Introduction to Dynamic Spin Chemistry: Magnetic Field Effects upon Chemical and Biochemical Reactions (World Scientific Lecture and Course Notes in Chemistry, Vol. 8)

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Introduction to Dynamic Spin Chemistry Magnetic Field Effects on Chemical and Biochemical Reactions

WORLD SCIENTIFIC LECTURE AND COURSE NOTES IN CHEMISTRY Editor-in-charge: S. H. Lin

VOl. 1:

Monte Carlo Methods in Ab lnitio Quantum Chemistry B. L. Hammond, W. A. Lester, Jr. & P. J. Reynolds

VOl. 2:

Quantum chemistry Aided Design of Organic Polymers: An Introduction to the Quantum Chemistry of Polymers and Its Applications J. -M. Andre, J. Delhalle & J. -L. Bredas

VOl. 4:

The Physical Chemistry of Biopolymer Solutions: Application of Physical Techniques to the Study of Proteins and Nuclei Acids R. F. Steiner& L. Garone

VOl. 5:

Theoretical HeterogeneousCatalysis R. A. van Santen

Vol. 6:

Density Matrix Method and Femtosecond Processes S. H. Lin, R. Alden, R. Islampour, H. Ma & A. A. Villaeys

VOl. 7:

Spectroscopy and Dynamics of Orientationally Structured Adsorbates V. M. Rozenbaum & S. H. Lin

voi. a:

Introduction to Dynamic Spin Chemistry: Magnetic Field Effects on Chemical and Biochemical Reactions H. Hayashi

World Scientific Lecture and Course Notes in Chemistry - Vol. 8

Introduction to Dynamic Spin Chemistry Magnetic Field Effects on Chemical and Biochemical Reactions

Hisaharu Hayashi RII<EN,The Institute of Physical and Chemical Research, Japan

r pWorld Scientific N E W JERSEY * L O N D O N

-

SINGAPORE

SHANGHAI

*

HONG KONG * TAIPEI * BANGALORE

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

INTRODUCTION TO DYNAMIC SPIN CHEMISTRY: MAGNETIC FIELD EFFECTS ON CHEMICAL AND BIOCHEMICAL REACTIONS Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts rhereoj may nor be reproduced in any form or by any means, elecrronic or mechanical, including photocopying, recording or any information storage and retrieval sysrem now known or lo be invented, without wrirten permissionfrom the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-423-5

Printed in Singapore by World Scientific Printers (S)Pte Ltd

Contents Preface 1

Magnetic Properties of Electron and Nuclear Spins 1.1 Scalar and Vector Products 1.2 Lorentz Force 1.3 Potential Energy of a Magnetic Moment 1.4 Orbital Magnetic Moment 1.5 Spin Magnetic Moments

2

Introduction to Electron Spin Resonance and Nuclear Magnetic Resonance 2.1 Photochemical Primary Processes 2.2 ESR (Electron Spin Resonance) 2.3 NMR (Nuclear Magnetic Resonance). 2.4 The Hyperfine Coupling (HFC) in ESR 2.5 The Spin-Spin Coupling in NMR References

3

4

1 1 1 2 4 5

9 9 11 14 15 17 19

The Radical Pair Mechanism 3.1 Chemical Reactions through Radical Pairs 3.2 Singlet and Triplet States of Radical Pairs and Their Mixing 3.3 Kaptain's Theory for the S-ToConversion 3.4 Approximations of Recombination Probabilities for S, T, and F-Precursors References

21 21 22 25

Chemically Induced Dynamic Nuclear Polarization (CIDNP) Discovery of Chemically Induced Magnetic Polarization and ESR Spectra of Radical Pairs 4.2 Theoretical Interpretation of CIDNP by the Radical Pair Mechanism 4.3 Examples of CIDNP References

35

29 30

4.1

V

35 39 42 45

vi

5

6

7

8

Chemically Induced Dynamic Electron Polarization (CIDEP) 5.1 Historical Introduction 5.2 Theoretical Interpretation of CIDEP by the Radical Pair Mechanism 5.3 Theoretical Interpretation of CIDEP by the Triplet Mechanism 5.4 Examples of CIDEP due to the RPM and the Triplet Mechanism 5.5 CIDEP due to Spin-Conelated Radical Pairs References Magnetic Field Effects upon Chemical Reactions due to the Radical Pair Mechanism (RPM ) 6.1 Historical Introduction 6.2 Classification of Magnetic Field effects due to the RPM 6.3 Magnetic Field Effects on Product Yields due to the RPM 6.4 Magnetic Field Effects on Thermal Reactions 6.5 Magnetic Field Effects on Radiation Reactions 6.6 Magnetic Field Effects on Photochemical Reactions 6.7 Magnetic Isotope Effects References Magnetic Field Effects due to the Relaxation Mechanism 7.1 Magnetic Field and Magnetic Isotope Effects Observed in Micellar Solutions 7.2 Proposal of the Relaxation Mechanism 7.3 Magnetic Field Effects on Chemical Reactions through Radical Pairs Involving Heavy Atom-Centered Radicals 7.4 Theoretical Analysis of Relaxation Rates References Magnetic Field Effects on Chemical Reactions through Biradicals 8.1 Historical Introduction 8.2 Magnetic Field Effects on Thermal Reactions through Biradicals 8.3 Magnetic Field Effects on Photochemical Reactions through Biradicals References

47 47 47 54 56 61

64

75 75 77 80 83 85 88 94 95 97 97 101 103 108

112

117 117 119 120 125

vii

9

Magnetic Isotope Effects (MIEs) 9.1 Introduction 9.2 MIEs of 13C/12C 9.3 MIEs of I5N/l4N 9.4 MIEs of 170/160, I8O 9.5 MIEs of 29Si/28Si,30Si 9.6 MIEs of 33S/32S,34S,36S 9.7 MIEs of 73Ge/70Ge,72Ge,74Ge,76Ge 9.8 MIEofSn 9.9 MIEs of 235U/234U, 238U References

127 127 127 131 131 132 133 134 136 136 136

10

Triplet Mechanism (TM) 10.1 Introduction 10.2 Magnetic Field Effects due to the TM 10.3 CIDEP due to the d-type TM References

139 139 141 149 154

11

Theoretical Analysis with the Stochastic Liouville Equation 11.1 Density Matrix Method 11.2 Density Matrix Treatment for S-ToConversion of Radical Pairs 11.3 Effects of Spin Relaxation on Dynamic Behavior of Radical Pairs 11.4 Theoretical Analysis of MFEs, CIDNP, and CIDEP with the Density Matrix Method References

157 157

12

Effects of Ultra-High Magnetic Fields upon Chemical Reactions 12.1 Historical Introduction 12.2 Effects of Ultra-High Magnetic Fields due to the AgM 12.3 Effects of Ultra-High Magnetic Fields due to the RM 12.4 Effects of Ultra-High Magnetic Fields upon Reactions of Kramers Doublet Species References

159 165 169 171 177 177 177 184 193 195

viii

13

Effects of Magnetic Fields of High Spin Species 13.1 Magnetic Field Effects of Luminescence 13.2 CIDEP of High Spin Species 13.3 Magnetic Field Effects of High Spin Species References

197 197 20 1 204 21 1

14

Optical Detected ESR and Reaction Yield Detected ESR 14.1 Historical Introduction 14.2 ODESR Measurements of Radical Pairs 14.3 RYDESR Studies of Radical Pairs 14.4 ODESR Studies of Radical Pairs References

217 217 219 22 1 227 232

15

Magnetic Field Effects upon Biochemical Reactions and Biological Processes 15.1 Historical Introduction 15.2 Magnetic Field Effects on Enzyme Reactions 15.3 Effects of Low Magnetic Fields on Chemical Reactions through Radical Pairs References

Index

233 233 234 239 245 249

Preface A radical is any molecule or atom which possesses one unpaired electron. Radicals are of great importance since they often appear as intermediates in thermal, radiation, and photochemical reactions. It is noteworthy that radicals are usually produced in pairs through the above-mentioned reactions. Such a pair of radicals has been called “a radical pair”. Thus, radicals and radical pairs play very important roles in chemical and biochemical reactions. For example, they appear in such important processes as polymerisation and combustion reactions, radiation curing and lithography, photosynthesis in plants and bacteria, autooxidation and aging of organic molecules, polymers, and living organisms, and stratospheric ozone depletion by freons. This is a book about reaction dynamics of radicals and radical pairs in the presence and absence of an external magnetic field and its applications to many related phenomena in chemistry, physics, and biology. It grew out of a lecture which was given every year from 1997 to 2001 by the author to the Department of Electronic Chemistry, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology. Electron Spin Resonance (ESR) or Electron Paramagnetic Resonance (EPR) is one of the most powerful methods for investigating the structure and reaction dynamics of radicals and radical pairs. In ESR experiments, an external magnetic field is applied to radicals and radical pairs, which are split into their individual spin sub-levels by the field. A resonance microwave is also applied to them and induces the transitions between their individual spin sub-levels. At first, most ESR measurements were carried out under steady-state conditions because usual ESR apparatus adopted 100 kHz field-modulation in order to improve their sensitivity. This means that the time resolution of such measurements with field-modulation should be longer than 0.1 ms. Afterwards, ESR measurements were also made during chemical reactions with much shorter time resolutions of up to 10 ns. The intensities of ESR spectra of some reacting systems were found to show emission and/or enhanced absorption, but other characteristics such as line frequencies and line width to be normal. Such peculiar ESR signals have been called “Chemically Induced Dynamic Electron Polarization (CIDEP)”. Similar anomalous signals were also observed for nuclear magnetic resonance (NMFt) spectra when they were measured during chemical reactions and have been called “Chemically Induced Dynamic Nuclear Polarization (CIDNP)”. CIDEP and CIDNP are due to nonequilibrium populations in the electron spin sub-levels of reacting radicals and in the nuclear spin sub-levels of reaction products, respectively. CIDEP and CIDNE’ have been successfully explained by the radical pair mechanism, according to which the singlet and triplet states of radical pairs can be mixed with each other through the Zeeman interaction ix

X

between radicals and the external magnetic field and the hyperfine interaction between electron and nuclear spins inside the radical pairs. Because the populations of electron and nuclear spin sub-levels can be changed by ordinary magnetic fields through the radical pair mechanism, not only the yield of reaction products generated through radical pairs but also the reaction rates of radical pairs were also expected to be affected by the fields. Indeed, such effects have been observed for many chemical and biochemical reactions through radical pair, being called “Magnetic Field Effects (MFEs)” upon chemical reactions. Discoveries of CIDEP, CIDNP, and MFEs in chemical reactions through radical pairs brought about the advent of a new research field, which has been called “Dynamic Spin Chemistry”. Dynamic spin chemistry encompasses not only basic theoretical and experimental investigations of all phenomena in which free radicals and other species possessing unpaired electron spins occur but also many important applied researches including photosynthesis in plants, biochemical reactions in human and animal bodies, and various industrial processes. Let us show two interesting problems concerning MFEs on biological processes. The homing of pigeons has been considered to be due to a magnetic compass inside the brain of pigeons. There have been many scientific papers reporting that environmental electromagnetic radiation causes children cancer. In my opinion, these two important phenomena have neither been proven by reproducible experiments nor been explained by reliable theories. Many interesting problems including the above-mentioned ones have been left for further investigations of dynamic spin chemistry in the 21st century. Because the radical pair mechanism is well understood and based on established science, I can say that dynamic spin chemistry provides a good guidance for researchers of MFEs on biological processes as well as other processes. This is not the first book on CIDEP, CIDNP, and MFEs in chemical and biochemical reactions, but most of the other books were written for specialists of dynamic spin chemistry without explaining how to derive its basic principles. This book differs from others in the emphasis placed on making it a learning text for those with minimum knowledge in quantum mechanics. At the same time, this text serves a secondary purpose of showing how the field of dynamic spin chemistry has been established from a veil of secrecy, historically important papers being introduced to the beginners of this field. The first half of this book explains the basic principles in magnetic properties of electron and nuclear spins, ESR and NMR, the radical pair mechanism, CIDNP, CIDEP, and MFEs upon chemical reactions due to the radical pair mechanism. The second half describes typical results on dynamic spin chemistry, including MFEs due to the relaxation mechanism, MFEs on chemical reactions through biradicals, magnetic isotope effects, triplet mechanism, theoretical analysis of dynamic spin chemistry, effects

xi

of ultra-high magnetic fields upon chemical reactions, MFEs on chemical reactions through high spin species, optical detected ESR and reaction yield detected ESR and MFEs upon biochemical reactions and biochemical processes. I hope that this book will be useful not only for graduate and senior undergraduate students of pure and applied chemistry, physics, and biology but also for academic and industrial researchers in the fields of dynamic spin chemistry, photochemistry, photophysics, photobiology, magnetic resonance, electromagnetism, environmental science and nano-scale technologies. The author expresses his sincere thanks to Prof. Saburo Nagakura, President of Kanagawa Academy of Science and Technology, for his encouraging the author to write this book, to Prof. Sheng Hsien Lin, President of Institute of Atomic and Molecular Sciences, for his inviting the author to publish this book at World Scientific Publishing Company, and to the editors of World Scientific Publishing Company for their editorial efforts. The author is also thankful to those publishers which kindly permitted him to use their figures and tables in this book. Tokyo, January 2003

Hisaharu Hayashi

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1

1. Magnetic Properties of Electron and Nuclear Spins 1.1 Scalar and Vector Products In this volume, we will often use the scalar and vector products of two vectors. Thus, we will start with a review of vector algebra. The scalar product of vectors a and b is defined as a* b = axbx+ ayby+ azbz= abcosB. (1-1) Here, B is the angle between a and b as shown in Fig. 1-l(a). The scalar product is also called the inner product, which means the degree of the overlap between a and b. The vector product (a x b) of vectors a and b is defined as (1-2a) (a x b)x= arbz- a,by, (I -2b) (a x b)Y= a$, - a&,, (a x b)z = axby- a,&. (1-2c) Here, 6 is the angle between a and b as shown in Fig. l-l(b). The vector product is also called the outer product, which meaqs that its value is the space (5') spanned between a and b and that its direction is the right-handed screw one turning from a to b. 1 a x b =&sin@. (1-3)

1

bcos 6 Fig. 1-l(a). The scalar product of a and b.

Fig. 1-l(b). The vector product of a and b.

1.2 Lorentz Force In the 19" century, H. A. Lorentz found experimentally that a charge with q moving with a velocity of v in a magnetic field (B) felt the following force (F): (1) The magnitude of F was found to be proportional to qvBsin 0 . (2) F was found to be directed at right angles to the plane of spanned by v and B as shown in Fig. 1-2. Thus, F can be represented by the vector product of v and B.

F = aqv x B. (1-4) Here, a is the proportional coefficient. If a is assumed to be a dimensionless constant having a=l,B can be defined as follow: F = q v x B.

(1-5)

4 Fig. 1-2. The Lorentz force.

2

The dimension of B defined by Eq. (1-5) is called Tesla (T). Thus, we can obtain the dimension of T ([TI) with those of F ([F] = "1 = [kg m s-~]),q ([q] = [C] = [A s]), and v ([v] = [m s-I]), using Eq. (1-5).

(1-6a) [kg m s - ~=] [A sl [m s-ll [TI, (1-6b) [TI = [kg s - A"]. ~ T = lo-' mT. Conventionally, the unit of gauss (G) is also used for B, where 1 G = More generally, the force induced on a charge in the presence of both the electric (E) and magnetic fields is given by

F=q(E+vxB). This force is called the Lorentz force. 1.3 Potential Energy of a Magnetic Moment Let us consider a charge with its charge density of p which moves at its velocity of v as shown in Fig. 1-3. In this case, its current density (j) becomes

j = pv. (1-8) Thus, the force (f) induced to this unit volume by a magnetic field of B is given from Eq. (1-5) by f = pv x B = j x B. (1-9) When the cross section of this wire is S, the force (dF) induced to the wire with its length of dL becomes as follows: (1-10) dF = fSdL = j x B SdL. Because the total current of this wire (I) is Sj, dF can be represented as (1-11) d F = I x BdL =ZdL x B.

(1-7)

dF

1' Fig. 1-3. The force induced by B to I in a wire.

Here, it is obvious that IdL = ZdL, because I N dL as shown in Fig. 1-3. Then let us consider the forces induced by B to a rectangular coil (ax b) with a current of I as shown in Fig. 1-4.

B

F2 Fig. 1-4. Forces induced by B to a rectangular coil (ax b) with a current of I.

3

FI and FZ are given from Eq. (1-11) by (1-12) F I = IF11 = IF1'1= BZa, (1-13) Fz = IF21 = JF2')= BZ~COSB. The torque (N) induced by F1 and F1' to this coil can be obtained as shown in Fig 1-4(b). N = F,bsinB= BIabsin8. (1-14) Thus, N can be represented in terms of a vector product as follows: N = Iabn x B. (1-15) Here, n is the unit vector perpendicular to the coil plane as shown in Fig. 1-4. Let us define an important vector in spin chemistry, which is the magnetic moment (m) induced by a moving charge. For this coil, m is given by m = Iabn. (1-16) Using m, we can write N as N=mxB. (1-17) From Eq. (1-17), we can see that the torque tends to turn m toward B. This means that the loop (8= 0) shown in Fig. 1-5(a) is more stable than that ( 8 > 0) shown in Fig. 1-5(b). In other words, N tends to decrease 8 and the magnitude of N at @isrepresented by N ( 9 = IN(e)l= mBsin8. (1-18) In order to keep the coil at 0, the external force (Next(@)) must be put on the loop. Next(@= - N(8) = - m x B. (1-19) When this coil is rotated from 8 t o 8+ 68,the work (su)done in this rotation is 6U(@ = lNext(8)l68= mBsin868 = -mBG(cosB). ( 1-20) Thus, the total work necessary for the rotation of this coil from 8= 0 to 8 = u i s obtained as U

U

U (a)= Iscl(8) = - JmBS(cos8) = - mBIcos81~= - mBcosa 0

+ const.

(1-21)

0

bl Fig. 1-5. A coil with the magnetic moment of m is placed in a magnetic field of B: (a) 8 = 0 and (b) 8 > 0.

4

This is called the potential energy of this coil, which is defined without the constant term of

Eq. (1-21) as U ( a ) = -mBcos(a)

= -m

B.

(1-22)

From Eq. (1-22), it can clearly be seen that the potential energy at a=Ois smaller than that at a#O (O
An electron moving in an orbit as shown in Fig. 16 has its magnetic moment (b) from Eq. (1-16). b = ISn. ( 1-24) S i n Eq. (1-16) is ab, but S is m2in Eq. (1-24). Z is given as I = q1T = -ev/2m. (1-25) Here, q is the charge of an electron (-e) and T is the time for its single revolution (T = 2mlv). Thus, b can be given as follows: = -(ev/2m)(m2)n = -(evr/2)n. (1-26)

-6 V

q = -e

clo

Fig. 1-6. The orbital motion of an electron.

Problem 1-1. Show that &] = [A m2]. From quantum mechanics, the angular momentum (Z) of an electron moving as shown in Fig. 1-6 is expressed by l=rxp. (1-27) Because p is mev where me is the electron mass, 1 can be rewritten by 1 = r mevn. (1-28) Problem 1-2. Show that [Z] = [J s]. Note that this is the same dimension as A (=h/27r). Thus, can be represented by 1 as (1-29) = -(el2 me)1 = -(e A I2 me)(llA) = -PEL. Here, ,& is called Bohr magneton and L is the dimensionless operator corresponding to the orbital angular momentum of an electron. They are defined as (1-30) PE = (e A 12 me), L = (ZIA). (1-31) Problem 1-3. Confirm that ,UB = 9.2740~10-"J T-'. It is very convenient to use L, which is a dimensionless operator instead of I which has the unit of A . According to quantum mechanics, these operators obey the following relations: z ~ ~m z ,p = Z(Z + 1) A 21~, m p , (1-32a)

Z$, m p = mI A IZ, m p ,

(1-32b)

PIZ,m p = JZ(Z + 1) - m,(m, + 1) A IZ,ml+l>,

(1-32~)

tl~, m p = JZ(Z + 1) - m, (ml - 1) A 1 , m,-l>,

(1-32d)

and (1-33a)

5 (1-33b) (1-33~)

(1 -33d) Here, (1-34a) 1, = ( l f + 1 - ) / 2 ,

(1-34b)

1, = ( I + -Z-)/2i.

(1- 3 4 ~ )

The same relations as Q. (1-34) are also applied to L. In the right sides of Eqs. (1-32) and (1-33), 1 and L are called the orbital quantum number and they are integers of 0, 1, 2. .... The mI and r n L values are called the magnetic quantum numbers and they take on the values -1, 1+1,.....1-1,l and -L, -L+l,..... G I , L, respectively.

1.5 Spin Magnetic Moments According to quantum mechanics, the electron and many other elementary particles such as proton and neutron have their own angular momentum, which is independent of the angular momentum. This independent angular momentum is called "spin" because spin is similar to ( but not the same as) the rotation of the earth. For the electron, the magnetic due to its spin (s and S) is given by moment (k)

= -gr%'(s/ fi = -gp#.

(1-35) Here, g is called the g-factor, which is 1 for the orbital magnetic moment as shown in E@. (129) but about 2 for the spin magnetic moment of a free electron as shown below. The operators for the electron spin angular momentum obey the following relations, which are similar to those shown in Eqs. (1-32) and (1-33): (1-36a) s'Is, ms> = S(S + 1) IS, ms>, (1-36b) (1-36~) S~S,ms> = JS(S

+ 1)- m, (m, - 1) IS, m r l > .

(1-36d)

S*,S,, and S, are also defined by the relations similar to those given in Eq. (1-34). In the right side of Eq. (1-36), S is called the spin quantum number and ms the magnetic quantum number of electron spin. For a single electron, S = 1/2 and ms = f 1/2. Combining quantum mechanics and the principle of relativity, Dirac introduced theoretically the existence of electron spin, which had the g-factor of 2 for a free electron. Experimentally, its g-factor (g,) has been measured very precisely as ge = 2.002319304. (1-37) This value can be explained theoretically in terms of quantum electrodynamics. If a molecule has an odd electron, this is called a free radical. The g-factor of a free radical is generally different from g,. This difference is due to the spin-orbit interactions of component atoms. As the electron has the magnetic moment due to its spin, many nuclei such as 'H and I3C have the moment ( p ~due ) to nuclear spins (i and 1).

6 PN = gNpA(i/A ) = g N p N I . Here, g N is the g-factor of a nuclear spin and the nuclear magneton mass of proton (mp) as follows:

@N)

(1-38) is defined with the

p~ = (e A /2 mp). (1 -39) Problem 1-4. Confirm that p~ = 5 . 0 5 0 7 9 ~ 1 0J- ~T-I. ~ It is noteworthy that there is no minus factor in Eq. (1-38) for nuclear spins in contrast to Eq. (1-35) for electron spin because nuclei have positive charge. It is also noteworthy that pN is as small as about of ,&I because mp is much larger than me. The operators for the nuclear spin angular momentum obey the following relations, which are similar to those shown in Eqs. (1-32), (1-33), and (1-36): I’V, m p = I(I + 1) (I, m p , (1-40a) (1-40b) I#, ml> = ml)I,m 3 ,

PII,m p = J I ( I + 1) - m, (m, + 1) 1 , m+l>,

(1- 4 0 ~ )

r(I,ml>= J ~ ( ~ + l ) - m , ( m , - l ) ~ , m ~ l > .

(1-40d)

Table 1-1 shows the g N values of typical nuclei together with their I values and natural abundance. It is worth while to remark from this table that many nuclei such as I2C and I6C have no spin ( I = 0). The isotopes with and without spin, therefore, are called “magnetic and non-magnetic isotopes”, respectively. Even now, it is very difficult to explain theoretically the observed I and g N values. This is one of the frontiers of modem physics.

Solutions to the Problems 1-1. From Eq. (1-24), b ] = [g [Sl= [A m21. 1-2. [I] = [ r m,v] = [m] [kg] [ d s ] = [kg m2 s-*] [s] = [J s]. 1-3.

,uB

=-

eA

2m.

=

1 . 6 0 2 1 8 ~ 1 0 - ’ ~*C1 . 0 5 4 5 7 ~ 1 0 Js - ~ ~= 9.2740x10-24 Jcs/kg. 2 9.10939 x kg

Using [C] = [As], [JCsikg] = [JAs’ikg]. Using [A] = [kg/s2Tl from Eq. (1-6b), [JAs2/kgl = [Jl [kg/s2Tl [s2/kgl = [J T-’]. Thus, ,UB = 9 . 2 7 4 ~ 1 0J-T-I. ~~

1-4.

If‘m, of the above p~ calculation is replaced by mp (= 1.67262 X lO-”kg), the pNvalue can be obtained to be 5 . 0 5 0 7 9 ~ 1 0 ~ ~T-I. ’J

7

Table 1-1. Nuclear spin properties. Isotope

Natural

Spin

(4

(gN)

99.985 0.015

112 1

5.58570 0.85744

98.90 1.10

0

-

Abundance (%)

'H *H I2C

l3C

__

g-factor

112 1 112 ______

1.40482 0.40376 4.56638

I4N 'N

99.634 0.366

l6O

99.762 0.038 0.200 100

0

-

512 0 112

4.75752

92.23 4.67

0 112

-

-1.1106

3.10

0

-

0 312

0.42921

0

-

170

I8O .___I

I9F

28si 29~i 30~i

32s 33s 34s,36s __-___________ 35~1 37~1 55

Mn

__

95.02 0.75 . . . . .

312 312

75.77 24.23 100

_.__ -._.___.__ ~

73Ge 7.73 .__.___ 70Ge,72Ge,74Ge,76Ge, 2 3 4 ~ 0.0055 2 3 5 ~ 0.7200 23SU 99.2745

5.25774

__

0.54791 0.45608

512 __ 1.38748 912 -0.19544 0 -._.___-.-I-.---.-...-

_ _ I _ _ _ _ _ _

0 712 0

-

-0.109 -

__ __

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9

2. Introduction to Electron Spin Resonance and Nuclear Magnetic Resonance 2.1 Photochemical Primary Processes In dynamic spin chemistry, photochemical reactions in solution have mostly been studied. Thus, we will start with a review of photochemical primary processes in solution [ 11. Fig. 21 shows the various intramolecular processes initiated upon photon absorption by a molecule in the ground singlet state. Here, we deal with a molecule which contains an even number of electrons without orbital degeneracy. The lower energy levels of such a molecule are shown in this figure: the ground singlet state and the first and second excited singlet states are denoted by So, S1, and Sz, respectively and the first and second excited triplet states by TI and T2, respectively. Various primary processes induced upon photon absorption by this molecule are also shown in Fig. 2-1. The photon absorption processes associated with the vibrationalelectronic transitions from SOto S I and Sz are represented by SO +Sl Abs. and SO -& Abs., respectively. By internal conversion (IC) we mean a radiationless process between two different electronic states of the same spin multiplicity. In Fig. 2-1, IC from S:! to S1 and IC from S1 to SO are shown. Usually, the rate constants of S:! +S1 IC and Sl +So IC are more than 10” s-’ and 106-1012 8,respectively. By intersystem crossing (ISC) we mean a radiationless process between two different electronic states of two different spin multiplicities. In Fig. 2-1, SI+TI ISC and T1+So ISC are shown. The rate constants of the

1 T2

H T1

Ill

Y

SO Fig. 2-1. A diagram of the lower energy levels of a molecule which contains an even number of electrons without orbital degeneracy and the primary processes induced upon photon absorption by this molecule.

10

former and latter processes are 104-101*s/ and 10"-105 s-', respectively. By vibrational relaxation (VR) we mean a radiationless process from higher vibrational levels to vibrationless one. In Fig. 2-1, VR processes within Sl, TI, and SOare shown. The rate constants of such VR processes are usually more than 10'' s-'. It is noteworthy from Fig. 2-1 that photon absorption by a molecule which contains an even number of electrons without orbital degeneracy produces quasi-stable vibrationless S1 and T I levels. By fluorescence (Fluo.) we mean an emissive process between two levels of the same spin multiplicity. In Fig. 2-1, SI+SO Fluo. is shown, its rate constant being 106-109 s-'. By phosphorescence (Phos.) we mean an emissive process between two levels of two different spin multiplicities. In Fig. 2-1, TI+& Phos. is shown, its rate constant being 10-2-104 s-'. Because TI lifetime is fairly long, the TI+T, absorption(TI+T, Abs.) is often observed. In Fig. 2-1, Tl+TZ Abs. is drawn as an example. Similarly, the SI+S, absorption (S143, Abs.) can also be observed in many molecules, but it is not shown in Fig. 2-1 for simplicity. Apart from such radiation and radiationless processes as described above, chemical reactions also occur from various excited states such as Sl and TI. Typical photochemical reactions [2] is shown as follows: (1) Photodecomposition reactions

(2-la)

In reaction (2-la), the 0-0 bond cleavage of benzoly peroxide occurs from S I , giving a singlet radical pair involving two benzolyloxy radicals. In reaction (2-lb), however, the P-C bond cleavage of triphenylphosphine occurs from T I , giving a triplet radical pair involving the diphenylphosphinyl and phenyl radicals. In the latter case, the SI+TI ISC of triphenylphosphine is much faster than the reaction from SI.

(2) Photoinduced electron transfer reactions

(2-2a)

In reaction (2-2a), the electron transfer reaction occurs from Sl of pyrene to N,Ndimethylaniline, giving a singlet radical ion pair involving the pyrene anion and N,Ndimethylaniline cation radicals. In reaction (2-2b), the electron transfer reaction occurs from

11

(2-2b)

CH3 TI of 10-methylphenothiazine to dicyanobenzene, givining a triplet radical ion pair involing the 10-methylphenothiazine cation and dicyanobenzene anion radicals. In the latter reaction, the &+,TI ISC of 10-methylphenothiazine is much faster than the reaction from S1.

(3) Photoreduction reactions

&..-

s1-ISC

T1-RH 3 [ / $ 5

/

IRI

(2-3a)

In each of reactions (2-3a) and (2-3b), the hydrogen abstraction reaction occurs from T, of benzophenone or naphthoquinone, giving a triplet radical pair involving the ketyl or semiquinone radical and the radical fromed from a hydrogen donor (RH). In these reactions, the S,+T, ISC of carbonyls and quinones is very fast ( in the ps-region) and their TI lifetime is very long without RH (in the ps-region). Thus, these reactions occur from TI.

2.2 ESR (Electron Spin Resonance) As shown in reactions (2-l), (2-2), and (2-3), many radicals are produced in photochemical reactions. Electron spin resonance (ESR) is a good method to detect radicals. Especially, short-lived intermediate radicals have recently been measured with various timeresolved ESR (TRESR) techniques [3]. In this section, an introductory review of ESR will due to an electron spin ( s or S) be given. As shown in Eq. (1-35), the magnetic moment (JLS) is given by = -g&(s/h ) = -g/4&. (2-4) In the presence of a magnetic field (B).the potential energy (Uj' of the spin can be represented from Eq. (1-22) as follows:

U = -JLS B = - ( - g B S ) B = gPBs B . (2-5) If the direction of B is taken as the z-axis in the laboratory frame, the Hamiltonian (Hs)of the spin is written by Hs = g,U&

B = g,UB(sxBx+ s& + s $ z ) = g,U&B.

(2-6)

12

Here, B, = By= 0 and Bz = B. According to quantum mechanics, the energy (Es) of one electron spin can be given as

ES = < S, rnslHsP, ms> = < S, rnsl gpBszB P, ms> = gpBB < S, rnsPzlS, rns>

s,

= gp& s, msl rns Is, ms> = gpBB rns< rns Is, rns> = gpBB rns. (2-7) For obtaining the last term of Eq. (2-7), we used Eq. (1-36b) and the fact that < S, rns IS, rns> = 1. Because rns = +1/2 for one electron spin, its energy levels in the absence and presence of a magnetic field of B are shown by Fig. 2-2. As shown in this figure, the spin with ms = +1/2 is called "a-spin" and that with rns = -1/2 "P-spin". Fig. 2-2. Energy levels of an electron spin in the absence and presence of a magnetic field of B. The ESR transition between P 7 Es(mF-1/2) the two levels occurs at the B>OT microwave frequency of v. The energy splitting of the a- and P-spins is called "Zeeman splitting" and the energy difference between them "Zeeman energy". The ESR transition between the two levels occurs at the microwave frequency of v.

a*Es(mF+1/2)

hv = Es(rns = +1/2) - Es(rns = -1/2) = gp~B/2-(- gp~B/2)= gpBB. (2-8) When B = 1 T, the Zeeman splitting (hv) and the frequency of the ESR transition (v) for a free electron (g = 2.002319) are given from Eq. (2-8) as follows: hv = 2.002319~9.2470~10-~~J T ' x l T = 1.85695~10-~~J = 0.935cm-' = 1.345K3, (2-9) v = 1.85695x10-23J/6.62608x10-34J~ = 2.8025~10'~~-' = 28.025GHz. (2-10) We can see from Eqs. (2-9) and (2-10) the following interesting results: (1) the Zeeman splitting of an electron spin at B =1 T is about 1 cm-' (about 3 cal/mol) which is much smaller than the thermal energy at room temperature (about 600 cal/mol) and the activation energy for chemical reactions (usually about 10 kcal/mol). (2) The ESR transition frequency of an electron spin at B = 1 T is about 28 GHz, which correspond to microwave. The most popular ESR apparatus uses so-called X-band microwave, whose frequency is about 9 GHz. When v = 9 GHz, the ESR transitions of radicals with gl = 2.0023 and gz = 2.0123 occur at magnetic fields of BI and Bz, respectively, as shown in Fig. 2-3.

BI = hV/gIpB = 6.62608~10-~~J~~9~10~~-~/2.0023X9.2740~10-~~J T I = 0.321 15T, (2-11) T'= 0.31955T. (2-12) Bz = hV/g2pB = 6.62608X10-34J~~9~109~-'/2.0123X9.2740~10-24J Thus, the difference between BI and BZ is 1.60 mT, which is 16.0 G because 1T = lo4 G.

B,=3195.5G

BI=3211.5G

16.OG d3190G

3200G

3210G

B

Fig. 2-3. ESR transitions with Xband microwave (v = 9 GHz) for g l = 2.0023(Bl) and gz = 2.0123(B2).

13

Table 2-1. Anisotropic(g,, radicals.

gb,

and gc) and isotropic(gi,, = (g&b+g,)/3)

g-values of typical

Problem 2-1. K-band microwave (v-25 GHz) and Q-band one (v-35 GHz) are also popular in ESR measurements. Calculate the fields (B, and B2) of the ESR transitions of radicals with gl = 2.0023 and g2 = 2.0123 when v=25 and 35 GHz. Obtain also the values of B, - B2. From ESR measurements of radicals in single crystals, the three principal values (g,, gb, and g,) of their g-tensor can be obtained, but such measurements in single crystals have not been applied to most radicals. For most radicals, only the average value (g=(g&gb+gc)/3) of the three principal values can be obtained in solution. The former and latter g-values are called anisotropic and isotropic ones, respectively. Such g-values observed for typical

14 radicals are listed in Table 2-1. From observed g-values, the electronic structure of radicals can be studied.

2.3 NMR (Nuclear Magnetic Resonance) Nuclear magnetic resonance (NMR) is similar to ESR in principle. At present, NMR is much more popularly used in chemistry and biology than ESR. As shown in Eq. (1-38), the magnetic moment ( p ~due ) to a nuclear spin (i or I) is given by P N = gNpN(uh = gNpNI. (2-13) In the presence of a magnetic field (B), the potential energy (U) of the spin can be represented from Eq. (1-22) as follows:

(2-14) u = -PN B = -(gNpNI) B = -gN,&I B. If the direction of B is taken as the z-axis in the laboratory frame, the Hamiltonian (HN)of the spin is written by (2- 15) HN= -gN,hNI B = -gNpN ( I $ X + I y B y + I$?) = -gN,UNI$. Here, B, = By = 0 and B, = B. According to quantum mechanics, the energy (EN)of one nuclear spin can be given with the similar method used for the electron spin in Eq. (2-7). EN = -< I, mdHNlI, mi > = --< I, mi( g~ ,uNI~B11,mi> = - g N ~ N (2- 16) = - g N ~ N B -I,<mil mill, ml> = - g N ,UNBmr< I, 11,mi> = - g N ,UNBmi. The energy splitting of a nuclear spin in a magnetic field is represented by Eq. (2-16). This splitting is also called the Zeeman splitting of the nuclear spin, its energy being called its Zeeman energy. For proton and 13C,their I value is 1/2. For such a nuclear spin, its energy levels in the absence and presence of a magnetic field are shown by Fig. 2-4. As shown in this figure, the spin with mi = +1/2 is also called "a-spin" and that with ml = -112 "P-spin". It is noteworthy from Fig. 2-3 that the energy of P-spin is higher than that of a-spin. On the other hand, for an electron spin, we can see from Fig. 2-4 that the energy of a-spin is higher than that of P-spin. Fig. 2-4. Energy levels of a nuclear spin with 1=1/2 in the absence and presence of a b, E~(ml=-1/2) magnetic field of B. The NMR transition occurs at the radio B=OT ' a, &(mi=+ 112) frequency of v.

::lhv B>OT

When B=lT, the Zeeman splitting (hv) and the frequency (v) of the NMR transition for a proton (gp5.58570) are given from Eq. (2-16) as follows: hV

= gNpNB = 5.58570~5.05079xlO-*~JT~'xlT = 2.8212x10-26J,

v = 2.8212x10-26J/h= 42.58MHz.

(2-17) (2-18)

Problem 2-2. Calculate the fields (B, and Bb) of the NMR transitions of a proton at v = 100 and 700 MHz. In ESR, the g-values of radicals reflect their electronic structure. In NMR, especially in proton-NMR, chemical shifts are used for different transition fields due to different situations of different spins. We consider that chemical shifts are due to the shielding of an external magnetic field (B) by electrons in a molecule. When the shielding constant of a proton is written by 0, the field felt by this proton becomes B(1-0). We usually adopt

15

tetramethysilane (TMS) as the standard of chemical shifts, its shielding constant being written by o0. For protons in organic molecules , their shielding constants (01)are smaller than 00. For protons in metalhydrides, their shielding constants (02) are larger than 00. Thus, the resonance fields of the proton in TMS (Bo), a proton in an organic molecule ( B I ) , and the proton in a metalhydride (Bz) can be obtained from Eq. (2-17). (2- 19a) (2-19b) (2- 1 9 ~ ) (2-20a) (2-20b) Here, the fact that the o values are much smaller than 1 is used.

In proton NMR, chemical shifts (6,i = 1 or 2 ) are defined as follows: Si = (Bg-Bi)/ Bo = (Oo-Oi)/( l+Oo) = 00-Oi. In Fig. 2-5, the regions of proton NMR spectra are shown.

(J1<00

00<02

B 1
BO
I

I

I

6 =10 ppm

(2-21)

6 =O ppm

6 =-lo ppm

2.4 The Hyperfine Coupling (HFC) in ESR In section 2.2, we considered ESR spectra of radicals having one electron spin (S). In this section, we will see how their ESR spectra are changed by nuclear spins in them. At first, let us consider a radical which has only one nuclear spin (I). In this case, its Hamiltonian is written by H s = g p n S * B -gN/.LNI*B + A S 0 1 = g,Un&B - gN,UNIzB +

c,

(2-22)

AS,Ii.

Here, the spin-spin interaction between an electron spin and a nuclear spin can be represented by the isotropic term (AS I ) when the radical rotates very fast in solution. This term is called the hyperfine coupling (HFC) and the coefficient ( A ) the (isotropic) HFC constant. When it is fixed in a crystal or solid matrix, however, the interaction becomes anisotropic, but the latter case is not considered in this section. The energy of this system (E(ms,ml)) can also be obtained by the similar way as Eqs. (2-7) and (2-16).

E(ms, md = < S, msl< I, mr(HslS, ms>lr, m, > = < s, ms(<6 ml( g,UnS,B - gN,UdZB +

c,

A S c I lIs, ms>P9 ml>

16

= < s, msl g,UBS;B Is, Ws>< I, mil11, mi > - < s, mslls, ms>< 1, mi(gN,UNI:BII,ml>

+ < S,msl< 1, mll

c,

AS, I,IS, ms>ll,

= g,uBBms - gNpNBml+ Amsml. The ESR transition between the a and B spins can be given as

(2-23)

h v = E(ms= 1/2, ml) - E(rns=- 112, m1) = g,UBB + Am/. In the following part of this book, we will write ms and simplicity.

(2-24) rn1

as m and M , respectively, for

hV(M)=g/@+AM.

(2-25)

When the microwave frequency ( v ) is fixed, the field position (B(M)) where the ESR transition occurs for each M is given as follows: (2-26)

B(M) = h u'g,& - AM/g,UB = Bo - aM. Here, Bo = hu'gpB and a = A/gpB. The unit of a is also T or G. positions for I = 1/2 and 1 are shown.

BO

BO

(4

16131 B> pl

a>O: M=1/2' M=-1/2 a
M=1/2

p*l

(b) M=l M=t) M=-1

In Fig. 2-6, the field

Fig. 2-6. Field positions for ESR transitions of a radical involving a nuclear spin with (a) 1=1/2 and (b) I=1.

M=-1 M=O M=l

As shown in Fig. 2-6, the field positions for ESR transitions are split to a doublet for 1=1/2 and a triplet for I=I, respectively, from the original position (Bo). Such split structures are called HF structures. It is noteworthy that the sign of a can not be determined from ESR measurements at room temperature and even at 77 K. The sign can be obtained experimentally from chemically induced dynamic nuclear polarization (CIDNP) which will be explained in Chapter 4. In general, there are many nuclear spins (Ik) in radicals. In such cases, the field positions for the ESR transitions can be represented with different HFC constants ( a k ) for different nuclear spins as follows: (2-27) B(Mk) = Bo - ZkUkMk. If there are only N-equivalent nuclear spins in a radical, Eq. (2-27) is simplified as (2-28) B(Mk) = Bo - U(&Mk ). Here, a is the HFC constant for the equivalent nuclear spins. Fig. 2-7 shows the HF structures of a radical with two different or equivalent nuclear spins with Ik=1/2. As shown in Fig. 2-7(a), the signal position at Bo for a radical with two different nuclear spins is split by the first one with its HFC constant of at, and then each of the split lines is split again by the second one with its HFC constant of a2. In this case, four signals with an intensity ratio of 1:l:l: 1 appear as shown in Fig. 2-7(a). As shown in Fig. 2-7(b), the signal position at Bo for a radical with two equivalent nuclear spins is split by one of them with its HFC constant of a, and then each of the split lines is split again by the second one with the same HFC constant of a . In this case, three signals with an intensity ratio of 1:2:1 appear as shown in Fig. 2-7(b).

17

(b)

I

Bo

.........'j.: ........ ...... . . . . . . i

.

:

.

r(d i

...-..

rd"4

....

:

.

.

'B

....

1

>B

.../.., ..... j .............. ...........,/..'-.. : .........

B

1\

1\

1\

B

M2= 112 MZ=- 112 Fig. 2-7. Field positions for ESR transitions of a radical involving two different or equivalent nuclear spins with Ik=1/2 (kland 2). (a) This case corresponds to a radical with two different nuclear spins with positive HFC constants (al>a2>0). (b) This case corresponds to a radical with two equivalent nuclear spins with the same positive HFC constant (al=a2=a>O).

Problem 2-3. Obtain the number of ESR signals and their intensity when a radical has Nequivalent nuclear spins with 1=1/2 and I for N = 1-6. Problem 2-4. Obtain the general formulas for the number of ESR signals and their intensity when a radical has N-equivalent nuclear spins with 1=1/2. 2.5 The Spin-Spin Coupling in NMR

In section 2.3, we considered NMR spectra of isolated nuclear spins. In this section, we will see how their NMR spectra are changed when there are interactions among the spins. Such an interaction between two spins is called their spin-spin coupling. At first, let us consider two nuclear spins (11 and 1 2 ) which are coupled with the coupling constant of Jlz. In this case, its Hamiltonian is written ,as the case of the HFC between an electron spin and a nuclear one (see Eq. 2-22), by (2-29) HS= - gNIpNI1 B - gN2pNI2 B iJIZ 11 12. In this section, let us consider Proton-NMR and I3C-NMR correspond to this case. Other I larger than 15121, the energy of cases can similarly be treated. If I gNlpNB - ~ N Z ~ NisBmuch the nuclear levels (E(Ml, M2)) can similarly be obtained as Eq. (2-23).

E W I , Mz)= < 11,Mil< 12, M z l H s l 4 , MI>JIz,Mz> (2-30) = - gNi,&BMi - ~ N Z ~ N+BJ12MiMz. M~ Thus, the NMR transition between the a and B spins ( k l or 2) can be given as follows: (2-31a) hQ = E(M1=-1/2, M2) - E(M1=1/2, .v2)= gf,r,,UNB - JlzM2, (2-31b) hW = E(M1, M2=-l/2) - E(M1, M2=1/2) = gNIpNB - J12MI. If the radio-frequency is fixed (Q = ~2 = u), the field positions of the NMR transitions ( B , and B2) become P N+ J 1 2 M2, Bi(M2 ) = h d gNiPN -t J I ~ M ~ ~ N=I BI

B2(M1 ) = hzJ ~ N $ N

+ J I & ~ I / ~ N I ~=NB2 +

J IM~I .

(2-32a) (2-32b)

18

2-8 shows how the NMR Here, BIZ hvl g N i j L N , B2= hvl ~ N $ N , and j,*= J 1 2 / g ~ j ~ ~Fig. . spectra are changed by the spin-spin coupling between two nuclear spins with 11=12=1/2.

(a)

B1

6

B2

B ,..'?..

.:.: j ...,... .i :. .. .,..' j. "....

B> J12>0:

M2=-1/2 M2=1/2

M1=-1/2 M1=1/2

J12<0:

M2=1/2 M2=-1/2

M1=1/2 M1=-1/2

Fig. 2-8. NMR spectra of a system where two nuclear spins belong to the same isotope having I1=12=1/2 (a) without and (b) with the spinspin coupling.

When the i-th nuclear spin with Ii=1/2 is coupled with many nuclear spins signal is split by the spin-spin coupling (Ji,) as follows: B~(M,) = Bi +

C jj i j M j .

(4), its NMR (2-33)

and jij= J i / gNi,& . If the i-th nuclear spin with 43112 is coupled with Here, Bi= hvl gN&N N-equivalent nuclear spins (I,), its NMR signal is split as follows: (2-34) Because Eq. (2-34) is similar to Eq. (2-28), the splitting of the NMR signals for the i-th nuclear spin becomes similar to the HF structures of a radical which is coupled with Nequivalent nuclear spins. Such NMR spectra split by other nuclear spins are usually seen in NMR data books. If I g N i , u N B - gf@NBI is not much larger than IJcl, NMR spectra become very complicated. Such complicated spectra are often explained in NMR textbooks, but this problem is beyond the scope of this book.

19

References [l]"Organic Molecular Photophysics", J.B. Birks, Ed.,Wiley, London, 1973,Vol. I.

[2] "Modern Molecular Photochemistry", N. J. Turro,Benjamin, Melon Park, California, 1978. [3] H. Murai and H. Hayashi, in "Dadiation Curing in Polymer Science and Technology, Vol. II",J. P. Fouassier and J. F. Rabek, Eds., Elsevier Applied Science, London, 1993,Chap. 2.

Solutions to the Problems 2-1. When v=25 GHz, the BI and B2 values can be calculated with Eqs. (2-11) and (2-12). BI = hV/glpB= 6.62608x10-34J~x2Sx109~-'/2.0023~9.2740~10-24J T'= 0.89207T, B2 = hV/g$B = 6.62608x10-34J~x2Sx109~-'/2.0123~9.2740~10-24J T'= 0.88764T. Thus, the BI - Bz value becomes 0.00443T= 44.3G. When v=3S GHz, the BI and Bz values can similarly be calculated as = 6.62608x10-34J~x35x109~-'/2.0023~9.2740~10~24J T'= 1.24890T, BI = hV/g/pB B2 = h ~ / g 2 = p 6.62608x10~34J~x35x109~-'/2.0123x9.2740~10~24J ~ T'= 1.24270T. Thus, the BI - B2 value becomes 0.00620T= 62.0G. 2-2. When v = 100 and 700 MHz, the fields(B, and Bb) of the proton-NMR transitions become from Eq. (2-17)as follows: B, = hV/gN&, = 6.62608x10-34JsX108~-'/5.58570~5.05079~10~27JT~1=2.34866T, Bb = hV/gN,L& = 6.62608~10~34J~~7~108~~L/5.S8570~S.05079~10~27JT~'=16. 2-3.

N

I = 1/2

I= 1

1 1 1:l 1:l:l 2 1 :2: 1 1:2:3:2:1 3 1:3:3:1 1:3:6:7:6:3:1 4 1:4:6:4:1 1:4:9:16:19:16:10:4:1 5 1:5:10:10:5:1 15:14:29:41:1:41:29:14:5: 5 1 6 1:6:15:20:1S:61: 1:6:20:48:841 :21:133:121:84:48:20:61: 2-4. or 0

1

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21

3. The Radical Pair Mechanism

3.1 Chemical Reactions through Radical Pairs In Section 2.1, typical photochemical reactions through singlet and triplet excited states are shown. In these reactions, radical pairs are often produced as short-lived intermediates through decomposition, electron transfer, and hydrogen transfer fkom these excited states. It is noteworthy that radical pairs are also generated at the initial stage of photosynthesis, which provides almost all energy to living creature. Radical pairs also appear in many useful industrial reactions such as polymerization and lithography. Thus, radical pairs are very important in both pure and applied science. On the other hand, reactions through radical pairs have recently attracted the attention of scientist because such reactions have been found to be greatly influenced by ordinary magnetic fields [l]. Here, ordinary magnetic fields mean those induced by usual electromagnets, which can produce fields less than 2 T. From the energetic viewpoint of chemical dynamics, however, we cannot expect the occurrence of appreciable magnetic field effects on chemical reactions fiom the following considerations: (1) As shown in Section 2.2, the Zeeman splitting of radicals at B = 1 T is as small as about 1 cm-'. (2) As shown in section 2.3, the Zeeman splitting of nuclear spins at B = 1 T is as small as -10" cm-'. (3) The thermal energy at room temperature is -200 cm". (4) The activation energy for usual chemical reactions is -3000 cm-' or more. At first, chemical reactions through radical pairs were found to show anomalous intensities in ESR and NMR spectra. Such anomalous intensities arise fkom radicals and reaction products which are formed with non-equilibrium spin polarization. These phenomena are called chemically induced dynamic electron or nuclear polarization (CIDEP or CIDNP). From studies on CIDEP and CIDNP, populations of electron and nuclear spin states were found to be changed by external magnetic fields andor nuclear spins. It was, therefore, expected that the sum of such populations could also be changed by them. This was the initial strategy for exploring magnetic field and magnetic isotope effects (MFE and MIF) on chemical reactions through radical pairs. In this chapter, we wiU see the basic theory for radical pairs [l], which is called the radical pair mechanism (RPM). In the next chapters, we will see the detailed explanations on CINDP, CIDEP, MFE, and MIF. Let us consider chemical reactions through radical pairs as shown in Fig. 3-1. Radical pairs are usually produced fkom singlet and triplet excited states. These reaction precursors are called S- and T-precursors, respectively. Radical reactions also occur fkom flee radicals, which is called F-precursors. Because the dynamic behavior of radical pairs fkom Fprecursors is similar to that fkom T-ones, we omit discussion of reactions fkom F-precursors for simplicity's sake in this book. As shown in Fig. 3-1, radical pairs generated fkom S- or T-precursors are surrounded by solvent molecules, called "solvent cage", and retain the spin multiplicity of their precursors. Two radicals are produced as nearest neighbors. Such pairs are called "close pairs". Sometimes recombination (or disproportionation) reactions occur fkom S- and T-close pairs immediately after the formation radical pairs. Such reactions are called "primary recombination" and the products of such reactions "cage products". From S-close pairs, cage products in the ground singlet state are produced due to the spin preservation during chemical reactions. From T-close pairs, however, cage products in the excited triplet state are produced if close pairs have enough energy for producing excited molecules. Because the latter reactions are much rarer than the former ones, we will not consider the latter ones in this chapter for simplicity.

22

4 Singlet Precursor

JI

SO Interaction

A

Cage Product (singlet state)

Close Pair

S-T Conversion

V

'1x9

r Triplet Precursor

Cage Product (triplet state)

!$Y1

Escape Radicals

u Xe + Ye

Fig. 3-1. Reaction scheme of radical pairs generated from singlet and triplet precursors. Singlet and triplet radical pairs are represented by ' [ X p by] and '[X ++Y], respectively. (Reproduced from Ref. [2] by permission from The Chinese Chemical Society in Taipei)

In close pairs involving only light atom-centered radicals such as C-, N-, and 0-radicals, no spin conversion occurs between their singlet and triplet states as will be seen in the next section. On the other hand, singlet-triplet (S-T) conversion is possible for close pairs involving heavy atom-centered radicals such as S- and Ge-radicals due to their spin-orbit (SO) interaction. In this chapter, therefore, we will not consinder the S-T conversion of close pairs for simplicity. As shown in Fig. 3-1, two radicals in each close pair which refrains from cage recommbination start to diffuse from each other, forming a separated pair. In separated pairs, the S-T conversion becomes possible through such weak magnetic interactions as the Zeeman and HF ones of radicals as will be seen in the next section. Two radicals in some separated pairs approach each other, forming close pairs again. On the other hand, two radicals in other separated pairs continue to diffuse with each other, forming free radicals. Such radicals escaped from solvent cage are called "escape radicals", which loose the spin memory of their precursors. Escape radicals react with each other or with solvent molecules. Products from such reactions are called "escape products".

23 3.2 Singlet and Triplet States of Radical Pairs and Their Mixing

A radical pair consists of two weakly coupled radicals and its spin Hamiltonian (Hw) in solution can be represented by the exchange (a,) and magnetic (Hmag) terms [ 11,

HRP= H e x +

(3-1)

Hmag.

where

Hex = -J(2SiSz

+ 1/2),

(3-2) (3-3)

In Eq. (3-2), J i s the value of the exchange integral between two electron spins (Sl and S2) and the 1/2 term is put for convenience although it is not used in many textbooks. Eq. (3-3) is just the sum of spin Hamiltonian of one radical given by Eq. (2-22), but the nuclear Zeeman terms are omitted in Eq. (3-3) because their magnitude is much smaller than those of the electron Zeeman term and the HFC one. In Eq. (3-3), ga and gb are the isotropic g-values of two component radicals (radicals a and b) in a radical pair, respectively, and Ai and Ak are the isotropic HFC constants with nuclear spins (11and I k ) in radicals a and b, respectively. The combination of two electron spins (S1 and Sz) with SI = S, = 1/2 in a radical pair generates the singlet (S) and triplet (T,,; n = 21 and 0) states which can be represented by the product of the electron- and nuclear-spin wavefuctions.

I

(a)Singlet- and triplet electron-spin wavefunctions (IS) and T n )):

Is)= (la;P*)-JPla,a;))/Jz~

(3-4)

ITO) = (la,Pz)+lP,a;))/JZ.

(3-5)

IT+1>=l~ls)7

(3-6)

IT-l>=laP&

(3-7)

(b)Nuclear-spin wavefunctions (IX,)

Ix,>=n;IMi)n:l~J '

For simplicity, we will write

I

X N

) = IM i

(3-8a)

Ix,) as (3-8b)

Mk).

In the absence of an external field, the energies of the singlet and triplet radical pairs can be represented as follows:

E(S)=(SI

HexIS)

=J,

E(Tn)=(TnIH e x I T , ) =-J.

(3-9a) (3-9b)

In Eq. (3-9), the contribution from the HFC terms is neglected because of its smallness. Problem 3-1. Prove Eq. (3-9) with the relation of S2= (Sl + S#. When the distance between radicals a and b is written by r, the r-dependence of J can approximately be given by J ( r j = J Oexp(-&). (3-10)

24

Since JO is usually negative for radical pairs consisting of neutral radicals, the r-dependence of the energies of such singlet and triplet radical pairs in the absence of an external magnetic field is schematically depicted in Fig. 3-2(a). This figure shows that the energy of the triplet state is higher than that of the singlet one by 214 and that their energy separation decreases with increasing r. When r becomes large enough, IT,) (n = +1,0, and -1) and IS) are almost degenerated with each other, such r-region being represented by Region I in Fig. 3-2(a).

I /

II

Fig. 3-2. Dependence of radical pair energy on the distance ( r ) between two component radicals when J(r) is negative. (a) in the absence of an external field; (b) in the presence of the field. The S-T, conversion occurs at Region I in the absence of the field. In the presence of the field, however, the S-To and S-T., conversions occur at Regions I and II , respectively. (Reproduced kom Ref [la] by permission kom Gakkai Shuppan Center) In the presence of an external field of B, the energies of the singlet and triplet radical pairs can be represented as follows:

I

I

E ( S ) = ( S ,x N Hex+Hmg S, x N ) = J,

(3-lla)

Here, n = +1,0, and -1; g = ( g o+ g , ) / 2 . Problem 3-2 Prove Eqs. (3-1 l a and b) with Eqs. (3-2)-(3-7). The r-dependence of the energies of the singlet and triplet radical pairs in the presence of an external magnetic field is schematically depicted in Fig. 3-2(b). This figure shows that the three triplet states (IT,)(n = +1, 0, and -1)) are split by about gp,B due to the Zeeman

interaction and that IT,) and IS) become degenerated in Regions I and 11.

In Region I,

ITo) and IS) are degenerated as in the case without a field. In Region 11, IT-,) and IS) are crossed at r=rLC for a negative J-value at a level-crossing field (BLc).

25 BE = 2p(rE)v

(3-12)

gpB '

If J is positive, a similar level-crossing occurs between IT,,) and IS). When the energy separation between ITn) and IS) becomes nearly zero, the spin conversion between them occurs through the following off-diagonal matrix elements:

(3- 13b)

) ; f l ( f o r I x ~ ) = I M ; , M , ) ) ,and M i = Here, A g = g a - g , , Mi(for I X ~ ' ) = I M ; ' , M ~ I=) M

Mk . Problem 3-3. Prove Eqs. (3-13a and b) with simailar methods used in Problem 3-2. From Fig. 3-2 and Eqs. (3-13a and b), the following features of the spin conversion in radical pairs can be obtained: (1) In the absence of a magnetic field, the S-T conversion between ITn) (n = +1,0, and -1) and IS) through the HFC terms of Eqs. (3-13a and b) at large r (Region I

of Fig. 3-2(a)). In the presence of a field, the S-To conversion occurs through hQN of Eq. (313a) at large r (Region I of Fig. 3-2@)). (3) In the presence of a field, the S-T-I conversion occurs through the HFC term of Eq. (3-13b) at r=m (Region II of Fig. 3-2(b)) when J is negative. When J is positive the S-T+I conversion occurs similarly.

3.3 Kaptain's Theory for the S-To Conversion As shown in Fig. 3-1, two radicals in close pairs which refrain from cage recommbination start diffuse with each other, r in Fig. 3-2 becoming larger and larger. When r becomes sufficiently large (Regions I and II of Fig. 3-2), the S-T conversion occurs through the offdiagonal elements given by Eqs. (3-13a) and (3-13b). In this section, we describe the results of the S-TOconversion in detail because this conversion can be understood intuitively. The results of the S-Trl conversion will not given here, but can be treated similarly. Kaptain obtained the time evolution of the wave function ( Y (t) ) of a radical pair during the S-To conversion [3], solving the Schrodinger equation, (3-14)

dt Here, Y (t) is given by

(3-15)

y ( t ) = csN(t)lS,~N)+ c m ( t > l ~ o r ~ N ) . The coefficients of Eq. (3-15) are obtained as follows: C,(t)=

( (

''

C,(O) cosuNt-i-sinuNt uN

Here,

- iC,(O)-sinu,t, QN .

'

(3-16a)

uN

QN . u N t. uNt - iC, (0)-sin

C , (t) = C, (0) cos u N t+ i-sin uN'

1 1

uN

(3-16b)

26

3 = JIA, w, =

(3-17)

@x.

(3-18)

Problem 3-4. Prove Eqs. (3-16a and b). For reactions which occur from S-precursors, IC;, (Of =1 and lC;rN(0)l2= 0 in Eqs. (316a) and (3-16b).

Thus, the singlet and triplet character at t (IC&(t)l' and IC;,(t)I*) are

given by

IC;,(~)I~

,;cI

=I-(Q, / u , ) 2 s i n 2 u N t ,

(t)I2 = (Q, / w ,

)2

(3- 19a)

sin2w,t.

For reactions which occur from T-precursors, IC;,

(3-19b) (0)l2= 0 and lC& (O)/'= 1/3 in Eqs. (3-16a)

and (3-16b). Thus, the singlet and triplet character at t ( IC;N (t)I2 and IC;, (t)I2) are given by 1 Jc;, (1112 = :(Q,

/wN)' sin2w,t,

1 Icf,(t)l2 =-[I-(Q,

/w,)' sin2w,t].

3

(3-20a) (3-20b)

It is noteworthy from Eqs. (3-19) and (3-20) that the S-To conversion rate is given by w,, which is 108-109s-' for usual radical pairs as shown in Problems 3-5(a) and 3-5(b).

Problem 3-5. (a) Confirm that w,= 4.4Ox1O8s-' when 9 = 0 s-', Ag = 0.01, B = 1 T, and A, lgp, = A, lgp, = OrnT . (b) Confirm that w, = = 4 . 4 0 ~ 1 0 when ~ ~ ~ '3 = 0 s-', Ag = 0, AiI gpB= 10 mT (Mi=1/2), and A, I gp, = OrnT . As shown in Fig. 3-1, some radicals in separated radical pairs re-encounter their partners within the solvent cages, but others escape from the cages, forming "escape radicals". Since the time scales for the secondary recombination and the S-TOconversion rates are 10-'0-10-7 and 10-9-10-8s-I, [3] respectively, the change in the S-TOconversion rate by an external magnetic field and/or the HFC term can influence the yield of cage and escape products. Let us consider a reaction where the cage recombination occurs only through the singlet close pair but not through the triplet close pair as shown in Fig. 3-1. In such a reaction, Noyes [4] showed that the probability (f(t)) of the first re-encounter between t and t+dt for a pair, separating from an encounter at t=0, is given from the theory of random flights as (3-21) This equation was derived from the following conditions:

-

(1)For long times, f(t) mt-3/2. (2) The exponential factor ensures that f(t) becomes zero for t = 0. The exact behavior at short times is not so important. In Eq. (3-21), p is the total probability at least one re-encounter.

27

(3-22) Problem 3-6. Prove Eq. (3-22). Both p and m can be expressed in terms of the basic quantities p (the encounter diameter), o (the root mean square displacement for relative diffusion motion), and v (the frequency of relative diffusion displacements) as [3] p=1-(1/2+3p/20)-',

(3-23)

rn = 1 . 0 3 6 ( 1 - p ) z ( p / ~ ) 2 ~ ~ 1 ' 2 .

(3-24)

For small radicals in ordinary solvents, v is about 10"s.'. Because o i s equal to p or may be smaller than p , the p and rn values becomes as follows: 0.5 5 p < 1 ,

(3-25)

rn

(3-26)

z=

10-6s1'2.

(a) Probability of product formation during a first re-encounter: P N I( t )

t=w2

-b

1N ( t )

(b) Probability of product formation during a second re-encounter: P d t ) t=O

1N ( t l )

1

(c) Probability of product formation during a third re-encounter: p N 3 ( t )

Fig. 3-3. Probabilities of product formation during first, second, and third re-encounters ( P N I ( t )P, ~ 2 ( t )and , P ~ 3 ( t )after ) the formation of radical pairs in solution.

28

Fig. 3-3 shows how two radicals in radical pairs recombine with each other or separate fiom each other. Let us consider the probability for recombination @’At)&)in the interval (I, t + dt). Fig. 3-3(a) shows the chance of product formation during a first re-encounter (P,,(t)dt)in the interval (t, t + dt). When the probability for recombination during a singlet encounter is expressed by A,P ~ l ( tbecomes ) the product of f(t), lCsly(t)1’,and A . pM(t) =

a lCs~(t)12 fit) = R ~ ( t fit>. )

(3-27)

for Here, A,,,(t)= R ICsN(t)12was used for simplicity. Thus, the total probability (RN,(~)) recombination during a first re-encounter in the interval (0, t ) is given as, RNl(0

=

j; PNI (W = j; A N (t)f(W.

(3-28)

Radicals of a pair that fails to react during the first re-encounter start again their random walk. The total probability (QM(~))for such radicals is given as, (3-29) Fig. 3-3(b) shows the chance of product formation during a second re-encounter (P,(t)dt) in the interval (t, t + dt) after a first re-encounter at tl. With a similar method used for a first R N ( t- t l ) , f(t-t~). re-encounter, Pm(t) becomes the product of QNI(~I),

f‘m(4

= Q N I ( ~ I )2,

=

(t -

f(t-tJ =

(1- 4,(tl > ) f ( t l ) 44, ( t - I , ) f(t-td

j(W,(t - - ~ 1 ) ( 1 - ~ , ( ~ l ) } f ( t- 4 ) f Q I ) .

(3-30)

Eq.(3-30) is the same as the second term of the right side of Eq. (21) in Ref. [l]. Thus, the total probabilities for recombination and non-recombination (RN2(t)and Qm(t)) during a second re-encounter at t=t2 are given as, Rm@)= j;PNz(t2)4 = [4[24~N - ~ l( ) ~t1z- 4 f ( ~ I ) } f- (4~) fz( t I ) >

1, ;1

Qm(t)= dt2 dt, (1-

-

- 1, (ti ) } f ( t z - > f ( 4).

(3-31) (3-32)

Fig. 3-3(c) shows the chance of product formation during a third re-encounter (pN3(t)dt) in the interval (t, t + dt) after a second re-encounter at t2. With a similar method used for a second re-encounter, Pm(t)becomes the product of Qm(tl),A, ( t - t 2 ) ,f(t-t2). P M ( ~=) Q d t d =

j; W

& ( t - t z ) f(t-td

N (t

- t, > f ( t 12 , I’4 (1- A N (12 - t, )HI - 2, (4

m z - )f@,1. tl

(3-33)

Eq. (3-33) is the same as the third term of the right side of Eq. (21) in Ref. El]. The probabilities for recombination at fourth, fifth, etc. re-encounters can similarly be obtained. Thus, the probability (PN(t)dt)for recombination in the interval (t, t+dt) can be given from Eqs. (3-27), (3-30), and (3-33) as (3-34) PN(t)= PNl(t) + Pm(t) + pM(t) + ... ... . This is the same as Eq. (21) of Ref. [3].

29

3.4 Approximationsof Recombination Probabilities for S, T, and F-Precursors Let us consider approximations of Eq. (3-34) for the cases of S, T, and F-precursors. (a) S-Precursor In this place we will consider the case where both R and IC,,

(t)lzare close to

unity for

small radicals. Here, we can neglect all but the first term in Eq. (3-34). In this case, the total fraction of pairs (PN’),with nuclear state N that recombines, is given by

PN’

=

lomp,i(t)dt = %R,(t)f(t)dt

=

1 2 IC&(f)i2f(r)df

(3-35)

Because IC&(t)/2and f(t) are given from Eqs. (3-19a) and (3-21), Eq. (3-35) becomes

=

A$?- ThQi I .

(3-36)

m,

Problem 3-7. Prove that I =

in Eq. (3-36).

Thus, the PNsvalue can approximately be given by

2 )R(p - X,). PN’ R(p - m d / z Q i w ~ / = Here, X, is defined as X, = m n - 1 1 2 Q ~ & ~ 3 / 2 . @) T- Precursor

(3-37) (3-38)

30

-

I a z”zhQ; 3(1- p ) ~ ;

3(1- p ) ~ : ’

X

N

3(1- p ) ‘

(3-42)

9.

It is noteworthy from (3-42) that XN enters in P” with a positive sign. In Eq. (3-37), however, XNenters in PN wth a negative sign. (c) F-Precursor The case of uncorrelated free-radical encounters is not so easy to understand as the S and

T cases. During their first encounter, a fraction of singlet pairs ( / C ,(Of = 1/4) combines with a rate of A and the remainder has a chance of meeting again. In Ref. [11, Kaptain showed that the total fraction of pairs (P;), with nuclear state N that recombines, would be given by the following relation with a similar treatment used for Pz: (3-43) When A = 1 , Eq. (3-43) becomes very simple.

PN” q 4 l+3-].

(3-44)

We can see from Eqs. (3-42) and (3-44) that P z and P z behave similarly in the presence of an external magnetic field.

References [ 11 (a) H. Hayashi, in “Molecular Magnetism”, K. Itoh, Ed., Gakkai-Shuppan-Center, Tokyo, 1996, Chap, 5. (b) H. Hayashi, in “Dynamic Spin Chemistry”, S. Nagakura, H. Hayashi, and T. Azumi, Eds., Kodanshfliley, 1998, Chap. 2. [2] H. Hayashi, J. Chinese Chem. Soc.,49 (2002) 137. [3] R. Kaptain, J. Am. Chem. SOC.94 (1972) 625 1. [4] R. M. Noyes, J. Am. Chem. SOC.,78 (1956) 5486.

31

Solution to the Problems

32

Because S o 1 = SJ,

+ SJy + &Iz = -1( S + r +

2 34c), we can get the following result:

Sr')+ &Iz from Eqs. (1-34a), (1-34b), and (1-

33 From Eq. (3-14), we can write as follows: i A D = ifi dt

i~

(i.,(t)p, + i., (t)p, ) = H~ w t ) = H~ (c,( t ) p , + c, ( t ) p ,I

(& (t)pS+ &-(w,) = H~ (c,(t)vS+ c, WP, >.

Integration of the above equation with qi and 9; gives the following results:

A - 3~ C,(t). ihi.r(t)= Q ~ c,(t)

If we take C, ( t ) = AseP and C, ( t )= AreP, we can obtain the following solutions from the above two equations: p = fiw, = kid=, A, =-- 3 + w A,. QN

Thus, C,(t) and C,(t) can be represented as follows when simplicity:

WN

is written as

w for

C, ( t )=Cleiw+ CZe-jn',

3-w,

--

C,e-'"'

.

QN

Using

eiw = cos ux

+ i sin WL and e-jw = cos m - i sin on ,the above two equations become as

C , (t) = (C, + C,)cos uNt+ i(C, - C,)sin uNt= Acos u N t+ Bsin WNt,

(lN

Because C, (0) = A and C, (0) = - -A QN

B =--C,

UN

9 (0)- -CS WN

(0)

+ -3-B i N

)

,B becomes as

34 Substituting A and B by CdO) and CdO) in the above equations for Cdt) and Cdt), we can prove Eqs. (3-16a) and (3-16b). 3-5. (a) From Eq. (3-18), wN = QN. Because QNis given by Eq. (3-13a), W,

1 = QN = -AgpBB= 0 . 0 1 ~ 9 . 2 7 4 0 ~ 1 0 ~ ~ ~ J T ~ ~ * l T / 2 * 1 . 0=54.40~10*~-’ 457~1O~~J~

2ti (b) From a similar procedure as shown above,

*1/2 =4,40x108s~l, 1 10mT*gpBM,~10mT*2~9.2740x10-24JT-‘ = Q N =- A i M i = 2ti 2ti 2 1.05457~10-”Js 3-6. When x is used as x = lit, dx/& becomes as dx/& = - f 2 W,

Right side of Eq. (3-22) =

[f(t)dt

=

$mt-”’

ex(%).

3-7. In Eq. (3-36), the exponential part becomes 0 and 1 for t respectively. Thus. The integral can be approximated as follows: I = $sin’ ~ , t ( t ) - ~ / ’

= [sin’ w,t(t)

-

0 and t

+

m,

-312

dt

= F.

In order to get the value of F, we can use the relation of (f g)’ = f g + f 8’. Integration of the both sides of this equation gives $(fg)’dt

=

$f’gdr+ lornfg‘dt.

Because f = sin’ mNt and g’=t-312in

OUT

case,

y=2(sinwNt)(coswNt)wN=(sin2w,t)wN

and g = -2t-’I2. The above integration becomes as fg ;1

=

irf’gdt + F.

Because fg(t = 0) = 0 and fg(t = m) = 0, F becomes as

F = -jmf’gdt

=

-$(sin2cu,t)oN(-2t-’’*)dr=

20,c

sin 20,t

2 20,

35 4. Chemically Induced Dynamic Nuclear Polarization (CIDNP) 4.1 Discovery of Chemically Induced Magnetic Polarization and ESR Spectra of Radical

Pairs In 1963, Fessenden and Schuler [l] found during irradiation of liquid methane (CH4 and CD4) at 98 K with 2.8 MeV electron that the low-field signals (a1 and b l ) for both hydrogen and deuterium atoms appeared inverted (emissive signals) and that the central deuterium atom signal (b2) was very weak as shown in Fig. 4-1. Although the cause of such anomalous ESR spectra was not clear at that time, similar anomalous ESR signals have been observed in many reactions and have been called Chemically Induced Dynamic Electron Polarization (CIDEP)”. CIDEP should be due to non-equilibrium electron spin state population in radicals. ‘I

:: 19.9aS; .1

“v” +-+I

17.24G

4

;d

506.6G

Ib3 Fig. 4-1. Second derivative ESR spectra observed for hydrogen atom (a1 and a2) and deuterium atom @I, b2, and b3) with 2.8 MeV electron irradiation of liquid methane ( and deuteromethane) at 98 K. The observed spacings and the shifts from the first-order positions are indicated. The first-order positions of both hydrogen and deuterium atoms center at a field (indicated by the arrow) which corresponds to g = 2.00223. The tallest signal indicated by c is due to CD3. (Reproduced from Ref. [ 11by permission from The American Institute of Physics) On June 11, 1965, the author (H. Hayashi) and Dr. K. Itoh visited Dr. Y. Kurita at his office in The Basic Research Laboratory of Toyo Rayon Company, Ltd. and saw his beautiful ESR spectra of radical pairs (“J” and “K’) in single crystals of dimethylglyoxime irradiated by X-rays at 77 K 121. Here, the radical pairs “J” and “K’ are symmetric and asymmetric pairs, respectively, as shown in Fig. 4-2. The typical ESR spectra observed for the radical pairs “J” and “ K are shown in Fig. 4-3. The author noticed from Fig. 4-3(b) that the central three lines of the nine hyperfine (HF) lines due to two nitrogen atoms of “K” were not equally spaced [3], but that there is no anomaly in the HF lines of “J” as shown in Fig. 4-3(a). We found that the anomalous HF lines of “K’ could be explained by the mixing of the singlet and triplet states of a radical pair in the complete Spin Hamiltonian of the pair developed by Dr. Itoh [3]. This theory has been called “the radical pair mechanism”.

36

Cr - O H

\ CH3

B

1OOG Ic--.--;w

I

I

+

Fig. 4-2. The crystal structure of dimethylglyoxime projected on the (001) plane. The configurations of the radical pairs ”J” and “K’ assigned by Kurita [2] are shown. (Reproduced from Ref. [3b])

Fig. 4-3. Second derivative ESR spectra of the radical pairs (a) “J” in y-irradiated single crystals of dimethylglyoxime-d* and (b) “K’ of dimethylglyoxime, respectively [3]. The external magnetic field is applied along the a’ axis [2].’ M, J, and K denote the ESR signals due to the mono-radical and the radical pairs “J” and “K’,

respectively.

In 1967, Bargon et al. [4] and Ward and Lawler [5] found independently that the intensities of NMR spectra of reacting systems showed emission (E) and enhanced absorption (A), but that other characteristics such as line frequencies and line width were normal. Bargon et al. studied the thermal decomposition of dibenzoylperoxide and di-p-chloro dibenzoylperoxide. Fig. 4-4(a) shows NMR spectra (100 MHz) taken during the thermal decomposition of dibenzoylperoxide at 110°C in cyclohexane: At t=O min., the sample has just been transferred into the NMR probe and the spectrum shows the normal absorption signals of dibenzoylperoxide. During its decomposition reactions, these signals decrease in their intensities and an emissive signal appears at 6=7.31 ppm. The intensity of this signal reaches a maximum at t=4 mins., then decreases. At r=7 min., this signal reappears in an absorptive phase and reaches a constant absorptive intensity at the end of the reaction. Fig. 4-4(b) shows the time dependence of

37 this signal, which is assigned to benzene (Ph-H) formed through hydrogen abstraction from the solvent (RH) by the phenyl radical (Ph'): (PhCOO-)*+ heat + '[PhCOO' 'OOCPh] Ph'

+ RH

--f

+ '[PhCOO'

'Ph]

+ C02,

Ph-H + R'.

(4-1) (4-2)

Here, '[A' 'B] represents a singlet radical pair consisting of A and B radicals.

I Idrnin

r

0

l

,

5

10

15 mm

Fig. 4-4. (a) NMR spectra (100 M H z ) taken during the thermal decomposition of dibenzoylperoxide at 110°C in cyclohexane. (b) Time dependence of the signal (6=7.31 ppm) assigned to benzene. (Reproduced from Ref. [4a] by permission from The Verlag der Zeitschrift fur Naturforschung)

38

d

c

d

Fig. 4-5. NMR spectra (60 MHz) taken during the thermal reaction of n-butyl bromide (n-CdHgBr) with nbutyllithium (n-C4HgLi) in hexane. (a) NMR spectrum of I-buten. (b) NMR spectrum taken at the beginning of the reaction. (c) NMR spectrum taken during the reaction. (d) NMR spectrum taken at the end of the reaction. (Reproduced from Ref. [5a] by permission from The American Chemical Society)

b

a I

I

6

5 PPm

Ward and Lawler studied the thermal reactions of n-butyl bromide with n- and tbutyllithiums [5]. Fig. 4-5 shows NMR spectra (60 MHz) taken during the thermal reaction of n-butyl bromide (n-C4HgBr) with n-butyllithium (n-C4HsLi) in hexane: (1) Spectrum b was taken at the beginning of the reaction. Here, there was no signal. (2) Spectrum c was taken during the reaction. Here, signals indicated by A showed NMR polarization in emission (E), those by B the polarization in enhanced absorption (A), and those by C the polarization of emission followed by enhanced absorption from a low to a high magnetic field (WA). (3) Spectrum d was taken at the end of the reaction. This spectrum is consistent with that of 1-buten shown in spectrum a. Polarized signals observed during this reaction is assigned to 1-buten formed by the following processes: n-GH9Br + n-GH9Li --t ‘[n-C4‘H9 n-C;H9]+ LiBr, + n-C4Hlo.

(4-3) (4-4) Such anomalous NMR spectra as observed in the above reactions have been called “Chemically Induced Dynamic Nuclear Polarization (CIDNP)”. CINDP should be due to nonequilibrium nuclear spin state population in reaction products. At first, the mechanism of CIDNP was tried to be explained by the electron-nuclear cross relaxation in free radicals in a similar way to the Overhauser effect [4b, 5b]. In 1969, however, the group of Closs and Trifunac [6] and that of Kaptain and Oosterhoff [7] showed independently that all published CIDNP spectra were successfully explained by the radical pair mechanism. CIDEP could also be explained by the radical pair mechanism as CIDNP. In this and next chapters, we will see how CIDNP and CIDEP can be explained by the radical pair mechanism, respectively. 2 n-C4‘H9 -+ CH$H$ZH=CH

39

4.2 Theoretical Interpretation of CIDNP by the Radical Pair Mechanism In this section, we will see how CIDNP can be explained by the radical pair mechanism, In Chapter 3, we obtained following the pioneer paper written by Kaptein [7b]. approximately the recombination probabilities from S, T, and F-precursors (PN' from Eq. (337), PN' from Eq. (3-42), and PN" from Eq. (3-44) as follows: pNs=

(4-5)

a(p-xN)>

Here, XN is defined as X, = r n ~ ' ~ ~ Q ~ w ; ~ ' ~ . The escape probabilities from S, T, and F-precursors (EN', EN'

=1-

PN'

(4-8)

r = S, T, and F) are given as

(r= S, T, or F).

(a> INM > 0

(4-9)

(b) INM < 0

Fig. 4-6. NMR signals due to spin i (a) in absorption and (b) in emission. The intensity of the NMR transition (ZNM) due to nucleus i from I x , ) ( = I M ; , M ~ ) ) to

Ix,)

=

( 1 ~ ;- 1 , ~ ~ is) )given as

(4-10) zNM = yN' - YM' (Y = P or E, r = S, T, or F). As shown in Fig. 4-6, an absorptive signal corresponds to a positive IN,+, value and an emissive one a negative ZNM value. From Eqs. (4-5) - (4-9), the ZNM value can be represented by

INM

0~

P&(XN

-XM

1.

Here, p is negative from S-precursors and positive from T- or F-precursors and for cage products and negative for escape products.

(4-1 1) E

is positive

The &value in Eq. (4-8) is given by Eq. (3-18).

w, =

Jm.

(4-12)

CIDNP occurs in the region of radical pairs where 3' >> Qi . From Eqs. (4-8) and (4-1 l), the INM value can be expressed as

INM 0~

-Qh 1.

(4-13)

40 FromEq. (3-13a), the

QN

is given as

1 Qry= -[ Ag,LLBB + C P A i M i 2h Thus, the INM value becomes ~ N M0~

P ~ Q -; Q i 1=

P(QN

Zz AkM, ] .

(4-14)

= E([Ag,uBB+AiM+ i T A , M , -C:A,M,] 4A pfi -[ Agb!,B+

(Mi

(4- 15a)

- Q M X Q N +Q M )

-l) +

f:

p

2

-x:

I*)

(4-15b)

p#i

(4-1%) = E A i [ A g p , B + f : A , M , - C t A , M , + A i ( M i-1/2)]. pfi 2A Here, nucleus p is located on the same radical (a) as nucleus i is, but nucleus k is located on the counter radical (b). Eq. (4-15) is the fundamental equation for CIDNP and can be classified into three typical cases as follows: Case 1: The first term in [ ] of Eq. (4-1%) is the most important. In this case, the INM value becomes

INM OC P& Ai AgPBB I . (4-16) This means that the sign of INM can be determined by the product of ,U& Ai Ag . Thus, the sign of the net polarization of nucleus i (r,(i) ) is given by the product of four signs (4-17) Here, A and E represent enhanced absorptive and emissive signals, respectively. Case 2: The second term in [ ] of Eq. (4-1%) is the most important. In this case, the value becomes (4-18a)

INM oc P& Ai [ F A p M p l . pfi

If there are several equivalent nuclei p, Eq. (4-16a) is simplified as (4- 18b) B = Bi (with [C$p]=O) (a) Jip > 0

CO) Jip < 0 Fig. 4-7. NMR signals with (a) positive and (b) negative .Iipvalues.

41

From Fig. 2-8, we can see that the NMR signals appear as shown in Fig. 4-7. Let us consider the case when the product of p & A i Ap is positive. In this case, emissive signals appear at lower fields than Biand enhanced absorptive ones at higher fields than Bi if Jip is positive as shown in Fig. 4-7(a). This is denoted by E/A. If Jip is negative, the reversed A E signals can be observed as shown in Fig. 4-7@). Thus, the phase of the multiplet effect of nucleus i coupled with several nuclei p which are located on the same radical (r,,,, (i, p ) ) is given by the product of five signs. (4-19)

3 of Eq. (4-15c) is the most important. In this case, the INM

Case 3: The third term in [ value becomes INM

Oc

-pE Ai

[xfiAkMk 1.

(4-20a)

If there are several equivalent nuclei k, Eq. (4-20a) is simplified as INM OC

-p& Ai Ak [

x:

Mk

1.

(4-20b)

Similarly, the phase of the multiplet effect of nucleus i coupled with several nuclei k which are located on the counter radical (r,,,, (i, k ) ) is given by the product of five signs. (4-21) The sign rules represented by Eqs. (4-19) and (4-21) can be generalized when qj is introduced. l-,,,e(i,j ) = m i A jJ , Gj {

+

EIA,

- A/E.

(4-22)

Here, qjis positive when nuclei i and j are located on the same radical, but q is negative when nuclei i and j are located on the different radicals. The fourth term [ ] of E y (4-15c) is not so important because it vanishes when Mi is 112. This is the case for proton-, 3C-, and F-NMR. Eqs. (4-17) and (4-22) are called Kaptain’s rules, which are very useful for the interpretation of CIDNP spectra.

42

4.3 Examples of CIDNP In this section, several typical CINPD spectra will be shown. These spectra can be explained by Kaptain's rules. Typical net absorptive and emissive CINDP signals were observed during the thermal decomposition of acetyl peroxide (AP) in hexachloroacetone at 110 "C as shown in Fig. 4-8. Here, enhanced absorptive signals were observed for CH3CI and CH4 and emissive ones for CH3COOCH3 and CH3-CH3. (CH,COO!,

IN HCA. 110 OC

1

f

3

4

2

1

0 & P P ~

Fig. 4-8. 60 MHz 'H-CIDNP spectrum of the thermal decomposition of acetyl peroxide in hexachloroacetone. (Reproduced from Ref. [8] by permission from Kluwer Academic Publishers) This reaction occurs from the following scheme: (CH3C00)2, Acetyl peroxide (AP)

1 heat '[CH3COO' '00CCH31

(4-23a)

1 CO;?+ '[C'H3 '00CCH3]

+

CH3COOCH3 (Cage product)

(4-23b)

1 2C02 + '[C'H3 C'H3]

+

CH3-CH3 (Cage product)

(4-23c)

1

CH3CI, CH4 (Escape products). (4-23d) In this reaction, the radical pair consisting of the methyl and acetoxy radicals, '[CH3. .00CCH3], is important for the CIDNP signals of its product. The phase of CIDNP signals can be explained by Eq. (4-17), which needs the g-values of the methyl and acetoxy radicals and the HFC constant of the methyl radical.

43 Table 4-1. The g values and HFC constants ( A a ,A,, and A ,values) obtained for some radicals by ESR studies. a The HFC constants of o-H, m-H, and p-H of the phenyl radical are

The g values and HFC constants (A o , A,, and A .values) obtained for some radicals by ESR studies are listed in Table 4-1. It is noteworthy that the positive and negative signs of the HFC constants could be obtained from molecular orbital calculations for the radicals listed in Table 4-1 [12]. The emissive CIDNP observed for CH3COOCH3 can be explained by Eq. (4-17) and Table 4-las follows: r,(C&COOCH,) = ~(S-precursor)~(cage)A,(CH~.)[g(C~H~)-g(CH~COO')] = (-)(+)(-)(-) = (-) E. (4-24) It should be noted that the A,(C& COO') value was not used in Eq. (4-24) because this value is negligibly small. The emissive CIDNP observed for CH3-CH3. can also be explained as follows: rn(C&-C&) = ~(S-precursor)~(cage)A,(CH~~)[g(C'H~)-g(CH~COO')J = (-)(+)(-)[-I = (-) E.

(4-25)

It should be noted that [g(C'H3)-g(CH3COO')] is used for the Ag value in Eq. (4-25) because this radical pair is the most important in this reaction. Such phenomena is called "the memory effect." Similarly, we can explain the enhanced absorptive CIDNP observed for CH3CI and CH4 during this reaction and the emissive CIDNP observed for benzene during the thermal decomposition of dibenzoylperoxide shown in Fig. 4-4. Problem 4.1. Prove the enhanced absorptive CIDNP observed for CH3CI and CH4 during the thermal decomposition of acetyl peroxide (AP) in hexachloroacetone. Problem 4.2. Prove the emissive CIDNP observed for benzene during the thermal decomposition of dibenzoylperoxide shown in Fig. 4-4. Typical multiplet CINDP signals were observed during the thermal decomposition of propionyl peroxide in hexachloroacetone at 110 "C as shown in Fig. 4-9. Here, A/Esignals were observed for CH3CH2CI. This reaction occurs through the following scheme: (CH3 CH2C00)2 + heat +2C02

+ '[CH3 C'H2

C'H2CH31-

C4HIo (cage product)

1 RCI CH3CH2CI (escape product). The A/E signal of the methyl proton at & = I S ppm can be explained by Eq. (4-22).

(4-26)

44

Fig. 4-9. 60 MHz 'H-CIDNP spectrum of the thermal decomposition of propionyl peroxide at 110 "C in hexachloroacetone. (Reproduced from Ref. [8] by permission from Kluwer Academic Publishers)

4.0

3.0

2.0

1.0

6PPM

rm,(C&CH2CI) = p(S-precursor)~(escape)A,(C€I3C'H2) AJ(CH3C'&)J(C&CbCl)~iJ = (-)(-)(+)(-)(+)(+) = (-1 : A/E. It is noteworthy that the sign of J(C&C&Cl) is positive. ethyl proton at L 3 . 5 ppm can be explained by Eq. (4-22).

(4-27) Similarly, The A/E signal of the

Problem 4.3. Prove the A/E signal of the ethyl proton at 6=3.5 ppm in Fig. 4-9. Although CIDNP signals were first found during the thermal reaction of n-butyl bromide (n-C4HsBr) with n-butyllithium (n-C4HgLi) in hexane [ 5 ] , their interpretation has not been clear. This is due to the fact that the NMR spectrum of I-butene is very complex as shown in Fig. 45(a). Anyhow, the NMR signals around 6-4.8 ppm are due to protons at the 1-position and those around 6-5.8 pprn due to proton at the 2-position. From Fig. 4-5(c), both protons at the 1- and 2-positions showed the EIA patterns, when radical scavengers were added to this reaction. On the other hand, no CIDNP was observed without the scavenger. This means that the observed CIDNP is due to the cage recombination. Thus, the observed EIA patterns can also be explained by Eq. (4-22). T,ne(CH3CH2CH=C&) =~(S-~~~CU~SO~)E(C~~~)A,(CH AJ(CH3CH2C&C'H2) ~CH~CH~C'&)

J(CH~CH~CH=C&)CJ,, = (-)(+)(-)(+)(+)(+)= (+) : EIA.

(4-28)

and r,ne(CH3CH2C€J=CH2) = p(S-precursor)~(cage)A,(CH3CH2C&C'H2)A,(CH3CH2CH2C'&)

J(CH~CH~CFJ=C&)CJ,, = (-)(+)(+)(-)(+)(+) = (+) : EIA.

(4-29)

Although the HFC constants of CH3CH2CH2CH2-are not listed in Table 4-1, the sign of A, should be negative and that of A,positive. The J(CH3CH2CH=C&) value is also considered to be positive.

45

References [ l ] R. W. Fessenden and R. H. Schuler, J. Chem Phys., 39 (1963) 2147. [2] (a) Y. Kurita, J. Chem. Phys., 41 (1964) 3926; (b) Y. Kurita and M. Kashiwagi, ibid, 44 (1966) 1727. [3] (a) H. Hayashi, K. Itoh, and S. Nagakura, Bull. Chem. SOC.Jpn., 39 (1966) 199; (b) K. Itoh, H. Hayashi, and S . Nagakura, Mol. Phys., 17 (1969) 561. [4] (a) J. Bargon, H. Fischer, and U. Johnsen, Z. Naturforswch., A, 22 (1967) 1551; (b) J. Bargon, H. Fischer, ibid, 22 (1967) 1556. [5] (a) H. R. Ward and R. R. Lawler, J. Am. Chem. SOC.89 (1967) 5518; (b) R. R. Lawler, ibid, 89 (1967) 5519. [6] (a) G. L. Closs, J. Am. Chem. SOC.91 (1969) 4552. (b) G. L. Closs and A. D. Trifunac, J. Am. Chem. SOC.92 (1970) 2183,2186. [7] (a) R. Kaptain and L. J. Oosterhoff, Chem. Phys. Lett., 4 (1969) 195, 214. (b) R. Kaptain, J. Am. Chem. SOC.94 (1972) 6251. [8] R. Kaptein, in “Chemically Induced Magnetic Polarization”, L. T. Muus, P. W. Atkins, K. A. McLauchlan, and J. B. Pedresen, Eds., D. Reidel, Dordrecht, Holland, 1977, Chap. 1. [9] P. H. Kasai, P. A. Clark, and E. B. Whipple, J. Am. Chem. SOC.92 (1970) 2640 [ 101 R. LoBrutto, E. E. Budzinski, and H. C. Box, J. Chem. Phys., 73 (1980) 6349. [ l l ] S. Yamauchi, N. Hirota, S. Takahara, H. Sakuragi, and K. Tokumaru, J. Am. Chem. SOC.

107 (1985) 5021. [I21 J. A. Pople, D. L. Beveridge, and P. A. Dobosh, J. Am. Chem. SOC.,90 (1968) 4201. Solution to the Problems 4.1. From Eq. (4-17) and Table 4-1, the enhanced absorptive CIDNP of CH3C1 can be explained as follows: Tn(CH3Cl and CH4) = y(S-precursor)~(escape)A,(C‘H~)[g(C’H3)-g(CH3 COO’)] = (-)(-)(-)[-I = (+) : A. 4.2. From Eq. (4-17) and Table 4-1, the emissive CIDNP of benzene can be explained as follows:

rn(&-H)

= y(S-precursor)E(escape)A,(Ph’)[g(Ph’)-g(PhCO0’) = (-)(-)(+)[-I = (-) E.

Here, the most important radical pair in this reaction is ‘[PhCOO. ‘Ph]. It is noteworthy that the A,(Ph.) values are shown to be positive from Table 4-1.

4.3. From Eq. (4-22) and Table 4-1, the A/E signal of the ethyl proton at 6=3.5 ppm can be written as follows: r,,,,,e(CH3C€&CI)= p(S-precursor)E(escape)Al(CH3C&) A,(C&C’H~)J(CI~~CH~CI)G~, = (-)(-)(-)(+)(+)(+)= (-) A E .

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47

5. Chemically Induced Dynamic Electron Polarization (CIDEP) 5.1 Historical Introduction In 1963, Fessenden and Schuler [l] found during irradiation of liquid methane (CJ& and CD4) at 98 K with 2.8 MeV electron that the low-field line for both hydrogen and deuterium atoms appeared inverted (emissive signals) and that the central deuterium atom line was very weak. Although the cause of such anomalous ESR spectra was not clear at that time, similar anomalous signals have been observed in many reactions and have been called " Chemically Induced Dynamic Electron Polarization (CIDEP)". CIDEP should be due to non-equilibrium electron spin state population in radicals and could also be explained later by the radical pair mechanism as CIDNP. In 1960s, CIDEP was less popular than CIDNP because CIDEP did need much faster measuring techniques than CIDNP. This is due to much faster relaxation times (usually less than 1 LL s) of polarised electron spins than those (usually a few second for protons) of nuclear spins. In 1968, Smaller et al. [2] observed a population inversion for the cyclopentyl radical with a 2-MHz ESR apparatus coupled with a 15 MeV electron beam with pulse duration of 0.5 -4.0 s. The response time of the system corresponded to a time constant of 1.6 LL s. In 1970, Atkins et al. [3] obtained the photo-CIDEP for the ketyl radical from benzophenone in paraffin solvents with a 2-MHz ESR apparatus coupled with a 20-ns laser flash. Under favorable chemical conditions, Wong and Wan [4] demonstrated that the photoCIDEP for some semiquinone radicals in alcohol solvents could be observed with a commercial ESR spectrometer having a 100-kHz modulation unit and a custom-designed rotating sector giving light pulses. Although such a 2-MHz ESR apparatus was very sophisticated, its time resolution was not enough for measurement of CIDEP. In 1973, Fessenden [5] found that the direct ESR measurement without field modulation improved the time resolution, observing CIDEP signals in solution with pulse radiolysis. This method was applier' for laser-photolysis measurements in solids [6] and in solution [7]. A spin-echo ESR technique was also found to be useful for CIDEP [8]. Since then, CIDEP experiments with cw-ESR and pulsed-ESR spectrometers without field modulation have become much more popular than before. Through such transient ESR measurements, CIDEP due to not only the radical pair mechanism but also several other mechanisms have been observed in many chemical reactions including biologically important ones such as photosynthesis reactions . In this chapter, we will show several mechanisms for CIDEP with several typical examples.

5.2 Theoretical Interpretation of CIDEP by the Radical Pair Mechanism In this section, we will see how CIDEP can be generated from the radical pair mechanism. The spin polarization (P) of ESR transition is represented as shown in Fig. 5-1. Here, P is given as follows:

The population of the i-th level is denoted by ni (i = ct or p), which is represented by

(5-2)

nu = N ( 1 / 2 + ( S , ) ) ,

n p =N(1/2-(Sz))

.

(5-3)

48

a,Es(ms=+l/2)

p, Es(ms=- 112) (a) P < 0

(b) P > 0

Fig. 5-1. ESR signals in (a) absorption and (b) in emission.

In Eqs. (5-2) and (5-3), N is the total number of electron spins and dz> is the net zcomponent of the spins. Eqs. (5-2) and (5-3) are reasonable because n, (np) becomes N o r 0 (0 or N) when d,> is 1/2 or -112. From Eqs. (5-1) - (5-3), P can be given as foIlows:

P =2dz> (5-4) Problem 5-1. Calculate the P values in the thermal equilibrium (P,,) at room temperature (20 "C) for the ESR transitions of an electron at the X-, K-, and Q-bands (9, 25, and 35 GHz). CIDEP is observed for the case of P < P,, (enhanced absorption) or P > 0 (emission) as shown in Fig. 5-1. (a) CIDEP due to the S-To mixing The theoretical interpretation of CIDEP is much more complex than that of CIDNP. As is shown below, the total polarization (PI + P 2 ) of two radicals generated from a radical pair is zero. Thus, there is no CIDEP when the two radicals have their ESR signals at the same position. On the other hand, CIDEP is possible when the two radicals have their signals at different positions. In this section, therefore, we will consider the latter case and calculate the P I and P2 values separately when the S-To mixing is the most important. Pl(t) = 2<s1,>, (5-5a) P 2 ( t ) = 2<s2,>. (5-5b) value of the N-th nuclear state Using Eq. (3-15), the dlz> <SlZ>N

= (W)JS,,

(Ix,))is obtained as follows:

JW))

(5-6a)

(c;,(t)(s,x,(+Ck(t)(T>XNI)slZ(CSN(t)lS,X,)+CTN(t)lT,jl,)) = (c;,wc, ( t )+ C k ( t ) C , ( t ) ) l 2 . =

(5-6b) (5-6~)

Problem 5-2 Prove Eq. (5-6c) from Eq. (5-6b). value of the N-th nuclear state (Ix,))is obtained as follows: Similarly, the dzz> <s2z>N

= ( y ( t )IS,

Iy(t))= - (cl, (t)c,( t ) c; (t)cpq( t ) ) /2 =

-<slz>N

.

(5-7)

Problem 5-3. Prove Eq. (5-7). From Eqs. (5-6) and (5-7), one can see that < S l z > ~+ <s2z>N is zero as stated at the beginning of this section. At first, let us consider the polarization induced at the first reencounter from t = 0 to t = t' PI@')= 2 d I , > =

(c;,(t')c,(1') + c;, ( t ' ) C ,

@I)).

From Eqs. (3-16a) and (3-16b), the Pl(t') value becomes as follows:

P l ( t ' )= {C,(O)C~(O)+C~(O)C,(O)}{cos2w,t'+2(Q, /w,)' sin2~ , t ' }

(5-8)

49

+ (is/ w , {c,( 0 ) ~(0) : - C; (o)c, ( 0 ) )sin 2w,t' + ( 2 Q N 3 / u ; ){ ICs(0)Iz- ~ C , ( 0 ) ~ 2 } s i n 2 ~ N t '

(5-9)

Problem 5-4. Prove Eq. (5-9). Eq. (5-9) shows that the I Pl(t')I value is too small to give appreciable CIDEP from the following reasons: (1) Because Ci (O)C, (0) and C; (O)C, (0) are zero for reactions from Sand T-precursors, the first and second terms of Eq. (5-9) become zero. (2) Because the 3 value is nearly zero during t = 0 - t ', the third term of Eq. (5-9) becomes also zero. From Eq. (5-9), CIDEP is not induced at the time of the first reencounter for t'-0 although CIDNP and magnetic field effects occur at this time. CIDEP, however, is developed when the 13 I value keeps much larger than the lQ4 one during the second reencounter for t--t ' as follows: Pl(t-f')= {C,(t')C,'(t')+C~(t')Cs(t')}(cos2w,(t-r')+2(Q, /w,)' sinZw,(t-t')}

+ (i3/0,)

{~,(tl)~~(t')-~;(t')~,(t')}sin2w,(t-t')

+ (2Q,S/w;){

(Cs(0)~'-~C,(O)~'}sin'~,(t-t').

(5-10)

The first term of Eq. (5-10) becomes zero because fC,(t')Ci(t')+ C;(t')Cs(tt))is zero from Eqs. (5-8) and (5-9). The third term of Eq. (5-10) is also negligible because 1 2 Q , 3 / 0 ; 1 ((1. The second term of Eq. (5-lo), however, can induce CIDEP because of the following relation:

Problem 5-5. Prove Eq. (5-11). Thus, Pl(t-t') is given from Eqs. (5-10) and (5-1 1) as follows:

In Eq. (5-12), wN is replaced by 131 during the second reencounter for t-t '. When zc (=t-t ') is represented by the time range of the second reencounter with 13 l>>lQd, the spin polarization can be given as

[i

P,(Z~) = f(t')rI(t-t')dt'= [o-mt'-3'z<(t-t')dt'

Problem 5-6. Show that the integral of I (= ~0mmt'-3'2 sin 2 I QN 1 t ' d f ) is given by

50 I = 2m&i Thus, Eq. (5-13 is transferred into

& where z,

Because rn is nearly equal to

is the average time between diffusive steps

(z, = 10-'2s)[9] , Eq. (5-14) becomes as follows:

In the case of strong collisions where I 3 I z, >>I, the average value of [sin 2 I 3 I z, ] is zero. In the case of weak collisions where ( 3(z, <<1, however, the average value of [sin 2 1 3 I z, ] is not zero, but can be assumed to be about 0.1 [ 101. In the former case, no CIDEP is produced, but appreciable CIDEP can be expected for the latter case. By writing the exchange interaction with an exponential dependence on inter-radical separation (r),

3 ( r ) = 3, exp(-Ar) .

(5-16)

Monchick and Adrian [ l 11 obtained an asymptotic solution for the polarization under the condition of slow STOmixing [12]: (5-17) Here, d is the distance of closest approach. Eq. (5-17) gives important results that the degree of CIDEP is proportional to lQNz, 11/' and that the sign of CIDEP is determined by

{ lc, (0)(2- ICT ( o r }

QN30 N O

.

Similarly, the polarization of the other electron can be

expressed from Eqs. (5-7) and (5-17) as follows: (5- 18)

Because

QN

is given from Eq. (3-13a) as

51

(5-19) the qualitative results for the phase of CIDEP are summarized in two simple sign rules: (i) For the nest effect where Ai = A, = 0 J, the sign (r,)of -PI is defined as

I-,,= signof ( - P I ) = pJAg.

(5-20)

Here, p is negative for radical pairs from S-precursors and positive for T- or F-precursors as shown in the case of CIDNP. From Fig. 5-1 and Eqs. (5-17), (5-18) and (5-19), a positive value of r,, implies a CIDEP spectrum in absorption (A) and a negative value, one in emission (E). (ii) For the multiplet effect where Ag = 0 , the sign of this effect (I-,,,) is defined as

(5-21) From Fig. 5-1 and Eqs. (5-17), (5-18) and (5-19), for a positive r,,,, the low-field half of a CIDEP spectrum is in absorption and its high-field half in emission (AE). For a negative I-,, , however, the low-field half of a CIDEP spectrum is in emission and its high-field half in absorption (E/A). (iii) For the mixed effect of the net and multiplet cases where Ag # 0 and Ai,Ak # 0 J, mixed spectra of the two cases appear. If the mixed effect of E and E/A phases appear, for example, the observed spectrum shows an E*/A phase, where the asterisk denotes a predominance.

Problem 5-7. Confirm the sign rule represented by Eq. (5-20) in the following case: (1) The reaction occurs from a T-precursor, (2) the sign of J is negative, (3) gl = 2.0123, (4) g2 = 2.0023, and ( 5 ) Ai = A,, = 0 J. Problem 5-8. Confirm the sign rule represented by Eq. (5-21) in the following case: (1) The reaction occurs from a S-precursor, (2) the sign of J is positive, (3) gl = 2.0023, (4) g2 = 2.0023, ( 5 ) there is only one nuclear spin with I = 1/2 in each of radicals a and b, and (6)AP = A t = A > O J. Problem 5-9. Confirm the sign rule represented by Eq. (5-21) and the relative CIDEP signals for each HF line in the following case: (1) The reaction occurs from a S-precursor, (2) the sign of J is positive, (3) gl = 2.0023, (4) gz = 2.0023, ( 5 ) there is three equlivalent nuclear spins with I = 1/2 in radical a (AT = A; = A; = A > 0 J), and (6) there is no nuclear spin in radical b ( Akb= 0 J).

(b) CIDEP due to the S-TtI mixing In this section, CIDEP through the S-T,I mixing is explained qualitatively. From Eq. (313b), the S-T,, mixing occurs through the following off-diagonal matrix element:

52

Since J(r) is usually negative as shown in Fig. 3-2(b), the S-T.1 mixing is more important than the S-T+l mixing. Thus, let us consider the effect of the S-T.1 mixing at first. As shown in this figure, this mixing occurs in region II where r = r,, . Thus, a significant CIDEP is only generated in the following two cases: (i) a very large HF interaction of a radical pair produced a rapid S-T.1 conversion andor (ii) the slow diffusion of radicals in a radical pair enables the pair to remain a relatively long time in the region of the S-T.1 level crossing. As shown in Q. (3-13b), the S-T.1 mixing needs the following selection rules:

AMs =,1

and AM, =T1.

(5-22)

Here, Ms and MI are the magnetic quantum numbers of electron and nuclear spins, respectively.

In order to understand how CIDEP appears through the S-T.1 mixing, one should consider the following simple example: (1) radical a in a radical pair has one large HF interaction (A, > 0 J) with P or H (Zi= U2). (2) radical b in the pair has no HF interaction. (3) the radical pair is produced from a T-precursor. In this case the S-T.1 mixing occurs from the T., level of this radical pair (p+(l)p(2)) to its corresponding S level ([a(I)p(2)- p-(I)a(2)]/2”*). Here, p+ denotes the spin state having a down electron-spin and an up nuclear spin. a and phave similar meanings. This mixing decreases the populations of p+(l) and p(2) and increases those of a-( l), p-( l), and a(2)as shown in Fig. 5-2(b). On the other hand, the ST.1 mixing does not occur from the other T.l level of this radical pair (p-(l)P(2)) to its corresponding S level because the selection rule of AM, =-1 cannot be fulfilled. The above-mentioned results on the S-T.1 mixing shows that this model system gives emissive signals for both of radicals a and b as shown Fig. 5-201). It is noteworthy that the high field signal of radical a is missing in this figure. If a reaction proceeds from an S-precursor for this model system, the S-T.1 mixing occurs from the S level ([a(l)p(2)- P-(l)a(2)]/2’”) to its corresponding T.I level (p+(l)p(2)). This mixing decreases the populations of =(I), p(l), and a(2) and increases those of p+(I) and p(2). These results on the S-T.1 mixing shows that this model system gives enhanced absorption signals for both of radicals a and b as shown Fig. 5-2(c). When J(r) is positive, the S-T+I mixing occurs and the similar results can also be obtained for the CIDEP spectra. Thus, the sign rule for the S-T,1 mixing can be expressed as

r, = pJ

(positive: A, negative: E).

Problem 5-10. Confirm the sign rule of Eq. (5-23) for the S-T+l mixing.

(5-23)

53

(a) Thermal eq.

n I I

-

Abs.

I

I I I

Radical a

P+<1)

(b) S-T-l mixing (T-precursor)

Em.

P(2)

Radical b

a(2)

I I I

Abs.

I

I

-Qm&J-l I P+U) Radical a

Em.

P(2)

Radical b

(c) S-T.l mixing (S-precursor)

Radical a

Radical b

Fig. 5-2. Phase pattern of CIDEP spectra through the S-T.1 mixing (J < OJ) for the following simple example: (1) radical a in a radical pair has one large HF interaction with Ai > 0 J and I; = 1/2. (2) radical b in the pair has no H F interaction. (3) The pair is generated from (b) a Tprecursor and (c) an S-precursor, respectively. The number of circles schematically shows the population of each sub-level and the black circle indicates the transfer of the population through the S-T.1 mixing.

54 5.3 Theoretical Interpretation of CIDEP by the Triplet Mechanism There are many other mechanisms for generating CINEP. In this section, the Triplet Mechanism (TM), which is the oldest among them, will be explained. CIDEP due to the TM appears in the reactions which proceed through triplet-excited states. Thus, the reactions through T-precursors often show CIDEP spectra which have contributions due to both the PRM and the TM. The spin-Hamiltonian (HT)of a triplet state can be represented as follows:

HT=HZFS +

( 5-24)

Hmagr

where

HZFS = DS, + E ( S , - S, Hmag

1,

(5-25)

= gTP8 S.B.

(5-26)

represents the term of the zero-field splitting ( Z F S ) of the triplet state, D and E are the ZFS-parameters, and X, Y, and Z denote the principal axes of the ZFS as shown in Fig. 5-3. Here, the low-lying energy levels of benzophenone and pyrazine and the principal axes of represents that of the magnetic interactions. their ZFSs are illustrated for example. Hmag Here the HF coupling one is not shown for simplicity, but the Zeeman one is only shown. The g value of the triplet state is represented by g T , which is about 2 for usual organic molecules. At a zero magnetic field, the triplet sub-levels (T,,, rn = X, Y, and Z) can be labelled by the molecular frame (X, Y, and Z). In the presence of a magnetic field, the sub-levels (Ti, i = +1, 0, and -1) can be labelled by the laboratory frame and Ti can be represented with T,, as follows: HZFs

(5-27) Here, Ci, is a cofficient which depends on ZFS, B, and the Euler angles relating the molecular and laboratory frames. At a high field limit, all the I C,; values become 1/3 for fast tumbling molecules. On the other hand, when the field is not high enough ans/or when the molecule rotation is not so fast, the I Ci, values do not necessarily become 1/3. In low-temperature solids, the spin-polarization of the lowest triplet state has often been observed at a zero field for many molecules [13]. This phenomena occurs through the selection rule in the intersystem crossing (ISC) from the lowest excited singlet state (S') to T,, as shown in Fig. 5-3. Here, the rate constant of the ISC (kac) can be given by the spin-orbit interaction (Hso) as follows:

12

12

(5-28) Since Hso is totally symmetric, the S-T ISC should occur between S' and T,, having the same symmetry as shown in Fig. 5-3. Thus, one or two triplet sub-levels of the lowest triplet state are populated preferentially in many molecules [ 131. As shown in Fig. 5-3, for example, the top sub-level of the lowest triplet state is selectively populated for benzophenone [14], but the bottom one for pyradine [ 151.

55

'nn* (A2)

3~~*(A1)

I I

Z(A2)

I

Y(B1)

I

WB2)

I

ZFS(in GHz)

j,

3n~*(A2)

j

ZFS(in GHz)

3n~*(B3u)

I

3.663

- 1.0474

Y ( J 3 2 )

-2.0895

X ( B 1 )

/

X(A,)

I

I

I

-6.902

f

Y(B1u)

Z

Fig. 5-3. Main routes of the ISC from the lowest singlet state to the lowest triplet one for (a) benzophenone [14] and pyrazine [15].

(a) Spin-PolarizedTriplet State

radical a

radical b

000000

B=OT (b) Thermalized

Triplet State

B = 0.33 T

No CIDEP

Fig. 5-4. CIDEP of two radicals a and b induced through the TM from a spin-polarized triplet state. The rate constants the reaction from the triplet state and its spin-relaxation of , are denoted by kR and k s ~respectively.

56 For ESR measurements in solution, two effects should be taken into consideration: those of an applied magnetic field and molecular rotation. For a sufficiently high magnetic field and fast enough rotation, all of the 1 Cim, values of Eq. (5-27) become 113. In this case, there is no spin-polarization for the lowest triplet state. Actually, the Zeeman energy of X-band ESR spectrometer ( ~ 9 . 5GHz) is not much larger than the ZFS (several GHz) of the lowest triplet state [13] and the molecular rotation in usual solvents is not so fast even at room temperature. Thus, spin-polarization can usually be preserved for the lowest triplet state at B = 0.33 T, which is the central field of the X-band ESR. The spin-polarization of the lowest triplet state, however, vanishes through spin-relaxation. Its rate constant ( k s ~=: 10’ s-’) can be comparable to the constant &)of the reaction which forms two radicals (a and b) from the lowest triplet state. If the k~ value is larger or comparable to the ~ S Rvalue, the spinpolarization of the triplet state can be transferred to radicals a and b as shown in Fig. 5-4 (a). If the k~ value is much smaller the k S R value, on the other hand, no polarization occurs for two radicals as shown in Fig. 5-4 (b). Fig. 5-4 shows that the spin-polarization in the sublevels above the center of gravity of a spin-polarized triplet state induces emissive CIDEP spectra. On the other hand, the polarization in the sub-levels below the center of gravity gives enhanced absorption. Thus, the CIDEP due to the TM has either a totally emissive (E) or absorptive (A) phase pattern and the relative intensities of its HF lines are not distorted from those observed for radicals in thermal equilibrium. Because this type of CIDEP is induced by selective populations of the triplet sub-levels, such CIDEP is also called that due to “the p-type TM”. Later, another type of CIDEP due to selective decays of the three sub-levels of triplet exciplexes and triplet radical pairs was found. This second type of CIDEP is called that due to “the d-type TM’, which will be explained in Chapter 10. In this chapter, however, the p-type TM is only written as “the TM’, for simplicity.

12

5.4 Examples of CIDEP due to the RPM and the TM Trifunac et al. measured CIDEP spectra for the photoreduction of benzophenone (Ph2CO) in 2-propanol [7] as shown in fig. 5-5. The spectra are composed of the signals due to the two different radicals, the 2-hydroxy-2-propyl (signals 2 in fig. 5-5) and benzophenone ketyl (signal 1 in this figure) radicals. These radicals are produced by the following reactions: (5-29) 3Ph2CO* + (CH&CHOH + Ph2C’OH + (CH3)2C’OH. Their spectral pattern are EIA, or slightly E*/A. The slightly net emissive polarization can be explained by a slight contribution from the TM. The main E/A pattern can be explained by the multiplet effect due to the S-To mixing as given by Eq. (5-21).

r,,,=luJ.

(5-21)

The observed E/A pattern means that I-,,, is negative. Because p is positive for a T-precursor, the J value should be negative. This negative J value has often been observed for neutral radical pairs. From Problem 5-11, the HF intensity ratio of the ClDEP spectrum of the 2hydroxy-2-propyl radical in this reaction is obtained to be - 1:-2& : -5& : 0 : 5& : 2& : 1 , which is -1:-3.46:-8.66:0:8.66:3.46:1. This ratio is quite different from that for the thermal equilibrium, 1:6:15:20:15:6:1, as shown in Problem 2-3.

Problem 5-11. Calculate the HF intensity ratio of the CIDEP spectrum of the 2-hydroxy-2propyl radical for the multiplet effect where Ag = 0 and p is positive.

57

n

.I -.6

pscc

I\

Fig. 5-5. CIDEP spectra of radicals formed from the photoreduction of benzophenone (-0.5 M) in 2-propanol at two time windows after the laser pulse. (Reproduced from Ref. [7] by permission from Elsevier Science B. V.) Miyagawa et al. added diethyl aniline (DEA) to the cyclohexane (CHX) solution of benzophenone and measured its CIDEP spectra. They found that the spectra changed dramatically as shown in fig. 5-6 [16]. The whole spectra observed in CXH showed strong emissive patters as shown in fig. 5-7. The spectra are composed of the two signals due to the benzophenone ketyl radical (central part of (a) and (b)) and the neutral radical of DEA. The latter radical has a structure of hydrogen abstracted at the methylene group. (5-30) 3Ph2CO* + (CH3CHz)zNPh ---t PhZC'OH + CH3C'HNPh(CH3CHz). The strong emissive CIDEP is explained by the TM, because the rate of reaction (5-30) is much more enhanced than that of reaction (5-29). The spectrum of the ketyl radical altered in the different concentrations of DEA as shown in figs. 5-7(c) and (d). This is due to the change of the HFC constant of the hydroxy proton in the different solvent polarities.

58

4

OEA :CHX

Em.

DEA : ANL =1:5

1 : 40

4-

2.0 mT

Em.

T

1:s

En.

Fig. 5-6 (left). CIDEP spectra (delay time: 1.3 ps) of radicals formed from the photoreduction of benzophenone in mixed DENCHX solvents: (a) 1 5 , (b) 1:40. Spectra (c) and (d) are the expanded spectra of the central part of (a) and (b), respectively. (Reproduced from Ref. [16] by permission from The Japanese Chemical Society) Fig. 5-7 (right). CIDEP spectra (delay time: 2.0 ps) of radicals formed from the photoreduction of benzophenone in mixed DENCAN solvents: (a) 1 5 , (b) 1:40. Spectra (c) and (d) are the expanded spectra of the central part of (a) and (b), respectively. (Reproduced from Ref. [16] by permission from The Japanese Chemical Society) When acetonitrile (ACN) was used as a solvent, the CIDEP spectra were found to change as shown in fig. 5-7. Under the high concentration of DEA, the signal from the DEA radical disappeared. This implies that the hydrogen-abstraction reaction proceeds through an electron transfer followed by a proton transfer. (5-31a) 3Ph2CO* + (CH3CHz)zNPh + PhZCO- + (CH3CH2)2NPh+', (5-31b) Ph2CO- + (CH3CH2)2NPh+'-* Ph2C'OH + CH3CSHNPh(CH3CH2).

59 Because the cation radical of DEA transfers the electron to the neutral DEA molecule, the emissive polarization of the cation radical may be quenched during the electron exchange. (5-32) (CH3CH2)2NPh+' + (CH3CH2)2NPh + (CH3CH2)zNPh + (CH3CH2)2NPhf'.

/

I

1Om Q

D I

UI

so'@

Frrpurncy IHHII

Fig. 5-8. CIDEP spectra measured in ethanol at -28 "C for photogenerated DQ- as function of time delay after laser excitation. Absorption signals point up and emission ones down. (Reproduced from Ref. [ 171 by permission from The American Chemical Society) Prisner et al. measured the time dependence of CIDEP spectra in the photooxidation of zinc tetraphenylporphyrin (ZnTPP, 5 ~ 1 0M) - ~with duroquinone (DQ, 5 ~ 1 0M) . ~ in ethanol at -28 "C with FTESR [17]. The observed spectra are shown in fig. 5-8. Here, the signals with sharp HFS are due to DQ- and the underlying broad ones are due to ZnTPP". ZnTTP + hu --t3ZnTTP*, 3ZnTTP* + DQ + ZnTPP+'

+ DQ-.

(5-33a) (5-33b)

60

As shown in Fig. 5-8, the signals are absorptive at early time region ( t < 0.1 ps), but are gradually changed to an E/A pattern (0.1 ps < t < 1 ps). This E/A pattern stays constant from 1 ps to 10 ps , while the overall signal intensity drops gradually. Then the signal pattern changes to an absorptive one at later time region (10 ps < t < 80 ps). The initial absorptive polarization is due to the TM and the following E/A one is due to the multiplet effect due to the S-To mixing. In this reaction, the J value is also proved to be negative. The absorptive signals observed at later time region are due to the usual ESR ones at thermal equilibrium. I

Fig. 5-9. (a) The experimental spectrum recorded 0.5-1.0 ps after radical creation through photolysis of acetone in diethyl phosphite. (b) The calculated stick spectrum employing only the STOpolarization. (c) The calculated stick spectrum employing only the ST.1 polarization. (d) A spectrum calculated by adding spectra (b) and (c) in almost equal proportions, with some added net absorption. (Reproduced from Ref. [18] by permission from Elsevier Science B. V.) Buckley and Mclauchlan measured CIDEP spectrum upon photolysis of acetone in diethyl phosphite as shown in fig. 5-9(a) [18]. The reaction occurs as follows: (CH3)ZCO + hV -+3(CH3)zC0*,

(5-34a)

61

(5-34b) 3(CH3)2C0 * + HPO(OCzH& + (CH&C 'OH + OP'(OC2H&. In Fig. 5-9(a), the central signals are due to (CH&C'OH and the outer E*/A ones due to OP'(OC2H5)2. The HF splitting of the latter radical is much larger than those of the former one. This is due to a much larger HFC of 31P(1=1/2) than those of protons. When they employed only the STo mixing, they obtained a nearly EIA spectrum as shown in Fig. 5-9(b). The distortion of the central signals is due to the difference of the g-values of (CH3)2C'OH and OP'(OCzH&. When they employed only the ST.1 mixing, they obtained the spectrum as shown in fig. 5-9(c). This spectrum agrees with that predicted by the ST.1 mixing for reactions from T-precursors as shown in fig. 5-2(c). Here, the central signals show net emission, the low-field one also shows net emission, but the high-field one vanishes. Spectrum shown in fig. 5-9(d) was calculated by adding spectra (b) and (c) in almost equal proportions, with some added net absorption. Here, the net absorption is due to the TM. It is noteworthy that CIDEP spectra due to the ST.1 mixing have often been observed for phosphorus radicals even in non-viscous solutions.

5.5 CIDEP due to Spin-Correlated Radical Pairs In 1984, the author's group found an anomalous CIDEP spectrum in the photoreduction of benzophenone in as SDS micellar solution as shown in fig. 5-10 [19]. The reaction occurs through the following scheme: 3Ph2CO* + RH Ph2C'OH + R'. (5-35) Here, RH represents a micellar molecule. In Fig. 5-10, the signal positions of Ph2C'OH and R' are shown by the stick diagrams denoted by K' and R', respectively. Although the signal positions are the same in spectra (a) and (b), the phase pattern of (a) is completely different from that of (b) as clearly seen in Fig. 5-10. Spectrum (b) has an EIA pattern, which is very similar to those observed in 2-propanol as shown in Fig. 5-5. This EIA pattern can be explained by the STo mixing. On the other hand, an alternating EIA phase pattern was observed for each of the HF lines of R' in spectrum (a). Such a peculiar phase pattern could not be explained by the ordinary CIDEP theories which had ever been presented before this measurement. The authors' group also found a similar phase-pattern in the photoreduction of xanthone with triethylgermanium hydride in micellar solutions [20]. Because such clear anti-phase doublets were found in the above-mentioned reactions, CIDEP spectra somewhat distored by similar anti-phase doublets which had been observed in 1977 with a pulse radiolysis of some aqueous micellar systems was reconsidered to be due to the same mechanism 1211. On the other hand, they had been interpreted by the ST.1 mixing. The findings of such peculiar phase patterns attracted considerable attention because they could not explained by the ordinary CIDEP theories ( the TM, STo mixing, and ST+]mixing). Buckley et al.[22] and Closs et al.[23] independently interpreted the these peculiar patterns in terms of spin-correlated radical pairs. It is noteworthy that ESR signals due to radical pairs were found to be directly observable in solution even at room temperature. Since then, CIDEP due to this spin-correlated mechanism (SCM) has often been obtained not only in micellar solutions but also in other confined systems such as in viscous solutions 1241, in linked systems [25], and in bacterial photosynthesis systems [26]. ---f

62

Abs.

I

Em.

Ti:

Abs.

t

B

Em.

Fig. 5-10. CIDEP spectra observed for a micellar SDS (8x10.’ m ~ l d m - ~solution ) of benzophenone ( 2 ~ 1 0 .m~~ l d m - at ~ ) a time delay of (a) 1.2 ps and (b) 3.0 ps after laser excitation at room temperature. (Reproduced from Ref. [ 19a] by permission from Elsevier Science B. V.) Let us consider the SCM, taking a radical pair with gl # g2 and AiI gp, = A, I gpB = 0 T. Here, QN of Eq. (3-13a) becomes Q. Q = ( 8 , -g,)PBBI2A. (5-36)

(I

Since the STo mixing occurs through Q in this model, the eigenstates p ) ) and eigenvalues (,?,)of a spin-correlated radical pair become as follows: El = - J + Eo, (5-37a) 11) = IT+,)?

12)

=COSX/S)

13) = -sinxlS)

+sinxl~,),

E2

=

W,

(5-37b)

+ cosxlT,),

E3

= -w,

(5-37c)

Ed

= -J - Eo.

(5-37d)

14) = IT-,)>

Here, EO,x ,and w can be represented as follows:

63 (5-38)

Eo = (8, + g 2 ) p B B / 2 , tan& = Q/3 (Here, 3 = J / A from Eq. (3-17) ), w = Aw

(Here, w 2= Q 2

+ 3'

(5-39) (5-40)

from Eq. (3-18) ).

Problem 5-12. Prove Eqs. (5-37a) - (5-37d).

(a) J > 0 J (w > 0 J)

(b) J < 0 J (w < 0 J)

I2L

'13

I3L

'12

I3L

'12

124

'13

Fig. 5-1 1. The STo mixing of a radical pair with gl # g2 and Ai1gpB= A, / g p B = 0 T and its possible ESR transitions whose signal intensities (Zpq) can be calculated from Eq. (5-43). Here, the radical pair is assumed to be produced from a T-precursor and the initial population of each sub-level is schematically represented by a circle. (Reproduced from Ref. [27] by permission from CRC Press) Fig. 5-1 1 shows the energy diagrams of this model for (a) J > 0 J (w > 0 J) and (b) J < 0 J (w< 0 J). Here, the radical pair is assumed to be produced from a T-precursor and the initial population of each sub-level is schematically represented by a circle. The four possible ESR transitions have energy differences (Epq= Ep - Eq) and transition probabilities (P,,& given by E12= E o - w - J , (5-4 1a) ~ 1 =2 (sinx)*, E 3 4 = Eo- w + J , (5-4 1b) P34 = (cosx)2.

- J,

= (cosx)2, (5-41~) = (sinx)'. (5-4 1d) When the triplet sub-levels are equally populated, the initial populations of 11) and 14) are

E13= Eo+ w E24= Eo+ w

+J,

PI2

P24

3 ( C O S X ) ~respectively. /~, both 1/3, while those of 12) and 13) are ( s i n ~ ) ~ /and initial populations (&), the intensities (I,,) of the four transitions become Ipq

= C(Nq - Np) (Ppq>>

124

=

134

= - 112 = - 1 ~ 3= ~ ( s i n ~ x ) ~=/ 1C(Q/13)~/12. 2

With these (5-42) (5-43)

64

Here, C i s a positive instrumental constant. Problem 5-13. Prove Eq. (5-43). Fig. 5-11 also shows the CIDEP spectra calculated with this model (a) J > 0 J (w > 0 J) and (b) J < 0 J (w < 0 J). As clearly seen in this figure, each spectrum has two anti-phase doublets with an A/E (E/A) pattern for J > 0 J (J < 0 J). Similarly, one can show that it has an E/A (AE)one for a radical pair generated from an S-precursor for J > 0 J (J < 0 J). The splitting of each anti-phase doublet corresponds to 23. This is a novel method to determine the J values of radical pairs and biradicals in solution. Let us consider the conditions for detecting the CIDEP of spin correlated radical pairs. When 191(= ) J /hl) is much larger than lQl, little polarization can be obtained because I Zp41 becomes very small from Eq. (5-43). When IJlhl) is smaller than the line width of each ESR line, the CIDEP signals also vanishes because the anti-phase components of each doublet cancel out with each other. For intermediate 131(=lJ/hl), CIDEP due to the SCM becomes intense. Each of the anti-phase doublets of Spectrum A of Fig. 5-10 has an E/A pattern. Because this reaction occurs from a T-precursor, the J value of this radical pair was proved to be negative. This is the case for most radical pairs consisting of neutral radicals. The splitting of each anti-phase doublet of this spectrum is about 5 G. Thus, its J value can be obtained as follows:

PI(=

J = - gpB(5G/2) = - p B (SG) = - 9 . 2 7 4 0 ~ 1 0 ~ ~ ~ J T ~=~-4.6~10-*~J. ~S~lO~~T

(5-44)

References [I] R. W. Fessenden and R. H. Schuler, J. Chem Phys., 39 (1963) 2147. [2] B. Smaller, J. R. Remko, and E. C. Avery, J. Chem Phys., 48 (1968) 5174. [3] (a) P. W. Atkins, I. C. Buchanan, R. C. Gurd, K. A. McLauchlan, and A. F. Simpson, Chem. Commun, (1970) 513. (b)P. W. Atkins, K. A. McLauchlan, and P. W. Percival, Mol. Phys., 25 (1973) 281. [4] S. K. Wong and J. S. K. Wan, J. Am. Chem. SOC.,94 (1972) 7197. [S] R. W. Fessenden, J. Chem Phys., 58 (1973) 2489 [6] S. S. Kim and S. I. Weissman, J. Magn. Reson., 24 (1976) 167. [7] A. D. Trifunac, M. C. Thurnauer, and J. R. Norris, Chem. Phys. Lett., 57 (1978) 471. [8] (a) A. D. Trifunac and J. R. Norris, Chem. Phys. Lett., 59 (1978) 140. (b) A. D. Trifunac, J. R. Norris, and R. G. Lawler, J. Chem. Phys., 71 (1979) 4380. [9] R. Kaptein, in “Chemically Induced Magnetic Polarization”, L. T. Muus, P. W. Atkins, K. A. McLauchlan, and J. B. Pedresen, Eds., D. Reidel, Dordrecht, Holland, 1977, Chap. 1. [lo] F. J. Adrian, J. Chem. Phys., 54 (1971) 3918. [ 111 L. Monchik and F. J. Adrian, J. Chem. Phys., 63 (1978) 4376. [12] C. D. Buckley and K. A. McLauchlan, Mol. Phys., 54 (1985) 1. [13] M. Kinoshita, N. Iwasaki, and N. Nishi, Appl. Spectr. Rev., 17(1981)1. [14] S. Yamauchi and D. W. Pratt, Mol. Phys., 37(1979)541. 1151 D. M. Burland and J. Schmidt, Mol. Phys., 22(1971)19. [16] K. Miyagawa, Y. J. I’Haya, and H. Murai, Nippon Kagaku Kaishi, (1989) 1357.

65 [17] T. Primer, 0. Dobbert, K. P. Dinse, and H. van Willigen, J. Am. Chem. SOC.,110 (1988) 1622. [18] C. D. Buckley and K. A. McLauchlan, Chem. Phys. Lett., 137 (1987) 86. [19] (a) Y. Sakaguchi, H. Hayashi, H. Murai, and Y. J. I’Haya, Chem. Phys. Lett., llO(1984) 275.(b) H. Murai, Y. Sakaguchi, H. Hayashi, and Y. J. I’Haya, J. Phys. Chem., 90(1986) 113. [20] Y. Sakaguchi, H. Hayashi, H. Murai, Y. J. I’Haya, and K. Mochida, Chem. Phys. Lett., 120(1984) 401. [21] A. D. Trifunac and D. J. Nelson, Chem. Phys. Lett., 46(1977) 346. [22] C. D. Buckley, D. A. Hunter, P. J. Hore, and K. A. Mclauchlan, Chem. Phys. Lett., 136(1987) 307. [23] G. L. Closs, M. E. D. Forbes, J. R. Norris Jr., J. Phys. Chem., 91(1987) 3592. [24] K. Tominaga, S. Yamauchi, and N. Hirota, J. Chem. Phys., 92(1990) 5175. [25] G. L. Closs and M. E. D. Forbes, J. Am. Chem. SOC.,109(1987) 6185. [26] P. J. Hore, in “Advanced EPR-Application in Biology and Biochemistry”, A. J. Hoff, Ed. Elsevier, Amsterdam, The Netherlands, 1989, p.405. [27] H. Hayashi and Y. Sakaguchi, in “Lasers in Polymer Science and Technology: Applications”, J.-P. Fouassier and J. F. Rabek, Eds., CRC Press, Boca Raton, Florida, USA, 1990, Vol. II,Chap. 1.

66 Solutions to the Problems 5-1. From Eq. (2-8), the Zeeman splitting is equal to the energy of an applied microwave (hv)in the resonance condition of ESR. Thus, the n, / np ration can be given as

From this ratio, Peqbecomes as follows:

which is 4.04474~ At room temperature, the @value is 1.38067~1O~~~JK~'(273.15+20)K, 10-21J. On the other hand, the hv value of the X-band ESR is 6.62608x10~34Jsx9x109s~'= 5.963472~10-'~J.Thus, its Peqvalue becomes -0.5~5.963472xlO-~~J /4.04474~10-~'J = -0.000737. Similarly, the Peqvalues for the K- and Q- bands become -0.00205 and -0.00287, respectively. 5-2. From Eqs. (3-4) and (3-5), the S and To states are given as follows:

Is) = (la;P2)-IP1..,))/Jz,

(3-4)

67

= (1/4)

[cfcm ((aiP*)-(Piaz( ( P , a z ) ) + G v c s N IIaiPz)-(Piaz IIaaz))3 = -(1/4)[c;N c,, (2) + cg c, ( 2 )1 = -(1/2)[ cg c,, + c; c, I. 5-4. For simplicity, t is used fort’. From Eqs. (5-8), (3-16a), and (3-16b), the Pl(t) value is represented as follows:

(

Pl(t) = [ c,*(o) coswNt+i-sinw,t

3 ’

@N

)+

iC,*(O)-sinw,t QN

1

uN

The coefficients of Ci (O)C, (0) and C; (O)C, (0) becomes as follows: (coswNt)2f 2 (cosuNt)

= (coswNt)2 - (sinw,t)’

+ 2(-sinwNt)’ QN WN

* Z (coswNt) i-sinuNt (‘N‘

The coefficients of Ci (O)C, (0) and C;(O)C, (0) becomes as follows:

)

68

= +(2QN31wk)sin2u N t .

Thus, P l ( t ) becomes as follows: PI (t) =

{ c, (0)c; (0)f c;(0)C.y( 0 ) ){COS 2WNf -k 2(&

/ WN)' Sin2w ~ t }

+ ( i 3 / w N ){C,(O)C,'(O)-C;(O)C,(O)}sin2wNt + (2QN3/w;)(ICS(O)1*-ICT(0)12}sin2wNt. 5-5. From Problem 5-4, C, (t')C,'(t')- C; (t')C,(t') is given as follows when t is used for t' for simplicity.

[

c , ( t ) c i ( t ) - C f ( t ) C , ( t ) = { c,*(o) cosuNt+i-sinuNt

"

wN

1

+ iC,*(O)-sinw,t QN wN

1

69 At the last part of the above equation, wN(=J3’ + Q t from Eq. (3-18)) can be replaced by

IQA because 3’ is much smaller than Qifrom t=O to t=t ’ 5-6. From a book of mathematics, one can find the following formula:

lomy!&

mc7-I

=

~ ( a ) s i n ( a/z2) ‘

In this case, the integral (4 can be calculated as follows: I = 6 m t - 3 ’ 2sin2 I QN

I t‘dt’=

m z ( 2 I QN I)”’ 2 r ( 3 / 2 ) sin(3z / 4)

Here, r(3/2) is defied as follows: r

Thus,

5-7. From Eq. (5-18), the QNvalue becomes

1 O.Olpu,B QN = - hgpBB = ___ 2fi 2fi

From Eq. (5- 17), the PIvalue becomes

From Eq. (5-20), the sign of

rs= pJAg

r, becomes

= (+)(-)(+) = (-)

: E from Eq. (5-20).

From Eq. (5-18), the P2value becomes as (in this case Ag = 0.01)

& I 1’ =-(=) hgpBBrD 2fi

From Eq. (5-20), the sign of

< 0 : A from Fig. 5-1.

r, becomes (in this case hg = -0.01)

70

r, = pJAg = (+)(-)(-)= (+)

: A from Eq. (5-20)

5-8. From Eq.(5-18), the QNvalue becomes Q -1

- 2A

k q -c:A,M,)= ~ ~ ~1 (AM,” ~ AM,^)=

From Eq. (5-17), the P I value with

-

A (M; - M , ~ ) .

2A

28

=1/2 becomes

A(1/2+1/2)

>O:EfromFig. 5-1.

Similarly, the PI value with M : =-1/2 becomes

+ A(-1 / 2 + 1/ 2)

A(-1/2+ 1/2)2,

lA(-l/2+1/2)l

< O :AfromFig. 5-1

Because the A value is positive, the ESR signal with M,‘=1/2 appears at a lower field than that with M j R=-1/2. This means the CIDEP signals in this case show an E/A phase. From Eq. (5-21), the sign of

r,,becomes

71

r,, =

iuJ

= (-)(+) = (-) : E/A from Eq. (5-21).

Similarly, the Pz(l/2) and P2(-1/2) values can also be shown to be positive and negative, respectively. This means that its CIDEP signals also show an E/A phase. 5-9. From Problem 2-3,the usual ESR spectrum of this case shows HF lines with a relative intensity ratio of 1:3:3: 1. Here, the HF line at the lowest field corresponds to the nuclear state with M " = M P + M ;+M,"= 3 / 2 , that at the second lowest one with M a =1/2, that at the

second highest one with M a= -1/2 and that at the highest one with M n= -3/2. From Eq. (518), the QNvalue becomes A A A A Q~(3/2)=-(3/2),Q~(1/2)= -(1/2),Qp~(-l/2)= -(-1/2),Q~(-3/2)= -(-3/2). 2A 2A 2A 2A

From Eq. (5-17), the PI values with M n=3/2, 1/2, -1/2, and -3/2 become

(T)l~l [2) : 1 1 1' [2) ;1 [$1 : 1 /1 1 [2) :1 (g) -(x)1 ~ 1

= ( ~ ] { l - O ] ( + l ) l T3Az, l

112

= &?r

PI (1/2)= 3 - [l-O}(+l)-

PI

(-U2) = 3

PI(3/2) =

=3

3ATD

I12

>O:EfromFig.5-1.

- -r 2 > 0: E from Fig. 5-1.

- {l-O)(-1)- --3 - -(I"< 0 : A from Fig. 5-1. { l-O}(-l)lTl 3A z,

112

=

&?r

3AZD

112

< 0 : A from Fig. 5-1.

Thus, the CIDEP signals of this case show an EIA phase.

From Eq. (5-21), the sign of

r,,becomes

r,,,= ruJ = (-)(+) = (-) : E/A from Eq. (5-21). The CIDEP spectrum ofthis case shows HF lines with a relative intensity ratio of -P1(3/2) : P I (1/2) : -PI(-1/2) : -PI (-3/2). Thus, the ratio becomes -&:-3:+3:+&. It is noteworthy that this ratio of CIDEP (-1: -&:+& :+1) is quite different from that of the usual ESR (1:3:3:1). 5-10. In order understand how CIDEP appears through the S-T,, mixing, one should consider the following simple example: (1) radical a in a radical pair has one large HF interaction (A > 0 J) with P or H (Zi = 1/2). (2) radical b in the pair has no HF interaction. (3) the radical pair

72

is produced from a T-precursor. In this case the S-T+1mixing occurs from the T,i level of this radical pair (a( l)a(2)) to its corresponding S level ([a+( 1)p(2)- pi( l)a(2)]/21’2). This mixing decreases the populations of ~ ( 1 and ) a(2) and increases those of a+(l),p(2), and P,(l). On the other hand, the S-T+I mixing does not occur from the other T+I level of this l)a(2)) to its corresponding S level because the selection rule of AM, = +1 radical pair (a+( cannot be fulfilled. The above-mentioned results on the S-T,, mixing shows that this model system gives enhanced absorption signals for both of radicals a and b as shown Fig. 5-2(c). On the other hand, Eq. (5-23) gives the following result:

5-11. From Problem 2-3, the usual ESR spectrum of this case shows HF lines with a relative intensity ratio of 1:6:15:20:15:6:1. If the HFC constant (A) of the six equivalent methyl-protons is positive, the HF line at the lowest field corresponds to the nuclear state with M u = Z p M p = 3, that at the second lowest one withM“ = IpMMp = 2, that at the third lowest one with M a = XpMMp = 1, that at the central one with M y = C p M p = 0, that at the third highest one with M a = XpMMp = -1, that at the second highest one with M a = X p M P = -2, and that at the highest one withM“ = x y M p = -3.

From Eq. (5-18), the QN value

becomes 3A 2A &(+3) = f -, QN(&?)= f -, Q ~ ( k 1 = ) 2A 2A

A *, QN(O)= 0. 2A

From Eq. (5-17), the Pi values with M a = +3, &2,A , and 0 become from a T-precursor and a negative J value as follows:

1/2

1/2

73 Thus, the ratio of the PI values of this case becomes as follows from the low field to high field: +&: +6& : +15: 0: -15: -6& : -&. Because a positive P value means an emissive signal, the ratio of the CIDEP signals as follows: -1: -3.46: -8.66: 0: 8.66: 3.46: 1. Here, a negative (positive) intensity means an emissive (absorptive) signal. If the HFC constant (A) of the six equivalent methyl-protons is negative, the same procedure as described above gives the same result for the ratio of the CIDEP signals. Here, the HF line at the lowest field corresponds to the nuclear state with M a = E p M : = -3.

5-12.

Because there is no mixing in

(=I1)) and IT-,)(= 14)), their energies ( E l and

E4) can be given by Eq. (3-1 lb). El = E ( T + , ) = - J + ( g , + g , ) , ~ ~ B = / 2- J + E o ,

(5-37a)

E ~ = E ( T . I=) - J - (gl + g , ) , ~ ~ B =/ 2- J - Eo.

(5-37d)

On the other hand, there this mixing between IS) and ITo). Their energies can also be give by Eq. (3-llb). E(S) = +J = AS, E(To) = -J = - AS . The energy of the S TOmixing is represented by AQ from Eq. (3-13a). Thus, the energies of the mixed states ( E = E2 and E3) can be obtained from the following equation:

(J-E)(-J-E) - ( AQ )' = 0,

E2 = J2 + ( AQ )'.

Thus, the E values are obtained as E = k A , / m = f Am(= kw),where wis represented by Eq. (3-18). If the eigenfuctions for E2 and E3 are written by 12) (= cosxl S) -sinxlS)

+ cosxlT')),

+ sinxlT,) ) and (3)(=

the x value is given by the following relation:

( 2 ( H R , ( 3 )= -(cosx)(sinx)E(S) + (cosx)(sinr) E(T0) + [(cosx) - (sinx) ' 3 AQ = 0.

Using sin(A & B)=sinAcosB f cosAsinB and cos(A f B)=cosAcosBT sinAcosB, this equation is changed as -(sin2x)J + (cos2x) AQ = 0, (sin2x)J = (cos2x) AQ ,t a n h = AQ / J = Q/3 .

5-13. From Eq. (5-42), the Ip4value is given as follows: Ipq = C(N, - Np) (Ppq). Let us obtain the 124 value. 124 = C(N4 - 7%) (&) = C(U3 - (sinx) '13) (sinx) = C( 1 - (sinx) ) (sinx) /3 = C(cosx) (sinx) /3 = C(sin2x) /12 = C(Q/w)'/12.

Here, tan& = AQ N = Q/3 and Let us obtain the 113 value.

d = Q 2+ 3

are used. 134 is similarly obtained.

74

75

6. Magnetic Field Effects upon Chemical Reactions due to the Radical Pair Mechanism (RF'M) 6.1 Historical Introduction In 1976, Atkins [ 11 wrote a short review entitled "Magnetic field effects" and described at its beginning "The study of the effect of magnetic fields on chemical reactions has long been a romping ground for charlatans". Until then, so many papers had reported having found magnetic field effects (MFEs) on chemical and biochemical reactions. Almost all of such studies, however, lacked reproducibility andor theoretical interpretation. Thus, most scientists at that time believed that ordinary magnetic fields could not exert appreciable influence on chemical and biochemical reactions. Here, ordinary magnetic fields mean those less than 2 T, which can be generated by usual permanent and electric magnets. This view seems to be reasonable if one compares magnetic energies of molecules with those related to chemical reactions: (1) The Zeeman splitting of an electron spin at 2 T is about 2 cm-I. (2) The Zeeman splittings of nuclear spins are much smaller than that of an electron spin, that of proton at 2 T being about 0.002 cm-I. (3) The thermal energy at room temperature is about 200 cm-l. (4) The activation energies for chemical reactions are usually much larger than the thermal energy. Thus, MFEs on chemical and biochemical reactions seemed to be impossible from a thermodynamic consideration, unless extraordinarily large fields were used. The only exception to the above discussion was the MFE on the predissosiation of 12. In 1913, Steubing discovered that the visible emission of iodine vapor was quenched by magnetic fields below 2.2 T [2]. This MFE was interpreted in terms of a magnetically state [3]. Because the iodine emission was induced predissociation of the excited B3II(OU+) completely quenched by magnetic fields above 7 T, Falcaner and Wasserman found a 30 % increase in the vapor-phase iodine-photosensitized isomerization of cis- to trans-butene-2 A similar MFE was also observed for the chemiluminecence with an 8.5 T field [4]. intensity of the BZC+- A*Z*(O,O) band of CN in a flame [5]. This MFE was explained by the level-crossing of the short-lived BZC+(v= 0) levels with the long-loved A211(v = 10) levels through the rotational perturbation, but was not applied to MFEs on chemical reactions. In 1933, Karkas and Sachsse found that the conversion of para-hydrogen to orthohydrogen was induced by paramagnetic molecules and ions in aqueous solution 161. This phenomena was explained in terms of the inhomogeneity of the magnetic field during a collision of hydrogen with paramagnetic species 173. Because this inhomogeneity is of the size a hydrogen molecule, it is impossible for us to generate such an inhomogeneous field with any magnet at all. In this book, we are concerned with the control of chemical and biological reactions by external magnetic fields. In the case of the conversion of parahydrogen to ortho-hydrogen, the conversion is induced by the internal magnetic interaction with paramagnetic species but not by external magnetic fields. In 1968, however, Misono and Selwood found that the parahydrogen conversion in the gas phase at room temperature by paramagnetic catalysts was influenced by homogeneous external magnetic fields below 0.8 T [8]. It was noteworthy that the catalysed parahydrogen conversion rate over certain rare earths was found to be appreciably decreased by such very low fields as the Earth's magnetic field [8b]. At first, they had no theoretical interpretation for these interesting MFEs. In 1977, Selwood presented a systematic interpretation for these MFEs, but there has been no experimental study finding MFEs on the primary processes of his interpretation. In 1968, Fujiwara et al. [9] found the MFEs on polarography in the presence of magnetic fields below

76 1.8 T. Although this result was applied for the magnetic control of chemical reactions [lo], these MFEs concerning electrolysis ions in solution are beyond the scope of this book. In 1967, Johnson et al. found that the delayed fluorescence of anthracene crystal could be modulated by external magnetic fields below 2 T 1111. The magnitude of the MFEs was dependent on both the field strength and the orientation of its single crystal. These MFEs were successfully interpreted in terms of triplet exciton spin Hamiltonian [12]. The delayed fluorescence is induced from the lowest excited singlet state ('Sl) by the annihilation of two lowest triplet excited states (3T~). 3T1 + 3TI 1.335 3 ( TI ,3T1), (6-1) 1.3.5 3 ( TI , 3T1) +IS1 + 'SO(with a rate constant of ks), (6-2) 6-31 IS1 IS0 + h v ,(Delayed Fluoresence). ---f

Here, 'SOdenotes the ground state and the ks value was shown to be changed by magnetic fields. Similar MFEs in various crystals were also found for the delayed fluorescence due to the triplet-doublet quenching and the hyperfine interactions. In 1969, Faulkner and Bard found MFEs on the electrogenegated chemiluminescence of some anthracenes in solutions [13]. These MFEs could also be explained by the same T-T annihilation mechanism. Although these MFEs were strongly expected to be extended to the control of chemical reactions by external magnetic fields, this idea was not realized soon. In 1972, Gupta and Hammond [14] reported that with the isomeric stilbenes and pipexylenes as substrates (R) and a number of ketones as sensitizers (S) a magnetic field of 0.8-1.0 T changed both initial quantum yield and the composition of the photostationary states, the fomer being reported to be reduced by 2-17 % with the field. They suggested that the field influenced the relative rates of the non-radiative decay paths of triplet exciplexes (3(S,R)*). If the decay of triplet exciplexes is enhanced by the field, the energy transfer from 3(S,R)* to 3R* should be reduced by the field. Because the solvents were not written in this paper, the author of this book (H. Hayashi) asked the authors of this paper what solvents they had used, obtaining their answer that they had used benzene or isooctane for solvents. In such non-polar solvents, exciplexes should be formed but ion radical pairs should not. Immediately after this paper, Atkins [I51 presented a possible theoretical interpretation to these interesting results. The triplet exciplexes ( 3 w ( 3 ( D , A ) * ) can ) be represented by a and the charge-transfer (CT) state combination of the locally excited state (3v/(3D*,1A)) ( 3&D','A-)).

' w ( ~ ( D , A ) *=) 'w('D*,'A) + 2 'y/('D+,'A-).

(6-4)

Atkins proposed that the conversion from the triplet exciplexes to singlet ones might be enhanced by a magnetic field through the CT triplet and singlet states. This interpretation seems not to be appropriate because the energy difference between the triplet and singlet exciplexes should be much larger than the magnetic interaction energies of the cation and anion radicals.

In 1973, Sagdeev et al. [16] reported that the NMR intensities of several products after thermal reactions of substituted benzyl chlorides with n-butyllithium in solution were appreciably changed by magnetic fields less than 2.5 T. They explained the MFEs of these thermal reactions by the HFC mechanism of the radical pair mechanism. This interpretation was more plausible than the above-mentioned one because CIDNP had been observed in this type of reactions [17]. In 1974, Brocklehurst et al. [18] observed MFEs on the intesities of fluorescence and transient absorption in pulse radiolaysis of fluorene in squalane at room temperature. They found that the fluorescence intensity and the singlet yield observed 100 ns

77

after the pulse were increased by magnetic fields below 0.7 T. They explained these MFEs by the HFC mechanism of the radical pair mechanism because the excited states are produced by ion recombination in non-polar solvents. Although ion radicals were only produced in pulse radiolysis, it was noteworthy that the MFEs on primary reaction processes could be detected for the first time by this technique. After reading Ref. [15], Prof. S. Nagakura felt interest in MFEs on chemical reactions because he had long been studying the electronic structures of CT complexes and their reactions. He proposed the author (H. Hayashi) and Dr. Y. Tanimoto to start research in MFEs on chemical reactions through CT complexes and exciplexes in solution. For a few years, we tried many such reactions with magnetic fields below 1.26 T, but we could not find any MFE on those reactions. In 1975, Sakuragi et al. examined the MFE on the singlet and triplet photosensitized decomposition of dibenzolyperoxide, because CIDNP had already been observed not only in this reaction but also in many other photochemical reactions [ 191. They, however, observed no appreciable effect in the presence of magnetic fields of 1 - 1.4 T [20]. Thus, we re-examined this reaction in the presence of much higher fields of up to 4.3 T with a small super-conducting magnet and succeeded in observing external MFEs on the yields of cage and escape products in this reaction. Our results were published in 1976 [21]. Here, we could quantitatively interpret the observed MFEs by the Ag mechanism of the radical pair mechanism. In this study, we could undoubtedly establish the MFEs on photochemical reactions in solution from both experimental and theoretical aspects. In 1976, four other groups also reported MFEs on photochemical reactions in solution. With a nanosecond laser-photolysis method, Michel-Beyerle et al. [22] measured the transient absorptions of the pyrene triplet and pyrene anion produced in the quenching of the singlet excited state of pyren with N,N-diethylaniline in methanol at room temperature in the presence of magnetic fields below 0.02 T. They observed magnetically induced decreases (increases) in the triplet (anion) yield and explained these MFEs by the HFC mechanism of the radical pair mechanism. Schulten et al. [23] observed similar MFEs in the reaction of the singlet excited state of pyren with 3,5-dimeyhoxy-dimethylanilinein the presence of magnetic fields below 0.05 T. Buchachenko et al. [24] found that the 13C enrichment was decreased by magnetic fields below 0.053 T in the photodecomposition of dibenzyl ketone at room temperature in benzene and hexane. These MFEs seemed plausible from both experimental and theoretical viewpoints and have been confirmed by other studies. Hata reported the MFEs on the photochemical isomerization of isoquinoline N-oxide to lactan in ethanol in the presence of magnetic fields below 1.3 T [25a]. These MFEs and those obtained in other alcohol were explained by the level-crossing mechanism of the radical pair mechanism [25b]. It is noteworthy that the MFEs reported in Refs. [14], [16], and [25] have not yet been confirmed by any other group. Because photochemical reactions in solution and confined systems are various, many photochemical and photobiological reactions including photosynthesis ones have been shown to be affected by magnetic fields since 1976.

6.2 Classification of Magnetic Field Effects due to the RPM In Chapter 3, the energies of the singlet and triplet radical pairs in the absence and presence of an external magnetic field were obtained as follows:

E(S)=(S,X~IH,,+H,~~S.X~) =J,

(3-lla)

78

Here, n = +1,0, and -1; g = ( g , + g , ) / 2 . The dependence of the radical pair energies on the distance (r) between two component radicals is shown in Fig. 3-2. In Chapter 3, the offdiagonal matrix elements between the singlet and triplet radical pairs were also obtained as follows: (3-13a)

Here, A g = g g n - g b ,M;(for I X ~ ' ) = I M , ' , M ~ =' )M ) , T l ( f o r IxN)=IM,,Mk)), and M i = Mk . From Eqs (3-11) and (3-13), the S-T conversion of radical pars was found to be influenced by the following terms: (a) the Zeeman term which is characterized by Agp,B, (b) the hyperfine coupling (HFC) terms which are characterized by A, and Ak , and (c) the exchange term which is characterized by J. Thus, the MFEs on chemical reactions through radical pairs can be classified by the following typical mechanisms: (a) The Ag mechanism (AgM):

This mechanism is applicable when J = 0 J, Ag # 0, and A, = A, = 0 J. (b)The HFC mechanism (HFCM): This mechanism is applicable when J = 0 J, Ag = 0, A, (c) The level-crossing mechanism (LCM): This mechanism is applicable when J # 0 J, A,

#

#

0 J, and/or A,

0 J, and/or A,

#

#

0 J.

0 J.

Fig. 6-1 shows how the S-T conversion rate is influenced by an external magnetic field in each mechanism as follows: (a) In the case of the AgM, there is no S-T conversion in the absence of a magnetic field. In the presence of the field, however, the S-To conversion is induced through the A g p g term of Eq. (3-13a). This means that the S-T conversion rate increases with increasing AgB. (b) In the case of the HFCM, the S-T conversion at a zero field is induced between the singlet state and all the three triplet sub-levels through the HFC terms of Eqs. (3-13a) and (3-13b). In the presence of the field, the S-TOconversion is only induced through the HFC term of Eq. (3-13a) and no S-T,] one is induced because of the Zeeman splitting of these levels. This means that the S-T conversion rate decreases with increasing B. It is noteworthy that this decrease is saturated at higher fields than the degree of the HFC interaction ( = A / g p , ) . It is also noteworthy that the S-T conversion rate depends on the magnitude of the HFC interaction. This is the origin of the magnetic isotope effect (ME). (c) In the case of the LCM, there is no S-T conversion at a zero field because the singlet and triplet states are separated by the exchange energy (2M) as shown in Fig. 6l(c). In the presence of a magnetic field, the level-crossing occurs between the singlet state and T.1 (or T,]) at the level-crossing field (BE).

BLC= 2 W g p ~ . (6-5) Thus, the S-T conversion rate increases suddenly at the level-crossing field (B = BLC)through the HFC term of Eq. (3-13b).

79

(a) The Ag mechanism (Am): J = O J, Agf 0, and A, = A ,

S

Q

=

0 J.

T+I

T

(b) The hyperfine coupling mechanism (HFCM): J = 0 J, Ag = 0, A, # 0 J, andor A, + 0 J.

Tc I

T

-s HFC

TO

HFC

T-I

(c) The level-crossing mechanism (LCM): J # 0 J, A, + 0 J, andor A, + 0 J.

T+I

I -

........................................

TO

HFC

B=OT

B>OT

Fig. 6-1. Magnetic field effects on the singlet-triplet (S-T) conversion of radical pairs: (a) The Ag mechanism (AgM); (b) The hyperhe coupling mechanism (HFCM); (c) The levelcrossing mechanism (LCM). (Reproduced fiom Ref. [34] by permission fiom The Chinese Chemical Society)

80

6.3 Magnetic Field Effects on Product Yields due to the RPM In Section 6.2, we saw how the S-T conversion rate of radical pairs is influenced by an external magnetic field for the AgM, HFCM, and LCM. These MFEs on the S-T conversion also affect the yield of cage and escape products (Ye and YE), which are formed through radical pairs as follows: (a)TheAgM(J=OJ,Ag#O,and A, = A , = OJ.) First, let us consider the case when a radical pair is produced from an S-precursor. In this case, a singlet radical pair is initially generated. From Fig. 3-1, we can obtain the following MFEs on the yield of reaction products: Because the S-T conversion rate increases with increasing B due to the AgM as shown in Fig. 6-l(a), the yield of the cage product from the singlet pair (YcS)decreases with increasing B from Eqs. (3-37) and (3-38).

Ycs = P N s =A(p-mn”ZQiw;3/2) = A ( p - X , ) .

(3-37)

Here, XN is defined as

X , = mn1/2Qiw;3/2.

(3-38)

Because MFEs are induced in the region where Q i >> 3 ’,the B-dependence of YcS is represented by = A ( p - mlzAgp,B/ 2A11/2). Ycs(B) = A(p - mn’/zlQ, (B)1112)

(6-6)

Eq. (6-6) shows that the yield of the cage product decreases with increasing B. It is noteworthy that the magnetically induced decrease in Eq. (6-6) is proportional to B”’. On the other hand, the yield (YE’@)) of the products from the triplet pair such as escape radicals, escape products, and the triplet state, if any, increases with increasing B. yEs(B) = 1 - yCs(B) = (1 - &)+ /2mldgpflB/2h11/2.

(6-7)

Thus, the MFEs on product yields in this case can be illustrated as shown in Fig. 6-2. Second, let us consider the case when a radical pair is produced from a T-precursor. In this case, a triplet radical pair is initially generated. From Fig. 3-1, we can obtain the following MFEs on the yield of reaction products: Because the T-S conversion rate increases with increasing B due to the AgM as shown in Fig. 6-l(a), the yield of the cage product from the singlet pair (YcT)also increases with increasing B from Eqs. (3-42) and (3-38).

Eq. (6-8) shows that the yield of the cage product increases with increasing B and that the magnetically induced increase in Eq. (6-8) is proportional to B’”. On the other hand, the yield ( Y E ~ ( Bof ) ) the products from the triplet pair such as escape radicals, escape products, and the triplet state, if any, decareses with increasing B. YET@) = 1 - YcT(B) = 1 - -rnlzAgp, B / 2A1“‘. 3(1- P)

(6-9)

Thus, the MFEs on product yields due to the AgM can be illustrated as shown in Fig. 6-2(a).

81

( c ) The AgM + T h e HFCM

+

I

Y(B) - Y(0T) * - - - - - -

(d) The LCM Y(B)- Y(0T)

d

0 -

0

/

I

I

'X

I

0

-

TB

Fig. 6-2. Theoretical prediction of the magnetic field dependence on the product yield (Y(B)) in the reactions through radical pairs: (a) the Ag mechanism (AgM), (b) the HFC mechanism (HFCM), (c) the mixed effect of the AgM and the HFCM, and (d) the LCM. The full curves indicate the magnetic field dependence of cage (escape) products produced from S-(T-) precursors. The broken curves indicate the dependence of escape (cage) products produced from S-(T-) precursors. In this figure, cage products mean those produced from singlet radical pairs. The curves for triplet states are omitted for simplicity, but they show similar dependence as those of escape products. (Reproduced from Ref. [34] by permission from The Chinese Chemical Society)

82

(b)The HFCM ( J = 0 J, Ag = 0,Ai# 0 J, and/or A,

#

0 J).

First, let us consider the case when a radical pair is produced from an S-precursor. In this case, a singlet radical pair is initially generated. From Fig. 3-1, we can obtain the following MFEs on the yield of reaction products: Because the S-T conversion rate decreases with increasing B due to the HFCM as shown in Fig. 6-l(b), the yield of the cage product from the singlet pair (Ycs(B))increases with increasing 8 and the yield (YES@))of the products from the triplet pair such as escape radicals, escape products, and the triplet state, if any, decreases with increasing B. According to Weller et al. [26],the magnetically induced changes due to ) are much higher than the half-field value the HFCM should be saturated at fields ( B H which (Bl12) defined as

(6-10) Here, the B I Dvalue can be expressed by the sum of the HFC in each radical (B,). (6-1 1)

B , / , =2(B: + B i ) / ( B 1+ B , ) . where

(c,

(6-12) B, = A, I, (I, + 1))”’. Since the Bln value are less than 10 mT for most organic radical pairs, the magnetically induced changes due to the HFCM are usually saturated below 0.1 T. It is noteworthy that no analytical prediction of the magnetic field dependence of YcS(B)and YES@) is possible in the case of the HFCM. On the other hand, the quantitative YcS(B)and YE’@) values can only be obtained by numerical calculations with the stochastic Liouville equation [ 2 7 ] . Second, let us consider the case when a radical pair is produced from a T-precursor. In this case, a triplet radical pair is initially generated. From Fig. 3-1, we can obtain the following MFEs on the yield of reaction products: Because the T-S conversion rate decreases with increasing B due to the HFCM as shown in Fig. 6-l(b), the yield of the cage product from the singlet pair (YcS(B))decreases with increasing B and the yield (Y,S(B)) of the products from the triplet pair such as escape radicals, escape products, and the triplet state, if any, increases with increasing B. The magnetically induced changes due to the HFCM should also be saturated at the high field region (BH>>Bllz). Thus, the MFEs on product yields due to the HFCM can be illustrated as shown in Fig. 6-2(b). It is noteworthy that the direction of the MFEs due to the HFCM is opposite to that due to the AgM as shown in Figs. 6-2(a) and 6-2(b). (c)The mixed effect of the AgM and the HFCM ( J = 0 J, Ag # 0, A,

0 J). This is a general case where J = 0 J, but the Ag, A,, and/or Ak values are not zero. In this case, the MFEs due to the AgM and the HFCM appear simultaneously. As shown in Figs. 62(a) and (b), the MFEs due to the AgM are predominant at high fields, but those due to the HFCM at low fields. Because the direction of the MFEs due to the HFCM is opposite to that due to the AgM, the Ycs(B)and YES@) values increase (or decrease) with increasing B at low fields due to the HFCM and they decrease (or increase) with increasing B at high fields due to the AgM. Thus, the MFEs on product yields due to the mixed effect of the AgM and the HFCM can be illustrated as shown in Fig. 6-2(c). (d)The LCM ( J # 0 J, A, # 0 J, and/or A, # 0 J). In this case, the S-T conversion suddenly occurs at the level-crossing field (BLc)as shown in Fig. 6-l(c). When a reaction occur from an S-precursor, the S-T conversion rate is increased by a magnetic field at BLC. Thus, the Y 2 ( B ) value is decreased by BLC and the When a reaction occur from a T-precursor, the T-S YES@) value is increased by it. #

0 J, and/or A,

#

83 conversion rate is increased by BE. Thus, the Ycs(B) value is increased by BLC and the YEs@)value is decreased by it. From these results, the MFEs on product yields due to the LCM can be illustrated as shown in Fig. 6-2(d). It is noteworthy that The MFEs due to the LCM appear suddenly at the level-crossing field.

In the following part of this chapter, typical MFEs due to the RPM observed in thermal, radiation, and photochemical reactions will be explained. 6.4 Magnetic Field Effects on Thermal Reactions In 1973, Sagdeev et al. [16] reported MFEs on thermal reactions of substituted benzyl chlorides (ACI) with n-butyl lithium (BLi) in hot hexane and cyclohexane. Although there were many serious misprints in this paper, this was the first report for MFEs on chemical reactions through radical pairs. The reactions occur through the following scheme: (6-13a) LiCI, ACl + BLi -+ '[A' 's] iCage product: A-B, (6-13b) Escape products: A-A, B-B, and A-B.

(6-13~)

They measured the ratio (R) of the NMR intensities (I) of the unsymmetrical (AB) and symmetrical (AA) products. R (B) = I ( A B ) / I ( A A ) . (6-14) Their results are summarized in Table 6-1. They did not describe the solvent and temperature of Runs 1 and 2 in their paper [16], but they described afterward in their book [28] that the reactions had took place in a 1.5 T magnetic field in boiling hexane (68.7-C). The magnetic field dependence of the R (B) values was also illustrated in Fig. 5.1 of their book [28]. Unfortunately, the author could not obtain the permission to reproduce this famous figure in the present book. This figure shows that the R (B) value observed for the reaction of pentafluorobenzyl choride with n-butyl lithium in boiling hexane increases with increasing B from the earth's magnetic field (0.05 mT) to 0.1 T and that the value attains a saturated value with increasing B from 0.1 T to 1.65 T. Thus, we can see from Table 6-1 that there are serious misprints in the R(0.05 mT) and R(1.5 T) values of Run 1 in their paper 1161. As shown in reaction (613), the unsymmetrical product is derived mainly from the cage, whereas the symmetric one is an escape product. Fig. 5.1 of Ref.[28] shows that the Ycs(B)/ YEs@) ratio in each of the reactions of pentafluorebenzyl chloride and p-fluorobenzyl chloride with n-butyl lithium increases with increasing B from 0.05 mT to 0.1 T and that the ratio attains a saturated value with increasing B from 0.1 T to 1.65 T. This magnetic field dependence can be explained by the HFCM as shown in Fig. 6-2(b). Fig. 5.1 of Ref.[28] also shows that the ratio in the reaction of decafluorodiphenyl chloromethane with n-butyl lithium increases with increasing B from 0.05 mT to 0.1 T, but that the ratio decreases with increasing B from 0.1 T to 1.65 T. This magnetic field dependence can be explained by the mixed effect the AgM and the HFCM as shown in Fig. 6-2(c) because the Ag value is not zero in this radical pair. Sagdeev et al. also pointed out that larger MFEs could be obtained in more viscous solvents from Runs 3-5 of Table 6-1, because the viscosity of cyclohexane is three times higher than that of hexane. This is due to the fact that the radical pair lifetime is increased by the increase of solvent viscosity. From measurements of CIDNP in similar thermal reactions, it has also been believed that the main reason of the appearance of considerable MFEs on this kind of reactions is the interaction of the buthyl radical (B') associated with the hexamer of n-butyl lithium (BLi) with the benzyl radical (A'). Although

84

n-butyl lithium is considered to be hexameric in aliphatic hydrocrobon solvents at the concentrations usually employed for chemical reactions [19], it is unclear whether such hexamer formation is proved in boiling hexane. Table 6-1. Measurements of the R (B) (= Z(AB)/Z(AA)) values in the reactions of substituted benzyl chlorides (ACI) with n-butyl lithium (BLi) [16]. The author feel that this paper seems to have many serious misprints and unclear expressions.

Because no other group has reconfirmed the MFEs reported by this Russian group for several thermal reactions, the author’s group recently started to re-examine their MFEs. Although the Russian group only measured the ratio (R= Z(AE%)/Z(AA)) of the NMR intensities (Z) of the unsymmetrical (AB) and symmetrical (AA) products for each reaction, we tried to measure the yields (Ym and Y u ) of AB and AA separately with a gas chromatograph [29]. In order to make experimental condition as clear as possible, we carried out the following two procedures: (1) We prepared a degassed hexane solution of pentafluorebenzyl chloride (0.3 rnol dm”) and a degassed one of n-butyl lithium (1.52 rnol dm-3). Then we quickly added the latter solution (100 pJ) to the former one (2.0 ml). In this case, reaction occurred at 23 f 3 “C. (2) We prepared a degassed hexane solution of pentafluorebenzyl chloride (2.86 rnol dm-3) and a degassed one of n-butyl lithium (1.52 mol dm-3). Then we quickly mixed the latter solution (1.3 ml) with the former one (0.7 ml). In this case, reaction occurred at temperatures from 23 to 68.7 “C. In both cases, we have not yet obtained any appreciable MFE in the YABand Y , values with a 1.5 T magnetic field. On the other hand, the Russian group canied out the following procedure: (3) They added ACI to a frozen solution of n-butyl lithium; the latter concentration is unclear, but it may be 2 or more mol dm”. If the MFEs reported by the Russian group were true, they may be due to heterogeneous interfaces with frozen solution and/or deposited salt, where the radical pair lifetime becomes much longer than that in homogeneous solution.

85

6.5 Magnetic Field Effects on Radiation Reactions In 1974 Brocklehurst et al. [I81 reported MFEs on a reaction of a pair of cation and anion ion radicals generated from irradiation of an aromatic molecule (M) in a hydrocarbon solvent (S) by fast electrons. The process of ion-radical generation and recombination follows the scheme: s + electron-beam '[s+.+ e-I, (6-15a) -+

S+'+ M S +M+', (6-15b) e - + M + M", (6-1%) '[M+'+ M"] ++ 3[M+*+ M-'I, (6- 15d) 'M* +M, (6- 15e) '[M+.+ M-'] (6-150 3[M+.+ M-'1 -+ 3M* +M. Here, 'M* and 3M* are the excited singlet or triplet states of M, respectively. As shown in reaction (6-15a), a singlet radical pair is initially produced in this reaction. The ionradicals produced in very viscous nonpolar solvents cannot leave for the bulk but recombine with the probability nearly equal to unity because the initial intercharge distance (5 to 15 nm) is less than the Onsager radius for nonpolar solvents (30 nm). The S-T conversion (reaction (6-15d)) is expected to occur though the HFCM. Thus, the S-T conversion rate should be reduced by magnetic fields. Brocklehurst et al. employed squalane as S and fluorene as M. They measured the time profile of fluorene fluorescence during and after pulse radiolysis and found that the fluorescence intensity was increased by a 0.3 T magnetic field as shown in Fig. 6-3(a). They also measured the time dependence of the magnetic field enhancement of the fluorescence intensity as shown in Fig. 63(b). This figure shows that the MFE is very small or zero during the pulse, but that it rapidly reaches an apparent plateau (40 % increase) after about 1 0 0 ns. This is due to the fact that the MFE grows in several tens ns, which is the order of the S-T conversion due to the HFCM as shown in Chapter 3. Brocklehurst et al. found that the MFE on the fluorescence intensity at 200 ns after the pulse increase with increasing B from 0 T to 0.1 T, but that the MFE shows a saturated value (40 % increase) with increasing B from 0.1 T to 0.5 T as shown in Fig. 6-3(c). Such a MFE on the singlet yield can be explained by the HFC from an S-precursor as shown in Fig. 6-2(b). According to the HFCM, the triplet yield should be increased by the fields of 0.1 - 0.5 T, but such a MFE on the triplet yield was not clear in this reaction. Later, such MFEs on the triplet yield were found in photochemical reactions as shown in section 6.6. Similar results were also found in cyclohexane, but the observed MFEs were less than those observed in squalane. In benzene, there was no detectable MFE on the fluorescence intensity. This solvent effect can be explained by the effect on the lifetime of the generated ion-radical pairs. This means that the more viscous the solvent is the longer the radical pair lifetime becomes. ---f

-+

86

(4

,

0 0

0.1

0.2

0.3

0.4

0.5

Magnetic Field. Tesla

60

.

C

0

I

40

w C .-

IaJn

m

y

20

U

-

. 0

0

I

I

I

I

200

400

600

800

1000

Time. ns

Fig. 6-3. Magnetic field effects observed in the radiation reaction of a squalane (S) solution of fluorene (M) for pulse radiolysis with a 4-MeV electron accelerator. The reaction temperature is not described in the present papers, but may be room temperature. (a) Time profile of fluorine fluorescence during and after pulse radiolysis of a squalane solution: (1) at the minimum field less than 0.05 mT, where the residual field of an electromagnet is cancelled by passing a small reverse current through the magnet's coils; (2) at 0.3 T. (b) The time dependence of the magnetic field enhancement of the fluorescence intensity: (A)15-ns pulse; (w) 50-ns pulse. (c) The MFE on the increase in fluorescence intensity at 200 ns after the pulse. (Reproduced from Ref. [18b] by permission from The American Chemical Society)

87

la1

I

Fig. 6-4. (a) Experimental values of R = Z(0.16 T)/Z(O T) versus time after the peak of a scintillation pulse : ( 0 ) terphenyl-hl4 in delaline; ( 0 ) terphenyl-44 in delaline; (x) terphenyl-h14 in benzene. (b) Theoretical values of R of the amounts of signlet character in the ionpair wavefbnction [30]: (-) terphenylh14; (---)terphenyl-&. (Reproduced from Ref. [30] by permission from Elsevier Science B.

v.)

tlnr

AI I

(PbPPC

0. L

0.3 0.2 0.1

0

& ' :@ &ik@

Fig. dependence 6-5. Time of

88 As shown in Eqs. (3-19) and (3-20), the singlet and triplet radical pair of one nuclear state oscillate between them. I f there are many nuclear states, their oscillations interfere with one another and the S-T oscillation cannot be observed as shown in Fig. 6-3(b). Brocklehurst measured the scintillation pulse shape (Z) for solutions of para-terphenyl and obtained the following interesting MFEs ( R = Z(0.16 T)/Z(O T) ) as shown in Fig. 6-4(a) [30]: (1) For terphenyl-h14and terphenyl-di4in delaline, each of their R ( f ) values increases after the peak of a scintillation pulse and then attain the maximum value. After the maximum, each of their R(t) values decreases and then attain a plateau. (2) Although the plateau values for terphenylhi4 and terpher1y1-d~~ are the same, the time proflles of their R(t) values before the plateau are quite different fiom each other. This is due to the smaller HFC constants of terphenyl-dl4 than those of terphenyl-hi4. This is a typical example of MIE of D.

Klein and Voltz lirst observed clear oscillation of MFEs on the relative fluorescence intensity (&/I> for a cyclohexane solution of bis-2,5-(4-butylphenyl)-1,3,4-oxadiazole (2bPPD) at room temperature as shown in Fig. 6-5 [31]. Here, &/I is defined as follows: (6-16) d l / Z = [Z(B)-Z(0T)]/ Z(0 T). On the other hand, they could not find such an oscillation for a cycloheane solution of 2,5diphenyl-l,3-oxazole (PPO) as shown in Fig. 6-5. Because two equivalent nitrogen nuclei dominate the HF structure of the 2bPPD anion and cation radicals, the spin levels of their radical pair may be much simpler than those of the radical pair generated fiom PPO. In the case of PPD, one of the nitrogen nuclei of 2bPPD is replaced by a C-H group leading to a more complicated HF structure. Thus, the above-mentioned damped oscillation was only observed for 2bPPD.

6.6 Magnetic Field Effects on Photochemical Reactions As mentioned in section 6.4, Sagdeev et al. studied some thermal reactions in the absence and presence of ordinary magnetic fields and lirst reported MFEs on chemical reactions through radical pairs in 1973 [16]. Although many scientists were interested in this paper, most of them were suspicious of the paper, because there were many problems in the paper as described in section 6.4,. Moreover, Sagdeev et al. made no direct measurement of the yield of reaction products, but they only measured the ratio of the NMR intensities of unsymmetrical and symmetrical products. As mentioned in section 6.5, Brocklehurst et al. studied some radiation reactions in the absence and presence of ordinary magnetic fields and found MFEs on the intensity of fluorescence generated through cage recombination of ionradical pairs in 1974 [18]. They succeeded in observing MFEs on the time prolile of such fluorescence intensities as well as MFEs on the oscillation of fluorescence. Thus, there was no objection to the results of their MFEs. Although they tried to find MFEs on the triplet yield, they could find no clear result of such MFEs. Moreover, no MFE was obtained for product yields in their radiation reactions. In 1976, however, two Japanese and two German groups independently found such MFEs on product and triplet yields in photochemical reactions. Since 1976, therefore, a big bang of the studies of MFEs on chemical reactions has been started. In 1975, Sakuragi et al. examined the MFE on the singlet and triplet-photosensitized decomposition of dibenzoylperoxide (DBP), because CIDNP had already been observed not only in this reaction but also in many other photochemical reactions [19]. They, however, observed no appreciable effect in the presence of magnetic fields of 1 - 1.4 T [20]. Thus, the author’s group re-examined the singlet-sensitized reaction in the presence of much higher fields of up to 4.3 T with a small super-conducting magnet [21], trying to observe MFEs on

89 the yields of cage and escape products in toluene at room temperature (20 f 2 “C). Here, we used chrysene as a singlet sensitizer (Sens.). (6-17) Chrysene (Sens.) + hu -+ ‘Chysene*(ISens.*). The scheme of this reaction is illustrated in Fig. 6-6. As shown in this figure, a singlet radical pair (‘RP,) consisting of two benzoyloxy radicals is initially produced from the singlet excited state of DBP (‘DBP*). Since the lifetime of ‘RP, is very short, ‘RP, gives a secondary singlet radical pair (‘RPb), which involves the benzoyloxy and phenyl radicals. Because the g-value of the benzoyloxy radical (2.0123) is much larger than that of the phenyl radical (2.0024), the S-T conversion of this radical can be enhanced by external magnetic fields. Thus, the yield of phenyl benzoate (PB: cage product) is expected to be decreased by the fields. On the other hand, the yields of escape radicals and escape products are expected to be increased by the fields. Here, o-, m-, and p-methylbiphenyl (MB) and 1,2dipenylethane (DPE) are typical escape products [20]. We measured the yields (Y(E)) of PB, o-MB, and a mixture of p-MB and DPE separately with a gas chromatograph spectrometer. The observed relative change (R(E)) of each yield is plotted against E’” in Fig. 6-7. Here, AR(B) is defined as AR(B) = [Y(B)-Y(OT)]/Y(OT). (6-18) Fig. 6-7 shows that the AR(4.3 T) value of PB was observed to be -8 & 3 % at the maximum field. Thus, we could undoubtedly conclude that the PB yield was decreased by the field. Furthermore, it is clear from this figure that the AR(B) value of PB is proportional to the square root of B. On the other hand, the AR(4.3 T ) values of o-MB and a mixture of p-MB and DPE were measured to be +2 3 %. Although the increase in AR(4.3 T) is smaller than the standard deviation, we may safely recognize from Fig. 6-7 that the AR(B)values of o-MB and a mixture of p-MB and DPE have a general tendency to increase in the presence of magnetic fields. From Eqs. (6-6) and (6-18), the M(B)value of PB (AR p~ (B)) can be expressed as follows: AR

PB

(B) = [ Ycs(E) - Ycs(OT)j/ Ycs(OT) = - ml~AgpBE/2A1”*/p.

(6-19)

The observed square root dependence of AR PB (E) on B can be well interpreted by Eq. (6-19), which is derived from the AgM. From the slope of the observed AR PB (B) versus line, d p was determined to be 1.04 ~ 1 0s .” ~~ . Since 0.5
Problem 6-1. Prove that d p is 1.04 ~ 1 0s .” ~ . From Eqs. (6-7)and (6-IS), the AR(E) value of the escape products (RE (B)) can be expressed as follows: AR E (B) = [ YEs@)- YEs(oT)j/ YEs(OT)= - [Ycs(B)- Ycs(OT)]/[l - Ycs(OT) ] (6-20) = - { Ycs(OT) /[I - Ycs(oT) 1 ) PB (B). The AR PB (4.3 T) was 0.08 % and the yield of PB at 0 T (YcS(OT))was determined to be 0.12 [32]. Because the Ycs(0T) /[I - YcS(OT)]value of Eq. (6-20) is 0.136, the AR E (4.3 T) value is estimated to be +0.011. This value also agrees with the observed one (+2 f 3 %). Here, we could quantitatively interpret the observed MFEs by the AgM of the radical pair mechanism. In this study, we could undoubtedly establish the MFEs on photochemical reactions in solution from both experimental and theoretical aspects.

90

J

(PB: Cage Product)

(Escape Radicals)

(Solvent)

(MB, DPE, etc.: Escape Products) Fig. 6-6. The reaction scheme of the singlet-sensitized decomposition of dibenzolyperoxide (DBP) in toluene. In 'RPb, the g-value of the benzoyloxy radical (2.0123) is much larger than that of the phenyl radical (2.0024). (Reproduced from Ref. [34] by permission from The Chinese Chemical Society)

91

AR(B) = [Y(B)-Y(OT)]/Y(OT)

B1/2/Tl" Fig. 6-7. The external magnetic field dependence of- AK(B) observed for the singletsensitized decomposition of DBP in toluene at room temperature: (*) for PB, ( 0 ) for o-MB, and (x) a mixture of p-MB and DPE. The standard deviation of each M(B) is 3 %. (Reproduced from Ref. [21a] by permission from Elsevier Science B. V.) Michel-Beyerle et al. [22] and Shulten et al. [23] independently studied MFEs on the quenching reactions of singlet pyrene ('A*) with amines (D) in polar solvents.

A+hu+'A*, 'A* + D ---* '[D"+ A-'1 , [D+'+ A-' ] H [D" + A-'1 ,

(6-21a) (6-21b)

'[D+'+ A-'] 3[D+'+ A-']

(6-21d) (6-21e)

'

--f

+

A+D, 3A* + D,

'.3[Mc'+ M-'] + D+'+ A-'.

(6-21~)

(6-210

With the aid of ns-laser flash photolysis techniques, they found that the yield of triplet pyrene (3A*) had been decreased by magnetic fields less than 50 mT and that the yield of the pyrene anion (A-*)had been increaded by them as shown in Fig. 6-8. Werner et al. measured solvent, isotope, and magnetic field effects in the geminate recombination of radical ion pairs [23, 261. They found similar MFEs in reaction (6-21) in Their typical acetonitrile (ACN), dimethylformamide (DMF), ethanol, and 2-propanol. results on the magnetic field dependence of the pyrene triplet yield in acetonitrile are shown in Fig. 6-9. This figure shows that the yield of each reaction decreases with increasing B from 0 T, but that the increase is saturated at 65 mT. Such MFEs can be explained by the HFCM, where the S-T conversion rate in (6-21c) is reduced by magnetic fields. The B1/2 value of the HFCM can be expressed by Eqs (6-1 1) and (6-12).

B,,* =2(B: + B i ) / ( B ,+ B 2 ) ,

(6-1 1)

92

Fig. 6-8. Time evolution of the pyrene anion (A-') and triplet pyrene (3A*)of the reaction (6-21) in methanol measured in the absence and presence of a magnetic field of 50 mT . For each wavelength and magnetic field, 8-10 transients were averaged with a transient digitizer ( time resolution - 0.5 ns). (Reproduced from Ref. [23b] by permission from The American Institute of Physics)

E l

CH30H 0.12 0.00

:;/ I

0.0 6

20

00

60

40

100

Timtlns

Fig. 6-9. Magnetic field dependence of the pyrene triplet yield (a@)/ (0))in acetonitrile as derived from delayed fluorescence measurements: Pyrene-dldp-dicyanonenzened4, Bin = 8 G, Pyrenel p-dicyanonenzene,

Pyrene+DCNB in MeCN

0

200

100

650

7 Pyrene+DMT in

1 aeo

,j

I 100

,

I , 200

MeCN

+ 650

Magnetic Field Strength B/Gauss

Bin = 17 G ; Pyrene/N,N-dimethyl-ptoluidine, Bln = 59 G. (Reproduced from Ref. [26] by permission from Elsevier Science B. V.)

93 where

Werner et al. found that a plot of the observed values against the calculated B I l 2values gave a strait line going through the origin with a slope of unity [26]. This fact clearly confirmed the reliability of Eq. (6-1 1). It is also noteworthy from Fig. 6-9 that the observed I3112 value observed for the reaction of natural system is much larger than that of perdeuterated system as shown in Fig. 6-9. This is a typical M E due to the difference in H- HFCs and DHFCs. Table 6-2. Solvent effect on the geminate singlet and triplet recombination yields ( $s and b)and the yield of the ion radicals (@) for = 6.176 k n s and KT = 18.53 k n s . Relative diffusion coefficients defined by D = D(Py) + D(DMA) and dielectric constants (E) for various solvents are employed by the values at 25 "C. (Reproduced from Ref. [27] by permission from The American Institute of Physics)

Werner et al. analysed their MFEs with a stochastic Liouville equation [27] . Their typical results are summarized in Table 6-2. The results shown in Table 6-2 show that the singlet yield increases with strongly with decreasing solvent polarity, increasing viscosity, and decreasing temperature, but that the triplet yield is much less influenced by the solvent. This result can be explained by the competition between the singlet recombination (6-21d), occumng shortly after pair generation, and the triplet recombination (6-21e), starting only after the HFCM has succeeded to convert the radical pair from the singlet to the triplet electron spin state. The observed MFEs on & and @ can be explained by this theory based on the HFCM. It is noteworthy that the relative MFE on the triplet yield (MB+oo)/b(O.T)) is insensitive with respect to solvent polarity and viscosity. Hata studied the photoisomerization of isoquinoline N-oxide (QNO) in ethanol (HOR) under magnetic fields less than 1.7 T [25]. He reported that the yield of lactam (isocarboatyrill) was -67 % at low and high fields (0 T 5 B 50.5 T and 1.5 T 5 B 5 1.7 T ), but that the yield showed a remarkable decrease (-53 %) at B = 1 T as shown in Fig. 6-10. He proposed the following scheme for this reaction: QNO---HOR

+ hu -+'QNO*---HOR,

'QNO*---HOR

3

'[NQ+O' "HOR] '[NQ'O' -HOR] 1,3[NQ+0. 'HOR]

-

'[NQ'O'

-+

-+

(6-22a)

"HOR],

3[NQ'O'

(6-22b) (6-22~)

"HOR],

Lactam, NQ+O' + R O K '

+

QNO---HOR.

(6-22d) (6-22e)

94

If the S-T conversion (6-22c) is enhanced by the LCM, the yield of the cage product (lactam) can be reduced at the level-crossing field (BLC- 1 T) as shown in Fig. 6-2(d). His group also studied MFEs on this reaction in various alchohols and reported the results shown in Fig. 610. This figure shows that the order of the B w value becomes as follows: BLC(tert-butyl alcohol) < Bm (2-propanol) = BLC (ethanol) < BLC(methanol). Hata et al. proposed that the distance ( r ) between the radicals in each radical pair might decrease with decreasing pK, of alcohol as follows: r (tert-butyl alcohol) > r (2-propanol) = r (ethanol) > r (methanol). On the other hand, the value should increase with decreasing r from Eq. (5-16). Thus, it is expected that the BK value increases as the pK, value decreases. Although this is the first report of the MFE due to the LCM, there has not yet been any report for its reconfirmation.

70

-s

-

60 50

D

F 40 30 20

0

15

10

5

Field

Strength

(kG)

Fig. 6-10. h4FEs reported for the photoisomerization of isoquinoline N-oxide (QNO) in ethanol (HOR) under magnetic fields less than 1.7 T: (a) in ethanol, (b) in 2-propanol, (c) ten-butyl alcohol, and (d) in methanol. (Reproduced from Ref. [25b] by permission from The Japanese Chemical Society) 6.7 Magnetic Isotope Effects The magnetic isotope effect ( M E ) is one of the most important techniques which have been developed in the course of studies of MFEs on chemical reactions. It is noteworthy that the M E is a new type of isotope effect: This effect comes from the difference in nuclear spin, but not in nuclear mass. According to the HFCM, the S-T conversion of radical pairs depends on the HF interaction between nuclear and electron spins in the component radicals, even in the absence of an external magnetic field. Therefore, it is possible for MIEs to appear in most reactions which show MFEs.

In 1976, Buchachenko et al. found a M E of I3C in the potodecomposition reaction of dibenzyl ketone (DBK) in benzene and hexane[24]. Since this reaction occurs through the triplet state of DBK, the triplet radical pairs is initially produced as shown in scheme (6-23). DBK

+ hu

---f

'DBK*

+ 3DBK*,

3DBK* -+ '[PhCH2C'O 'CH2Ph1, 3[PhCH2'3C'0 'CHzPh] + [PhCH213C'0 'CH2Ph],

(6-23a) (6-23b) (6-23~)

95 '[P~CHZ'~C'O%H2Ph] 3[PhCH2'2C'0 %H2PhJ

+ ---*

DBK, 2PhCH2

(6-23d)

+

"CO,

(6-23e) (6-230

2PhC'Hz + PhCHz CHZPh. If the carbonyl group of the phenacetyl radical contains 13C,the generated triplet radical pair would quichly changes into the singlet pair through the large HF interaction of I3C as shown in scheme (6-23c). It is easy for the singlet pair to recombine to the starting molecule (DBK) within a solvent cage. If the carbonyl group of the phenacetyl radical contains I2C, on the other hand, the generated triplet radical pair would keep its spin multiplicity for a while because "C has no nuclear spin. The phenacetyl radical is so instable that it decomposes into the benzyl radical and CO, generating the triplet pair containing two benzyl radicals as shown in scheme (6-23e). From the escaped benzyl radicals, diphenylethane is produced as an escape product. Thus, the concentration of 13C in the carbonyl carbon of DBK starts to increase as the decomposition of DBK proceeds. Actually, Buchachenko et al. found that the I3C enrichment increased from 6.5 to 17.6 % when the decomposotion proceeded from 75 to 98 %. They also found that the I3C enrichment decreased from 9.5 to 6.0 % when an external magnetic field of 53 mT was applied. Although the enrichment was decreased by magnetic fields, this MFE on the enrichment is a direct evidence that the enrichment is due to a M E . In 1978, T w o and Ktaeutler found that the MlEs of the photodecomposition of DBK and related ketones can be enhanced greatly if radical pairs are generated in confined systems such as micelles [33]. The details of MIEs will be explained in Chapter 9.

References [ l ] P. W. Atkins, Chem. in Britain, 12 (1976) 214. [2] (a) W. Steubing, Verh. dtsch. Phys. Ges., 15 (1913) 1181; (b) W. Steubing, Ann. d. Phys.(4), 58 (1919) 55. [3] E. 0. Degenkolb, J. I. Steinfeld, E. Wasserman, and W. Klemperer, J. Chem. Phys., 51 (1969) 615 and references cited therein. [4] W. E. Falconer and E. Wasserman, J. Phys. Chem., 45 (1966) 1843. [5] H. E. Radford and H. P. Broida, J. Chem. Phys., 38 (1963) 645. [6] L. Fakas and H. Sachsse, 2.phys. Chem., 23B (1933) 1, 19. [7] P. W. Atkins and M. J. Clugston, Mol. Phys., 27 (1974) 1619 and references cited therein. [8] (a) M. Misono and P. W. Selwood, J. Am. Chem. SOC.,90 (1968). (b) P. W. Selwood, Nature, 228 (1970) 278. [9] S. Fujiwara, H. Haraguchi, and Y. Umezawa, Anal. Chem., 40 (1968) 249. [lo] T. Watanabe, Y. Tanimoto, T. Sakata, R. Nakagaki, M. Hiramatsu, and S. Nagakura, Bull. Chem. Soc.Jpn., 58 (1985) 1251. [ 111 R. C. Johnson, R. E. Memfield, and P. Avakian, Phys. Rev. Lett., 19 (1967) 285. [12] R. E. Memifield, J. Chem. Phys., 48 (1968) 4318. [ 131 L. R. Faulkner and A. J. Bard, J. Am. Chem. Soc., 91 (1969) 209. [I41 A. Gupta and G. S. Hammond, J. Chem. Phys., 57 (1972) 1789.

96

[Is] P. W. Atkins, Chem. Phys. Lett., 18 (1973) 355. The electron donor and acceptor (D and A ) in this paper correspond to the energy acceptor and donor (R and S) of Ref. [14], respectively. In Ref. [14], however, R and S were represented by A and D, respectively. [I61 R. Z. Sagdeev, Yu. N. Molin, K. M. Salikhov, T. V. Leshina, M. K. Kamha, and S. M. Shein, Org. Magn. Resonance, 5 (1973) 603. [ 171 Ref. [4] of Chapter 4.

[I81 (a) B. Brocklehurst, R. S. Dixon, E. M. Gardy, V. J. Lopata, M. J. Quinn, A. Singh, and F. P. Sargent, Chem. Phys. Lett., 28 (1974) 361. (b) F. P. Sargent, B. Brocklehurst, R. S. Dixon, E. M. Gardy, V. J. Lopata, and A. Singh, J. Phys. Chem., 81 (1977) 815. [ 191 For example, “Chemically Induced Magnetic Polarization,” A. R. Leply and G. L. Closs, Eds., John Wiley, New York (1973). [20] H. Sakuragi, M. Sakuragi, T. Mishima, S. Watanabe, M. Hasegawa, and K. Tokumaru, Chem. Lett., (1975) 231. [21] (a) Y. Tanimoto, H. Hayashi, S. Nagakura, H. Sakuragi, and K. Tokumaru, Chem. Phys. Lett., 41 (1976) 267. (b) H. Hayashi and S. Nagakura, Bull. Chem. Soc.Jpn., 51 (1978) 2862. [22] M. E. Michel-Beyerle, R. Haberkorn, W. Bube, E. Steffens, H. Schroeder, H. J. Neusser, E. W. Schlag and H. Seidlitz, Chem. Phys., 17 (1976) 139. [23] (a) K. Schulten, H. Staerk, A. Weller, H.-J. Werner, and B. Nichel, Z. Phys. Chem. NF, 101 (1976) S.371. (b) H.-J. Werner, H. Staerk, and A. Weller, J. Chem. Phys., 68 (1978) 2419. [24] A. L. Buchachenko, E. M. Galimov, V. V. Ershow, G. A. Nikiforov, and A. D. Pershin, Dokl. Acad. Nauk SSSR, 228 (1976) 379. [25] (a) N. Hata, Chem. Lett., (1976) 547. (b) N. Hata, Y. Ono, and F. Nakagawa, ibid, (1979) 603. [26] A. Weller, F. Nolting, and H. Staerk, Chem. Phys. Lett., 96 (1983) 24. [27] H. -J. Werner, Z. Sehulten, and K. Schulten, I. Chem. Phys., 67 (1977) 646. [28] “Spin Polarization and Magnetic Effects in Radical Reactions,” Yu. N. Molin, Ed., Elsevier, Amsterdam (1984).P.246. [29] M. Wakasa and H. Hayashi, Chem. Phys. Lett., to be published. [30] B. Brocklehurst, Chem. Phys. Lett., 44 (1976) 245. [31] J. Klein and R. Voltz, Can. J. Chem., 55 (1977) 2101. [32] K. Tokumaru, A. Ohsima, T. Nakata, H. sakuragi, and T. Mishima, Chem. Lett., 571 (1974). [33] N. J. Turro and B. Kraeutler, J. Am. Chem. Soc.,100 (1978) 7432. [34] H. Hayashi, J. Chinese Chem. Soc.,49 (2002) 137. Solution to the Problems 6-1. The AR PB (4.3 T) values was observed to be 8 % and the Ag value is 0.0099. Thus, Eq. (6-19) gives the following relation: AR p~ (4.3 T) = -0.08 = -mbg/.1,B12h1~/~/p= - ( m / p ) ( ~ / 2 ) 1I/A ~~/.I,B/~J~/~ = - ( d p ) ( 1.5708)”Z(0.0099~9.2740~10~24JT~1/105457x1O~34Js)1’2

= - (dp)(1.2533)(6.11854~10~~~’~).

Here, the d p value is obtained to be 1.04x106s1’2.

97

7. Magnetic Field Effects due to the Relaxation Mechanism 7.1 Magnetic Field and Magnetic Isotope Effects Observed in Micellar Solutions In 1981, the author's group found large MFEs and MIEs on the radical pair lifetime ( T ~ ~ )

and the escape radical yield (YE) under magnetic fields of 0 - 70 mT with an ns-laser photolysis at room temperature for the photoreduction reactions of benzophenone (BP), benzophenone-dlo (BP- dlo), and [carb~nyl-'~C]benzophenone (BP-I3C) in micellar sodium dodecyl sulfate (SDS) solutions [ 11. Here, micellar molecules act as hydrogen donors (RH). The scheme of such photoreduction reactions of the benzophenone isotopes (XCO) can be represented by the following reaction scheme:

XCO + hu + 'XCO*

+3 ~ ~ 3XCO* + RH + 3[XC'OH 'XCO*

3[XC'OH R'] 3"[XC*OH R']

H

~

R']

'[XCOH R']

+ XC'OH + R'

*

Photo-Excitation,

(7-la)

Intersystem Crossing(kac),

(7- 1b)

Generation of Radical Pairs (kc),

(7-lc)

Singlet-Triplet Conversion ( k s ~ ) ,

(7-ld)

Escape of Radicals ( k ~ ) ,

(7-le)

Cage Recombination (kp), (7-1f) '[XC'OH R'] + XC(R)OH The triplet and singlet radical pairs are represented by 3[A*B'] and '[A' B'], respectively, and ki shows the respective rate constant. It is noteworthy that the MFEs and MIEs on TRP and YEcan be measured with ns-lasers for the reactions in such confined systems as radical pairs in micelles as shown in Fig. 7-1. In this figure, the time profiles of the transient absorbance (A(t) curves) are observed for the generated ketyl radicals (XC'OH). The T~~ value can be obtained from the initial decay of the A(t) curve and the YEone from the nearly constant component observed after the initial decay. Table 7-1. Magnetic field and magnetic isotope effects on the radical pair lifetime (zRP) observed for the photoreduction reactions of benzophenone isotopes in SDS micellar solutions at room temperature under magnetic fields less than 70 mT [ 11 and under the fields of 0.1 - 1.34 T. (Reproduced from Ref. [2b] by permission from The Japanese Chemical Society)

98

0.2

0 0

2 TIME/ps

A

Fig. 7-1. The time profiles of the transient absorbance (A(t)curves) observed at 525 nm for a micellar SDS solution of (A) BP, (B) BP-dlo, and ( C ) BP-I3C: (a) in the absence of a magnetic field; in the presence of a magnetic field of (b) 10 mT, (c) 20 mT, (d) 40 mT, and (e) 70 mT. (Reproduced from Ref. [lb] by permission from The American Chemical Society)

99

Fig. 7-2. Plots of values proportional to the escape radical yield (YE)against magnetic fields ( B ) observed at 525 nm for a micellar SDS solution of ( 0 ) BP, ( 0 ) BP-dlo, and ( 0 ) BP-13C. (Reproduced from Ref. [lb] by permission from The American Chemical Society)

3 0 200 400 600

0’ 0

I

0.5

Fig. 7-3. Plots of the escape radical yield (YE)against magnetic fields ( B ) observed at 525 nm for a micellar SDS solution of ( 0 ) BP, ( 0 ) BP-dlo, and ( 0 ) BP-”C. (Reproduced from Ref. [23] by permission from The Chinese ChemicalSociety)

B,10-4T

I

1.o

1.5 B/T

From the A(t) curves shown in Fig. 7-1, we obtained remarkable MFEs and MIEs on the and YE values under magnetic fields less than 70 mT as shown in table 7-1 and Fig. 7-2 Because their MFEs and MIEs were not saturated at 70 mT [I], we extended the [I]. maximum field to 1.34 T with an electromagnet and obtained their MFEs and MIEs as shown From these results, we can see the following new MFEs in Table 7-1 and Fig. 7-3 [Z]. and MIEs under high fields of up to 1.34 T:

TRP

(1)

For the reactions of the BP isotopes in the SDS micellar solution, the ‘ t ~ pand YE values increases with increasing B from 0 T to 1.34 T. These MFEs can not be explained by the AgM, because the MFEs due to the AgM for reactions from triplet precursors should decrease with increasing B from 0 T as shown in Chapter 6.

a (2) These increases in the MFEs seem to approach their saturated values at B = 1.34 T. It is noteworthy that these increases do not show saturation at B 0.1 T, where the MFEs due to the HFCM should be saturated as shown in Chapter 6.

-

(3) Clear MIEs on the T~~ and YE values can be observed for present reactions. At each field, we can see the following order for the MIEs: BP-I3C< BP < BP- dlo. (4) These MIEs are very small at B = 0 T , but they increase with increasing B fiom 0 T to 0.2 T. With increasing B fi-om 0.2 T to 1.34 T, however, these MIEs decrease again. It is noteworthy that these MIEs can not be explained by the HFCM, because the MIEs due to the HFCM should decrease with increasing B ffom 0 T as shown in Chapter 6.

Because the observed MFEs and MIEs on zRPand YE at such high fields (0-1.34 T) could not be explained by the ordinary AgM and HFCM, the author's group proposed the relaxation mechanism in 1984 and succeeded in interpreting such novel MFEs and MIEs 121. In 1997, however, Tanimoto et al. reported that they had found no MIE on TRP for the same reactions of the benzophenone isotopes [ 3 ] . They have recently realized their mistakes and confirmed our results on the existence of the MIEs in these reactions 141.

Fig. 7-4. Schematic diagram of the relaxation mechanism for the singlet (S) and triplet (T,,, n 1, 0, -1) states of a radical pair with J = 0 and rate constants (a)concerning the states: (a) In the absence of a magnetic field; (b) In the presence of a magnetic field. (Reproduced fiom Ref. 1231 by permission fi-om The Chinese Chemical Society)

=

101 7.2 Proposal of the Relaxation Mechanism

It is noteworthy that the S-To conversion rate is given by QNfor a radical pair with J-0 J as shown in Chapter 3. When J - 0 J,Ag = 0.01, B = 1 T, and AJgpB = A&pB = 0 T, the rate becomes 4.4 x 10’ s-l from problem 3-5. If such S-To conversion rate is comparable to the escape rate of two radicals from a solvent cage, appreciable MFEs and MIEs can be observed. In some cases, this condition can be satisfied in homogeneous solvents. If two radicals are confined with membranes, micelles, or chemical bonds, the escape rate of the two radicals becomes much smaller than the S-TOconversion rate. In this case, the S and TOstates attain equilibrium and the T,I-To and T,I-S relaxations become important under sufficiently high fields as shown in Fig. 7-4@). In 1984, the author’s group proposed the relaxation mechanism (RM) [2] in order to explain MFEs and MIEs on chemical reactions in confined systems. When kST(0 T), ksT(B) >> kp in the RM as shown in Fig. 7-4, the rate equations of the populations of the singlet and triplet radical pair ([Sland [Tn]for n = +1,0, and -1) produced from a triplet precursor can be represented as follows [2]: (a)B = 0 T: (7-2a) d[Slldt = ~sT([T+II + [To1 + lT-11 - (3 k T + kp + kE) [Sl, d[T,J ldt = kST[q - (kST + kE) [T,] (n = +I, 0, and -1). (7-2b) ( b ) B > 0 T: (7-3a) d[Slldt = ks~[Tol+kR ‘([7’+11+ [ T - I ]) ( ~ S+Tkp + k~ + 2kR ‘1 Is], d[Tolldt = ksT [q + kR ([T+I]+ iT.11) - (ksT + kE + 2 k )~[To], (7-3b) d[Tn]ldt= kR ‘[SJ + k~[To]- (kE+ kR + kR ‘) [T,] (n = +I and -I). (7-3c) Here, the rate constants in Eqs. (7-2) and (7-3) are represented in fig. 7-4. Let us solve Eqs. (7-2) and (7-3). In Case (a), the populations of [Sl and [T,] (for n = +1, 0, and -1) can be assumed to be equilibrium with one another because kST(0 T)>> kp. Thus, and [Tn]can be written as x and Eq. (7-2) becomes as follows: d([q + [T+I]+ [To]+ [T.,])/dt = d(4~)/dt = 3ks~X- (3 ksT + kp + k ~ ) x+ 3ks~X- 3 ( k s ~ + kE) X = -( kp + 4 k ~X.) (7-4) Eq. (7-4) is rewritten as &ldt = -( kpf4 + kE) X. (7-5) When RP(t) (= [q+[T+I]+ [TO]+ [T.I] = 4x) is used for the radical pair population, it is given as

[a

RP(t)= Roexp(-k& (7-6) Here, its rate constant is represented by = kp/4 + kE. (7-8) In Case (b), we can assume that the populations of [Sland [TO]are equilibrium with each other ([A = [To] = x ) and that the populations of [T+I]and [T.]] are equal to each other ([T+]] = [T-I]= y ) . Eqs. (7-3a) and (7-3b) become as follows: d([g + [To])/& = d(k)/dt = 2ks~X+ ( 2 k ~ + 2 k ~‘ ) y- ( 2 k s ~+ kp + 2 k +~ 2 k +~ 2 k ‘)X. ~ (7-9) Eq. (7-9) is rewritten as dxldt = (kR + kR ‘ ) y - ( kp/2 + kE + kR + kR ‘)X, (7-10a) dddt = (kR + kR ‘ ) y - kF. (7-lob)

102 Here, kF is given as follows: kF = kpl2 + ks. (7-1 1) From Eq. (7-3c), the following relation can be obtained: d([T+,] + [T.,] )ldt = d(2y)ldt = ( 2 k ~+ 2 k ')X ~ - ( 2 k ~+ 2 k + ~2 k ~ ')y. (7-12) Eq. (7-12) is rewritten as (7-13) dyldt = (kR + kR ' ) x - ks y . Here, ks is given as follows: (7-14) ks = kR + kR' + kE. When RP(t) (= [fl+ [T,,] + [TO]+ [T.,] = 2x + 2y) is used for the radical pair population, it is approximately given as (7-15) RP(t) =: R@Xp(-kFt) + Rsexp(-kst). Problem 7-1. Prove Eq. (7-15). From the above results, the decay of the radical pair population in the absence and presence of a magnetic field can be represented by Eqs. (7-6) and (7-15) as follows: (7-6) (a) B = 0 T: RP(t)= Roexp(-kot), (b) B > 0 T: RP(t) R@Xp(-kFt) + Rsexp(-kst). (7-15) Here, the rate constants are defined by Eqs. (7-1 1) and (7-14). (7-1 1) kF = kpl2 + ks, (7-14) ks = kR + kR' + kE. From the above results, we can see the fact that kp is much larger than ks, that kp and k~ are independent of B, but that kR and kR' are affected by B. The exact expressions for k R and kR' are given in Section 7.4, but they are symbolically written in this chapter as (7-16a) (7-16b) Here, each V, term is due to each of the dipole-dipole interaction between two components radicals, the anisotropic Zeeman of the two radicals, and their anisotropic HFC terms, 2, is each rotational correlation time, and w,is given by g p B B I A . Thus, the following characteristics of the MFEs due to the RM can be obtained: (1)In the absence of a magnetic field, the radical pair decay can be represented by a single exponential function as shown by Eq. (7-6). Here, the decay rate constant is given by ko. From lZq. (7-8) , the value should show no M E . This prediction explains well results (4)shown in Section 7.1. ( 2 ) h the presence of a magnetic field, the radical pair decay can be represented by a combination of two exponential functions as shown by Eq. (7-15). The fast component can not often be observed because the transient absorbance due to radicals is often overlapped with that of their T-precursor. Thus, the slow component can usually be observed. From Eqs. (7-15) and (7-16), the kR + kR' value was found to decrease gradually with increasing B from 0 T to 1.34 T [2]. This means that the TRP value should

103

increase gradually with increasing B from 0 T to 1.34 T. This prediction explains well the MFEs on the TRP value shown by results (1) and ( 2 ) in Section 7.1. (3) In the presence of a magnetic field, the k~ and k ~ values ’ include the anisotropic HFC terms. From Eqs. (7-15) and (7-16), the more these values become, the more the HFC ’ can be obtained as follows: BPinteractions are. Thus, the order of the k~ and k ~ values I3C > BP > BP- dlo. This means that the order of the TRP value becomes as follows: BPI3C< BP < BP- dlo. This prediction explains well the MIEs on the TRP value shown by result (3) in Section 7.1. (4) When the k~ + k ~ value ’ decreases with increasing B from 0 T to 1.34 T, the YEvalue should increase with increasing B from 0 T to 1.34 T because of Eq. (7-14). Here, the YE value can be obtained as (7-17) YE = k~ /( k R + k ~ +’ k ~ ) . This prediction explains well the MFEs on the YEvalue shown by results (1) and ( 2 ) in Section 7.1. (5)In the absence of a magnetic field, the YE value should show almost no MIF because the k~ value should show no M E . In the presence of a magnetic field, the YEvalue should show MIEs of the following order: BP-I3C < BP < BP- dlo. This is due to the fact that the order ’ can be obtained as follows: BP-I3C > BP > BP- dlo. These preof the k R and k ~ values dictions explain well the MIEs on the YE value shown by results (3) and (4) in Section 7.1. Thus, the author’s group succeeded in explaining the new MFEs and MIEs by the RM. It is noteworthy that the RM could also explain similar MFEs and MIEs observed in reactions of biradicals as shown in Chapter 8. 7.3 Magnetic Field Effects on Chemical Reactions through Radical Pairs Involving Heavy Atom-Centered Radicals At the initial stage of the studies of MFEs and MIEs, the studies were confined to the reactions of radical pairs and biradicals involving only light atom-centered radicals such as Cand 0-radicals. On the other hand, MFEs and MIEs on the reactions of heavy atom-centered radicals such as Si-, S-, and Ge-radicals were believed to be too small to be observed beyond experimental errors from the following reasons: (1) In a radical pair or a biradical involving such a heavy atom centered radical, the S-T conversion occurs also during a closed pair through the spin-orbit (SO) interaction of the heavy atom. Because the S-T conversion of a closed pair should be independent of an external field, its MFE and MIF were considered to be negligible. (2)In a radical pair or a biradical involving such a heavy atom centered radical, its S-T relaxation of a separated pair becomes much faster due to the anisotropic Zeeman interaction of the heavy atom-centered radical than that involving only light atomcentered radicals. The author’s group, however, succeeded in finding h4EFs and MIEs in many reactions of radical pairs involving heavy atom-centered radicals, pursuing necessary conditions for the appearance of their MFEs and MIEs. Our results on MFEs will be described in this section and those on MIEs in Chapter 9. Most of the MFEs of heavy atom(E)-centered radicals have been observed in the following reactions: (1) Photo-decomposition reactions

RE-X + hu -+ %E-X* + 1,3[RE*X’]. ( 2 )Photo-induced electron transfer reactions A (or D) + hu + Is3A*(or Is3D*),

(7-18)

(7-19a)

104

l33A* + D + Iz3[A-' D+*],

(7-19b)

I33D* + A + 133[A-'Df'],

(7-19~) Here, A and D represent electron acceptor and donor, respectively, one of which contains at least one heavy atom-centered radical. (3)Photoreduction of carbonyl compounds (XC=O) XC=O + hu 3XC=O*, (7-20a) (7-20b) 3XC=O* + REH 3[RE? 'COHX]. Because the MFEs of heavy atom-centered radicals are usually much smaller than those of light atom-centered radicals, micellar solutions and oil emulsions have been used for obtaining MFEs of heavy atom-centered ones until quit recently. Recently, we have also observed MFEs of heavy atom-centered radicals even in usual non-viscous solvents. Some of such MFEs of heavy atom-centered radicals can be explained by the RM, but others by other mechanisms. In 1983, the author's group found MFEs on the yield of the escaped ketyl radicals from the radical pairs formed through the photoreduction of acetophenone and xanthone with triethylgermane in SDS micellar solutions [5]. This is the first observation of the MFEs of the reactions of a heavy atom-centered radical. Since then, we have been observing MFEs on many reactions of Si-, P-, S-, Ge-, and Sn-radicals not only in micellar solutions but also in non-viscous organic ones as listed in Table 7-2 . In 1987, the author's group found MFEs on the photodecomposition of phenacyl phenyl sulfone in an SDS micellar solution (Reaction S1 in Table 7-2) [lo]. In 1991, however, Scaiano's group tried to find MFEs in the photoreduction of benzophenone with thiophenol (PhSH) in an SDS micellar solution (Reaction S-2a in Table 7-2), but they could detect no appreciable MFE at 540 nm under magnetic fields of less than 0.32 T [ll]. -+

-+

0-3

0.2 0.1 0

0

10

20 Time / ns

30

4Q

Fig. 7-5. A(t) curves observed at 470 nm at room temperature under magnetic fields of (a)O T, (b) 1.0 T, and (c) 10 T for an SDS micellar solution containing benzophenone and thiophenol. (Reproduced from Ref. [I21 by permission from The Japanese Chemical Society)

105

In 1993, we also tried to find MFEs on the same reaction and found appreciable MFEs as shown in Fig. 7-5 [12]. The reasons why we could find MFEs on this reaction may be due to the following facts: (1) We monitored the transient absorbance (A(t)) at 470 nm which corresponds to the absorption due to PhS’ and observed its MFEs as shown in Fig. 7-5. (2) We also monitored the A(t) curve at 530 nm which corresponds to the absorption due to X(0H)C‘ and found that the MFEs at this wavelength were smaller than those at 470 nm because the absorption due to the T-T absorption of benzophenone overlapped that of the ketyl radical. (3) Because Scaiano’s group monitored the A(t) curve at 540 nm, their MFEs might be also very small. (4) We found that the peak intensity of the A(t) curve decreased with increasing B from 0 T to 10 T. This MFE could be observed with a combination of a super-conducting magnet with an accurate ns-laser flash photolysis apparatus in our group. ( 5 ) If the peaks of the A(t) curves observed at different fields were normalized as usual, their MFEs would become too small. It is noteworthy that the observed MFEs can be explained by the ARM [ 121 and that the MFEs could also be observed in non-viscous organic solvents

Table 7-2. MFEs observed in the reactions of Si-, P-, S-, Ge-, ai

Sn-radicals. Bnlara)[Ref.] 1.35 T [6] 1.35 T [7] 1.35 T [7] 1.2T [8] 1.75 T [9] 1.2T [lo] 0.31 T [ll],’ 10 T [12]

1.7T [I31 lOT [14]

1 0 T [15, 161 3 0 T [17, 181 1.0T [5] 1.0 T [5] 1.35 T r6i 1.35 T [6]

a)B,ax:maximum field. b’DCNB:p-dicyanobenzene. “’TCNB: 1,2,4,5-tetracyanobenzene. d’Ar:2,4,6-tnmethylphenyl grou elsee the reference(s) cited in this column. ONo MFE was observed. g’XnC=O: xanthone. ‘klC=O: 4-methoxybenzophenone. ‘)AcC=O:acetophenone. ”The magnetic field dependence of the M E was observed.

106 Each of the observed MFEs listed in Table 7-2 except Reaction P-2 can be explained by a combination of the RM and the AgM. Because the MFEs due to the RM can be quenched by the addition of paramagnetic ions [19], we can separate the MFEs due to the RM from those due to the AgM as shown in Fig. 7-6.

1.08

0

4 6 0 Magnetic Field i T

2

0

.

.

.

.

I

.

0.5

.

.

.

10

. . . . ,...

I

1

1.5

2

Magnetic Field / T Fig. 7-6. Magnetic field dependence of the yield of the escaped radical (Y(B))in the absence and presence of Gd3+,where R(B)= Y(B)I Y(0 T): (a) Observed in Reaction S-3 for (m) M, and ( 0 ) [Gd3'] = 2.0 x [Gd3'] = 0 M, ( 0 ) [Gd3'] = 0.5 x M, (A) [Gd3'] = 1.0 x M. (Reproduced from Ref. [ 141 by permission from The American Chemical Society (b) Observed in Reaction S-4. (Reproduced from Ref. [ 161)

107

Figure 7-6(a) shows the observed magnetic field dependence of the yield of the escaped radical (Y(B))in Reaction S-3 [14]. Here, R(B)=Y(B)I Y(0 T). In the absence of Gd3+,R(B) was found to increase with increasing B from 0 T to 1 T, but to decrease with increasing B from 1 T to 10 T. By the addition of Gd3+to this solution, R(B) was reduced as shown in Fig. 7-6(a), but kept to be larger than 1. Because the quenching of R(B) by Gd3+was uniform, it can be concluded that the observed MFEs are due to the RM. This is reasonable because the radical pair generated in this reaction includes two equivalent thiyl radicals. This means that the A g value of this radical pair is zero. Figure 7-6(b) shows the observed magnetic field dependence of R(B)in Reaction S-4 [16]. In the absence of Gd3+,R(B) was found to increase with increasing B from 0 T to 0.1 T, but to decrease with increasing B from 0.1 T to 1.7 T. Figure 7-6(b) also shows that the magnetically induced increases in R(B) observed at low fields ( B < 0.2 T) decreased with increasing [Gd3’] from 0 mM to 6.9 mM and that almost all of the increases were quenched by the addition of Gd3+of 6.9 mM. On the other hand, the magnetically induced decreases in R(B) observed at high fields ( B > 0.3 T) still remained with increasing [Gd3’] from 0 mh4 to 6.9 mM. This means that the magnetically induced decreases in R(B) observed in the presence of of Gd3+of 6.9 mM is due to the the AgM. This can be explained by the fact that the MFE due to the AgM is much less quenched by paramagnetic ions than that due to the RM. Here, the radical pair involving the xanthone ketyl and p-aminophenylthiyl radicals has a large A g value (0.0047) [ 161. The magnetically induced increases in R(B) observed in the absence of of Gd3+canbe explained by the RM. The MFEs due to the AgM on Reaction S-5 observed in usual organic solvents [17, 181 will be treated again in Chapter 12.

0

L

0.3 0.5 0.7

1.5T

-

0

7

I

200

I

I

I

LOO

I

I

SO0 Timelns

I

I

800

0

10

20

30

(Field/mT)”2

40

Fig. 7-7. (a): A(r) curves observed at 330 nm for Reaction P-2 in 2-propanol at room temperature under magnetic fields of 0,0.3,0.5,0.7, and 1.5 T (from top to bottom). (b): Magnetic field dependence of the absorbance (A(200 ns)) observed at 330 nm in Reaction P-2 in 2-propanol. The right-hand ordinate indicates the absorbance at B relative to the value at 0 T. (Reproduced from Ref. [9] by permission from Elsevier Science B. V.)

In Reaction P-2, the author’s group found peculiar MFEs in non-viscous organic solvents as shown in Fig. 7-7(a) [9]. Here, the A(t) curves observed at 330 nm in 2-propanol under various magnetic fields are displayed. Each of the curves is composed of a fast decay within 30 ns after excitation and the following slow decay. The former corresponds to the excite singlet andor triplet states of triphenylphosphine and the latter to the escaped diphenylphosphinyl radical. This figure shows that the yield of the escaped radical decreases with increasing B, but that the fast decay is too fast to obtain its MFE. Figure 7-7(b) shows the magnetic field dependence of R(B), which correspond to Y(B)I Y(0 T). It is noteworthy from this figure that the observed MFEs are very large (R(1.0-1.75 T)-0.75) even in 2-propanol.

108

Because R(B) decreases with increasing B, the observed MFEs can not be explained by the RM. Because the observed magnetically induced decrease has no lineu relationship with B’” as shown in Fig. 7-7(b), the observed MFEs can neither be explained by the AgM. Moreover, the present MFEs show a peculiar solvent dependence as follows: R(1.5 T) = 74 +. 3% in 2-propanol (q = 2.379 cP) (7-21a) > R( 1.5 T) = 64 f 3% in p-dioane (q = 1.322 cP) (7-21b) - R(1.5 T) = 65 +. 2%in cyclohexane (q = 0.9751 cP) (7-2 1C) < R(1.5 T) = 84 f 5% in n-hexane (q = 0.3216 cP). Such MFEs can be explained by the d-type TM, which can be applied to the case that the radical pair lifetime is much shorter than that of the usual case. MFEs due to the d-type TM will be explained in Chapter 10.

7.4 Theoretical Analysis of Relaxation Rates The spin Hamiltonian of a single electron (Hs)consists of two parts; the Zeeman energy in a steady external field (Ho), and a random perturbation term due to local fieds ( V ( t ) ) , (7-22a) Hs = H o + V( t), Ho= gj& B = g,UB&B = fiws,. (7-22b) Here, the external field is applied along the z-axis of the experimental coordinate. The matrix elements of V ( t )is represented as follows:

The off-diagonal elements depend on the x and y components of the local field, which contain many fluctuating components’oscillating at different frequencies. The parts which oscillate at the ESR frequency ( w ) induce transition between the a- and p-states. By making a Fourier analysis of V(t) and using time-dependent perturbation theory [22], the transition probability between the a- and p-states (Pap) is given by (7-24) Here, the integrand is an average of the off-diagonal matrix elements taken over all values of the time t and over a large number of samples of the molecular motion. The rotational correlation time is represented by z,, which is defined by 2,

4qr3 3kT

=-.

(7-25)

Here, r and 77 are the radius of the radical and the solvent viscosity, respectively.

Problem 7-2. Prove Eq. (7-24). Problem 7-3. Calculate the 2, value for a water molecule ( ~ 0 . 1 5 n m in ) water at 20°C. The nonradiative transitions between the a- and p-states is called the spin-lattice relaxation, the time ( T I )of which is given by (7-26)

Problem 7-4. Prove Eq. (7-26).

109

The line width of the ESR transition (T2) depends on the lifetimes of the a- and j3-states, but it is also affected by fluctuations in the energy difference between them. Thus, T2 is expressed by 1 1 1 -=-+-. (7-27) T2 2T T2 Here, T2' is defined as a time averaged product of the diagonal element of V:

-=IL-(vm-vBB)z 1 . Ti

(7-28)

A2

7'1 and Tz are also called the longitudinal and transverse relaxation times, respectively. At first, let us consider a free radical which has an anisotropic g-tensor (g>k)andtumbles rapidly. The anisotropic term can be represented by

V ( 0 = p d ( g b s,+g'q s,+g '22 SZ) = V,(G Now, the matrix elements of V are

s+,

V,(t) s,+ V Z ( G

Vas=(V,-iV,)/2, V@

sz.

(7-29) (7-30a)

=v,,

(7-30b)

so that TI andT2 can begiven by

(7-31a) (7-3 1b)

Problem 7-5. Prove Eqs. (7-30a) and (7-3Ob). As shown in Ref. [22], TI and T2 are expressed by the so-called inner product of the tensor with itself 2 (7-32) (g':g') = (gl- giscJ2 + (g2 - giso) + (g3 - giso) ' . Here, g, (i=1,2, and 3) is each principal value of the g-tensor and giSois the isotropic g-value (7-33) giso = (81 + g2 + 83)/3. If the g-tensor is axial symmetric (g, = g, and g, = g, = gL),(g':g') becomes /9.

(g':g') = 6(g, - gJ

(7-34)

Problem 7-6. Prove Eq. (7-34). The required averages are 2 (gUY = - - ( g ' : g ' ) , 15

(7-35a)

- -

1 g'). 10 From Eqs. (7-3 1)- (7-39, we can get 7'1 and T2 as follows: (g,)2

= (g,)2

= -(g':

-1_ - ( g ' : g ' ) p i B 2 TI

30A2

62, 1+w2r; '

(7-35b)

(7-36a)

110

(7-36b)

' the probabilities of the TeI-To and As shown by Eqs. (7-15) and (7-16), kR and k ~ are T,I- S transitions, respectively. Thus, they can be expressed by (7-37) Here, Tia and TI a are the longitudinal relaxation times of radicals a and b, reapectively, in a radical pair. Problem 7-7. Prove Eq. (7-37). From Eq. (7-37), we can obtain the following equation for the anisotropic g-tensors:

(7-38) Here, ga' and gb' are the anisotropic g-tensors of radicals a and b, respectively, and 29 and q, are the rotational correlation times of radicals a and b, respectively. Let us turn to the electron relaxation by anisotropinc g - and HFC-tensors. For simplicity, we will consider a radical with a single nuclear spin with I and m,. In this case, its spin Hamiltonian is expressed by

H o = gBS,B

+ AI,&

(7-39a)

sy+g',

v(t)= /LBB(g', Sx+ g,'

S,) + C I j A ' ,

s,

j,k

(7-39b) = Fx(t)sx+FJt) s+ , F,(t) s,. Here, A' is the anisotropic part of the HFC-tensor. Let us consider a transition between la,m,) and P,m,). Among Fj(t)0' = x, y. and z), the perturbation terms V;(t)) during this transition can be written as follows:

I

f,(t> =PBBg',+m,A',

(7-40a)

>

f y ( t )= PBBg'Zy+m,A',,

(7-40b) (7-40~)

f , ( t ) = PBBg'U+m,A', . Because the relaxation rate depends on the mean squared values of contributions to l/TI and l/Tz are proportional to

L, fy, and f i , their (7-41a) (7-41b) (7-41~)

In the above part, we considered the transition between la,m,) and IP,m,), but there are other contributions to l/T1 and 1/T2 from those between Ia,m,) and [ P,m, k 1). For

111

simplicity, we will consider the transition between Ia,l/2)and [/3,-1/2) for a nuclear spin with 1=1/2. The anisotropic HFC-tensor can be written as

The first term of the right side of Eq. (7-42) can be transformed to

1 . -1( ~ l l A ' j x - i ~ I k A ' b ) S=+ -[-zI+A',+~I~(A'_+A',)+ 2 1

2

k

Iz(A'u-iA'zy)]S+.

(7-43)

2

Thus, the matrix element of this transition can be given by

1 i Vap=(a,1/2) - ( ~ I , A ' , - i z I k A ' , ) S + l / ? - l / 2 ) = --A'?, 2 j k 2

(7-44a)

Similarly, the matrix element of the transition between )a,-1/2)and I/3,1/2) for a nuclear spin with k 1 / 2 can be given by 1

1

(7-44b)

V 'US =(a,-1/21 - ( ~ I , A ' , - i ~ I k A , ) S ' I ~ , 1 / 2=) -(A',+A,). 2 ,

We can assume that

kR

4

k

+ kR'

is the mean value for the transitions from IT+,,112) and

,-1/2), it is obtained as kR

+ kR'

=:

1 1 1 1 -[-{--(q =1/2)+-(rn, T," 2 2 T

+

- 3(g', : g ' , ) p i B 2 (AIa: A', ) -

30A2

1 1 =-1/2)}+-(-(m/ 2 qb

1 =1/2)+-(rn,

T2

=-1/2)]]

rO 1 +0 2 2 :

(7-45)

Problem 7-8. Prove Eq. (7-45). The spin relaxation of radical pairs also occurs through the dipole-dipole interaction (HD) of the component radicals (a and b). Its matrix element between IT+,)and ITo) is similar to that of two protons given by Eq. (66) in Chapter 11 of Ref. [22]

Here, 8 and 4 are the polar angles between radicals a and b and r is the distance between them as shown in Fig. 3-1 of Ref. [22]. Thus, the relaxation rate due to Ho is given by

(7-47a)

112

where G~is the rotational correlation time of the vector directing radical a to radical b. It is noteworthy that there is no matrix element between IT,,) and IS). This means that the following equation holds: kR’

(T+,IHDIS)1’ = 0.

(7-47b)

References [I] (a) Y. Sakaguchi, S. Nagakura, A. Minoh, and H. Hayashi, Chem. Phys. Lett., 82 (1981) 213. (b) Y. Sakaguchi, H. Hayashi, and S. Nagakura, J. Phys. Chem., 86 (1982) 3177. 121 (a) Y. Sakaguchi and H. Hayashi, Chem. Phys. Lett., 87 (1982) 539. (b) H. Hayashi and S. Nagakura. Bull. Chem. SOC.Jpn., 57 (1984) 322. [3] Y. Fujiwara, K. Yoda, T. Aoki, and Y. Tanimoto, Chem. Lett., 435 (1997). [4] Y. Fujiwara, Y. Taga, T. Tomonari, Y. Akimoto, T. Aoki, and Y. Tanimoto, Bull. Chem. SOC.Jpn., 74 (2001) 237. [5] H. Hayashi, Y. Sakaguchi, and K. Mochida, Chem. Lett., (1984) 79. [6] M. Wakasa, Y. Sakaguchi, and H. Hayashi, J. Am. Chem. SOC.,114 (1992) 871. [7] M. Wakasa, Y. Sakaguchi, J. Nakamura, and H. Hayashi, J. Phys. Chem., 96 (1992) 9651. [8] H. Hayashi, Y. Sakaguchi, M. Kamachi, and W. Schnabel, J. Phys. Chem., 91 (1987) 3936. [9] Y. Sakaguchi and H. Hayashi, Chem. Phys. Lett., 245 (1995) 591. [lo] H. Hayashi, Y. Sakaguchi, M. Tsunooka, H. Yanagi, and M. Tanaka, Chem. Phys. Lett., 136 (1987) 436. [ I l l C. Bohne, M. S. Alinajjar, D. Griller, and J. C. Scaiano, J. Am. Chem. SOC.,113 (1991) 1444. [12] M. Wakasa, Y. Sakaguchi, and H. Hayashi, Chem. Lett., (1994) 49. [13] M. Wakasa and H. Hayashi, J. Phys. Chem., 100 (1996) 15640. [14] M. Wakasa, H. Hayashi, Y. Mikami, and T. Takada, J. Phys. Chem., 99 (1995) 13181. [15] M. Wakasa, Y. Sakaguchi, and H. Hayashi, Chem. Phys. Lett., 215 (1993) 631. [16] M. Wakasa, Y. Sakaguchi, and H. Hayashi, Mol. Phys., 83 (1994) 613. [17] M. Wakasa, K. Nishizawa, H. Abe, G. Kido, and H. Hayashi, J. Am. Chem. SOC.,120 (1998) 10565. [18] M. Wakasa, K. Nishizawa, H. Abe, G. Kido, and H. Hayashi, J. Am. Chem. SOC.,121 (1999) 9191. [19] Y. Sakaguchi and H. Hayashi, Chem. Phys. Lett., 106 (1984) 420. [20] M. Wakasa, H. Hayashi, K. Ohara, and T. Takada, J. Am. Chem. SOC.,120 (1998) 3227 [21] M. Waksa and H. Hayashi, Chem. Phys. Lett., 229 (1994) 122. [22] A. Carrington and A. D. McLauchlan, “Introduction to Magnetic Resonance”, Harper & Row, New York, 1967. [23] H. Hayashi, J. Chinese Chem. SOC.,49 (2002) 137.

113

Solutions to the Problems 7-1. We should solve the following simultaneous differential equations: duldt = (kR + k R ' ) y - kG, (7-lob) dyldt = (kR + k R ' ) X - ks y. (7-13) According to textbooks of mathematics, the above equations have the following solutions:

x = Aexp(-p), y = Bexp(-pt). Thus, Eqs. (7-lob) and (7-13) are transferred as -pAexp(-pt) = ( k +~kR ') Bexp(-pt) - kFAexp(-pt),

-p Bexp(-p) = (kR + kR ') Aexp(-p) - ks Bexp(-pt). Elimination of exp(-p) from the above equations gives the following simultaneous equations: (pkF )A + (kR + kR ') B = 0, (kR+kR ' ) A + @ - k s ) B = 0 . The p value can be obtained by

p-k, k, +k,'

k,+k,' 1=0. p-k,

@- kF ) @- ks ) - ( k + ~ kR

')2

= 0.

p2-(k, +k,)p+k,k,

-(kR + k R ' ) *= O .

p=-((k, 1 2

+k,)2 -4(k,k,

+k,)+J(k,

-(kR +k,')'))

=-((k, 1 +k,)fJ(k, -k,)* +4(k, + k , ' ) 2 ) . 2 From Eqs. (7-1 1) and (7-14), the following relation can be obtained: k,

-

k, = k , I 2 >> k,

+ k,

I.

Thus, the p value can approximately be expressed as 1 p=-((k,+k,)f(k, -k,)) = k ~ o ks. r 2 Therefore, RP(t) (=2(x + y)) can approximately be represented by a linear combination of exp(-kF t ) and exp(-ks t ) as shown by Eq. (7-15). 7-2. See Appendix E of Ref. [22].

7-3. The

O - ~(= 1.00 cP), where value of water at at 20°C was measured to be ~ . O O X ~Pa.s [Pas] = [Nsm-'] = [kgm-'s-']. Thus, the 2, value is given by 4qr3 - 4 ~ ( 1 . 0 0 x kgm-'s-')(1.5 x 10-'0m)3 =3,5 x l o - l ~.s - ~ 3kT 3(1.38x JK-')(293K)

2, =-

7-4. The rate of change of population of the a-state ( N a) is given by the equation

dNa - N a p p , - Napaa= P ( N a - N , ) . dt

--

114

When a new variable, the population difference ( n = N p - N , ) is introduced, N

a

and N p are

represented by

N

a=

( N + n)/2,

Np = ( N - n)/2,

where N = N , + N p. Thus, the rate equation is written by

This equation gives the result that 1/T1 = 2P.

7-5. In order to prove Eq. (7-30a), we can use the following equation: vx(t)Sx+V y ( t ) S y =V,(S+ +S-)/2-iVY(S+- S - ) 1 _ - (v,- i v y > ~+-(v, + +ivy)S-.

2 2 Thus, we can get the equation

Vap= (V, -iVy)/2. In order to prove &. (7-30b),we can use the following equation:

v,, - vpp=v(2- (-v,/2) = v,. 7-6. In order to prove Eq. (7-34), we can use the following equations: giso

= (gl + g2 +

gI-giso=

= ( g / / +2g,)/3

1

2(g//-g,)/3,

g2 - giso = g3 - giso = - ( g / / - g1)/3,

Thus, (g':g') is given by (g':g') = (4/9+1/9+1/9) (g" - g , ) * = 6(g,

- 8,)'

19.

7-7. Because kR and k ~ 'are the probabilities of the T,I- To and T+1- S transitions, respectively, the T+1- To and T+1-S transition probabilities (P+oand P+s)are obtained from Eq. (7-24) as follows: P+o= P+s 1 -

= - I(T+II [ V 1 " * ( t + z ) + V ~+z)]lToorS)(T+, (t ~[Vl"(t)+V~(t)]~ToorS)exp(im)dz. A' __

Here, vj" represents the random perturbation term of thej-the electron (j = 1 or 2) in the k-the radical ( k is a or b). The T+1,To, and S states are represented by

IT+l)= Imm) ?

Thus, P+oand P+sare given by

115

1 +J(a(2) 2A2 __

I v ~(t + 2>1~(2))(a(2)IvZ"(ill ~ ( 2 ) ) e x p ( i r n)dz2 , -

1 2A

7-8. From Eqs. (7-41) to (7-44), the following equation can be obtained for a radical: 1 1 1 -[-(mi = 1 / 2 ) + - ( m i = - 1 / 2 ) ] 4 T T, - --I [ L ( p j B 2 ( g 'g: ' ) + 2 -1p u , B ( g ' : A ' ) + 1- ( A ' :A ' ) ) + -1- ( 1 A':A')]-

42A2 10

+ -1- [ -1{ p j 2B 2 ( g ' : 42A2 10

2

4

g ' ) - 2 - p1 U , B ( g ' :A ) + - (1 A ' : A ' ) ) + - -1( A2' : 2 4 16 15

+

= 1 - [ 3 , ~ @ ~ ( g g' :' ) ( A : A ' ) ) ] - 22,

30A

4 10

1 + w22: .

22, 1 + w%,2

A')]- 22, 1 + w2z;

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117

8. Magnetic Field Effects on Chemical Reactions through Biradicals

8.1 Historical Introduction CIDNP, CIDEP, and MFEs on chemical reactions through radical pairs have been reviewed in the former chapters, but there is another group of reactions which can give similar phenomena. Because biradicals (or diradicals) have two radical centers in one molecule, the reactions through biradicals are also expected to be influenced by external magnetic fields as those through radical pairs. It is noteworthy that biradicals differ from radical pairs in the absence of diffusive separation of the component radicals. From this point, the reactions through biradicals have the following characteristics: (1) Because the lifetime (rBR) of biradicals are much larger than that ( r ~ pof) radical pairs in fluid solutions, MFEs of biradicals are expected to be much larger than MFEs of radical pairs in fluid solutions and similar to MFEs of radical pairs in micellar solutions. (2) Because the distance between two radicals in a biradical can be changed artificially, different MFEs can be expected in the reactions of biradicals than those of radical pairs. In 1973, Closs and Doubleday studied the photochemical a-cleavege of alicyclic ketones in chloroform at room temperture in the presence of magnetic fields below 2.3 T [la]. Their reactions are illustrated in Fig. 8-1.

Fig. 8-1. Reaction scheme of the photochemical a-cleavege of alicyclic ketones (Ia-e and IV). (Reproduced from Ref. [la] by permission from The American Chemical Society) Irradiation of ketones Ia-e and IV yieded the corresponding alkenals. Each irradiation was carried out for 50 sec at the desired field (B) and the sample was transferred quickly to the probe of a 100 MHz NMR spectrometer. For each ketone, the relative integrated intensity of aldehyde CIDNP signal is plotted as a function of B. The obtained results of such measurements are shown in Fig. 8-2.

118

B/100mT Fig. 8-2. Intensity of aldehyde proton emission signals of ma-e (C7- C11) and of aldehyde derived from IV,as functions of B. The intensities are in arbitrary units and not normalized among the different compounds. (Reproduced from Ref. [la] by permission from The American Chemical Society) The curves in Fig. 8-2 show the following striking features: (1) The CIDNP intensity of each compound except IV increases with increasing B from 0 mT to a field (Bmax),where the intensity shows the maximum value. (2) The CIDNP intensity decreases with increasing B value from B,, to higher fields. (3) As the length of the biradical increases, the B,,, decreases. The distance dependence of the B,,, value is listed in Table 1. (4) As the length of the biradical increases, the curve width decreases drastically. (5) For IV, the B,,, value could not be determined due to limitation of experimentally available magnetic fields. Features (1)-(3) can be explained when we consider a fictitious, totally rigid biradical with a S-T splitting of U. In this model, MFEs due to the LCM are possible to occur, where the magnetic field dependence of reaction products is shown in Fig. 6-2(d). The B,, value corresponds to the BE value given by Eq. (6-5). &ax = BLC= ~ M / ~ P B . (8-1) Because the IJ1 value of a biradical decreases as the length of the biradical increases, Feature (3) can be explained by the LCM. Feature (4) can be explained by the distribution of the IJ1 value because a real biradical is constantly changing its conformation over its lifetime. Feature (5) can be explained by the fact that its BE value is too large to be determined from this study. This is the first determination of the average M values in highly reactive biradicals. Although the MFEs on the CIDNP intensities were only measured in this study, it became also very hopeful that the MFEs on reaction rates and yields could be detected for the reactions through biradicals. In 1979, Doubleday examined the reactions of similar biradicals with a superconducting magnet, but he found no B,,, value for a biradical with n=4.

119

Problem 8-1. Obtain the S-T energy splitting when BE = 1 T

"N is the number of methylene groups which separate two radicals in a biradical. 8.2 MFEs on Thermal Reactions through Biradicals In 1979 Turro et al. studied MFEs on the thermolysis of endoperoxdes of aromatic compounds [2], where its reaction scheme is illustrated in Fig. 8-3. This figure shows that an endoperoxide decomposes predominantly either by the biradical pathway (a, b, c, and d in Fig. 8-3) or by the concerted pathway (e and f i n Fig. 8-3). In general, the yield of singlet oxygen through the concerted pathway is larger than that through the biradical one. The MFE can be expected for the biradical one but not for the concerted one, because the S-T splitting for the former one is much smaller than that of the latter one.

t

1.

'BR I

'1 r

f

3BR

Id

Fig. 8-3. Reaction scheme for the thermolysis of endoperoxdes of aromatic compounds through a biradical pathway (a, b, c, and d ) or a concerted pathway (0. Here, 'BR and 3BR represent the singlet and triplet biradicals, respectively. (Reproduced from Ref. [2b] by permission from The American Chemical Society)

120

Turro et al. studied the thermolysis of 9,lO-diphenylanthracene 9,lO-endoperoxide (l),

1,4-dimethyl-9,1O-diphenylanthracene1,4-endoperoxide (2), 1,4-dimethylnaphthalene 1,4endoperoxide (3), and 1,4&trimethylnaphthalene 1,4-endoperoxide (4). The yield of the generated singlet oxygen ( ' 0 2 ) was measured with singlet molecular oxygen acceptors such as tetracyclone and 9,lO-dimethylanthracene. The singlet oxygen yields observed in 1,4dioxane under various magnetic fields are listed in Table 8-2. Table 8-2. Singlet oxygen yields (%) observed under various magnetic fields (B). Here, the measurements in the earth's magnetic field are represented in the column at B = 5 ~ 1 0 - ~ T .

The data in Table 8-2 show that the thermolysis of 1 had the largest magnetic field dependence, but the thermolysis of 2 had no appreciable MFE. The singlet oxygen yield for 1 was found to decrease with increasing B from the earth's magnetic field to 1.30 T, the magnetically induced change being proportional to B'". Thus, the observed MFEs in this study can be explained by the AgM, where the S-T conversion (process b of Fig. 8-3) is enhanced by an external magnetic field. It is noteworthy that the &values of the present biradicals estimated from the g-values of the benzyl ( ~ 2 . 0 0 2 and ) peroxyl ( ~ 2 . 0 1radicals ) should be much larger than those of usual biradicals and radical pairs involving only Cradicals. From the obtained activation entropy (AS' =: +10 eu), 1 was proposed to undergo decomposition predominantly via a biradical. Thus, the largest MFE was observed for the thermolysis of 1. On the other hand, 2 was proposed to undergo decomposition predominantly via a concerted fragmentation from the obtained activation entropy (AS' =: -2 eu). Thus, the external magnetic field should have no significant effect on the singlet oxygen yield from 2. Because 3 and 4 were proposed to represent intermediate cases from their activation entropies (AS'= +2 eu for 3 and AS' FZ +3 eu for 4), the MFEs observed for 3 and 4 were also intermediate cases between 1 and 2. Turro et al. also found MIEs in the present reactions, but their MIEs will be reviewed in Chapter 9.

8.3 MFEs on Photochemical Reactions through Biradicals In 1984, Weller's group reported MFEs on the primary photochemical processes of polymethylene-linked compounds containing electron acceptor and donor groups (A and D) with a nanosecond-laser photolysis technique in the presence of magnetic fields below 0.3 T [3a]. Their reaction scheme is represented as follows: 'A*-(CH&D, A-(CH&D + hu -+ 'A*-(CH&-D '[A-'-(CH&D+'], '[A-'-(CH2),-D"'] tf 3[A-'(CH&D''],

(8-2a) (8-2b)

3[A-'-CH21n-D+']

(8-2d) (8-2e)

-+

'[A-'-(CH&,-D'']

-+ -+

3A*-(CH2)n-D, A-(CH&D.

(8-2~)

121

This scheme is quite similar to that given for radical pairs by reactions (6-21a)-(6-21f), but there is no escape of component radicals in the case of biradicals. For each polymethylenelinked compound with A=pyrene and D=dimethylaniline in acetonitrile at room temperature, the group measured the intensity (ET(B))of the transition absorption due to 3A*. Their typical results are shown in Fig. 8-4.

+ 0'-

260 L60 600 magnetic-fieid strength, BIG Fig. 8-4. Plot of relative methylpyrene triplet extinctions (I?@)/ ~-

~

field strength for the linked compound of A-(CH&-D. permission from The Royal Society of Chemistry)

800

&(OT)) against magnetic (Reproduced from Ref. [3a] by

As shown in Fig. 8-4, Weller's group found the following MFEs: (1) For each of the compounds with 7 5 n 5 11, the intensity (ET(B)) of the transition absorption due to 3A* increased with increasing B from 0 T, attained a maximum value at the maximum field ( 2 ) From these results, the (B,,,), and started to decrease with increasing B from B,,,. observed MFEs should be due to the LCM and the observed B,,, values correspond to the Bw ones. The observed B,,, values are listed in Table 8-1. (3) The observed B,,, values (4) There was no MFE were found to decrease with increasing n as shown in Table 8-1. for each of the compounds with n 5 6. This is due to fact that its BE value is much larger than 0.3 T. (5) For each of the reactions of biradical with n 3 12, the intensity of the transition absorption due to 3A* was found to only decrease with increasing B from 0 mT. This MFE was similar to that observed for the corresponding radical pair and could be explained by the HFCM. Weller's group also measured the MFEs on the exciplex fluorescence intensity (I@)) observed in acetonitril at room temperature for the same polymethylene-linked compounds with A=pyrene and D=dimethylaniline [3b]. The exciplex fluorescence (hu ), is emitted from the following processes: '[A-'-(CH2),-D"] '[A--(CH&Df]ex

++ +

'[A--(CH2),-D'],,, A-(CH2),-D

+ hu ex.

(8-20 (8-2g)

122

Here, '[A--(CH;?),-D+],, represents the singlet exciplex which is in equilibrium with the singlet biradical as shown by process (8-20.

1.5 1.4

16 14 12

2

1.3

11

53

1.2

10

k! 4

1.1

9 1.0

8 n

0.9-, 0

800 1000 Magnetic Field Strength 8I Gauss

200

400

600

Fig. 8-5. Magnetic field dependence of the relative exciplex fluorescence intensity (Z(B)/Z(OT)) observed in acetonitril at room temperature for the polymethylene-linked compounds with A=pyrene and D=dimethylaniline. (Reproduced from Ref. [3b] by permission from Elsevier Science B. V.) Their results are shown in Fig. 8-5, from which the following MFEs could be obtained: (1) For each of the compounds with 8 5 n 5 11, the intensity (Z(B)/Z(OT)) decreased with increasing B from 0 T, attained a minimum value at the minimum field (Bmin),and started to increase with increasing B from B,,,,,,. ( 2 ) Because the singlet exciplex is in equilibrium with the singlet biradical as shown by process (8-20, the observed MFEs should be due to the LCM and the observed &,, values correspond to the BE ones. These Bh,, values are very similar to the B, ones obtained from the intensity (ET(B))of the transition absorption due to 3A* as listed in Table 8-1. (3) For each of the compounds with n 2 12, the intensity (Z(B)/Z(OT)) was found to only increase with increasing B from 0 mT. This MFE was similar to that observed for the intensity (I?@)) of the transition absorption due to 3A* and could be explained by the HFCM. Tanimoto et al. also observed similar MFEs on the exciplex fluorescence intensity for the polymethylene-linked compounds with A=phenanthrene and D=dimethylaniline [4]. Their BE values are also listed in Table 8-1.

123

SCHEME I

3

3

2n+1

3,

+ diphenylcycloalkanes + disproportionation SCHEME II

4n

+ disproportionation

Fig. 8-6. The schemes of the reactions which proceed through biradicals. from Ref. [6] by permission from The American Chemical Society)

(Reproduced

With a ns-laser photolysis apparatus, Turro et al. measured the MFEs on the biradical generated by the reactions shown in Fig. 8-6 [5, 61. Their lifetimes ( z B R ( B ) = I/~BR(B)) typical results observed for the biradicals generated through Scheme I of Fig. 8-6 in methanol ) of each at 296 K are also shown in Fig. 8-7 [6]. Fig. 8-7 shows that the ~ B R ( Bvalue biradical increases with increasing B from 0 T to B,,, and that the value decreases increases with increasing B from B,,, to 0.2T, which was the maximum field of their studies. Each B,, value corresponds to its BLCone, which is listed in Table 8-1. It was also found that the BE value decreased with increasing N. It is noteworthy that the observed MFEs on k B R ( B ) T ) of the biradical with n=14 in Fig. were very large. For example, the ~BR(B,,,) l k ~ ~ ( 0 value

124

8-7 was found to be 1.35 and its k ~ ~ ( 0 . 2 Tl k) ~ ~ ( 0 T value ) to be as small as 0.1. The magnetically induced increase in ~ B R ( Bis) due to the LCM and the magnetically induced decrease in k B R ( B ) should be due to the RM. As shown in Fig. 8-7, the MFEs of the biradicals with n = 6 and 5 were too small to be detected at the fields of 0 - 0.2 T. This is due to the fact that their BE values should be much larger than 0.2 T.

1.5

1 .o

0.5

0

500

1000

1500

2000

magnetic field (Gauss) Fig. 8-7. Magnetic field dependence of the decay rate (kBR(B)/kBR(OT) = TBR(OT)/ZBR(B)) of the biradicals generated through Scheme I of Fig. 8-6. (Reproduced from Ref. [6] by permission from The American Chemical Society) Tanimoto et al. found MFEs on the biradical lifetimes (zBR(B))in the intramolecular hydrogen abstraction of ~-(xanthone-2-carbonyloxy)-o(xanthene-2-carbonyloxy)alk~es (XO-n-XH, n = 2 - 12), whose molecular structure for n=12 is illustrated in Fig. 8-8(a). [7] They obtained the magnetic field dependence of ZBR(B)for the reaction of XO-12-XH in acetonitrile at room temperature as shown in Fig. 8-8(b). This figure shows that the z ~ R ( B ) value increases steeply with increasing B from 0 T to 0.05 T and that it increases more gradually with increasing B from 0.05 T to 0.8 T without saturation at 0.8 T. This magnetic field dependence of ZBR(B)can be explained by the RM. They also measured the similar dependence for the reaction of each of the compounds with n = 2 - 12. Their results about the n-dependence of the zBR(0.8T)/ZBR(OT)value are shown in Fig. 8-8(c). This figure shows that the zBR(0.8T)/ZBR(OT)value increases with increasing n. Nakanura et al. also found similar MFEs on the biradical lifetimes in the electron transfer reactions in porphyrinviologen linked systems (ZnPCnV, n = 4,6, and 8) in aqueous acetonitrile at room temperature [8]. Their MFEs are due to the rates of reverse electron transfer in the photogenerated intramolecular ion radical pairs and can also be explained by the RM.

125

in CH3C.N

12

14

1

I

?I

i

I6

18

I

I

XO-n-XH 20-

10

I

0' 0

0.2

1

I

04

0.6

BIT

O

;

I

k

i

l

:

l

;

0.8

It

Fig. 8-8. (a) Molecular structure of XO-lZXH, (b) magnetic field dependence of the biradical lifetime (zjjR(B)) observed for the reaction of XO-12-XH, and (c) n-dependence of the zjBR(0.8T)/ZBR(OT)value observed for the reaction of XO-n-XH. (Reproduced from Ref. [7] by permission from Elsevier Science B. V.)

References [l] (a) G. L. Closs and C. E. Doubleday Jr., J. Am. Chem. SOC.,95(1973) 2735. (b) C. E. Doubleday Jr., Chem. Phys. Lett, 64(1979) 67. [2] (a) N. J. Torro and M. -F. Chow, J. Am. Chem. SOC.,lOl(1979) 3701. (b) )N. J. Torro, M. -F. Chow, and J. Rigaudy J. Am. Chem. SOC.,103(1981) 7218. [3] (a) A. Weller, H. Staerk, and R. Treichel, Faraday Discuss. Chem. Soc.,78(1984) 271. (b) H. Staerk, W. Kiihnle, R. Treichel, and A. Weller, Chem. Phys. Lett., 118(1985) 19. [4] Y. Tanimoto, N. Okada, M. Itoh, K. Iwai, K. Sugioka, F. Takemura, R. Nakagaki, and S. Nagakura, Chem. Phys. Lett., 136(1987)42. [5]M. Z. Zimmt, C . Doubleday, Jr., and N. J. Turro, J. Am. Chem. SOC.,107(1985) 6726. [6] J. Wang, C. Doubleday, Jr., and N. J. Turro, J. Phys. Chem., 93(1989) 4780. [7] Y. Tanimoto, M. Takashima,K.Hasegawa, and M. Itoh, Chem. Phys. Lett., 137(1987) 330. [8] H. Nakamura, A. Uehata,A. Motonaga, T.Ogata, and T. Matsuo, Chem. Lett., (1987) 543.

126

Solution to the Problems 8-1. Because the S-T energy splitting corresponds to 21.4, the following relation can be obtained from Eq. (8-1): 2M = Bm g&= 1T x 2.0023 x 9.274~10.'~ JT-' = 1.857 x ~ O - ' J~ = 1.857 x m 5 ~ 1 0 J" ~ = 1.857 x I O - ~x 50341 cm-' = 0.9348 cm-I. Here, g is assumed to be 2.0023.

127

9. Magnetic Isotope Effects

9.1 Introduction In Section 6.7, the discovery of the magnetic isotope effect (ME) of 13C was introduced. In this Chapter, a more detailed review of ME% will be given. Because the chemical isotope effects (CEs) of hydrogen is much larger than those of heavier elements, their MIEs will be dealt with in this Chapter. Let us consider the following reactions from a starting molecule (A) through a radical pair (RP):

A RP

+

RP

+

N

Formation of a RP with a rate constant of kl, Formation of escaped products from RP with a rate constant of

(9-1) k2,

(9-2)

Recombination of a RP to A with a rate constant of k.1.

RP -+ A (9-3) Here, N represents escaped products. We can see from the radical pair theory that reactions (9-1) and (9-2) are independent on nuclear spins inside the RP, but that reaction (9-3) is dependent on the nuclear spins. If a starting molecule contains a magnetic isotope such as I3C, it will be represented as A'. On the other hand, if A contains no such magnetic isotope, it will be represented by A. In this case, the isotope ratio ( 6) is given by

6 = [A#]/[A]. (9-4) When 6 0and 6 are the ratios measured before and after an enrichment reaction, respectively, the isotope enrichment (S)in the starting molecule is represented as follows:

s = 8/60. The rates of the decreases of (A] and [A 1' can be represented by -d[A]/dt = kl[A](I-P),

(9-5) (9-6)

(9-7) -d[A#]/dt = kl[A#](l-P#). Here, P and P are the probabilities of regeneration of A and A ', respectively, through reaction (9-3). Thus, the one-step enrichment coefficient ( a ) is defined as follows:

'

a = (1- P)/(l- P # ) .

(9-8)

If ' P is larger than P, a becomes larger than 1. Since S is given by

S can be related with the chemical conversion (F)of A as follows: F = 1- [A]/[A]o, logs= (1 - I/a)[- log(1-F)]. Problem 9-1. Prove 3. (9-11).

(9-10) (9-11)

Eq. (9-11) means that the avalue can be obtained experimentally from the observed linear relation between logs and [- log(1-F)]. 9.2 MIEs of 13C/'2C As shown in Section 6.7, Buchachenko et al. found a MIF of I3C (1.10%, 1=1/2) in the photodecomposition reaction of dibenzyl ketone (DBK). The reaction scheme of DBK is shown in Fig. 9-1.

128

@H2-C

II

- C H 2 G

0

(DBK)

1

h'

Recombination

@H2-C-CH20

-

(lDBK*)

II

0

1

ISC

3[ @ 3 3 2 - f - C H ~ G 0

(3DBK*) 3[ O C H 2 - f t

(3w 0

1 b

1

2

a

l2CO

2x ( 3 - - C H 2 t + co

1

-

Escaped Products

Fig. 9-1. Reaction scheme of DBK.

As shown in Fig. 9-1, the photodecomposition of DBK occurs from its triplet excited state, which gives a triplet radical pair (3RP) involving the phenacetyl and benzyl radicals. Among the HF coupling constants of all nuclear spins inside these radicals, that of I3C in the carbonyl group of the phenacetyl radical (A/gp~=12.5mT)is much larger than others. Thus, 3RP involving I3CO can be converted its singlet state ('RP) much more quickly than 3RP involving I2CO. The cage recombination occurs through 'RP, but not through 3RP. Thus, 13C0can be enriched on the starting DBK as its decomposition proceeds. On the other hand, l2C0 can be enriched on the escaped products such as diphenylethane. Fig. 9-2 shows the observed linear relation between logs and [- log(1-F)]. From the slope of this line, the vale of 1 - l / a can be obtained. This figure shows that the a value observed in a micellar solution at room temperature in the absence of an external magnetic field (B = 0 T) is much larger than that observed in benzene, but the a value observed in a micellar solution at room temperature in the presence of an external magnetic field (B = 1.5 T) is smaller than that in the absence of the field. Fig. 9-3 shows the magnetic field dependence of the a value. As clearly shown in this figure, the a value was found to increase with increasing B from 0 T to 30 mT and to decrease with increasing B from 30 mT to 1.4 T.

129 A

-Earth’s magnetic field - _ _15.000 _ Gauss

Fig. 9-2. The observed linear relation between logs and [- log(1F)]in the photodecomposition of DBK at room temperature in benzene and micellar HDTCL solutions in the absence and presence of an external magnetic field of 1.5 T . Here, HDCL indicates hexadecyltrimethyl-

0.3 -

0.05m HDTCl

ammonium chloride. (Reproduced from Ref. [I] by permission from The American Chemical Society)

a = 1.47 i O . 0 2

-

0.2 v)

-

0 0

/

o

1I ’ /’

Fig. 9-3. The observed magnetic field dependence of the a value for the photodecomposition of DBK in a HDTCL micellar solution at room temperature. It is possible to measure the a value by two different methods: ( 0 ) mass spectral analysis and ( x ) quantum yield measurements. (Reproduced from Ref. [ 2 ] by permission from The American Chemical Society)

/-

./-

a = 1.027t0.001 -

,//BENZENE

,

0

/

-

c

I

*

1

0

-109 ( 1 - 1 ) 1.53 Z 0.04

1.1

-

1.0 ’

1.Ki 0.04

I

I

I

I

1

I

I

,

130

Table 9-1. 13C one-step enrichment coefficient (u) observed for DBK in the absence and presence of an external magnetic field (B)

Typical a values observed for DBK in the absence and presence of an external magnetic field ( B ) are listed in Table 9-1. As clearly seen in this table, the u value observed in the HDTCL micelle at B=O T is much larger than those observed in organic solvents at B=O T. This increase is due to the fact that the lifetime of the generated radical pair becomes much longer in such a confined system as the micellar solution than those in usual organic solvents. On the other hand, it is not clear whether the observed isotope effects for DBK in organic solvents are due to MIE or CIE. In order to clarify this point, we should measure the magnetic field effect (MFE) of the M E . Indeed, Turro et al. measured the MFE on the a value for DBK in the micellar solution and obtained the results shown in Fig. 9-3 and Table 9-1. This clear MFE is a strong evidence for the fact that the I3C enrichment observed for DBK in the micellar solution is due to the M E . The MFE of the M E observed for DBK in the micellar solution can be explained by the HFCM as follows: (1) At B < BI/z(RP), the T-S conversion of 'RP to 'RP occurs from all of three triplet states (To and T,'). (2) At B - Bl/z(RP),the T-S conversion from the T,' states starts to be suppressed by the Zeeman splitting of the T+1 states. (3) At B > BIR(RP),the T-S conversion of 3RP to 'RP occurs only from TO,but the conversion from T ,1 is completely suppressed by the Zeeman splitting. On the other hand, the BIR(RPwith I3CO) and Bl/z(RP with "CO) values can be obtained to be 17.9 mT and 9.0 mT, respectively.

Problem 9-2. Prove that B I / ~ ( R P with I3CO) is 17.9 mT and that B I I ~ ( R with P "CO) is 9.0 mT,. From the B1/2(RP with I3CO) and BI/z(RP with "CO) values, the magnetic field dependence of the M E observed for DBK in the micellar solution can be explained as follows: (1) At B < BIR(RPwith I2CO), the T-S conversion of 3RP with I3CO and with I2CO to 'RP occurs from all of three triplet states (To and T,'). But, the T-S conversion rate of 3RP with I3CO to 'RP is much larger than that of 3RP with "CO. Thus, there is a M E of I3C at this range. (2) At B - B1/2(RP with "CO), the T-S conversion of 3RP with I3CO to 'RP occurs from all of three triplet states (TOand T,'). But, the T-S conversion of 'RP with I2CO to 'RP from the T+1 states starts to be suppressed by the Zeeman splitting of the T,1 states. Thus, the M E of I3C at this range becomes larger than that at lower fields. (3) At B BIl2(RP with I3CO), the T-S conversion of 3RP with "CO to 'RP from the T,, states starts to be suppressed by the Zeeman splitting of the T+1 states. But, the T-S conversion of 3RP with "CO to 'RP from the T,1 states is much more suppressed by the Zeeman splitting of the Trl states. Thus, the M E of I3C at this range may be similar to that at B - B I I ~ ( R P with 12 CO). (4) At B > B1/2(RP with "CO), the T-S conversion of 3RP with I3CO and with "CO to 'RP occurs only from TO,but the conversion from T,l is completely suppressed by the Zeeman splitting. Because the T-S conversion occurs only from TO,the M E of I3C at this

-

131

range should be smaller than that at B=O T. ( 5 ) The gradual decrease in the a value at much higher fields is due to the AgM.

9.3 MIEs of "N/14N Although "N (0.366%, Z=1/2) has a different nuclear spin compared with 14N(99.634%, Z=l), there has been no report on the M E of I5N for any chemical reaction. Concerning the M E of lsN/I4N, the magnetic field dependence of the delayed fluorescence and of the injection current was investigated for an anthrecene crystal with dye molecules (rhodamine B) absorbed on the surface. The details of the M E of 15N/14Nwas reported in Ref. [4].

9.4 MIESof l 7 o P 0 , 1 8 0 It is very difficult to enrich I7O (0.038%, Z=5/2) from I6O (99.762%) and I 8 0 (0.200%) with the CIE, because the mass of 170locates between I 6 0 and I8O. On the other hand, it is very hopeful that I7Ocan be enriched with the M E , because l60and I8O have no nuclear spin. Turro and Chow found the M E of I7O for the thermolysis of 9,lO-dipenylanthracene 9,lOendoperoxide (DPA-02) in CHC13, dioxane, and benzene [5]. The reaction scheme of DPA0 2 is shown in Fig. 8-3, where a singlet biradical ('BR) is initially generated through reaction (a). From 'BR, singlet oxygen is produced through reaction (c). In competition with reaction (c), S-T conversion (reaction (b)) occurs, generating a triplet biradical (3BR). From 3BR, triplet oxygen is produced through reaction (d). This S-T conversion is expected to be enhanced by an external magnetic field and the HFC of I7O. T w o ' s group synthesized DPA-02 enriched in I7O (3% I6O, 37% I7O, and 60% I8O) and DPA-02 enriched in '*O (4% I604% , I7O, and 92% I8O) and found the following results [5]: (1) The yield of singlet oxygen from DPA-I702 (0.34&.01 in CHC13) was smaller than those from DPA-l6O2 and DPA-I802 (0.37H.01 in CHC13) in the absence of an external magnetic field. (2) The singlet oxgen yield from DPA-1702was decreased by an external field of 1.OT (0.31&.01 in CHCl3). (3) The singlet oxgen yields from DPA-I602 and DPA-"02 were decreased by external fields of - 1 T (0.32M.02 and 0.31M.01 for the former and latter, respectively, in CHCl3) (4) Similar MFEs and M E s were observed in dioxane. ( 5 ) On the other hand, no appreciable MFE and MJE could be observed in benzene. These MFFis and MIEs observed in CHCl3 and dioxane can be explained by the fact that the S-T conversion of the biradical is enhanced by the HFC of I7O in the absence of an external field and the fact that the S-T conversion is also enhanced by a magnetic field of - 1 T through the AgM as explained in Chapter 8. It is noteworthy that the M E observed in the absence of an external field was decreased by a magnetic field of 1 T. No MFE and M E in benzene can be explained by the fact that the S-T energy splitting of this biradical in benzene becomes much larger than that in CHC13 and dioxane. Turro et al. also found "0 isotope enrichment in the thermolysis of DPA-02. They measured 1 7 0 composition of nontrappable 0 2 , which is maily due to triplet oxygen generated from 3BR. When DPA-02 containing 36.9% 1 7 0 was thermolyzed in degassed CHC13 at 90 "C in the absence of an external field, the composition of I7Oin the nontrappable oxygen became 0.380&.005. But, the composition at 1.0 T was not changed (0.368a.002)from the starting molecule. These results showed that I7O could be enriched in triplet oxygen in the absence of the field by the enhancement of the S-T conversion of the biradical though the HFC of I7O, but that the M E of I7O was quenched by the external magnetic field through the AgM. The observed MIEs for DPA-02 were so small that no a value could be obtained for them.

-

132

Buchachenko et at. found the M E of I7O in the thermal oxidation of powders or thin films of polymers [6]. Here, the chain oxidation of organic compounds occurs via repetitive sequence of two kinetic chain propagation reactions: R' + 0 2 + RO;, (9-12a) R 0; + RH + ROOHR + R.. (9- 12b) Here, RH is the organic substrate, R. is the alkyl radical which is produced by 6oCoirradiation, and ROz. is the peroxy radical. The chain termination reaction includes the recombination of peroxy radicals which is expected to be spin selective and , therefore, an isotope-sorting reaction: (9-12~) R 0; + R 02' * 133[R02' ' 0 2 R], (9-12d) '[RO; 'OzR] tf '[RO; '02R1, '[RO; '02R] + ROAR + 02+products. (9- 12e) In the encounter RP,the ratio of singlet and triplet spin state populations is 1:3 as shown by reaction (9-12c). The MIE arises almost completely from the T-S conversion of tripler RP. Thus, the recombination probability of peroxy radicals with terminal 170atoms is higher than that of radicals with terminal l60or l80atoms. In the case of polypropylene, the a value of 1.060kO.005 was obtained for I7O, but 1 . 0 1 5 ~ . 0 1 0for '*O. This result certified that a(MIE) >> a(CE) for reaction (9-12).

9.5 MIESof 2 9 ~ i / 2 83~0i ,~ i Many MFEs and MIEs have been observed in reactions of light atom-centered radicals such as C- and 0-centered ones. On the other hand, MFEs and MIEs have been believed to decrease drastically with increasing atomic number of the nucleus where one of the unpaired electrons in a RP is mainly localized. This is due to the magnetic-insensitive spin-orbit (SO) interaction of heavy atoms, which enhances the spin conversion of RPs. As shown in Chapter 7, however, many MFEs have been observed in the reactions of heavy atom-centered radicals such as Si-, S-, Ge-, and Sn-radicals if some suitable reaction systems are designed for such radicals. Using such reaction system which show MFEs, we will be able to find MIEs of such heavy isotopes. The 29Si(4.67%, 1=1/2) MIE was found by Step et al. in the photolysis of silyl-containing ketone (PhCHz-COSi(CH3)zPh) sensitised by triplet acetopenone (3ACP*) in SDS micelles

"71. ACP + hv + 'ACP* + 3ACP* , (9- 13a) 3ACP* + PhCHz-COSi(CH3)zPh + ACP + 3PhCH2-COSi(CH3)2Ph*, (9- 13b) 3PhCH2-COSi(CH3)2Ph* '[RC'Hz 'COSi(CH3)2Ph], (9-13~) 3[RC'H2 'COSi(CH3)zPhI tf '[RC'Hz 'COSi(CH3)2Ph], (9-13d) '[RC'H2 'COSi(CH3)2Ph] + PhCH2-COSi(CH&Ph. (9-13e) In this reaction, Ph(CH&SiOC' is not an Si-centered radical, but its odd electron is somewhat de-localized on Si. Although the HF coupling constants (Algpe) of Ph(CH3)zSiOC' have not yet reported, typical acyl o-radicals have values of 12-13 mT for the I3CO-constantand lower values for the 29Si-constant. In this reaction, Step et al. obtained the a value of 1.086 for I3C, but the value of only 1.023 for 29Si. The author's group tried to find the "Si M E in the direct photolysis of methyluiphenylsilane (Ph3CH3Si) in Brij 35 micelles at room temperature [8]. ---f

133 Ph3CH3Si + hv + 'Ph3CH3Si* + 3Ph3CH3Si*,

(9-14a)

[PhzCH3Si' 'Ph],

(9-14b)

3Ph3CH3Si*+

3[Ph2CH3Si' 'Ph]

c-f

'[Ph2CH-,Si'

'[PhzCH3Si' 'Ph]

+

Ph3CH3Si,

'Ph],

(9-14~) (9-14d)

In this reaction, the diphenylmethylsilyl radical is produced. Although its 29SiHF coupling constant has not yet been measured, it may be similar to that of the triphenylsilyl radical (A/g,uB = 7.96 mT). Typical a values for the 29Si-enrichmentobserved for this reaction in the absence and presence of external magnetic fields are listed in Table 9-2. Table 9-2. Typical one-step enrichment coefficients (a) observed for the MJEs in the absence and presence of an external magnetic field ( B ) IsotoDe

1 Radical Pair

Solvent

a(B)

Ref.

l 3 C (1=1/2)

3[PhCH2CO''CHzPh]

Benzene

l.O4(OT)

131

l 3 C (1=1/2)

'[PhCHzCO' %H2Ph]

HDTCL rnicelle

[2]

170(1=5/2)

3[R02' 'OzR] a)

Film

1.37(0T), 1.53(15mT), 1.16(1.4T) 1.06(OT)

29Si(l=1/2)

3[PhMe~SiCO''CHzPh] 3[PhzMeSi' 'Ph]

SDS micelle Brij 35 micelle

.

29S1(1=1/2)

33

S (1=3/2)

3[PhCOC'H2 'S02PhI

SDS micelle

33

S (1=3/2)

3 [ k s ' ' s k ] b,

SDS micelle

Ge(Z=9/2)

3[Ph2MeGe' 'Ph]

Brij 35 micelle

73

Brij 35 micelle SDS micelle SDS micelle

1.023(OT) 1.003(OT) 1.005(2OmT) 1.003(0.1T) 1.015(OT)

[61 [71 [81

191

1.009(OT) @I 1.015(10mT) 1.008(50mT) 1.006(OT) ~ 3 1 1.015(20mT) 1.009(0.1T) 1.078(0T) PI 1.005(OT), 1.053(50mT) [I41 1.020(0T) ~ 7 1

"'RH is of polypropylene. b'ArS-SAr is p-aminophenyl disulfide. "Ar'OH is 2,6-diphenyl4-stearoylphenol.

9.6 MIEs of 33S/32S, 34S, 36S Step et al. tried to find the 33S (0.75%, 1=3/2) M E in the direct photolysis of sulfurlcontaining ketone (PhCOCH2-S02Ph)in SDS micelles [9]. PhCOCH2-SO2Ph + hv +

PhCOCHz-S02Ph* +

PhCOCHz-S02Ph*-+ 3[PhCOC'H2 'SOzPh], 3[PhCOC'H2 'S02PhI '[PhCOC'H2 'S02PhI

c-f

+

'[PhCOC'Hz 'S02Ph1, PhCOCH2-SO2Ph.

PhCOCH2-S02Ph*,

(9- 15a) (9-15b) (9- 1 5 ~ ) (9-15d)

134

In this reaction, the benzenesulfonyl radical is produced, where its 33SHF coupling constant had been obtained to be A/g,uB = 8.32 mT. Because the author's group had already observed the MFE in this reaction [lo], this was one of the candidates for detecting the M E of 33S. Indeed, Step et al. found the a value for the 33Si-enrichmentobserved for this reaction in the absence of an external magnetic field as listed in Table 9-2. The author's group tried to find the 33S M E in the direct photolysis of is p-aminophenyl disulfide (ArS-SAr) in SDS micelles at room temperature [8]. ArS-SAr + hv +

ArS-SAr*,

ArS-SAr* -+

(9-16a)

Ars-sAr*--t 3[Ars''SAr], 3[Ars''SAr] * "ArS' 'SAr],

(9-16b) (9-16c)

"ArS' ' S A r ] -+ Ars-SAr, (9-16d) In this reaction, two p-aminophenylthiyl radicals ( A r S . ) is produced, but its 33SHF coupling constant has not yet been obtained. Because the author's group had already observed the MFE in this reaction [ 113, this was another candidate for detecting the M E of 33S. Indeed, we found the a values for the 33Si-eMchment observed for this reaction in the absence and presence of external magnetic fields as shown in Fig. 9-4(a) and Table 9-2. The Bdependence of this M E of 33S can be explained by the HFCM.

, _ . . , . . . , . .,... ~

(a) !

1

7'

=s

0

CI

1.m:

0

200 400 600 8 0 0 1 O o O 1 m - W I G

MagndcFleldIG

Fig. 9-4. Magnetic field dependence of the one-step enrichment coefficients (a) observed at room temperature for the MIEs of (a) 33S in reaction (9-16) (Reproduced from Ref. [18] by permission from Kodanshfliley) and (b) 73Ge in reaction (9-17) (Reproduced from Ref. [13] by permission from The American Chemical Society).

9.7 MIEs of 73Gd70Ge,72Ge,74Ge,76Ge There are many isotopes in Ge, but its magnetic isotope is only 73Ge (7.73%, 1=9/2). Because the author's had found many MFEs on the reactions of Ge-radicals [12], we tried to find its MIF using some of them. The first reaction was the direct photolysis of methyltriphenyl-germane(Ph3CH3Ge) in Brij 35 micelles at room temperature [ 131. Ph3CH3Ge + hv ---* 'Ph3CH3 Ge* 3Ph3CH3Ge*, (9-17a) --f

3Ph3CH3Ge*-* 3[Ph2CH3Ge' 'Ph], 3[Ph2CH3Ge' 'Ph]

* '[PhzCH3Ge' 'Ph],

(9-17b) (9- 17c)

135

[PhzCH3Ge' P h ]

+

Ph3CH3Ge.

(9-17d)

In this reaction, the diphenylmethylgermyl radical is produced. Although its 73Ge HF coupling constant has not yet been measured, it may be similar to that of the triphenylgermyll radical (AlgpB = 8.32 mT). Typical a values for the 73Ge-enrichment observed for this reaction in the absence and presence of external magnetic fields are listed in Table 9-2. The B-dependence of this M E of 73Ge is shown in Fig. 9-4(b) and can be explained by the HFCM. A larger M E of 73Ge was found in the photolysis of dimethyldiphenylgermane (PhzMezGe)sensitised by triplet xanthone (3Xn*) in Brij 35 micelles [S]. X n + h v - + 'Xn* -+ 3Xn*, (9-18a) 3 Xn* + PhzMezGe + Xn + 3Ph2Me~Ge*, (9-18b) 3 PhzMe2Ge*-+ 3[PhMe2Ge' 'Ph], (9-18~) (9-18d) 3[PhMezGe' 'Ph] tf '[PhMezGe' 'Ph], '[PhMezGe' 'Ph] + PhzMezGe. (9-18e) The a value was observed for reaction (9-18) only in the absence of an external field as listed in Table 9-2, but it was much larger than the corresponding value observed for reaction (9-17). Another M E of 73Ge was found in the photoreduction of benzophenone (PhZCO) with triethylgermane (Et3GeH) in SDS micelles [14]. PhzCO", (9-19a) PhzCO + hv + Ph2CO* PhZCO* + Et3GeH + 3[Et3Ge' 'COHPhz], (9-19b) 3[Et3Ge' 'COHPh2] t) '[Et3Ge' 'COHPh2J, (9-19~) (9-19d) '[Et3Ge' 'COHPhz] -+ Et3Ge-COHPhz.

'

-a

4

$ 9 P;

4.0

-

xg

3.0

-

h

a

3

-+

-

136

Because the rate of the T-S conversion (reaction 9-19c) in a RP with 73Geis much faster than that in RPs with other Ge isotopes, it is expected that 73Geis enriched in the cage product (Et3Ge-COHPhz).

In Fig. 9-5, the c S ( ~ ~ and G ~ )6(72Ge)values observed after photolysis of the SDS micellar solutions containing Ph2CO and Et3GeH under 95 % conversion in the magnetic field range of 0 - 1 T are shown. AS clearly shown in this figure, 73Gecan be enriched in the cage product of this reaction. The typical a values obtained in this reaction are also listed in Table 9-2. As shown in Fig. 9-5, the 6(73Ge)and 6(72Ge)values observed in this reaction were found to give peculiar magnetic field dependence as follows: (1) In the absence of an external magnetic field, the 6(73Ge) and S(72Ge) values show no appreciable M E beyond the experimental error. (2) With increasing B from 0 mT to 50 mT, the 6(73Ge)value increases, but the 6(72Ge)value is not changed. (3) With increasing B from 50 mT to 1 T, the 6(73Ge) value decreases without showing saturation, but the 6(72Ge) value is not changed. The magnetic field dependence of the the 6(73Ge)value is quite different from that of the a (I3C) value as shown in Fig. 9-3. The latter field dependence can be explained by the HFCM, but the former one by the RM, where the T,l-S conversion of a RP with 73Gestill exists even at B > B I l 2of this RP. Here, the B1/2 (RP with 73Ge)value and the (RP with a non-magnetic Ge) one can be calculated as 82.8 mT and 1.83 mT, respectively. Problem 9-3. Calculate the B1/2 (RP with 73Ge)value and the B1/2 (RP with a non-magnetic Ge) one. Here, the following HF coupling constants can be used: For Et3Ge., AlgpB (CH3)= 0.056 mT, A/gpB (CH2)= 0.475 mT, and A/gpB (73Ge)- 8.47 mT ,which is the value for the trimethylgermyl radical. For COHPh;?,A/gpB (o-H)= 0.321 mT, A/gpB (m-H)= 0.123 mT, A/gpB (p-H)= 0.364 mT, andd/gpB (OH)= 0.291 mT. 9.8 MIE of Sn There are several magnetic isotopes in Sn: "'Sn (0.34%, 1=1/2), Ii7Sn (7.68%, 1=1/2), and "'Sn (8.58%, 1=1/2). Although CIDNP and MFE have been observed for the reactions of Sn-centered radicals, no MIE has been obtained for Sn. Podoplelov et al. reported a M E on a Sn-centered radical in 1979 [15], but this paper was cancelled afterward [16]. 9.9 MIEs of 235U/234U, 23*U Among uranium isotopes, 235U(0.72%, 1=7/2) can only be used for nuclear fuel. Thus, enrichment of 235Uis one of the goals of M E . Although it seems very difficult because of a large SOC of U, there have been some report of MIEs on 235U. In the photolysis of uranyl peroxide salt, U02(C104)2. in a D20 micellar SDS solution in the presence of 2,6-diphenyl-4stearoylphenol (Ar'OH), Khudyakov and Buchachenko [17] found that the a value of 235U enrichment became 1.020 as shown in Table 9-2, but that the a value of 238Uwas 0.994. Because MIE was greater and opposite in sign to CIE, M E shows its superiority over CIE even for U.

References [ l ] N. J. Turro and B. Kraeutler, J. Am. Chem. SOC.,100 (1978) 7432. [2] N. J. Turro, D. R. Anderson, M.-F. Chow, C-J. Chung, and B. Kraeutler, J. Am. Chem. SOC.,103 (1981) 3892.

137

[3] N. J. Turro and B. Kraeutler, Acc. Chem. Res., 13 (1980) 369. [4] W. Bube, M. E. Michel-Beyerle, R. Haberkorn, and E. Steffens, Chem. Phys. Lett., 50 (1977) 389. [5] (a) N. J. Turro and M.-F. Chow, J. Am. Chem. SOC.,102 (1980) 1190. (b) N. J. Torro, M. -F. Chow, and J. Rigaudy J. Am. Chem. SOC.,103(1981)7218.

[6] A. L. Buchachenko, Chem. Rev., 95 (1995) 2507. [7] E. N. Step, V. F. Tarasov, and A. L. Buchachenko, Chem. Phys. Lett., 144 (1988) 523. [8] M. Wakasa and H. Hayashi, unpublished data. [9] E. N. Step, V. F. Tarasov, and A. L. Buchachenko, Nature, 345 (1990) 25. [lo] H. Hayashi, Y. Sakaguchi, M. Tsunooka, H. Yanagi, and M. Tanaka, Chem. Phys. Lett., 136 (1987) 436. [ I l l M. Wakasa, Y . Sakaguchi, and H. Hayashi, J. Phys. Chem., 97 (1993) 1733. [I21 M. Wakasa, Y. Sakaguchi, and H. Hayashi, J. Am. Chem. SOC.,114 (1992) 8171. [I31 M. Wakasa, H. Hayashi, T. Kobayashi, and T. Takada, J. Phys. Chem., 97 (1993) 13444. [I41 M. Wakasa, H. Hayashi, K. Ohara, and T. Takada, J. Am. Chem. SOC.,120 (1998) 3227. [15] A. V. Podoplerov, T. V. Leshina, R. Z. Sagdeev, Yu. N. Molin, and V. I. Gol’danskii, JETP Lett., 29 (1979) 380. [I61 A. V. Podoplelov, V. I. Medvedev, R. Z. Sagdeev, K. M. Salikhov, Yu. N. Molin, V. M. Moralev, and I. N. Misko, Proc. Int. Conf. Chemically Induced Spin Polarization and Magnetic Effects in Chemical Reactions, R. Z. Sagdeev Ed., Novosibirsk, USSR, (1981) 81. [I71 I. V. Khudyakov and A. L. Buchachenko, J. Chem. SOC.,Mendeleev Commun., 3 (1993) 135. [18] H. Hayashi, in “Dynamic Spin Chemistry”, S. Nagakura, H. Hayashi, and T. Azumi. Eds., KodanshalWiley, Tokyo, 1998,p.39.

138 Solutions to the Problems 9-1. From Eqs. (9-10) and (9-ll), the following relations can be obtained: lOg(l-F) = l~g([A]/[A]o)= -kl(l-P)t, logS=-kl(P-P#)t=-kl[-(l - P ) + ( I - P # ) ] t =-log(1-F)+[(l -P#)/(l-P)]log(l-F) = (1 - l/a)[- log(1-F)]. 9-2. The I3C HFC-constant (AlgpB) of the phenacetyl radical was reported to be 12.5mT [3], but the ‘H HFC-constants of the phenacetyl radical were too small to be observed [3]. From Eq. (6-12), the B1 (the phenacetyl radical with I3CO) and B1 (the phenacetyl radical with 12 CO) values become 10.8 mT and 0 mT, respectively. B1 (the phenacetyl radical with I3CO) = 12.5 mT(&/2) = 10.8 mT. The ‘H HFC-constants of the benzyl radical were reported to have the following values [3]: 1.6 mT for -CH2-, 0.5 mT for o-protons, 0.2 mT for m-protons, and 0.6 mT for p-proton. From Eq. (6-12), the B2 (the benzyl radical) value is given by B2

(the benzyl radical) = (1.6mTx2+0.5mTx2+0.2mTx2+0.6mT) ( & / 2 ) = 4.5 mT.

From Eq. (6-1 l), the BIR(RPwith I3CO) and B1/2(RP with ”CO) values can be obtained as follows:

BI&P

with I3CO)= 2

(10.8mT)’ + (4.5mT)’ = 17.9mT, 10.8mT+ 4.5mT

9-3. In calculation of the B1/2 (RP with 73Ge)value and the B112 (RP with a non-magnetic Ge) one, the following HF coupling constants can be used: For EtsGe’, A/gpB (CH3)=0.056 mT, A/gpB (CH2)= 0.475 mT, and A/gpB(73Ge)- 8.47 mT ,which is the value for the trimethylgermyl radical. Thus, the A1 values without and with 73Gebecome as follows: 13 A1 (Et3Ge without 73Ge)2= --{9(0.056mT)’ + 6(0.475mT)Z}=1.03648(mT)2, 22 9 11 A1 (Et3Ge.with 73Ge)2= 1.03648(mT)2+ --(8.47mT)’= 1776.6237(mT)’. 2 2 For ‘COHPh2, A/gpB(o-H)= 0.321 mT, A/gpB (m-H)= 0.123 mT, A/gpB @-€I)= 0.364 mT, and A/gpB(OH)= 0.291 mT. Thus, the &value becomes as follows: 13 A2 (‘COHPh2)2= --{4(0.321mT)’ + 4(0.123mT)’ + 2(0.364mT)’ + (0.291mT)’) 22 = 0.6167647(mT)2. From Eq. (6-1 I), the BIR(RPwithout 73Ge)and BI/~(RP with 73Ge)values can be obtained as follows: 1.0364805+ 0.6167647 mT = 1.833 mT, BII~(RP without 73Ge)= 2 1.018+0.7853 1776.6237+ 0.6167647 mT = 82.78 mT, B1/*(RPwith 73Ge)= 2 42.15 +0.7853

139 10. Triplet Mechanism

10.1 Introduction In 1975, b u n g and El-Sayed reported on a very interesting observation that the rate of the biphotonic photochemistry of pyrimidine in benzene at 1.6 K had been found to decrease when the system had been exposed to microwaves in resonance with its zero-field (ZF) transitions or to a static magnetic field [l]. The effect of resonant microwaves and static magnetic field (1 T) on decreasing the rate of the photochemical disappearance of pyrimidine is shown in Fig. 10-1. W e can see from this figure that these perturbations cause a decrease in the value of the rate constant of the observed photochemical change by a factor of 1/2 1/3.

MICROWAVE ON

V

21EI

MICROWAVE OFF k = (7.9 t0.2)

(a) I

5 4

-

I .

(b)

k I

I

I

I

I 10-3

1

min-'

I

I

MICROWAVE OFF (7.5 2 0.3) x 10-3 min-'

-

I

1

I

I

-

H 10 KGwrr

MAGNETIC FIELD OFF k

-

(7.7 t 0.41x 10-3 min"

-I

Fig. 10-1. The effect of microwaves in resonance with ZF transitions or I T static magnetic field on the rate of photochemical reaction of the triplet state of solid pyrimidine in benzene at 1.6 K, Here, the vertical axis represent the relative phosphorescence intensity of this system. (Reproduced from Ref. [ 11 by permission from The American Chemical Society)

140

Because the spin-lattice relaxation between the three ZF levels (Tj, j = x, y, e) is much slower than other photophysical processes at 1.6 K, a set of rate equations may be derived for the following biphotonic mechanism:

a, I so #

rjr

Ti

+Product.

(10-1)

Pi

The rate equations for populating the ZFJ-level of the lowest triplet state (T,) and the ground state (SO)are given by (10-2a) (10-2b) Here, x, y, and z are the three orthogonal ZF levels of the lowest triplet state. Since the reaction is much slower than the other photochemical processes, these rate equation can be solved under a pseud-steady-state approximation. The rate constant for this photoreaction at certain light level, k(I), is found to be Y12 k(Z) = I+" where

Y =C j n j y j l

x

=

C j n j= CjaI/Pi.

(10-3a)

(10-3b) (10-3~)

y ZF transition is applied, the population If a microwave field in resonance with the x of the two levels tends to equalize. Under this condition, the quantities k,, Y,, and X , are obtained by substituting the new steady-state population

a. +a..

(10-4)

for n, and n, in Eqs. (10-3). It is noteworthy that resonant microwaves could affect the rate of the photochemical reaction if either the populations ( n j = a jf P j ) or the photoreactivities ( r j ) of its ZF levels are unequal or both. An external magnetic field can similarly affect the rate. Eq. (10-3a) can be transformed as (10-5) Y = K(I)( 1 + X1>f12. If a microwave field in resonance with the x c* y or x * z ZF transition is applied, the following relations can be derived from Y f l and YJY :

(10-6b)

141

Since the n j = ai/ p j values had been determined as listed in Table 10-1 [2], the relative f i values could he obtained from several experiments with different excitating light intensities and Eqs. (10-6a) and (10-6b) [ 13 “JX

:yy :yz = 1:0.70~0.13:0.5+0.1.

(10-7)

Table 10-1. ZF splittings and rate constants determined with the technique of microwave induced delayed phosphorescence for individual triplet sub-levels of pyrimidine in benzene at 4.2 K [2]. ZF level

IZF splitting1

aj

Pi

x-level y-level z-level

5.640 BHz 4.716 GHz 0 GHz

1.4 1.4 1.o

3.08 s-’

34

60.2 s-’

2.0

84.7 s-’

1.o

nj

Each */i value contains the product of the probabilities of both the absorption of the second photon and the nonradiative processes leading to the observed photochemistry. Spin selectivity obtained as Eq. (10-7) could arise if the different ZF levels have different probabilities for either or both of these two processes. For example, it is very promising that the second photon could excite the molecule to a higher energy triplet state which, via spinorbit interaction, radiationlessly and spin selectively crosses to a photochemically active singlet state or to a singlet state of product. Although this was a very interesting paper on the spin selectivity of a triplet molecule and the associated effects of microwaves and external magnetic fields on this reaction, no further investigation has not yet been carried out for determination of its primary reaction processes and reaction products. On the other hand, much more magnetic field effects (MFEs) have been observed for chemical reactions through triplet states at room temperature as shown in the next section because the primary processes of photochemical reactions at room temperature can be detected much more easily than those at liquid helium temperatures.

10.2 MFEs due to the TM The results obtained in the previous section show that chemical reactions from triplet states at room temperature can also be affected by external magnetic fields if there are anisotropies in aj,pi,and ”/j values of the states. Of course, the rate of the spin relaxation processes, which randomise the anisotropies, becomes comparable at room temperature to those of other processes. If some of the anisotropies remain at room temperature, the succeeding reactions are possibly influenced by magnetic fields and generated radicals from such reactions possibly show CIDEP. Such MFEs and CIDEP have been called those due to the triplet mechanism (TM). Selective populations of three sub-levels of a triplet excited state are generated during the ISC through the a, anisotropy. If the rate of a chemical reaction from the triplet state is faster than or comparable to the rate of spin relaxation within the triplet sub-levels, CIDEP can be observed in the generated radicals from this reaction as shown in Chapter 5. It is noteworthy that decay rate of each triplet level (p,)is usually much slower than the reaction rates as shown in Table 10-1. This mechanism is called “the p-type TM.” Indeed, many reactions have been found to show CIDEP due to the p-type TM. If a reaction of a radical pair occurs through the To-S conversion, the yield of this reaction should be proportional to the initial population of TO,p(B), which is in general affected by magnetic fields. Such an example of a MFE will be dealt in Chapter 11 (see Eq. (11-54a)). Because such MFEs appear mostly at the initial population of radical pairs or escaped radicals, it is

142

much more difficult to detect such MFEs than those due to other mechanisms. The latter MFEs can more easily be obtained from the decay of radical pairs or the relative yield of escaped radicals. Thus, there has been no established paper for the discovery of any MFE due to the p-type TM. There is the second type of TM. If the decay rate of each triplet level @,) becomes faster than or comparable to the reaction rate from the triplet state, MFE and CIDEP are also possible for this case. Such enhancement of the P j values can sometimes be realized by heavy atom effect. This mechanism is called "the d-type TM." In 1979, Steiner reported MFEs on the radical yield of electron-transfer reactions between a dye triplet (3A') and heavy-atom-substituted electron donors (D) in methanol [3]. Although the experimental temperature was not described in his early papers, it may be room temperature. The scheme of the reactions can be shown as follows: (10-8a) A' + D+', 'lj 3

+*

A

+ D + 3(AD')* A+

+

D.

(10-8b)

Here, 3(AD)' is a triplet exciplex (T), ~j (j=x. y, z) is the population efficiency of each triplet sublevel, k, is the rate constant for exciplex dissociation into radicals, and kj O=x, y, z) is the ISC rate from the triplet exciplex. The k, value can be assumed to be equal for the T,, T,, and T, levels, where z-axis is taken to be perpendicular to the molecular plains of A+ and D as shown in Fig. 10-2. As shown later, the k, value of T, can be assumed to be equal to the ky value of T,, but the k, value of T, to be much smaller than others.

tz

Fig. 10-2. Assumed geometry of the triplet exciplex between p-substituted halogen aniline with thionine. Here, only p-orbitals of halogen substituent are indicated. (Reproduced from Ref. [4a] by permission from The Royal Society of Chemistry) With an ns-laser photolysis apparatus, Steiner measured the escape radical yield (@,) in the absence and presence of an external magnetic field below 0.414 T, exciting thionine (TH') at 500 nm and observing its semiquinone (TH2+')absorption at 780 nm. He obtained the relative MFE (AR) defined as (10-9)

143

Fig. 10-3 shows the obtained B-dependence of AR(B) for the reaction of thionine with p iodoaniline in methanol. As clearly seen from this figure, the AR value decreases with increasing B from 0 to 0.414 T. Using other donors, he also measured the dR(B) values for their reactions. The obtained AR(0.414 T) values are plotted against OP(OT) in Fig. 10-4. This figure shows that no appreciable MFE was observed for the reaction of aniline, but that the dR(0.414 T) value increases with decreasing QP(O T).

0

-2

-4 -6 -8

- 10 - 12 I

2

magnetic field,

3

4

kGauss

Fig. 10-3. Relative magnetic field effect AR (cf. Eq. (10-9)) as a function of magnetic field for the triplet exciplex thioninelp-iodoaniline. Circles with error bars represent the experimental results obtained in Ref. [3]. Lines are calculated with an equation similar to Eq. (10-17) and the different values for kp indicated in the diagram. (Reproduced from Ref. [4a] by permission from The Royal Society of Chemistry) At first, Steiner considered the heavy atom induced sublevel-selective ISC in triplet exciplexes as follows [4a]: The rate constant (krsc) of ISC from 3(AD)' to I(A+D)o will be given by the approximation (10-10) krsc = Fei x FFC, where Felis the electronic factor and FFCthe Franck-Condon factor. Steiner assumed that triplet exciplex and singlet ground state are directly coupled by the spin-orbit interaction. Thus, F,l is proportional to the square of the corresponding matrix element (10-11)

He also assumed that the exciplex is of a sandwich type structure as shown in Fig. 10-2. This structure provides good overlap between the n-electron systems of acceptor and donor component. If the triplet exciplex in methanol can be approximated by a pure charge transfer state, Eq. (10-1 1) becomes

144

(10-12)

Here, a is the lowest unoccupied molecular K-orbital of the acceptor and d is the highest occupied molecular n-orbital of the donor. Finally, Steiner obtained the heavy atom induced sublevel-selective ISC in triplet exciplexes as follows [4a]:

k , = k , = <'c2,

(1 0-13a)

k, = 0 ,

(10- 13b)

where 6 is the atomic spin-orbit coupling constant and c is the MO-coefficient of the atomic valence p,-orbital of the halogen atom. Problem 10-1. Prove Eqs. (10-13a) and (10-13b).

I

0.1

0.3

0.5

0.7

0.3

Fig. 10-4. Relative magnetic field effect dR (cf. Eq. (10-9)) observed at 0.414 T as a function of radical yield at zero field ('BP(OT)). The numbers refer to different donors in the exciplex with thionine triplet: 1 aniline, 2 p-Br-aniline, 3 rn-I-aniline, 4 o-I-aniline, 5 p-1aniline. The solid line is calculated with an equation similar to Eq. (10-17) and kp = l.lx109s-'. (Reproduced from Ref. [4a] by permission from The Royal Society of Chemistry)

145

‘x kx

kY

TY

kP

Fig. 10-5. General reaction scheme for the d-type TM. (Reproduced from Ref. [4b] by permission from CRC Press) MFEs due to the d-type TM are based on the general reaction scheme shown in Fig. 10-5, which represents population and decay of an excited molecular triplet state. Here, the energies (E,) of triplet sublevels (Tx, Ty, and T,) at zero field can be represented by (10- 14a) Ex = Dl3 + E, Ey= Dl3 - E, (10-14b) E, = -2013. (10-14~)

In Fig. 10-5, q,, qy, and q, denote the population efficiencies of the triplet sublevels. These will, in general, be different if the population is due to ISC from an electronically excited singlet state, but may be equal if it is due to a reaction from a relaxed triplet state. In Fig. 105 , k,, ky, and k, denote the decay rates of the triplet sublevels. These will, in general, be different as shown by Eqs. (10-13a) and (10-13b) because the decay is due to the ISC to the singlet ground state. In Fig. 10-5, kp denotes the rate of a chemical product formation. This is usually independent of the triplet sublevels. According to the RPM, its MFEs are due to the selection rules for RP spin sublevel decay. According to the d-type TM, its MFEs are due to symmetry selection rules governing SOC. Another important factor in the p-type T M is the relaxation among the triplet sublevels. It is a stochastic process with rate constant w in Fig. 10-5. MFEs on the product yield (QP)arise from a MFE on the effective value of w. All the other parameters are considered to be magnetic field-independent. As will be shown in problem 10-2, w increases monotonically with increasing B. Problem 10-2. Show that w increases monotonically with increasing B. At first, we will consider a simple case, for a clear understanding of the d-type TM, as follows: (1) Population is not sublevel selective (qx = qy = qz = U3). (2) The k, value is assumed to be equal to the ky one and they are assumed to be much larger than the k, one as shown by Eqs. (10-13a) and (10-l3b) (kx= ky = kx,y>> kz). Then, let us specify Opfor w+O and w-m. For w+O, the decays of the triplet subleves are independent of one another. Thus, the Opvalue for w-0 can be expressed as

2 k @p(w+o) = -A + -1. kP 3 kx,y+ k p 3 k , + k ,

(10-1 5a)

146

For w+m, however, the relaxation between the triplet subleves becomes much faster than their decays. Thus, the Opvalue for w-+m can be expressed as CDp(w-'m)

=

k, 3kP 2 1 2k,,, + k , -kx,y + - k , + k , 3

+ 3kp

(10-15b) '

3

From Eqs. (10-15a) and (10-15b), the difference between the two limiting values becomes OP(W"O)

- Op(w-tm)

= -___ k, + l k- A p 3%

3 kX,,+ k ,

3 k,

+k,

2k,.,

+ k L+ 3k,

~ X_ k ,Z ~ ( X - Z ) ' - k, ~ Z ( ~ X + Z ) + X ( ~ X + Z ) - 20.

_-

3

+Z )

XZ(2X

3 XZ(2X + Z )

(10-16)

As shown in Problem 10-2, w increases monotonically with increasing B. Thus, the Opvalue may decrease with increasing B. From Eqs. (10-15a), (10-15b), and (10-16), we can see that MFEs due to the d-type TM appear only if the decay is sublevel selective. On the other hand, selective population of the triplet sublevels is neither necessary nor sufficient to produce MFEs due to the d-type TM. This is a difference from the p-type TM in producing CIDEP in radical reactions, where selective population is a sufficient condition. Theoretical analysis of MFEs due to the d-type TM has been carried out with the aid of stochastic Liouvile equations (SLEs). Although exact treatments with SLEs are beyond the scope of this book, an introduction to SLEs will be given in Chapter 11. Applying a SLE to R(B) of Eq. (10-9) for a uniform population of triplet exciplexes (ox = vy = qz), Serebrennikov and Minaev obtained the following result [5]: (10-17) with (10-18a)

W = gp,BIA,

k, = k,

+ ( k , +k , + k , )1 3 ,

(10-18b)

k=ko+11r2,

(10-18c)

D, = ( k , + k , )I 2 - k , ,

(10-18d)

and

2 1/T, =-(0' 15k

(

+3E2)

k2

~

k2+ w 2

4k2

'm).

(10-180

Here, D and E are the parameters of zero field splitting (ZFS) defined by Eqs. (10-14a) - (1014c) and r2 is the orientation relaxation time of the axial second-rank tensor [ 5 ] . We can see from Eq. (10-17) that the Op value decreases with increasing B as shown in Fig. 10-3. It is noteworthy from Eq. (10-17) that, contrary to ClDEP due to the p-type TM,

147

MFEs due to the d-type TM do not depend on an nonvanishing ZFS as follows: (0’+ 3 E 2 )+ 0 , the dR(B) value of Eq. (10-17) approaches

Letting

(10- 19)

\ I 1

Z

b

10

20

u)

w o z a o s o o

lmo

w Fig. 10-6. Relative magnetic field effect AR (cf. Eq. (10-9)) as a function of the Larmor frequency (w in log scale) observed for the triplet exciplex (thioninelp-iodoaniline) in methanol. Points with error bars represent the experimental results obtained in Ref. [6]. Lines are calculated with an equation similar to Eq. (10-17) and the different values for dR(B+m) indicated in the diagram. (Reproduced from Ref. [6] by permission from The American Chemical Society) Fig. 10-6 shows the observed dR(B) values in the reaction from the triplet exciplex (thioninelp-iodoaniline)and the theoretical prediction of dR(B) with the different values for dR(B-m). As clearly seen from this figure, the MFE of this reaction is saturated at about 2

148

T and its saturated value is about -20 % even in a fluid homogeneous solution. It is noteworthy that confinement of radical pairs in micells and biradicals is not necessary for the appearance of MFEs due to the d-type TM.

In 1995, the author's group found similar MFEs on the photodissociation of triphenylphosphine (Ph3P) in some fluid homogeneous solutions at 293 K[7]. Its reaction scheme has been proposed as follows: Ph3P + hU + 'Ph3P* + 3Ph3P*, 3Ph3P*+Ph3P, 3Ph3P*+ 3[Ph2P' 'Ph], 3[Ph2P' 'Ph] t-) '[PhZP' 'Ph],

(10-20a) (10-20b)

(10-2Oc) (10-20d)

(10-20e) '[Ph2P' 'Ph] + Ph3P (cage product), (10-200 3"[Ph2P' 'Ph] + PhzP' + Ph' (escaped radicals). Fig. 7-7(b) and Fig. 10-7 show the magnetic field dependence of R(B) =Y(B)IY(O T), where Y(B) represents the yield of the escaped diphenylphosphinyl radical (Ph2P.) observed at 330 nm. The MFEs observed for this reaction in some solvents had the following characteristic features which could not be explained by the usual RPM: (1) The R(B) ratio was not changed by magnetic fields below 0.1 T, but started to decrease with increasing B from 0.1 T to 1.75 T. (2) The magnetically induced changes in the R(B) ratios observed in the employed solvents were not proportional to B''2, but they were almost saturated at 1.75 T. (3) The observed MFEs at 1.5 T in the employed fluid homogeneous solvents ranged from 15 % to 35 %. These values were much larger than those due to the RPM in fluid homogeneous solvents. The latter MFEs are usually less than 10 % below 1.5 T. (4) If we arranged the R(1.5 T) ratios in the order of solvent viscosity (77) at 293 K, the following peculiar relation was obtained: R(1.5 T) = 74&3% in 2-propanol (7 = 2.379 cP) >

R( 1.5 T) = 64k3 % in p-dioxane (77 = 1.322 cP) R( 1.5 T) = 65k2 % in cyclohexane (77 = 0.975 1 cP) < (10-21) R(1.5 T) = 84+5 % in n-hexane (11 = 0.3216 cP). These MFEs can be explained by the d-type TM because Features (1) - (3) correspond well to the theoretical curves in Fig. 10-6. We can safely conclude that the decay of 3Ph3P* is enhanced by the SOC of P and that the 3Ph3P* decay becomes comparable to its decomposition represented by Process (10-2Oc).

149

0.90 0.80 0.70

0.60

I

0.50

'r3

-2

-1 0 Log(Field/T)

1

,. -..

..I.

nnnc]

2

Fig. 10-7. Magnetic field dependence of the absorbance (A(t) ) observed at t = 100 or 200 ns after laser excitation for some phosphinyl radicals in cyclohexane at room temperature. This figure is reproduced from Ref.[8].

10.3 CIDEP due to the d-type TM As shown in Chapter 5 , many reactions have been found to give CIDEP due to the p-type TM. In 1987, Serebrennikov and Minaev [5] also predicted that CIDEP due to the d-type TM (P(B)) could be given as follows: 1

(10-22)

In 1994, Tero-Kubota's group first found CIDEP due to the d-type TM as shown in Fig. 10-8 [9]. They studied the electron transfer reaction from triplet xanthene dye (3Xn2-*)to p quinones (Q)in 1-propanol at room temperature. (10-23) 3 ~ n 2 -+*Q + Xn" + Q-'. Fig. 10-8 shows the observed CIDEP spectra for the reaction of triplet eosin Y (FlBrz-) with duroquinone. In this figure, CIDEP spectra of the duroquinone radical anion were only observed. The spectra of Xn-' were not observed because of its fast spin relaxation. As clearly shown in Fig. 10-8, the initial spectrum measured at 60 ns after the laser excitation showed an emissive polarization, which was due to the usual p-type TM. This polarization was found to change as the delay time was increased. The spectrum measured at 200 ns after the excitation showed a strong absorptive polarization, which was proposed to be due to the d-type TM. Similar polarization changes were also observed for such dyes as erthrosin B (F1b2-)and dibromofluorescein (FlBr?.) which contain heavy atoms. On the other hand, an emissive polarization was only observed for the reaction of fluoresein ( F f ) , which contain no heavy atom. From these results, Tero-Kubota et al. concluded that the strong absorptive

150

polarization was induced by the SOC of heavy atoms such as Br and 1. This was the first observation of CIDEP due to the d-type TM.

200 ns

-40

- 20

20 Offset frequency / MHz

0

40

Fig. 10-8. CIDEP spectra of duroquinone anion radical generated by the photoinduced electron transfer from triplet eosin Y in I-propanol at room temperature. The delay times of the CIDEP measurement after the laser excitation (Nd:YAG at 532 nm) are shown. (Reproduced from Ref. [9] by permission from The American Chemical Society)

151

timels Fig. 10-9. Time profiles of CIDEP signals of the central HF line for DQ-' observed from the photoexcitation of DQ in the presence of (a) DMA, (b) 4BrDMA, (c) 3BrDMA, and (d) 2BrDMA in 1-propanol at room temperature. (Reproduced from Ref. [lo] by permission from The American Chemical Society) His group also studied the electron transfer reactions from triplet duroquinone (3DQ*) to N,N-dimethylaniline @MA) and its halogen substituents (D) [10,11]. (10-24) 3DQ* + D + DQ-' + D". Fig. 10-9 shows the time profiles of CIDEP signals of the central HF line for DQ-- observed from the photoexcitation of DQ in the presence of DMA and Br-substituted DMAs. As seen from curve a in this figure, an emissive polarization was found to grow for the reaction of DMA and to approach the absorptive signal due to the thermal equilibrium. This time profile can be explained by CIDEP due to the p-type TM. As seen from curve d in Fig. 10-9, an absorptive polarization was found to grow for the reaction of 2BrDMA and to approach the absorptive signal due to the thermal equilibrium. This time profile can be explained by CIDEP due to the d-type TM. In the reactions of 4BrDMA and 3BrDMA, their time profiles were found to be initial emissive polarization due to the p-type TM followed by absorptive polarization due to the d-type TM. Tero-Kubota et al. also found that the time profile observed for the reaction of 4IDMA was similar to curve d in Fig. 10-9 and that the profile for 4CIDMA to curve c in Fig. 10-9. These results show that the order of SOC in the present reactions can be expressed as follows: 4IDMA, 2BrDMA > 4BrDMA > 3BrDMA, 4ClDMA > DMA.

(10-25)

152

Problem 10-3. Explain the effect of the Br-positions in Relation (10-25).

DQ+D

<

The scheme for the photoinduced electron transfer reactions between Fig. 10-10. Here, A# duroquinone (DQ) with N,N-dimethylaniline and its halogen substituents ( D ) . shows a spin-polarized triplet state or radical. The time profiles obtained in Fig. 10-9 can be explained by the scheme shown in Fig. 1010, where the rate constant of each process is indicated. Tero-Kubota et al. determined some important parameters in this scheme, assuming pseudo-first-order reactions [ 10, 111. Table 10-2 shows the following parameters: (1) The kq[D] value was determined with [D] = 3 ~ 1 0 mol . ~ dm-3. Thus, the value was obtained to be 8.3 or 6 . 7 ~ 1 0mol-' ~ dm3 s-I. These values are slightly larger than the diffusion-controlled rate in 1-propanol. (kd = 8RT/37 = 6.50~10'mol-' dm3 s-'/( 7 /cP) = 3 . 0 ~ 1 mol-' 0 ~ dm3 s-') This may be due to the fact that the electron transfer occurs at a larger distance than the sum of the effective radii of two molecules. ) were obtained to be 1.2-1.6 ps. These values are smaller than (2) The TIR (= 1/ k ~ values the TIRvalue reported in 2-propanol at 20 "C ( 6 . 7 ~ ~ )The . TIRvalues in this case may be reduced by long-range interactions with ion radicals. (3) The @,, value corresponds to the yield of the escaped radical, which is defined by (10-26) @sc = kesc /( kesc + kb). Here, the ke,, value was assumed to be 1x10' s-'. This table shows that the &, value decreases with increasing SOC of the halogen atom. This means that the kb value is comparable to the k,,, value or larger than it for 4BrDMA and 4IDMA due to SOC. (4)The kb value consists of the isotropic back electron transfer rate (kiso,b) and the anisotropic one (kani,b), which is due to SOC of the halogen atom. kb = kiso,b kani,b. (10-27) k,,, ratio as listed in this table. This table shows that Tero-Kuhota et al. obtained the kani,d the ratio increases with increasing SOC of the halogen atom

153 (5)Tero-Kubota et al. also determined the intrinsic enhanced factor (VSOCM= PSOCM / Peq)due to the d-type TM as listed in this table. Here, P s o c ~and Peq denote the polarization due to the d-type TM and that due to the thermal population. This table shows that the value increases with increasing SOC of the halogen atom. Table 10-2. Fitting parameters for CIDEP signals of DQ-• generated by the photoinduced electron transfer from the donor in 1-propanol at room temperature. (Reproduced from Ref. [111 by permission from The American Chemical Society)

R

I

0

Em.

I

II b

Fig. 10-11. CIDEP spectrum observed at a delay time of 0.25 p after laser excitation of a 2-propanol solution of Ph3P at room temperature. Stick diagrams a and b indicate the signal positions due to the 2-hydroxy-2propyl and diphenylphosphinyl radicals, respectively. (Reproduced from Ref. [7] by permission from Elsevier Science B. V.)

In 1995, the author's group also measured CIDEP spectra for the photodissociation of triphenyl-phosphine (Ph3P) in some fluid homogeneous solutions at 293 K [7]. Fig. 10-11 shows a typical spectrum among them. In this figure, absorptive signals due to the 2hydroxy-2-propyl and diphenylphosphinyl radicals ((CH3) 2C'OH and Ph2P.) were observed at a delay time of 0.25 ps after laser excitation. The latter radical is produced by Reaction (1020e), but the former one by the secondary reaction from the phenyl radical generated from Reaction (10-20e) as follows: Ph' + (CH3) 2CHOH -+ PhH + (CH3) 2C'OH. (10-28)

At 0.85 ps after excitation, absorptive signals due (CH3) 2C'OH were observed, but the signals due to Ph,P* disappeared completely. The existence of its optical signal at 0.85 ps indicates that this disappearance is due to the electron spin relaxation. Consequently, the absorptive polarization is not due to thermally equilibrated signals but due to CIDEP caused by the p-type TM andor the d-type TM.

154

In 1997,Paul’s group also found an usual net emissive CIDEP spectrum of 2-cyano-2propyl radicals when generated from 2,2’-azobis[isobutyronitrile] (AIBN) by triplet sensitization with thermally equilibrated triplet acetone (3S*) [ 121. Here, this reaction occurs as follows: 3S* + AIBN -+ S + 3AIBN*, (10-29a) 3AIBN* -+ N2 + Z(CH3) 2C’CN. (10-29b) They also obtained a similar CIDEP spectrum by triplet sensitisation with triplet benzophenone [13]. The polarization was found to consist of two net component, one reflecting the polrization of the triplet sensitizer due to the p-type TM, and the other being attributable to the d-type TM. References [l]M. Leung and M. A. El-Sayed, J. Am. Chem. SOC.,97(1975)669. [2]D. M. Burland and J. Schmidt, Mol. Phys., 22(1971) 19. [3] U.Steiner, Z.Naturforsch., 34a(1979) 1093. [4](a) U. Steiner, Ber. Bunsenges. Phys. Chem., 85 (1981)228. (b) U.Steiner and H.-J. Wolff, in “Photochemistry and Photophysics Vol. IV”,J. F. Rabek, Ed., CRC Press, Boca Raton, 1991,Chap. 1. [5]Yu.A. Serebrennikov and B. F. Minaev, Chem. Phys., 114 (1987)359. [6]T. Ulrich, U.E. Steiner, and R. E. Foll, J. Phys. Chem., 87 (1983)1873. [7]Y.Sakaguchi and H. Hayashi, Chem. Phys. Lett., 245 (1995)591. [8]Y.Sakaguchi and H. Hayashi, unpublished data. [9]A. Katsuki, K. Akiyama, Y. lkegami, and S. Tero-Kubota, J. Am. Chem. SOC.,116 (1994) 12065. [lo]S.Sasaki, A. Katsuki, K. Akiyama, and S. Tero-Kubota, J. Am. Chem. SOC.,119 (1997) 1323. [ 111 S. Sasaki, Y.Kobori, K. Akiyama, and S. Tero-Kubota, J. Phys. Chem. A, 102 (1998) 8078. [I21A. N. Savitsky, S. N. Batchelor, and H. Paul, Appl. Magn. Reson., 13 (1997)285. [13]A. N.Savitsky and H. Paul, Chem. Phys. Lett, 319 (2000)403. Solution to the Problems 10-1. Triplet exciplexes (3(AD’)*) fromed in the reaction of thionine triplet (3A’*) with halogen anilines (D) undergo ISC to the singlet ground state (‘(A’D)) mainly due to the influence of the haligen substituent in the aniline part. The rate constant of ISC (kj) can be expressed by kj = FeIUj) FFC, the Frank-Condon factor. We can asume that the where Fel(T,) is the electronic factor and FFC triplet exciplex and singlet ground state are directly coupled by the SOC Hamiltonian (Hso), so that FeI(Tj)is proportional to the square of the corresponding matrix element:

155

We assume that the exciplex is of sandwich type structure providing good overlap between the 7c-electron systems of the acceptor and donor components as shown in Fig. 10-2. Since the solvent is strongly polar, we shall approximate the triplet exciplex as a pure charge transfer state. In order to calculate the matrix element of Fel(Tj), we re resent the exciplex state by a two-electron wavefunction corresponding to the configuration (ad), where a is the lowest unoccupied molecular 7c-orbital of the acceptor and d is the highest occupied molecular n-orbital of the donor. The exciplex ground state configuration will be correspondinglt represented by '(dd), Thus, Fe1(T,) becomes:

P

I(

FedTj)

3 ( u 4j Iffso

1 (W)lz. I

We approximate the SOC operator by the contribution of the halogen atom

(c)

(tl& +{ziz)*(s, +s,>/2 + (t,&-~zzz)*(sl 4 ) / 2 The operator + izcommutes with S2,the total spin operator, and so it cannot mix states of =

s, s,

different multiplicity. However, the operator - does not commute with S2, and so this component of the operator is the one that is responsible for singlet-triplet mixing. The triplet and singlet states ( 3 @ ~and @o) can be expressed by 1 1 = -{~(l)d(2)-d(l)a(2))-(a,P, +PIaz>.

Jz

Jz

For the z-component of SOC, the spin operator is Slr - S,, and its effect is

Thus, the remaining orbital operator part of the SOC Hamiltonian is

= ( 5 1 ( 4 1 > ) ~ z ) d ( l ) )+) /(52(a(2)IL,IJd(2)))/2 2 = S(a)L,Jd).

Thus, Fel(Tz)is given by FedTz) -15(aJLzld)lz. Because the main term of the d-orbital is the p,-orbital of the halogen atom, F,l(T,) becomes C 2i

Fei(Tz)

S(al~~l~~> 1'3

where C is the coefficient of the p,-orbital in the d-orbital. Because L , ( p , ) = 0, Fel(Tz) becomes nearly zero. Similarly, F,,(T,) and Fe1(T,) can be expressed by FdTJ

-

C2 I~(a(L,(pz)lz = c21 ~ ! ( a I ( P ~ ) 1 ~ ,

156

FeI(Ty) m C’ I S ( a l ~ ~ lI’ ~=, C ) ’ I t ( a l l ~ , 1’). Thus, we can get the following relation:

FedTx) FedTy) >> FedTz) 0. 10-2. Eq. (35) of Ref. [4] shows that w can be expressed by the following equation: w = 60,

1+ 5xZ+ 4x4 1+3x2 + 4 x 2 / 5 ’

Here, 60, (= 3kT3) is the inverse of the molecular rorational correlation time ( I / 2, ) where 4wa q a n d a are solvent viscosity and the molecular hydrodynamic radius of the exciples. Assuming a = 0.4 nm, we obtain for methanol at room temperature 1/ z, = 25 x lo9s-’ . x = w/6D,where wis defined by Eq. (10-18a). Thus, we can see that w increases monotonically with increasing B.

10-3. The results obtained in Problem 10-1 can be used in this problem. kj = FeI(Tj) FFC, and FeI(Tz)

m

FedTx)

0~

FeI(Ty)

OC

C’ I S ( a l ~P,) ~ 1 I’

0,

C’ I S(alL,l~z)I’ = C2 I S(a11 P Y

)

1’3

c2lS(a(LylPJl’ = c2It(allPx)IZ

We can assume that C2 is proportional to the density of the carbon atom connecting the halogen atom in the highest occupied molecular 51-orbital of the donor. The 51-densities of the carbons of DMA calculated for the planar structure are shown in this figure. From this figure, the order of the n-density becomes as follows: 4BrDMA>ZBrDMA>3BrDMA. The SOC order of 4BrDMA > 3BrDMA shown in Relation (10-25) can be explained by the above order, but the order of 2BrDMA > 4BrDMA shown in Relation (10-25) can not be explained. The latter order may be explained by the twist of the nitrogen p-orbital of the dimethylamino group from the the aromatic ring-C-Br plane.

CH3 0.49

0.19

(This figure was reproduced from Ref. [ 101 by permission from The American Chemical Society.)

1 57

11. Theoretical Analysis with the Stochastic Liouville Equation

11.1 Density Matrix Method In Chapter 3, we calculated the time evolution of the wavefunction ( Y ( t )) of a radical pair during the S-To conversion, solving the Schrodinger equation, iti f l ( t )= HRP Y(t). dt Here, Y ( t ) is given by ~ ( t=) c , ( ~ ) ~ s , x N )

(3-14)

(3-15)

+ c,(t>l~o,~N).

In this chapter, we will develop a more elegant method for the theoretical analysis of radical pairs [l]. At first, let us consider a case where Y ( t ) can be expressed by two base functions as follows: Y(q,t ) = a,

@MI ( 4 )+ a2 w 4 2 ( 4 ) .

(1 1-1)

Here, Y ( t ) is rewritten as Y(q,t ) and q represents internal coordinates. When an observable property of the system R is measured, its value can be given by

(11-2)

Here, k(q) represents an operator involving q and the denominator of the right-hand side of Eq. (11.2) is usually normalized to be unity. From Eqs. (11-1) and (11-2), (R(t))can be expressed by

( R W ) = la, (t)12 R,, + la2(0l2R,, + al*(Oa, ( 0 4 2 + a,’(t)a,(t)R21

1

(11-3)

where &k is given as follows: R,k =

--J4,(q,f)&7)4Jk *

(q7

.

(11-4)

Problem 11-1. ProveEq. (11-3). We can define a matrix D, thejk-element of which is given as follows: D,k

= a,

(11-5)

(?)’

Using Eq. (1 1-5), we can rewrite Eq. (I 1-3) as

(w)= DIIRII +D22R22 +DZIR12

+D12R21

= DIIRII +D12R21 fD2lRl2 +D22R22 = (DR)II + (DR)22 = Tr[DR]. (11-6) Here, the symbol Tr[X] (read “trace of X’) means the sum of the diagonal elements of the square matrix [X] with elements x,k. The time-dependent Schrodinger equation like Eq. (3-14) is generally written as

158

(11-7) Let US consider the case when Y ( q , t ) is given by Eq. (11-1). rewritten as

In this case, Eq. (11-7) is (11-8)

Both sides of Eq. (11-8) may be multiplied by The result is i A ( y =) a,H,,+a,H,,

4,' from the left and then integrated over q.

.

(11-9a)

Similarly, the next result can also be obtained. ifi -.?-.

=a,H,, +a,H,,

(11-9b)

To find a differential equation for a particular matrix element of D, the definition in Eq. (115) may be used: (11-10) From Eq. (11-9a), its complex conjugate, and Eq. (11-lo), the following relation can be obtained:

(-

ifi

= ( a , H , , + a 2 H , , ) a , * - a , ( a , * H , , *+ a 2 * H , , * )

1 a :)

= [ ( H l l ~ +H12D2,) ,, - ( D l l H , , +D12H21)1,

(11-11)

where the Hermitian conditions (H,' = H , and Hike = H , ) are used. The right-hand side of Eq. (1 1-11) may be recognized as the matrix element of the product of two matrices:

(a:L) = [(HD)II-(DH)11].

iA

(11-12)

The right-hand side of Eq. (1 1-12) may be rewritten by a matrix element of the commutator of the operators H and D: (1 1-13) More generally, (11-14) It is customary to rewrite Eq. (11-14) as a relationship between the operators D and H themselves, (11-15)

159

In a canonical ensemble consisting of N identical systems, the n-the system has internal coordinate qn and a wavefuction Y(q, t ) = a,'"'(fV1(4" 1+ 9

(11-16)

for the the n-the system with elements defined as in Eq. (11-5) may be

An operator D,, obtained:

D,,'"'

4") ( t ) & (4" 1 .

= aj'"'(t)a,'"'*(t) .

(11-17)

This operator can be used as in Eqs. (11-6) - (11-15). For example, the quantity represented by R can be expressed as Eq. (11-6) by ( R ( t ) ) , ,= Tr[D.Rl.

(11-18)

The ensemble average of N systems is given by (11-19) n=l

The right-hand side of Eq. (11-9) may be rewritten with Eq. (1 1-18) as . h l

(11-20) The following distribution law holds for the trace of matrices: Tr[AB] + Tr [AC] = Tr[A(B+C)] . The right-hand side of Eq. (1 1-20) may be rewritten with Eq. (1 1-21) as

(11-21)

(11-22) Eq. (11-22) means that the ensemble-quantum averaged value of R is the trace of the product of the matrices of R and p, where .the matrix of p is the linear average of the D,, matrices: (11-23) The operator p is called "the density operator" and its matrix "the density matrix." right-hand side of Eq. (1 1-22) may be rewritten with Eq. (11-23) as

(R(f))N= Trlp(t)RI.

The (11-24)

We can see that the equation of motion for p is the same as that for D (Eq. (1 1-15)

i h ( y ) = [H, p].

(11-25)

An equation like Eq. (1 1-25) occurs in classical statistical mechanics, where it is called "the Liouville equation."

11.2 Density Matrix Treatment for S-To Conversion of Radical Pairs In Section 3.3, we considered the S-To conversion of radical pairs with the method of Kaptain. In this section, we will reconsider the S-To conversion with the method of the density matrix [2]. In Eq. (11-25), we take the two basic functions as follows:

160

Here, I S , x N ) and lTo,xN)weregiven in Section 3.3. Thejk- element p is given as

---aj"'(t)a,'"'*(t).

(11-27) = --y(D,), I N = I N N n=l "=I Because N radical pairs are equivalent with and independent of one another, we will simply write (p),kas if it is the element for one system involving ( S , x N )and 1 T O , x N ) .Thus, all the (P)jk

elements can be expressed as follows: (P)II

l N

= N-ya""'(t)al'"''(t) = a l ( t ) a l * ( t = ) pss,

(1 1-28a)

n=l

l N (p),z = ---ya,'"'(t)a,'"''(t)

= a,(t)a,*(t) =

pw,

(1 1-28b)

"=I

l N (p)21 = -Ca2("'(t)a,'"''(t) = a z ( t ) a l * ( t= ) pros. N n=l (PI22

=

FI cN a,'"'

(t)a,'"'*(t)

a2

(t)a,*( t ) =

(1 1 - 2 8 ~ ) (1 1-28d)

hOT0,

"=I

Thus, the time profile of the S-TOconversion of a radical pair can be described from Eq. (1 125) as

In Chapter 3, the matrix elements of HRPwere obtained as follows: (11-30) Thus, the time evolution of p(t),k can be expressed as

(11-31)

Problem 11-2. Prove Eq. (1 1-31). Because the population of this radical pair is constant, the following condition should be held:

a Z b S S

+PTOTO)

(11-32)

= O.

1.

From Eqs. (1 1-3 1) and (1 1-32), the following equation can be obtained:

['

at

]

Pss - PTOTO

Ps~o PTOS

-2Q

2Q

=

-23

When the following vector R is defined as

][

Pss - P T o T o

PSTO PTOS

(11-33)

161 Pss - PTOTO

(11-34) The both sides of Eq. (1 1-33) are transferred as (11-35)

Problem 11-3. Prove Eq. (1 1-35). The solution of Eq. (1 1-35) can be obtained by standard methods A’

A

Q . -sin(Rt)

R(t) =

A

Q9 -cos(Rt)] --[I

A cos(Rt)

9 .

-sin(Rt)

A’

3

- -sin(Rt)

A

Q‘

R(t=O),

(11-36)

g2

-+,cos(Rt)

where D is defined as

0 = 2 A = 2dQ2 + S 2

(11-37)

Problem 11-4 Prove Eq. (1 1-36). We have derived Eq. (1 1-36) with mixing of the singlet and triplet states of a radical pair in mind, but it is quite general for the time evolution of two levels separated by an energy under a stationary perturbation. In order to get the dynamic behavior of a radical pair, we should add a diffusion operator D and an operator K for the chemical reaction to Eq. (1 1-29),

a

I

- p = --[H,pJ+

D p + Kp. (11-38) at A This equation is called “ the stochastic Liouville equation.” We cannot obtain the general analytical solution of Eq. (11-38) because we cannot solve the coupled equations of the diffusion of radicals and the distance dependence of J. Eq. (1 1-38) can onIy be solved numerically. In order to get some sort of physical picture, we will try to find approximate treatments of Eq. (11-38). When two radicals in a radical pair react with each other, the d, distance (r) between them is denoted by r = d , which is usually about 0.4 nm. At r the exchange integral becomes much lager than the S-To mixing ( 9’>> Q2).This region is called the exchange region, where Eq. (1 1-36) can approximately be expressed as

-

(1 1-39)

This result shows that there is no S-TOmixing in the exchange region, the radius of which may be denoted by ro . Typically, ro = 0.6-0.8 nm. During a diffusive excursion of two radicals, they spend most of their time outside the small exchange region (r > ro ). In this region, the S-TOmixing becomes much larger than the exchange integral (3’ << Q 2 ) because p(r)l drops steeply with increasing r from ro. In this region, Eq. (1 1-36) can approximately be expressed as

? 62

L

cos(2Qt) -sin(2Qt)

R(t) =

sin(2Qt)

cos(2Qt) 0

I

0 0 R(t=O), 1

(11-40)

This result shows that the S-TOmixing occurs in this region and that the third element of R can be decoupled with its first and second one. Usual reactions of radical pairs occur through the following scheme: (1) Two radicals of each pair are initially created at r < ro, where geminate recombination may take place. (2) Two radicals which escape from recombination start to separate from each other to the region r > ro, where the S-TOmixing occurs. (3) Two radicals have a chance of meeting at r=d again at some later period of time from t to t+dt. Let us denote the probability of this event byf(t,d)dt, With this probability, we can calculate the average density matrix (fi) for an ensemble of radical pairs, where every pair has either completed a single diffusive excursion or separated forever. If we neglect the dwell time ofiradicals in the exchange region against the time during which they keep outside this region, R is given by -s

R(t)= p[z 0

c 0

0 0) R(t=O), 1

(11-41)

where c and s are related to the cosine and sine transformations of f(t,d) at the angular frequency of the S-TOmixing,

1"

c = - Icos(2Qt)f (t,d)dt , Po

1".

s = - Ism(2Qt)f (t,d)dt ,

(11-42)

(11-43)

Po andp is the total probability of at least one re-encounter,

-

P=

I f 0.d)dt .

(11-44)

0

Problem 11-5. If there is no significant attractive or repulsive interaction between two radicals, the probability of first encounter at r=d from the initial distance at r=ro is expressed as (1 1-45) With Eq. (1 1-45), verify the following relations:

p = d /ro,

( 11-46a)

c = cos(z)exp(-z),

(1 1-46b)

s = sgn(Q)sin(z)exp(-z), (sgn(Q)=l for Q>O and sgn(Q)=-1 for Q
( 11 - 4 7 ~ )

Here, the parameter z is defined as ( 11-47d)

163

Because the geminate recombination occurs from the singlet radical pair, it will therefore be more convenient to consider the dynamic equation for p, = p s s, pb = pToro, and

p, = i(pTos- p S r O ) .Outside the exchange region, a density matrix from Eq. (1 1-41) as

Wt))can

be derived

(11-48)

0 is defined as at)=

e(0) . Problem 11-6. With use of the relation that (pss+ pToro)(t) = (pss+ proro)(0), prove Eq.

where the transform matrix

(1 1-48). When the radicals encounter at r=Q, the singlet pairs react with probability of A. The surviving radicals diffuse apart to Fro and start a new reencounter cycle. By defining & as the average density matrix at the n-th reencounter and B' as the average density matrix of surviving radical pairs at the n-th reencounter, we obtain from Eq. (11-48)

en( t ) = &Lpn-i'(0).

(11-49)

The fraction of radical pairs that recombine at the n-th reencounter (F,) is given as

[::3

a =(I

Fn= (I 0 0) 0 0 0 Thus, ,a' becomes

[ ,"

1-a

&'=@-&,a=

0

0 ) A a = (1

0 0)AMpn.l'(O).

;I&.

;o o

(11-50)

(11-51)

By defining F as the total fraction of radical pairs that recombine in the first encounter and all reencounter we can obtain it as follows: F=(l 0

O ) A g [M (E- &I"&.

(1 1-52)

"=O

Here, the n=O term of the right-hand side of Eq. (11-52) has a value of Ap, , which represents the probability of the geminate recombination within the primary cage. By summing the series of the right-hand side of Eq. (1 1-52), we can arrive at the result

F=(l 0

O)A[E-

M ( E - &I-'&.

(1 1-53)

Problem 11-7. Prove Eq. (11-53). Because the dimension of the matrices in Eq. (11-53) is only three, it is possible to carry out the matrix inversion and put Eq. (11-53) into an analytical form. From Eqs. (11-48), (1 1-5I), and (1 1-53), we can get

a

F = pss(0)--[1- p(1+3c)/2 + p 2 c / 2+ pZ(s2 + c 2 ) /21 N

a

+ proro(O)-L(lN

c ) / 2 + p2(1- c ) / 2 - p 2 ( 1- s 2 - c2)/2],

( 11-54a)

164

where

N = I - p[i +

- a ( i + c ) / 2 ] +p z

is2 + c z + 2c - iqsZ+ c 2+ 3c )/2] - p 3(1-

+cz). (11-54b)

Problem 11-8. Prove Eqs. (11-54). We can use two parameters, A and F*, to rewrite Eqs. (11-54) to a more useful form. A is the spin-independent probability of the reaction from a singlet precursor. This means Q=O, c=l, and s=O in Eqs. (1 1-54) A = F(S; ~ = 0=) a/(i- p +

pa).

(11-55)

F * is defined as the probability of forming geminate product from a triplet precursor for &I. This measures the conversion from the triplet radical pair to the singlet one and vice versa. Eqs. (1 1-54) lead to the following expression for F*: F * = F(To; =I) =

p('+ pz[(I- (l- s 2 - '* 11 2(1- p)' + 3p(l- c) - p'[(l- c )+ (1 - s 2 - 2)j.

(11-56)

Thus, we can rewrite Eqs. (11-54) as (1 1-57a) F(S) - A = -

A(I - A)F * l+F*(l-A)'

A 1 A2F* F ( FR) - - = 2 2l+F*(l-A)

(11-57b) (11-57~)

Here, the spin multiplicity of the precursor is written in parentheses; (a) for To-precursor pss(0)= 0 and proTo(0)=l,and (b) for S-precursor pss(0)= 1 and pToTo(0)= 0. FR means that the reaction occurs from free radicals (F-precursor) where pss(0) = proTo(0) =1/2. Problem 11-9. Prove Eqs. (1 1-57). It is noteworthy that the expression for F * can be simplified when z <<1. By expansion of c and s in power of z, substitution in Eq. (ll-56), and retaining only the first order terms, we obtain the result (11-58)

Problem 11-10. Prove Eq.(11-58). This asymptotic dependence of F * on Q'" is well known. Freed also proposed that F * could well be approximated by the following analytic form [3]:

(11-59)

165 11.3 Effects of Spin Relaxation on Dynamic Behavior of Radical Pairs In the previous section, we learned how the recombination probability of a radical pair depended on the coherent mixing of the spin states caused by the S-To mixing. On the other hand, it also depends on the incoherent mixing caused by transversal and longitudinal relaxation. In this section, we will show that one can obtain a qualitative understanding of these effects by considering the spin evolution described by the Bloch equations. From textbooks of Quantum Mechanics, any Hermitian operator F has the following property:

d I; = L[H,F].

A Here, H is the Hamiltonian of the considered system.

dt

(11-60)

Problem 11-11. Prove Eq. (11-60). From Eqs. (2-4) - (2-6), the Hamiltonian of an electron in an external magnetic field is given by H = - h * B= g/.!,S,B AUS,. (2-6) Here, the external magnetic field is applied along the z-axis. we have the following relations:

From Eqs. (1 1-60) and (2-6),

d (1 1-61a) dt A d 1 - S - - [H, Sx] = ia;i S,, Sx] = iw is, = -as,, (1 1-61b) dt ' - A i d (11-61~) - S,=--[H,S,]=ia;[S,,S,]=i~- is,)=&,, dt Problem 11-12. Prove Eqs. (11-61). BecauseS,=S,kiS,,wecanalsogetfromEqs. (11-61b)and(ll-61c)as d -s, =w(i2s,f i s , ) = + i o ~ * . (1 1-61d) dt Since the S,, S,, and S,, values of the non-interacting radicals in a radical pair at the thermal equilibrium are zero, their spin mixing can be described by Bloch equations d 1 (11-62a) - SZ(t) = --Sz(l), dt TI

- S, = I [H, S,] = ia;i S,, S,] = 0,

d

1

-s, (I)= ( f i w - -)S*

(11-62b) (t). T2 Here, TI and T2 are the longitudinal and transversal relaxation times, respectively. It is convenient to start with the direct product basis, which consists of the following 4 basis states:

dt

I1)=Iolp), 12)=1Pa), I 3 ) = l a a ) , and 14)=1PP).

(11-63)

The rate equation for the density matrix p i s block diagram, indicating that not all matrix elements are coupled. Two of the elements @12 and pzl) are completely uncoupled from the S2+ rest. Let us consider the rate equation of SI.

166

= (-W, - i2Q)Sl_S,+.

(11-64)

Here, the constants are defined as follows: (1 1-65a)

Q=(UA-UB)/2, W,, = UTZv(for

v = A and B),

(11-65b)

w, = w,,

+WZB. The average values of the both sides of Eq. (1 1-64) can be rewritten as

(11-6%)

(1 1-66a) <(-W, - i2Q)S1-Sz+> = Tr[p (-Wz - i2Q)S1-S,+ 3 = pI2(-W, -i2Q). Problem 11-13. Prove Eqs. (1 1-66a). Thus, Eq. (11-64) becomes Piz

( t ) = (-W2 - i2Q)pIz ( t ).

(11-66b)

(1 1-67)

Eq. (11-67) is easily solved

PI2(t) = exp(-W,t)exp(- 2iQt)Plz(0) = exp(-W2tXcos(2Qt) -isin(2Qt)XRepl, (0) +iImp,,(O)}.

(1 1-68a)

Similarly, the rate equation of S1+ S2. gives the result PZ1 0) = exp(-W,t)exp(+ 2iQt)pz1(0)

+

= = exp(- WZtXcos(2Qt) isin(2Qt)XRe pZ1 (0)+ i ImpZ1(0)).

(11-68b)

Summing Eqs. (11-68a) and (1 1-68b), we can obtain

bI2 + pZl)
plZ(O) + 2sin(2Qt)Impl,(0)}. Here, we used the relations Re pI2(0) = Re pZ1(0) and Im plz(0) = -Im pZ1(0).

(11-69)

The remaining 4 matrix elements satisfy the rate equation

(11-70)

which only contain the individual and total longitudinal relaxation rates

W,, = lIT,, (for v = A and B),

(1 1-7la)

w,=Wla+WIB.

(11-7lb)

Problem 11-14. Prove Eqs. (11-70). We can see from Eq. (11-70) that the diagonal matrix elements are time independent in the absence of longitudinal relaxation. Eq. (1 1-69) illustrates the difference between coherent mixing (Q) and relaxation (WZ). The former causes an oscillatory behaviour of the offdiagonal elemebts, while the latter causes an irreducible decay of the elements. In order to illustrate how this time dependence causes transitions between initial triplet and singlet spin

167

states, we express the populations of these states in terms of elements in the direct product basis. The resulting expressions are 1

PTOTO

=~

1

Pss

=

%$b

+ P2l + P22

(1 1-72a)

- PI2 - P a + P 2 2 ) .

(1 1-72b)

l +lPI2

Here, the TOand S states are given by (11-73a)

(11-73b)

Problem 11-15. Prove Eqs. (1 1-72). From Eqs. (1 1-72), the following relation can be obtained: PTOTO ( t ) - P S S ( t )

=

(PI2 + P2l

(11-74)

)<'I '

When a reaction occurs from a T-precursor, only the real part of p,,(O) appears in Eq. (1169) and the right-hand side of Eq. (1 1-69) gives PToTo(t)

- PSS 0)= co~(2Qt)ex~(W2t).

(1 1-75)

This equation shows that the coherent mixing causes as oscillation between the TO and S states, but that the transversal relaxation tends to equilibrate the two states. In the coupled basis, the TO,T+1 (=IT+)= Ioa)) , used.

a -

Pro

PSrO

PTOS

Pros

at Proro

= (M1+%)

PT+T+

T.1

(=IT_)= (@)) , and S states are

(11-76)

PTOTO PT+T+

where

iv&=

iQ -iQ W2/2 0

-W212 W2/2 Wz/2 -Wz/2 -iQ iQ 0 0

-iQ iQ -W2/2 0

0 0 0 0

0 0 0 0

(1 1-77a)

168

0

0

-w,

w,

w,

-W, -W, 0

-W, -W,

0 0

-Awl -AW,

Awl AW,

0

-w,

w,

w,

W,

-Awl

-Awl

-2Wl

0

W,

AW,

AW,

W, W,

0

-2W,

f-q

' 1 -

I&=

4

0 0

-w,

(1 1-77b)

Here, we have introduced the notation

AW, = W,, -WIB.

(1 1-77~)

Problem 11-16. Prove Eqs. (1 1-77). The analytic formulas for Eq. (1 1-76) are expressed in terms of dimensionless variables defined as

d2 q=Q-, D

(1 1-78a)

dZ (1 1-78b) k j = Wj - (forj = 1,2), D where D = Dl + D2 is the diffusion coefficient for the relative motion of two radicals, and d is the distance of closest approach. First, let us consider the case where there is no longitudinal relaxation. In the absence of coherent Q-mixing, it is clear from Eqs. (11-75) and (11-77a) that the spin mixing occurs as a direct transition from S to To in which the ofdiagonal elements of p (in the coupled basis) play no role. The recombination yield of an unpolarized triplet precursor, induced solely by the transversal relaxation, is

(11-79) The yield cannot exceed one third since the T+1 and T.1 states are not coupled to the S state by the transverse relaxation and thus cannot contribute to the recombination. The coherent Q-mixing is seen to involve the off-diagonal elements of p between S and To states, but the expression for the recombination yield of an unpolarized triplet precursor, induced by coherent mixing, is identical in form to that for transversal relaxation, i.e.

-

Jq Y ( 4 )= -T

(1 1-80)

When both effects are present, the transverse relaxation interferes with the Q-mixing because the relaxation makes the off-diagonal elements decay. The expression for this case is

J"+

1 YT(k2,q)=-3 2+&'

(11-81)

where

Q, = , / m + k 2 / 2 When all spin mixing processes are included, the following expression is obtained:

(11-82)

169

-

1

1 y r ( k , , q )= 3 .

1+Jk, 1 I+-+l+&

2 l+JQ+

2

(11-83)

‘

I+&

Problem 11-17. Prove Eqs. ( I 1-79), ( 1 1-80), ( 1 1-81), and (1 1-83). 11.4 Theoretical Analysis of MFEs, CIDNP, and CIDEP with the Density Matrix Method The theoretical analysis of MEFs and CIDNP due to the S-To mixing can be carried out with Eqs. ( 1 1-57). With these equations, the theoretical calculation of MEFs and CIDNP due to the S-To mixing is possible. In Chapter 12, typical results of such an analysis of MFEs will be shown. The theoretical analysis of CIDEP due to the S-To mixing can also be carried out with the density matrix method as shown in the following part of this section [5]. The electron spin polarization (Pa)of radical A in a nuclear state (a) can be written as (11-84) Po = 2Tr[p(a)S~zl= d a ) , - p(a), .

Problem 11-18 Prove Eq. (1 1-84) It is convenient to introduce a quantity P a b = 2Tr[p(a,b)S~~l,

(Pab)

by the definition (11-85)

where p(a,b) is the electron spin density matrix for the pair of radicals A and B in specified nuclear spin states of the pair. The quantity P,b may be interpreted as the conditional electron spin polarization of radical A in a nuclear spin state a when radical B is known to be in a nuclear spin state b. The unconditional polarization Paof radical A in a nuclear state a is obtained by averaging Pat,over the nuclear spin states of B

11,

P, =-,

(11-86)

‘ob

Cbl

where the initial value of P,b is normalized to be unity. From Fig. 5-1, we can see that the ESR signals are in absorption and emission for negative and positive P , respectively. From problem 5-1, we can obtain the P values in the thermal equilibrium (Peq)at room temperature for the ESR transition of an electron spin at the X-, K-, and Q-bands.

In the following, p(a,b) is simply denoted p and the singlet and triplet states are used as a basis for the electron spin space of the pair. Because 2SAz can be expressed as 2sAz = (SAz - S B z -k (SAz + SBz )> Eq. (1 1-85) for the conditional polarization of radical A can be rewritten as Pab(t) =

brio ( t )

f

pros (t)]-k b r + r +( t )- Pr-r-

(f>1.

(11-87)

(1 1-88a)

Similarly, the polarization of radical B is Pbdf) =-

b,o(t) + Pros (r)l+ b r + r +( t ) - Pr-r- (t)l.

(11-88b)

Problem 11-19. Prove Eqs. (I 1-88), In high magnetic field, fi+T+(t) and h . ~ . ( t ) are time independent when the S-To mixing is only considered. Thus, the polarization is given by the first term of the right-hand terms of Eqs. ( 1 1-88). From Eqs. ( I 1-36) and ( 1 1 -88a), &(t) can be given by

170

Pab(t) =

[ p r o ( t )+ Pros ( t ) ]= (Q3/ A 2 )[I-

cos(~t)l(Pss- PToTo )(o) .

(11-89)

If the radicals stay in the exchange region before separation with a probability

exp(-t / 2 ) z the average polarization generated before separation is P(t) =

(11-90)

(11-91)

Problem 11-20. Prove Eq. (1 1-91). The polarization generated by this mechanism is sometimes called cage polarization because it is generated when the radicals A and B are together in the solvent cage. In liquid, the cage polarization is negligible compared with the reencounter polarization discussed below. For the photochemical processes in the solid state, however, the cage mechanism may be important and it appears to be responsible for the polarization observed from the primary processes of photosynthesis. The reencounter mechanism, usually just called radical pair mechanism, involves two steps, i.e. the radicals must separate and reencounter later. When the radicals are separated, the exchange integral becomes negligible (3’ << Q’). From Eq. (11-40), an off-diagonal element of the density matrix is developed i(P.m - Pros

k)= sM2Qt)bss - ProTo KO) ,

(1 1-92a)

but the electron polarization is not induced outside the exchange region from Eq. (1 1-40)

(Pro + Pros )
(11-92b)

From Eq. (11-41), the average value of the left-hand side of Eq. (11-92a) with respect to the duration of a reencounter can be expressed as i(Pn0 - Pros), = PS(Pss - Proro KO)

.

(11-93)

When the radicals reencounter, the element in Eq. (11-93) is converted to the electron polarization by the exchange interaction from Eqs. (11-39) and (11-93)

(Pro +Pros Kt) = sin@) = sin(@

PdP,

+no

- Pros), + cos(Qt) (Pro +Pros),

- Proro)
(11-94a)

, is zero from Eq. (11-92b). Here, it is noteworthy that the (pn0+ p r o s )value

Because 3* >> Qz at the exchange region, Eq. (11-94a) becomes as follows:

(Pro + pros)(t)= sgn(Q3)I~slsin(2131t)bsS - P T o T o )(O) .

(11-94b)

If the average time spent in the exchange region is given by Eq. (11-90), the average polarization developed in this single reencounter cycle can be given by (1 1-95)

Problem 11-22. Prove Eq. (11-95).

171

-

For small magnetic interactions (z << l), the characteristic function of s(z) given by Eq. (1 147c) is approximated as 1s(z)) z . In this case, Pat,is approximated as

(1 1-96a) Furthermore, in this limit, but only in this limit, the effect of multiple reencounter can be included simply by dividing the right-hand term of Eq. (11-96a) by (1-p) (11-96b)

References [ l ] “Dynamics During Spectroscopyic Transitions”, E. Lippert and J. D. Macomber Eds., Springer, Berlin, 1995. [2] J. B. Pedersen, J. Chem. Phys., 67 (1977) 4097. [3] J. H. Freed, in “Chemically Induced Magnetic Polarization”, L. T. Muus, P. W. Atkins, K. A. McLauchlan, and J. B. Pedersen Eds., D. Reidel, Dordrecht, Holland, 1977, p.309. [4] M. J. Hansen and J. B. Pedersen, RIKEN Rev., 44 (2002) 34. [5] M. J. Hansen, A. A. Anatole and J. B. Pedersen, Chem. Phys., 260 (2000) 125. [6] ‘Theories of Chemically hduced Magnetic Polarization”, J. B. Pedersen, Odense Univ. Press, Odense, 1979. Solution to Problems 11-1. From Eq. (1 1-2), (R(t)) is given as

-

(w) = ~Y’(%t)li(q)Y(%t)dy -

-

= j{al(041( 4 )+ a2 ( 0 4 2 (4)F &){a,

--

014, ( 4 )+ a* 0 1 4 2 (dl4

= the right side of Eq. (1 1-3).

11-2. From Eq. (1 1-29), the time evolution of (p),, ( = ps ) is obtained as

i

= --[(HRP)II ( P ) I I +(HRP)Iz(P)zI-((P)II(HRP)II-(P)I~(HR h = -i[Q(p)z~- Q(P)IzI= -~[QPTosQPswI. Similarly, the other elements can be obtained. 11-3. From Eq.(11-33), the first element of R are given as follows: p ~ -sp r ~ m= (-9(-2Q p m + 2Q ) = -2Q[i( Pros - BW11. Similarly, the other elements can be obtained.

172

-A 2Q

o

-2Q

0

-1 - 2 3 =O. 23

-a

2iQ x(O)=A, +-(B, Q

-C2),

173 2i3 ~ ( 0=)A, --(B2

n

- C,) .

From the above equations, A1, A3, B, -t C,, B, k C, and B, y(O), and z(0) values as B,

* C, can be expressed by the x(O),

+c,= Y ( O ) ,

B, - C , = Qx(0)-3z(0) (This is derived fromx(0) and z(0) with 2QA, = 23A,), iA

(B, f C , ) = 2ie(B2 T C , ) (The right-hand side can be expressed by x(O), y(O), and z(O).),

n

2i3 (B, k C,) = --(B,

n

T C,) (The right-hand side can be expressed by x(O), y(O), and Z(O).).

Finally, x(t), y(t), and z(t) can be expressed by

2iQ x ( t ) = A l +-(B,

n

2iQ -C,)cosQt+i-(B,

+C,)sinnt

n

y ( t ) = A , + ( B , +C,)cosQt+i(B, -C,)sinQt = (cosQt)y(O) + i(sinQt) Q 4 0 ) - %(O)

iA

Q = -((sinQt)x(O) + (cosQt)y(O) A

2i3 z(t) = A, --(B,

n

3 .

- -(smRt)z(O),

A

2i3

- C,) cosnt - i-(B,

n

+ C,)sin Qt

11-5. When t is rewritten as t = x 2 , dt becomes to be dt = 2xdx. Thus, c and s can be expressed by the following definite integrals:

174

Eq.( 1 1-46a) can be proved as Problem 3-6. 11-6. From Eq. ( 1 1-41), the following relation is obtained: bss

- Proro >(t)= C(P,

With this relation and (p,

-

- PTOTO 10) s Pc ’

+ proro)(t) = (p,, + pTorokO), we can derive

For other elements, we will be able to obtain similar results. 11-7. See page 334 of Ref. [l]. 11-11. Suppose we have a pair of wave functions f i t ) and Rt),both of which are solutions of the same Schrtidinger equation:

Let us have an Hermitian operator Fthat has no explicit time dependence. Then

$-jD

* m d r + j @ * F -dY d z = L j @ * ( H F - F H ) Y d r . dt A 11-12. The commutator relations for the spin operators S =( S, ,S, ,S, ) are represented, for example, as follows: dj@*mdr dt

=

=(S,(S++ S - ) - ( S ++ S - ) S z ) / 2 .

[S,,S,]=S,S,-S,S,

I

From Chapter 1, [S,,S, ] S , m, ) becomes

IS,>S,llS>ms> = 1 2

-(.JS(S+ 1) - m, (m,+ 1)1 S , m, + 1 ) - &S + 1) - m, (m,- 1)1S , 312,

- 1)) .

On the other hand, S,I S , m, ) is given by 1 S,(S,m,)=-(S+ 2i

-S-)IS,m,)=

1

-(Js(s + 1) - m, (m,+ 1>1S , m, + 1) - JS(S + 1) - m, (m,- I>~s, m, - 1)) 2i

Thus, we can prove [S,,S,]= is,. We can also prove similar relations. 11-13. The non-vanishing matrix element for S,-S,+is only (21Sl-Sz+11), which is given by (2lS*-&+l1) = (PalSI-S2+1@) = (pa(sI-Iaa)(aa(S,+IaP) = 1,

175 where Eqs. (1-36c) and (1-36d) are used. 11-14. From Eq. (1 1-62a), the average values of S I , can be rewritten as

11 = -I( A .

d dp(t) S1,(t) > = Tr[ (-)(Slz dt dt

<-

2

- P22+ P33- PM)

(1) Eq. (1) becomes the following differential equation:

(PI1 - P 2 2

=

+P33 -P44)

- K 4 ( P l l -P22 +P33

(2)

-Pa).

From Szz,we can obtain a similar equation as

(-PI1 + P 2 2

= -wlB(-pll

+P33

a)

'p22 +P33 -P44)

From SlzSzz, we can obtain the following relation:

(4) From Eq. (4), we can obtain the following equation: =

(-P11-Pz+P33+P44)

d dt

From a unit operator (-(I)

(5)

-W,(-P,,-P22+P33+P44).

= 0), we can obtain the following equation:

(6)

=''

(PII+P22+P33+P44)

From Eqs. (2), (3), (5), and (6), we can prove Eq. (11-70). For example,

P1, is given as

1

PI1 = T ( - y p l l

fW1Bp33

+%AP44).

11-15. When a function @(t)is expressed in the direct product basis, it is given by 4

C(t> =

Cqt>Jj). j=l

From Eqs. (1 1-73), the triplet and singlet spin states are expressed as ITO) =

Is)=

1 $ I

4 I Pa)) +

1

z1 (1.P-)I B.% 1

11) = lap) and 12) = Pa) can be transferred as 1

Il)=laP) = ~ ( I T n ) + I S ) ) . 1 2 ) = ) P a )= q G ) - ~ s ) ) .

J?s

Thus,

@ ( t )becomes as

(7)

176

From the above result, prorocan be expressed as

11-21. From Eqs. (1 1-28b) and (11-28c), TO (0) and p r o s (0) are shown to be zero for both of the reactions from S- and T-precursor as follows: p T 0

(0) = a, (O)U,*(O) = 0,

p r o s (0)= u2(O>a,* (0) = 0.

11-22. This problem can also be solved as Problem 11-20.

177

12. Effects of Ultra-High Magnetic Fields upon Chemical Reactions 12.1 Historical Introduction Ordinary magnetic fields less than 2 T can easily be generated by electromagnets. In order to get higher fields than 2 T for measurements of MFEs on chemical reactions in a usual laboratory, one can use a super-conducting magnet with a large room temperature bore. In 1976, the author’s group [ l ] first introduced a super-conducting magnet for such measurements as mentioned in Chapter 6. Our maximum field was 4.3 T, which was induced by a small magnet for an MCD apparatus. In 1982, Boxer et al. [2] first made an nslaser photolysis apparatus with a super-conducting magnet, the maximum field of which was 5 T. In 1993, the author’s group [3] extended the maximum field to 10 T for a ns-laser photolysis apparatus. Usual super-conducting magnet should be cooled by liquid helium and the maxinium field for such magnets has arrived at about 20 T. Recently, He-free superconducting magnets have been developed in Japan. Their maximum field has attained to 10 T and the diameter of their large room temperature bore has exceeded 10 cm. In Japan, USA, and Europe, there are several magnetic laboratories, where many gigantic magnets have been installed. Among them, Bitter type magnets are useful for measurements of MFEs on chemical reactions at room temperature under steady fields. In 1982, Turro et al. [4] measured MFES at one high-field value of 14.5 T produced by a Bitter type magnet. In 1994, Steiner’s group [ 5 ] performed a ns-laser photolysis study with a similar magnet, varying the field gradually from 0 T to 17.5 T. Much higher steady fields of up to 35 T can be induced by hybrid magnets composed of super-conducting and Bitter type magnets, but there has been no investigation of MFEs with such a hybrid magnet. This may be due to the fact that it is very difficult to get its machine time sufficient for the measurement of MFEs. Higher fields than 20 T can also be obtained by pulsed magnets, the maximum field of which has attained to 80 T at 77 K. Because usual pulsed magnets should be cooled by liquid nitrogen, they have been mainly used for the studies of solid state physics. It is also possible for pulsed magnets to be operated at room temperature for measurements of MFEs on chemical reactions, although their maximum field should be much reduced from 80 T. In 1987, Ferraudi and Argiiello [6] first used a pulsed magnet for ns-laser photolysis measurements at room temperature. The maximum field of their magnet was 5 T and the diameter of its bore was 10 mm. In 1993, Tanimoto’s group [7] made a ns-laser photolysis apparatus with a pulsed magnet. The maximum field of their magnet was 14 T and the diameter of its bore was 16 mm. Because the repetition rate of their magnet was 1/30 sec, they could not accumulate transient signals. In order to extend the maximum field to about 30 T, the author’s group [8] tried to develop a pulsed magnet for a ns-laser photolysis apparatus. The maximum field of our magnet was 29.6 T and the diameter of its bore was 23 mm. Because we could attain the repetition rate of 1/3 sec for our pulsed magnet, we could realize very accurate measurements of MFEs with signal accumulation together with a double-beam probe system. In this chapter, we will show several typical MFEs on chemical and biochemical reactions which have hitherto been observed under ultra-high magnetic fields with various magnets mentioned above. 12.2 Effects of Ultra-High Magnetic Fields due to the AgM As mentioned in Section 6.6, the author’s group studied MFEs on a singlet-sensitized photodecompostion reaction of dibenzoylperoxide in toluene at room temperature with a small super-conducting magnets, whose maximum field was 4.3 T [ 11. We found that the field of 4.3 T decreased the yield (Ycs)of the cage product (phenyl benzoate) by 8+3 % and

178

that the magnetically induced decrease in the yield (AR C ) was proportional to B’12. We also found similar increases in the yield of escape products (YE’). We interpreted these MFEs in terms of the AgM, where the magnetically induced changes in the yields can be given by Eqs (6-19) and (6-20) as AR c (B) = [ Ycs(B)- Ycs(OT)]/ Ycs(OT) = - m l ~ 4 g pB, / 2hI1”/ p ,

AR E (B) = [ YE’@)

-

(12-1)

y~’(oT)]/ YE’(OT) = - [Ycs(B)- Ycs(OT)]/[1 - Ycs(OT) ]

= - ( Ycs(OT) /[1 - Ycs(OT) 3 ) AR c (B).

(12-2)

As shown in Section 6.6, the observed MFEs on the yields of the cage and escape products in this reaction could quantitatively be explained by Eqs (12-1) and (12-2). It is noteworthy, however, that the effect of the AgM on the S-T conversion is smaller than other reaction processes. This means that its effect was so small that the magnetically induced decrease in the yield was proportional to B’”. At that time, there was a theoretical and experimental question of how chemical reactions could be effected by much higher fields. As mentioned in Chapter 11.2, the general results obtained by Pedersen for MFEs in high magnetic fields for freely diffusing systems [9] are given by F * which is defined by Eq. (1156) as the probability of forming geminate product from a triplet precursor for h=l

F* = F(To; h =1) =

p(l-c)+ pq(1-C) -(1- s2 -c2,1 2 ( 1 - ~ +3p(l-c)-pz[(l-c)+(l-s2 )~ -c2)]’

(12-3)

This measures the conversion from the triplet radical pair to the singlet one and vice versa. Freed [lo] proposed that F * could well be approximated by the analytic form expressed by Eq. (11-59) as

(12-4)

Here, q is denoted by

q=lQld2/D

(12-5)

The expression for F* can be simplified as

%

F*=: 2.

(12-6)

I+;& When q <<1,

a.(12-6) can further be simplified as (12-7)

Eq. (12-7) shows that the magnetically induced changes are proportional to B’” for low fields ( q <
(12-8)

179

For a typical situation (D=10~5cm2s~' and d=0.6nm), MFEs due to the AgM can almost be saturated at the following field: AgB = 1.OxlOT.

(12-9)

Problem 12.1. Prove Eq. (12-9). Eq. (12-9) shows that MFEs due to the AgM for Ag=O.Ol are saturated at extremely high magnetic fields of the order of lo3 T. Schulten and Epstein also obtained a similar result for the saturation of MFEs due to the AgM [ 1 11. Table 12.1. Dependence of F* on (Qld' I D for small values of J. (Reproduced from Ref.

Boxer's group [2] first made a ns-laser photolysis apparatus with a super-conducting magnet. The sample was excited at 532 or 600 nm with a frequency-doubled YAG pumped dye laser (8ns, fwhm) and was probed at 860 nm with a laser diode. The maximum field of their magnet was 5 T. With this apparatus, they measured the quantum yield of triplet states (6) detected optically in quinone-depleted photosynthetic reaction centers (RCs) from R. spheroids, R-26 mutant, as a function of applied magnetic strength and temperature. The reaction scheme for qinone-depleted RCs is shown in Fig. 12.1. Here, the singlet and triplet radical-ion pair (RP)are represented by '[D+* A-'1 and 3[D+* A-'1, respectively, and the A-'1, and the rate constants of the S-T conversion of RIP, the recombination from 'ID" recombination from 3[D+0 A-7 are denoted by ST, ks, and k ~respectively. , The observed @ values at 100 and 293 K are shown in Fig. 12.2. This figure shows that the % value at each temperature decreases with increasing B from 0 T to 0.1 T but the value increases with increasing B from 0.1 T to 5 T. Because the generated RIPS are aligned in membranes, the component radical ions do not diffuse with one another. Thus, the RIP lifetime is determined by ks and k ~ . When kST is much smaller than k ~ the , magnetically induced decrease in @ with increasing B from 0 T to 0.1 T can be explained by the HFCM, through which ksT should be decreased by such low fields. On the other hand, the magnetically induced increase in with increasing B from 0.1 T to 5 T can be explained by AgM, through which kST should be increased by such high fields. As ksT approaches kT, the ( ks)), which is shown by a dotted increase in % should approach the limiting value ( k ~ /k+ line at each temperature in Fig. 12.2. The k~ values at 293 and 100 K were found to be 3.1~10'and 1.2x108s~',respectively [2]. As proved in Problem 12-2, the ST value becomes 2.42x108s-' when 9 = 0 s-', Ag = 0.0011, B = 5 T, and Ai l g p , = A, / gp, = OmT . It is noteworthy that the tendency of such saturation of 6 was found to appear at 5 T in RCs, where the Ag value is as small as 0.001 1. This filed of 5 T is much lower than the saturation

? 80

fields (the order of lo4 T) predicted for the MFEs due to the AgM (for Ag=O.OOl) in freely diffusing systems. Problem 12-2. Confirm that w,, = 2 . 4 2 ~ 1 0 ~when ~ ' ' 3 = 0 s-', Ag = 0.0011, B = 5 T, and Ai f gp, = A, f gpB = 0mT .

,

I[D+- A-'1 <

Fig. 12.1.

3[D+*

ksT

I

Reaction scheme for quinonedepleted photosynthetic reaction centers (RCs).

kT

3D+A

D+A 10

,

I

I

1

I

I

Fig. 12.2.

09

Quantum yield of triplet states (&) detected optically in quinonedepleted photosynthetic reaction centers (RCs) from R. spheroids, R-26 mutant, as a function of applied magnetic strength and temperature. The limiting value (kT/( k+ ks)) of & for high fields at each temperature is represented by a dotted line. (Reproduced from Ref. [2] by permission from The American Chemical Society)

0 8

07 06 0 5

04 03

02 01

0

10

20

30

40

50

Bl 1OOmT The author's group [12] tried to find saturation behavior of the MFEs due to the AgM in fluid solutions with our pulsed magnet and found that the MFEs on the escape radical yield (YE(@) observed for the photoreduction of 4-methoxybenzophenone with thiophenol (Reaction S-5 in Table 7-2) were almost saturated by the fields of -30 T. The isotropic gvalues of the thiyl and ketyl radicals have been determined to 2.0082 and 2.0027 so Ag=0.0055 [121. From ns-laser photolysis measurements with our electromagnet, superconducting magnet, and pulsed magnet, we observed the time profiles of the transient absorption (A@) curves) of the ketyl radical and obtained the MFEs (R(B)=YE(B)IYE(O T)) on the yield. The R(B) values obtained at room temperature in 2-methyl-1-propanol are plotted

181

against B"* in Fig. 12-3. As is clear from this YE(B)/YE(OT)-B'12 relationship, we can see that the magnetically induced change (dR(B)= [YE@) - YE(O T)]/YE(O T)) starts to deviate from the linear relationship between R(B) and B'" above 4 T and that the change attains a saturated value above 20 T. This is the first observation of the saturation of the MFEs due to the AgM in fluid solutions.

h

oc . . . . .*. .a . . .m

. .

Fig. 12-3. Magnetic field dependence of the escape radical yield (R(B)=YE(B)/YE(OT)) observed at 293 K for Reaction S-5 in Table 7-2 in 2-methyl-lpropanol with ( 0 ) an electromagnet, (x) a superconducting magnet, and )(. a pulsed magnet. (Reproduced from Ref. [12b] by permission from The American Chemical Society) Fig. 12-4. Simulated R(B) curves by Eq. (12-10) with (..........................) p = 1/3 and () p = 0.43. (Reproduced from Ref. [12b] by permission from The American Chemical Society)

0.9 0.8

0.7 0.6 0

5

10

15

20

25

30

B/T

At first, the magnetic field dependence of R(B) shown in Fig. 12-3 was analysed by the following empirical equation proposed by Freed for freely diffusing systems [lo]: (12-10) R ( B ) = 1- p ( B ) F * . Here, p(B) is the initial population of To and F* is given by Eq. (12-4). Fig. 12-4 shows the simulated R(B) curves by Eq. (12-10) withp = 1/3 andp = 0.43, respectively. We could see from this figure that the simulation withp = 1/3 did not reproduce the observed results, but that the simulation with p = 0.43 did. Because the latter parameter was not realistic, we tried to analyse the magnetic field dependence of R(B) with the following relation: 1 (3/2)Z (12-11) P ( B ) = - - Ps 3 B Here, p~ is the polarization factor and Z is the zero-field splitting of the triplet precursor. The 2 value was found to be 1.8 GHz from our ESR measurement of triplet 4-methoxy-

182

benzopheneone in methylcyclohexane at 77 K [12b]. The simulated R(B)curve with Eqs. (12-10) and (12-11) in 2-methyl-1-propanol is shown in Fig. 12-5. This figure shows that the simulated R(B) curve does not agree with the observed values at both low and high field regions. Especially, the nearly saturation of the observed R(B) values at high fields can not be explained by the simulated curve. Here, the saturated value may be 213, which is represented by a dotted line in Fig. 12-5.

Fig. 12-5. The observed R(B) values ( 0 ) for Reaction S-5 in Table 7-2 in 2-methyl-1-propanol and the simulated R(B) curve ( j with Eqs. (12-10) and (12-11). The saturated value of R(B) is represented by a dotted line. (Reproduced from Ref. [ 12b] by permission from The American Chemical Society) 10

5

0

15

20

25

30

BIT

Because Eq. (12-10) was derived from the S-T conversion due to only the AgM, Pedersen and the author made a cooperative investigation on this phenomena, taking the effects of the transversal and longitudinal relaxation [13]. The result was shown in Chapter 11 and R(B) is represented by R ( B ) =1-(113)YT(kj,q). (12- 12) Here, YT(kj,q)wasgivenby Eq. (11-83) as

I l+Ji;; Y T ( k j . 4 )= 1 I+-+I+&

2

3--

1+& 2

'

(12-13)

1+&

where the parameters in Eq. (12-13) are given in Chapter 11 as

Q = (wA

112

3

W,, = 1/TZv(for v = A and B), w2

+w,)j

= wzA

1

(1 1-65a)=(12-14a) (1 1-65b)=(12-14b) (1 1-65~)=(12-14~)

W,, = l/T," (for v = A and B),

(1 1-71a)=(12-15a)

wI

(1 1-7lb)=( 12-15b)

+w]B>

=WIA

d2 q=Q-,

(11-78a)=(12-16a)

d2 k j = W, - (forj = 1,2),

(1 1-78b)=(12-16b)

D

D

183

(1 1-82)=(12-17)

Fig. 12-6. The observed R(B) values (+) for Reaction S-5 in Table 7-2 in 2-methyl-lpropanol and the simulated R(B) curves with Eq. (12-13): q mixing (dotted line (a)), T2 relaxation (dash-dot line @)), q and T2 (solid line (c)), and q, T2,and TI (solid line (d)). This figure is reproduced from Ref. [ 141. We assumed that the triplet radical pair is not initially polarized (p(B)=1/3) and that the relaxation occurs though only the anisotropin Zeeman terms (Sg) given by Eqs. (7-36a) and (7-36b).

(7-36b)=(12-1%) Here, Sg is defined as ( 12-19) (Sg)2 = (g’: g’)/3. Using these assumptions, we successfully simulated the observed R(B) values. The typical simulation for the values in 2-methyl-1-propanol (2-Me-1-PrOH) is shown in Fig. 12-6 [13, 141. Here, we first found that the contribution of the transversal relaxation (W2) to the effective To-S mixing given by Eq. (12-17) became similar, in magnitude, to that of the

184

coherent To-S mixing ( 4 ) at ultrahigh fields (B-30T). This is the reason why the MFEs in this reaction are almost saturated at B-30T against the prediction given by Schulten and Edstein [ I l l . On the other hand, we also found that the contribution of the longitudinal relaxation (WJ to the MFEs was much smaller than that of the coherent To-S mixing ( 4 ) even at ultrahigh fields (B-30T). This means that the contribution of the longitudinal relaxation (Wl) is negligible to the MFEs of radical pairs in fluid solutions, although it is very important for the MFEs of radical pairs in micellar solutions as shown in Chapter 7. Table 12-2. Obtained values of the fitting parameter y, using the common value for Sg. (Reproduced from Ref. [ 13bl) Solvent

&

Y

qlcP

2-Me-1-PrOH 2-PrOH EtOH MeOH

5 x1O4 5 x10“ 5 x10“ 5 x1O4

0.39

3.30 2.04 1.08

0.19

0.13 0.07

0.55

When the same value was used for Sg, the observed R(B) values in all solvents can well be simulated with the parameters listed in Table 12-2. Here, the parameter y is defined by

y=-. A‘wu,d2 2DA If the Stokes-Einstein relation is introduced for D

(12-20)

(12-21) y is given by 3zAg,uBd3 (12-22) 77 ’= 4kTh Eq. (12-22) means that y is expected to be linear in the viscosity q, which is indeed seen in Table 12-2. The value of the anisotropic factor (6gz = 5 x 1 0 4 ) listed in Table 12-2 is slightly larger than the reported value of 1x1O4 for the phenyl thiyl radical, which seems not to be so reliable. The obtained value for D in 2-Me-1-PrOH from Eq. (12-20) was 4.6x10”cm2 I s when it was assumed that d=0.6 nm. This value is also similar to that predicted by the Stokes-Einstein relation, which is 2.2 x 10” cm2 / s from Eq. (12-21).

12.3 Effects of Ultra-High Magnetic Fields due to the RM Using our superconducting and pulsed magnets, the author’s group studied the effects of ultrahigh magnetic fields on the TRP and YEvalues at room temperature for the photoreduction of carbonyl and quinone compounds (XCO) in micelles. We have used benzophenone (BP), decafluorobenzophenone (DFBP), and 1,4-naphthoquinone (NQ) for carbonyl and quinone compounds. We have used sodium dodecyl sulfate (SDS) and a-dodecyl-o hydroxypolypoly(oxyethy1ene) (Brij 35) for micelles. The scheme of such photoreduction reactions of XCOs can be represented by reactions (7-la)-(7-10. Typical results observed with our ns-laser photolysis apparatus, super-conducting magnet and pulsed magnet are shown in Figs. 12-7 and 12-8.

185

Fig. 12-7. A(t) curves observed with a ns-laser photolysis apparatus and a superconducting magnet at 320 nm in the photoreduction reactions of (a) BP and (b) DFJ3P in Brij 35 micellar solutions at 293 K. (Reproduced from Ref. [151)

Time / us

-'b

5

10

15

20

25

30

Time I p s

. 0

!!

2 8

0

10

15

Fig. 12-8. A(t) curves observed with a nslaser photolysis apparatus and a pulsed magnet at 525 nm in the photoreduction reactions of BP (a) in an SDS micellar solution and (b) in a Brij 35 micellar solutions at 293 K. (Reproduced from Ref. [8]by permission from Elsevier Science B. V.)

186

From the A(t) curves shown in Figs. 12-7 and 12-8 for the generated ketyl or semiquinone radicals (XHCO'), we obtained the following new MFEs on the e R p and YE values under ultrahigh magnetic field of up to 30 T: (1) Each of the ks values in the above-mentioned systems was found to decrease with increasing B from 0 T to 2-3 T , but to increase with increasing B from 2-3 T to 29.6 T as shown in Fig. 12-9. Here, the ZRP value corresponds to the llko(0 T) or Ilks@) one. The ko and ks values were given by Eqs. (7-8) and (7-14). = kpl4 + kE. (7-8)=( 12-23) ks = kR + kR' + kE (7-14)=(12-24) (2) For the reactions of BP and NQ in the SDS micellar solution, the magnetic field dependence of each of the ks values shows a shallow reversion as shown in Figs. 12-9(A) and 9(C). (3) For the reaction of BP the Brij 35 micellar solution, the magnetic field dependence of the ks value shows a medium reversion as shown in Fig. 12-9(B). (4) For the reaction of DFBP in the Brij 35 micellar solution, the magnetic field dependence of the ks value shows a deep reversion as shown in Fig. 12-9(D). (5) Each of the YEvalues in the above-mentioned

I

5

I

(A) BP/SDS 4 -

-

7 0

$3

- t D

I;

0

-:

:2

22

~W

1

____.-------- _---

.-~------------.--LI---^-_-------I

*--___--------I

0

I

0-

..--------.__-___________.________ .--------------__._.__.__ 0

I

0

lo B I T

I

20

0

30

I

0

10

I

Bl T

20

30

Fig. 12-9. Magnetic field dependence of the ko(0 T) and ks(B) values obtained for (A) BP(benzophenone)/SDS(sodium dodecyl sulfate) system, (B) BP/ Brij 35(ol-dodecyl-whydroxypolypoly(oxyethy1ene) system, (C) NQ(naphthoquin0ne)lBrij 35 system, and (D) DFBP(decafluorobenzophenone)Brij 35 system. The Bi values estimated with Eq. (12-30) are indicated by the arrows in this figure. The broken lines show the magnetic field dependence of kR+ kR' and the dotted lines show the kE values. (Reproduced from Ref. [ 16b] by permission from The Chinese Chemical Society)

187

From Eqs. (7-45) and (7-47a), k R + kR' can be expressed as follows: kR + kR' = ka + kb + kab.

(12-25)

Here, k,, kb, and kab correspond to the relaxation rates due to the alkyl radical (R'), and the ketyl or semiquinone radicals (XHCO'), and their dipole-dipole interaction, respectively. These rate constants are given by (12-26a)

(12-26b) (12-26c) Because wis the Zeeman splitting of the triplet radical pair, wis proportional to B. At low and high fields, kj ('j= a and b) and kab have the following limiting values: kj

(B-OT)

=

(AIj : A ' j ) 30h

'

zj

I

(12-27a) (12-27b)

(12-27~) kab (B-+d')

(12-27d)

= 0 S-'.

At intermediate fields, each of k,, kb, and kab changes drastically at each of l/&, l/&, and l/zgb, respectively. Because R' is derived from a micellar molecule, its zg value should be similar to the ?& value of the dipole-dipole interaction. On the other hand, the & value should be much smaller than zg because XHCO' is much smaller than R'.

z,, > 2, >> 7,.

(12-28a)

The observed principal values of g- and HFC-tensors of typical radicals are listed in Table 123 and 12-4. From the observed HFC constants of R' and XHCO', the anisotropy of the HFC of R' should be much larger than that of XHCO'. (AIa : A', ) >>

(Alb

:A', ).

(12-28b)

There is no data for the anisotropy of the g-tensors of R' and XHCO', but that of XHCO' may be slightly larger than that of R' because the odd electron of XHCO' is delocalixed on oxygen. (g'a g',

'

< ( g ' b g ' a 1.

(12-28~)

From the above consideration, the wdependence of k,, kb, and kab can be obtained schematically as shown in Fig. 12-10. From this figure we can see that the kR + kR' value should decrease with increasing B from 0 T to high fields, but that the value should show reversion at higher fields.

188

Table 12-3. Principal values ( g l . g2, and g3) of g-tensors of typical radicals and their isotropic g-values (g,so=(gl+g2+g=J3).

a)phenothiazinecation radical. Table 12-4. Principal values (Al, A*, and A3) of HFC-tensors of typical radicals and their

189

Fig. 12-10. Schematic diagram for the magnetic field ( w = g,uBB / h ) dependence of ka, kb, and kab. Eq. (12-25) shows that k R + kR' is the sum of ka, kb, and kab. (Reproduced from Ref. [ 171 by permission from Elsevier Science B. V.)

'h

4

61

L

!

.-__

............................ E' _----

I

,' kR+kR'

I I

/

-\

/I' I /

0

10

B/T

0

10 Bfr

Fig. 12-1 1. Classification of the magnetic field dependence of the rate constant ( k ) of radical pair decay. Here, the k value at 0 T is represented by ko and the value above 0.04 T by k,. (Reproduced from Ref. [ 171 by permission from Elsevier Science B. V.)

190

The observed magnetic field dependence of the rate constants (ko and ks) of radical pair decays shown in Fig. 12-9 can be explained qualitatively by the classification shown in Fig. 12-11. Case (a) corresponds to the reactions where k~ >> k~ + kR' holds. In this case, the magnetic field dependence of radical pair decay shows a shallow reversion. The reactions of BP and NQ in SDS micellar solutions (Figs. 12-9A and 9C) belong to this case. Case (b) corresponds to the reactions where k E - k~ + kR' holds. In this case, the magnetic field dependence of radical pair decay shows an intermediate reversion. The reaction of BP in Brij 35 micellar solution (Fig. 12-9B) belongs to this case. Case (c) corresponds to the reactions where k~ << k R + k ~ holds. ' In this case, the magnetic field dependence of radical pair decay shows a deep reversion. The reaction of DFBP in Brij 35 micellar solution (Fig. 12-9D) belongs to this case. The observed magnetic field dependence of the rate constants (ko and ks) of radical pair decays shown in Fig. 12-9 can be explained quantitatively by the following procedures: (1) We can see from by Eq. (12-24) that ks is represented by the sum of k~ and k~ + k R ' . (2) The magnetic field independent k~ values for the reactions in micellar solutions can well be determined with an optical-detected ESR technique, which will be explained in Chapter 14. In this chapter, we can use its results as shown in Fig. 12-9, where the k~ values are represented by dotted line. (3) The reversion of the k R + kR' values at higher fields than 2 T is mainly attributable to the anisotropic g-tensor of XHCO' as

(12-29) Here, the g-tensor of XHCO' is assumed to be axial symmetry with its principal values of g// and g, . From the observed field dependence of ks above 2 T and Eq. (12-29), the lg//- g, I and % values were simulated with the least square method for the BP/Brij 35 and DFBP-Briji 35 systems [16]. The obtained values are listed in Table 12-3. For the BP/SDS and NQ/SDS systems, the observed MFEs of ks above 2 T so small compared with those for the BPBrij 35 and DFBP-Briji 35 ones that the lg// - g, I and q,values could not be simulated with the above method. On the other hand, the magnetic field (BJ at the inflection point of the observed field dependence of ks above 2 T can be estimated numerically. For the systems where Eq. (12-29) is valid, Bi can be expressed by B . =--,1 A ' &gPB'b

(12-30)

Problem 12.3. Prove Eq. (12-30). In Fig. 12-9, each Bi value obtained by the observed field dependence of ks above 2 T for each system is indicated by an arrow. With this Bi value and Eq. (12-30), the corresponding q, value was estimated as listed in Table 12-5.

191

Table 12-5. Obtained Ig,/ - 8,) and G, values simulated with Eqs. (12-29) and (12-30) for the reactions shown in Fig. 12-9.

We can see from Table 12-5 that the estimated ]g//- g, I values are reasonable from the observed values for some radicals and that the estimations with Eqs. (12-29) and (12-30) give similar G, values. Because the b/, - g, I values have been measured for few radicals, this is a new method for its estimation. It is noteworthy that the obtained G, values with this method are much smaller than the rotational correlation time given by Eq. (7-25) 47~772 z, =-. 3kT

(7-25)=(12-31)

Using adequate parameters for the BP/Brij 35 system (r-0.45 nm, 7-0.03 Pas), we can obtain ~ = ns3 with Eq. (12-3 1). Eq. (12-3 1) can be derived from the assumption that the size of a solute molecule is quite larger than the solvent one. This assumption may not be held for XHCO' in micellar solutions. Similar MFEs have been observed for reactions of biradicals by Tanimoto's group. A typical result of their MFEs is shown in Fig. 12-12. They studied the reverse electron transfer process in an a-cyclodextrin inclusion complex of phenothiazine-viologen chainlinked compound [ 181. They measured the MFEs of the lifetime (ZBR) of the generated biradical involving the phenothiazine and viologen cation radicals (Ph+' and V+*)asshown in Fig. 12-12(a) with a pulsed magnet, the maximum field of which was 14 T.

A ( t ) = A , , exp(-t/z,)+A,.

(12-32)

Here, ABRand AF are the absorbance of the biradical and the free mono-radical generated from the biradical. They found the MFEs on Z ~ Rand AF as shown in Fig. 12-12(b). We can see from this figure that ZBR and AF increase with increasing B from 0 T to 1 T (ZBR(OT)= 140 ns and ~jj~(1T) = 6.6 ps), but that they decrease with increasing B from 1 T to 13 T ( z ~ ~ ( 1 3 = T) 2.18 p). This reversion of ZBR is much clearer than those of the lifetimes of radical pairs in micellar solutions. This fact can be explained qualitatively from the following consideration. Because there is no escape of component radicals in biradicals, the decay rate (kBR) of biradicals can be given by kR' + kF. kBR = k R i(12-33) Here, k~ is the field independent rate constant of the other reaction processes from biradicals. If k~ is much smaller than k~ + kR', very clear reversion can be observed for kBR as shown in Case (c) of Fig. 12-11. Tanimoto et al. also made quantitative simulations for their MFEs with Eqs. (12-25) and (12-26). Using adequate parameters for this biradical, they obtained the simulation curve for ZBR as shown in Fig. 12-12 [ 181, where the rotational correlation times of Ph", Vf*, and their dipole-dipole interaction were taken to be 204 ps, 38.3 ps, and 374 ps, respectively. These G values were calculated from Eq. (12-31). As clearly seen

192

from Fig. 12-12, the simulated curve shows no reversion for ZBR. If a much smaller value was used for V+' (G=lps), the simulation curve was found to explain well the observed reversion of ZBR [Is]. This situation is very similar to that of radical pairs in micellar solutions. Fig. 12-12. (a) Generated biradical from an a-cyclodextrin inclusion complex of phenothiazine-viologen chain-linked compound.

10.4

81

. . .. ,.. ..

__-

0

3 w 0

-h

7

2 ;CD

F n

1 0'

m

0

2

4

6

8101214 / T

Magnetic F i e l d

(b) Observed field dependence of ( 0 )ZBR and ( 0 )AF for the triplet biradical shown by Fig. 12-12(a). The full curve shows a simulation with adequate parameters for )ZBR of this biradical. (Reproduced from Ref. [ 181 by permission from Elsevier Science B. V.)

193

12.4 Effects of Ultra-High Magnetic Fields upon Reactions of Kramers Doublet Species Steiner's group [ 5 ] measured MFEs on photooxidation quantum yields of Ru"tris(bipyridine) type complexes (RuL~~')in water at room temperature with a Bitter type magnet at the High Magnetic Field Laboratory of MPYCNRS at Grenoble. Their reactions occur as follows:

+ 'hu RuL~~ 3' R u L P *

(12-34a)

+ 3'R~L32+*3

-

+ MV2++ 3'[R~L33' MV"]

3'[RuL33+ MV"]

" [ R u L ~ ~ 'MV"]

" [ R u L ~ ~ 'MV"] + R u L +~MV2+ 3'[RUL33+ MV"] -+ *RUL? + 2MV+'

1'.

kq[MV2'l,

(12-34b)

ksr,

(12-34c)

kbet,

(12-34d)

kE.

(12-34e)

Here, MV2+ is methylviologen and the reaction rate of each process is denoted above. The MFEs on the formation of free radicals (YE) was measured by photostationary illumination in a continuous flow system. The observed relative MFEs (~R(B)=[YE(B)-YE(OQ]I Y,(Or)) on the yield of photoinduced MV" radical with the four complexes (la-ld) is depicted in Fig. 12-13. This figure shows that the YE value of each complex decreases with increasing B from 0 T to 15 T and that it is almost saturated above 15 T. It is noteworthy that the ldR(B)l value decreases regularly for every replacement of a bipyridine ligand by a phenanthroline one. The effective spin state of the radical pairs (3'[R~L2f MV+']) may be represented in the direct product basis of two doublet, whereby the MV" radical is represented by a normal spin doublet ( a and p), but the RuL?' moiety by a strongly spin-orbit coupled Kramers doublet (a' and p'). The lowest JSramers doublet of a Ru(III) complex of D3 symmetry may be expressed in the general form

at=sin(x)la, I + cos(x)l~-I,

(12-35a)

PI=sin(x)lii,l+ cos(x)le+I.

(12-35b)

The extent of mixing of different spin-orbital states expressed by the angle ( x ) is determined are lower by the ratio of the trigonal ligand field splitting parameter A (positive if ( a , and (a,

1

in energy than /e+land 1Z-l) and the one-electron SOC constant tan( 2x) =

J2 112-Al<.

1

(0of the Ru 4d shell (12-36)

The ratio of d/r can be determined experimentally from the g-tensor component of the Ru(III) complex g, = g L L= 2[sin2(x)-(~+k)cos2(x)l,

(12-37a)

g, = g, = gVy= 2[JZksin(x)cos(n)+sin'(x)l.

(12-37b)

Here, k is the dilution factor. The g-tensor values were determined experimentally for all complexes in low-temperature solid sulphuric acid/acetonitrile matrices. It turned out that these values are essentially invariant ( g, = 2.6 and g, = 1.0 ) through the series of compounds. They correspond to a mixing angle of x=67.5".

194

0.00 -0.05 -0.10 -0.15 -0.20

-0.25

-0.30 -0.35

'

[Ru"(bpy,]'+

a

1b [Ru"(bpy),(phen)12+ [Ru"(bp~)(~hen)~l~ + lc [Ru"(phcn),J2

+

'

d

15

10

5

0

20

25

Fig. 12-13. bpy

m

phen

MV2+

Observed relative MFEs ( W B ) = [ Y E ( B ) -YE(OT)Y YE(OT)) on the yield of photoinduced MV" radical with the four complexes (laId). The curves are the results of theoretical simulation with the parameters given in Table 12-6. (Reproduced from Ref. [5]by permission from WEEY-VCH Verlag)

The observed MFEs were analysed in terms of the density matrix (p) of the primary product pair ([Ru(L,lL'3.J3+ MV+']) represented in the basis of the four degenerated spinorbit states (TI?, T'o, and 5').

T I +=lava)= s i n ( x ) T + ( a , r ) + c o s ( x ) [ T , ( e - r ) - S ( e _ r ) ~ / J Z ,

(12-38a)

T I -=

(12-38b)

=sin(x)~,(a,r)+cos(x)[~,(e+r)-~(e+r)l/JZ,

/PIP)

T I ,= (Ia'p)+lp'a))/JZ = sin(x)T,((a,r)+cos(x)[T+(e+r)-T(e_r)l/JZ, S'=

(Ia~~)-Ip'a))/Jz = sin(x>S(a,r>+cos(x>[T+(e+r) -T(e-r>l/Jz.

(12-38~) (12-38d)

The time dependence of p is given by the stochastic Liouville equation (SLE): d@dt = -i[H, p ] h + R p - [K, pJJ2.

(12-39)

Here, H is the spin Hamiltoniam of the radical ion pair (RIP), R is the relaxation super operator, and K is the reaction operator. In H, the effect of the Zeeman interaction within the Ru3+-moietyis most efficient in pair spin state mixing due to the strong anisotropy of the g-

195

Ru2+-complex

g,

g //

la lb

2.60 2.60 2.54 2.53

1.18

lc Id

1.18 0.95 0.90

kd1o ~ ~ -&,JI~ o'Os-' 2.3 2.2 1.9 I .5

7.8 6.6 4.6 3.2

%IPS

26.7 23.5 21.3 19.2

References [I] Y. Tanimoto, H. Hayashi, S . Nagakura, H. Sakuragi, and K. Tokumaru, Chem. Phys. Lett., 41 (1976) 267. [2] S. G. Boxer, C. E. D. Chidsey, and M. G. Roelofs, J. Am. Chem. SOC.,104 (1982) 1452. [3] Y. Sakaguchi and H. Hayashi, Chem. Lett., (1993) 1183. [4] N. J. Turro, C.-J. Chung, G. Jones 11, and W. G. Becker, J. Phys. Chem., 86 (1982) 3677. [5] D. Burher, H.-J. Wolff, and U. E. Steiner, Angew. Chem. Int. Ed. Engl., 33 (1994) 1772. [6] G. Ferraudi, G. A. Argiiello, and M. E. Frink, J. Phys. Chem., 91 (1987) 64. [7] M. Mukai, Y. Fujiwara, Y. Tanimoto, and M. Okazaki, J. Phys. Chem., 97 (1993) 12660. [8] K. Nishizawa, Y. Sakaguchi, H. Hayashi, H. Abe, and G. Kido, Chem. Phys. Lett., 267 (1997) 501. [9] J. B. Pedersen, J. Chem. Phys., 67 (1977) 4097. [lo] J. H. Freed, in Chemically Induced Magnetic Polarization, Eds. L. T. Muus, P. W. Atkins, K. A. Mclauchlan, and J. B. Pedersen, D. Reidel Publishing Company, Dordrecht-Holland, 1977, p.309. [113 K. Schulten and I. R. Epstein, J. Chem. Phys., 71 (1979) 309. [12] (a) M. Wakasa, K. Nishizawa, H. Abe, G. a d o , and H. Hayashi, J. Am. Chem. SOC.,120 (1998) 10565. (b) M. Wakasa, K. Nishizawa, H. Abe, G. Kido, and H. Hayashi, J. Am.

196

Chem. SOC.,121 (1999) 9191. [13] (a) M. J. Hansen and J. B. Pedersen, RIKEN Rev., 44 (2002) 34. (b) J. B. Pedersen, M. J. Hansen, A. A. Neufeld, M. Wakasa, and H. Hayashi, Mol. Phys., 100 (2002) 1349. [14] J. B. Pedersen and H. Hayashi, to be published. [15] M. Igarashi, Q.-X. Meng, Y. Sakaguchi, and H. Hayashi, Mol. Phys., 84 (1995) 943. [16] (a) K. Nishizawa, Y. Sakaguchi, H. Abe, G. Kido, and H. Hayashi, Mol. Phys., (2002). (b) H. Hayashi, J. Chinese Chem. SOC.,49 (2002) 137. [17] Y. Nakamura, M. Igarashi, Y. Sakaguchi, and H. Hayashi, Chem. Phys. Lett., 217 (1994) 387. [18] Y. Fujiwara, T. Aoki, K. Yoda, H. Cao, M. Mukai, T. Haino, Y. Fukuzawa, Y. Tanimoto, H. Yonemura, T. Matsuo, and M. Okazaki, Chem. Phys. Lett., 259 (1996) 361. [19] (a) P. Gilch, F. Pollinger-Dammer, C. Musewald, M. E. Michel-Beyerle, and U. E. Steiner, Science, 281 (1998) 982. (b) C. Musewald, P. Glich, G. Hartwich, F. PollingerDammer, H. Scheer, and M. E. Michel-Beyerle, J. Am. Chem. SOC.,121 (1999) 8876.

Solutions to the Problems 12.1. For the AgM, Q is given as 1 Q =-AgP,B. 2A Thus, the following equation could be obtained from Eq. (12-8): 320AD AgB = 7 = 1.OlXlOT. PBd 12.2. From Eq. (3-18), o,,= Q,. Because QN is given by Eq.(3-13a), 1 w,, = Q N =-AgpBB = 0.01 1*9.2740~10~"JT~'*5T/2*1.05457~10~~~Js = 2.42~1O~s-l. 2A 12.3. From Eq. (12-29), k, can be written as

k , =- aB2 +const. l+bB2 Taking k= ks and x=B, we can obtain the following equations: dk 2ax - -dx (1+bx2)' ' d k - 2 4 1 + bn2)(1- 3bx2) d2x (1 + b X 2 l 4

--

Since the inflection point (xi=B,)satisfy the relation d2Wd2x= 0, this point is expressed by

197

13. Effects of Magnetic Fields of High Spin Species 13.1 MFEs of Luminescence In 1967 Johnson et al. [I] discovered that the intensity of triplet-exciton annihilation luminescence in anthracene cryatal at room temperature increased in weak magnetic fields up to a maximum increase of 5 % at 35 mT and that the intensity decreased in higher fields. The intensity was found to level off at 80 70of the zero-field value at B > 0.5 T. Their typical result is shown in Fig. 13-1.

W

0

z

0.6

v)

w

e

0 3

0.4

J

LL

s>

0.2

d 0

0

I

5.0

10.0

I

d

15.0

20.0 Bl 1OOmT

Fig. 13-1. The influence of a magnetic field on the delayed fluorescence intensity from an anthracene crystal with 15 ms triplet lifetime at room temperature. The magnetic field was applied in the ac plane of the crystal in the direction at -17" with respect to the LZ crystal axis as shown in the insert. The dashed lines in the insert indicate the field direction which produce the greatest diminution in delayed fluorescence intensity. (Reproduced from Ref. [ 11 by permission from The American Physical Society) The delayed fluorescence can be produced by the following scheme: 'So + hZ)E .+ IS1

a:

(13- 1a)

Is1

4sc.

(13-lb) (1 3- lc)

. +

3 ~ 1

3T1+ 3TI + 1*325(T1 Ti)

ki ,

198

1.3,5(~l T’) + 3 ~ +1 3 ‘(TI TI) -+ IS1 + ‘So

~ 1

k-I, ks ,

(13-ld) (13-le)

IS1 +’So + hVDF P. (13-lf) In principle, the magnetic field could be affecting all of the above-mentioned processes. Johnson et al. demonstrated the absence of any magnetic field effect (for fields up to 2 T) on intensity of fluorescence from singlet excitons generated directly with uv light. This eliminated the possibility of MFEs of a a n d /. They also measure the MFEs at 35 and 240 mT with a pulsed-magnet. Because the time dependence of the delayed fluorescence followed the time dependence of the fields, they demonstrated that magnetic fields effected the triplet-triplet annihilation probability (n)given by [2]

(13-2) Here, S; is the singlet character of nine triplet pairs at their collision. If N pair states have equal singlet character, Eq. (13-2) is rewritten as (13-3) We can see from Eq. (13-3) that a decrease (or an increase) of N by a magnetic field brings about a decrease (or an increase) of YT. This is a qualitative explanation of the MFEs of the delayed fluorescence intensity. At first, let us consider the zero-field case. There are three triplet excitons with wavefunctions of Ix>, Iy>, and Iz> and nine possible combinations for the pair state. These are Ixz>, Jzx>, Iyz>, and Izy>. (13-4) Ixx>, Iyy>, Izz>, I x p , where, x, y, and z correspond to the zero-field states which are splitted by zero-field splitting. These states make a singlet (IS>), three triplet (IT>), and five quintet (IQ>) states. The details of the eigen-functions and eigen-values for these states will be treated in Problem 13.4. From its solution, the singlet state is represented by

IS> = 3-”’[(xx>+~yy>+(zz>]. (13-5) Thus, N(OT)=3. Next, let us consider the case when the Zeeman splitting is much larger than zero-field splitting. In this case, the triplet excitons are described, to a good approximation, by the wavefunctions of lo>, 1+1>, and I-1> where 0, +1, and -1 correspond to the standard magnetic sublevel quantum numbers. The nine possible pair states in this case are

loo>, 1+1-1>, 1-1+1>, lob, 110>, 1-10>, lO-l>, 1+1+1>, and 1-1-1>.

(13-6)

The singlet, triplet, and quitet states are represented by

IS> = 3-’/’[ )00>-1+1-1>-1-1+1>], IT+’> = 2-’/’[ 1+10>-10+I>], ITo> = 2-’12[I+1-1>-I- 1+1>I,

(13-7a) (13-7b)

(T.1> = 2~’”[I-10>-~0-1>],

(13-7d)

IQ2> = 1+1+1>, IQ1> = 2-’/’[ 1+10>+10+1>],

(13-7e) (13-7f)

IQo> = 6-’/’[2100>+1+1-1>+1-1+1>],

(13-7g) (13-7h) (13-7i)

IQ-1> = 2-1’2[)-10>+10-1>],

IQ-2> = 1-1-l>.

(13 - 7 ~ )

199

In this case, two pairs from the possible nine pairs of l3q. (13-6) have the singlet character. It is noteworthy that only the symmetric combination of )+1-1> and )-1+1> has the singlet character but not the anti-symmetric one. Thus, N(B>O.IT)=2. This is the reason why a magnetic field higher than 0,lT decreases the delayed fluorescence intensity. Em and Menrifield [3] found that paramagnetic defects in anthracene crystals quenched triplet excitons and changed the field dependence from a negative to a positive gradient. They interpreted these MFEs in terms of the triplet-doublet quenching expressed by 3T1+ 'Do + 234(T~ Do) ki , (13-8a)

2 , 4 ( ~D1 ~ + ) 3 ~ +1 2 '(TI Do)--* 'SI + 'Do

~ o

k-1, ko.

(13-8b) (13-8~)

The quenching efficiency ( ~is )also given by yD =-k1 6

2kD

' k-, + Zk,

/N

(13-9) '

At zero-field, there are six combinations at the collision of a triplet exciton and a doublet quencher (13-10) 1~+1/2>,Ix-1/2>, ly+1/2>, $1/2>,1~+1/2>, and 12-1/2>. These states make two doublet and four quartet states. As shown by Scheme (13-8), the quenching occurs through the doublet states but not through the quartet ones. From the solution of Problem 13-5, the doublet states are expressed by (13-11) [ D,,,,) = -[(1 z k 1/2) k ( xT 1/2)+ i( y T 1/2)].

43

Thus, N(OT)=6. In the case where the Zeeman splitting is much larger than zero-field splitting, there are six combinations at the collision of a triplet exciton and a doublet quencher (+1+1/2>, (+1-1/2>, (0+1/2>, (0-1/2>,1-1+1/2>, and 1-1-1/2>. (13- 12) The doublet and quartet states are represented by (13-13a)

IQ,,,,)

=

If

1f 112),

(13-13b) (13-13~)

Thus, N(B>O.lT)=4. This also shows that a magnetic field higher than 0.1T decreases the triplet-doublet quenching. In fluid solutions, Faulkner and Bard first found the MFEs on the electrogenerated chemiluminescence of anthracence triplet-triplet annihilation [4]. The general features of the MFEs were similar to those observed in crystals: the intensity of delayed fluorescence was found to decrease with increasing B from 0 T to 0.8 T and to approach asymptotically a value which was about 4 % below the zero-field intensity. They also found the MFEs on the anthracene fluorescence in the presence of doublet species such as Wurster's Blue perchlorate [ 5 ] : the fluorescence intensity was found to be quenched by the doublet species, but the quenched intensity was found to be increased by magnetic fields. Their typical results are shown in Fig. 12-2. Although these MFFis observed in fluid solutions were smaller than those in crystals due to rotational motion of triplet molecules, the MFEs in solutions could also be explained by Eqs. (13-3) and (13-9). The magnetically induced decrease in the

200

delayed fluorescence intensity without a doublet quencher is due to the fact that yis decreased by magnetic fields, but the magnetically induced increase in the delayed fluorescence intensity with doublet quenchers due to the fact that is decreased by magnetic fields. Atkins and Evans presented a more exact theoretical analysis of such T-T annihilation and TD quenching in fluid solutions [6]. Although these MFEs on luminescence were strongly expected to be applied to the control of chemical and biochemical reactions by external magnetic fields, the MFEs on reaction yields due to the T-T and T-D processes had not been observed for long years after the discovery by Johnson et al. On the other hand, however, MFEs on reaction yields due to the RPM were observed much earlier. Fig. 13-2. Magnetic field effects on delayed fluorescence from methylene chloride solutions: (a) 3~10.~ M anthracene, (b) 8x10” M anthracene, and - ~ anthracene and (c) S X ~ O M 1 . 8 ~ 1 0M . ~ Wurster’s Blue perchlorate. (Reproduced from Ref. [5] by permission from The American Chemical Society)

e

0

0

4 6 FIELD STRENGTH, h 9

2

4 6 FIELD STRENBTH, kG

e

8

20 1

13.2 CIDEP of High Spin Species Paul's group [7] and Obi's group (81 found a new type of C D E P during the T-D quenching. Some of their typical results are shown in Fig. 13.3, where emissive signals are observed for TEMPO (2,2,6,6-tetramethyl-l-piperidinyloxyI)in the presence of excited benzophenone and pyruvic acid.

a)

Fig. 13-3.

A b S.

I

CIDEP spectra of TEMPO (0.60 nM) in (a) TEMPO and benzophenone (55 mM) and (b) TEMPO and pyruvic acid (72 nM) at room temperature in benzene by 308-nm excitation. (c) CW-ESR spectrum of TEMPO. (Reproduced from Ref. [8] by permission from The American Chemical Society)

r":

Em.

4b2

At first, such CIDEP was considered to be due to the polarization transfer from an excited triplet molecule to a radical. The initial spin polarization of triplet benzophenone is emissive, but that of triplet pyruvic acid is absorptive. Similarly, the spin polarization of the radicals was always emissive regardless of the spin polarization of the triplet states [8]. Thus, such emissive signals were attributed to the T-D quenching as shown by Processes (13-8) 3 T+ ~ 'Do -+ Z'4(TlDo) ki , (13-14a) 2 . 4 ( ~D1 ~ -+)

+

3 ~ 12 ~ o

k-1,

'(TI Do) 4 IS1 + *DI

kD,

2DI +2Do

P.

(13-14b) (13- 1 4 ~ ) (13- 14d)

202

Here, 'D1 is the lowest excited doublet state of a radical. Because the T-D energy transfer ~ be a from an excited triplet molecule to an excited doublet one is very fast ( k should diffusion-controlled value), it makes a major contribution to CIDEP generation in T-D systems. is given by those of a triplet The spin Hamiltonian of a triplet-radical system (HTR) molecule ( H T ) , a radical (HR), and their exchange interaction (Hex)

HTR= Hex

+

HT+HR,

(13- 15a)

where

(13- 15b)

Hex= -WSTSR, H T =gTpu,ST,B+xrA,STI, HR

= gRPBSRzB+ xIAkSRIk

-k

'

HD,

(13- 1 5 ~ ) ( 13- 15d)

Here, HDis the Hamiltonian of the zero-field splitting (ZFS) of a triplet molecule with principal axes (X, Y, and Z) and principal values ( X , Y, and z ) 1 (1 3-16a) HD= - X S & -YS& - Z S k = D ( S k - - S 2 ) + E ( S & - S & ) , 3 where ( 13- 16b) D = -3212 = ( X + Y)/2-Z, (13- 1 6 ~ ) E = - ( X - Y)/2. Problem 13.1. Prove Eqs. (13-16b) and (13-16c). As shown in the preceding section, the quartet and doublet states are produced at collision between a triplet molecule and a radical, where He,gives the energy splitting between the quartet and doublet state

( Q ~ L ~ Q= -J) ,

(13-17a)

(DlfLlD) = 2.r.

(13-17b)

Problem 13.2. Prove Eqs. (13-17a) and (13-17b). At a high magnetic field limit where the Zeeman splitting is much larger than the ZFS, the energy levels of the quartet and doublet states can be expressed by

E(Qnr) = -J + gTpu,mB,

+

(13- 18a)

(1 3- 18b) g,pBnB, where the smaller splitting due to the HFC interactions and the ZFS is not shown in Eqs. (1318a) and (13-18b) for simplicity. The energy diagram for the closed and separated pairs of such levels are schematically illustrated for the case of a negative J in Fig. 13-4. The CIDEP due to the T-D quenching occurs at a high field limit from the following procedures: (1) There are six combinations at the collision of a triplet molecule and a doublet radical (13-12) 1+1+1/2>, 1+1-1/2>, 10+1/2>, 10-1/2>,1-1+1/2>,and 1-1-1/2>. At the first collision (t=O) where the Q-D conversion is prohibited at the close pairs because of a large IJI value, the quartet and doublet pairs have equal population @) (13-19) m,,,(t=O.) = hn(t=0.) = 1/6. E(Dd = 2J

203

T+I+D+I /2 ->T+I+D-

112,

Closed pair (J
Separated pair (J
>

I

R(trip1et-radical) R-a, Fig. 13-4. Energy diagram and state mixing of spin states of a triplet-radical encounter complex for a negative J. (2) At the first collision, the T-quenching occurs at the close pairs from only the doublet pairs due to the fast processes (13-14c) and (13-14d). After the T-quenching, the population of the doublet pairs becomes smaller than that of the quartet pairs (1 3-20) fi,,,(t'O+) = 1/6 > po,,(t=o,) = (1-&)/6.

(3) The doublet radical and unquenched triplet molecule in the close pairs start to diffuse with each other, forming separated pairs where the Q-D conversion becomes possible through the following processes as shown in Fig. 13-4:

lQ-3/2)-l~-l/2),

(a)level crossings between lQ_3,,)-lDl/2), (b) Q-D fixings between

I

Ql/,)

-

I&)

and

I

Q-1/2

) - I&/2)

and

lQ-,,z)-I~I/2).

.

These Q-D conversions mainly occurs through the ZFS of the triplet molecule, because the matrix elements due to the ZFE are much larger than those due to the Zeeman and HFC interactions. Here, the matrix element of (Tm,IHDITm)was given by Eq. (3.10) of Ref. [9]. (c) From Eqs. (13-13a) and (13-13b),

I

Q-312)

1

(13-21a)

= - 1- 11 2) ,

(13-21b) Thus, the level crossings between

IQ-, ,)

- IDll2) and

lQ-3/2) - lD-l/2)produce the

Q-D

conversion which decreases the P-radical population and increases the a-radical population. From Eqs. (13-13c) and (13-13a)

204

1 Q-,,, ) = E l 0 - 112) + $1

d:

+1

-1

d:

(13-21~)

2) ,

I D,,,) = - -1 0 + 1/ 2) + -1 + 1- 1/ 2).

(13-21b)

IQ-, )

- IDll2)does not change the a- and P-radical Because the level-crossing between populations, this Q-D conversion gives no CIDEP. Similarly, a combination of the Q-D conversions through the l Q,,,)-lDI l2)and lQ-,,,)-lD-l,2)mixing gives no CIDEP. The QD relaxation also occurs at the separated pairs, but the spin relaxation also gives no CIDEP because the relaxation effects of the Qn, and D states cancel the effects of the Q-mand D.n states. (4)Some of the doublet radical and unquenched triplet molecule in the separated pairs continue to diffuse further with each other and become free radical and triplet molecule. This On the other hand, the other doublet radical and component produces no CIDEP. unquenched triplet molecule in the separated pairs re-encounter with each other and become the closed pairs again. At the second encounter, the population of lD,,,,)is increased and

and IQ-,,,)that of IQ-,,,)is decreased through the level crossings between ~Q-3,z)-~Dl,z)

ID-,,,) at the separated pairs. (5) At the second encounter, the T-quenching also occurs from the doublet pairs. Thus, net emissive polarization is generated in the radical for the case of negative J values. After the T-quenching at the closed pairs, the same processes occur successively from (3). This is the main reason why a new type of CIDEP is produced through the radical triplet pair mechanism (RTPM). Many variations of CIDEP due to the RTPM have been observed. For example, net absorptive CIDEP was observed in the T-D quenching for positive J values and IQ,,,) - ID,,,) produce such [lo]. Here, the level-crossings between IQ,,,) CIDEP. Kawai and Obi [ l l ] found that the S-D quenching also gave CIDEP, the phase of which was inverse to that due to the T-D quenching. In the S-D quenching case, the first encounter gives equal populations among the doublet pairs but no population for the quartet pairs. Thus, D-Q conversion at separated pairs gives CIDEP. Hyperfine-dependent CIDEP was also obtained for the RPTM [8]. Here, the D-Q mixing through the Zeeman and HFC terms is important as in the case of the S-T mixing for the RPM. @,I,,

I

- H D + H R D,,,,

)=

* yJZ[ ( g r - g ,)rU$

(c T

+

R

AiMi-

A,M,

)I .

(13-22)

k

Problem 13.3. Prove Eq. (13-22). Thus, the phase patterns of the CIDEP signals due to the RTPM can be summarized as listed in Table 13-1. Table 13-1. Phase patterns of the CDEP signals of radicals due to the RTPM. ~

Sign of J J>OJ J
T-D Quenching

S-D Quenching

Abs. + AiE Em. + EIA

Em. + EIA Abs. + AIE

13.3MFEsofhighSpinSpecies B e c a u s e

s h o w n i n

205

similar systems. Because the ground state of molecular oxygen is of a triplet state ( 3 0 2 ) , the triplet quenching by 3 0 2 was the first candidate for such MFEs on reaction yields. Indeed, Tachikawa and Bard [12] found MFEs on oxygen quenching of delayed fluorescence of anthracene and pyrene in N,N-dimethylformamide (DMF). They found that the delayed fluorescence intensity (0 without oxygen was decreased by magnetic fields below 0.8 T (Z(0.8T)/Z(OT)-O.98), but that the intensity with oxygen was found to increased by the fields. The maximum increase (Z(0.8T)/Z(OT)) arrived at - 1.06. Such magnetically induced increases in Z(B) can be explained by the magnetically induced decrease in the rate of the oxygen quenching of triplet molecules. On the other hand, there are many peculiar phenomena in the oxygen quenching of triplet molecules. First of all, Tachikawa and Bard found no MFE on oxygen quenching of delayed fluorescence of anthracene and pyrene in acetonitrile. Such a solvent dependence on the MFE of the oxygen quenching has not yet been investigated extensively, but the role of the solvent may be important in these MFEs. Secondly, Geocintov and Swenberg [ 131 examined the MFE on the phosphorescence decay profiles of chrysene, a,h-dibenzanthracene, and coronene on a polystyrene matrix in the presence of oxygen with a Bitter magnet at the National Magnet Laboratory. Because they found no appreciable MFE within the experimental error of 1%-2% by a magnetic field of 14.5 T, they tentatively explained the null effect assuming either (1) that the kfi.1 ratio is small or (2) that the effect of charge transfer mixing in the collision complex is significant. The other possible way of finding the MFE on the oxygen quenching is to measure the MFE on the decay profiles of the T-T absorption with laser photolysis techniques. To the best of the author's knowledge, however, there has been no report of such a MFE. The author has the following comments on the MFEs on the oxygen quenching: (1) The null effect observed on the polystyrene matrix may be due to the fact that there is no reencounter process on such a solid matrix. (2) The solvent dependence of the MFE on the oxygen quenching in fluid solutions should be studied further because the effect of charge transfer mixing in the collision complex and the effect of solvent in the dynamics of oxygen and triplet molecules are very important for appearance of the MFE on the oxygen quenching. (3) Because the MFE on the oxygen quenching in fluid solutions should be less than 10 %, it is very difficult to find appreciable effects on the decay profiles of T-T absorption in the presence of oxygen. (3) Because the ZFS of the ground triplet state of oxygen (101-4 cm-I) is much larger than that of usual triplet molecules (PI-0.1cm-'), the theoretical treatment of the oxygen quenching of triplet molecules at a low field region @
206

Fig. 13-5. Schemes for the reactions involving three odd-electrons: (a) the electron transfer reaction of triplet 10-methylphenothiazine (3D*)with 4-(4-cyanobenzoyloxy)TEMPO ('AR1.) in 2-propanol at 293 K and @) the reactions of triradicals generated by photolysis of benzophenone(BP)-dipheylmethane(DPM)-nitroxide(R~~) trifunctional compounds at 293 K. (Reproduced from Ref. [ 151 by permission from The Japanese Chemical Society)

207

As shown in this figure, the quartet and doublet states consisting of the cation radical (2D'') and the anion biradical (3AL-R1') are generated after the electron transfer. From both of the quartet and doublet states, the radical and the biradical escape from solvent cages, forming the escape radical and biradical. On the other hand, the cage recombination only occurs from the doublet state, forming D and 'A-R,'.

2

1.8

F

1.6

!t 8

1.4

*m

1.2

e

1 0.6

0

2

4

6

8

10

6

8

10

BIT

1.7 1.6

F

e

??

8 M

1.6 1.4

1.3

la 1.1

1 0

2

4

BIT Fig. 13-6. Magnetic field dependence of the escape yield (YE@)/ YE(OT))observed for (a) the electron transfer reaction of triplet 10-methylphenothiazine (3D*) with 4-(4cyanobenzoy1oxy)TEMPO (2A-R1') in 2-propanol at 293 K in the presence and absence of R1' (Reproduced from Ref. [ 151by permission from The Japanese Chemical Society) and (b) the electron transfer reaction from the excited trip-sextet (6Tl)state of chloro-(3m e t h y l i m i d a z o 1 ) - ( n e s o - t e t r a p h e n y l p o r p h y r (C2"P) to viologens (V") in acetonitrile at 293 K (Reproduced from Ref. [16]).

208

We measured the A(t) profiles of the cation radical under magnetic fields of up to 10 T. From the profiles, we obtained the MFEs on the YE@) values. In Fig. 13-6(a), we plot the YE@)/YE(OT) ratios observed with and without R1'. In the reaction without RI', we can see from Fig. 13-6(a) that the YE@)value increases with increasing B for 0 T to 0.2 T and that the value decreases with increasing B from 0.2 T to 10 T. The maximum YE@)/YE(OT) value is 1.09 k 0.02 at B = 0.2 T. The magnetically induced increase and decrease in Y#) can be explained by the HFCM and the AgM, respectively. In the reaction with RI', we can see from Fig. 13-6(a) that the YE@)value increases with increasing B from 0 T to 2 T and that the value decreases with increasing B from 2 T to 10 T. The maximum YE@)/YE(OT)value is as large as 1.90 f 0.08 at B = 2 T. These MFEs observed with R I can be explained in principle by the T-D quenching mechanism presented by Atkins and Evans [9]. The magnetically induced increase in YE(B) with increasing B from 0 T to 2 T can be explained by the RM, where the spin relaxation between the quartet-doublet states plays an important role during the lifetime of the pair states as follows: (1) The radical pairs are formed in both the quartet (Q+3/2and Q+1/2)and doublet (D+1/2)states as shown in Fig. 13-5(a). (2) In the absence of a magnetic field, the Q-D conversion occurs among all spin states at separated pairs, where the J value is very small. (3) In the presence of a sufficiently large field, the Q+1/2 and D+1/2 states can still mix with each other, but the Q+3/2 ones are energetically separated from the D+1/2 ones. (3) The Q*3/2-4+1/2conversion and Q*3/2-D+112 one occur through the spin relaxation by the spin-spin interaction of the anion biradical and their rate may show the magnetic field dependence which is similar to that of the T+l-To conversion and T+l-Sone in micellar solutions. It is noteworthy that the MFEs through the RM can be observed for this Q-D system even in non-viscous solvents. This is due to the fact that the spin-spin interaction of the anion biradical is much larger than the anisotropic interactions of the T-S systems. On the other hand, the magnetically induced decrease in Y@) with increasing B from 2 T to 10 T can be explained by the AgM, the Q-D mixing of which is represented by Eq. (13-22). We also studied the MFEs on reactions of triradicals generated by photolysis of benzophenone(BP)-dipheylmethane(DPM)-nitroxide(R;) trifunctional compounds at 293 K radical. The scheme [ 171. Here, R; is the PROXYL (2,2,5,5-tetramethylpyrrolidin-l-oxyl) of the reactions for BP-C,-DPM- R; ( n = 2 and 8) is shown in Fig. 13-5(b). In this figure, we can see that a triradical is generated for each reaction. From the A(t) profiles at 345 nm, we observed the MFEs on the decay rate of each triradical (km). For the triradical generated from BP-Cs-DPM- R; its km value was found to decrease with increasing B from 0 T (kTR(0 T) = 1.05~ lo7 s-I) to 2 T (km(2 T) = 2 . 0 lo6 ~ s-') and to be nearly constant with increasing B from 2 T to 10 T. We also found that the ~ T Rvalue observed in the reaction of BP-C2-DPMR; decreased with increasing B from 0 T ( k ~ ~ T) ( 0= 4 . 5 ~ 1 0s-') ~ to 1.7 T (km(1.7 T) = 2 . 6 lo6 ~ S - I ) and that the value increased slightly with increasing B from 1.7 T to 10 T ( k ~ ( 1 0T) = 3 . 0 lo6 ~ s-I). These magnetically induced decreases in km can also be explained by the RM, where the spin relaxation between the quartet-doublet states plays an important role during the lifetime of the triradicals. The decay rate of the triradicals at B = 2 T is about 2 x lo6 s-', which is much larger than the decay rate of the corresponding biradicals at B = 2 T (about 1x lo5s-'). This increase is due to the enhancement of spin relaxation by R2'. Because there has been no clear study of the MFE on any chemical reaction involving a higher spin state than 3/2, we recently tried to get clear MFEs on the electron transfer reaction from the excited trip-sextet (6T1) state of chloro-(3-methylimidazol)-(mesotetraphenylporphyrinato)chromium(m) (CrmP) to viologens (V2') in acetonitrile at 293 K [ 161. The scheme of this reaction is shown in Fig. 13-7.

209

d

>

P

+

is

+ h

ach; ---m

h

i%

rrr

Fig. 13-7. Scheme for the electron transfer reaction from the excited trip-sextet (6T1) state of chloro-(3-methylimidazol)-(meso-tetraphenylpo~h~inato)chromium(~ (CrmP) to viologens (V"). (Reproduced from Ref. [ 161)

210

This figure shows that this reaction occurs as follows: (1) The ground (4S~)state of Cr"'P consists of the quartet state of Cr"' and the singlet state of P. (2) The direct excitation of 4 S ~ produces the excited quartet (4Sl) state consisting of the quartet state of Cr'" and the excited singlet state of P ('P*). (3) The internal conversion from 4S1 gives the excited trip-quartet (4TI) state consisting of the quartet state of Cr"' and the excited triplet state of P (3P*). (4) The intersystem crossing from 4T1 gives the excited trip-sextet (6T1) state consisting of the quartet state of C?' and 3P*. (5) The electron transfer reaction from 6 T ~to V2+ produces a geminate ion-pair in the sextet (Sx) state t['(Cr'"PP'+) 'V"]), which decays through either the escape from the solvent cage to give the free ions or the spin conversion to the quartet (Qa) state (4['(CrmP'+) 'V"]). (6) The Qa state decays though either the escape from the solvent cage to give the free ions or the backelectron transfer to give the ground state of CrmP and V2+. (7) Because the Sx-Qa conversion rate of the geminate ion-pair can be influenced by an external magnetic field, the free ion yield (YE) is expected to be changed by the fields. Fig. 13-6@) shows the YE@)/ YE(OT) values observed upon excitation under magnetic fields of 0 - 10 T [16]. From this figure, we can clearly see that the YE@)/ YE(OT) value increases with increasing B from 0 T to 4 T, that its maximum value is as large as 1.6 at 4 T, and that the value decreases slightly with increasing B from 4 T to 10 T. Similar MFEs were also observed for the reactions with l,l'-dimethyl-4,4'-bipyridinium (MV") and 1,l'diheptyl-4,4'-bipyridinium (C7V"). To the best of our knowledge, this is the first clear It is study of the MFEs on chemical reactions concerning higher spin states than 312. noteworthy that no MFE was observed for the electron transfer reaction in acetonitril at 293 K from the excited triplet state of meso-tetraphenylporphyrinatozinc(Zn(tpp)) to BV2+,where a Thus, the geminate ion-pair in the triplet state (3[2(Zn(tpp) ") 2V"]) was produced. magnetically induced increase in YE@)/ YE(OT) for the reaction of 6(Crm3P*)with BVZ+with increasing B from 0 T to 4 T can be explained by the RM, where the Sx-Qa conversion rate of the geminate ion-pair is reduced by the ZSF of '(Cr'"P'+). The absence of any M E in the reaction of 3(Zn(tpp))* is due to the fact that the anisotropic HFC and Zeeman interactions of The magnetically induced 3[2(Zn(tpp)'+) 'V'+] are much smaller than the ZSF of '(Cr"P'+). decrease in YE@)/ YE(OT) for the reaction of 6(Crm3P*)with BV2+with increasing B from 4 T to 10 T can be explained by the AgM. From the studies shown in this section, we can get the result that the MFEs on chemical reactions concerning higher spin states than 1 are much larger those due to the S-T or T-S conversion. Thus, remarkable MFEs can be obtained for the former reactions even in fluid solutions. We can predict from this fact that ordinary magnetic fields are possible to control such biochemical reactions as high spin states are involved.

Problem 13.4. The zero-field Hamiltonian (HzF) of two-interacting triplet molecules can be represented by

HZF= Hex+ HD, where

Hex = US IS^, H D =C ( - X , S i -TSf-Z,S,',). i=1,2

Calculate the eigen-functions and eigen-values for Hm in the case when the exchange energy (Hex) is much larger than the dipolar one (HD). Problem 13.5. The zero-field Hamiltonian (HzF) of a pair of a triplet molecule and a radical can be represented by

HZF= He,

+ HD,

21 1

where

Hex = -~JSTSR, H D =- X S & -YS& -ZSh. Calculate the eigen-functions and eigen-values for HZF in the case when the exchange energy (Hex) is much larger than the dipolar one (HD).

References [l] R. C. Johnson, R. E. Merrifield, P. Avakian, and R. B. Flippen, Phys.Pev. Lett., 19 (1967) 285. [2] R. C. Johnson and R. E. Merrifield, Phys. Rev. B, l(1970) 896. [3] V. Em and R. E. Memfield, Phys. Rev. Lett., 21 (1968) 609. [4] L. R. Faulkner and A. J. Bard, J. Am. Chem. SOC.,91 (1969) 6495. [S] L. R. Faulkner and A. J. Bard, J. Am. Chem. SOC.,91 (1969) 6497. [6] P. W. Atkins and G. T. Evans, Mol. Phys., 29 (1975) 921. [7] C. Blattler, F. Jent, and H. Paul, Chem. Phys. Lett., 166 (1990) 375. [8] A. Kawai, T. Okutsu, and K. Obi, J. Phy. Chem., 95 (1991) 9130. [9] P. W. Atkins and G. T. Evans, Mol. Phys., 27 (1974) 1933. [lo] A. Kawai, Y. Watanabe, and K. Shibuya, Mol. Phys., 100 (2002) 1225. [ l I] A. Kawai and K. Obi, J. Phys. Chem., 96 (1992) 56. [12] H. Tachikawa and A. J. Bard, J. Am. Chem. SOC.,95 (1973) 1672. [13] N. E. Geacintov and C. E. Swenberg, J. Chem. Phys., 57 (1972) 378. [14] Y. Mori, Y. Sakaguchi, and H. Hayashi, Chem. Phys. Lett., 286 (1998) 446. [15] H. Hayashi, Y. Sakaguchi, and M. Wakasa, Bull. Chem. SOC.,74 (2001) 773. [I61 Y. Mori, M. Hoshino, and H. Hayashi, Mol. Phys., 100 (2002) 1089. [17] Y. Mori, Y. Sakaguchi, and H. Hayashi, Chem. Phys. Lett., 301 (1999) 365. [IS] A. Carrington and A. D. McLauchlan, “Introduction to Magnetic Resonance”, Harper & Row, New York, 1967. Solutions to the Problems 13.1. From Eq. (13-16a), the following relations can be obtained -X= -013 i- E, -Y = -013 - E, -Z = 2013. Thus, D = -3212 = -212 - Z = (X i- Y)/2 - Z, E = -(X - Y)/2, where we used a relation X + Y + Z = 0. 13.2. Because the total spin S is the sum of STand SR,the following equations can be obtained:

212

(QIHexlQ)= ( Q ~ ~ / ~ I - ~ J S , S R ~(Q-t2/31-J(S2 Q , ~ , ~ ) = -<S: +Si)lQ+2/3) = - J ( Q ~ s ’ ~ Q+) J(+ 1 f 11215’;

+ Silk 1 k 1 / 2) = -J(3/2)(5/2) +J[(1)(2)+(1/2)(3/2)] = -J,

(DIffex[ 0 )= -J( 1/2)(3/2) +J[(1)(2)+(1/2)(3/2)] = 2J. 13.3. The left side of Eq. (13-22) becomes (Q*i/2IH,

- H , +HRID+i/*)

t t Jz

= -(+ 3

tJz- t

-(klT1/2JH, - H , + H , (- -IOf1/2)+

= ( -(0+1/21+

Jz

I IH, - H,lf 1) - -(+ 3

1 / 21HRIf 1/ 2) + -(+ 3

-1+1T1/2))

1 / 2 1 4 IT 1 / 2)

13.4. The eigen-functions of each triplet molecule at zero-field are represented by ITLI),IT,), and

ITz)( i=

L2).

(1)

Thus, a pair of two triplet molecules can be represented by the following nine functions: ITPTzq) @,q=x,y, and z).

(2)

for the nine functions, Let us calculate the matrix element of Hex

I

(TpTzq

(3)

H~XITJ~,.).

In order to calculate the matrix elements, we can use the following formulae for the effect of various spin operators on the triplet wavefunctions [ 181: S,pJ

= o , S&)

S,IT,)

= il T, , S, IT,) = -il T x ) ,and S , ITz)= 0 .

= ipz) SJT,) = -ilT,), 3

S,IT,)=-ilT2), S,lT,)

= o , S,IT2) = ilTx),

)

(4)

Using Eq.(4), we can easily prove the following result: (

~

p

I

~H e zx I T~p T * p )

=

- u ( ~ , p ~ z~ p(

+ ~ 1 , ~ 2+,

S ~ ~ S Z X

sizszZ )IT~T~~)

= 0 (forp=x,y, and z). Similarly, we can easily obtain the following result:

(TA

I

=

( T J ~ ,I

~ e x 1 ~ ~ 2 , )

(5)

I

= (T,,T~,

H ~ ~ I T Z T ~ ~ )

H ~ X I T J ~ ~ )

Thus, the 3x3 matrix spanned by IqpTzp)(forp=x,y, and z ) is given by

The eigen-values ( x ) can be obtained by the following equation:

= 2~.

(6)

213

-X

2J

2J

25 2J

-X

2J =O,

2J

-X

Similarly, the other triplet and quintet states may be obtained

and

X i + y + Z j = O (for i = 1 and Z), Using Eqs. (13) and (141, we can get the E values as follows: E(S)=(S/HZFIS)= 4J+ (113) c ( X j +Y, + Z j ) = 45, i=1.2

214

1

E(T~=(T, H Z F J T )

=

2 - m + z2)/2,

(15b)

E(T~)=(T,IHZFIT,)

=u-(xi+ x z ) ~

(15c)

E(T3)=(T3 I H z F ~ T = ~ )~ - ( Y I+ y2)/2,

I E(Q~)= (Q, IHZF IQ, ) = -u-(zl

(154

E(Ql)=(Q, HZFIQ,) = -U+(X1+X2+Y1+Y2+4Z1+42~)/6 = -2J +( ZI + Z2)/2,

+ z2)/2,

(150

E(Q3)=(Q3 IHzF(Q~) = -2J-(ZI + Z2)/2,

(15g)

I

I

We)

E(Qd=(Q, HZF Q, ) = -u-(XI + Xd/2,

U5h)

E(Q5)=(Q5IHmlQ,) = - ~ - ( Y I + Y2Y2,

(151)

13.5. The eigen-functions of the triplet molecule and the radical at zero-field are represented by

IT^), 1

~ ~ )ITz),and ’ 1*1/2).

(1)

Thus, a pair of the triplet molecule and the radical can be represented by the following six functions:

IT, f 1/ 2) (for p=x,y, and z ) .

(2)

Let us calculate the matrix element of Hex for the six functions,

1

( q p M R H e x 1 T , , , M t R )(for Mp, and M’R = *l/2).

(3)

In order to calculate the matrix elements, we can use Eq. (4) of Problem 13.4 for STand the following formulae for S R : 1

1

+S,)(+1/2)=-(-1/2), 2 1 1 - 1/ 2) = -(S,’+ S,)I - 1/ 2) = 1 / 2) , 2 2

s,(+1/2)=-(s; 2

s,

I

-I+

s , 1+ 1I 2) = -21(S ;; - s, )I+ 1I 2) = -2i 1- 1/ 2) , i s, I- I / 2) = =(s; 2

- S,)I

-

1/2) =

-I+-2i

1/ 2).

a (4b) (4c) ad

Using Eq.(4), we can easily prove the following results: (T,

HexIT, f l / 2 ) = -U(T,

( S T I S , +STySRy+STZS,)lT, i112)

= 0 (for p=x,y, and z),

(5a)

and

(T, f1121 He,IT, T1/2) = O(forp=x,y, andz).

(5

The non-vanishing matrix elements can be obtained as follows:

(TI +1/21 H,,)Ty +1/2) = +iJ,

a)

(T, +1/2) H e x ) T zT1/2) = T J ,

(6b)

215

(r,f1/2I H,,\T,

&1/2) = T i J ,

(6c)

(T, +1/2) H,,IT, T1/2) = i J ,

(6d)

(Ti 51/21 H,,)T, T1/2) = 3z J ,

(6e)

(T, -t1/2( H,,(T, Tl/Z) = -iJ,

(60

From IT, - 1/ 2), ITy - 1/ 2 ) ,and (Ti+ 1/ 2), we can get the following equation for the eigenvalues (x):

-x

-iJ

iJ J

-x

J iJ = 0 ,

-iJ

--x

(7a)

- 2 + 3 J 2 x + 2 J 3= - ( x - ~ J ) ( x + J ) ’

=O.

(1

(7b)

(I

for x = 2J) and two quartet states Q,) and Thus, we can obtain one doublet state D1),

1 Q,) ,for x = -J)as follows: IDI>= -(ITi 1

43

IQ,) = -(21Tz 1

45

] Q ) - -(ITx 1

-JZ

+ 1 / 2 ) + ( T x-1/2)+ilT, - m ) , +1/2)-lT, -1/2)-i[T, -1/2), -l/Z)-iIT,

From IT, +1/2), IT,

-1/2).

(8a) (8b) (8c)

+ 1/2), and IT, - 1/2), we can obtain one doublet state ( I D 2 ) ,for x =

2 4 and two quartet states (IQ,) and IQ,), for x = -J) as follows:

ID ) - IT^ 1 , A - 1 / 2 ) - 1 ~ +~ 1 / 2 ) + i J ~+1/2), IQ,) = -(21~, 1 - 1 / 2 ) + 1 ~+~1 / 2 ) - i J ~ ,+1/2), JZ lQ

) - -(-(T,+1/2)-ilT,+1/2). 1

-Jz

(9a) (9b) (9c)

The energies ( E ) of the zero-field Hamiltonian of the doublet and quartet states can be calculated as follows:

E(Di)=(D, IHmID,) =2J,

(W

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217

14. Optical Detected ESR and Reaction Yield Detected ESR 14.1 Histvrical Introduction Although the upper state involved in thc phosphorescence of many organic molecules in rigid solvents was proposed to be the lowest triplet state in 1444, numerous attempts to observe these triplet states directly by ESR had given ncgative results for many years. In 1958, Hutchinson and Mangum [ 11 succeeded in obscrving ESR signals due lo the triplet state of naphthalene in a single crystal of durene at 77 K. Their results showed that the naphthalene triplet state was already split into three states in the absence o f a magnetic field and that the resulting anisotropy of the magnetic splitting was tw great for one to observe the ESR signal due.to the h = & l transition for randomly oriented molecules at any given magnetic field. The Spin Hamiltonian (HT) of a triplet molecule is given by

HT=gp&aB+ Ho. (14-1) Here, HDis the Hamiltonian of the zero-field splitting (ZFSj of a triplet molecule with principal axes (X, Y, and Z) and principal values ( X , Y , and 2) 1 (14-2a) H D =-XS& - Y S h - Z S & = D ( S k - - S 2 ) + E ( S & -S&), 3 where (14-2b) D = -3Z2 = ( X + Y)/2- Z, (I 4-2~) E = -(X-y1/2. Hutchinson and Mmgum determined the following values for the naphthalene triplet state in durcne: s= 1, (14-3a) g = 2.0030+0.0004 (isotropic), (14-3b) (14-3~) D / hc = kO.1003 k 0.0006 cm-’, (14-3d) E / hc = 70.0137 k 0.0002 cm-I. Here, the X-, Y-, and Z-axes were assigned to be parallel to the long and short axes of naphthalene, and to be perpendicular to its molecular plane, respectively. It is noteworthy that the sign of D could not be determined from the present work. It was determined to be positive from relative intensity measurements of ESR signals at much lower temperatures [ 2 ] . Because the parameters of triplet molecules were determined from measurements in single crystals, ESR signals of triplet states due to the h = + 2 131 and h = ? l [4] transitions could also be observed in randomly oriented systems. The signal-to-noise ratio (S/N) for usual ESR detection of an excited state depends on its population (hr, and hence is proportional to the lifetime (.t> of the state. On the other hand, the signal intensity for the optical experimcnt depends on the number of photons emitted from the excited state which is proportional to NW. Herc, W is the optical transition probabiIity which is proportional to l/z. Therefore, the S/N ratio for the optical experiment is independent of 2. Using this strategy, Geschwind et al, [ 5 ] found, in 1959, the ESR signals due to the excited Z(’E) state of Cr3+in A1203 with optical detection (OD) methods of ESR at liquid He temperatures. Applying such ODESR techniques to organic triplet states, Shamoff [6] first observed, in 1967, an ODESR signal due to the &=*2 transition of the triplet state of naphthalene-& in a biphenyl crystal at 1.8 K, monitoring the change of the phosphorescence intensity induced by the X-band microwave transition. Soon after the

218

Sharnoff s paper, Kwiram [7] and Schmidt et al. 181 detected the ODESR signals due to both of the h = & 2 and h = ? l transitions for phenanthrene-dlo in a biphenyl crystal and quinoxalined6 in a durene crystal, respectively, at liquid He temperatures. Because such triplet states as observed above with ODESR methods are of x- x* type, they had already been observed with usual ESR ones at 77 K. It is noteworthy, however, that new information such as the assignment of the lowest triplet state and the determination of the population and decay rates of the state can be obtained, in principle, with ODESR methods. Sharnoff also observed ODESR signals due to the &=&2 and h = * l transitions of the n- x* triplet state of pyrazine in a crystal of paradichlorbenzene at 1.3 K [9]. This was the first observation of the ESR transition for an n- x* triplet state, which had not yet been detected with usual ESR methods because of the much shorter lifetimes of the n- a* triplet states than those of the x- x* ones. It is also noteworthy that the absolute D and E values could be determined with the ODESR method as follows:

D = -(0.309G0.001)cm-', E = +(0.006G0.002)cm-'.

(14-4a) (14-4b)

Here, the z-axis was taken to the axis of greatest splitting of the &=A transitions and the yaxis to be the axis of smallest splitting. The x-axis was found to be the axis perpendicular to the molecular plane. The ODESR results showed that the order of the triplet sublevels should be

Ez Ey < En (144) and that the Tz level should be the optically active component polarized along the y-axis. Thus, the y-axis was unambiguously identified with the N-N axis of pyrazine. It is also noteworthy that the absolute D value of pyrazine was found to be approximately twice of that of benzene. The sign and magnitude of the D values of the n-x* triplet states could be explained by the spin-spin dipolar interaction. The author explained such anomalous sign and magnitude of the D values of pryrazine and other n- x* triplet states in terms of the mixing between the n- x* and n- x* triplet states through the spin-orbit interaction [ 101. In 1968, Schmidt and Van der Waals [ l l ] succeeded in optical detection of zero-field transitions in phosphorescent triplet states. In this technique, the system was irradiated continuously with ultraviolet light, in order to obtain a steady-state population of the triplet state. A microwave field was then swept slowly through one of the zero-field transitions and the concurrent variation in phosphorescence intensity recorded by a photomultiplier. The experiments were often performed at a temperature of 4.2 K where relaxation between the spin sub-levels turned out to be so fast that amplitude modulation of the microwaves could be used together with phase sensitive detection techniques at frequencies much higher than the decay rates of the individual levels. Relative to conventional zero-field spectroscopy, this optical detection method offered the advantage of circumventing the difficulties inherent in wide-band microwave detection methods. As compared to the usual ESR experiments performed in the presence of a high magnetic field, the absence of the field eliminated the anisotropy introduced by such a field and the scrambling of zero-field sub-levels with different modes of decay. In the zero-field method, therefore, the ZFS parameters could be determined much easily with randomly oriented systems than in conventional ESR methods where the angular-dependent measurements with single crystals had been indispensable. Because the relaxation rates of zero-field sub-levels were found to be much reduced by lowering the temperature from 4.2 K, the spin sub-levels in many molecules could be isolated from each other at 1.25 K. During the decay of such an isolated triplet spin system, drastic changes could be induced in the phosphorescence intensity by the sudden irradiation of a microwave transition between a pair of levels. Through such microwave-induced delayed

219

phosphorescence signals, the dynamics of populating and depopulating the individual spin sub-levels of the phosphorescent triplet states could be investigated precisely [ 121.

14.2 ODESR Measurements of Radical Pairs As shown in Section 14.1, the optical detection ESR (ODESR) technique was first employed to study the ESR spectra of isolated triplet excited molecules in molecular crystals at low temperatures. The microwave-induced transitions between the triplet sub-levels result in changes in the phosphorescence intensity because the light emission probabilities are unequal for different sub-levels. In the 1970's, the ODESR technique was applied to observe the ESR spectra of light-induced short-lived pairs of ion-radicals and triplet exitons in molecular crystals, where the resonance microwave radiations between the triplet sub-levels result in changes in the fluorescence intensity from the singlet excited state generated through the cage recombination within the singlet radical pair. In 1979, Molin's group [13] first applied the ODESR technique to observe the ESR spectra of short-lived radical pairs in solutions at room temperature. Aromatic ion-radical pairs (M+.---M-*) were produced in nonpolar solvents (S) by ionizing radiation as follows: (14-5a) S + positrons ("Na) + S+'+ e-, e- + M --+ M-*, (14-5b)

S+' + M --+ M+', (14-5~) M+'+ M-*--+ M* + M. (14-Sd) The excited molecule (M*) can arise either in the singlet state and thus emit fluorescence, or in the tridet state which is nonluminescent. depending on the multiplicity of the M+'---M-' pair at the moment of recombination. Fig. 14-1. Optical detected ESR spectra of the ion-radical pair (naphthalene)-/ (naphthalene)' for 1 . 1 510.' ~ M naphthalene solution in squalane at room temperature at various microwave fields ( H I ) . (Reproduced from Ref. [ 13b] by permission from The American Chemical Society)

609G

B

Fig. 14-2. Energy level diagram and S-T transitions in a radical pair at a high external magnetic field (a) without and (b) with the resonance microwave field; hfi is S-To mixing due to hyperfine interactions; ESR is resonance microwave transitions. (Reproduced from Ref. [13b] by permission from The American Chemical Society)

220

Fig. 14-1 shows the ESR spectra of the naphthalene+’---naphthalene-’ pairs detected optically in squalane at room temperature under various microwave powers. The maximum ESR signal (at 3.8 W) corresponds to a 5 % decrease in the fluorescence intensity. The initial M+’---M-’ pair is a singlet-born one because the unpaired electron spins were paired in the parent solvent molecule (S). According to the RPM, it is known that prior to recombination the multiplicity of this pair can be changed from singlet to triplet due to the HFCM. The HFCM mixes the S and TOlevels in a high external magnetic field as shown in Fig. 14-2. The T+1 and T.1 levels are off resonance with the S one because of the Zeeman splitting and hence cannot be populated due to the HFCM. The time average depopulation of the S level can amount to 50 % due to this S-To mixing in radical pairs containing many magnetic nuclei. If resonance ESR transitions are induced by microwave irradiation, all the triplet levels can then be populated as shown in Fig. 14-2. Hence, the depopulation of the S level achieve 75 % in the extreme case of equal average population of all the coupled levels. Thus, the probability of singlet recombination and hence the fluorescence intensity may decrease substantially under microwave irradiation which makes it possible to detect the ESR signal optically. From Fig. 14-1, the ESR signal intensity is seen to decrease both with increasing and decreasing microwave power from its optimum value (3.8 W). This behavior can easily be explained qualitatively. At low microwave power, the ESR intensity drops since the rate of spin flips becomes insufficient. When the microwave power considerably exceeds the hyperfine splitting, all the spins precess about the microwave magnetic field (in rotating frame) with nearly the same frequency. The ESR intensity drops again as this synchronous precession does not destroy the initial spin correlation of the radical pair. Because the generation rate of aromatic pairs and their lifetime can be estimated to be 2x1OSs-*and IO-’s [13], respectively, the average concentration of the radical pairs is about 20 per sample with the S/N value of 10/1. This sensitivity of this ODESR method can be compared with that of conventional ESR spectrometers equal to 10” spins per sample at a line with of 1 G. Although no HF structure could be observed for the naphthalene system [13a], the HF structure could be observed in the ODESR spectrum of diphenyl system [13b]. Thus, the ODESR method was found to be a unique and powerful method in studying the nature, reactions, and molecular dynamics of short-lived radical pairs. Since the pioneer work of radical pairs in solution by Molin’s group [13] with an ODERS method, many researcher have investigated the structure and dynamics of radical pairs with various ESR methods other than the conventional one. As for ODESR, phosphorescenceand fluorescence-detected ESR were explained at Sections 14.1 and 14.2, but ESR spectra can also be measured by detecting the optical density due to the intermediate radicals and excited triplet states. The latter method is called absorption-detected ESR (ADESR), but is often included in ODESR. There are many other ESR techniques to detect such intermediate species as radicals and excited triplet states without direct microwave detection. Because the yield of reaction products and their CIDNP intensity are also be changed by ESR transitions in the reactions of radical pairs, reaction-yield-detected ESR (RYDESR) and CIDNP-detected ESR have also been applied to the investigations of radical pairs and biradicals. Because conductance of ions generated from radical pairs is also changed by ESR transitions, conductance-detected ESR is also possible for the detection of radical pairs. The name “Reaction-Yield-Detected Magnetic Resonance” (RYDMR) is sometime used to refer to the above-mentioned ESR methods without direct microwave detection in general. A good review was written by Okazaki [14] for RYDMR and its applications to control chemical reactions. In the next two sections, therefore, we will explain only the typical results obtained by RYESR and ADESR.

221

14.3 RYDESR Studies of Radical Pairs The author's group studied MFEs on the photoreduction reactions of carbonyl and quinone molecules (XCO) in micellar solutions. The reaction scheme was shown in Chapter 7 as follows: Photo-Excitation, (14-6a) XCO + hu -+ 'XCO* 'XCO* j 3 x ~ ~ *

Intersystem Crossing(kIsc),

(14-6b)

3XCO* + RH -+3[XC'OH R']

Generation of Radical Pairs (b),

(14-6~)

3[XC'OH R']

To-S Conversion ( k s ~ ) ,

(14-6d)

Escape of Radicals ( k ~ ) ,

(14-6e)

Cage Recombination (kp).

(14-60

'.'[XC'OH

R']

'[XC'OH R']

t)'[XCOH

R']

+ XC'OH + R' + XC(R)OH

Here, RH is a micellar molecule. As shown in Chapter 7, we have found large MFEs on both the lifetimes of radical pairs and the yields of escape radicals in the above-mentioned reactions, measuring the transient absorption of intermediate radicals with an ns-laser photolysis method. Okazaki and Shiga [15] observed similar MFEs on the yield of R'in the reaction of 2-methyl-l,4-naphthoquinonein a micellar SDS solution, measuring the yield of spin adducts with a conventional ESR method. Here, the escape radical reacts with a spin trap (TNO) and generates a spin adduct (T(R)NO') as follows:

TNO+ R']+

T(R)NO'

Formation of Spin Adduct(kTrap)

(14-6g)

T

Fig. 14-3. PYDESR spectra observed for the photoreduction of 2-methyl-] ,4naphthoquinone in a micellar SDS solution at 23kl"C with (a) PBN and (b) DMNS as spin traps. The microwave irradiation was performed at 9488 MHz and 160 mW. (Reproduced from Ref. [I51 by permission from Nature

I

334

I

336

I

338

340

342

Magnetic field (mT)

Okazaki and Shiga [15] further observed the effect of ESR transitions on the spin-adduct yield. They irradiated the de-aerated sample solution with a 500-W ultra-high-pressure mercury lamp and the X-band microwave for 30 s at 23kl"C. After a specified waiting time (1 .O min), they recorded the ESR spectrum of the spin adduct. They used a flow system to change the solution without changing the filling factor of the sample solution in the ESR

222 cavity. The experiments were repeated with a new solution using a different magnetic field, which was changed point by point to scan the field. Thus, they obtained the reduction in the yield of the spin adduct during this reaction as a function of the external magnetic field with each of phenyl-tert-butylnitrone (PBN) and perdeuterio-2,4-dimethyl-3-nitrosobenzenesulphonete (DMNS) as a spin trap. Such PYDESR spectra as observed with the abovementioned method are shown in Fig. 14-3. The observed spectra could be assigned by the 1:l mixture of the semiquinone and SDS radicals as shown in Fig. 14-4. The decrease in the spin adduct yield could be explained by the enhancement of the T+l-Toand T-l-Toconversions induced by the resonance microwave transitions. Although the procedures for measuring the PYDESR spectra with this method were very complicated and time-consumptive, Okazaki and Shiga demonstrated that the ESR spectra of short-lived radicals could be detected with a mercury lamp and a conventional ESR apparatus. It is also noteworthy that this was the first experiment to change the product yield by ESR transitions. From this experiment, a technique called “spin-manipulation” has been developed.

4&

332

I

336

I

I

I

I

338 340 Magnetic field (mT)

I

I

342

w 345

Fig. 14-4. A calculated ESR spectrum for the 1:1 mixture of the semiquinone and SDS radicals. Stick diagrams under the simulated spectrum indicate the signal positions of the semiquinone and SDS radicals. (Reproduced from Ref. [ 151 by permission from Nature) As the first step of spin manipulation, Okazaki and Tonyama [ 161 studied the microwave power dependence of the RYDESR spectra for the photoreduction of anthraquinone in SDS and perdeuteriated SDS micellar solutions. Their observed spectra are shown in Fig. 14-5, together with the simulated spectra under various powers. Fig. 14-5(A) shows the RYDESR spectra in an SDS solution at the microwave power of 4.0 and 1000 W. A relatively sharp spectrum was observed at 4.0 W, where the HF structure should remain from the simulated spectrum. The spectrum obtained at the microwave power of 1.0 kW is broadened over the range of 40 mT, showing a minute structure at the center. Fig. 14-5(B) shows the RYDESR spectra in a perdeuteriated SDS micellar solution at the microwave power of 16 and 1000 W.

223 The spectrum obtained at the microwave power of 1 .O kW is also broadened, showing a large reversed peak at the center. Such broadening and reversed peaks are due to spin locking.

g 100 90

80 Y

.e,

7 .g r/) Y

70

60 50 40

30 2031s

320

325

330

335

340

345

350

355

340

345

350

355

Bo J mT 100

3

90

oao

315

320

325

330

335

B, / mT Fig. 14-5. PYDESR spectra detected for the photoreduction of anthraquinone at room temperature: (A) in a non-labeled SDS micellar solution at the microwave power of (a) 4.0 W and (b) 1.0 kW. (B) in a deuteriated SDS micellar solution at the microwave power of (a) 16.0 W and (b) 1.0 kW. The simulated spectra are shown by solid curves. (Reproduced from Ref. [ 161 by permission from The American Chemical Society) Okazaki et al. showed some examples for spin-manipulation [14]: (1) Because the yield of spin adducts produced from escape radicals was found to be decreased by the resonance microwave irradiation, the yield of cage (escape) products was found to be increased (decreased) with a high pressure liquid chromatography (HPLC) technique. (2) Because the resonance microwave irradiation induces a transition between two definite nuclear spin-levels, the enrichment of this nuclear spin could be realized. Indeed, the ratio of 13C/12Cwas

224

changed by the resonance microwave irradiation. (3) Because the spin locking behavior was found to be different for deuteriated and non-labeled SDS micellar solutions as shown in Fig. 14-5, the D/H separation was also shown to be possible with the resonance microwave irradiation. With pulse laser and microwave, Okazaki et al. [17-191 further demonstrated that some kinetic information on radical pairs could be obtained with the pulse-RYDESR method. They employed the three types of pulse sequences as shown in Fig. 14-6. Using this pulseRYDESR method, they studied the photoreduction reactions of carbonyl and quinone Fig. 14-6.

EXP.l

Timings (z)between the laser ignition (narrow) and the gating pulse for the PIN switch (shaded) to pulse the microwave (MW). In Exp. 1, the MW-width was typically 50 ns. In Exps. 2 and 3, the MWwidth was fixed to 20 ps. (Reproduced from Ref. [ 191 by permission from Springer Verlag KG)

EXP.2 EXP.3

a

0

0

2

2

6

4

6

4

w s

8

8

10

10

Fig. 14-7. Reductions in spin adduct yield in the three pulse experiments for the photoreduction of (a) anthraquinone and (b) benzophenone in SDS micellar solutions. The concentration of spin trap was 1.0 mM. Laser irradiation was made for 20 s at 10 mJ (1= 335 nm). Calculated dependence on the delay time (z)is shown for each pulse experiment: Exp. 1 (broken curves), Exp. 2 (solid curves), and Exp. 3 (dotted curves). (Reproduced from Ref. [ 191 by permission from Springer Verlag KG)

225 molecules (XCO) in micellar solutions. Their typical results are shown in Fig. 14-7. Here, reductions in spin adduct yield observed in the three pulse experiments for the photoreduction of anthraquinone (AQ) and benzophenone (BP) in SDS micellar solutions are shown. They simulated their results by integrating a set of differential equations for a detailed kinetic model using the Runge-Kutta method [17-191, but their simulation can be explained by the following procedures. According to the RM reviewed in Chapter 7, the respective rate constants shown in Scheme (14-6) and Fig. 7-4 have the following order: (14-7) klsc >> kST >> kc kp >> kE kR - kR’ k~,JTrap] Thus, the growth and decay of [MJ(=[S]+[ToJ)and [T,](=[T+lJ+[T.l]) in the presence of a high magnetic field can be given by the following equations:

-

-

-

[TJ = 2- k, [exp(-k,t) - exp(-k,t)] 3 k, -ks

.

(14-8b)

Here, kF was given in Chapter 7 (Eq. (7-1 1))as follows: (14-9a) kF = kp/2 + ks, and ks of Eq. (14-8b) is given by the sum of ks of Eq. (7-14) and k~,~[Trap]as follows: ks = kR + kR’ + kE + k~,~[Trapl. (14-9b) The time profiles of [MJ and [TJ are schematically iIlustrated in Fig. 14-8.

Population

m

tIp Time profiles of radical pair population: [MI = [S]+[ToJ and [T*J = [T+1]+[T.l 1. kc a n d b 1= -k G . Here, a = -3 k,-k, 3 k, -kF Fig. 14-8.

226 In Exp. 2, the microwave pulse is applied from the laser excitation (t=O ns) to t=z. Thus, the population of [TJ generated from t=O ns to t = z i s transferred to the M level and this change gives the cage product. The observed growth in the time profiles (solid curves in Fig. 14-7), therefore, corresponds to the generation of the cage product. Thus, the kc and kp values can be obtained from the growth of the solid curves. In Exp. 3, the microwave pulse is applied from t= z to the end of the reactions. Thus, the population of [T+]remained at t= z is transferred to the M level. The observed decay in the time profiles (dotted curves in Fig. 14-7), therefore, corresponds to the decay of [TJ Thus, the ks value can be obtained from the deca of the dotted curves. Here, the kTmP value of Eq. (14-9b) was determined to be ’ were calculated from the 2 . 0 2 ~ 1 0M -1 s-1 with changing [Trap] and the k~ and k ~ values main term of the spin-lattice relaxation of the radical pairs as follows:

7

(14- 10) where Bo, BI,, are the external magnetic field for ESR, the local field of the anisotropic HFC of the alkyl radical generated from the micellar molecule, and the rotational correlation time of the alkyl radical, respectively. In Exp. 1, the microwave pulse is only applied at t=z after laser excitation. Thus, the population of [T+]at t= z is transferred to the M level. The observed growth and decay in the time profiles (broken curves in Fig. 14-7), therefore, corresponds to [TJ-[MI at t=z. Okazaki et al. also used the transition probability due to the ) an additional parameter for the simulation of the broken curves. microwave pulse ( k ~ was The kinetic parameters obtained from the above-mentioned simulation are listed in Table 141 Table 14-1. Kinetic parameters (in lo6 s-’) obtained for the photoreduction of quinone and carbonyl molecules (XCO) in micellar (RH) solutions at room temperature with an X-band PYDESR method (Bo-334mT) [17-191.

XCO: AQ for anthraquinone, AQS03- for anthraquinone-2-sulfonate,BP for benzopheneone, I for 1,5-diphenyl-l,4-pentadiyn-3-one, and 11 for 1,3-diphenyl-2-propyn-l-one.The k R and kR’ values are calculated from Eq. (14-10). From Table 14-1, the following results could be found for the kinetic parameters of various radical pairs: (1) The hydrogen abstraction rate (kc)had already been measured from the decay of triplet precursors with transient optical absorption methods. Because the obtained kG values listed in Table 14-1 were found to be similar to those measured with optical absorption methods, the present simulation in PYDESR was found to be correct. (2) The cage product formation rate (kp) had not been determined with other methods including optical absorption ones before this PYDESR one. Thus, it is noteworthy that this was the ~ first method to determine the k p values. (3) The escape and the relaxation rates ( k and kR+kR’) had already been measured with transient optical absorption methods, but the

227

accuracy in determining these values with optical methods had not been so good. Although the PYDESR method was found to be a new one to determine these values, this method was not enough to obtain these values solely through experimental procedures.

14.4 ODESR Studies of Radical Pairs Although the PYDESR method developed by Okazaki et al. was very unique in obtaining the kinetic parameters in photoreduction of carbonyl and quinone molecules in micellar solutions, its experimental procedures were very complicated and their method could not determine all the parameters solely through experimental procedures. Thus, much easier and much more accurate methods than the PYDESR one have been pursued for obtaining kinetic parameters of radical pairs and biradicals. One of such candidates is the ODESR method, where ESR transitions are monitored through transient optical absorption of radical pairs and escape radicals. Maeda et al. [20] made such an X-band ODESR apparatus with a pulse laser (Nd :AG laser, il=266 nm) and pulse microwave (employed pulse width of 2 ps and maximum output power of 2 kW) and studied a biradical generated in the photolysis of the polymethylene linked xanthone and xanthene system. Although they could observe its time-resolved ESR spectra by changing point to point of an external magnetic field (Bo), they could not determine its kinetic parameters with their ODESR method. Under such a situation, Sakaguchi et al. [21] tried to improve the X-band ODESR apparatus with a Nd:YAG laser and pulse microwave (shortest pulse width of 15 ns and maximum output power of 1 kW). They succeeded not only in measuring time-resolved ESR spectra of radical pairs by sweeping Bo but also in determining their kinetic parameters by monitoring the time profile of the transient absorption ( A @ ) of radical pairs and escape radicals with and without resonance microwave pulse. Fig. 14-9 shows the definition of parameters and pulse sequences applied in Ref. [21]. Fig. 14-10 shows some typical A(t) curves observed in the photoreduction of 2-methyl-I ,4-naphthoquinone (MNQ) in as SDS micellar solution at room temperature under Bo=331 mT. As clearly seen in this figure, the A(t) curve without a microwave pulse was found to be much decreased by the application of a short or long microwave pulse. This is due to the increase in the cage recombination after the microwave-induced transfer of the radical pair population from [T,] to [MI. Using such A(t) curves observed in the absence and presence of a various microwave pulse, Sakaguchi et al. obtained ESR spectra of radical pairs as shown in Fig. 14-11 and determined their kinetic parameters as shown in Fig. 14-12.

Laser (t=O ns)

Microwave Pulse .......................

;;;;:;l ..... .il. I

A ( t ) Curve I

tD

AtMW

I

Time Window

Fig. 14-9. Definition of parameters and pulse sequences applied in Ref. [21]. Here, the A(t) values are averaged during the time window.

228

e0 KJ

P

0.2-

m

0.0

-1 ..

I

I

I

I

I

,

t/ps

Spectrum c of Fig. 14-1 1 was obtained with the A(t) curves observed in the absence and presence of a long microwave pulse (t~=-40ns,dtMw+s, and B1=0.04&.01mT). Here, the A(t) curves were averaged in the time window from 3 to 8 ps after the laser pulse. Because this spectrum was measured under a low BI value, one can clearly distinguish the signals due to the alkyl and semiquinone radicals ( Diagrams a and b in Fig. 14-11, respectively). Spectrum c of Fig. 14-11 was much better resolved than the spectra shown in Fig. 14-3. It is noteworthy that the former spectrum could be measured with much easier procedures than the latter one. Spectrum e of Fig. 14-11 was obtained with the A(t) curves observed in the absence and presence of a stronger microwave pulse which was the same that was used for measuring Curve c of Fig. 14-10. Here, the A(t) curves were averaged in the time window from 3 to 8 ps after the laser pulse. One can see that the width of Spectrum e was increased to more than 10 mT from that of Spectrum c and that additional inverted features appeared in the center of Spectrum e. The resolution of the inverted signals became better in the spectrum observed between 200 and 400 ns after the laser excitation (Spectrum d in Fig. 1411). Okazaki et al. had predicted such inverted signals from their theoretical calculation as shown in Fig. 14-5, but their experimental results shown by points in Fig. 14-5 had not been able to separate two dents at the center of their spectra. It is noteworthy that these dents were ascribed to the level-crossing phenomena [22], although these inverted features had conventionally been referred to a typical phenomena due to the spin-locking effect.

229

r

300

310

330 34-0 350 magnetic field/ rnT

320

360

Fig. 14-11. ESR spectra observed in the photoreduction of 2-methyl- 1,4naphthoquinone (MNQ) in as SDS micellar solution at room temperature. (a) and (b): Stick diagrams indicating the signal positions of the alkyl and semiquinone radicals, respectively. (c): ODESR spectrum obtained with the A(t) curves observed in the absence and presence of a weak microwave pulse (a=40ns, dtMW=5psLs,and B1=0.04+0.01 mT). . (d) and (e): ODESR spectra obtained with the A(t) curves observed in the absence and presence of a strong microwave pulse (t~=-40ns,d t ~ w = 5 P S , and B1=2f10.3mT),where the time windows for observing the A(?) curves are (d) 200 - 400 ns and (e) 3 - 8 ps, respectively. (Reproduced from Ref. [211 by permission from Elsevier Science B.

v.1

When a short microwave pulse ( d t ~ w = l k Sis ) applied at f = to, a part of the population at the T, state (C([T+kt,)- [ M k t , ) ) ) is transferred to the M state. [TiNt, + E ) = [ T + I ( f D ) - C([T+l(t,)-[Mkt,)),

( 14- 11a)

[MktD+ E ) = [Mkto 1+ C([T,kto 1- [MktD1) .

(14- 11b)

Here, the time profiles of [MI and [T+] are given by Eqs. (14-8a) and (14-8b), respectively. But, ks is represented as follows: ks = k R + kR’ + kE. (14-12) , a of Fig. 14-10 was found to be After the application of the microwave pulse at t = t ~Curve changed to Curve b of Fig. 14-10. The b-a difference is illustrated by Curve d in Fig. 1412(a). This curve shows that a-b value gradually increases immediately after the microwave pulse irradiation and that the value slowly decreases after its maximum at t = t ~ l . The initial ) by Eq. (14increase corresponds to the disappearance of [MI with its rate constant ( k ~given In principle, this rate constant for radical pairs generated by the photoreduction of 8a). carbonyl and quinone molecules in micelles can be obtained from their A(t) profiles with nslaser photolysis measurements. In the actual measurements, however, this rate constant has never been obtained from their A(t) profiles because there are many components due to other species such as triplet precursors and reaction products in the profiles. From the present ODESR measurement, Sakaguchi et al. could purely generate the mixed M state with the microwave pulse and succeeded in direct measurement of the k~ value for the first time. From the decay of the a-b value, the ks value can be obtained. They, therefore, could determine the k p value for the first time from their ODESR method.

230 kp = 2(kp - ks). (14-13) After the decay of the a-b value, it attains a constant value at t = t E as shown in Fig. 14-12(a). This value corresponds to the yield of the escape radical. Thus, the rate constant of the spin) also be relaxation and that of the escape of radicals from a radical pair (kR+kR’ and k ~ can obtained with Eq. (14-12) and the following relations: (I 4-14) [a-bl(tE>/[a-bl(tMI) = kid ks.

Fig. 14-12. (a) Curve d is the difference between Curve b and Curve a shown in Fig. 14-10. (b) The difference of the A(t)

‘p -0.06

n

-0.08

curves (dt~w=15ns,and B , = l d . l m T ) with and without a short microwave pulse (Aesc) was averaged in the time window from 6 to 8 ps after tha laser pulse. The solid curve was calculated with the kinetic parameters obtained in this measurement. (Reproduced from Ref. [21] by permission from Elsevier Science B. V.)

-0.10 -0.12

-0.14

time/ ps

I

I

tM1

tE

a

-* 4

6

0 4

4 2

0 0

1

3

2

4

5

tD /rs

Sakaguchi et al. also obtained the kinetic parameters from the following method. When they measured the [a-b](t= t ~value ) by changing the t~ value, they obtaind the curve shown in Fig. 14-12(b). The following features can be obtained from Fig. 14-8: If to is negative, the [a-b](t= t E ) value becomes zero, because there is no radical pair at t = to. When tD is increased from 0 ns, the [a-b](t = t ~ curve ) shows a fast rise and a slow decay after its maximum at t = t M 2 as shown in Fig. 14-12(b). If the generation of the radical pairs is much faster than the disappearance from its mixed M state ( k > > k p ) , the initial part of the rise corresponds to the generation process with the rate constant of kc. During this stage, the [M]/[T,] ratio is kept approximately constant. When the generation of the radical pairs is slowed down due to the consumption of the triplet precursor, the [h4/[Tt] ratio is decreased by the cage recombination from the M state. On the other hand, the decay from the T, state is much slower than that of the M one. Because the [Ml/[T+]ratio becomes zero at the maximum point ( t = tM2),the decay part of the [a-b](t= t ~curve ) corresponds to the decay of [ T J . From its decay, therefore, the ks value can also be obtained. This method for the

231

determination of the ks value is much better than that from the a-b curve mentioned before if the k~ value is much larger than the k ~ + k ~one. ' Table 14-2. Kinetic parameters (in lo6 s-') obtained for the photoreduction of quinone and carbonyl molecules (XCO) in micellar (RH) solutions at room temperature with X-band (Bo-331mT) and Ku-band (Bo-622mT) ODESR methods [21,23,24].

XCO: MNQ for 2-methyl-l,4-naphthoquinone,BP for benzopheneone, and DFBP for decafluorobenzophenone. It is noteworthy that all of the rate constants for the photoreduction of carbonyl and quinone molecules in micelles can accurately be determined from this ODESR method. From the usual laser photolysis method where only the A(t) profiles are used, the k~ value has not yet been measured and its accuracy in the determination of the other rate constants is much worse than that with this ODESR method. Sakaguchi et al. have been carrying out such measurements for many reactions with this X-band ODESR method and have determined all of their rate constants very precisely. Their typical results [23] are shown in Table 14-2. Here, the rate constants determined at room temperature with the ODESR method at B = 331 mT for the photoreduction of 2-methyl-l,4-naphthoquinone(MNQ), benzophenone (BP), and decafluorobenzophenone (DFBP) in micellar SDS and Brij 35 solutions are listed. From this table, we can see the following features in the spin dynamics of these reactions: (1) The rates of hydrogen abstraction (kc)of MNQ and DFBP are about ten times larger than those of BP. This is due to the fact that the n, K* characters of 3MNQ* and 3DFBP* are larger than that of 3BP*. (2) The rates of cage recombination (kp) in the SDS micellar solution are about three times larger than those in the Brij 35 solution. This is due to the fact that the SDS micelles are smaller than the Briji 35 ones. (3) The rates of triplet spin relaxation ( k +~ kp, ') of DFBP are slightly larger than those of MNQ and BP. This is due to the fact that the anisotropic Zeeman and HFC terms of the three kinds of radical pairs are similar to one another. The fact that the k~ + k~ ' values of DFBP are slightly larger than those of MNQ and BP may be due to the effect of ten F-atoms. (4) The rates of radical escape (kE) have the increasing order of ~E(MNQ)> k,(BP) > ~E(DFBP). This can be explained by the increasing oder of the hydrophilicity of the MNQ semiquinone, BP ketyl, and DFBP ketyl radicals. The MFEs on the TRP and YE values under ultrahigh fields described in Chapter 12 correspond well to features 3 and 4 obtained in this section. The above-mentioned results clearly show that the ODESR method gives unique and accurate knowledge on spin dynamic of radical pairs. A weak point of ODESR is the fact that its measurements have been limited to the X-band microwave (U - 9 GHz, B - 0.3 T). This means that the spin dynamics can only be measured at B - 0.3 T. At present, it is very difficult for us to construct ODESR apparatus at various microwave frequencies because the microwave amplifier is only available at X and Ku band regions. Sakagucihi has recently

232

-

developed a Ku-band ODESR apparatus (at B 0.6 T) for the first time [24]. His preliminary result on the k~ +kR’ value is also listed in Table 14-2.

References [l] C. A. Hutchison, Jr. and B. W. Mangum, J. Chem. Phys., 29 (1958) 952,34 (1961) 908. [2] A. W. Homing and J. S. Hyde, Mol. Phys., 6 (1963) 33. [3] J. H. van der Waals and M. S. de Groot, Mol. Phys., 2 (1959) 333. [4] W. A. Yager, E. Wasserman, and R. M. R. Cramer, J. Chem. Phys., 37 (1962) 1148. [51 (a) S. Geschwind, R. J. Collins, and A. L. Schawlow, Phys. Rev. Lett., 3 (1959) 545. (b) S. Geschwind, G. E. Devlin, R. J. Collins, and S. R. Chinn, Phys. Rev., 137 (1965) A1087. [6] M. Sharnoff, J. Chem. Phys., 46 (1967) 3263. [7] A. L. Kwiram, Chem. Phys. Lett., 1 (1967) 272. [8] J. Schmidt, I. A. M. Hesselmann, M. S . De Groot, and J. H. Van der Waals, Chem. Phys. Lett., 1(1967) 434. [9] M. Sharnoff, Chem. Phys. Lett., 2 (1968) 498. [lo] H. Hayashi and S. Nagakura, Mol. Phys., 24 (1972) 801. [ 111 J. Schmidt and J. H. Van der Waals, Chem. Phys. Lett., 2 (1968) 640. [12] (a) J. Schmidt, D. A. Antheunis, and J. H. Van der Waals, Mol. Phys., 22 (1971) 1. (b) D. M. Burland and J. Schmidt, Mol. Phys., 22 (1971) 19. [13] (a) 0. A. Anisimov, V. M. Grigoryants, V. K. Molchanov, and Yu. N. Molin, Chem. Phys. Lett., 66 (1979). (b) Yu. N. Molin, 0. A. Anisimov, V. M. Grigoryants, V. K. Molchanov, and K. M. Salikhov, J. Phys. Chem., 84 (1980) 1853. [14] M. Okazaki, in “Dynamic Spin Chemistry”, S. Nagakura, H. Hayashi, and T. Azumi. Eds., Kodanshfliley, Tokyo, 1998, Chapter 8. [15] M. Okazaki and T. Shiga, Nature, 323 (1986) 240. [16] M. Okazaki and K. Toriyama, J. Phys. Chem., 99 (1995) 17244. [17] N. E. Polyakov, Y. Konishi, M. Okazaki, and K. Toriyama, J. Phys. Chem., 98 (1994) 10558. [18] N. E. polyakov, M. Okazaki, K. Toriyama, T. V. Leshina, Y. Fujiwara, Y. Tanimoto, J. Phys. Chem., 98 (1994) 10563. [19] M. Okazaki, N. Polyakov, Y. Konishi, and K. Toriyama, Appl. Magn. Reson., 7 (1994) 149. [20] K. Maeda, Y. Araki, Y. Kamata, K. Enjo, H. Murai, and T. Azumi, Chem. Phys. Lett., 262 (1996) 110. [21] Y. Sakaguchi, A. V. Astashkin, and B. M. Tadjikov, Chem. Phys. Lett., 280 (1997) 481. [22] K. M. Salikhov, Y. Sakaguchi, and H. Hayashi, Chem. Phys., 220 (1997) 355. [23] J. R. Woodward and Y. Sakaguchi, J. Phys. Chem. A, 105 (2001) 4010. [24] Y. Sakaguchi, Mol. Phys., 100 (2002) 1129.

233

15. Magnetic Field Effects upon Biochemical Reactions and Biological Processes 15.1 Historical Introduction It has long been a matter of dispute in biology whether magnetic fields can give appreciable influence to biological reactions or organisms. Although there has been no report of the immediate death of living things, there have been many reports of bioIogical effects of magnetic fields [l]. Almost of such reports, however, were lacking for experimental reproducibility and theoretical background. As an example, MFEs on enzymatic reactions have long been examined. But there have been many negative reports of MFEs on enzymatic reactions with a few exceptions as will be explained in Section 15.2. As shown in the preceding chapters, MFEs on chemical and biochemical reactions through radical pairs, biradicals, triradicals, and higher spin intermediates have been established in high magnetic fields (B > 10 mT), and they have been explained successfully in terms of the radical pair mechanism and related ones. Here, high magnetic fields mean that they are higher than typical HFCs (1-10 mT). As an example, MFEs on the photosynthetic reaction center (PIX) have been established in the framework of the radical pair mechanism [11. Here, P is the primary electron donor (chromophore), I is an intermediate electron acceptor, and X is the first stable acceptor. The reaction scheme of PIX is shown as follows:

'[P+'IX-']. (15-1) P I X + h u - + 'P*IX -+ '[P+'T'X] The absorption of a photon by P produces the lowest excited singlet state of P ('P*) that irreversibly donates an electron to I and leads to formation of a singlet radical pair ('[P+'r 'XI ) in about 2.8 ps. In the normal biological reaction, the '[P+'T*X] radical pair transfers an electron to the quinone acceptor (X) to produce '[P+'IX-'] in about 200 ps. The lower free energy of l[P+*IX-*] makes this an irreversible process. Because the above-mentioned rates are much faster the rate of the S-T conversion of the radical pairs, no appreciable MFE can be observed in the normal reaction. Instead, if X is chemically reduced (X') to prevent it from accepting an electron, the reaction of such a system occurs as follows: PIX-' + hu -+ 'p*IX-' -+ '[P+*I-*]X-*H 3[P+*r']X-*+ 3p*m-' . (15-2) ---f

In this case, the lifetime of '[P+'I-'] X-*is extended and can transfer to the triplet radical pair (3[P"r'] X-*)that can recombine to yield the triplet state of P (3P*). The yield of 3P* was found to decrease by about 50 % at 50 mT. Such a MFE can be explained by the HFCM of the radical pair mechanism. Some interesting effects of high magnetic fields due to other mechanisms have also been investigated [2]. The perhaps most striking and useful effect at the supermolecular level is the alignment of biopolymers, proteins, viruses, large assemblies such as retinal rods and membranes when suspended in solvent. Usually, very high fields (1T < B < IOT)) are necessary for this effect, which is due to the anisotropy of the diamagnetic susceptibility of the constituent groups and bonds. Ferrofluids respond to much lower fields because they consist of suspensions of usually monodomain ferromagnetic particles several tens of nanometers, which are surrounded by a repelling polymeric coating preventing agglomeration. There have been many investigations concerning the influence of the Earth's magnetic field (-50 pT) on the orientation behavior of living organisms. The magnetotactic behavior of some bacteria and algae is the well-explained orientation process based on the Earth's field. It is based on chains of ferromagnetic particles inside the organism, where their alignment in the Earth's field against thermal agitation forces the organism to swim along the field lines. Similar particles are known to exist in various higher animals, such as insects, fish, birds, and mammals, but the definite role of these particles is still unclear. Although the fact that these

234

animals can feel the Earth's field and use this information as a magnetic compass has been widely believed, its mechanism has not yet been established.

In 1979, Wertheimer and Leeper [3] reported an increase in childhood cancer for individuals living near electric power lines. Whether environmental electromagnetic fields generated by electric power lines (50-60Hz), portable telephone (about 1-2 GHz), etc. are sources of illness in human beings, however, remains a matter of dispute. In the framework of the radical pair mechanism, it is far less clear how a magnetic field comparable in strength to the Earth's field, and weaker than typical HFCs, can influence chemical and biochemical reactions through radical pairs. In Section 15.3, a typical experimental result on such a low magnetic field effect and some theoretical calculations of the low field effect will be introduced. 15.2 Magnetic Field Effects on Enzyme Reactions In 1994, Harkins and Grisson [4] reported MFEs on the reactions of the coenzyme BL2dependent enzyme ethanolamine ammonia lyase. Although there had been many attempts to find MFEs on enzyme reactions before 1994 [I], no significant effect had been observed except for a few noteworthy exceptions. In 1967, Harberditzl [5] reported the effects of uniform (variation of 3 % over the sample length of 4 cm) and non-uniform (variation of 30 % over 4 cm) magnetic fields on the catalytic enzyme activity. His experiments were carried out with a watercooled solenoid of 5 cm inner diameter, the maximum field of which was 8 T. He studied the oxidation of 2-oxoglutarate by glutamate dehydrogenase (GDH) and the degradation of hydrogen peroxide by catalase in probably water solution at 25°C. In the reaction of GDH, the reduction effects of uniform magnetic fields of 5.0-7.0 T (from -5% to 12%) and non-uniform fields of 6-7.8T (from -12% to -93%) were observed. In the reaction of catalase, the acceleration effects of a uniform field of 6.0 T (from +5% to +9%) and a nonuniform field of 6.0 T (from +16% to +52%) were observed. He proposed the diamagnetic alignment mechanism for the effects of uniform fields and the magnetic force mechanism for the effects of non-uniform fields, respectively, but there has been no direct evidence to prove his proposal. Although Haberdizl and Muller [6] reported no MFE on the reaction of catalase with fields of up to 2.8 T in 1965, Molin's group [7] studied MFEs on the hydrogen peroxide decomposition rate induced by catalase and a complex of EDTA with ferric ion in 1978. Molin et al. reported that the rate acceleration had increased with increasing B up to 0.8 T, attaining a 20% increase for catalase and 24% for [Fe3'(EDTA)]2. On the other hand, Vanag and Kuznetsov [8] reviewed the current literature to 1986 and reported no known h4FE on enzymatic reactions. In 1989, Hummel et al. [9] reported no MFE on the decomposition of hydrogen peroxide by catalase at 20°C and the oxidation of L-ascorbic acid by oxygen in the presence of ascorbate oxidase at 25°C in buffered water solutions under a uniform field of 1.05 T. There is no direct evidence for participation of radical pairs in the reaction of catalase, but a radical pair is known to be formed through thermal or photochemical homolysis of alkylcob(III)alamins (RCblm) from ps-laser photolysis experiments. The structure of adenosyl-cob(III)alamin (AdoCblm) and metylcob(III)alamin (MetCbl'") and the radical pair produced from RCbl" are shown in Fig. 15-1. Vitamin B12 is a cofactor for many enzymatic reactions in its various forms. The common structural element is the macrocyclic corrin ring that holds Co3+in a square-planar coordination geometry as shown in Fig. 15-l(a) [I]: The form that is a cofactor for methyl transferase reactions is MetCbl'", the form that is a cofactor for about a dozen enzymes that catalyse 1,2-migrations is AdoCbl"', and the form found in

235 nutrition supplements is cyanocob(m)alamin. Thus, Grison's group [lo] tried, at first, to find MFEs on photochemical reactions of RCbl"'. R

AdoCbl" R=

MetCbl'" R= CHaOH

S

hv

(Diffusion)

Escape

l -

I

[b] 1 Escape

Fig. 15-1. (a) The structure of adenosyl-cob(III)alamin (AdoCblm) and metylcob(III)alamin (MetCblm). (Reproduced from Ref. [lo] by permission from the American Chemical Society) (b) The radical pair produced from photoexcitation of RCblm. (Reproduced from Ref. [ 11by permission from The American Chemical Society) As an example, the MFEs observed for the anaerobic photolysis of AdoCbI" in various solvents by Grison's group [ 101 will be introduced below. For continuous-wave photolysis, they irradiated AdoCblm in 75%(v/v) glycerol in water (q/qo = 30), 20%(w/v) Ficoll-400 (q/qo = 30), and water (q/qo = 1) at 20°C with an Art laser (514nm) and measured the quantum yield (@) of the decomposition of AdoCbl" as a function of B by monitoring the absorbance of AdoCbl? Fig. 15-2 shows the observed MFEs on @. From this figure, they found that the @ values in glycerol and Ficoll-400 decreased by nearly 2-fold as B was

236 increased from 0.05 mT to 50 mT, but that the @ value in water was nearly invariant with magnetic fields below 0.2 T. The observed MFEs can be explained by the HFCM, where the S-T conversion is reduced by magnetic fields below 0.2 T. In viscous solvents, separation of the radical pair is restricted by a slower rate of diffusion than in water. This is the reason why MFEs were observed in glycerol but not in water. The MFEs observed in Ficoll-400 are not due to restricted diffusion because molecular diffusion is decreased only 2.7-fold in 20% Ficoll-400 relative to the rate in water, but due to less escape from the cage structure formed by Ficoll-400 because its microviscosity is much larger than water.

4.4

e ~-

4.4

0

0.05

0.1

0.15

0.2

0.25

0

0.05

0.1

Q.15

0.2

0.25

Fig. 15-2. Observed MFEs on the quantum yield (40) of the decomposition of AdoCbl" for its anaerobic photolysis at 20°C with an Ar' laser (514nm): (A) in 75%(v/v) glycerol in water (q/qo = 30), (B) in 20%(w/v) Ficoll400 (q/qo= 30), and (C) in water (q/qo = 1). (Reproduced from Ref. [lo] by permission from The American Chemical Society)

i MIONETIC WELD FLUX (TESIA)

Chagovetz and Grissom [ 101 also made laser-photolysis measurements of AdoCbl"' in 75%(v/v) glycerol in water (v&, = 30) and water (q/qo = 1) at 20°C with a ps-laser. Photolysis at 532 nm was accomplished with a frequency-doubled Nd:Yag mode-locked laser with a pulse width of 30 ps. Cob(n)alamin formation occurred within the 30-ps photolyzing

237

pulse and decayed with pseudo-first-order kinetics. Fig. 15-3 shows the observed MFEs on the first-order rate constant (krec)of the geminate recombination. In contrast to steady-state photolysis experiments, the k,,, value in both water and glycerol solutions increased with increasing B from 0.05 mT to about 0.1 T, but the value decreased with increasing B from about 0.1T to about 0.18 T. The magnetically induced increase and decrease in the k,,, value can be explained by the HFCM and the AgM, respectively. Because the k,,, value at 0.05 mT was insensitive to solution microviscosity, the k,, value was confirmed to be the rate constant for true geminate radical pair recombination. It is noteworthy that the observed MFEs on the @ value is not due to the geminate recombination but due to secondary recombination. 1

5)

0

0.04

0.08

0 ' 0

I

I

0.04

0.08

0.12

I

0.12

0.18

Fig. 15-3. Observed MFEs on the k, value following photolysis of AdoCbl"' in (A) water (q/qo = 1) and (B) 75%(v/v) glycerol in water (q/qo= 30) at 20°C with a ps-laser (532 nm). (Reproduced from Ref. [ 101 by permission from The American Chemical Society)

I

0.18

MAGNmC FIELD FLUX (TESU)

In a unimolecular enzymatic reaction, it converts a substrate (S) to a product (P). Product formation can only occur from the enzyme-substrate (ES) complex, where the reaction mechanism is expressed by E+S-+ES kl , (15-3a) ES+E+S k2, (15-3b) ES+E+P k3. (15-3~) The first step is binding of E and S to form ES. Since P can only be formed from ES, d[P]/dt= k3 [ E q . Under initial velocity conditions, [P]=O and the reverse reaction from P to S does not occur. In the conditions of a typical in vitro assay, [E]<<[S], and the steady-state assumption can be employed to describe [ES]: d[ESl/dt=O. With these assumptions, the

238 kinetic rate expression that describe the rate of product formation can be expressed by the Michaelis-Meten equation [ 1, 111 as follows: (154a) Here,

(15-4~)

Problem 15-1. Prove the Michaelis-Meten equation expressed by Eqs. (15-4a)-( 15-4c). If ES involves a radical pair, the recombination rate of ES (kz) is possible to be influenced by an external magnetic field. On the other hand, kl and k3 should be independent of the field. Harkins and Grisssom [4] studied MFEs on the conversion of unlabeled and deuterated ethanolamine to acetaldehyde and ammonia in bacteria by ethanolamine ammonia lyase. In this reaction, AdoCblm acts as a coenzyme and a radical pair is easily generated through the enzyme-induced homolysis of the C-Co bond. The escape 5'-deoxyadenosyl radical from the pair initiates the conversion reaction. They measured MFEs on the Vm, and V,,lK, values at 25°C and obtained the results as shown in Fig. 15-4. The V,, value was independent of B up to 0.25 T. This is reasonable because k3 should be independent of B. On the other hand, the Vm,IKm values of the unlabeled and deuterated systems exhibited decreases of 25 % (at 0.1 T) and 60 % (at 0.15 T), respectively. These magnetically induced deceases can be explained by the HFCM, where kz should be increased by such low fields as 0.1-0.15 T. At higer fields, the V,,,IK, values were found to increase from their minimum values. These magnetically induced increases can be explained by the AgM, where kz should be decreased by higer fields than 0.1-0.15 T. The larger MFEs in the deuterated system than the unlabeled one can be explained by the fact that the k3 value of the deuterated one may be smaller than that of the unlabeled one.

'

3 2.50 2.00

(

:p-;&b +,?*I-

+pf-+

9 1.o

0.00

C

0.05

0.10

0.15

0.20

Magnetic leld flux B (T)

0.25

Fig. 15-4. Observed MFEs on the V,,, and V,,lK,,, values for the conversion of unlabeled and deuterated ethanolamine to acetaldehyde and ammonia in bacteria by ethanolamine ammonia lyase at 25°C. (A) V,, with unlabeled ethanolamine, (B) V,,,IK, with unlabeled ethanolamine, and (C) Vm,,IKm with deuterated ethanolamine. (Reproduced from Ref. [4] by permission from Science)

239 15.3 Effects of Low Magnetic Fields on Chemical Reactions through Radical Pairs Although effects of low magnetic fields have been observed for a few reactions in solution, their effects are usually much smaller than those observed in higher magnetic fields ( B > 10 mT). As an example, Batchelor et al. [ 121 investigated MFEs on the electron transfer reaction from the the lowest excited singlet state of pyrene ('Py*) to 1,3-dicyanobenzene (DCB) in solution. (15-5a) Py + hu -+ I@*, 'Py* + DCB '[Py'"DCB-']

-+ cf

'[@''DCB-'],

(15-5b)

'[Py"DCB-'],

( 15-5~)

'[Py" DCB-']-+ Py + DCB,

(15-5d)

Y(B)- Y(0mT) /

Fig. 15-5. A schematic representation of the experimentally observed yield of escape radical (Reproduced from Ref. [13] by (Y(B)-Y(0mT)) in the pyrene/l,3-DCB system [12]. permission from Taylor & Francis Limited) They found that the yield of escape radicals was changed by magnetic fields as schematically shown in Fig. 15-5 [13]. This figure shows that the yield (Y(B))increases with increasing B from 0 to about 0.3 mT, but that it decreases with increasing B from 0.3 to 5 mT. At 10 mT, the yield attains a saturated value. The magnetically induced decrease in the escape radical yield observed under magnetic fields from 1 to 10 mT can be explained by the HFCM. Although the magnetically induced increase (about a few %) in the yield observed under low fields below 0.8 mT is much smaller than the MFE at 10 mT, the increase can not be explained directly by the ordinary HFCM. If it is assumed that any increase in the radical concentration in the human body is harmful, then since biological reactions occur from Sprecursors, we can say that the effect of ordinary magnetic fields ( 10 mT < B < 1 T) on radical reactions of the type expected in biology would be protective rather than harmful. At higher fields than 1 T, the magnetically induced increase in the escape radical yield can appear through the AgM. At very low fields, including those of the order of the Earth's field, it is also possible that more radicals survive during reactions through radical pairs than at

240

either higher or zero field. If the thesis that a small increase in radical concentration may be significant in biology is correct, the radical pair mechanism provides a possible one for very low magnetic fields to affect biological processes. Although the low field effects on chemical reactions through radical pairs had been explained by the LCM, T h e 1 et al. [14] proposed that the so-called low field effects arose also fiom coherent superpositions of degenerate electron-nuclear spin states in a radical pair in zero field. They made some model calculations for their mechanism At first, let us consider the case of a radical pair with a single spin-112 nucleus, e.g., a proton. When the exchange can be expressed fiom Eq. (3-3) as term is not included ( J = 0 J), its spin Hamiltonian (a) H = gp,S,,B +ASPI+ gpBS2ZB, Ad,,

=

(15-5a)

+ AaSpI + fid,,.

( 15-5b)

Here, the g-values of radicals a and b are taken to be the same value (g), and S1, S2,and I are the spin angular momentum operators of the two electrons and the nucleus in radical a. The eigenfunctions (In>) and eigenvalues (En) of this Hamiltonim in zero field (B=OT) can be expressed by (15-6a)

(15-6d)

I5, = 1

16)

P NPI)I22

)

E5 =Aa/4,

(15-6e)

= "I4,

(15-60

E, =Aa/4,

(15-6g)

E,

(15-6h)

1 =z

I 7, = 1 )1' =

[ l a N f l l ) +

aNal)I

IPNal)lla2),

P2)

laNal)(a2),

E6

= AaI4.

Here, a~ and PN denote the m ~ W states 2 of the nucleus and a,and states of the i-th electron (i= a or b). Problem 15.2. Prove the results shown by Eqs. (15-6a) - (15-6h).

p, denote the ms=fl12

The energy diagrams of this radical pair in zero field and in the presence of a weak (w << lal) magnetic field are illustrated in Fig. 15-6. In the presence of an external magnetic field, the e i g e h c t i o n s (In'>) and eigenvalues (Ens) are changed fiom those in zero field Il')=cos
El,=-Aal4-Aw/2-AR/2, 2,

(15-7a)

+ Awl2 - h R l 2 ,

(15-7b)

12') = cos
E3, =fia/4-Aw,

14') = -sin <[I) + cos
(15-7~) (15-7d)

24 1

15‘)= 15),

E,. = A u / 4 ,

(15-7e)

] 6) = -sin 612) + cos 616),

E,. = -ha 14 + Awl 2 + AQI 2 ,

(15-70

17‘)= 17),

E,, = f i u / 4 ,

(15-78)

18) =Is),

E,. =AaI4+Aw.

(15-7h)

Here, 0and tan26 are given as follows:

a=JZ7,

(ma)

tan2<=0/a.

(15-8b)

Problem 15.3. Prove the results shown by Eqs. (15-7a) - (15-7h).

8

6

.., . . .

.. -. .. ... . . . . ... . . . A

3-8 -)::..

.

5,7

Fig. 15-6. The energy diagrams of a one-proton radical pair in zero field (left) and in the presence of a weak (w<< IaI) magnetic field (right). The energy separations are illustrated in unit of h. . (Reproduced from Ref. [141 by permission from Taylor & Francis Limited)

3 a

T

2

In Chapter 11, the equation of motion for p is given by (15-9)

Thus, we may obtain a formal solution of Eq. (15-6) as P(t) = e-(i/n)Hfp(0)e(i/”)H’~

Problem 15.4. Prove Q. (15-10) .

( 15-10)

242

By utilizing the fact that Hln>=EJn>, and using the power series expansion of the exponential operator, we get

-E&l(kIP(0)1m)

(kIPWlm) = exp[fQ.

(15-1la)

>

Ido)m) I.

= exP[i(wm - w k )tl(k lp(o>l m, = exp[iwdtl(

( 1 5- 1 1b)

The fraction @s(t))of radical pairs in the singlet state at time t is given by Eq. (1 1-24)

Pdtl =TrrP(Om = C(kk(r)lm)(m(PSIk).

( 15-12)

h

Here, Ps is the singlet projection operator. In this model system, we get ps =

1

1 + I 'PN

1

NP'()

(15-13)

'

Using Eqs. (15-6a) - (15-6h), the (mlPs(k) values in the eigenbase of the zero field Hamiltonian can be obtained as follows: Il> 12> /3> 14> 15> IF> 17r IS>

0 114 0 0 0 114 -11245 0

0 -114 0 0 0 0 0 114 0 -11247 0 0 0 0 0 0

1/24? 0 0 -11245 112 0 0 0

0

0

114 -11247 0 0 0 0 0 0 114 -1/2& - .I1245 112 0 0

0

0 0 0 0 0 0 0

(15-14)

Problem 15-5. Prove (15-14). Because we are considering reactions from a S-precursor, we can find the following result fromEqs. (15-llb) and(15-12): (15-15) Because ps(t) is a real quantity, we can choose the initial condition as shown by Eq. (15-15). In order to explore limiting cases to discover how large the low field effect could be under the right conditions, Timmel et al. [ 151 adopted an approximation that the singlet and tiplet radical pairs disappear with first-order kinetics with a rate constant of k In this is given by approximation, the singlet product yield

(as)

Qs = k[p,(t)exp(-kf)dt

.

(15-16)

From Eqs. (15-14) and (15-15), the singlet yield becomes (15-17) where kZ f(x) =k2+x2

(15-1 8)

243

is the result of integrating kcos(xt)exp(-kt) over the range 0 5 t 5 00 diagonalization of the radical pair Hamiltonian gives the exact result: 1 + -[1 --If(-+w 8 a

a 2

w a 1 + -1 [1+ -1w f (-a - + -) + -[1+ -Ifw

(-

3 lo2 c& = -+ --+

laZ -,f(Q)

8R

8 8Q2 8

Q

2 2 2 8 Problem 15-6. Prove (15-19). 1.0

i

2

0.8

0.5 /

0.3

0.6

I1 I

'0.005

0.2

k la

0.4

8

a

w n + -- -) . 2 2 2

0.4

0.0

1 + -[1 --If(--o

a

k /a = 2.0

1

n 2

-)

a 2

w

n

2

2

this model,

-- -)

(15-19)

I

I

I

w -+ 2

. In

Fig. 15-7. (A) Magnetic field dependence of the of a singlet recombination yield one-proton radical pair for various rate constant (Wa), calculated using Eq. (15-19). (B) Properties of the low field minimum in the singlet recombination yield for a one-proton radical pair. The right and left hand axes plot the field position (%inla) at which the minimum occurs and the depth of the minimum (QS(m0) respectively, as a function of the recombination rate constant (Wu). . (Reproduced from Ref. [14] by permission from Taylor & Francis Limited)

(a)

a(%")),

0.6

Fig. 15-7(A) shows the MFEs on the singlet recombination yield (a)of a one-proton radical pair for various rate constant (Wa),calculated using Eq. (15-19). Fig. 15-7(B) shows the field position (&i,/a) at which the minimum occurs and the depth of the minimum (@(mO) - a(&")) as a function of the recombination rate constant (Wa). Here, a is assumed to be positive. From these figures, we can see the following striking features: (1) The abrupt drop in is produced by even a tiny magnetic field when the recombination is

a

244

slow. (2) In the limit of k << a, the zero-filed singlet yield of 0.625 falls to 0.375 when a weak field is applied. This means that the magnetically induced decrease of 40 % in c& is due to the low field effect and that the yield of escape radicals at zero field (0.375) rises to 0.625. (3) The gradual increase in @ as the field becomes larger than the HFC ( m a ) is the normal MFE due to the HFCM. In the limit m>a>>k, 0 s tends to 0.5. (4)As k is increased the depth of the minimum in c& decreases and eventually disappears when k > 0.71~1. The position of the minimum in @ increases with increasing Wa from 0 to 0.27, but it decreases with increasing Wa from 0.27 to 0.7 ( 5 ) For values of k larger than a, @ rises towards unity because recombination occurs before significant hyperfine-driven S-T conversion can take place. Although the one-proton case (case a) nicely demonstrates the existence and properties of the low field effect, it is important to realize that this model is atypical. Timmel et al. [ 151 made some other model calculations for the low field effect of 0 s . For radicals each with a single nuclear spin (case b), it is possible to obtain an explicit form of Eq. (15-14), but the general expression for @ is too lengthy to describe here. The calculated results in case b for were very similar to those obtained in case a. When the recombination is the MFEs on slow, the low field effect on C& was found to attains to 20-35 % for almost all of the radical The above results may be extended readily to radicals bearing several pairs in case b. equivalent nuclear spins (case c). The calculated low field effects in case c were similar to those in case b. When one or both radicals has more than one HFCs (case d), it is usually impossible to obtain an analytical expression for the low field effect because of the difficulty of diagonalizing the spin Hamiltonian, especially, in the presence of an external magnetic field. Timmel et al. [8] made some approximate calculations for case d and found a general picture that three seem to be few cases in which the low field effect is smaller than 20 % in the limit k < < w < a and that small effects occur only in special pairs. From the above-mentioned results, Timmel et al. [14] obtained the following features for the low field effect on chemical reactions through radical pairs in solution: (1) Even a feeble applied magnetic field can in principle increase the yield of escape radicals and suppress recombination by 10-40 % when the recombination is slow. (2) The magnitude of the low field effect does not depend strongly on the number of magnetic nuclei, their spin quantum numbers or HF interactions, or on whether the nuclei are equivalent or inequivalent. (3) As the recombination is increased, the magnitude of the low field effect abruptly drops. (4) In principle, therefore, the low field effect seems to be a general effect, potentially observable for almost any radical pair, provided its lifetime is long. (5) This is an important result in the light of current interest in the possibly harmful effects of weak magnetic fields in biology and medicine. (6) At last, some remarks should be added to the low field effect: (a) The reactions through radical pairs in solution occur on a timescale of ns - ps. This means that radical pairs see any field oscillating at a frequency of less than 1 MHz as static. Thus, electric wires’ frequencies of 50-60 Hz are static on this timescale. Recently, Woodward et al. [15] reported new experiments in which weak radiofrequency fields (B1i 300pT) in the frequency range of 1-80 MHz gave appreciabIe effects on reaction (15-5). (b) In biological systems, radical pairs from F-precursors are more likely to arise. The low field effect in Fpairs is unlikely to be of significance in biology because the probability of the two radicals remaining in proximity is very small. Also if the low field effect could occur to a nonnegligible extent, it would now lead to a decrease in the radical concentration in solution.

245

References [l] C. B. Grissom, Chem. Rev., 95 (1995) 3. [2] G. Maret, N. Boccara, and J. Kiepenheuer, Eds., “Biophysical Effects of Steady Magnetic Fields”, Springer-Verlag, Berlin, 1986. [3] N. Werthheimer and E. Leeper, Am. J. Epidemiol, 109 (1979) 273. [4] T. T. Harkins and C. B. Grissorn, Science, 263 (1994) 958. [5] W. Haberditzl, Nature, 213 (1967) 5071. [6] W. Haberdizl and K. Muller, Z. Naturforsch., 20b (1965) 517. [7] L. M. Vainer, A. V. Podoplelov, T. V. Leshina, R. Z. Sagdeev, and Yu. N. Molin, Biofizika, 23 (1978) 234. [8] V. K. Vanag and A. N. Kuznetsov, Izv. Akad. Nauk SSSR, Ser. Biol., (1988) 215. [9] K. Hummel. K. Martl, and M. G. Martl, J. Mol. Catl., 54 (1989) L1. [lo] A. M. Chagovetz and C. B. Grissom, J. Am. Chem. SOC.,115 (1993) 12152. [ l l ] J. I. Steinfeld, J. S. Francisco, and W. L. Hase, “Chemical Kinetics and Dynamics, 2”d ed.” ,Prentice Hall, Upper Saddle River, New Jersey, 1999,pp159-163. [I21 S. N. Batchelor, C. W. Kay, K. A. McLauchlan, and I. A. Shkrob, J. Phys. Chem., 97 (1993) 13250. [13] B. Brocklehurst and K. A. McLauchlan, Int. J. Radat. Biol., 69 (1996) 3. [14] C. R. Timrnel, U. Till, B. Brocklehurst, K. A. McLauchlan, and P. J. Hore, Mol. Phys., 95 (1998) 71. [15] J. R. Woodward, C. R. Timmel, P. J. Hore, and K. A. McLauchlan, Mol. Phys., 100 (2002) 1181.

Solutions to the Problems 15-1. The sum of [El and [ES] and that of [ S ] and [PI are fixed [El, = [El + [ESI, [Slt = [Sl + [PI. Thus, the rate equations for [ES], [S], and [PI are given by d [ES] / dt = -(k, + k, )[ES] + k, [El [SJ , d [ S ] / d t =-k,[E][S]

+ k,[ES],

d[P]/dt = - d [ S ] / d t .

From the steady-state condition of d[ES]/dt = 0 , [ES]becomes

Thus, d [ P ] l d t can be obtained as follows:

15-2. We can see that the total electron plus nuclear spin angular momentum (J = J a + Jb) and its projection (Jz) onto an arbitrary axis are conserved at zero field. Here, Ja = S1 + I and

246

= Sz. Thus, the eigenfunctions (In>) can be expressed by Eqs. (15-6a) - (15-6h). The diagonal elements can be calculated easily. For example, El and E4 are obtained as follows:

Jb

1-1 -111- 1 A -3a = -(-), 2 2 2 2 2 2 2 2 2 Here, the following relation was used: A

= -(a--+a--+a-+a-)

A a -(-).

2 2

s-I= 3i = s,zz + (s+z-+ s-I, I 2 . All of the off-diagonal elements should be vanished. For example, the following element can be proved to be zero:

J z ~ ( ~ N ~ l l - ( ~ N ~ l ~ ~ ( ~ ~ I ~ ~ ~ ~ l f ~1/ l ~ l ~ N ~ l ) + I ~ N ~ l ) ~ I ~ Z )

<11H 14>= 1

A

=-((%PI 2 laSIqaNP1)- ( P N . . ,

J ~ S l i J P N a ;-) ( P N S JaS,flcr,P,))

I ~ V I P N a ; ) + ( ~ N P I

= 0.

15.3 Among the off-diagonal elements for the Zeeman terms (Hz) in the eigenbase of the zero field Hamiltonian, the following ones are only non-vanishing: (11Hz14)= (41HzIl)= (21Hz16)= (61Hz12)=-Am/2. Thus, the results shown by Eqs. (15-7a) - (15-7h) can be obtained from the two-levels’ problem, which often appears in quantum mechanics. Here, the following relations are useful. (l’IH14‘)= -sin [cos [(l IHI 1) + sin [cos [(4 [HI4) + (cos’ [- sin [)(1IHI 4) = ha sin [cos [ + (I - 2sinz [)(-Am/ 2) = (ha I2)sin 2[

- (Am/

2)cos 2 5 = 0,

2sin
15-4. If Eq. (15-10) is correct, the following relation can be obtained:

Thus, the left side of Eq. (15-9) becomes

47)

-

= Hp(t) p ( t ) H = [H, p].

15-5. As an example, (51Ps11) is obtained as follows:

247

-1 1 = (-)(-)(-)

-1

JZ&&

= -. 1 2&

The other elements can also be obtained. 15-6. Using Eqs. (15-7) and (15-14), the contribution from the term with m=l’ and n=1’ to the right side of Eq. (15-17) becomes

(E$(’f(w,.,.) = (cos2<(1/4) + 2cosCsin 6(-1/4) + sin? [(1/4)( ’ f ( 0 ) = (I / 4) ’(1- W / Q(?(1) = (1/ I 6)(1- w / Q I 2 .

Here, we used 2cos 6 sin 6 = sin 2 5 = w / Q . The contribution from the term with m= 1’ and n=4’ to the right side of Eq. (15-17) becomes

JP,;,)’ f(w1.4.) =

1-

coscsin 6(1/4) + (cos’

= (1/ 16))l- 2 sin

5 - sin‘ 6)(-1/4) + sin 6 cos 4-(1/4)1*S(Q)

6)’f (Q) = (1/ 16)11- (1 - cos 26)l’ f (Q) = (1/ 16)(a/ 0)’f (Q)

Here, we used 2sinz 6 = 1-cos25 = 1- u / Q . The contribution from the term with m=l’ and n=5’ to the right side of Eq. (15-17) becomes

~ < < , l f~ ( w I m 5=, ) I c o s 5 ( 1 / 2 ~ i+) sin {(-I/

2&)I2f(a / 2 + w / 2 + SL/ 2)

= ( 1 / ~ ) J c o s-sin c c l ’ f ( a / 2 + w / 2 + Q /2) = (l/S)ll- 2cos 6 sin ( l f ( a / 2 + w / 2 + s2/2) = (1/8)[1- w / Q ] f ( a/ 2 + o / 2 + Q / 2) .

Similarly, the contributions from the other terms can be obtained. gives the right side of Eq. (15-19).

The sum of all the terms

This page intentionally left blank

249

Index A a (see one-step enrichment coefficient) a-spin electron spin 12

decafluoro- (DFBP) 184,231 4-methoxy- 180 benzyl chlorides 83 biradical 117, 191

nuclear spin 14 absorption photon 9 SI--tS, 10 Ti-+T, 10 acetyl peroxide 42 ADESR (see electron spin resonance)

Bitter type magnet 177, 193 Bohr magneton bB) 4 BP (see benzophenone) Brij 35 133,184,231 n-butyl bromide 38 n-butyllithium 38, 83

A/E

'3C-enrichment 94, 127 cage product 21,89 cancer 234 catalase 234 chemical shift 14 chemically induced dynamic electron polarization (CIDEP) 21 due to d-type TM 142,149 due to RTPM 204 due to spin-correlated radical pairs 61 due to S-TOmixing 48,60 due to S-T,l mixing 51,61 due to T-D quenching 201 due to triplet mechanism (TM) 54,60 electron irradiation 35 photo-CIDEP 47 photolysis 60 photooxidation 59 photoreduction 56,57 theoretical analysis 169 chemically induced dynamic nuclear polarization (CIDNP) 21 in biradical reaction 1 17 multiplet CIDNP signal 43 photochemical reaction 77 theoretical analysis 169

CIDEP 5 1 signal 41,43 alicyclic ketones 117 alkylcob(III)alamin (RCbl"') 234 anisotropic HFC term 102 anisotropic g-value 13 anisotropic Zeeman term 102, 183 anthracene 197, 199 anthraquinone (AQ) 225 anti-phase doublets 61,64 AQ (see anthraquinone) A- signal (see enhanced absorotive signal) A(t) curves (see time profiles of the transient absorbance) 2,2'-azobis[isobutyronitrile] (AIBN) 154 B P-spin electron spin 12 nuclear spin 14 B (see magnetic field) B I / value ~ 91,120 benzophenone (BP) 56,61,97, 184,225, 23 1 BP-dlo 97 [~arbonyI-'~C] (BP-I3C) 97

C

250 thermal decomposition 36,42,43 thermal reaction 38 CIDEP (see chemically induced dynamic electron polarization) CIDNP (see chemically induced dynamic nuclear polarization) close pair 21 D G(see isotope ratio) delayed fluorescence 76, 197, 199 density matrix 157 density operator (p) 159

AgM (see Ag mechanism) Ag mechanism (AgM) 77,78, 120, 177, 237 dibenzoylperoxide (DBP) 37, 88 dibenzyl ketone (DBK) 94 diethyl phosphite 60 dimethylglyoxime 35 dipole-dipole interaction 102, 187 duroquinone 151

E EIA CIDEP 51 signal 41 pattern 44,56, 60, 61 E*IA 56 phase 5 1 signal 61 electron spin resonance (ESR) 1 1 absorption-detected ESR (ADESR) 220 CIDNP-detected ESR 220 CW-ESR47,201 2-MHz ESR 47 optical detected ESR (ODESR) 217,219, 227,232 pulsed-ESR 47

spin-echo ESR 47 triplet state 217 endoperoxides 119 enhanced absorptive (A) signal 40,42 emissive (E) signal 40, 42 environmental electromagnetic field 234 enzyme reaction 2 18 enzyme-substrate complex 237 eosin 149 escape product 22,89 escape radical yield 97,99, 181, 186, 193, 239 E-signal (see enhanced emissive signal) ESR (see electron spin resonance) exchange integral (J)23,64, 118,202,204 exchange region 161 exciplex fluorescence 122 triplet 76, 142

F F-precursor 21,51, 164 fluorene 86 fluorescence 10 G Gd3+107 73Ge-enrichment 134 g-factor free electron (g,) 5 nuclear spins (gN) 6, 7 glutamate dehydrogenase (GDH) 234 g-tensor 109, 188 g-value anisotropic 13 isotropic 13, 188, 188 radical 43

H

reaction yield detected ESR (RYDESR)

HFC (see hyperfine coupling)

217,220

HFCM (see hyperfine coupling mechanism)

251 hyperfine coupling (HFC) 15 anisotropic term 102 constant 15,23,43 tensor 110, 188 term 23 hyperfine coupling mechanism (HFCM) 77,78,220,237

I

Lorentz force 1

M magnetic field ( B ) 1 low 239 ultra-high 177 magnetic field effect ( M E ) 21 due to AgM 79,81,89, 106, 120, 177 due to d-type TM 142

IC (see internal conversion)

due to HFCM 79,81,91

ISC (see intersystem crossing) internal conversion (IC) 9

due to LCM 79,81, 118, 121 due to RM 97, 101, 106, 184 due to RPM 7.5,77 due to Q-D conversion 208 due to Sx-Qa conversion 210 due to T-T annihilation 76, 198,200 in micellar solution 97 on biochemical reaction 233 on biological process 233

intersystem crossing (ISC) 9, 54,97 p-iodoaniline 147 isoquinoline N-oxide 93 isotope enrichment (S) 127 isotope ratio (6)127, 135

J J (see exchange integral) K Kaptain's rule 27 theory 25 K-band microwave 13 Kramers doublet species 193 Ku-band ODESR 232 L laser A f 235 Nd:YAG 227 ns- 47,77, 91,97, 120, 123, 142, 177, 179, 180, 184 PS- 195,237

LCM (see level-crossing mechanism) level-crossing field (&-) 24, 78, 119 level-crossing mechanism (LCM) 78, 118, 121,240 lifetime biradical

(ZBR)

123, 124, 191

radical pair ( ZRP) 97, 186

on biradical lifetime (ZBR) 123, 124, 191 on biradical 117 on '3C-enrichment 77 on chemiluminescence 75,76, 199 on delayed fluorescence 76, 197, 199 on enzyme reaction 218 on escape radical yield ( Y E )99, 181, 186, 193,239 on exciplex fluorescence intensity 122 on fluorescence intensity 85 on Ge-radicals 105 on heavy atom-centered radicals 103 on high spin species 197 on Kramers doublet species 193 on photochemical isomerization 77,93 on photochemical reaction 88, 139 on photoreduction reaction 97, 180, 184 on photosensitised decompostion 77, 88 on photosynthesis reaction center 179, 233 on P- radicals 105 on predissociation of 12 75

252

on product yield 80 on pulse radiolysis 76 on radiation reaction 85 on radical pair lifetime (zip)97, 186 on S- radicals 105 on singlet recombination yield 243 on Si- radicals 105 on Sn- radicals 105 on theoretical analysis 169 on thermal reaction 76, 83 on triplet yield 91 on triradical 208 magnetic isotope effect ( M E ) 21 due to RM 101 in micellar solution 97 94, 127 on 13C/12C on escape radical yield 99 on 73Ge/70Ge,72Ge,74Ge,76Ge 134 on ’5N1’4N131 on ‘70/’60, I 8 0 131 on radical pair lifetime 97 on 33S/32S,34S,36S133 on 29Si/28Si,30Si 132 on Sn-radical 136 on 235U/ 234U,238U136 magnetic moment nuclear 5 orbital 4 spin 5 methane 3.5 10-methylphenothiazine 205 methylviologen (MV”) 193 MFE (see magnetic field effects) micellar solution Brij 35 133, 184,231

HDTCL 129,131 SDS 97, 131, 184,221,227,231 micelle 95 Michaelis-Meten equation 238

M E (see magnetic isotope effects) MNQ (see 1,4-naphthoquinone) N l5N/I4N-ME 131 naphthalene 217 1,4-naphthoquinone 184 2-methyl- (MNQ) 221, 227,231 NMR (see nuclear magnetic resonance) nuclear magnetic resonance (NMR) 14

0 ODESR (see electron spin resonance) 170-enrichment 131

one-step enrichment coefficient (a)127, 130, 133 P phosphorescence 10 photochemical primary processes 9 photochemical reaction 10 CIDEP 56 CIDNP 77 magnetic field effect 88 photodecomposition reaction 10 photoinduced electron transfer reaction 10 photoreduction reaction 11,97, 180, 184, 22 1 photosynthesis reaction center (RSc) 179 polymethylene-linked compound 120 precursor free radical (F-) 21, 51, 164 singlet (S-) 21,26, 51, 164, 242 triplet (T-) 21, 26, 51, 164, 168 propionyl peroxide 44 pulse radiolysis 86 pulse microwave 224,227 pulsed magnet 177, 180 pyrene (Py) 91,239 pyrimidine 139

Q Q-band microwave 13

253

Q-D conversion (see quartet-doublet conversion) quartet-doublet (Q-D) conversion 202

R p (see density operator) radiation reaction CIDEP 35 magnetic field effect 85 radical pair mechanism (RPM) 35 radical escape 22 g-value 13 Ge- 105, 135 0- 132 P- 105 S- 105, 134 Si- 105, 132 Sn- 105,136 CIDNP 39 CIDEP 47 magnetic field effect 75 radical pair 21, 77,97, 157, 159,234, 239 ESR spectra 35 decay 189 ion- 85,91,219 lifetime 97, 186 ODESR studies 227 RYDESR studies 221 singlet 22,23,97,242 triplet 22,23,97,242 radical triplet pair mechanism (RTPM) 204 random perturbation 108 RSc (see photosynthesis reaction center) recombination cage 25,97 primary 21 probabilities 29 secondary 26 re-encounter 26

relaxation mechanism (RM) 97, 100, 184, 225

RM (see relaxation mechanism) RPM (see radical pair mechanism) rotational correlation time 102, 108, 191 RTPM (see radical triplet pair mechanism) Ru"-tris(bipyridine) complex 193 RYDESR (see electron spin resonance) S 33S-enrichment133 S-precursor 21, 26, 51, 164,242 scalar product 1 SCM (see spin-correlated mechanism) SDS (see sodium dodecyl sulfate) separated pair 22 sextet-quartet (Sx-Qa) conversion 210 29 Si-enrichment 132 singlet state ground (SO)9 first excited (S,) 9 second excited (S2) 9 singlet-triplet (S-T, S-To, or S-T,,) conversion 22, 25, 78, 97, 157, 159, 182 conversion rate 26 SLE ( see stochastic Liouville equation) sodium dodecyl sulfate (SDS) 97, 131, 184, 221,227,231 SO interaction (see spin-orbit interaction) solvent cage 21 spin electron 1 Hamiltonian 23, 108, 217, 240 nuclear 1 spin-correlated mechanism (SCM) 61 spin-lattice relaxation 108 spin-orbit (SO) interaction 22, 54, 103 spin-spin coupling 17 squalane 86 S-T, S-To, or S-T+Iconversion (see singlet-

254 triplet conversion) stochastic Liouville equation (SLE) 93, 146, 157, 161, 194 Stokes-Einstein relation 184 super-conducting magnets 77, 177, 179, 180,184 He-free 177 Sx-Qa conversion (see sextet-quartet conversion)

T Ti 108, 165, 183 T2 109, 165, 183 T-D quenching (see triplet-doublet quenching) T-precursor 21,26,51, 164, 168 2,2,6,6-tetramethyl-1-piperidinyloxyl (TEMPO) 20 1,205 theoretical analysis 158 CIDEP 169 CIDNP 169 MFE 169 thermal reaction CIDNP 36 magnetic field effect 83 thionine 142 time profiles of the transient absorbance (A ( t)curves) 97, 104, 107, 180, 186, 227 time-resolved ESR (TRESR) 11 TM (see triplet mechanism) transient probability 108 triphenylphosphine 148, 153 triplet-doublet (T-D) quenching 199,201 triplet exciplex 76, 142 triplet mechanism (TM) CIDEP 28, 141, 149 d-type 142

p-type 141 triplet state ESR 217 first excited (TI) 9 second excited (T2) 9 trip-sextet TI) state 208 triradical 208 triplet-triplet (T-T) annihilation 76, 198, 200

U 235U-enrichment136

V vector product 1 vibrational relaxation (VR) 10 vitamin B12234

x X-band microwave 12,221,227,231 Y yield cage product 80,83,89, 177 escape product 80,83,89, 178 escape radical 97,99, 106, 181, 186, 193, 239 singlet state 85 singlet oxygen 120 triplet pyrene 9 1 triplet state 179

Z Zeeman anisotropic term 102, 183 energy 12,21, 108 splitting 12,21,75, 187 term 23 zero-field splitting (ZFS) 54, 141, 202,217 zinc tetraphenylporphyrin 59

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Chemical Reactions Copyright © 2008 by Infobase Publishing All rights reserved. No part of this book may be reproduce...

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Chemical Reactions Copyright © 2008 by Infobase Publishing All rights reserved. No part of this book may be reproduce...

Chemical Reactions (Essential Chemistry)

Chemical Reactions Copyright © 2008 by Infobase Publishing All rights reserved. No part of this book may be reproduce...

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