. ( 1) can be numerically solved by discretizing t and r by employing the CrankNicholson implicit integration scheme [ 181, (l),
ap(r,t)lat=[p(r,,t,+I)-p(r;,tj)l/~t,
n
+
{Tr[P(r,t)]rdr. 0
0
(2)
a2dr, Wr2= [P(r,+,,tj+l)-@(r,, $+I) +p(ri--I,~~+~)+P(r~+,,
The symbols have their usual meanings [6]. J(r) is the exchange interaction between the unpaired electron spins of the pair. We employ an usual exponential decaying form, J(r) 4, exp [ --A( r- d) ] with r,, =A-‘5 In 10 giving the range over which J(r) decays to 1O-5 of its initial value [6]. d is a distance of the closest approach. r is a relative diffusion operator of the radical pair, defined as
fph1,
lj)-@(ri,
fj)
tj)1/2Ar2,
(5)
At and Ar are finite differences of t and r, respectively. r is divided into N points from r, to rN.Ar is taken to be small in the region where J(r) is large
e
T.qT.1
QkDV[V~+ (l/U)
pVU(r)]
, _____ I?’
or
___._._ Rd
(3) where p( r, t ) E rp ( r, t) and D is the mutual diffusion constant equal to the sum of the diffusion constants of the two radicals and U(r) is the potential acting between the pair radicals. In this case U(r) approximately equals to the spin exchange interaction, J(r),
which is dependent on the spin state. The third term in eq. ( 1) represents the relaxation between the spin states induced by two effects, the fluctuating field, R’, and the dipole-dipole interaction between the two unpaired electrons, Rd. The chemical reactions are taken into account in the last term, which includes both the geminate recombination in the singlet pair when the pair is in contact with each other, K”, and 36
Fig. I. A schematic illustration of the matrix elements of the terms of eq. ( 1) except for the diffusion term.
Volume 179, number I ,2
(r= 0.2 a for T< 2r,, ), and to be large at large r. The maximum value of Ar is 30 A. N= 100, and rNis fmed to 650 A which guarantees the convergence of the solutions of eq. ( I ). In the finite difference technique, the diffusion operator is replaced by matrix W with the elements, WFi=D
+
t
We assume that only the S-T0 mixing is important. This assumption is adequate in a high magnetic field, but is not valid in low and zero fields. The spin Hamiltonian is then expressed in the matrix form as = - &,SS’= &iro-ro = HToToSTo
- &,,, = &sTos
2
-
12 April 1991
CHEMICAL PHYSICS LETTERS
Ari Ari_i
=
-
H ST&To
r,+l Fa(~i+I)-ri_IFa(ri-r)
(6a)
Yi(AY/+Ari-1)
2 I#‘;,+, =D * ri Ar, ( At+, + Ar,_ 1) (
= HTOSSS
H~osTr,ro =
-
=
-
&o~oTos
=
Q,
HT&ToS=2J(r),
where 4?= ( 1/2fi)PH(g, --A .
F”(ri) + Art + Ari_1 ’
(6b) The matrix elements of the relaxation term were obtained by de Ranter et al. [ 191 in terms of the Redfield theory [ 201. As for the fluctuating field, the interaction Hamiltonian is
2 Ari_ I( Ari + Ari- I I
F”(ri) %= - (P/h) ]&Si ‘HI (l)
Arit Ari_1> ’
wN,N
=
-
[ V(N-
1 > / V(N)
WN,N- I = - [ VN-
1 wN-
1) / V(N)
-[V(N-1)/V(N)lWN-2,,-,
1.N)
(t>
1,
(10)
(6d)
where H, (t ) and H2( t) are the fluctuating local magnetic fields of radicals 1 and 2, respectively. The matrix element is, for example,
(6e)
R ;SToTo
(60
Here G= ;p’I HI (t ) ( 2, and rCis a correlation time. The dipole-dipole interaction Hamiltonian is
(W
& = (g2f12/fi2)
1 wN- I.N- I ,
+g2sz*H2
1 +4J(r)2tz]
= ~GT,/[
X [ ($*S2)/r3-3(S,~r)(&.r)/r’]
(11)
(12)
where a indicates the spin state and V(i) is the volume element at ri, given by V(i) =Ariri. F(r) = ( 1/kT)XY( r) /ar and in this case,
and the representative matrix element is
&s(r) = -&T,,(r)
= -FT+,~+,(~)
R$T~T~T~=
CWWJ(r)
where w0=gpH/h. We consider the geminate recombination in the singlet pair when the radicals are in contact. The matrix for K*has non-zero elements only for i= 1 (rid) expressed as
3 (gB4 =-&-IT-, &,,(d
(r)
=&dT)
= =O.
,
(7)
An absorbing wall condition, eqs. (6f) and (6g), is adopted in order to avoid the influence of the reflection at the wall [ 61. The matrix elements of eqs. (6a)- (6g) are derived from the particle conservation requirement [ 6,8 1, T V(i) wj,j=o *
(8)
J&s+,
- --~
5 h2r6
7c
ltw 0r2’ C
(13)
=- k ,
K”SToSTo.i= I --GJsTos,r=, = - fk 1
(14)
The homogeneous reaction is such that the radical undergoes a first-order reaction at any ri; 37
Volume 179, number 1,2
=KhT+lT+IT+IT+l
-Kh -
CHEMICAL PHYSICS LETTERS
T-IT-IT-IT-I
=
-
kh
12April 1991
.
(15)
The solution given by eq. ( 1) ‘now becomes a matrix equation,
= [It f ( W+H+RftRd+Ks+Kh)]PO’)
. (16)
The vector space in which p(j) is defined is the 6Ndimensional space formed from the product of the six-dimensional spin state and N-dimensional radial space. The matrix elements of Ware given by eq. (6). The matrices H, Rf, Rd, KS, and K” block diagonal, where the blocks are given by eqs. (p.),, ( 11), ( 13), (14), and (15). The formula of eq. (16) is recurrently solved under the initial condition as (17) p( tj) is numerically obtained in the finite-difference form, I
+PT+lT+ltfj>
ri)
+PT_Cl_I
ttj,
ri)
1v(i) *
(1’8)
We set At to be 100 ps. The calculation was performed by the HITAC M-680H and S-820/80 computer in the Computer Center of the Institute for Molecular Science.
3. Result and discussion Levin et al. recently reported MFE on the transient absorptions of the radical pair composed of the benzophenone ketyl radical and the radical produced by the abstraction of the hydroxyl hydrogen of pcresol [ 3 1. The decay curves of the transients at H= 0 and 3400 G are shown in fig. 2. Here we apply our method to simulate the results described above. We use the following parameters; d= 7 A, G= 1X 10” s-~, r,= lo-t2 s. The value of G was chosen so that the spin-lattice relaxation time of the separated radical is 10-5-10-” s [ I5 1. As an approximation Ag and Q are fixed to be 0 and 5 G, respectively [21]. 38
Fig. 2. The results obtained by Levin et al. [3] on the decay of the benzophenone ketyl radical in glycerol under H=O (lower) and 3400 G (upper) are indicated by dots. Calculated curves are obtained with Jo= IO9rad ST’, r,,=4 A, Is= IO9s, and kh=10' s-'; (a) H= 3400 and (b) 0 G in the normal diffusion process and (c) H=3400and (b) 0 G in the model diffusion process with r,= 14.8A and DC=D_/ IO, which are defined in the figure.
We employ the reported value of D ( 1.2 X 1O-l2 S- ’ m*) [ 31. The decay curves for H=O and 3400 G simulatedwithJ0=109rads-‘,r,,=4A,~=109~-’, and k,,= 10’ s-’ are shown in fig. 2. The exchange parameters, Jo= lo9 rad s-l and r,,= 4 A are taken to be the same as those calculated by Adrian for neutral radicals [ 51. The chemical reaction parameters, k, and k,,, are taken as lO”Oand lo5 s-‘, respectively. k,, is considered to be reasonable for the decay of the neutral radical in solution at room temperature [ 31. k, corresponds to the recombination rate constant of the geminate radical pair when they are in contact [ 61. The calculation reproduces decays of correct magnitudes both at H=3400 G and H=O, despite the fact that the S-T* , mixings are neglected at H= 0. However, the initial decays are much faster than those observed. Since the result calculated without the relaxation term produces MFE much smaller than observed, it is cqnsidered that the spin relaxation is considered to be important in causing MFE in the present case as suggested by Hayashi and Nagakura previously [ 141. In fig. 3 we show the result of the calculation obtained for various values of the parameters. It is seen that decay is very sensitive to the parameters. Es-
\ I+
Volume 179, number I ,2
a. e,
CHEMICAL PHYSICS LETTERS
1.0
B..
‘;::.-_y._._ --.--_:_--_-
_ ____
_‘__,__.J
1J4cxJG
0.5
b.. Fig. 3. Calculated decay curves for H= 3400 G. (a) The reaction parameters are fixed (k,,= 10’ s-l and k,= 1O’Os-l). Jo and r,, are lO”and4(-), 109and6(---),and 109rads-‘and4A (-,-.-). (b) The exchange interaction parameters are fixed (J,,1,=10’~rads-‘andr,,=4A).k,andk,are lOsand 101”(-), lOSand ICI’*(---), and 2~ lOSand ICI”s-l (-.-.-).
pecially the exchange and reaction kinetic parameters have drastic effects on the decay profile. Larger Jo values produce slower initial decays. One may obtain valid values for the parameters of the system by comparing the experimental results with those of the simulation carefully. We next examine the modified model of the diffusion process which was used to interpret the temperature dependence of the CIDEP of the acetone ketyl radical in i-propanol [ 171. Levin et al. concluded from a semi-quantitative analysis of their decay profile that the coefftcient of the mutual diffusion of the radicals of the pair is smaller than the sum of the macroscopic diffusion coefficients of the individual species [ 31. It is worthwhile to examine their claim by our method. Our model is that the diffusion coefficient for r< r, (see fig. 2), DC,is much smaller than that observed macroscopically, D,,, [ 17 1. The dashed lines in fig. 2 are obtained with r,= 14.8 A and D,= D,,,/ 10. As seen from the figure, this model
12 April 1991
gives very different time profiles from those obtained with the normal diffusion model, and gives a more satisfactory agreement with the experimental result in the early time (tc0.4 ps). This may indicate that the radicals in the pair stay together for a longer time than expected from the normal diffusion process. In conclusion, we have given a method to solve the stochastic-Liouville equation numerically in the time domain. We have demonstrated that this calculation reproduces the observed decays of the transient radicals in glycerol reasonably well. Accurate values of the parameters such as D, and Jo may be obtained by simulating the results measured under various conditions. This method can be used to interpret not only the decays of the transient absorptions but also the time profiles of CIDEP and CIDNP [ 22-241, especially the development of the spin-polarization. The analyses of MFE and CIDEP could offer information about RP that is complementary to each other. The present treatment is, however, not adequate in treating the phenomena in zero and low magnetic fields because of the neglect of the S-T+, mixings. A basis set including the nuclear spin states should be employed in order to take account of the contribution of the S-T,, mixings properly. Such a basis set, would have 64 elements of the density matrix, &&, where i andj refer to the electron spin states, S, T,,, T,, or T_,. a and h refer to the nuclear spin states, a or p. Furthermore, to be more precise the results should be summed over all the nuclear states instead of using a representative value for the hyperfine coupling constants.
Acknowledgement The authors thank the Computer Center of the Institute for Molecular Science, for. the use of the HITAC M-680H and S-820/80 computer.
References [ I] U.E. Steiner and T. Ulrich, Chem. Rev. 89 ( 1989) 5 1,
39
Volume 179, number I,2
CHEMICAL PHYSICS LETTERS
[2] Y. Sakaguchi and H. Hayashi, J. Phys. Chem. 88 (1984) 1437; Y. Tanimoto, K. Hasegawa, N. Okada, M. Itoh, K. Iwai, K. Sugioka, F. Takemura, R. Nakagaki and S. Nagakura, J. Phys. Chem. 93 (1989) 3586. [3]P.P. Levitt, I.V. Khudyakov and V.A. Kuzmin, J. Phys. Chem. 93 ( 1989) 208. [4]L.T. Muus, P.W. Atkins, K.A. McLauchlan and J.B. Pedersen, eds., Chemically induced magnetic polarization (Reidel Dordrecht, 1977). [5] F.J. Adrian, J. Chem. Phys. 57 (1972) 5107. [6] J.H. Freed and J.B. Pedersen, Advan. Magn. Reson. 8 (1976) 1. [7] R. Haberkorn, Chem. Phys. 26 (1977) 35. [8] H.-J. Werner, 2. Schulten and K. Schulten, J. Chem. Phys. 67 (1977) 646. [9] K. Lendi, Chem. Phys. 20 (1977) 135. [lo] J. Tang and J.R. Norris, Chem. Phys. Letters 92 (1982) 136. [ 111R.G. Mints and A.A. Pukhov, Chem. Phys. 87 ( 1984) 467. [ 121K. Luders and K.M. Salikov, Chem. Phys. 98 (1985) 259. [ 131AI. Shushin, Mol. Phys. 64 (1988) 65.
40
12 April 1991
[ 141H. Hayashi and S. Nagakura, Bull. Chem. Sot. Japan 57 (1984) 322. [ 151K. Tominaga, S. Yamauchi and N. Hirota, J. Chem. Phys. 88 (1988) 553. [ 161K. Tominaga, S. Yamauchi and N. Hirota, Chem. Phys. Letters 149 (I 988) 32. [ 171K. Tominaga, S. Yamauchi and N. Hirota, J. Chem. Phys. 92 (1990) 5175. [ 181R.D. Richtmeyer and K.W. Morton, Difference method for initial-value problems (Wiley-Interscience, New York, 1967). [ 191F.J.J. de Kanter, J.A. den Hollander, A.H. Huizer and R. Kaptein, Mol. Phys. 34 (1977) 857. [20] A.G. Redlield,Advan. Magn. Reson. 1 (1965) 1. [2 1 ] Landolt-Bumstein, Magnetic properties of free radicals (Springer, Berlin, 1980). [22] A. Angerhofer, M. Toporowicz, M.K. Bowman, J.R. Norris and H. Levanon, J. Phys. Chem. 92 (1988) 7164. [23] M. Pluschau, A. Zahl, K.P. Dime and H. van Willigen, J. Chem. Phys. 90 (1989) 3153. [24] G.L. Closs and O.D. Redwine, J. Am. Chem. Sot. 107 (1985) 6131.
CHEMICAL PHYSICS LETTERS
12Aprill991
Analysis of the radical decay in a magnetic field by stochastic-Liouville equation Keisuke Tominaga ‘, Seigo Yamauchi * and Noboru Hirota Department of Chemistry,Faculty ofScrence, Kyoto University,Kyoto 606, Japan
Received 22 March 1990;in final form 7 January 1991.
A method for the simulation of the decay of the transient radical in a magnetic field is given by solving the stochastioLiouville equation numerically. The method is applied to analyze the decay of benzophenone ketyl radical in glycerol reported by Levin et al. It is suggested that the analysis of the decay by this method will provide information about the microscopic details of the geminate radical pairs.
1. Introduction The magnetic field effect (MFE) on the kinetics of chemical reactions is a subject of current interest [ 11. MFE has made a success in proving the existence of ,the spin-correlated geminate radical pairs (RP) and clarifying the details of reaction mechanisms. By the use of pulsed UV lasers it has now become possible to explore the details of MFE in the time domain with time resolution of nanoseconds [ 2,3 1. Such time-resolved studies have been made in micelles, biradicals, and homogeneous solutions [ I 1. MFE has its origin in the same mechanisms as those of the related phenomena of chemically induced nuclear and electron spin polarization (CIDNP and CIDEP) of which theories have been developed extensively during the last two decades [ 4131. The most general theoretical treatments are based on the stochastic-Liouville equation (SLE) which considers the effects of the diffusion, the spin Hamiltonian, the relaxation, and the reaction simultaneously [ 61. Since MFE takes place when the radicals in the pair are close enough to interact with each other, it is expected that the analysis of the time’ Present address: Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, USA.
’ Present address: Chemical Research Institute of Non-Aqueous Solutions, Tohoku University, Katahira 2-l-1, Sendai 980, Japan. Elsevier Science Publishers B.V. (North-Holland)
resolved MFE provides information about the microscopic details of RP, e.g. the interaction between the radicals, the diffusional and rotational motions, the spin relaxation, the geminate recombination, and so on. Such information should be obtained by solving time dependence of SLE. There have been a few studies to examine MFE from this point of view [ 7101, but they lack in generality. Furthermore, there is no study which includes the effect of the electron spin relaxation in the time dependence studies, though this is considered to be one of the main mechanisms of MFE [ 141. However, for a proper understanding of MFE a detailed analysis of time dependence of MFE based on SLE is desirable. Recently we have been concerned with CIDEP of the acetone ketyl radical in i-propanol to elucidate the microscopic properties of RP [ 15- 171. Wehave performed the simulation of the EPR spectra of RP and the temperature dependence of the spin polarization of the separated radicals (SR) by solving SLE numerically. It has been shown that the experimental results can be reproduced reasonably well in terms of a modified diffusion model which presuppose existence of the microscopic solvent structures [ 171. In this Letter we first present a method to solve SLE in the time domain numerically, and then demonstrate that this method can reproduce the result reported by Levin et al. [ 3 ] who studied MFE on the geminate recombination kinetics of the radical pairs 35
Volume 179,number 1,2
CHEMICALPHYSICS LETTERS
formed by the photolysis of benzophenone in glycerol.
12Aprill991
the homogeneous reactions, Kh. The concentration of the radical at time t, p(t), which is obtained from the experiment, is given by m
2. Theory and calculation
p(t)= rTr[p(r,i)]dr=
The spin motion, diffusion, spin relaxation, and reaction of radical pairs are described by the method of the density matrix p( r, t), which is the solution of SLE 161, ap(r,t)/at=-i[,n,p]+rp+Rp+~.
(1)
This equation implies p (r, t) to be spherically symmetric. The first term of eq. ( 1) describes the eleo tron spin motion of the radical pair which is governed by the spin Hamiltonian,
1 a2,Z2,S,-J(r)(2S1S2+f). m
(4)
We rewrite the stochastic-Liouville equation, eq. into the matrix form in the following fashion. and First, we choose AS, PsT~,~~s,~~,,~+~T+~~ h_,=_, as a basis set [ 111. All the terms in eq. (1) important in a high magnetic field are schematically illustrated in fig. 1 in this basis set except the diffusion operator. Eq. ( 1) can be numerically solved by discretizing t and r by employing the CrankNicholson implicit integration scheme [ 181, (l),
ap(r,t)lat=[p(r,,t,+I)-p(r;,tj)l/~t,
n
+
{Tr[P(r,t)]rdr. 0
0
(2)
a2dr, Wr2= [P(r,+,,tj+l)-@(r,, $+I) +p(ri--I,~~+~)+P(r~+,,
The symbols have their usual meanings [6]. J(r) is the exchange interaction between the unpaired electron spins of the pair. We employ an usual exponential decaying form, J(r) 4, exp [ --A( r- d) ] with r,, =A-‘5 In 10 giving the range over which J(r) decays to 1O-5 of its initial value [6]. d is a distance of the closest approach. r is a relative diffusion operator of the radical pair, defined as
fph1,
lj)-@(ri,
fj)
tj)1/2Ar2,
(5)
At and Ar are finite differences of t and r, respectively. r is divided into N points from r, to rN.Ar is taken to be small in the region where J(r) is large
e
T.qT.1
QkDV[V~+ (l/U)
pVU(r)]
, _____ I?’
or
___._._ Rd
(3) where p( r, t ) E rp ( r, t) and D is the mutual diffusion constant equal to the sum of the diffusion constants of the two radicals and U(r) is the potential acting between the pair radicals. In this case U(r) approximately equals to the spin exchange interaction, J(r),
which is dependent on the spin state. The third term in eq. ( 1) represents the relaxation between the spin states induced by two effects, the fluctuating field, R’, and the dipole-dipole interaction between the two unpaired electrons, Rd. The chemical reactions are taken into account in the last term, which includes both the geminate recombination in the singlet pair when the pair is in contact with each other, K”, and 36
Fig. I. A schematic illustration of the matrix elements of the terms of eq. ( 1) except for the diffusion term.
Volume 179, number I ,2
(r= 0.2 a for T< 2r,, ), and to be large at large r. The maximum value of Ar is 30 A. N= 100, and rNis fmed to 650 A which guarantees the convergence of the solutions of eq. ( I ). In the finite difference technique, the diffusion operator is replaced by matrix W with the elements, WFi=D
+
t
We assume that only the S-T0 mixing is important. This assumption is adequate in a high magnetic field, but is not valid in low and zero fields. The spin Hamiltonian is then expressed in the matrix form as = - &,SS’= &iro-ro = HToToSTo
- &,,, = &sTos
2
-
12 April 1991
CHEMICAL PHYSICS LETTERS
Ari Ari_i
=
-
H ST&To
r,+l Fa(~i+I)-ri_IFa(ri-r)
(6a)
Yi(AY/+Ari-1)
2 I#‘;,+, =D * ri Ar, ( At+, + Ar,_ 1) (
= HTOSSS
H~osTr,ro =
-
=
-
&o~oTos
=
Q,
HT&ToS=2J(r),
where 4?= ( 1/2fi)PH(g, --A .
F”(ri) + Art + Ari_1 ’
(6b) The matrix elements of the relaxation term were obtained by de Ranter et al. [ 191 in terms of the Redfield theory [ 201. As for the fluctuating field, the interaction Hamiltonian is
2 Ari_ I( Ari + Ari- I I
F”(ri) %= - (P/h) ]&Si ‘HI (l)
Arit Ari_1> ’
wN,N
=
-
[ V(N-
1 > / V(N)
WN,N- I = - [ VN-
1 wN-
1) / V(N)
-[V(N-1)/V(N)lWN-2,,-,
1.N)
(t>
1,
(10)
(6d)
where H, (t ) and H2( t) are the fluctuating local magnetic fields of radicals 1 and 2, respectively. The matrix element is, for example,
(6e)
R ;SToTo
(60
Here G= ;p’I HI (t ) ( 2, and rCis a correlation time. The dipole-dipole interaction Hamiltonian is
(W
& = (g2f12/fi2)
1 wN- I.N- I ,
+g2sz*H2
1 +4J(r)2tz]
= ~GT,/[
X [ ($*S2)/r3-3(S,~r)(&.r)/r’]
(11)
(12)
where a indicates the spin state and V(i) is the volume element at ri, given by V(i) =Ariri. F(r) = ( 1/kT)XY( r) /ar and in this case,
and the representative matrix element is
&s(r) = -&T,,(r)
= -FT+,~+,(~)
R$T~T~T~=
CWWJ(r)
where w0=gpH/h. We consider the geminate recombination in the singlet pair when the radicals are in contact. The matrix for K*has non-zero elements only for i= 1 (rid) expressed as
3 (gB4 =-&-IT-, &,,(d
(r)
=&dT)
= =O.
,
(7)
An absorbing wall condition, eqs. (6f) and (6g), is adopted in order to avoid the influence of the reflection at the wall [ 61. The matrix elements of eqs. (6a)- (6g) are derived from the particle conservation requirement [ 6,8 1, T V(i) wj,j=o *
(8)
J&s+,
- --~
5 h2r6
7c
ltw 0r2’ C
(13)
=- k ,
K”SToSTo.i= I --GJsTos,r=, = - fk 1
(14)
The homogeneous reaction is such that the radical undergoes a first-order reaction at any ri; 37
Volume 179, number 1,2
=KhT+lT+IT+IT+l
-Kh -
CHEMICAL PHYSICS LETTERS
T-IT-IT-IT-I
=
-
kh
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.
(15)
The solution given by eq. ( 1) ‘now becomes a matrix equation,
= [It f ( W+H+RftRd+Ks+Kh)]PO’)
. (16)
The vector space in which p(j) is defined is the 6Ndimensional space formed from the product of the six-dimensional spin state and N-dimensional radial space. The matrix elements of Ware given by eq. (6). The matrices H, Rf, Rd, KS, and K” block diagonal, where the blocks are given by eqs. (p.),, ( 11), ( 13), (14), and (15). The formula of eq. (16) is recurrently solved under the initial condition as (17) p( tj) is numerically obtained in the finite-difference form, I
+PT+lT+ltfj>
ri)
+PT_Cl_I
ttj,
ri)
1v(i) *
(1’8)
We set At to be 100 ps. The calculation was performed by the HITAC M-680H and S-820/80 computer in the Computer Center of the Institute for Molecular Science.
3. Result and discussion Levin et al. recently reported MFE on the transient absorptions of the radical pair composed of the benzophenone ketyl radical and the radical produced by the abstraction of the hydroxyl hydrogen of pcresol [ 3 1. The decay curves of the transients at H= 0 and 3400 G are shown in fig. 2. Here we apply our method to simulate the results described above. We use the following parameters; d= 7 A, G= 1X 10” s-~, r,= lo-t2 s. The value of G was chosen so that the spin-lattice relaxation time of the separated radical is 10-5-10-” s [ I5 1. As an approximation Ag and Q are fixed to be 0 and 5 G, respectively [21]. 38
Fig. 2. The results obtained by Levin et al. [3] on the decay of the benzophenone ketyl radical in glycerol under H=O (lower) and 3400 G (upper) are indicated by dots. Calculated curves are obtained with Jo= IO9rad ST’, r,,=4 A, Is= IO9s, and kh=10' s-'; (a) H= 3400 and (b) 0 G in the normal diffusion process and (c) H=3400and (b) 0 G in the model diffusion process with r,= 14.8A and DC=D_/ IO, which are defined in the figure.
We employ the reported value of D ( 1.2 X 1O-l2 S- ’ m*) [ 31. The decay curves for H=O and 3400 G simulatedwithJ0=109rads-‘,r,,=4A,~=109~-’, and k,,= 10’ s-’ are shown in fig. 2. The exchange parameters, Jo= lo9 rad s-l and r,,= 4 A are taken to be the same as those calculated by Adrian for neutral radicals [ 51. The chemical reaction parameters, k, and k,,, are taken as lO”Oand lo5 s-‘, respectively. k,, is considered to be reasonable for the decay of the neutral radical in solution at room temperature [ 31. k, corresponds to the recombination rate constant of the geminate radical pair when they are in contact [ 61. The calculation reproduces decays of correct magnitudes both at H=3400 G and H=O, despite the fact that the S-T* , mixings are neglected at H= 0. However, the initial decays are much faster than those observed. Since the result calculated without the relaxation term produces MFE much smaller than observed, it is cqnsidered that the spin relaxation is considered to be important in causing MFE in the present case as suggested by Hayashi and Nagakura previously [ 141. In fig. 3 we show the result of the calculation obtained for various values of the parameters. It is seen that decay is very sensitive to the parameters. Es-
\ I+
Volume 179, number I ,2
a. e,
CHEMICAL PHYSICS LETTERS
1.0
B..
‘;::.-_y._._ --.--_:_--_-
_ ____
_‘__,__.J
1J4cxJG
0.5
b.. Fig. 3. Calculated decay curves for H= 3400 G. (a) The reaction parameters are fixed (k,,= 10’ s-l and k,= 1O’Os-l). Jo and r,, are lO”and4(-), 109and6(---),and 109rads-‘and4A (-,-.-). (b) The exchange interaction parameters are fixed (J,,1,=10’~rads-‘andr,,=4A).k,andk,are lOsand 101”(-), lOSand ICI’*(---), and 2~ lOSand ICI”s-l (-.-.-).
pecially the exchange and reaction kinetic parameters have drastic effects on the decay profile. Larger Jo values produce slower initial decays. One may obtain valid values for the parameters of the system by comparing the experimental results with those of the simulation carefully. We next examine the modified model of the diffusion process which was used to interpret the temperature dependence of the CIDEP of the acetone ketyl radical in i-propanol [ 171. Levin et al. concluded from a semi-quantitative analysis of their decay profile that the coefftcient of the mutual diffusion of the radicals of the pair is smaller than the sum of the macroscopic diffusion coefficients of the individual species [ 31. It is worthwhile to examine their claim by our method. Our model is that the diffusion coefficient for r< r, (see fig. 2), DC,is much smaller than that observed macroscopically, D,,, [ 17 1. The dashed lines in fig. 2 are obtained with r,= 14.8 A and D,= D,,,/ 10. As seen from the figure, this model
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gives very different time profiles from those obtained with the normal diffusion model, and gives a more satisfactory agreement with the experimental result in the early time (tc0.4 ps). This may indicate that the radicals in the pair stay together for a longer time than expected from the normal diffusion process. In conclusion, we have given a method to solve the stochastic-Liouville equation numerically in the time domain. We have demonstrated that this calculation reproduces the observed decays of the transient radicals in glycerol reasonably well. Accurate values of the parameters such as D, and Jo may be obtained by simulating the results measured under various conditions. This method can be used to interpret not only the decays of the transient absorptions but also the time profiles of CIDEP and CIDNP [ 22-241, especially the development of the spin-polarization. The analyses of MFE and CIDEP could offer information about RP that is complementary to each other. The present treatment is, however, not adequate in treating the phenomena in zero and low magnetic fields because of the neglect of the S-T+, mixings. A basis set including the nuclear spin states should be employed in order to take account of the contribution of the S-T,, mixings properly. Such a basis set, would have 64 elements of the density matrix, &&, where i andj refer to the electron spin states, S, T,,, T,, or T_,. a and h refer to the nuclear spin states, a or p. Furthermore, to be more precise the results should be summed over all the nuclear states instead of using a representative value for the hyperfine coupling constants.
Acknowledgement The authors thank the Computer Center of the Institute for Molecular Science, for. the use of the HITAC M-680H and S-820/80 computer.
References [ I] U.E. Steiner and T. Ulrich, Chem. Rev. 89 ( 1989) 5 1,
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[2] Y. Sakaguchi and H. Hayashi, J. Phys. Chem. 88 (1984) 1437; Y. Tanimoto, K. Hasegawa, N. Okada, M. Itoh, K. Iwai, K. Sugioka, F. Takemura, R. Nakagaki and S. Nagakura, J. Phys. Chem. 93 (1989) 3586. [3]P.P. Levitt, I.V. Khudyakov and V.A. Kuzmin, J. Phys. Chem. 93 ( 1989) 208. [4]L.T. Muus, P.W. Atkins, K.A. McLauchlan and J.B. Pedersen, eds., Chemically induced magnetic polarization (Reidel Dordrecht, 1977). [5] F.J. Adrian, J. Chem. Phys. 57 (1972) 5107. [6] J.H. Freed and J.B. Pedersen, Advan. Magn. Reson. 8 (1976) 1. [7] R. Haberkorn, Chem. Phys. 26 (1977) 35. [8] H.-J. Werner, 2. Schulten and K. Schulten, J. Chem. Phys. 67 (1977) 646. [9] K. Lendi, Chem. Phys. 20 (1977) 135. [lo] J. Tang and J.R. Norris, Chem. Phys. Letters 92 (1982) 136. [ 111R.G. Mints and A.A. Pukhov, Chem. Phys. 87 ( 1984) 467. [ 121K. Luders and K.M. Salikov, Chem. Phys. 98 (1985) 259. [ 131AI. Shushin, Mol. Phys. 64 (1988) 65.
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[ 141H. Hayashi and S. Nagakura, Bull. Chem. Sot. Japan 57 (1984) 322. [ 151K. Tominaga, S. Yamauchi and N. Hirota, J. Chem. Phys. 88 (1988) 553. [ 161K. Tominaga, S. Yamauchi and N. Hirota, Chem. Phys. Letters 149 (I 988) 32. [ 171K. Tominaga, S. Yamauchi and N. Hirota, J. Chem. Phys. 92 (1990) 5175. [ 181R.D. Richtmeyer and K.W. Morton, Difference method for initial-value problems (Wiley-Interscience, New York, 1967). [ 191F.J.J. de Kanter, J.A. den Hollander, A.H. Huizer and R. Kaptein, Mol. Phys. 34 (1977) 857. [20] A.G. Redlield,Advan. Magn. Reson. 1 (1965) 1. [2 1 ] Landolt-Bumstein, Magnetic properties of free radicals (Springer, Berlin, 1980). [22] A. Angerhofer, M. Toporowicz, M.K. Bowman, J.R. Norris and H. Levanon, J. Phys. Chem. 92 (1988) 7164. [23] M. Pluschau, A. Zahl, K.P. Dime and H. van Willigen, J. Chem. Phys. 90 (1989) 3153. [24] G.L. Closs and O.D. Redwine, J. Am. Chem. Sot. 107 (1985) 6131.