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Chemically induced dynamic nuclear polarization
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
147
Glossary of Symbols Hyperfine coupling between electron 1 and nucleus a Population of nuclear spin state n in combination product and its equilibrium value Diffusively-separated radical pair New diffusively-separated pair formed by scavenging of D Relative diffusion constant for two radicals Diameter of encounter pair Encounter pair Probability of first re-encounter occurring at time t ;/-factors of radicals I and 2 Magnetic field of spectrometer and field in which reaction is carried out Components of nuclear spin angular momentum Intensity of NMR transition between state n and m in combination product and its value at equilibrium Electron exchange coupling constant Spin-spin coupling of nuclei A and B Rate constant for a reaction which competes with radical recombination Pseudo first-order rate constant for overall reaction Rate constant for radical scavenging Magnetic quantum number for set of magnetically equivalent nuclei a Probability that a radical pair will undergo at least one re-encounter Probability that two radicals will escape from their encounter without reacting Thermal electron polarization Thermal nuclear polarization Probability of finding radical pair in singlet state Population of nuclear spin state n in reagent which forms a radical pair z-component of electron spin Population of nuclear spin state n in scavenging product and its equilibrium value Relaxation time for population difference between spin states n and M in combination product Nuclear and electron relaxation times in a diffusively separated pair Enhancement factor for transition n-m Enhancement factor for an ESR hyperfine line with nuclear spins in state n Transition probability between states n and k Total number of nuclear spin states in combination product, C, and reagent, R Magnetogyric ratio Signature symbols for classifying high field and low field CIDNP and CIDEP respectively Nuclear spin dependent singletttriplet mixing coefficients Chemical shift differcncc between nuclei A and B Radical pair product type signature Spin function for two electrons in state 0 Probability that a radical pair in the singlet state will react during an encounter Radical pair precursor signature NMR frequency of nucleus A Spin density matrices Probability of initially finding radical pair in singlet or triplet states Signature symbol for relative location of nuclei A and B on two radicals Time between first and second radical pair encounters Lifetime for diffusive separation Lifetime for electron exchange Lifetime for radical scavenging Probability that reagent in nuclear spin state n forms combination or scavenging product in state rn Probability of finding combination product in state n if all states of reagent have the probability Z, ’ of being occupied Yield of combination product Nuclear spin function Larmor frequencies of electrons 1 and 2
CHEMICALLY
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1. Introduction
About 6 years ago it was discovered by accident UJ) that the NMR spectra of samples which are undergoing a chemical reaction may show extremely unusual and dramatic effects. These effects, which have been the subject of subsequent vigorous investigations on almost a worldwide scale, have come to be known as Chemically Induced Dynamic Nuclear Polarization (CIDNP).(3) The main experimental observations and the requirements so far found necessary for the production of CIDNP are the following: 1. Intense emission and enhanced absorption of lines may be observed in reacting samples with ordinary high resolution NMR equipment. 2. The effects always seem to occur in molecules which are the products of chemical reactions (although in some instances the chemical situation is such that these are the same as reactants).(4*5) 3. The enhancements can be extremely large, e.g. in excess of 1000 in favourable instances. This fact alone makes the phenomenon impossible to ignore, even if one is not interested in it per se. For example, a reaction product formed in only 0.1% yield may exhibit a transient spectrum as intense as that of the starting material! 4.’ From the evidence available to date, only the intensities of the lines appear to be significantly perturbed by the reaction. Changes in linewidths, i.e. T,, in the spin lattice relaxation time, T,, or line position shifts such as observed in the presence of chemical exchange seem to be negligible. 5. Because of competing destruction of the enhancement by thermal relaxation processes, the effects can be observed only during reactions which proceed at a rate not too much slower than the nuclear spin relaxation times. Experimentally, the effects are usually largest in reactions with half-lives less than 5 min, although occasionally they appear and persist for reactions which go to completion in more than 1 hr.@) 6. The only types of reaction so far found to produce CIDNP are those involving unpaired electrons, i.e. free radical intermediates. 7. The effects are largest when the reaction producing the products is of the initiation type in free radical chemistry where the product-forming step occurs by reaction of pairs of free radicals. A fairly typical example ofa CIDNP spectrum, obtained during the thermal decomposition of C,H,CO,O,CCH,CH,, is shown in Fig. 1. The purpose of the present article is to summarize the current status and scope of research on the CIDNP effect and present and discuss some representative examples of the phenomenon. The principal goal will be to present the kinds of effects on NMR spectra which are known to arise and to discuss their origins. It is not intended to give a comprehensive or critical review of all of the reactions which have so far been found to exhibit CIDNP or to emphasize the potential of the phenomenon as a tool in mechanistic organic chemistry. A survey of the CIDNP literature up to mid-1971 is availablec7) and the real and potential applications of the effect have received adequate attention recently.(*p9) The proceedings of an international colloquium entitled “CIDNP and Its Impact on Mechanistic Organic Chemistry” held in Brussels, 18-19 March 1971, have also been published.” ‘)
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R. G. LAWLER
Since most of the effects to date have been explained a posteriori rather than predicted, it is perhaps not surprising that several different versions of a recently-developed theory explaining these effects have arisen. They differ primarily in the way that their various proponents view the dynamic properties of reacting molecules in liquids. This is of course a very difficult problem which requires interpretation of experimental data from sources outside of magnetic resonance, to say nothing of outside of CIDNP. We will therefore make
FIG. 1. ‘H NMR spectrum decomposition
obtained at 60 MHz of C,H,COIOjCCHICH3
before (A), during (B) and after (C) thermal in o-dichlorobenzene at 120”.
no attempt to present all the sides of this still-developing story and will try to present instead what will be hopefully a coherent, reasonably accurate, but necessarily biased synthesis of the various versions of the theory. It should be pointed out that discussions of these separate points of view are also becoming available from their originators in several reviews prepared at approximately the same time as this one.18,9.11-14)Perhaps the most comprehensive and self-contained account so far available is the remarkable doctoral dissertation” 2, of Robert Kaptein of the University of Leiden. The present narrative owes much to the clarity and meticulous character of the work of the Leiden group.
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2. Radical-pair Theory
Perhaps the most significant barrier to the understanding of the origins of the CIDNP effect and its potential applications is the fact that the theory requires information nearly equally from the fields of organic free radical chemistry, the dynamics of rapid reactions in the liquid state, and magnetic resonance spectroscopy. Unfortunately, it also appears that there is little that one can learn from CIDNP spectra at present‘that does not require some familiarity with the theory. We will therefore devote an appreciable portion of this review to a presentation of background material from the above three disciplines and show how it is used to predict ultimately the observed effects, or, conversely, how proper analysis of CIDNP spectra may yield information of importance to these areas. Before beginning, however, we will list, in a manner analogous to that done for the experimental observations, the essential assumptions involved in the theory.” 5-35) 2.1. Qualitative Theory 1. It is believed at present that all CIDNP effects (and even related effects in electron-spin resonance) have their origins in interactions between pairs of radicals in solution. The present theory of CIDNP, first published by C10ss’~~’and independently by Kaptein and Oosterhoff,“h’ has in fact usually been designated the Radical-pair Theory to distinguish it from the first proposed explanations (36,37) of the effects which were based on the Overhauser Effect. Effects arising from the latter mechanism have so far never been observed unambiguously and are apparently much smaller than those due to radical-pair interactions. 2. It must be assumed that the ability of two free radicals to dimerize or form other stable combination products, such as olefins, derived from both radicals, depends on the combined spin multiplicity of the two unpaired electrons. In practice this means that only singlet-state pairs can produce stable combination products. 3. Somewhat related to (2) is the assumption that bond breaking in singlet or triplet radical-pair precursors proceeds with preservation of the correlation of the electron spins, thus making it possible to define a time at which the spins of a radical pair are correlated. The multiplicity dependence of combination reactions also introduces correlation even in pairs of independently generated radicals which undergo reactive collisions with each other. The idea of electron-spin correlation is essential to the theory because without it the terms “singlet” and “triplet” have no meaning. 4. Nuclear spins, and therefore NMR, enter the picture via the assumption that the internal magnetic fields due to the nuclei in a free radical pair with correlated spins can bring about a mixing of the singlet and triplet states, i.e. “intersystem crossing”, in the pair. 5. As a result of (4) the probability that a given pair of free radicals will form a combination product depends on the nuclear spin state of the pair. In effect this says that the chemical reactivity of a pair of free radicals depends on the nuclear spin state. As we shall see, this rather revolutionary assumption need have barely detectable chemical consequences in order to produce NMR enhancements of several thousandfold. 6. For virtually all reactions carried out in the magnetic fields of several thousand gauss usually employed for high resolution NMR singlet-triplet mixing may be induced by the nuclei without having any nuclear spins&. Thus the nuclear spin magnetic quantum numbers merely serve as unchanging labels for atoms which are carried by the chemical reaction from one kind of molecule to another. The observed intensity changes are then due
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R. G. LAWLER
simply to the fact that more molecules of product are formed for some nuclear spin states than for others. 7. Statement (6) breaks down for reactions examined at low magnetic fields(“~‘9*26’31’ and, apparently, for some rather special bases of intramolecular reactions proceeding via biradical intermediates.‘22s38,39)Very little experimental data are, however, yet available on these two aspects of CIDNP, although they represent especially promising directions for extending studies of the phenomenon. 2.2. Important Features of Radical-pair Reactions in’ Solution The first impression which might occur to one who begins to look at the abundant literature dealing with free radical chemistry(40) is that it is very complicated. In fact, two observations which sometimes make organic chemists suspect the intermediacy of free radicals in a reaction are an abundance of different products and poorly reproducible reaction kinetics. The former effect arises mostly because radicals as a class seem to be less selective in their reactions than are ionic intermediates, while the latter is a manifestation of the well-known sensitivity of radical reactions, especially those involving chain steps, to trace amounts of impurities. Fortunately, however, there are only a few general types of processes which take place during radical reactions in solutions. The four kinds required for production of CIDNP are discussed below by tracing the life history of a molecular fragment as it changes from reactant to free radical intermediate and ultimately to a molecule of a reaction product. 2.2.1. INITIALIZATION ANDCOMBINATION. In radical chemistry “initiation” usually is considered to be a reaction step which produces two radicals by thermal or photodissociation of a precursor in which the electrons are initially paired (or occasionally in the triplet state). For CIDNP, however, we will use the term “initialization” to mean a reaction step, proceeding with pseudo first-order rate constant k, which determines the overall rate of the chemical reaction, in which a reagent, R, first forms a pair of radicals which are close enough for the electron spins to interact with each other. Such a close pair, in which the two radicals are probably within a Van der Waals radius of each other. we will call an encoun&r pair, and give it the symbol E. Furthermore, there will be some probability, A, that such a pair will react to form a combination product, C, before the radicals move apart by diffusion, provided that the two electrons are in the singlet state. A “combination” step we therefore define as one which depends on the electron-spin multiplicity of a radical pair. The probability that the pair is in the singlet state, P,(t), will generally depend on time beca&e of the hypothesis that singlet-triplet mixing can occur. It should be stressed that as far as CIDNP is concerned the first reactive encounter of two independently-generated radicals may also be considered to be an initialization step. The initialization and combination steps we represent schematically by: Scheme I : Initialization R,-X-R, (R)
--&
R,. .R, 03
Scheme II: Combination R, .
.R, m
03
RI-_R, (Cl
CHEMICALLY INDUCEDDYNAMIC NUCLEARPOLARIZATION 153 where R1 and R, represent the molecular fragments, usually containing magnetic nuclei, which are carried through the reaction and a bar is used to represent radicals which may react with each other. Some examples of initialization steps in reactions where CIDNP has been observed are given below. Ia. Unimolecular decomposition Singlet precursor(41342) (CH&H,CO,), _* CH,CH, . . CH,CH, Triplet precursor(43) C,H&H,CHO (photoexcited) C,H,CH, . . CHO Ib. Bimolecular reaction Singlet precursor(44) CH,CH,Li Triplet precursor (4s) (C,H,),CO IC.
+ ICH,CH, -5 CH,CH,. .CH,CH, (photoexcited) + C,H,CH,’ (C,H,),COH.
.CH2C,H,
Diffusive encounter of independently-generated radicals’46’ .C,HS + CH,Cl, + CH,ClCOOH ---+ C,H, + .CHClCOOH + Cl,HC.Cl,HC. 9CHClCOOH
2.2.2. DIFFUSION.If an initially-formed encounter pair does not react after a few collisions between the partners, it will be separated by the insertion of solvent molecules between the two radicals. We designate such separated pairs by D. Diffusive separation by this mechanism will take place within an average time, rn, which will typically be lo-” to lo- I1 sec.‘47) If the pair of radicals never undergoes a re-encounter, CIDNP is not produced in pairs because the interactions between electrons and nuclei are too weak to have any effect in this short period of time. In general, however, the random motions of solvent molecules will drive the partners toward each other as well as away from each other so that there is a finite probability that the original partners will re-encounter each other at some later time. In fact, in the limiting case where only a single pair of radicals was formed in a finite volume of a completely inert medium the probability of a re-encounter would be unity. (One might, however, have to wait a very long time for it to occur!) This point ‘of view, which is based on the theory of random flights,‘48’ was applied to diffusion controlled reactions such as those of free radicals by R. M. Noyes about 18 years ago.(49) It differs from the original idea of a “cage effect” proposed by Rabinowitz”‘) in that the latter acknowledges only that molecules in solution may be held together in solvent “cages” for longer than they would be in the gas phase but still requires that a pair of radicals react within the time rn or not at all. Noyes has discussed in considerable detail how the probability of a re-encounter depends on time and various molecular parameters.” ‘) Thus he defines the quantity f(t) dt which is the probability that a pair which separated from a first encounter at t = 0 will undergo a re-encounter between t and t + dt. We may thus represent the process of diffusion of spin correlated radical pairs by Scheme III. Scheme III : Diffusion m
,
%$R,./.R,
(D) (E) where a solidus is inserted to represent intervening solvent molecules. At any instant one would expect most pairs of radicals in a dilute solution to be of the D type rather than the E type simply because the latter requires a close approach of two species which are present in low concentration.
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R. G. LAWLER
Although both Adrian(2”26) and Kaptein Cl291 *) have adopted a rather specific form for f(t) which varies with the inverse 3/2 power oft, it is more likely that it must be considered only as a rough guideline which seems to describe the diffusive motion of atoms in monoatomic liquids adequately (52)but may be grossly in error for complex molecules. About all one can say at present is that the re-encounter probability decreases with time less rapidly than exponentially but quite possibly(53’ more rapidly than inverse 3/2. Fortunately the qualitative and semi-quantitative appearance of CIDNP spectra does not seem to be very sensitive to the functional form off(t), but lack of knowledge of this quantity does constitute perhaps the greatest area of uncertainty for the quantitative understanding of the effect. As more accurate and reliable CIDNP intensities become available it should even be possible to use them to obtain information aboutf(t), which seems to be difficult or impossible to obtain at present in any other way. 2.2.3. SCAVENGING.Even though a pair of radicals may have diffused away from each other, the individual radical partners will still remain reactive and will eventually be lost by reactions with other molecules in the solution. Such reactions will take place predominantly with the more abundant D pairs and would not be expected to be affected by the overall electron spin multiplicity of the pair. Such a “scavenging” process is usually a reaction, such as shown in Scheme IV, between D and a scavenger molecule MY. We assume it proceeds with rate constant k, = zs ’ [MY] - l and gives a diamagnetic molecule S, derived from one of the radical fragments, and a new pair, D’; the two electron spins in the new D’ pair have the same wave function as in the pair D at the instant of formation of S. D’ may in some instances, in fact, live long enough to undergo a re-encounter to form a new combination product. Usually, however, the electron correlation in D’ will be lost before re-encounter takes place. Scheme IV : Scavenging R,./.R, + MY * R,Y + M./.R, (D) (D’) (S) For CIDNP, however, all that is necessary is that a scavenging process be one which (a) prevents re-encounter of a D pair with correlated spins and (b) does not depend on the electron-spin multiplicity of the pair. All of the processes given in the examples below (taken from the CIDNP literature) will therefore serve to “scavenge” pairs with correlated electron spins even though strictly speaking only the first is of the type shown in Scheme IV. IVa. Atom abstraction’42’ CH,CH,I + (CH3)2CH./.CH(CH3)2 CH,CH, ./. CH(CH,), + (CH,),CHI IVb. Unimolecular decomposition of one fragment’54’ C,H,./.C,H, + 13C02 C,H, 13C02./.C6H, IVC. Addition(55’ B./.B
+ C,H,CH
IVd. Recombination Naphthalene’/.
= I;(C,H,)O-
-
C,H,CHB
-
N(C,H,)O./.B B. = (CH,),(CN)C.
of one partner with another radicalc5@ NaphthaleneT + CH, + Naphthalene CH, + Naphthalene’ s
IVe. Thermal relaxation of electron and/or nuclear spins(46,57*58) 2&% equilibrium populations C,H,CH(OH)./.OCC,H,
CHEMICALLYINDUCEDDYNAMICNUCLEARPOLARIZATION
155
The last process fixes the upper limit of the length of time which electron spins may be correlated as approximately the relaxation time of electrons, T,,, and nuclei, T,,, in free radicals in solution. For most organic radicals this is probably 10e6 to 10e4 sec.(46,57*59) Thus all chemical scavenging processes involved in CIDNP must take place in times, rs, less than a few microseconds. Also it is only under special circumstances, such as the use of a high concentration of stable radical as a reactant shown in Example IVd, that the overall radical concentration in the solution is high enough for reactions between a pair and an independently-generated radical to compete with spin-lattice relaxation in the pair. This means that all “memory” of previous encounters will usually be erased by the time two independently-generated radicals diffuse together in an initialization encounter of the type illustrated in Example Ic.
2.2.4. SUMMARYOF KINETICPROCESSESNECESSARYFOR CIDNP.Thekineticprocessesjust discussed are all essential for the production of CIDNP from reactions of radical pairs. These are summarized in Scheme V in the general notation just described. Scheme V: Initialization
R k,
E
Combination
E 3
C
Diffusion
E =D ‘(‘jdr
Scavenging
D-Ir,S+D’
The quantities kR,f(t), 1, PJt) and rs are therefore the parameters in this model which play a role in determining CIDNP intensities. These will generally depend in a complex way on molecular structure and the time dependent details of diffusion and chemical reactions in solution. It is as well to point out, however, that while the time scale for the last three processes is measured in microseconds or less, the first step will proceed at the overall rate of the chemical reaction, i.e. the order of a few minutes under the usual CIDNP conditions. Furthermore, the relatively long nuclear relaxation times in diamagnetic product molecules C and S, in solution means that the NMR measurements are also carried out on the more leisurely time scale. Consequently the effects on the reaction products arising from relaxation or interaction with the radiofrequency field may be separated conveniently from the much faster events involving intermediate radical pairs. 2.3. Rrlutiorlship
hctwcvrl Rtrdicul-puir’ Rruction
Probabilities
und NMR
Intensities
We have so far said nothing about the connection between the foregoing kinetic processes and an experimentally-observable NMR spectrum. The connection is made in the following way. Consider that initially a reactant molecule, R,, contains the nuclei whose NMR spectrum we will eventually observe in a product and that the combined spin state of these nuclei is indicated by the subscript n. The reactant then undergoes the rapid series of processes which take the nuclei from R to E to D and eventually to products S or C. During this excursion the nuclei may remain in spin state n or may possibly undergo a selective transition to another state m. In any case, within a few microseconds the nuclei will either be in C or in S molecules or, through relaxation in the radical, they will have
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R. G. LAWLER
“forgotten” what their original spin state was. We may thus represent the net reaction as Scheme VI.: Cm,O S m,O
where &,,, and 4:,,, are the respective probabilities of finding nuclei which started out in R, in the spin state m of C (combination) or S (scavenging product) after the rapid radicalpair reaction sequence is over. The intensity, I:,,,,, of an NMR transition between states n and m, of a combination product, for example, under slow passage, non-saturating conditions, is G&~l~“lm)z(C, - C,)
(1)
where (C, - C,) is the population difference between the two states and (nlZ”lm)is the usual transition probability for NMR transitions involving the operator, I”, for the x-component of the nuclear spin. During the chemical reaction the population difference between the two levels of the product, C, according to Scheme VI obeys the equation @G - C,)/dt = k, C (4”, - &JR,
- C K,(C,
+
1 k
-
G,o
-
Ck
+
Ck,o)
k
k wmk(cm
-
%,O
-
ck
+
(2)
ck,O)
where the last two terms allow for relaxation of the populations to their equilibrium values, C,,,, etc., with time-independent transition probabilities, W,,. At thermal equilibrium the difference in population between two levels may also be expressed in terms of the polarization, p”, of the transition PO
=
(C,,,
-
C,,oMC,,o
+
G,,,)
=
WWBoPkT
(3)
5 x lo-(’ for protons in a 60 MHz spectrometer at room temperature, where y is the zagnetogyric ratio of the nucleus and B, is the magnitude of the applied magnetic field. If Zc is the total number of nuclear spin states available in a given molecule C (for example, 2” for x spin l/2 nuclei) and C is the total concentration of C in the sample, we have (C,,, - G.0) = 2P°CZ, 1
(4)
as the population difference responsible for NMR absorption in samples at equilibrium. These population differences are, however, usually much smaller than those arising from CIDNP and from now on we will ignore contributions to the spectrum from the equilibrium, or thermal, polarization of the spins. If this is done, all states of reactant R are assumed to have the same initial population, RZ, ‘, and equation (2) simplifies to d(C, - C,)/dt = k,RZ,‘@; - ;
- #$,) - T,;,(C,
1/2(wnk
-
wmk)
tc,
-
ck
- C,) -
ck
+
cd
(5)
where we have simplified the definition of@: to be 2, times the probability of finding the product C in spin state n if each spin state of R had a probability Z, ’ of being populated and T,,!,
= 2w,, + l/2
2 k#n,m
(w,, +
wmk).
(6)
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
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POLARIZATION
The first term of equation (5) then represents the driving of the transition intensity by the reaction, the second term an exponential decay of the population and the last term possible contributions, via cross-relaxation, of other transitions which are being chemically driven. The time dependence of C, - C,, and therefore the NMR intensity during CIDNP, based on equation (5) should therefore be the following. The driving term due to the reaction should cause an initial linear increase in intensity with time. As the population difference becomes appreciable, however. the second term in equation (5) should bring about a reduction in the rate of increase of intensity until the first and second terms become equal and an essentially steady state is reached. The last term may bring about contributions from
FIG. 2. Time dependence of CIDNP (40 MHz) from benzene during thermal 5 mol. y/, benzoyl peroxide in cyclohexanone” at I IO”.
decomposition
of
other transitions. but with a rate constant less than T<,!,,. As the concentration of reactant, R, begins to decrease, however, the first term will become less effective than the second and the intensity will decrease. If k, is less than both T&,, and the cross-relaxation rates then the rate constant for decrease will be simply k,. Finally, after a sufficiently long time the reaction will be over and the product will have relaxed to give an NMR spectrum with normal intensity. This effect was beautifully demonstrated in the first report”’ by Fischer et al. of CIDNP from the protons of benzene formed from the thermal decomposition of benzoyl peroxide in a hydrogen-donating solvent. This spectrum is shown in Fig. 2. For the purposes of quantitative discussion of spectra it is sometimes convenient to define an enhancementfactor, V,,, for a transition vLl = (El
- L,O)lEll,O
= [(C, - C,)/2p°CZ,
‘1 - 1.
(7)
V,, in a reacting system is, however, time dependent because of time dependence of concentrations in both the numerator and denominator of equation (7). For reactions in which k, is small compared with the relaxation rates, however, a steady state in (C, - C,) may be reached rapidly. Also in the early stages of reaction C increases approximately linearly with
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R. G. LAWLER
time and the concentration of R remains approximately’constant. Under these conditions, and ignoring the cross-relaxation last term in equation (5), we have C, = k,R&
@a)
(C, - CA = k,RZ,‘(4;
- 4:)T,,,,
1 + %tl(t) = (T,,“mMQO)- ‘M
- 4tJM3 @v-&d
(8b)
(84
= EmwEl,o@) where 4,” is the probability of formation of C from R averaged over all nuclear spin states; that is, it is the yield of C which one would measure chemically after the reaction had gone to completion. The experimental value of V,, at any time during a reaction may be obtained by noting the observed intensity of the enhanced line, rapidly stopping the reaction (for example, by cooling the sample or shutting off the light in a photochemical reaction) and then recording the intensity of the weak. unenhanced lines arising from the amount of product formed up to that time. To obtain the microscopically interesting parameter (4, - +,,,)/4,, it is then also necessary to determine the relaxation time, T,,,, for the transition and possibly also correct for the effects of cross-relaxation.‘8,61’ In those cases where this procedure has been used, (14,45,46,60)the observed values of (Z,/Z,)I/,,(t/T,,,,) range up to about 103. This in turn requires only that (4, - 4,,,)/4,, be 10 -2 or smaller, that is about 1% discrimination of the reaction according to nuclear spins. It is for this reason that one can still say that for practical purposes the yields of chemical reactions so far found to exhibit CIDNP do not depend appreciably on the nuclear spin state, in spite of the quite large enhancements sometimes observed. 2.4. Quantum Mechanical Calculation
qf Reaction Probabilities
From the above we see that the empirically-obtainable quantities determining CIDNP intensities,are the nuclear-spin dependent reaction probabilities, or yields, 4,. It is therefore necessary to relate these obiervable quantities to the microscopic parametersf(t), A, P,(t) and TV. We do this by recognizing how the probabilities are related to the wave function, $(t), for the total system consisting of E, D. C and SD’. We write v/(t) as a linear combination of products of the space part of the wave functions YE, Yy,. Yy, and Ys,, for each of the species, the spin functions for the two electrons, E_ and the spin functions representing the spin states of the nuclei, x.. That is (9) This separation of space and spin coordinates emphasizes our picture of the chemical reaction and motions of the liquid as simply moving the overall spin system, represented by the products EJ,, from one chemical species, represented by Yy,, Y, etc., to another. The squares of the expansion coefficients /Cz,,(t)(“,IC!$(t)l’, etc., are then simply the probabilities of finding the overall system in one of the four kinds of chemical environments and in the 0 electron spin state and n nuclear spin state. Furthermore, if YR is the space wave function for the radical pair precursor, R, we have
(10)
CHEMICALLYINDUCED DYNAMICNUCLEAR POLARIZATION Similarly, since E and D eventually disappear overall system, after a long time we have ;;
IC&)12
but the electrons
+ Ic:%)12
-
159
and nuclei stay still in the
Icm12 = 0.
(11)
The boundary condition in equation (11) is in effect just a special case of the condition that Y(t) remain normalized for all times. This conclusion is not changed even if we include relaxation in D, or D’, since this only redistributes the coefficients (probabilities) for a given species but keeps their sum the same. More explicitly, the spin functions E, and x,, are given by E, = 2-+(@
- /?a) = S,
singlet state
xn = II,, m,)lZ,, mb) . . . (product
over all nuclei)
(12b)
where c( and B are the usual electron spin functions and II, m) denotes the spin function for a nucleus with spin I and value m for its magnetic quantum number. We will consider a set of nuclei to be “equivalent”, and therefore lumped together with a single value of I, only if they are equivalent in all four of the species E, D, C and SD’. Using this notation, the definition of the squares of the coefficients in equation (9) as probabilities and the assumption that only the singlet state of the combination product, C, will be populated by the reaction, we find, in the absence of relaxation in the radicals, the following relationships between the reaction probabilities +E, 4.“, the probability P,,(t) of finding E in the singlet state and nuclear spin state n and the coefficients:
2.4.1. EVALUATION OF THE WAVE FUNCTION FOR THE SYSTEM.In principle it is possible to evaluate Y(t) by defining a general time-dependent Hamiltonian, H(t), which is a function of all of the space and spin coordinates for all four of the chemical species and solve the time-dependent Schroedinger equation i dY(t)/dt
= (2n/h)H(t)Y(t).
(14)
In practice, of course, this is impossible and a number of drastic simplifying assumptions must be made. Furthermore, since we are not concern&d here with Y(t) itself but only with the products of the coefficients of its expansion in a space and spin basis, it is more convenient to adop’t the density matrix notation P”,,,,.” .(t) = C!&(t)C!&(t), etc.
(15)
If we assume that the processes of reaction and diffusion manage to avoid producing nonzero products of coefficients such as C,“,(t)~$,,(t) between two different species, it is then permissible to consider the spin system of only one species at a time and allow for transfer of the spins from one species to another in a phenomenological way. This is, for example, very similar to the procedure which has been used in treating the effects of intermolecular spin2xchange in ESR spectra. (62,63) The principal differences between that case and the
160
R.G.LAWLER
one we are considering here is that our system is not in thermal equilibrium and we must also consider four different species instead of only two. The time dependence of the spin system under the influence of reaction and diffusion is then describable by the following scenario. first proposed by Adrian.(24 ~26) An encounter pair. E. forms from R for the first time at to and cvolvcs in time under the influence of the Hamiltonian H, until it separates by diffusion at t, to form the separated pair, D. The spin system of D then evolves in time under the influence of H, until the first re-encounter tq reform E at tz. E then forms D at t,, D undergoes a second re-encounter at t, and the diffusion process thus continues with the pth encounter beginning at t2p_2 and ending at t,,_ i. At the end of each encounter the combination product, C, also will have had the probability LPs(tlp _ 2) of forming in the singlet state. Similarly, in the relatively long interval (t,, - fzp_i) between encounters the process of scavenging will produce S and D’ and deplete the probability of finding the system describable as a D pair. In the density matrix formalism (64) the time evolution of the density matrices pE and pD under the influence of their respective Hamiltonian matrices HE and HD is 3pE/c?t = i(pEHE - HEpE); and the above scenario
may be described
8pD/8t = i(pDHD - HDpD)
by the following
equations:
t,,_ z < f < t,,_,dpE/dt = i(pEHE - HEpE) - (c?pE/?t),,,,; tzp-,
< t < tzp
(16)
dpD/dt = i(pDHD - HDpD) - TS ‘pD;
dpc/dt = (iipEl?t),,,,
(17a)
r/pSD’/dt = TS ’ pD
(17b)
where the partial derivative refers to the electron-spin dependent formation of combination product during an encounter. In the above equations the effects of the spin Hamiltonians H,, H, and H,, which are actually responsible for magnetic resonance, have been ignored because, as we have already stated, the above scheme will generally be over before they can produce appreciable effects. Furthermore, we have implicitly ignored relaxation in D by using the same rs for loss of D and gain of SD’ (see Section 4.2). 2.4.2. MODEL FORAPPROXIMATESOLUTIONOF MASTER EQUATIONS FOR CIDNP. Equations (17) with perhaps the inclusion of the effects of H,, etc., and explicit addition of approconstitute the master equations for priate terms for relaxation as in the Redfield theory,“” the time evolution of the spin system under the influence of the interactions responsible for CIDNP. Just as in the general theories of magnetic relaxation, however, it is easier to write down the master equations than it is to solve them. The following approximations have thercforc ken employed to give solutions which are both tractable and consistent with most of the experimental data so far obtained. 1. We will assume that the majority of pairs formed initially undergo at most one r~wwcow~ter and will therefore extend the above scenario only through the time t, when the second encounter of a pair ends. This sequence of events is shown in Schcmc VII. Scheme VII:
2. We include in HE only the interactions involving ignore entirely the nuclei. Conversely, in H, we assume
the two unpaired that the electrons
electrons and are sufficiently
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
161
well
separated from each other that we may ignore the interactions of the electrons with each other and consider only interactions of one electron with the nuclei on its particular radical. 3. We assume that both H, and H, are independent of time in the intervals during which they operate. This is probably a rather good approximation for H, which is essentially the spin Hamiltonian for two free radicals, but the approximation is certainly not strictly valid for H,. We may therefore anticipate that the effects of H, will have to be evaluated almost completely empirically. (The time dependence of H, has, in fact, been used explicitly by Fischer” ‘) and Glarum” ‘) in a mechanism for producing CIDNP by adiabatic level crossing. It seems unlikely, however, that E pairs ever live long enough for such processes to be important.‘24’) In writing down explicitly the spin Hamiltonians we will use subscripts 1 and 2 for the unpaired electrons on radicals R, and R, and subscripts a and b respectively for nuclei on the two radicals. In general, then, we will have the following Hamiltonians for E and D in a magnetic field. Encounter
Pair Hamiltonian: HE =,
HI,
+
H,,
+
H2,
+
H,,
+
H,,
+
H,,
+
H,,
+
HLR.
(18)
In equation (18) z refers to a Zeeman interaction between the applied magnetic field and a magnetic moment generated by a source of angular momentum. SL represents the spin orbit interaction, SR the spin rotation interaction, LR the orbit rotation interaction. All of these interactions are, of course, well known for molecules in the gas phase’““’ and also may contribute to line positions(67) and relaxation times(“) in ESR. Unfortunately, it is impossible at present to assess accurately these terms for a pair of radicals which are held within a molecular diameter of each other with variable orientation. About the best one can at present do is hope that the average effect of these interactions may be neglected. Furthermore, the electron Zeeman interaction in such a pair ‘is probably considerably smaller than the electronelectron interaction. H, I, Following Adrian’26’ we therefore will drop all but the first term in H,, The remaining H,, term consists of two parts: the magnetic dipole-dipole interaction between the electrons, and the exchange interaction. The former interaction, however, has been estimated”“’ by Freed to be smaller in magnitude than the latter for most radicals at the distances and for the times characteristic of encounters and we will ignore it. We are then left with H, z
-.!(I/2
+ 2S,.S,)
(19)
where the “exchange integral”, J, is,defined as one-half’the separaticrn in energy between the lowest singlet and triplet electronic states for the two electrons.““” Unfortunately, even J is a poorly understood quantity for complicated molecules. All one can say with reasonable accuracy is that (a) J falls off approximately exponentially Mit.11the average distance between the electrons, (b) the value of J for two hydrogen- atoms in the 1s state separated by6A (“,“)is comparable to the nuclear hyperfine splitting typical for organic free radicals and (c) for n-radicals of the type usually encountered in organic chemistry the sign of J varies with the orientation of the p-orbitals containing the unpaired electrons. One might therefore expect a smaller average value of J for such pairs than for hydrogen atoms at the same distand.
162
R. G.LAWLER
D#i~.siw/_v Separated Pair Ha+ltonian : We assume that H, is the sum of the individual and R,
spin Hamiltonians
for free radicals
R,
H,=H,+H,=H,,+CH,,+H2=+CHZh+CHaZ+CHbr+CH,,,+CHbb'. (20) In general, interactions
of course, one should include the spin orbit, spin rotation, orbit Zeeman, etc., for both electrons and nuclei. It is known from ESR, however, that for organic
free radicals in solution, the parts of these interactions which shift the energy levels of the spin system (i.e. the isotropic parts) simply modify slightly the coupling constants (g-factors, chemical shifts, hyperfine splittings) used to describe the interactions between the electrons, nuclei and the magnetic field. The anisotropic parts of these interactions, on the other hand, induce transitions between energy levels, but at the rate of the relaxation times which we have assumed is usually slower than the re-encounter and scavenging rates of radical pairs. Furthermore, the nuclear-nuclear couplings in the last two terms of H, will always be small compared with the hyperfine coupling terms H,,and H,,. It can also be shown’i2’ that the nuclear Zeeman interactions, ,which may be comparable with the hyperfine interactions at high field, do not have any effect on the elements of pD and may therefore be ignored. For these reasons, then, it. is permissible to omit all but the first four terms in H, to give
HD = ~1s; + 02s; + CA,,SfZi+ CA,&Zf, + 1/2CAi,(S:Z, a +
1/2
c
A,b(S;I,
b +
s;I,c)
+ S;IJ)
a (21)
b
where oi = gl/?(27c/h)\B0is the Larmor frequency of electron 1 which has the g-factor gi and similarly for electron 2, and the isotropic hyperfine splittings Ala, etc., are in radians per second. 2.4.3. HIGHFIELDSOLUTIONSOFTHEAPPROXIMATEMASTEREQUATIONS. Wenowundertake to solve for the time dependence of the density matrices of E, D, C and SD’ using the approximations above. In addition, however, it will be assumed that the magnetic field is high enough so that the Larmor frequencies of the electrons are much larger than the nuclear hyperfine splittings, or alternatively that the applied magnetic field is much larger than the hyperfine splittings, in gauss. Since typical values of A for organic free radicals are between 5 and 100 gauss, with most clustering around 25 gauss,‘72) this approximation will certainly be good for reactions run in fields in excess of 1000 gauss, which is still l&20 times smaller than the fields normally employed in high resolution NMR. Under high field conditions an extremely important simplification of the problem occurs: the last two terms in ff,, given in equation (21) may be ignored”‘,“” (see Appendix A.11). We find therefore that the operators S’ = ST + Sg and I’, for every nucleus, commute with both H, and H,. However, S2 commutes with H, but does not commute with H,. In
practice this means that the net component of the electron spins, and that of every nuclear spin. along the held direction are constants of motion but that singlet and triplet electronic states may still be mixed. This in turn means that (a) electron spin states E-Z,, and E+Z, in equation (I 2) do not mix with any other states and therefore P”~:,~,,(;c) = p:,.,,(O) and (b)Eo andc, may mix but only if they are combined with the same nuclear spin function, ln.
CHEMICALLY
INDUCED DYNAMIC
NUCLEAR POLARIZATION
163
Thus, components of the density matrices P&,~.~. vary with time only if n = n’, that is, in highfield no nuclear spinsjip. With the above simplifications we may now write the following solutions to the equations (17) to give & and 4:. The following initial conditions are also assumed : (a) all off-diagonal elements of p”(t,-J vanish, (b) the diagonal elements of pR(to) with the electron spin state, B, all have the same value, and (c) initially all three triplet states are equally populated (if they are populated at all). We therefore have r = tO:P:n.n’n’(rO) = P,“,,,,“&)
p+ = p- = po =
= 6,,,&“,Pnz,1;
(224
p7
= (1 - 4Ps-G1; PD = JPSG’; an,an@l) = P:“,o”(~,LPc,,,“@l) P~#d~,)= PK’ t = t, :P:,&Z) = Z;’ e-(tz-‘L)irs{(l - I)p, + [or - (1 - A)ps] sin2 &(t, - tr)} t =
t1
:&&I)
c PK&2) =
Z,‘(l
- e-(‘2-‘1)“S)[(1
Wb) (224
- J.)ps + 3p,]
d
where 6, =
VW,
-
(32
+
= singlet-triplet t=
tj:
4: = = A&l
zR&,.,(x~
c a
condition
-
c
A,,m,) b
mixing coefficient z
(23)
zRkt&,(f3)
+ (1 - A)e-(12-L1)i~s]
+ 1 e~(‘2-‘1)I’S[~r and the normalization
4a%
from equation
- (1 - A)pJ sin2 &(t,
(11) requires
- t,)
(24)
that
- 3Le-(fZ-f1)‘rS[PT - (1 - 3.)p,] sin’ s,(t,
- t,).
(25)
Consideration of equations (23 j(25) leads to the following conclusions: (a) CIDNP arises from the second term in equation (24) which is the only one which depends on the spin states of the nuclei; the first term is the same for all molecules and cancels when one considers population differences; (b) the expressions for the reaction probabilities above would be the observed ones only if all radical pairs spent the same time, t, - I,, between the first and second encounter; as mentioned before. however. it is necessary to average over a distribution of times between encounters; (c) the conclusion contained in equation (25) follows directly from the high field approximation which prevents mixing of one nuclear spin state with another, and the assumption that all nuclei which do not end up in C finally find themselves in the scavenging product S. Before discussing the results obtained by averaging equations (24) and (25) over the times between encounters we list the conclusions about NMR intensities of a transition between states n and /n in C and S products which are ind~)prntlent of the way in which the averaging is carried out.
164
R.G. LAWLER
3. Characteristics
of Observed Spectra
3.1. Relative Sigm of Enhancement of Combination and Scavenging Products From equation (25) we see that ($I;: - 4:) = -(& - 4:). Therefore, corresponding lilies in combination and scavenging products derived from the same radical pair should have enhanced intensities of opposite sign. Furthermore, in the case where there is only one
::
::
CH31
CH3COOCCC13
CH3CCl3
A
I
2.5
wm
FIG.3. (A) Proton NMR Spectrum (60 MHz) recorded 5 min after a solution and I2 in CCI, was warmed to 50 (B) Spectrum taken at 0’ immediately
of CH,CO,O,CCCI, following scan A.‘“’
scavenging product and no relaxation occurs before scavenging of the radicals which escape from the second encounter, the intensities should be equal and opposite. This is very strikingly demonstrated in Fig. 3 which shows the emission line from CH,CC13 .and the enhanced absorption line from CH,I which are obtained as the combination and scavenging products respectively obtained during the thermal decomposition of CH,CO,O,CCCl, in the presence of I2 .(9)
(‘HEMICALLY
INDUCXLI
DYNAMIC
NUCLLAH
POLAKIZATION
165
Scheme VIII: I2
CH,CO,
-2c0, 02CCC13 -
CH I CH J
+ I
.CCI,
~CCh~cH:cc,I
3.2. Dependence of Sign of Enhancement on Electron-spin Multiplicity of the Initially-formed Pair
The last term in equation (24) is multiplied by the factor [pT - (1 - A)p,] which depends on the probabilities of finding the pair initially in the singlet and triplet states. The difference in 4 for two different states, and therefore the intensity of each NMR line from a product, will be multiplied by this factor. There are three different kinds of initial conditions which are likely to be encountered. (i) Singlet-born pairs, s ps = 1; (ii) Triplet-born
PT = 0
pairs, t j+ = l/3
ps = 0;
(iii) Pairs born with uncorrelated
spins, u PS =
PT
=
1/4
On substituting the initial conditions into the multiplying factor we reach the conclusion that the enhancement of a product from a given radical pair born in a singlet state will be opposite to that obtained if the same pair is generated in the triplet state or by an encounter between two independently-generated rudicals with uncorrelated spins. One of the nicest
examples of this effect is that quoted by Closs~73~74~ who was able to produce the benzyldiphenylmethyl radical pair both by thermal decomposition of the unsymmetrical azo compound and by reaction of the triplet state of diphenylmethylene with toluene. Scheme IX: (GH,),CHN,CH,GH,
-
SC,H,),CH..
CH,C,H: 1 (C,H,),CH-CH,C,H,
-
(&H&C:
+ CH,C,H,
The near mirror image spectra obtained from the CH-CH, group of the coupling product obtained from the pair generated by each of these methods is shown in Fig. 4. It has also been found that products from the thermal and triplet photosensitized decomposition of acyl peroxides exhibit enhancements of opposite sign.‘75.7h’ Some especially-striking examples of the opposite signs of enhancements in singlet pairs and pairs with uncorrelated spins have been found in reactions between alkyllithium compounds and alkyl halides. ‘44.77) It is found, for example, that the CH,CH, group of
166
R. G. LAWLER
I
I
120
I
so
60 HZ
FIG. 4. ClDNP
spectra of 1.1,2-triphenylethane obtained from coupling of bcn/yl and diphcnylmc~h!l radic;~I> formed from thermal decomposition of :vo compound (upper) and from reaction of triplet state carhciic (lower). ‘-‘I The scale is in HI downfield of the CH, resonance of totucnc (60 MHrl.
n-butyliodide becomes enhanced during its reaction with n-butyllithium. The details of the reaction are rather complex and open to some question, but the steps responsible for CIDNP are most likely those given in Scheme X. Scheme X:
CEA,
RCH-CH; -LiI RCH, CH,I
+ LiCHzCHZR
II
s
RCH,CH,I
CA&, -
-2RCH,CH;
RCH, CH,i
+
RCH, Cl-l;
RCH, CH?I*
+
RCH, CH;
RCH,CH; ” l 2RCH,CH;
-
(El,
I RCH-0-l; (AE) -
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
167
In Scheme X we have used a star to indicate molecules with enhanced NMR spectra. The symbol EA has also been introduced to indicate an enhanced multiplet with both absorption and emission lines in which enhanced absorption is upjeld of emission and vice versa for AE. Reference to Fig. 5 shows that simply adding an alkyne, Z-hexyne, to the reaction mixture alters the contributions to enhancement arising from singlet and uncorrelated pairs and reverses the sign of every line from both the iodoalkane, which here is a product of a scavenging reaction, and the olefin formed as a combination product.
No
additive
:; Bu;LI
+ Bu” I
CH3CH2 CH= CH2
r
CH,CHP CH2 CH21
j
Added 2 - hexyne
fm FIG. 5. Effect of added alkyne in changing radical pairs giving CIDNP diffusive (lower) in reaction between n-butyliodide and n-butyllithium
from singlet-born (upper) to (‘H spectrum at 60 MHz).
In the special, and so far hypothetical, case where a system involving both radicals and dimerization products is in thermal equilibrium one would of course find no enhancement. One way of viewing this necessary requirement is that the enhancement generated in each dissociation of dimer to form a singlet pair is exactly cancelled by the enhancement of opposite sign generated by pairs of free radicals which have lived long enough to undergo spin relaxation and re-encounter partners from other dissociations (Scheme XI). Scheme XI :
,,
Rz _
2R. &
2R. __r
R,
It can be shown by considering the kinetics of the reactions that the expression in equation (24) for the reaction efficiencies of different nuclear spin states is quite consistent with this view.(‘,“) Such cancellation apparently may also occur even when the system is
168
R. ‘3. LAWLER
not in equilibrium. It is found, for example, that when the reaction in Scheme X is carried out with n-butylbromideinsteadofn-butyliodidenoenhancement is observed in the 1-butene rapidly formed unless a scavenger such as another olefin is added to upset the balance of enhancement from singlet and uncorrelated pairs.“’
Prohahility
i
It has been proposed “” that the dependence of the sign of enhancement on the multiplicity of the radical-pair precursor should be a useful tool in photochemistry where the multiplicity of product precursors must otherwise be established by rather laborious determination of the effect of quenchers and sensitizers on product yields. Ambiguity may, however, arise if both singlet and triplet states may give rise to radical pairs. In the case of highly reactive pairs (/1 close to l), it may be possible for the sign of [pT - (1 - i)p,] to be positive (and therefore lead to CIDNP consistent with a triplet precursor) even though /I,~ > 3/j,.. that is. most of the pairs ari
3.4. Drpcvdencr
of Sign
of Enhuncment
on the S -
7;) Mi.uing
Cotfficiewt,
6,
The quantity 6, which is defined in equation (23) is responsible for the nuclear-spin dependent mixing of the singlet and triplet states through its linear dependence on the quantum numbers m, and m,, for nuclei on the two radical fragments. The actual intensity of an NMR transition in a product will then depend on the difference 4, - $,, and therefore on sin2 s,(t, - t,) - sin’ s,(t, - tl), for two nuclear spin states which differ by one unit in the sums of the nuclear spin quantum numbers. To illustrate how the intensities depend on the quantum numbers, on hyperfine splittings and on w1 - o2 we will consider the commonly-occurring case of an NMR transition in a product molecule which to a first approximation Hips only the spin of one set of equivalent nuclei, A. These nuclei may also be coupled to another set of equivalent nuclei, B, on the same or the opposite radical. The transition is thus from spin state II,,. r~,)ll,,. ~1,~)to II,,. ~1, - I ))I,,. ~rl,,).where I, and I,, arc in general irreducible components of the coupled angular momcuL;1 of equivalent A and B
CHEMICALLY
nuclei. In a normal NMR
INDUCED
spectrum this transition
DYNAMIC
NUCLEAR
has a first-order
169
POLARIZATION
absorption
intensity
given by’79’ I(IA, MA3IS, q’
IA3 mA -
I. I,? md x
81A81,(1A
+
mA)(h
-
‘?4
+
I)
(26)
where g,, and gr, ‘are degeneracy factors. For n equivalent spin l/2 nuclei these are gr = n!(n - 2k + l)/k!(n - k + l)! k=
0,1,2 ,..., 0, L2,.
n/2
..,(n
-
I = (1/2)n - k n even
1)/2
(27)
nodd.
The line position, v, in frequency units, for transition mA to mA - 1, is given to secondorder by”” v = VA- JABmB+ (J~,/2~dUB - ‘%)tlB +
“*B
+
1)
+
(JiB/6AB)mAmB
(28)
whereh,, = (vA - vB)is the resonance frequency difference and J,, is the nuclear spin-spin splitting constant. In the first-order limit where JAB 4 a,, the last two terms in equation (28) are very small and the line positions for A transitions depend only on vA, JAB and the quantum number for the nuclei B. The last term, however, shows that when the nuclear spin coupling is not weak, transitions with different values of mA will have different line positions. In the usual field-swept NMR experiment, of course, a transition with a large value of v will occur at low field and vice versa. Substitution of the appropriate expressions for 6, in equation (24) for the transition above yields an expression for the difference in reaction efficiencies which depends only on mAand mg, i.e. is independent of I, and I,. Intensity cc fj m.&m”(r)- &, - 1.m”(T) = + AfJ . -‘,=[pT
- (I - /.)ss] (N f
M + P)
(29)
where N = sin(A,/2)r sin(o, - tozh cos(&m,)r
cos AA(mA- l/2)r
M = sin(A,/2)r sin(A,m,)r cos[(w, - w2) + A,(m,
-
1,‘2)]r
P = sin(A,/2)7 sin A,(m, - 1/2)r COS(W,- Wl).r cos(A,m,)r
(30a) (30b) (3Oc)
and T = t2 - t, is the time between the first and second encounter of a pair. The first ) sign is positive if the product is of the combination type, C, and negative if it is of the scavenging type, S. The second f. sign is positive if the B nuclei are the same radical as A and minus if it is on the other radical. Each of the three terms in parentheses in equation (29), which are defined in equations (30), give rise to CIDNP effects of a different type. In genera1 all three kinds of effects will occur simultaneously, but under certain limiting conditions each term alone may be responsible for the observed effects. These effects are discussed below. It is well to point out, however, that all three terms vanish if A is zero. This, therefore, emphasizes the requirement that enhanced lines arise only from transitions involving nuclei which are coupled to the unpaired electron in one of the radicals. 3.4.1. NET ENHANCEMENT. If we consider the case where there are no other nuclei coupled to A, in effect I, = mB = 0, and we see that M vanishes. Furthermore, all transitions of A occur at the same frequency regardless of mA. Finally, since mA - l/2 = I, - l/2,
170
R.G.LAWLER
I, - 312,. . . , -(I, - 3/2), -(I, - l/2) the sum over all mA of the sin A,(m, - 1/2)r term in P vanishes. Similar conclusions are also reached even when B spins are present, provided both that A, is zero and the spectrum is sufficiently first-order that the last term in equation (2X) is negligible compared with the NMR linewidth. Under either of these conditions CIDNP arises solely from the term N, which is non-zero only if wi # 02, i.e. only if the g-factors of the two radicals are different. When N is averaged over all m, one is still left with a net enhancement of the transitions due to A. The spectra shown in Figs. 2 and 3 are both of this type. In the first case the enhancement is generated in the radical pair consisting of the phenyl radical, C,H, . , and the benzoyloxy radical, C,H,CO, . , in which the g-factor of the former radical is less than that of the latter. (*O)The net enhancement in Fig. 3 arises in the CH,. . Ccl, radical pair.“’
3.4.2. MULTIPLET EFFECT.In the case where o, = o2 and the spectra are also first order the contributions from N and P vanish and leave M as the only possible source of CIDNP. M is non-vanishing, however, only if both A and B are coupled to one of the unpaired electrons in the radicals. The enhancement obtained under these conditions is such that a line in a multiplet with quantum number mB has an intensity opposite in sign to that for the line on the other side of the multiplet with -mB. This kind of enhancement has come to be known as the multiplet efSect. It is shown in the spectra in Figs. 4 and 5 where the enhancement is generated in pairs of hydrocarbon radicals which have almost identical g-factors (see Schemes IX and X). 3.4.3. HIGHER ORDER EFFECTS.(i) Second-order CIDNP. It is possible to envisage a case in which B may be coupled to A in a product but where o, = w2 and A, = 0. This happens if the radical pair involves identical radicals and the nucleus B is either in a part of the radical inaccessible to the unpaired electron, e.g. the CH, group in a radical of the type CHJH,CX,., where X has no nuclear spin, or is transferred to the radical from a diamagnetic molecule in a scavenging reaction. 2H,R.
=
H,RH,
In either case both N and M vanish. Since AA is not zero, however, P may still make a non-zero contribution to the spectrum provided that the last term in equation (28) exceeds the line width so that transitions with different values of mA appear at resolvably different positions. The signs of enhancement will be independent of the sign of A, and will be opposite for the oppositely-signed value of /nA - l/2. Each first-order line for a given lug will therefore exhibit a “multiplet effect”. So far the only clear cut example of this rather striking effect has been found by Kaptetn “2.‘x’ in the NMR spectra of the second-order components of the methyl group doublet of 2-cyanopropane formed in Scheme XII, which is of the second type mentioned above. Scheme XII :
(CH,),C(CN)N,(CN)C(CH,), -
2(CH,),(CN)C. 5
(CH,),(CN)CH
(ii) intensity borrowing. The expression given in equation (26) for the intensity of a transition arising from A is strictly valid only in the limit where J,, 4 6,, so that transitions are essentially pure A or pure B transitions. We have already mentioned the line shifts which may arise when this condition breaks down and may expose strongly-enhanced transitions of opposite sign which otherwise cancel. The observation of these second-order
CHEMICALLY
INDUCED DYNAMIC NUCLEAR POLARIZATION
171
effects, however, still requires that the nuclei whose spins flip in the transition be coupled to the unpaired electron on one of the radicals. It is also possible to have an apparent enhancement of nuclei which are not coupled to an unpaired electron, provided that they become strongly coupled to another set of nuclei which are strongly enhanced. Under these conditions what appear to be pure A transitions, for example, are really a mixture of A and B transitions and may therefore reflect the enhancement of B. If B is very strongly enhanced, A intensities may be affected even though the amount of intensity borrowing in the normal spectrum is rather small. A rather nice example of this effect is given by Lehnig and Fischer@i’ in the spectra of chlorobenzene and para-dichlorobenzene formed from the phenyl and paru-chlorophenyl radicals respectively. In these radicals the ortho protons are more strongly coupled to the unpaired electron than are the metu protons. In the non-first-order spectra of these products, however, the lines nominally due to transitions of the metu protons appear in emission, probably because of intensity borrowed from the strong emission lines arising from the ortho protons. 3.5. Rules for Predicting the Sign of Enhancement Equations (26H30) contain, in principle, all of the information required to calculate the intensity of first-order CIDNP NMR lines in an A,B, spin system provided that one knows how to average over a distribution of re-encounter times, z. Even in the absence of such averaging, however, it is possible to predict the sign of enhancement of such a transition by considering the way in which the two + signs in equation (29) and the signs of the parameters A,, A,, o1 - o2 and m, (which in turn requires knowing the sign of J,,) affect the expression for the intensity. The effects of the f signs are trivial and given by definition. Furthermore, only the sine factors in equations (30) will have an effect on the overall sign of the intensity, provided that the distribution of re-encounter times is a monotonically decreasing function of time. Rules based on these considerations for determining whether net enhancement will be enhanced absorption (A) or emission (E) and whether the multiplet effect will be EA or AE have been recently provided in tabular form (14)and in rather convenient formulae.‘*.“’ The formulae given by Kaptein “‘)for the net enhancement, r,,, and multiplet effect, r,, are net enhancement: r”, = F(SI
- SzM*
+=A
-=E
multiplet effect:
rm, = wWBJAB~~~~
+ =
EA
- = AE
where p, E and g’ABare signature symbols given by the following rules: p=
+ when the radical pair is formed from a triplet (t) or by an encounter of free radicals with uncorrelated spins (u). i - when the pair is formed from a singlet (s) precursor. + for combination
E=
products
( - for products of scavenging. + if nuclei A and B belong to the same radical uAB
=
-
if nuclei A and B belong to different radicals.
172
R.G.LAWLER
As an illustration of these rules we consider the spectrum of n-butyliodide shown in Fig. 5. The nuclear spin system is the RCHy’CH\*‘I formed as a scavenging product of the radical RCH,CH, . group. Both the top and the bottom spectra are of the pure multiplet effect type. It is now well known (82) that for alkyl radicals of the above kind the hyperfine coupling constants of the protons c(and fi to the unpaired electron are negative and positive respectively. We therefore have A, = -, A, = +, E = +, J,, = +, cAB = + from which we conclude that the multiplet effect should be AE if the radical pair precursor were singlet and EA if it arose from encounters of free radicals with uncorrelated spins. (The possibility of a triplet radical precursor is highly unlikely for a thermally-induced reaction such as this.) The vicinal proton coupling constant in RCH,CH,I is also known to be positive.‘83’ From the chemical standpoint the spectra in Fig. 5 lead us to the rather remarkable conclusion that the addition of an alkyne to the solution of an organolithium compound and alkyliodide specifically intercepts those radical pairs formed in the initial reaction between the two reagents. It has been proposed that this occurs because alkynes may form complexes with alkyllithium compounds and therefore be held near the first formed singlet-state radical pair.‘44’ The above rules may be considered absolute as long as (a) only two sets of equivalent nuclei are involved, (b) the radical-pair precursor fits neatly into one of the categories s, t or U, i.e. is not formed in an unequal mixture of s and t, (c) the effects are either pure multiplet effect (gl = g2) or pure net enhancement (J,, or A, = O), (d) the spectra are first order. Although the rules undoubtedly still sometimes hold even when these restrictions are relaxed, their range of validity has not yet been explored sufficiently to guarantee that misinterpretation will not result. When the rules are not applicable the signs of enhancement: the higher order effects discussed above and others as well, may be taken into account by numerical solution of the expressions for NMR intensities and frequencies, suitably modified to include changes in populations, in a manMer analogous to that now routinely applied to ordinary spectra. (84) Before discussing computer simulation, however, we must consider different schemes for averaging equation (24) over a distribution of re-encounter times, t.
3.6. Quantitative
Calculation
of CIDNP
Intensities
3.6.1. PROBABILITY OF DIFFUSIVE RE-ENCOUNTER OF PAIRS WITH CORRELATED SPINS.In
ELECTRON
Section 2.2.2 the opinion was expressed that at present there is considerable doubt about the proper time dependence of the function f (t)dt which describes the probability of a re-encounter of a pair taking place between t and t + dt. One can at least make a reasonable guess, however, that it will have the following properties: (a) After some short time after a pair separates f (t) will decrease monotonically with increasing time, (b) the functional dependence will be approximately t-“, (c) each degree of freedom for a random walk separation of the pair will make a contribution of l/2 to the power qc4’) (d) the number of degrees of freedom will lie somewhere between 3, for the three relative translational modes for the pair of radicals, and 9, which would include the addition of three rotational degrees of freedom for each radical, and therefore that n will lie between 3/2 and 9/2. Furthermore, it is probably valid to. define a characteristic “encounter lifetime”, zD, which is the time required for one diffusive step in the random walk, and before which ,f’(t) is small. We can also define the probability, p = j,“,,f’(t)dc that a pair
CHEMICALLY INDUCED DYNAMIC NUCLEAR POLARIZATION
will undergo a re-encounter approximation forf(t)
sometime between Q, and infinity. This gives us the following o
f(t) = 0 f(t)
173
= (n -
5D ‘< t
l)z;;‘pt_”
(31) 312 d
n d
912.
The parameters n, p and Q, are, unfortunately, all highly subjective gnd depend on how we define an encounter. In general an “encounter” will require that the pair be in close enough proximity and have the r.ight orientation of the two radicals for there to be a probability i that they react before separating. If the rate of rotation of the fragments is much faster than the reciprocal of the encounter lifetime, TV),the rotational degrees of freedom will not count in the random walk and one should find approximately
a t-3’2
dependence for f( t).
For a three-dimensional
random walk TVmay be approximately zc,
g
defined as
2 WD,,,
where dz is a mean square “encounter diameter” and Dre, is the diffusion coefficient for relative translational displacement of the two radical fragments.‘85’ For small molecules in relatively nonviscous liquids Dre, is approximately 5 x lo-’ cm’ set- ‘. Therefore an encounter diameter of 5 A would give a value of.T, = lo- ’ 1 sec. To calculate the observable value of 4 we must integrate the expression in equation (24) over all intervals T = t, - t,, using an appropriate form forf(z).’ 4 = IT
(32)
Wf(z)dz.
0
The integration is simplified if we recognize that typical values of 6, are 5 x lo8 rad set- ‘, while most scavenging and relaxation processes probably take place with characteristic lifetimes, ~~~longer than lo-’ sec. Under these conditions the sine-squared term in equation (24) undergoes many oscillations before scavenging takes place. It is therefore permissible to ignore the exponential in the integral. In the case of inverse 3/2 power re-encounter probability the averaging of equation (24) yields 4: = &[l
+ p(1 - ;1)] + L7t*(p/2)z#2[p. - (1 - Q&J+
= 1 - 4;.
(33)
The conclusion 4: = 1 - & is unaltered by the averaging, provided that scavenging product is formed before relaxation in the radical takes place. It should be pointed out that the dependence of the reaction probability on the square root of the mixing coefficient, 6,, is a peculiarity of the inverse 3/2 power dependence assumed forf(t). For example, if the unlikely assumption is made that the pairs disappear predominantly via the exponential decay in equation (24) the expression for I#$ analogous to equation (33) varies as the square of fi,,.(16,21*28,34) A ch aracteristic of the early stages of evolution of the radical-pair model was theinclusion of a non-zero exchange interaction J, for D pairs. Cl8,3s) This may also lead to a 6: dependence for J larger than the hyperfine splittings. It now appears, however, that virtually all known high-field CIDNP spectra may be adequately explained by a S$ dependence, (*) although the predicted intensities are in many cases quite similar for the two different functional forms of 6,. 3.6.2. COMPUTER SIMULATION OF CIDNP SPECTRA.The singlet-triplet mixing coefficient, 6,,, is a function of nuclear spin quantum numbers in a representation, x,,, consisting of
174
R. G. LAWLER
products of functions for individual spin Hamiltonian for a diamagnetic C. will be of the form
sets of equivalent nuclei [equation (12b)]. The nuclear product molecule, for example,.a combination product,
HC = 1 Vilf + C Jijlilj I
where vi coupling. generally states xn.
(34)
i
is the Larmor frequency of the nucleus i and Jji is the nuclear spiny spin In the case of non-first-order spectra. however, the spin functions z,, will not be eigenfunctions of H, and the transitions in C will involve mixtures of the If we write the eigenfunctions, xy, of H, as linear combinations of xn we have
The intensity,
I,.,, of a transition
between
two eigenstates
1’and p of Hc is given by (36)
where the resonance frequency of the spectrometer, v,, , satisfies the conditions v0 = v, - vp. The diagonal density matrix elements in the xy representation are, however, related to those in the x,, representation by
where U,, and (U-I),, are the elements of the unitary matrix which transforms xn into xy, as in equation (39, and its inverse respectively. The intensity of the transition is therefore related to the reaction efficiencies 4,. e.g. as given in equation (33), by
The transition probabilities (x,,lC I&,)’ and the matrix U are both calculated in the course of the usual numerical simulation of normal NMR spectra.@5’ These programs may be relatively easily modified for simulating high field CIDNP spectra by providing for weighting of the level populations as in equation (38) before the intensities are calculated. including magnetic equivalence The programs NMRIT”” and LAOCOON,‘57’ factoring, Cl*,l 9, have now been modified to perform such calculations. Consequently simulations are now beginning to appear in the literature as a standard part of reporting CIDNP results (5’,60,81,8’) An example of a simulated high-field CIDNP spectrum calculated by using equations (33) and (38) is given in Fig. 6 which shows the spectrum of the n-propyl sidechain of n-propylbenzene formed from the coupling of ethyl and benzyl radicals produced during the reaction of ethyllithium with benzylchloride.(88) Scheme XIII : C,H,CH,Cl
+ LiCH,CH,
----+
C,H,CH,
. . CH,CH,
-
C,H,CH,CH,CH,
A rather reliable computer simulation can be made in this case because all of the hyperfine splittings in the free radicals are known from ESR measurements(89) and the NMR spectrum of the product has been analysed completely to give chemical shifts and spin-spin splittings of the protons. (90,9 ‘) The small deviations which appear between the simulated and experimental spectra are probably due to a combination of effects such as variations of Tl from line to line, cross-relaxation between enhanced transitions. partial breakdown in the
CHEMICALLY +cH2
FIG. 6. Experimental n-propylbenzene
INDUCED
Cl t Li CeCH,
(upper) and simulated formed in the coupling
-
DYNAMIC
NUCLEAR
POLARIZATION
175
#GHz CHz CH,
(lower) CIDNP ‘H NMR spectrum (60 MHz) from reaction between benzylchloride and ethyllithium.
assumptions involved in deriving equation (33) and, perhaps most significantly, the fact that lines in different parts of the spectrum were of necessity scanned at different times during the chemical reaction. In fact, since all of these effects undoubtedly do occur to some extent in all CIDNP spectra, the agreement between calculated and observed spectra in this and other cases is quite remarkable.
4. Dynamic
Effects
4.1. Ej>cts of’Rela.uation
OH CIDNP
Intensities
(a) Tl the same for all lines in the spectrum. In Section 2.3 we discussed qualitatively the time dependence of the CIDNP signal. In the special case where all lines in a spectrum have the same T,, as they might for example if all nuclei relaxed by interaction with an added paramagnetic substance, the relative intensities of lines are unaffected by relaxation. If the overall rate constant for the chemical reaction, k,, is much smaller than T;’ the observed intensity is simply reduced by the factor k,T, as shown in equation (8b). If the two rate constants are more nearly of the same magnitude, however, the expressions for intensity as a function of time become more complicated. Buchachenko and coworkers”” ““ha ve derived extensive formulae for CIDNP intensities including relaxation.
176
K. G. LAWLER
Although some of their conclusions are independent of the mechanism by which the enhancement is produced, they have not incorporated the radical pair model into their discussions and their studies to date are of rather limited utility. Although it is possible, in principle, to analyze time-dependent CIDNP spectra to obtain both Ti and k, it is doubtful if either the precision, convenience or generality of the method can compare with the more conventional methods of obtaining these quantities. (b) A slowly relaxing nucleus coupled to a set of rapidly relaxing m&i. It has become generally well established that the relaxation time of a proton bound to a carbon atom containing no other magnetic nuclei (a methyne proton) is often an order of magnitude longer than if the carbon contains one or more additional protons which may relax by interaction. (lol) This means, for example, that for a the intramolecular dipole-dipole CH,CH or CH,CH group the nuclear spin levels which differ only in the spin state of the CH, or CH, protons will reach thermal equilibrium more rapidly than those which differ only in the spin of the methyne proton. The redistribution of populations for an AX, spin system exhibiting a multiplet effect and/or net enhancement and showing such selective relaxation of the X spins is illustrated in Fig. 7. The effects are as follows: (i) a pure multiplet effect disappears for both sets of nuclei and approximately with the relaxation time of the faster relaxing set, (ii) net enhancement also disappears for the fast relaxing set but is retained in the slower set, (iii) a spectrum exhibiting both multiplet and net effects relaxes with the faster relaxation time to give no enhancement of the fast relaxing nuclei and to destroy the multiplet effect components of the enhancement of the After
relaxation
of x
-“A
-“A
-“A +nA
-“A
+5
+“A
+nA
I
I
I
I
I
I
FIG. 7. Etfect on CIDNP intensities of fast relaxation of two nuclei. X2. coupled to a slow relaxing nucleus, A. Symbols for contributions to level populations given at the right of the energy levels to net effects, are: 11”= common to all levels z N,,,/8; m = multiplet effect: nA, )1Xare contributions assumed negative for A and positive for X. Pertinent quantum numbers for each level are given at left.
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
177
slow
set. The net enhancement of the slow set remains, however. Consequently, spectra corresponding to case (iii) appear with exaggerated net effects. Attempts to simulate such spectra when all the necessary g-factors and hyperfine splittings are known from ESR would therefore be expected to predict a larger amount of multiplet effect than observed. An example of such a failure of simulation is shown in Fig. 8 which gives a comparison of the expected and observed CIDNP spectrum of the methyne proton in C,H,C(CH,),H produced in the reaction”87’ in Scheme XIV.
FIG. 8. Disagreement (60 MHz) of methyne
between observed (A) and simulated (B) CIDNP proton NMR spectra proton septet in C,H,CH(CH,), brought about by destruction of multiplet effect by rapid relaxation of CH3 protons.
Scheme XIV : C,H,C(CH,),CHO
hv
C,H,C(CH&
. . HCO -
C,H,C(CH,),H
+ CO
In this example both the hyperfine splittings and g-factors necessary to simulate the spectrum of the product are known and lead to a simulation with an appreciable multiplet effect component in contrast to the experimental results. In cases where all of the hyperfine splittings and both g-factors are not known the above effect can lead to estimated g-factor differences which are too large.‘8’
178
R. G. LAWLER
In cases where only partial equilibration of the levels of the faster relaxing set of nuclei takes place, distortion (but not complete removal) of the multiplet effect is observed.‘“” Both steady state and transient relaxation effects in this intermediate region have been elegantly demonstrated recently by Muller and C10ss.‘~~’
4.2. Conrpc~titiori hetwwrl Scrrt’rn~~ir~yard Rr~bsrrtion of’ N uclcar Spins in Free Radicals The sequence of events by which enhancement shown in Scheme XV.
in a scavenging
Schcmc XV :
x”’
%
product,
S, is generated
is
T,
R-E-D-E-D-S-S, TO
- ?I
b
'c&, D
0
The times t, to t, are those discussed in Section 2.4.2 and it is assumed that both scavenging and relaxation are first-order kinetic processes describable by characteristic lifetimes. Using equations (25) and (8b) and the analogous relationship for the scavenging product, the observed population difference (S, - S,), for a transition between states n and tn of product S containing a set of nuclei which may also end up in’ a combination product under steady-state conditions is
6, - %a)= (UK =
-(cl
‘(4: - &,)T,,,,T, ‘/bi ’ + T&-1 + T&J - ctJ(Ts,,JG,,,)~S %G l + 4- l + ~i&?J.
(39a)
Wb)
In the special case where the rate of the scavenging reaction leading to S is much faster than either nuclear relaxation in the diffusively-separated pair, D, or competing reactions which give rise to other scavenging products, S’, corresponding transitions in S and the combination product, C, have relative intensities which are affected only by the relaxation times in the two kinds of diamagnetic products. If T, and Tc are the same, the CIDNP spectra of S and C will have opposite but equal intensities. Although these limiting cases are sometimes observed (see for example, Fig. 3) there are at least two reports of differences of intensities of CIDNP lines in scavenging and combination products which have been attributed to loss of intensity during the scavenging step through competition between the scavenging rate and the rate of relaxation in the intermediate radicals.(4h*57,58) Comparison was made of the intensity of net enhancement of protons in the products R-R’ and R-R formed according to Scheme XVI. Scheme XV 1: R-X-_-‘-R..
--
R’-R./.R’%rR
Aqzo1~-~’
-_R-_R
kd As=0
(see reference 46) R = . CHCl,, R’ = . CH,COCH, (see reference R = .CHZC,Hs, R’ = .CH(p-BrC,H,),
57)
In both cases the “scavenging” process was recombination of R. with a radical produced from another encounter, i.e. r; ’ = k,(R.). Under these conditions net enhancement is generated only in the non-identical pair and singlet-triplet mixing during the diffusive encounter between identical radicals, necessary to produce R-R products, may be ignored. Use of the simplified scheme above, which ignores other competing paths for loss of R.,
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
179
and estimation of k,(R.), leads to reasonable estimates of the order of 10e4 set for the proton relaxation time in benzyl and substituted methyl radicals. Relaxation has also been invoked by C10ss’~~) to explain variations in the relative CIDNP intensities of the ortho and aldehyde protons of p-chlorobenzaldehyde serving both as a scavenger and as a scavenging product .in the reaction sequence shown in Scheme XVII. Scheme XVII : p-CIC,H,CH,OH
+ “(I?-BrC,H,),C=O
&ClC~H,CH*OH-
-
,,_(.,$ 1(.Hot p-ClC,H,CH*O
.(p-BrC,H4)ZCOH
+ p-ClC6H4i3HOH
It was found that at high concentrations of added benzaldehyde the ratio of the aldehyde to ortho proton intensity in the benzaldehyde product approached that calculated for the radical-pair model as in equation (33). At lower concentrations, however, the (starred) aldehyde proton (which was benzylic in the radical) was of anomalously low relative intensity. This was attributed to competing nuclear relaxation in the hydroxybenzyl radical which is more effective for the benzylic than for the ring protons owing to the higher unpaired electron density at the former position. If TD for the benzylic proton was assumed to be 10m4 set the concentration dependence of the relative intensities was best fit using a value of 8 x lo4 M-’ set- ’ for the rate constant, kt,, for transfer of a hydrogen atom from the hydroxybenzyl radical to the corresponding aldehyde. This value is, however, very sensitive both to the estimate of TD and to possible changes in relative relaxation times of the two kinds of protons in the diamagnetic product brought about by changes in the medium. It has been found.““’ for example. that the addition of as little as 0.1 M molecular iodine as a scavenger during the decomposition of benzoyl peroxide causes up to a sixfold, but non-uniform, decrease in relaxation time in the various types of protons in reaction products.
4.3. scuucmjng
R~~~~ctims which cornprte with Rr-crJc~our7tc~r.s qJ’Rudictr1 Pairs
In the derivation of the nuclear-spin dependent combination efficiency given in equation (25) of Section 2.4.3 it was assumed that a D pair formed after the initialization step retains its identity during the time t, - t, between the first and second encounters of the pair. Consequently the nuclear-spin dependent mixing coefficient, 6,, remains constant between encounters. It is possible, however, to have a rapid reaction, or other rate process, which competes with the re-encounter frequency, changing the identity of one partner, but not destroying the pair. In such a case the new pair may undergo a re-encounter to form a new combination product, but one in which the unchanged radical partner has felt the influence of two mixing coefficients, 6, and Sk, and may “remember” its partner from the original encounter. The life history of such a pair is shown schematically in Scheme XVIII. g 7s’ Scheme XVIII : R-E-D
-D’--E’-C’
‘0
‘I t-t,+
‘1
‘2
180
R. G. LAWLER
Five such processes
are illustrated
below with reactions
where CIDNP
has been observed:
Scheme XIX : A. Fragmentation’102’ -
C,HsCO,O,CC,H,
-“Oi b C,H, . .CH,CH,
. CH,CH,
C,H,CO,
L C,H,CH,CH,
I C,H,CO,CH,CH, B. Rearrangement’io3’
(CH,),CHCIH,C(CH,), + LiC(CH,),
-
(CH,),CHCJH, . WH3)3
+ CH2=C(CH3)2
-
.CH2CH2CH=CH2.C(CH3),
(CH,),CHCH,Br CH3CH2CH=CH2 C. Conformation
change with exchange
of hyperfine
splittings”
2,
(A’ )
(A)
H\c-ci /H ,,’
‘R
D. “Pair substitution”‘102’ C,H,C02.
.C,H,
“‘HZ’H’K6H,C02.
E. Racemization
. CH,CH,
---+
C,H,CO,CH,CH,
+ C,H,I
1 C,H,CO,C,H, of pairs generated
from optically
active precursors(io4)
R
The expression analogous to equation (22~) for the probability of finding the pair D’ in the singlet state at time t, if it was formed suddenly for D at t; is (Appendix A.111) PF’~ n, n(t 1t’) = 2;’ If the mixing coefficients dependent recombination
r-“‘S((l
- 1.)~~ + [pr - (1 - A)pJ sin2[&t
- (6; - &)t’]).
(40)
for D and D’ are different the probability of an electron-spinoccurring at t, therefore depends not only on the time interval
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
181
POLARIZATION
t = t, - t, between encounters, but also on the difference t’ = t; - t, between the time the first encounter ended and the time D changed to D’. The latter time interval is, however, governed by the rate constant, k, for the pair reaction process. Thus the probability of reaction occurring between t’ and t’ + dt’ will be k exp( - kt’) dt’. The probability of combination product C’ in spin state n forming at the end of the second encounter then becomes, by analogy with equations (3 1) and 32),
42’ =
l’k
m ‘p&. ss0 0
(t,t’) e-““f(t)
dr’ dt.
(41)
If the rate constant k is less than both 6, and 6, -6: (typically lo8 set- ‘) but greater than the rate constant rs 1 for processes which scavenge both D and D’, the integral in equation (41), evaluated with the three-dimensional random walk expression for_/“(t)given in equation (3 l), becomes f#$’ = (X/2) pq
7c1’2 CO -4~s
+
PTIk”’ + CPT-(I - 4psl W/W, - 4Jl x
(42)
Thus the enhancement in the product C’ depends, as expected, on the mixing coefficients in both D and D’ pairs. The functional dependence is, however, more complicated than a simple weighted average of the mixing coefficients for the two kinds of pairs. This may lead to some rather striking intensity effects which deviate markedly from a simple superposition of net enhancement and multiplet effects even for first-order spectra in simple spin systems. Figure 9 shows the calculated and observed CH, quartet of ethylbenzene formed from benzoylpropionyl peroxide in the 2-step fragmentation process in Example A. In this particular example the difference in mixing coefficients, 6, - Sk,is nuclear-spin independent and reflects only the g-factor difference of the first pair. The rate constant k in equation (42) serves only as a scale factor for the intensity of each line in the spectrum of C’ and therefore does not affect the relative intensities of lines. It is in principle, however, possible to obtain a value for k by comparing the intensities of lines from C’ and those in a product C formed by combination of pairs which did not undergo the reaction but nevertheless underwent a reactive diffusive re-encounter. In Example A the product C is ethyl benzoate formed from the initially formed benzoyloxy-ethyl radical pair. Any such analysis of the relative intensities will inevitably, however, suffer from lack of information about the relative values of the parameters 1, p and zn for the D and D’ pairs. Nevertheless it is often very difficult to obtain even reasonable.estimates for rate constants in the range 10’ to lo9 see-’ which would be detected by this phenomenon. For example, the best estimate”05’ for the rate constant for decarboxylation of benzoyloxy radical under the conditions in Example A is about lo5 set- ‘. The mere appearance of a “memory effect” in ethyl benzene shown in Fig. 9 is good evidence that this estimate is at least two orders of magnitude too low. Examples B and C are in the same category as A, but have not yet been studied in
182
R. G. LAWLER
METHYLENE
GROUP
OBSERVED
OF ETHYL
BENZENE CALCULATED
FK. 9. Comparison of observed (A) and calculated (B) CIDNP proton NMR spectrum of -CH,quartet in ethylbenzene formed from decomposing C,H,CO,O,CCH,CH,.
(60 MHz)
sufficient detail to permit quantitative comparison between the experimental and calculated spectra. The “pair substitution” process given in Example D is invoked to explain cases in which combination products have been observed containing molecular fragments which were not in the original pair but which, by application of the rules in Section 3.5, must have been formed from singlet born rather than uncorrelated radical pairs. Pair substitution has also been invoked to explain the change in sign of the enhancement of HCCl, obtained during thermal decomposition of isobutyryl peroxide with high concentrations of added BrCCl,. Cl06) Kaptein (i2) has attempted to estimate the rate constant for the reaction Scheme XX: 2(CH,),CH.
Br’TA+ (CH,),CHBr
+ (CH,),CH.
.CCl,
-
HCCI,
by comparing the experimental dependence of the enhancement factor on BrCCl, concentration with that calculated using a model for the competition between diffusion and reaction similar to the above. Example E is included because of the observation of CIDNP from rearrangement reactions in which only partial racemization of an initially optically-active radical-pair precursor occurs. Cl04) The “rate constant” for the racemization process in this case, however, is simply the correlation time for rotational motion of the asymmetric radical. Since this is typically IO- ’ ’ set, rotation should be completely randomized before re-encounter of pairs which have lived long enough to develop appreciable nuclear spin dependent S-T mixing. This in turn leads to the prediction of equally intense CIDNP from the two
CHEMICALLYINDUCEDDYNAMICNUCLEARPOLARIZATION
183
enantiomers. This would, of course, be observable only if one could devise a suitably unsymmetrical environment, e.g. by the use of asymmetric paramagnetic shift reagents, to simultaneously observe the NMR spectrum of both species. Such an experiment has not yet been reported, but would provide a good opportunity for confirming the rather long times necessary for the production of CIDNP.
5. Variation of CIDNP Intensities with Magmetic Field 5.1. High Field .Eficts In one of the first reports of CIDNP by Bargon and Fischer,‘36’ it was stated that the intensity of the emission line from benzene formed from decomposing benzoyl peroxide depended on the strength of the magnetic field in which the measurements were carried out. At that time the field effect was interpreted to arise from the frequency dependence of relaxation times in the free phenyl radical and was taken as evidence in favour of the Overhauser mechanism for CIDNP. The fields used in this experiment, however, were only 10, 13 and 23 kilogauss because both the reaction and the NMR spectrum were run in the same field. In later work(81,107-1 lo) it h as been found that nuclear T1 in reaction products are sufficiently long that the reaction may be carried out in a variable field and the sample rapidly transferred to a spectrometer operating at constant field and still retain an appreciable portion of the CIDNP generated in the external field. This has greatlyincreased the ease with which the field dependence of CIDNP may be studied. 5.1.1.CHARACTERISTICSOFHIGHFIELDEFFECTS. Theabovefieldeffectonbenzeneemission and the results of subsequent investigations may now also be understood when viewed from the vantage point of the radical-pair model. In fields sufficiently high that the singlet state of the pair mixes only with the T, triplet state the variations of CIDNP intensities with field depends on the following qualitative factors: (a) Variations with magnetic field of enhancement generated in radical pairs occurs only if the radical pairs have different g-factors, i.e. if there is net enhancement. (b) The thermal polarization, p”, used to calculate the enhancement factor increases linearly with the field [equation (3)]. Therefore the enhancement factors for CIDNP spectra, such as the benzene emission spectrum above, in which the reaction and spectra are run in the same field, have this additional source of field dependence. In particular, even if the radical pair interactions produce no field dependent CIDNP, e.g. for identical radical partners (Ag = 0), the enhancement factor calculated by comparing the “normal” and CIDNP intensities at the same field will decrrasr with increasing field. (c) Since nuclear relaxation times and intensity borrowing may be field dependent, enhancement factors must be suitably corrected for these added sources of intensity variation with field.
5.12. EXAMPLL OF FIELD IXPI.SIWW~-.IN Tw: HIGH FIELD REGION: BENZENE FROM BENZOYL PI:KOXIIX.The
field dcpendcncc of the CIDNP spectrum of benzene formed from benzoyl
184
R. G. LAWLER
peroxide according to Scheme XXI has been studied extensively,by Fischer and Lehnig’lo8’ and by Slonim et al.” lo) Scheme XXI :
(C,H, CO,),
-
C6H5 CO,.
Since the ortho protons in the phenyl radical are most strongly coupled to the unpaired electron, the contribution of the other three protons to the enhanced spectrum may be ignored as a first-approximation. The net intensity of the tw.o transitions mortho= + 1 + 0 and 0 + - 1 of the two equivalent ortho protons as given by equation (33) then becomes 1 cc (4+1 - 4-i) K C(91 - S~W’~O
+ A? - C(s1 - SJWBO
- ‘a+
(43)
where A is the hyperfine coupling of the ortho protons and radical 1 is the phenyl radical. The intensity in equation (43) passes through a maximum which depends on both A and (gi - gz) and falls off at both high and low fields according to the. asymptotic relationships IAl
(gi - g&?, $ A(“high” field)
III @z
(gI - g&j
I11oc I91 - g$
e A(“low” field) By
I91 - gPf
= 2JA(gauss)l lgl - gJ_l.
IAl+
’ Wc)
In the present case g1 = 2.002,““’ g2 = 2.010,‘1’2’~ A = + 17 gauss.““’ According to equation (44~) ,r should occur at approximately 4000 gauss, in almost exact agreement with the experimental observation. (i lo) The functional dependence on field predicted by equations (44) in both the “high” and “low” field limits also agrees well with the results obtained by running the reaction in the variable field and observing in a 60 or 100 MHz spectrometer. In experiments where the spectrometer field is also varied, however, the enhancement factor is obtained by dividing equations (44a) and (44b) by B,. This leads to the prediction that the enhancement factor, measured, for example, by determining the amount of added unenhanced product needed to cancel a CIDNP emission signal,” l 3, should be independent of Ho when (gl - g2)Bo 6 A and fall off as BG” in the “high” field region. The latter field dependence predicts enhancement factors for benzene in the early work of Bargon and Fischerc3@ m the ratios 1: 2.4: 3.9 at 100,56.4 and 40 MHz respectively. These are in reasonable agreement with the observed ratios 1:2.6:6.8. 5.2. Low Field EfSects In the field region below approximately 1000 gauss the strength of the coupling of the unpaired electrons to the applied magnetic field becomes comparable in magnitude to the electron-nuclear hyperfine coupling. Under these circumstances the singlet electron-spin state of the radical pair mixes with all three of the triplet states and the mathematical simplification leading to equation (22) no longer occurs (see Appendix AIV.). Much of the t Assuming that the g-factors of C,H,CO, and D02CCH=CHC0,. are similar.
CHEMICALLYINDUCED
DYNAMICNUCLEARPOLARIZATION
18.5
qualitative understanding of CIDNP leading to the simple high field rules given in Section 3.5 is lost and it is necessary to resort to computer-assisted numerical calculation of nuclearspin state populations. (1g,3L,86)In the simple case of a radical pair with only one set of equivalent nuclei on one radical fragment and no nuclei on the other an analytical expression for &,&t) is m fact obtainable’86’ which is valid for all fields. It is, however, too complicated to be very useful in obtaining physical insight into the problem. As in the high field case, the shape of the field dependence in the majority of examples of low field CIDNP observed to date may apparently be simulated adequately using only hyperfine splittings and g-factors. Differences in g-factors, however, become unimportant in low fields. The parameters A, p and z,, which describe the reactivity and diffusive properties of radical pairs serve only as scaling factors in low field as well as in high. 5.2.1. SOMECHARACTERISTICS OF LOW FIELD CIDNP. The only practical way at present of determining the field dependence of CIDNP over a wide range of magnetic fields is to observe the spectrum at a constant high field, i.e. in a conventional NMR spectrometer, and run the reaction in a variable field. In order for the observed NMR spectrum to reflect the population differences built up in the auxiliary held, however, two conditions must be met: (a) The polarization must persist during the transfer of the sample between the two magnets. A sufficient condition for this is that the spin-lattice relaxation time must not be too much shorter than the time required for transfer. (b) The relative populations of the nuclear spin levels must not be changed during the time that the system spends in passing through the intermediate field between the variable magnet and the spectrometer. Requirement (a) has limited the observation of low field CIDNP to protons with 7” of several seconds or more. These have so far included benzene, CHCl, and the methyne protons in (CH,),CHI and C,H,CHClCH,CH,. Attempts to observe low field CIDNP from protons in methyl or methyiene groups have been generahy unsuccessful except when the enhancements are very large. There are three more important and subtle factors which determine as in (b) whether the process of transferring a sample between two magnets alters the relative populations of the nuclear spin states. Two of these are common to all spectra and are discussed below. The third pertains only to spectra exhibiting spin-spin multiplets and is discussed in the next section. (i) The relative enhancements of different lines observed in the spectrometer will reproduce accurately those in the variable field only if the relative Tl in the variable, intermediate and spectrometer fields are the same. This is a good approximation in nonspin-coupled systems for the usual relaxation mechanisms with correlation times of lo- lo to 10-l’ set because 7” are field independent except at very high fields.. A possible exception may, however, occur at very low fields if relaxation is dominated by a scalar relaxation process, such as chemical exchange or spin-spin splitting modulated by quadrupole relaxation, with a long characteristic time. (I 14, In multiplet spectra the troublesome effects of coupled relaxation discussed in Section 4.1 also complicate the interpretation of variable field CIDNP. (ii) In order for the populations in the external field to correlate directly with those for the sample in the spectrometer it is necessary that the nuclear spin system remain quantized along the instantaneous static field as the sample is transferred. This occurs
186
R. G. LAWLER
only if the length of time spent at any given field during the transfer is long compared to the Larmor frequency of the nuclei in the instantaneous field.““) In other words the adiabatic condition dB,/dt
=+yB;
(45)
must be satisfied. Since the lowest field usually experienced by the sample is of the order of the Earth’s field (0.5 gauss) in which the Larmor frequency for protons is about 2 kHz, the condition in equation (45) will be met if the time spent in this field region is longer than about 1 msec. This is clearly not a serious restriction considering that at least 1 set is required to transfer a sample by hand or in a practical flow system. The adiabatic condition also implies that the instantaneous field is followed by the system regardless of the field direction. Consequently, simply determining whether the observed CIDNP changes
Br
FIG. IO. Field dependence of CIDNP proton NMR signal from HCCI, formed in reaction between BrCCI, and t-butylmagnesium bromide. Solid line is computer simulation using known q-factor for ‘Ccl, andg-factor and proton hyperfine sphtting of (CH,),C. but ignoring Cl hyperfine splitting in Ccl,. Dashed line is simulation with Ag = 0.
with the relative orientation of the external and spectrometer way of verifying that the transfer is indeed adiabatic.
magnetic fields provides a
5.2.2. EXAMPLE OF LOW FIELD CIDNP FROM CHCl,. The arguments in (i) and (ii) above make it appear reasonable that the field dependence of a CIDNP line arising from a single set of equivalent nuclei should be insensitive to the rate of transfer of the sample between fields as long as the time required for transfer, and therefore the-degree of relaxation of the sample, is the same at all fields. An example of the agreement between the calculated and observed field dependence of the proton CIDNP line from HCCl, generated in the reaction in Scheme XXII is shown in Fig. 10.
CHEMICALLYINDUCEDDYNAMICNUCLEARPOLARIZATION
187
Scheme XXII : (CH,),CMgBr
+ BrCCl, -
(CH,),C:
. Ccl,
-
HCCl, + (CH,),C==CH,
This spectrum illustrates the features characteristic of low field CIDNP of a single NMR line arising from a set of equivalent nuclei: (a) The signal goes to zero, as expected, when the magnetic field in which the reaction is carried out, B,, is zero. (b) The enhancement changes sign in the region below about 100 gauss where B, is comparable to the nuclear hyperfine splitting. This change in sign does not occur, however, unless more than two equivalent protons are in the radical pair. (c) The sign of the enhancement at all fields depends on the multiplicity of the radical pair precursor and on whether the product molecule is formed in a combination or scavenging step (the parameters p and i respectively in the simple rules for high field CIDNP given in Section 3.5). (d) The low field enhancement is independent of the g-factor difference in the radical pair, and net enhancement is obtained in low field even when Ag = 0. The dotted line in Fig. 10 indicates the fall off of intensity expected if Ag were zero for the radical pair in Scheme XXII. (e) As the high field limit is approached the enhancement increases linearly with &, just as predicted in equation (44b). 5.2.3. Low FIELD MULTIPLET SPECTRA. A more restrictive effect than either (i) or (ii) above arises when a spin-coupled product molecule containing two or more nuclei with different chemical shifts is transferred through a region of magnetic field which varies rapidly with distance. Even though the spins remain individually quantized along the instantaneous field as required in (ii), the passage from a low field region where the nuclear coupling constant, J ABt is greater than the difference in Larmor frequency, (vA - v,), into a high field region where JAB < (vA - vB) will, if carried out slowly enough, lead to adiabatic transfer of the population of the lowest low field energy level to the lowest high field energy level with the same value of Z; + I;. This second kind of adiabaticity is more restrictive than the kind pointed out in (ii). It requires that the length of time spent in the “intermediate coupling” field region where J,, w (vA - ve) be longer than the reciprocal of JAB. For values of J,, of a few hertz this requires that passage through the region around 1000 gauss be accomplished in a few tenths of a second or longer. For smaller values of J,, the time for adiabatic transfer must be proportionately longer. The above effect will produce changes in the observed .spectrum only if the CIDNP populations of some levels with the same value of (I:, + 1;) = I’ are different from others. This condition is met when the field in which the reaction is run is much smaller than any of the electron-nuclear couplings in the radical. Under these conditions the Hamiltonian for a radical pair containing two magnetic nuclei A and B on one radical fragment and none on the other is simply H, = A,AS,‘I,
+ A,,&‘I,.
(46)
From the properties of coupled angular momenta it is easily shown”‘@ that this form of HD commutes with both the square of the total angular momentum of the system, F = S, + S, + IA + I, = S + 1. and with its component, F’. along the weak, static magnetic field. H, generally does not commute, however, with S2, S’, 1’ or I’. Consequently,
188
R. G. LAWLER
the singlet-triplet mixing of the electrons induced by H, and responsible for CIDNP must be accompanied by a simultaneous change in I in such a way that F remains constant. Therefore in products formed in very low fields the nuclear spin states with the same value of I’ but different values of Z will be affected differently by singlet-triplet mixing and will therefore have different populations.
b”e - b$,Q,,
I
I
I
I
FIG. 11. Energy levels, relative populations (indicated by widths of levels), and high field spectra for adiabatic (A)and sudden (B) transfer of two coupled protons from low to high field. It is assumed that the 1 = 1 states are preferentially populated in low field and that JAB > 0.
The above effect is illustrated by the energy level diagram in Fig. 11 for the case of a radical containing two non-equivalent protons for which I = 0 or 1 in zero field. It has been assumed that the I = 1 state of the product molecule was populated by the chemical reaction to a greater extent than the Z = 0 state. Furthermore, at zero field all three of the Z = 1 states must have the same population. Adiabatic passage from low to high field (Fig. 11~) produces equal and opposite enhancement only of the outer two lines of the fourline multiplet of the AB spin system in the product while sudden passage (Fig 11~) would produce enhancement of all four lines. Figure 12 shows the NMR spectrum obtained formed from thermal decomposiexperimentally (l 09)for 1-bromo-4-chloro-benzene-2,3-d, tion (Scheme XXIII) of 4-chloro-dibenzoylperoxide-2,3-d,.
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
189
POLARIZATION
Scheme XXIII :
The experimental to high field.
result is thus in complete
agreement
with adiabatic
passage
from low
A
FIG. 12. Proton
NMR spectrum at 60 MHz of p-bromochlorobenzene-2,3-d, after decomposition of the peroxide and (B) several minutes
taken immediately later.“04’
(A)
The appearance of spin-spin multiplets in AX, proton spectra of products formed in zero field and observed in high field was first discussed by Glarum.” ‘) He has pointed out that (a) the 12+ 1 lines normally exhibited by the A proton should reduce to n lines, (b) the highest field line of the low field multiplet and the lowest field line of the high field multiplet vanish, and (c) the remaming lines should have approximately the binomial intensities expected for coupling to n +_ 1 protons. The spectrum of the two protons in Fig. 12 obviously obeys this rule. An additional example is given in Fig. 13 which shows the
190
R. G. LAWLER
multiplet obtained in Scheme XXIV.
from the methyne
proton
of 2-iodopropane
enhanced
by the reaction”
07)
Scheme XXIV: I 2i-C,H,I
ZR-
2j-CsH,.
3
___ 2i-CaH;
i-c&i+
CH;_C_CH’; I!I*
when the reaction is run in the spectrometer (Fig. 13~) and in nearly zero field (Fig. 13~). The EA multiplet effect observed in the high field reaction thus goes over in low field to a net effect with only six of the expected seven lines of the methyne proton multiplet.
I
Me-b-Me
J
5
I
4 mm
FIG. 13. NMR spectrum of the methyne proton of 2-iodopropane benzoyl peroxide when reaction is run in a 60 MHz spectrometer
The above “n and breaks down are present, e.g. in from the benzylic xxv.
in the presence of decomposing (A) and in the Earth’s field (B).
1 multiplet effect” in zero field was derived only for AX,, spin systems either in the A,X, case or when more than two sets of coupled protons the AM,X, case. This is illustrated in Fig. 14 by the three-line multiplet proton in C,H,CHClCH,CH, formed in the reaction in Scheme
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
191
Scheme XXV : C,H,CHCl,
+ LiCH,CH,
-
C,H,CHCl.
.CH,CH,
-
C,H,-CH*ClCHyCH:
Since the 3-line multiplet arises only from the AM, subsystem of the six-proton sidechain, the n - 1 rule would predict enhancement only of the two lowest field lines. The observed spectrum shows, however, that the presence of the methyl group apparently affects the correlation of low-field and high-field energy levels in such a way that all three lines appear enhanced. The high field simulated and obscrvcd spectra are shown for comparison.
Cs t-I5 CHClCHzCH3
Obs.
-&em.BR= 14.1OOG
CdC. +H
-CHF
5-o
4-o
-a-l3 1.
I
3-o
20
10
....,
0
FIG. 14. Calculated and observed proton NMR spectra at 60 MHz from I-chloro-I-phenylpropane produced in the reaction ofr,r-dichlorotoluene with ethyllithium. The upper spectrum was obtained when the reagents were mixed in the NMR probe and the lower spectrum of the methyne proton (d = 4.6) was recorded after the reagents were mixed outside of the probe and transferred to it after the reaction was complete. Spin-lattice relaxation of the CH, and CH, protons destroyed the low field enhancement for these groups before the spectrum could be scanned.
In the above case it is, nevertheless, possible to simulate both the low-field CIDNP population scheme and the effects of adiabatic passage by using a digital computer both to solve the complete equations (Appendix A) for the populations and to calculate spin energy levels in the product as a function of field. This approach has been discussed in detail recently by Kaptein and den Hollander’19’ and has also been employed by Evans@‘j’ to simulate the above spectrum (Fig. 14).
R. G. LAWLER
192
5.2.4. QUALITATIVE RULES FOR LOW FIELD CIDNP. It would clearly be desirable to have a set of qualitative rules relating the observed sign of enhancement of low field lines to the initial electron spin multiplicity of the radical pair, signs of hyperfine splittings, etc., in the same way as the high field rules discussed earlier. Kaptein has, in fact, proposed a modification of the high field rule for multiplet effects l-k
= ~AAARJABflTAH + gives EA
- gives AE
that may be applied to the zero field enhancement of two sets of coupled protons if the symbol EA is taken to mean emission for the lowest field multiplet and enhanced absorption for the high field multiplet. (i7) Application of the rule to the low field spectra presented here for Schemes XXIII-XXV is presented below. .-. /I D
Br
Cl
\
H
4,
II,
J hll
c \I,
_
+
+
+
+
1.
+
_ ._
+ _
+ +
+ _
E
D
/
u-
Zero field enhanceInlent of proton A
6
(Fig. 12)
H
CH,CHAICH; C,H,CHAC‘IC‘H~CH,
(Fig. 131
+
(Fig. 14)
t _
Even though all three of these examples agree with the prediction of the rule, the rule itself has apparently not yet been proven for zero field in the same sense as was done in Section 3.4 for its high field counterpart. Until this is accomplished and its range of applicability has been explored further, the rule should be used cautiously in treating any of the six parameters (usually p, the parameter for electron-spin multiplicity) as unknowns to be evaluated from the observed sign of the enhancement.
6. Effects arising from Electron-Electron
Interaction in Radical Pairs
Until now we have been concerned with CIDNP effects which were explainable in terms of magnetic coupling parameters of isolated free radicals. In particular, neither the high or low field effects presented so far have required explicit consideration of the exchange interaction between the two unpaired electrons which is turned on in an encounter pair: H, = -J(l/2
+ 2s, .S,).
(19)
We will now discuss three kinds of effects which seem to come from radical pairs in which this interaction must be taken into account. In general the exchange integral, J, for a pair of radicals is a rapidly-decreasing function of the distance between the two species. (70,7’) Figure 15 shows a schematic representation of the way the energy of separation, 25, between the singlet and triplet states of a two-electron bond varies with distance. The blown-up region of the diagram shows the energy levels when the pair is in a magnetic field, contains one proton and
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
193
FIG. 15. Energy levels for two unpaired electrons and one I = I/2 nucleus in a magnetic field. The upper diagram shows the variation of singlet and triplet state energies with distance. The lower diagram is an expanded view of the region of fixed r where J - u)‘,.
has a value of J between 0 and the order of the electron Larmor frequency 0,. In the case where J = 0 the So and r, states are degenerate and the pair behaves as already described at length in our previous discussion of high field CIDNP. As long as o, is large compared to both J and the strength of the singlet-triplet mixing (of the order of the hyperfine splitting, lo* rad set-‘), the T+ and T_ states are effectively isolated from the other two states. When o, -c A, i.e. in low fields, all four states mix as we have seen, even when J = 0. In the case where 25 _ o,, however, the S, and T_ levels become nearly degenerate and S, mixes more strongly with T_ than with To or T+. The difference between S, - To and S, - T_ mixing is that the selection rule for the former is Am, = 0, Am, = 0, i.e. no spins flip, while the latter is accompanied by simultaneous electron and nuclear spin flips (Am, = f 1, Am, = f 1). This is produced by the off-diagonal part of the hyperfine interaction in equation (21). (These terms were, of course, included in the low field calculations already discussed, but in that case J = 0 and they produced essentially equal mixing of S, with T_ and T+ .) As a consequence of this selective exchange of angular momentum between the electron and nuclear spin systems the total numbers of spin up and spin down nuclei in the sample changes and a true nuclear polarization results. The result is similar
194
R. G. LAWLER
to that obtained in the Overhauser effect where combined electron and nuclear spin flips transfer thermal polarization from the electron-spin system to the nuclei.” ’ 7, 6.1. CID N P,from lntramolrcular
Biradicals
There is at present only one rather special example of the above model in which J remains roughly constant and of the order of magnitude of w, for a sufficiently long time for nuclear spin induced S, - T mixing to occur. This arises during the photolysis of compounds capable of forming “intramolecular radical pairs”, or biradicals, in which the portions of the molecules containing the two unpaired electrons are never able to diffuse very far apart. Gloss has found, for example, that irradiation of an alkyl-o-benzoyl-benzoate (Scheme XXVI) at high field in the probe of the spectrometer Scheme XXVI : G-k co
hv CO&H,
0
W’5
=d-b CH,CH(CH,),
HO ,
*COH CO2 CHZ CHZ i: (CH&--
I
1; 0
CO2
) / P Cyclization
’ /
C,H, \I
HCOH 0,
CH,
CH-C(CH,),
1: o-’
produces an intense CIDNP spectrum which exhibits emission for all of the lines of the cyclization and disproportionation products formed from the intermediate biradical.‘38’ In Fig. 16 is presented a qualitative energy level scheme which shows how emission arises from T_ - S, mixing in a biradical born in the triplet state. It is significant that the rate of S - T mixing varies as the absolute value of the mixing coefficient, H_ +,s_, and the population difference between + and - nuclear spin states is, unlike S, - TO mixing, independent of the sign of the hyperfine splitting A. Consequently, all NMR lines in biradical products will occur in emission provided that (a) the biradical is produced initially in the triplet state and (b) the triplet state of the biradical lies above the singlet. Condition (a) is, in addition, probably a necessary condition for even seeing CIDNP via biradical intermediates because singlet biradicals are likely to cyclize or disproportionate before sufficient time has elapsed for the necessary nuclear spin flips to take place. If the triplet energy were lower than the singlet [i.e. J > 0 in equation (19)], as is, in fact, usual for excited electronic S and T states, S, would mix most strongly with the T+ state and enhanced absorption of all lines would result. This has, however, so far not been observed. Kaptein has recently shown (’ ’ 8, that S, - T_ mixing may produce CIDNP of carbon- 13 nuclei, again completely in emission, during the photolysis of cyclic ketones. Scheme XXVII :
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
19.5
It is interesting that in this example the starting ketone also exhibits CIDNP, thus demonstrating the significance of the biradical cyclization as an effective pathGay for inter-system crossing of the photo-excited triplet to the ground singlet state.
hv
(triplet excitation)
/
FIG. 16. Schematic representation of the origin of a proton emission line in both scavenging and combination products when S,, - T- mixing dominates in a triplet-born radical pair or biradical.
S, - T_ mixing has also been invoked by C10ss’~~’ to explain obtained by Berson rt L/I.‘39’ from the dimers of two intermediate ground states.
6.2. Low Field CIDNPfiom
S -
the emission spectrum biradicals with triplet
Ti Mixing
our discussion of high field CIDNP it was pointed out that the enhancement expected for a combination product is opposite in sign to that of a scavenging product derived from the same radical pair. In the limiting case where both products were the same the contributions of each would cancel and no enhancement would be observed. Another way of stating this result is that every nuclear spin state of the reactant would have a unit probability of ending up in the product, so that no nuclear level population differences would arise. Garst rt al.‘56) have reported what is apparently an example of just such an effect in the formation of propane from the reaction of 2-iodopropane with naphthalene radical ion, N I, in the presence of a weakly acidic proton donor, RH. In
196
R. G. LAWLER
Scheme XXVIII : ICH(CH,),
J%
I- + N + . CH(CH,),
-&+ NL .CH(CH,), e”CO”“tel
3 2
3
H,C(CH,),
Virtually all of the 2-iodopropane is converted to propane via the 2-propylanion. As expected, therefore, no CIDNP is observed from the propane protons in high field because of the cancellation of the effects of the combination and scavenging steps.” ’ 9, Remarkably, however, rather intense emission lines are observed from both sets of protons in the product when the reaction is carried out in a magnetic field of approximately 100 gauss. The observed effect is reminiscent of that previously discussed for triplet-born intramolecular biradicals. The emission lines are consistent with the postulated triplet-like diffusive encounter pair and dominant S, - T- mixing. The illustration in Fig. 16 shows that under these conditions a proton would appear as emission in both combination and scavenging products. The actual physical situation in the above low field case is, of course, a good deal more complicated than that for the intramolecular biradicals. In low field o, and the hyperfine splittings are of the same order of magnitude and all four of the electron spin levels mix, but with some preference for S, - T- mixing. Furthermore, since a T- level may mix with any SOlevel in which the nuclear spin state differs by one unit of angular momentum, ihe mathematical problem generally involves mixing of more than two levels at a time. Adriancz6’ has attempted to treat the problem of low field S, - T+ mixing for a one-proton radical pair quantitatively, but was forded to consider mixing of only two levels at a time, a result which is strictly valid only for very short radical-pair lifetimes. His treatment of the problem does, however, employ the physically realistic assumption that H, is turned on only during an encounter. The computer simulations of Garst et ~1.~~~)for a one-proton pair confirm the qualitative idea that S, - T_ mixing requires a non-zero exchange interaction, but treats J unrealistically as a constant throughout the lifetime of a radical pair. It should also be emphasized that the low field CIDNP from S, - T_ mixing will generally be superimposed on that arising from pairs in which there is no exchange interaction. Although it is tempting to treat these two contributions to low field CIDNP as separable, there seems at present little theoretical justification for doing so. In limiting cases, such as Garst’s or that of intramolecular biradicals, where only the results of S, - Tt mixing are observed, one may, however, write a qualitative rule for the net enhancement expected from a set of protons: r;, = pj
+ gives A
- gives E.
The exchange integral, J, has the following property, J=
+ when triplet state lies below singlet r -when
singlet below triplet (usual case)
and p, as before, is positive when the radical pair is triplet-born or results from a diffusive encounter and negative if it is singlet born. Note that the sign of enhancement is the same for combination and scavenging products and is independent of the sign of the hyperfine splitting, for reasons already mentioned.
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
197
6.3. Chemically Induced Dynamic Electron Polarization (CIDEP) In 1963 Fessenden and Schuler,‘89’ m . the report of their classic investigation of the ESR spectra of transient radicals formed during radiolysis of hydrocarbons, showed a spectrum of hydrogen atoms in which the low field line of the doublet was in emission. No explanation was offered at that time for this remarkable observation and it was largely ignored until the advent of CIDNP. Since that time several more examples ofemission and enhanced absorption in ESR have been reported. These are summarized in Table 1. In all of these cases the effect is observed only while the sample is being rather violently disturbed, almost always by radiolysis or photolysis, and the lifetimes oft the radicals themselves are short. It therefore seems proper to call it chemically induced dynamic electron polarization (CIDEP). Fischerc3’ was the first to point out a possible connection between CIDEP and CIDNP, but Kaptein and Oosterhoff (l 5, first provided a satisfactory explanation in terms of the Radical-Pair Theory. Subsequently, Glarum’120’ and Fischer’27’ offered a modified explanation based on considerations of level crossing and Atkins et al.“21) extended somewhat the ideas of Kaptein and Oosterhoff. As in the case of CIDNP, however, it was Adrian”?2’ who recently supplied the most physically-realistic explanation of CIDEP by incorporating the proper diffusive characteristics of radical pairs into the model. The essential feature of his model is that CIDEP arises only in a radical which has undergone at least one unreactive re-encounter with its partner, but one in which the exchange interaction, J, is not zero. The sequence of events leading to CIDEP in a radical Ry in nuclear spin state n is shown in Scheme XXIX. Scheme XXIX : (D
(D)
R:--X-RR,k,_R;..R,-R;./.R,-R:.
CD) .R,-
(JZO)
'0
‘I
:. 0
Te
(E)
‘2
N, .
+
\ Ts
R,
R; X
‘3
This is identical to the scenario in Scheme VII which was employed in the derivation of the equations (22H24) for the reaction probabilities for high field CIDNP. Since virtually all CIDEP so far reported has been obtained in fields of about 3OtMgauss we need again consider only S - To mixing. The experimental observable in CIDEP is the expectation value of the z-component of the spin ofelectron 1 when the nuclear spins are in state n. We assume that the measurement takes place at time t, after the radical has escaped from the re-encounter which began at t,. That is, (S;(r3L
= Tr p%)&
= &~,0n(t3)
+ &n,m(t3)1 = Re ~~,On(h)~
(47)
It is easily shown, (l 22, however that PE,0n(r3) = PE,0n(t2)e-i(H~-H~O)(f3--fZ) = pkeo,(tz) e-2iJTE
(48)
where J is the average exchange interaction during the encounter and rE is the time during which it operates. The off-diagonal matrix element P:,~.(!~) may, however, be evaluated (Appendix A.1) in exactly the same way as the diagonal elements were evaluated in
a
/‘--\
OH
in butane
HO,CCH(OI’I)C.H(OH)CO,H
peroxide
+ I”,, HzOr
of alcohols.
of 2chlorobenzaldehyde in liquid paraffin of solutions of K. Rb and Cs in 1.2-dimethoxyethane of benLophenone at 4K
of aqueous
Photolysis
Photolysis Photolysis Photolysis
of di-r-butyl
Photolysis
solutions
of production
of bcnrene
(L’H,)CHOH
+ CH,CH~CO~H
of aqueous
solutions
of cyclopentanc
of aqueous
Ti(II1) + HL02
Photolysis
Radiolysis
Pulsed radiolysis
Electron radiolysis of hydrocarbons Pulsed radiolysis of H,O and DZO Steady electron irradiation of aqueous acids and esters
Method
of CIDEP
observations
t E.4 refers to hyperfine lines on low field side of ccntre in emission. high field in enhanced absorption I,” R. W. F-I.SSLNIXNand R. H. SCHLLLK, J. C‘hrrr~.Phy.t. 39, 2 147 (1963). lb) B. SMALLER, E. C. AVERY and J. R. REMKO, ibid. 55, 2414 (1971). (‘I E. EIBEN and R. W. FESSENDEN,J. PhJa. Chrm. 75, 1186(197 I). N) P. NETA, R. W. FESSENDENand R. H. SCHULER, ibid. 75, 1654 (1971). (<) B. SMALLER,J. R. REMKO and E. C. AVERY, J. Che,n. Phys. 48,5174 (1968). If) R. LIVINGSTONand H. ZELDES, J. Amer. Chrk. Sot. 88, 4333 (1966). (a) H. PAUL and H. FISCHER, Z. Naturforsch. 25a, 443 (1970). (hi P. J. KRUSIC and J. K. KOCHI, J. Amu. Chu. Sot. 90, 7155 (1968). “i R. LIVINGSTONand H. ZELDES, J. Chem. Phys. 53, 1406 (1970). tJ) P. W. ATKINS, I. C. BUCHANAN, R. C. CURD, K. A. M~LAUCHLAN and A. F. SIMPSON. Chum. Comm. 513 (1970) “I S. H. GLARUM and J. H. MARSHALL, J. Chrrn. Phys. 52, 5555 (1970). “I W. S. VCEMANand J. H. VAN IXR WAALS. Chew Phys. Lrttw$, 7, 65 (1970).
2-CICeH&HOH Solvated electron H. (trapped in solid benzophenone)
!.H(oH)
HOzCCH(OH)
CH&HCOO-
.CH,CH,COO-
(CH,),COH
\\_,
I
H
0 .
H., D. H.. D. H.
Radical
TAI~L~ I. Summary
E E
t
IL l (tentative, detected by monitoring benrophenonc phosphorescence intensity as a function of magnetic tield)
.
E
E,4 (very weak)
EA
EA (very weak)
E 4 (weak)
E :I
EA EA EA
Type of enhancement
J k I
h
d
e
a b cd
Ref.
F
p’ E
F
x
z X
CHEMICALLY
199
INDUCED DYNAMIC NUCLEAR POLARIZATION
the presentation of high field CIDNP. Substitution and averaging as before over a distribution of times between the first and secdnd diffusive encounters yields equation (49). (S;),
= 2p(7rr,)+(sin
[(i - Il)p, - pT]f$
252,) Z;’
16.1”
(49)
n
Z, is the total number of nuclear spin states for radical 1, radical 2 is assumed to have no magnetic nuclei, (sin 252,) is an average over all encounter pairs, and all other quantities have the same meaning as in equation (33) for CIDNP. The observed value, ST,, for the electron magnetization will also be influenced by relaxation and scavenging processes. In fact, if the processes producing (S;), occur much more rapidly than either relaxation or scavenging (i.e. t, < T,, us) we have dS;, ~ = -&J’s;, dt
- T&S;,
- s;;p)
(50)
where si;p = P,o[l - &(2 P, is the probability reacting and
that
radicals
1 and
- ;*)I (>-rrs = p;p@r-*‘rS 2 will escape
from
their
(5la) encounters
without
P,” = z, 1 (gB B,/2kT)
(51b)
is the thermal electron polarization for nuclear spin state II. If we let S;,(O) be approximately (S;),, the time-dependent solution to equation (50) is S;,(t) = pfp,e-f/Ts The ESR enhancement
+ [(Sl),
- p,op,] ,-(rs-‘+T,‘)f.
(52)
V:(t), then becomes
factor Vi(t) =
s;,w-
%B)
sf;p(t)
=
[%
-
1]e-tlT1c
(53)
The above transient decay of enhancement has in fact been observed by Smaller et al.,” 23) Atkins et .1.“24’ and Glarum and Marshall. (12’) This is certainly the most sensitive way to detect CIDEP, and the only one where large enhancements have been seen, since the effects disappear by relaxation with r,, - 10m5 to lop6 sec. Under steady-state conditions, such as employed in the majority of other reports of CIDEP, V!(t) must be integrated over a distiibution of radical lifetimes using a weighting function ZS ’ exp( - t/r,) dt. This yields v: = 7;’
[T;' +
T,']-'
If zs B T,,, as may often be the case, the steady state enhancement fraction of the maximum transient enhancement. 6.3.1. QUALITATIVE ESR
line to equation
gyric ratio requires
RULES FOR CIDEP.
In order
to relate
(49) we must realize that the negative that I ;SR cc ( - s&J.
(54)
v:(o).
may be only a small
the observed
intensity
sign of the electron
of an
magneto-
(55)
200
R. G. LAWLER
We also recall that for a radical pair with only one set of.equivalent 6” = i[(g1 - g2)pc1Bo
nuclei on radical 1
+ Am].
Furthermore, if A is positive, the ESR hyperfine lines with positive values of m occur on the low field side of the spectrum. Referring to equation (49) we may therefore derive the following qualitative rules for CIDEP: Net enhancement (occurs only when (gr - gz) 9 Am) ri, = ,&(g, - $,)
+ gives A
- gives E
where the signs of p, J and (gr - gJ are determined as described previously. As an example we consider the radical 2-ClC,H,CHOH reported by Atkins et al.‘124’ and most likely formed according to Scheme XXX. Scheme XXX : 2-ClC,H,COH
--%
0 I
2-ClC,H,-C-H
(triplet)
“ky’-H )
OH
t
2-ClC,H,CH(OH).alkyl Yl ’
-
I
2-Cl,H,C-H
92
If J < 0 during the re-encounter of the pair, then we see that ri, = + - + and emission is predicted (and observed). Multiplet efSect (for the special case of one set of nuclei and Ag = 0) rz, = PJA
+ gives EA
- gives AE.
The simplest example of a pure multiplet CIDEP effect is the EA spectrum observed for atomic hydrogen and deuterium under radiolysis conditions-(see Table 1). The hyperfine splitting for both atoms is positive. If we assume that J < 0 for any pair involving these atoms, then the observed EA effect requires that ,LL be positive, i.e. that H. is either formed from a dissociated triplet excited state or, as seems more likely, undergoes a diffusive encounter with another radical with a similar g-factor. Unfortunately, CIDE,P spectra do not usually fit cleanly into either of the two limiting categories above. In fact, equation (49) is much less successful’12” in predicting quantitatively the relative intensities of lines within an ESR spectrum than the corresponding equation (33) is in predicting CIDNP intensities. Part of the problem undoubtedly lies in the greater importance of relaxation during observation of CIDEP. It is also possible, however, that the approximate physical model behind equation (49) is not a completely adequate description of the phenomenon. (241,243)Additional and better,-albeit labo.riously obtained, data appear to be needed in the field of CIDEP even more than in CIDNP.
7. Guide to the Literature of CIDNP In the 6 years since the discovery of CIDNP more than 200 publications dealing directly with the phenomenon have appeared. We have so far quoted those references which have
CHEMICALLY INDUCED DYNAMIC NUCLEAR POLARIZATION
201
been most concerned with establishing the mechanisms producing CIDNP and illustrating its scope and utility. In the hope of making this wealth of information more readily available to the uninitiated, we present below the complete list of CIDNP and CIDEP publications up to December 1972. These are organized into six broad categories and a number of sub-categories. The order of listing within a given category is the approximate order in which the papers were listed as received by the journals. 1. REVIEWSANDCONFERENCEREPORTS References: 3, 96, 125, 141, 9, 14, 126, 142, 127, 10, 136, 13, 8, 99, 117, 7, 240, 246. 2. TYPESOFREAGENTSANDREACTIONSPRODUCING
CIDNP
A. Peroxides a. Benzoyl peroxide References: 1, 128, 129, 92, 130, 131, 107, 132, 81, 95, 133, 134, 135, 46, 76, 109, 102, 137, 138, 139, 140, 219. b. Other acyl peroxides References: 41, 92, 130, 132, 94, 143, 144, 97, 145, 146, 9, 106, 42, 147, 148, 149, 238. c. Other peroxides References: 150, 151, 152. B. Nitrogen-eliminating Compounds a. Azoalkanes References: 128, 74, 153, 154, 55, 57; 155, 156, 245. b. Diazonium salts References: 157, 158, 159, 160. c. Other nitrogen-eliminating compounds References: 162, 163, 134, 164, 165, 166, 167, 168. C. Organometallic Compounds a. Alkyllithiums References: 2, 169, 170, 4, 5, 171, 172, 173, 107, 174, 77, 44, 227. b. Grignard reagents References: 175, 176, 177. c. Organomercurials References : 178, 179. d. Radical anions References: 56, 119, 180. e. Alkylsodiums Reference : 181. f. Alkyllead Reference : 182 D. Rearrangements (R-X-Y R.X-Y. X-Y-R) References: 183, 184, 185, 186, 187, 188, 189, 190, 191, 104, 192, 193, 194, 195, 196, 197, 198, 199,200,201,202, 203,204,205.
202
R. G. LAWLkR
E. Divalent Carbon Species, RR’C References: 73, 74, 153, 206, 57, 167, 207, 208, 209, 210, 250 F. Photochemical Reactions (photo-CIDNP) a. Carbonyl compounds References: 211, 45, 212, 213, 58, 75, 76, 214, 215, 33, 60, 216, 217, 78, 218, 38, 220, 221, 61, 34, 222, 87, 251, 254, 242. b. Other compounds References: 73, 206, 57, 167, 2 IO, 223, 255, 257. 3. THEORY References: 36, 37, 173, 93, 20, 74, 15, 16, 29, 30, 27, 21, 24, 28, 25, 22, 32, 17, 26, 119, 23, 109, 108, 35, 61, 225, 14. 117. 100, 226. 18. 19, 252, 253. 4. Low FIELD EFFECTS References:
107, 56, 109, 110, 108, 119, 31, Il. 19. 244.
5. CIDNP FROMNUCLEIOTHERTHAN‘H A. 19F References:
128, 227,209, 180, 248, 256.
B. 13C References: 228, 54, 229,230, 12, 118, 247, c. 3’P References : 23 1, 249. D. 2H. References : 12, 238. E. 15N Reference : 230. 6. CHEMICALLY INDUCED
DYNAMICELECTRONPOLARIZATION
A. Experimental References: 89, 123, 232, 124, 120, 233, 234, 235, 236, 258, 259. B. Theoretical References: 15, 120,27, 121, 122, 108, 237, 241,243.
8. Conclusions
In the preceding pages an attempt has been made to present a unified version of the theoretical basis which has recently been developed for CIDNP as well as discussing the phenomena which it is capable of explaining. Although the theory has certainly been
CHEMICALLY
INDUCED DYNAMIC NUCLEAR POLARIZATION
203
successful in qualitative terms, it remains to be seen whether it will stand up to more quantitative tests. A major stumbling block in this endeavour is sure to be the adequately accounting for relaxation effects. Conversely, almost no use has been made of CIDNP as a way of preparing a nuclear spin system with non-equilibrium populations for studying relaxation effects. With a careful choice of the free radical precursor of a product of interest, however, it should be possible to drive specific sets of energy levels to an extent not attainable with radiofrequency fields in ordinary multiple resonance methods. There is the added advantage that CIDNP samples are usually automatically deoxygenated by the radical reaction. This eliminates the often appreciable intermolecular relaxation effects of dissolved oxygen. Multiple resonance on samples exhibiting CIDNP may, nevertheless, be employed both as spin of spectra and as a way of probing the CIDNP enhancedecoupling (54) for simplification ment of specific transitions when single resonance spectra are severely perturbed by relaxation effects.‘61’ Multiple quantum transitions in samples exhibiting CIDNP do not appear to have been reported, but their intensities should be predictable by the radical-pair model provided that only the populations of the nuclear spin levels are perturbed by the chemical reaction. It can be expected that the amount of CIDNP data for nuclei other than protons will increase markedly in the near future. Hyperfine splittings for the less abundant nuclei in reactive radicals should thus be easier to obtain by NMR, via CIDNP, than by ESR since the latter generally requires isotopically enriched samples. The wide availability of pulsed Fourier transform spectrometers should make ’ 3C CIDNP especially attractive. Proton CIDNP spectra may also be obtained up to two orders of magnitude more rapidly using pulsed methods. This will make it possible to accurately follow the course of even very rapid CIDNP-producing reactions. One can also expect increased use of on-line data processing by small computers in applications such as automatically subtracting unwanted, unenhanced lines from a spectrum. In short, the development of a satisfactory theoretical basis for CIDNP now makes it clear that the phenomenon belongs to both NMR spectroscopy and free radical chemistry. It is sure to find further valuable applications in both fields.
9. Acknowledgements It is a pleasure to acknowledge the co-operation of many of the research groups working in the field of CIDNP for providing preprints or otherwise indicating new results prior to publication. Most of the experimental examples used as illustrations were good-naturedly provided by the students at Brown University. Financial support was received from the Alfred P. Sloan Foundation and the National Science Foundation. I owe a special debt of gratitude, however, to four people: my colleague Harold R. Ward of Brown University with whom I have had the good fortune to collaborate for some 5 years; Glenn T. Evans also of Brown University and Jan A. den Hollander of the University of Leiden who patiently and constructively participated as I groped my way toward an understanding of the theoretical basis of CIDNP; and, finally, Professor L. J. Oosterhoff for his thoughtfulness and generosity during my stay in the Department of Theoretical Organic Chemistry of the University of Leiden where most of this review was written late in 1971.
204
R. G. LAWLER
10. Appendices A.I. Solution of the Master Equation (17b).for the Radical Pair Density Matrix between First and Second Encounters In the time region t, < t < t,, pD obeys the equation pD = i[pD, HD] - z; ‘pD. By a simple transformation,
however,
(A.11
we can write (A.2a)
p”(t) = P(t) exp( - t/T,) where by substitution it can be shown that the matrix representation as pD, obeys the equation
P(t), which is defined
in the same
P(t) = i[P(t), HD].
(A.2b)
Since HD, given by equation (21), is time independent, a representation which diagonalizes HD is
the solution
of equation
(A.2b) in
P&(t) = Pzl(0) exp[ - i(EF - Ey)t]
where EF and Ey are two eigenvalues of HD (in rad set-‘). serves to define a transformation matrix, B, such that
(A.3)
The matrix
of these eigenvalues
ED = B-‘HDB
(A.4)
where the elements of HD are given in the EJ, representation energy’matrix. The matrices P*(t) and P(t) then obey the relationship
and
ED is the diagonal
P(t) = BP*(t)B- ‘. By combining of pD
equations
t%(t) = Cexp( -
WI 1 C C 1 k
I
s
where cukl = EF - EF. The matrix elements tion are Diagonal elements: H:.sn = H&,, H’&+, Off-diagonal
~mk&1~Ps(0)41K~
element,
exp( - iok, t)
1
of HD from equation
&,(t),
(‘4.6)
(2 1) in the l,x,, representa-
(A.7a)
= 0
= +(1/2)c(w,
+ 02) + cc Alams + c (1 b
~d%)l.
(A.7b)
elements: HZ”,,, = (1/2)W,
- %) + (c Al,m, a
HP,,, = i8-“2i”iAi[Ii(Ii
HP,,, wher.
(A3
(A.5), (A.3) and (A.2a) we have for an arbitrary
he nuclear
- c
A,bmb)l
=
&I
(A.7c)
b
+ i) - mi(mi + 1)]“2
= 8-1’LA,[Zi(Zi + 1) - mi(mi + l)]“’
spin state x_+ = I.. . Ii, ‘Hi + 1,. . . ) corresponds
(A.7d) (A.7e)
to a change of one unit
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
205
in m, for the ith set of equivalent nuclei and I is + 1 or - 1 depending on whether the ith nuclei are on fragments 1 or 2 respectively. A.11. High Field Solutions In the special case where l(wi + oJl B Aili the diagonal elements of the Tilxn states are much larger than any of their off-diagonal elements and they are only weakly coupled to the T, and S states. To a first approximation, then
ED,,= H%,t, B *n,Lbl = B&W
= Sk,,,o,n,.
(A.8a) (A.8b)
Therefore by substitution in equation (A.6) P%,,,,, = &,&O)
exp( -t/Q.
(A.9)
The states r,x, and Sx,, on the other hand, are strongly coupled by the off-diagonal element 6, since their diagonal elements vanish. We therefore find by solving equation (A.4) using the required 2 x 2 matrices that (A.lOa)
=
B-‘.
Substitution into equation (A.6) using the elements of B and ED in equations (A.lO) and the initial conditions in equation (22b) gives the equation (22~) for p:,,,(t). A.111. Memory Effect in a New Pair, ID’,formed from D Equation (40) for pf&,,(tz, t;) for a new pair which was formed at time t; after the first encounter is obtained by first solving equation (A.6) for p$Jt;), &,,,(t;) and &,,$t;), using these in another equation of the same form as (A.6) for the matrix element &,,,sn(tJ of the new pair formed at t;, and employing the new values + Sk for the eigenvalues of ED’.
A.IV. Low Field Populations In the case where l(wr + 02)) < Aili all four electron spin states are mixed by HD and numerical methods must be employed to solve equations (A.4) and (A.6) for &&t). When several kinds of nuclei are involved a further complication arises because elements of pD may build up which are diagonal in s but off-diagonal in the nuclear spin states, i.e. n # n’. In this situation the description of the nuclear spin system in terms only of “populations” breaks down and the elements of the form &,, must also be included in equation (37) to properly simulate low field CIDNP.‘31’
206
R. G. LAWLER
11. References I. 2. 3. 4. 5. 6. 7. 8. 9. IO. Il. 12. 13.
14. 15. 16.
17.
18. 19. 20.
21. 22. 23. 24. 25. 26. 27. 28. 29.
30. 31. 32. 33.
34. 35.
36. 37. 38. 39. 40. 41.
42. 43. 44. 45. 46. 47. 48. 49.
50. 51. 52. 53. 54. 55.
56. 57.
J. BAHC;ON. H. FIS(.HI.K and U. JOHNSEN.Z. Ntrtl!rfi>r.sch. 22a, I55I (19671. H. R. WAHI) and R. G. LAwLtK, J. A~rlrr. C~PIII. Sot. 89, 551X (1967). H. FISCHER and J. BARGON, Accounts Chem. Rrs. 2, 1 IO (1969). H. R. WARD, R. G. LAWLER and R. A. COOPER, J. Amer. Chem. Sot. 91,746 (1969). A. R. LEPLEY and R. L. LANDAU, J. Ajnrr. Chem. Sot. 91,748 (1969). P. LIVANT, Unpublished results, Brown University (1970). H. R. WARD, in Free Radicals, Vol. 1, Ed. J. K. KOCHI, Wiley-Interscience, New York (1972). G. L. CLOSS, Spec. Lrct. XXIIIrd Int. Congr. Purr Appl. Chem. 4, 19 (1971). H. R. WARD, Accounts Chrm. Rrs. 5, 18 (1972). Ind. Chim B&J 36, 1051-89 (1971); P. BAEKELMANSand G. J. MARTENS, I&. J. Chrrn. Kinetics3, 375(1971). S. GLARUM, in Chrmically Induced Magnrcic Polarization, Ed. G. L. CLOSS and A. R. LEPLEY, Wiley, New York (1973). R. KAPTEIN, Doctoral Dissertation, University of Leiden (1971). H. FISCHER, Fortschr. drr Chum. For&. 24, 1 (1971). R. G. LAWLER, Accounts Chrm. Rrs. 5,25 (1972). R. KAPTEIN and L. J. OOSTERHOFF, Chrm. Phys. Lvtters 4, 195 (1969). R. KAPTEIN and L. J. OOSTERHOFF,Chem. Phys. Letters 4,214 (1969). R. KAPTEIN, Chrm. Cornm. 732 (1971). R. KAPT~IN, J. Amer. Chem. Sot. 94, 6251 (1972). R. KAPTEIN and J. A. DEN HOLLANDEK. J. Amer. Chem. Sot. 94,6269 (1972). G. L. CLOSS. J. Amer. Chrm. Sot. 91, 4552 (1969). G. L. CLOSS and A. D. TRIFUNAC, J. Amer. Chem. Sot. 92, 2183 (1970). ’ G. L. CLOSS. J. Amer. Chern. Sot. 93, 1546 (1971). K. MOLLEK, Chem. Commun. 45 (1972). F. J. ADRIAN. J. Chrm. Phys. 53, 3374 (1970). F. J. AIIRIAN. J. C’heln. Phys. 54, 3912 (1971). F. J. ADRIAN, Chrm. Phys. Letters IO, 70 (1971). H. FISCHER, Chrm. Phys. Letters 4, 61 I (1970). H. FISTHER, Z. Naturforsch. 25a, 1957 (1970). F. GFKHART and G. OSTERMAN. Trtrtrhedron Letfers 4705 (1969). F. GERHART, Tptrahrdron Letters 5061 (1969). J. 1. MORRIS, R. C. MORRISON, D. W. SMITH and J. F. GAKST, J. Amer. C~L’III.Sot. 94, 2406 (1972). M. TOMKIEWI~Z and M. COCIVERA, Chem. Phq’s. Letters 8, 595 (1971). M. TOMKI~WICZ, A. GROEN and M. COCIVEKA, Chew. Phys. Letters IO, 39 (1971). M. TOMKIEWICZ, A. GROEN and M. COCIVERA, J. Chrm. Phys. 56,585O (1972). J. M. DEUTCH, J. Chem. Phys. 56, 6076 (1972). J. BARGON and H. FISCHER, Z. Naturjorsch. 22a, 1556 (1967). R. G. LAWLER, J. Amer. Chrm. Sot. 89, 5518 (1967). G. L. CLOSS, Ind. Chim. Brig. 36, 1064 (1971). J. A. BEKSON, R. J. BCSHRY, J. M. MCBRIDE and M. TRAMELLING, J. Atner. Chrrn. Sot. 93, 1544 (1971). See, for example, Frrr Radicals, Ed. J. K. KOCHI, Wiley-Interscience, New York (1972). R. KAPTEIN. Chern. Phys. Letters 2, 261 (1968). R. A. COOPER, R. G. LAWLER and H. R. WARD. J. Amer. Chem. Sot. 94, 545 (1972). S. ROSENF~LU, Ph.D. Dissertation, Brown University (1972). R. A. COOPER. R. G. LAWLCR and H. R. WARD. in Chumicully Induced Magnetic Polarization, Ed. G. L. CLOSS and A. R. LEPLEY, Wiley, New York (1973). G. L. CLOSS and L. E. CLOSS. J. Amer. Chew. Sot. 91, 4550 (1969). M. LEHNIG and H. FISCHER, 2. Naturforsch. 25a, 1963 (1970). A. M. NORTH, Collision Theory of Chemicul Reuctions in Liquids, Methuen, London (1964). S. CHANI~RASEKHAR,Rrc. Mod. Phys. 15, 1 (1943). R. M. NOYI s. J. Chvm. Phys. 22, 1349 (1954). E. RABINOWITZ, Trans. Faraday Sot. 33, 1225 (1937). R. M. N0vr.s. Progr. Reaction Kinetics I, 131 (1961). C. A. EMIEISand P. L. FEHDEK, J. Amer. Chem. Sot. 92,2246 (1970). N. K. AILAWADI and B. J. BERNE, J. Chem. Phys. 54, 3569 (1971). E. LIPI’MAA, T. PEHK, A. L. BUCHA~HENKO and S. V. RYKOV, Chew. Phy.s. Lettws 5, 521 (1970). H. IWAHLKA and M. IWAML~KA,Tetrahedron Letters 3723 (1970). J. G. GAKST. R. H. Cox. J. T. BARBAS, R. D. ROBERTS, J. I. MORRIS and R. C. MORRISON. J. Amer. Chem. Sot. 92, 5761 (1970). G. L. CLOSS and A. D. TRIFUNAC. J. Amrr. Chrm. Sot. 92, 7227 (1970).
CHEMICALLY
58. G. 59. 60. 61. 62. 63. 64.
65. 66. 67. 6X. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
101. 102. 103. 104. 105. IOh. 107. 108. 109. I IO.
11I.
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
207
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208 I 12. 113. 114.
I IS. 116. 117.
118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 131. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159.
160. 161. 162.
163. 164. 165.
166.
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2650
(1970).
Ser. Khim.
CHEMICALLY 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177.
178.
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
209
H. D. ROTH, J. Amer. Chem. Sot. 93, 1527 (1971). J. HOLLAENDER and W. P. NEUMANN, Ind. Chim. Be/g. 36, 1072 (1971). H. R. WARD, J. Amer. Chem. Sot. 89,5517 (1967). A. R. LEPLEY, J. Amer. Chem. Sot. 90, 2710 (1968). H. R. WARD, R. G. LAWLER and H. Y. LOKEN, J. Amer. Chem. Sot. 90,7359 (1968). A. R. LEPLEY, Chem. Comm. 64 (1969). A. R. LEPLEY, J. Amer. Chem. Sot. 91,749 (1969). H. R. WARD, Ind. Chim. Belg. 36, 1085 (1971). H. R. WARD, R. G. LAWLER and T. A. MARZILLI, Tetrahedron Letters 521 (1970). H. W. BODEWITZ, C. BLOMBERG and F. BICKELHAUPT, Tetrahedron Letters 281 (1972). L. F. KASUKHIN, M. P. PONOMORCHLJKand Z. F. BUTEIKO, Zh. Org. Khim. 8,665 (1972). I. B. BELETSKAYA,V. B. VOKYEVA, S. V. RYKOV, A. L. BUCHACHENKOand A. V. KESSENIKH,IZU. Akad. Nauk
SSSR, Ser. Khim 454 (1971). 179. 180. 181. 182. 183. 184. 185.
186. 187.
188. 189.
190. 19 1.
192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224.
225. 226. 227.
228.
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210 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241. 242. 243. 244. 245. 246. 247. 24x. 249. 250. 251. 252. 253. 254. 255. 256. 257. 258. 259.
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INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
147
Glossary of Symbols Hyperfine coupling between electron 1 and nucleus a Population of nuclear spin state n in combination product and its equilibrium value Diffusively-separated radical pair New diffusively-separated pair formed by scavenging of D Relative diffusion constant for two radicals Diameter of encounter pair Encounter pair Probability of first re-encounter occurring at time t ;/-factors of radicals I and 2 Magnetic field of spectrometer and field in which reaction is carried out Components of nuclear spin angular momentum Intensity of NMR transition between state n and m in combination product and its value at equilibrium Electron exchange coupling constant Spin-spin coupling of nuclei A and B Rate constant for a reaction which competes with radical recombination Pseudo first-order rate constant for overall reaction Rate constant for radical scavenging Magnetic quantum number for set of magnetically equivalent nuclei a Probability that a radical pair will undergo at least one re-encounter Probability that two radicals will escape from their encounter without reacting Thermal electron polarization Thermal nuclear polarization Probability of finding radical pair in singlet state Population of nuclear spin state n in reagent which forms a radical pair z-component of electron spin Population of nuclear spin state n in scavenging product and its equilibrium value Relaxation time for population difference between spin states n and M in combination product Nuclear and electron relaxation times in a diffusively separated pair Enhancement factor for transition n-m Enhancement factor for an ESR hyperfine line with nuclear spins in state n Transition probability between states n and k Total number of nuclear spin states in combination product, C, and reagent, R Magnetogyric ratio Signature symbols for classifying high field and low field CIDNP and CIDEP respectively Nuclear spin dependent singletttriplet mixing coefficients Chemical shift differcncc between nuclei A and B Radical pair product type signature Spin function for two electrons in state 0 Probability that a radical pair in the singlet state will react during an encounter Radical pair precursor signature NMR frequency of nucleus A Spin density matrices Probability of initially finding radical pair in singlet or triplet states Signature symbol for relative location of nuclei A and B on two radicals Time between first and second radical pair encounters Lifetime for diffusive separation Lifetime for electron exchange Lifetime for radical scavenging Probability that reagent in nuclear spin state n forms combination or scavenging product in state rn Probability of finding combination product in state n if all states of reagent have the probability Z, ’ of being occupied Yield of combination product Nuclear spin function Larmor frequencies of electrons 1 and 2
CHEMICALLY
INDUCED DYNAMIC NUCLEAR POLARIZATION
149
1. Introduction
About 6 years ago it was discovered by accident UJ) that the NMR spectra of samples which are undergoing a chemical reaction may show extremely unusual and dramatic effects. These effects, which have been the subject of subsequent vigorous investigations on almost a worldwide scale, have come to be known as Chemically Induced Dynamic Nuclear Polarization (CIDNP).(3) The main experimental observations and the requirements so far found necessary for the production of CIDNP are the following: 1. Intense emission and enhanced absorption of lines may be observed in reacting samples with ordinary high resolution NMR equipment. 2. The effects always seem to occur in molecules which are the products of chemical reactions (although in some instances the chemical situation is such that these are the same as reactants).(4*5) 3. The enhancements can be extremely large, e.g. in excess of 1000 in favourable instances. This fact alone makes the phenomenon impossible to ignore, even if one is not interested in it per se. For example, a reaction product formed in only 0.1% yield may exhibit a transient spectrum as intense as that of the starting material! 4.’ From the evidence available to date, only the intensities of the lines appear to be significantly perturbed by the reaction. Changes in linewidths, i.e. T,, in the spin lattice relaxation time, T,, or line position shifts such as observed in the presence of chemical exchange seem to be negligible. 5. Because of competing destruction of the enhancement by thermal relaxation processes, the effects can be observed only during reactions which proceed at a rate not too much slower than the nuclear spin relaxation times. Experimentally, the effects are usually largest in reactions with half-lives less than 5 min, although occasionally they appear and persist for reactions which go to completion in more than 1 hr.@) 6. The only types of reaction so far found to produce CIDNP are those involving unpaired electrons, i.e. free radical intermediates. 7. The effects are largest when the reaction producing the products is of the initiation type in free radical chemistry where the product-forming step occurs by reaction of pairs of free radicals. A fairly typical example ofa CIDNP spectrum, obtained during the thermal decomposition of C,H,CO,O,CCH,CH,, is shown in Fig. 1. The purpose of the present article is to summarize the current status and scope of research on the CIDNP effect and present and discuss some representative examples of the phenomenon. The principal goal will be to present the kinds of effects on NMR spectra which are known to arise and to discuss their origins. It is not intended to give a comprehensive or critical review of all of the reactions which have so far been found to exhibit CIDNP or to emphasize the potential of the phenomenon as a tool in mechanistic organic chemistry. A survey of the CIDNP literature up to mid-1971 is availablec7) and the real and potential applications of the effect have received adequate attention recently.(*p9) The proceedings of an international colloquium entitled “CIDNP and Its Impact on Mechanistic Organic Chemistry” held in Brussels, 18-19 March 1971, have also been published.” ‘)
150
R. G. LAWLER
Since most of the effects to date have been explained a posteriori rather than predicted, it is perhaps not surprising that several different versions of a recently-developed theory explaining these effects have arisen. They differ primarily in the way that their various proponents view the dynamic properties of reacting molecules in liquids. This is of course a very difficult problem which requires interpretation of experimental data from sources outside of magnetic resonance, to say nothing of outside of CIDNP. We will therefore make
FIG. 1. ‘H NMR spectrum decomposition
obtained at 60 MHz of C,H,COIOjCCHICH3
before (A), during (B) and after (C) thermal in o-dichlorobenzene at 120”.
no attempt to present all the sides of this still-developing story and will try to present instead what will be hopefully a coherent, reasonably accurate, but necessarily biased synthesis of the various versions of the theory. It should be pointed out that discussions of these separate points of view are also becoming available from their originators in several reviews prepared at approximately the same time as this one.18,9.11-14)Perhaps the most comprehensive and self-contained account so far available is the remarkable doctoral dissertation” 2, of Robert Kaptein of the University of Leiden. The present narrative owes much to the clarity and meticulous character of the work of the Leiden group.
CHEMICALLY
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151
2. Radical-pair Theory
Perhaps the most significant barrier to the understanding of the origins of the CIDNP effect and its potential applications is the fact that the theory requires information nearly equally from the fields of organic free radical chemistry, the dynamics of rapid reactions in the liquid state, and magnetic resonance spectroscopy. Unfortunately, it also appears that there is little that one can learn from CIDNP spectra at present‘that does not require some familiarity with the theory. We will therefore devote an appreciable portion of this review to a presentation of background material from the above three disciplines and show how it is used to predict ultimately the observed effects, or, conversely, how proper analysis of CIDNP spectra may yield information of importance to these areas. Before beginning, however, we will list, in a manner analogous to that done for the experimental observations, the essential assumptions involved in the theory.” 5-35) 2.1. Qualitative Theory 1. It is believed at present that all CIDNP effects (and even related effects in electron-spin resonance) have their origins in interactions between pairs of radicals in solution. The present theory of CIDNP, first published by C10ss’~~’and independently by Kaptein and Oosterhoff,“h’ has in fact usually been designated the Radical-pair Theory to distinguish it from the first proposed explanations (36,37) of the effects which were based on the Overhauser Effect. Effects arising from the latter mechanism have so far never been observed unambiguously and are apparently much smaller than those due to radical-pair interactions. 2. It must be assumed that the ability of two free radicals to dimerize or form other stable combination products, such as olefins, derived from both radicals, depends on the combined spin multiplicity of the two unpaired electrons. In practice this means that only singlet-state pairs can produce stable combination products. 3. Somewhat related to (2) is the assumption that bond breaking in singlet or triplet radical-pair precursors proceeds with preservation of the correlation of the electron spins, thus making it possible to define a time at which the spins of a radical pair are correlated. The multiplicity dependence of combination reactions also introduces correlation even in pairs of independently generated radicals which undergo reactive collisions with each other. The idea of electron-spin correlation is essential to the theory because without it the terms “singlet” and “triplet” have no meaning. 4. Nuclear spins, and therefore NMR, enter the picture via the assumption that the internal magnetic fields due to the nuclei in a free radical pair with correlated spins can bring about a mixing of the singlet and triplet states, i.e. “intersystem crossing”, in the pair. 5. As a result of (4) the probability that a given pair of free radicals will form a combination product depends on the nuclear spin state of the pair. In effect this says that the chemical reactivity of a pair of free radicals depends on the nuclear spin state. As we shall see, this rather revolutionary assumption need have barely detectable chemical consequences in order to produce NMR enhancements of several thousandfold. 6. For virtually all reactions carried out in the magnetic fields of several thousand gauss usually employed for high resolution NMR singlet-triplet mixing may be induced by the nuclei without having any nuclear spins&. Thus the nuclear spin magnetic quantum numbers merely serve as unchanging labels for atoms which are carried by the chemical reaction from one kind of molecule to another. The observed intensity changes are then due
152
R. G. LAWLER
simply to the fact that more molecules of product are formed for some nuclear spin states than for others. 7. Statement (6) breaks down for reactions examined at low magnetic fields(“~‘9*26’31’ and, apparently, for some rather special bases of intramolecular reactions proceeding via biradical intermediates.‘22s38,39)Very little experimental data are, however, yet available on these two aspects of CIDNP, although they represent especially promising directions for extending studies of the phenomenon. 2.2. Important Features of Radical-pair Reactions in’ Solution The first impression which might occur to one who begins to look at the abundant literature dealing with free radical chemistry(40) is that it is very complicated. In fact, two observations which sometimes make organic chemists suspect the intermediacy of free radicals in a reaction are an abundance of different products and poorly reproducible reaction kinetics. The former effect arises mostly because radicals as a class seem to be less selective in their reactions than are ionic intermediates, while the latter is a manifestation of the well-known sensitivity of radical reactions, especially those involving chain steps, to trace amounts of impurities. Fortunately, however, there are only a few general types of processes which take place during radical reactions in solutions. The four kinds required for production of CIDNP are discussed below by tracing the life history of a molecular fragment as it changes from reactant to free radical intermediate and ultimately to a molecule of a reaction product. 2.2.1. INITIALIZATION ANDCOMBINATION. In radical chemistry “initiation” usually is considered to be a reaction step which produces two radicals by thermal or photodissociation of a precursor in which the electrons are initially paired (or occasionally in the triplet state). For CIDNP, however, we will use the term “initialization” to mean a reaction step, proceeding with pseudo first-order rate constant k, which determines the overall rate of the chemical reaction, in which a reagent, R, first forms a pair of radicals which are close enough for the electron spins to interact with each other. Such a close pair, in which the two radicals are probably within a Van der Waals radius of each other. we will call an encoun&r pair, and give it the symbol E. Furthermore, there will be some probability, A, that such a pair will react to form a combination product, C, before the radicals move apart by diffusion, provided that the two electrons are in the singlet state. A “combination” step we therefore define as one which depends on the electron-spin multiplicity of a radical pair. The probability that the pair is in the singlet state, P,(t), will generally depend on time beca&e of the hypothesis that singlet-triplet mixing can occur. It should be stressed that as far as CIDNP is concerned the first reactive encounter of two independently-generated radicals may also be considered to be an initialization step. The initialization and combination steps we represent schematically by: Scheme I : Initialization R,-X-R, (R)
--&
R,. .R, 03
Scheme II: Combination R, .
.R, m
03
RI-_R, (Cl
CHEMICALLY INDUCEDDYNAMIC NUCLEARPOLARIZATION 153 where R1 and R, represent the molecular fragments, usually containing magnetic nuclei, which are carried through the reaction and a bar is used to represent radicals which may react with each other. Some examples of initialization steps in reactions where CIDNP has been observed are given below. Ia. Unimolecular decomposition Singlet precursor(41342) (CH&H,CO,), _* CH,CH, . . CH,CH, Triplet precursor(43) C,H&H,CHO (photoexcited) C,H,CH, . . CHO Ib. Bimolecular reaction Singlet precursor(44) CH,CH,Li Triplet precursor (4s) (C,H,),CO IC.
+ ICH,CH, -5 CH,CH,. .CH,CH, (photoexcited) + C,H,CH,’ (C,H,),COH.
.CH2C,H,
Diffusive encounter of independently-generated radicals’46’ .C,HS + CH,Cl, + CH,ClCOOH ---+ C,H, + .CHClCOOH + Cl,HC.Cl,HC. 9CHClCOOH
2.2.2. DIFFUSION.If an initially-formed encounter pair does not react after a few collisions between the partners, it will be separated by the insertion of solvent molecules between the two radicals. We designate such separated pairs by D. Diffusive separation by this mechanism will take place within an average time, rn, which will typically be lo-” to lo- I1 sec.‘47) If the pair of radicals never undergoes a re-encounter, CIDNP is not produced in pairs because the interactions between electrons and nuclei are too weak to have any effect in this short period of time. In general, however, the random motions of solvent molecules will drive the partners toward each other as well as away from each other so that there is a finite probability that the original partners will re-encounter each other at some later time. In fact, in the limiting case where only a single pair of radicals was formed in a finite volume of a completely inert medium the probability of a re-encounter would be unity. (One might, however, have to wait a very long time for it to occur!) This point ‘of view, which is based on the theory of random flights,‘48’ was applied to diffusion controlled reactions such as those of free radicals by R. M. Noyes about 18 years ago.(49) It differs from the original idea of a “cage effect” proposed by Rabinowitz”‘) in that the latter acknowledges only that molecules in solution may be held together in solvent “cages” for longer than they would be in the gas phase but still requires that a pair of radicals react within the time rn or not at all. Noyes has discussed in considerable detail how the probability of a re-encounter depends on time and various molecular parameters.” ‘) Thus he defines the quantity f(t) dt which is the probability that a pair which separated from a first encounter at t = 0 will undergo a re-encounter between t and t + dt. We may thus represent the process of diffusion of spin correlated radical pairs by Scheme III. Scheme III : Diffusion m
,
%$R,./.R,
(D) (E) where a solidus is inserted to represent intervening solvent molecules. At any instant one would expect most pairs of radicals in a dilute solution to be of the D type rather than the E type simply because the latter requires a close approach of two species which are present in low concentration.
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R. G. LAWLER
Although both Adrian(2”26) and Kaptein Cl291 *) have adopted a rather specific form for f(t) which varies with the inverse 3/2 power oft, it is more likely that it must be considered only as a rough guideline which seems to describe the diffusive motion of atoms in monoatomic liquids adequately (52)but may be grossly in error for complex molecules. About all one can say at present is that the re-encounter probability decreases with time less rapidly than exponentially but quite possibly(53’ more rapidly than inverse 3/2. Fortunately the qualitative and semi-quantitative appearance of CIDNP spectra does not seem to be very sensitive to the functional form off(t), but lack of knowledge of this quantity does constitute perhaps the greatest area of uncertainty for the quantitative understanding of the effect. As more accurate and reliable CIDNP intensities become available it should even be possible to use them to obtain information aboutf(t), which seems to be difficult or impossible to obtain at present in any other way. 2.2.3. SCAVENGING.Even though a pair of radicals may have diffused away from each other, the individual radical partners will still remain reactive and will eventually be lost by reactions with other molecules in the solution. Such reactions will take place predominantly with the more abundant D pairs and would not be expected to be affected by the overall electron spin multiplicity of the pair. Such a “scavenging” process is usually a reaction, such as shown in Scheme IV, between D and a scavenger molecule MY. We assume it proceeds with rate constant k, = zs ’ [MY] - l and gives a diamagnetic molecule S, derived from one of the radical fragments, and a new pair, D’; the two electron spins in the new D’ pair have the same wave function as in the pair D at the instant of formation of S. D’ may in some instances, in fact, live long enough to undergo a re-encounter to form a new combination product. Usually, however, the electron correlation in D’ will be lost before re-encounter takes place. Scheme IV : Scavenging R,./.R, + MY * R,Y + M./.R, (D) (D’) (S) For CIDNP, however, all that is necessary is that a scavenging process be one which (a) prevents re-encounter of a D pair with correlated spins and (b) does not depend on the electron-spin multiplicity of the pair. All of the processes given in the examples below (taken from the CIDNP literature) will therefore serve to “scavenge” pairs with correlated electron spins even though strictly speaking only the first is of the type shown in Scheme IV. IVa. Atom abstraction’42’ CH,CH,I + (CH3)2CH./.CH(CH3)2 CH,CH, ./. CH(CH,), + (CH,),CHI IVb. Unimolecular decomposition of one fragment’54’ C,H,./.C,H, + 13C02 C,H, 13C02./.C6H, IVC. Addition(55’ B./.B
+ C,H,CH
IVd. Recombination Naphthalene’/.
= I;(C,H,)O-
-
C,H,CHB
-
N(C,H,)O./.B B. = (CH,),(CN)C.
of one partner with another radicalc5@ NaphthaleneT + CH, + Naphthalene CH, + Naphthalene’ s
IVe. Thermal relaxation of electron and/or nuclear spins(46,57*58) 2&% equilibrium populations C,H,CH(OH)./.OCC,H,
CHEMICALLYINDUCEDDYNAMICNUCLEARPOLARIZATION
155
The last process fixes the upper limit of the length of time which electron spins may be correlated as approximately the relaxation time of electrons, T,,, and nuclei, T,,, in free radicals in solution. For most organic radicals this is probably 10e6 to 10e4 sec.(46,57*59) Thus all chemical scavenging processes involved in CIDNP must take place in times, rs, less than a few microseconds. Also it is only under special circumstances, such as the use of a high concentration of stable radical as a reactant shown in Example IVd, that the overall radical concentration in the solution is high enough for reactions between a pair and an independently-generated radical to compete with spin-lattice relaxation in the pair. This means that all “memory” of previous encounters will usually be erased by the time two independently-generated radicals diffuse together in an initialization encounter of the type illustrated in Example Ic.
2.2.4. SUMMARYOF KINETICPROCESSESNECESSARYFOR CIDNP.Thekineticprocessesjust discussed are all essential for the production of CIDNP from reactions of radical pairs. These are summarized in Scheme V in the general notation just described. Scheme V: Initialization
R k,
E
Combination
E 3
C
Diffusion
E =D ‘(‘jdr
Scavenging
D-Ir,S+D’
The quantities kR,f(t), 1, PJt) and rs are therefore the parameters in this model which play a role in determining CIDNP intensities. These will generally depend in a complex way on molecular structure and the time dependent details of diffusion and chemical reactions in solution. It is as well to point out, however, that while the time scale for the last three processes is measured in microseconds or less, the first step will proceed at the overall rate of the chemical reaction, i.e. the order of a few minutes under the usual CIDNP conditions. Furthermore, the relatively long nuclear relaxation times in diamagnetic product molecules C and S, in solution means that the NMR measurements are also carried out on the more leisurely time scale. Consequently the effects on the reaction products arising from relaxation or interaction with the radiofrequency field may be separated conveniently from the much faster events involving intermediate radical pairs. 2.3. Rrlutiorlship
hctwcvrl Rtrdicul-puir’ Rruction
Probabilities
und NMR
Intensities
We have so far said nothing about the connection between the foregoing kinetic processes and an experimentally-observable NMR spectrum. The connection is made in the following way. Consider that initially a reactant molecule, R,, contains the nuclei whose NMR spectrum we will eventually observe in a product and that the combined spin state of these nuclei is indicated by the subscript n. The reactant then undergoes the rapid series of processes which take the nuclei from R to E to D and eventually to products S or C. During this excursion the nuclei may remain in spin state n or may possibly undergo a selective transition to another state m. In any case, within a few microseconds the nuclei will either be in C or in S molecules or, through relaxation in the radical, they will have
156
R. G. LAWLER
“forgotten” what their original spin state was. We may thus represent the net reaction as Scheme VI.: Cm,O S m,O
where &,,, and 4:,,, are the respective probabilities of finding nuclei which started out in R, in the spin state m of C (combination) or S (scavenging product) after the rapid radicalpair reaction sequence is over. The intensity, I:,,,,, of an NMR transition between states n and m, of a combination product, for example, under slow passage, non-saturating conditions, is G&~l~“lm)z(C, - C,)
(1)
where (C, - C,) is the population difference between the two states and (nlZ”lm)is the usual transition probability for NMR transitions involving the operator, I”, for the x-component of the nuclear spin. During the chemical reaction the population difference between the two levels of the product, C, according to Scheme VI obeys the equation @G - C,)/dt = k, C (4”, - &JR,
- C K,(C,
+
1 k
-
G,o
-
Ck
+
Ck,o)
k
k wmk(cm
-
%,O
-
ck
+
(2)
ck,O)
where the last two terms allow for relaxation of the populations to their equilibrium values, C,,,, etc., with time-independent transition probabilities, W,,. At thermal equilibrium the difference in population between two levels may also be expressed in terms of the polarization, p”, of the transition PO
=
(C,,,
-
C,,oMC,,o
+
G,,,)
=
WWBoPkT
(3)
5 x lo-(’ for protons in a 60 MHz spectrometer at room temperature, where y is the zagnetogyric ratio of the nucleus and B, is the magnitude of the applied magnetic field. If Zc is the total number of nuclear spin states available in a given molecule C (for example, 2” for x spin l/2 nuclei) and C is the total concentration of C in the sample, we have (C,,, - G.0) = 2P°CZ, 1
(4)
as the population difference responsible for NMR absorption in samples at equilibrium. These population differences are, however, usually much smaller than those arising from CIDNP and from now on we will ignore contributions to the spectrum from the equilibrium, or thermal, polarization of the spins. If this is done, all states of reactant R are assumed to have the same initial population, RZ, ‘, and equation (2) simplifies to d(C, - C,)/dt = k,RZ,‘@; - ;
- #$,) - T,;,(C,
1/2(wnk
-
wmk)
tc,
-
ck
- C,) -
ck
+
cd
(5)
where we have simplified the definition of@: to be 2, times the probability of finding the product C in spin state n if each spin state of R had a probability Z, ’ of being populated and T,,!,
= 2w,, + l/2
2 k#n,m
(w,, +
wmk).
(6)
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
157
POLARIZATION
The first term of equation (5) then represents the driving of the transition intensity by the reaction, the second term an exponential decay of the population and the last term possible contributions, via cross-relaxation, of other transitions which are being chemically driven. The time dependence of C, - C,, and therefore the NMR intensity during CIDNP, based on equation (5) should therefore be the following. The driving term due to the reaction should cause an initial linear increase in intensity with time. As the population difference becomes appreciable, however. the second term in equation (5) should bring about a reduction in the rate of increase of intensity until the first and second terms become equal and an essentially steady state is reached. The last term may bring about contributions from
FIG. 2. Time dependence of CIDNP (40 MHz) from benzene during thermal 5 mol. y/, benzoyl peroxide in cyclohexanone” at I IO”.
decomposition
of
other transitions. but with a rate constant less than T<,!,,. As the concentration of reactant, R, begins to decrease, however, the first term will become less effective than the second and the intensity will decrease. If k, is less than both T&,, and the cross-relaxation rates then the rate constant for decrease will be simply k,. Finally, after a sufficiently long time the reaction will be over and the product will have relaxed to give an NMR spectrum with normal intensity. This effect was beautifully demonstrated in the first report”’ by Fischer et al. of CIDNP from the protons of benzene formed from the thermal decomposition of benzoyl peroxide in a hydrogen-donating solvent. This spectrum is shown in Fig. 2. For the purposes of quantitative discussion of spectra it is sometimes convenient to define an enhancementfactor, V,,, for a transition vLl = (El
- L,O)lEll,O
= [(C, - C,)/2p°CZ,
‘1 - 1.
(7)
V,, in a reacting system is, however, time dependent because of time dependence of concentrations in both the numerator and denominator of equation (7). For reactions in which k, is small compared with the relaxation rates, however, a steady state in (C, - C,) may be reached rapidly. Also in the early stages of reaction C increases approximately linearly with
1%
R. G. LAWLER
time and the concentration of R remains approximately’constant. Under these conditions, and ignoring the cross-relaxation last term in equation (5), we have C, = k,R&
@a)
(C, - CA = k,RZ,‘(4;
- 4:)T,,,,
1 + %tl(t) = (T,,“mMQO)- ‘M
- 4tJM3 @v-&d
(8b)
(84
= EmwEl,o@) where 4,” is the probability of formation of C from R averaged over all nuclear spin states; that is, it is the yield of C which one would measure chemically after the reaction had gone to completion. The experimental value of V,, at any time during a reaction may be obtained by noting the observed intensity of the enhanced line, rapidly stopping the reaction (for example, by cooling the sample or shutting off the light in a photochemical reaction) and then recording the intensity of the weak. unenhanced lines arising from the amount of product formed up to that time. To obtain the microscopically interesting parameter (4, - +,,,)/4,, it is then also necessary to determine the relaxation time, T,,,, for the transition and possibly also correct for the effects of cross-relaxation.‘8,61’ In those cases where this procedure has been used, (14,45,46,60)the observed values of (Z,/Z,)I/,,(t/T,,,,) range up to about 103. This in turn requires only that (4, - 4,,,)/4,, be 10 -2 or smaller, that is about 1% discrimination of the reaction according to nuclear spins. It is for this reason that one can still say that for practical purposes the yields of chemical reactions so far found to exhibit CIDNP do not depend appreciably on the nuclear spin state, in spite of the quite large enhancements sometimes observed. 2.4. Quantum Mechanical Calculation
qf Reaction Probabilities
From the above we see that the empirically-obtainable quantities determining CIDNP intensities,are the nuclear-spin dependent reaction probabilities, or yields, 4,. It is therefore necessary to relate these obiervable quantities to the microscopic parametersf(t), A, P,(t) and TV. We do this by recognizing how the probabilities are related to the wave function, $(t), for the total system consisting of E, D. C and SD’. We write v/(t) as a linear combination of products of the space part of the wave functions YE, Yy,. Yy, and Ys,, for each of the species, the spin functions for the two electrons, E_ and the spin functions representing the spin states of the nuclei, x.. That is (9) This separation of space and spin coordinates emphasizes our picture of the chemical reaction and motions of the liquid as simply moving the overall spin system, represented by the products EJ,, from one chemical species, represented by Yy,, Y, etc., to another. The squares of the expansion coefficients /Cz,,(t)(“,IC!$(t)l’, etc., are then simply the probabilities of finding the overall system in one of the four kinds of chemical environments and in the 0 electron spin state and n nuclear spin state. Furthermore, if YR is the space wave function for the radical pair precursor, R, we have
(10)
CHEMICALLYINDUCED DYNAMICNUCLEAR POLARIZATION Similarly, since E and D eventually disappear overall system, after a long time we have ;;
IC&)12
but the electrons
+ Ic:%)12
-
159
and nuclei stay still in the
Icm12 = 0.
(11)
The boundary condition in equation (11) is in effect just a special case of the condition that Y(t) remain normalized for all times. This conclusion is not changed even if we include relaxation in D, or D’, since this only redistributes the coefficients (probabilities) for a given species but keeps their sum the same. More explicitly, the spin functions E, and x,, are given by E, = 2-+(@
- /?a) = S,
singlet state
xn = II,, m,)lZ,, mb) . . . (product
over all nuclei)
(12b)
where c( and B are the usual electron spin functions and II, m) denotes the spin function for a nucleus with spin I and value m for its magnetic quantum number. We will consider a set of nuclei to be “equivalent”, and therefore lumped together with a single value of I, only if they are equivalent in all four of the species E, D, C and SD’. Using this notation, the definition of the squares of the coefficients in equation (9) as probabilities and the assumption that only the singlet state of the combination product, C, will be populated by the reaction, we find, in the absence of relaxation in the radicals, the following relationships between the reaction probabilities +E, 4.“, the probability P,,(t) of finding E in the singlet state and nuclear spin state n and the coefficients:
2.4.1. EVALUATION OF THE WAVE FUNCTION FOR THE SYSTEM.In principle it is possible to evaluate Y(t) by defining a general time-dependent Hamiltonian, H(t), which is a function of all of the space and spin coordinates for all four of the chemical species and solve the time-dependent Schroedinger equation i dY(t)/dt
= (2n/h)H(t)Y(t).
(14)
In practice, of course, this is impossible and a number of drastic simplifying assumptions must be made. Furthermore, since we are not concern&d here with Y(t) itself but only with the products of the coefficients of its expansion in a space and spin basis, it is more convenient to adop’t the density matrix notation P”,,,,.” .(t) = C!&(t)C!&(t), etc.
(15)
If we assume that the processes of reaction and diffusion manage to avoid producing nonzero products of coefficients such as C,“,(t)~$,,(t) between two different species, it is then permissible to consider the spin system of only one species at a time and allow for transfer of the spins from one species to another in a phenomenological way. This is, for example, very similar to the procedure which has been used in treating the effects of intermolecular spin2xchange in ESR spectra. (62,63) The principal differences between that case and the
160
R.G.LAWLER
one we are considering here is that our system is not in thermal equilibrium and we must also consider four different species instead of only two. The time dependence of the spin system under the influence of reaction and diffusion is then describable by the following scenario. first proposed by Adrian.(24 ~26) An encounter pair. E. forms from R for the first time at to and cvolvcs in time under the influence of the Hamiltonian H, until it separates by diffusion at t, to form the separated pair, D. The spin system of D then evolves in time under the influence of H, until the first re-encounter tq reform E at tz. E then forms D at t,, D undergoes a second re-encounter at t, and the diffusion process thus continues with the pth encounter beginning at t2p_2 and ending at t,,_ i. At the end of each encounter the combination product, C, also will have had the probability LPs(tlp _ 2) of forming in the singlet state. Similarly, in the relatively long interval (t,, - fzp_i) between encounters the process of scavenging will produce S and D’ and deplete the probability of finding the system describable as a D pair. In the density matrix formalism (64) the time evolution of the density matrices pE and pD under the influence of their respective Hamiltonian matrices HE and HD is 3pE/c?t = i(pEHE - HEpE); and the above scenario
may be described
8pD/8t = i(pDHD - HDpD)
by the following
equations:
t,,_ z < f < t,,_,dpE/dt = i(pEHE - HEpE) - (c?pE/?t),,,,; tzp-,
< t < tzp
(16)
dpD/dt = i(pDHD - HDpD) - TS ‘pD;
dpc/dt = (iipEl?t),,,,
(17a)
r/pSD’/dt = TS ’ pD
(17b)
where the partial derivative refers to the electron-spin dependent formation of combination product during an encounter. In the above equations the effects of the spin Hamiltonians H,, H, and H,, which are actually responsible for magnetic resonance, have been ignored because, as we have already stated, the above scheme will generally be over before they can produce appreciable effects. Furthermore, we have implicitly ignored relaxation in D by using the same rs for loss of D and gain of SD’ (see Section 4.2). 2.4.2. MODEL FORAPPROXIMATESOLUTIONOF MASTER EQUATIONS FOR CIDNP. Equations (17) with perhaps the inclusion of the effects of H,, etc., and explicit addition of approconstitute the master equations for priate terms for relaxation as in the Redfield theory,“” the time evolution of the spin system under the influence of the interactions responsible for CIDNP. Just as in the general theories of magnetic relaxation, however, it is easier to write down the master equations than it is to solve them. The following approximations have thercforc ken employed to give solutions which are both tractable and consistent with most of the experimental data so far obtained. 1. We will assume that the majority of pairs formed initially undergo at most one r~wwcow~ter and will therefore extend the above scenario only through the time t, when the second encounter of a pair ends. This sequence of events is shown in Schcmc VII. Scheme VII:
2. We include in HE only the interactions involving ignore entirely the nuclei. Conversely, in H, we assume
the two unpaired that the electrons
electrons and are sufficiently
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
161
well
separated from each other that we may ignore the interactions of the electrons with each other and consider only interactions of one electron with the nuclei on its particular radical. 3. We assume that both H, and H, are independent of time in the intervals during which they operate. This is probably a rather good approximation for H, which is essentially the spin Hamiltonian for two free radicals, but the approximation is certainly not strictly valid for H,. We may therefore anticipate that the effects of H, will have to be evaluated almost completely empirically. (The time dependence of H, has, in fact, been used explicitly by Fischer” ‘) and Glarum” ‘) in a mechanism for producing CIDNP by adiabatic level crossing. It seems unlikely, however, that E pairs ever live long enough for such processes to be important.‘24’) In writing down explicitly the spin Hamiltonians we will use subscripts 1 and 2 for the unpaired electrons on radicals R, and R, and subscripts a and b respectively for nuclei on the two radicals. In general, then, we will have the following Hamiltonians for E and D in a magnetic field. Encounter
Pair Hamiltonian: HE =,
HI,
+
H,,
+
H2,
+
H,,
+
H,,
+
H,,
+
H,,
+
HLR.
(18)
In equation (18) z refers to a Zeeman interaction between the applied magnetic field and a magnetic moment generated by a source of angular momentum. SL represents the spin orbit interaction, SR the spin rotation interaction, LR the orbit rotation interaction. All of these interactions are, of course, well known for molecules in the gas phase’““’ and also may contribute to line positions(67) and relaxation times(“) in ESR. Unfortunately, it is impossible at present to assess accurately these terms for a pair of radicals which are held within a molecular diameter of each other with variable orientation. About the best one can at present do is hope that the average effect of these interactions may be neglected. Furthermore, the electron Zeeman interaction in such a pair ‘is probably considerably smaller than the electronelectron interaction. H, I, Following Adrian’26’ we therefore will drop all but the first term in H,, The remaining H,, term consists of two parts: the magnetic dipole-dipole interaction between the electrons, and the exchange interaction. The former interaction, however, has been estimated”“’ by Freed to be smaller in magnitude than the latter for most radicals at the distances and for the times characteristic of encounters and we will ignore it. We are then left with H, z
-.!(I/2
+ 2S,.S,)
(19)
where the “exchange integral”, J, is,defined as one-half’the separaticrn in energy between the lowest singlet and triplet electronic states for the two electrons.““” Unfortunately, even J is a poorly understood quantity for complicated molecules. All one can say with reasonable accuracy is that (a) J falls off approximately exponentially Mit.11the average distance between the electrons, (b) the value of J for two hydrogen- atoms in the 1s state separated by6A (“,“)is comparable to the nuclear hyperfine splitting typical for organic free radicals and (c) for n-radicals of the type usually encountered in organic chemistry the sign of J varies with the orientation of the p-orbitals containing the unpaired electrons. One might therefore expect a smaller average value of J for such pairs than for hydrogen atoms at the same distand.
162
R. G.LAWLER
D#i~.siw/_v Separated Pair Ha+ltonian : We assume that H, is the sum of the individual and R,
spin Hamiltonians
for free radicals
R,
H,=H,+H,=H,,+CH,,+H2=+CHZh+CHaZ+CHbr+CH,,,+CHbb'. (20) In general, interactions
of course, one should include the spin orbit, spin rotation, orbit Zeeman, etc., for both electrons and nuclei. It is known from ESR, however, that for organic
free radicals in solution, the parts of these interactions which shift the energy levels of the spin system (i.e. the isotropic parts) simply modify slightly the coupling constants (g-factors, chemical shifts, hyperfine splittings) used to describe the interactions between the electrons, nuclei and the magnetic field. The anisotropic parts of these interactions, on the other hand, induce transitions between energy levels, but at the rate of the relaxation times which we have assumed is usually slower than the re-encounter and scavenging rates of radical pairs. Furthermore, the nuclear-nuclear couplings in the last two terms of H, will always be small compared with the hyperfine coupling terms H,,and H,,. It can also be shown’i2’ that the nuclear Zeeman interactions, ,which may be comparable with the hyperfine interactions at high field, do not have any effect on the elements of pD and may therefore be ignored. For these reasons, then, it. is permissible to omit all but the first four terms in H, to give
HD = ~1s; + 02s; + CA,,SfZi+ CA,&Zf, + 1/2CAi,(S:Z, a +
1/2
c
A,b(S;I,
b +
s;I,c)
+ S;IJ)
a (21)
b
where oi = gl/?(27c/h)\B0is the Larmor frequency of electron 1 which has the g-factor gi and similarly for electron 2, and the isotropic hyperfine splittings Ala, etc., are in radians per second. 2.4.3. HIGHFIELDSOLUTIONSOFTHEAPPROXIMATEMASTEREQUATIONS. Wenowundertake to solve for the time dependence of the density matrices of E, D, C and SD’ using the approximations above. In addition, however, it will be assumed that the magnetic field is high enough so that the Larmor frequencies of the electrons are much larger than the nuclear hyperfine splittings, or alternatively that the applied magnetic field is much larger than the hyperfine splittings, in gauss. Since typical values of A for organic free radicals are between 5 and 100 gauss, with most clustering around 25 gauss,‘72) this approximation will certainly be good for reactions run in fields in excess of 1000 gauss, which is still l&20 times smaller than the fields normally employed in high resolution NMR. Under high field conditions an extremely important simplification of the problem occurs: the last two terms in ff,, given in equation (21) may be ignored”‘,“” (see Appendix A.11). We find therefore that the operators S’ = ST + Sg and I’, for every nucleus, commute with both H, and H,. However, S2 commutes with H, but does not commute with H,. In
practice this means that the net component of the electron spins, and that of every nuclear spin. along the held direction are constants of motion but that singlet and triplet electronic states may still be mixed. This in turn means that (a) electron spin states E-Z,, and E+Z, in equation (I 2) do not mix with any other states and therefore P”~:,~,,(;c) = p:,.,,(O) and (b)Eo andc, may mix but only if they are combined with the same nuclear spin function, ln.
CHEMICALLY
INDUCED DYNAMIC
NUCLEAR POLARIZATION
163
Thus, components of the density matrices P&,~.~. vary with time only if n = n’, that is, in highfield no nuclear spinsjip. With the above simplifications we may now write the following solutions to the equations (17) to give & and 4:. The following initial conditions are also assumed : (a) all off-diagonal elements of p”(t,-J vanish, (b) the diagonal elements of pR(to) with the electron spin state, B, all have the same value, and (c) initially all three triplet states are equally populated (if they are populated at all). We therefore have r = tO:P:n.n’n’(rO) = P,“,,,,“&)
p+ = p- = po =
= 6,,,&“,Pnz,1;
(224
p7
= (1 - 4Ps-G1; PD = JPSG’; an,an@l) = P:“,o”(~,LPc,,,“@l) P~#d~,)= PK’ t = t, :P:,&Z) = Z;’ e-(tz-‘L)irs{(l - I)p, + [or - (1 - A)ps] sin2 &(t, - tr)} t =
t1
:&&I)
c PK&2) =
Z,‘(l
- e-(‘2-‘1)“S)[(1
Wb) (224
- J.)ps + 3p,]
d
where 6, =
VW,
-
(32
+
= singlet-triplet t=
tj:
4: = = A&l
zR&,.,(x~
c a
condition
-
c
A,,m,) b
mixing coefficient z
(23)
zRkt&,(f3)
+ (1 - A)e-(12-L1)i~s]
+ 1 e~(‘2-‘1)I’S[~r and the normalization
4a%
from equation
- (1 - A)pJ sin2 &(t,
(11) requires
- t,)
(24)
that
- 3Le-(fZ-f1)‘rS[PT - (1 - 3.)p,] sin’ s,(t,
- t,).
(25)
Consideration of equations (23 j(25) leads to the following conclusions: (a) CIDNP arises from the second term in equation (24) which is the only one which depends on the spin states of the nuclei; the first term is the same for all molecules and cancels when one considers population differences; (b) the expressions for the reaction probabilities above would be the observed ones only if all radical pairs spent the same time, t, - I,, between the first and second encounter; as mentioned before. however. it is necessary to average over a distribution of times between encounters; (c) the conclusion contained in equation (25) follows directly from the high field approximation which prevents mixing of one nuclear spin state with another, and the assumption that all nuclei which do not end up in C finally find themselves in the scavenging product S. Before discussing the results obtained by averaging equations (24) and (25) over the times between encounters we list the conclusions about NMR intensities of a transition between states n and /n in C and S products which are ind~)prntlent of the way in which the averaging is carried out.
164
R.G. LAWLER
3. Characteristics
of Observed Spectra
3.1. Relative Sigm of Enhancement of Combination and Scavenging Products From equation (25) we see that ($I;: - 4:) = -(& - 4:). Therefore, corresponding lilies in combination and scavenging products derived from the same radical pair should have enhanced intensities of opposite sign. Furthermore, in the case where there is only one
::
::
CH31
CH3COOCCC13
CH3CCl3
A
I
2.5
wm
FIG.3. (A) Proton NMR Spectrum (60 MHz) recorded 5 min after a solution and I2 in CCI, was warmed to 50 (B) Spectrum taken at 0’ immediately
of CH,CO,O,CCCI, following scan A.‘“’
scavenging product and no relaxation occurs before scavenging of the radicals which escape from the second encounter, the intensities should be equal and opposite. This is very strikingly demonstrated in Fig. 3 which shows the emission line from CH,CC13 .and the enhanced absorption line from CH,I which are obtained as the combination and scavenging products respectively obtained during the thermal decomposition of CH,CO,O,CCCl, in the presence of I2 .(9)
(‘HEMICALLY
INDUCXLI
DYNAMIC
NUCLLAH
POLAKIZATION
165
Scheme VIII: I2
CH,CO,
-2c0, 02CCC13 -
CH I CH J
+ I
.CCI,
~CCh~cH:cc,I
3.2. Dependence of Sign of Enhancement on Electron-spin Multiplicity of the Initially-formed Pair
The last term in equation (24) is multiplied by the factor [pT - (1 - A)p,] which depends on the probabilities of finding the pair initially in the singlet and triplet states. The difference in 4 for two different states, and therefore the intensity of each NMR line from a product, will be multiplied by this factor. There are three different kinds of initial conditions which are likely to be encountered. (i) Singlet-born pairs, s ps = 1; (ii) Triplet-born
PT = 0
pairs, t j+ = l/3
ps = 0;
(iii) Pairs born with uncorrelated
spins, u PS =
PT
=
1/4
On substituting the initial conditions into the multiplying factor we reach the conclusion that the enhancement of a product from a given radical pair born in a singlet state will be opposite to that obtained if the same pair is generated in the triplet state or by an encounter between two independently-generated rudicals with uncorrelated spins. One of the nicest
examples of this effect is that quoted by Closs~73~74~ who was able to produce the benzyldiphenylmethyl radical pair both by thermal decomposition of the unsymmetrical azo compound and by reaction of the triplet state of diphenylmethylene with toluene. Scheme IX: (GH,),CHN,CH,GH,
-
SC,H,),CH..
CH,C,H: 1 (C,H,),CH-CH,C,H,
-
(&H&C:
+ CH,C,H,
The near mirror image spectra obtained from the CH-CH, group of the coupling product obtained from the pair generated by each of these methods is shown in Fig. 4. It has also been found that products from the thermal and triplet photosensitized decomposition of acyl peroxides exhibit enhancements of opposite sign.‘75.7h’ Some especially-striking examples of the opposite signs of enhancements in singlet pairs and pairs with uncorrelated spins have been found in reactions between alkyllithium compounds and alkyl halides. ‘44.77) It is found, for example, that the CH,CH, group of
166
R. G. LAWLER
I
I
120
I
so
60 HZ
FIG. 4. ClDNP
spectra of 1.1,2-triphenylethane obtained from coupling of bcn/yl and diphcnylmc~h!l radic;~I> formed from thermal decomposition of :vo compound (upper) and from reaction of triplet state carhciic (lower). ‘-‘I The scale is in HI downfield of the CH, resonance of totucnc (60 MHrl.
n-butyliodide becomes enhanced during its reaction with n-butyllithium. The details of the reaction are rather complex and open to some question, but the steps responsible for CIDNP are most likely those given in Scheme X. Scheme X:
CEA,
RCH-CH; -LiI RCH, CH,I
+ LiCHzCHZR
II
s
RCH,CH,I
CA&, -
-2RCH,CH;
RCH, CH,i
+
RCH, Cl-l;
RCH, CH?I*
+
RCH, CH;
RCH,CH; ” l 2RCH,CH;
-
(El,
I RCH-0-l; (AE) -
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
167
In Scheme X we have used a star to indicate molecules with enhanced NMR spectra. The symbol EA has also been introduced to indicate an enhanced multiplet with both absorption and emission lines in which enhanced absorption is upjeld of emission and vice versa for AE. Reference to Fig. 5 shows that simply adding an alkyne, Z-hexyne, to the reaction mixture alters the contributions to enhancement arising from singlet and uncorrelated pairs and reverses the sign of every line from both the iodoalkane, which here is a product of a scavenging reaction, and the olefin formed as a combination product.
No
additive
:; Bu;LI
+ Bu” I
CH3CH2 CH= CH2
r
CH,CHP CH2 CH21
j
Added 2 - hexyne
fm FIG. 5. Effect of added alkyne in changing radical pairs giving CIDNP diffusive (lower) in reaction between n-butyliodide and n-butyllithium
from singlet-born (upper) to (‘H spectrum at 60 MHz).
In the special, and so far hypothetical, case where a system involving both radicals and dimerization products is in thermal equilibrium one would of course find no enhancement. One way of viewing this necessary requirement is that the enhancement generated in each dissociation of dimer to form a singlet pair is exactly cancelled by the enhancement of opposite sign generated by pairs of free radicals which have lived long enough to undergo spin relaxation and re-encounter partners from other dissociations (Scheme XI). Scheme XI :
,,
Rz _
2R. &
2R. __r
R,
It can be shown by considering the kinetics of the reactions that the expression in equation (24) for the reaction efficiencies of different nuclear spin states is quite consistent with this view.(‘,“) Such cancellation apparently may also occur even when the system is
168
R. ‘3. LAWLER
not in equilibrium. It is found, for example, that when the reaction in Scheme X is carried out with n-butylbromideinsteadofn-butyliodidenoenhancement is observed in the 1-butene rapidly formed unless a scavenger such as another olefin is added to upset the balance of enhancement from singlet and uncorrelated pairs.“’
Prohahility
i
It has been proposed “” that the dependence of the sign of enhancement on the multiplicity of the radical-pair precursor should be a useful tool in photochemistry where the multiplicity of product precursors must otherwise be established by rather laborious determination of the effect of quenchers and sensitizers on product yields. Ambiguity may, however, arise if both singlet and triplet states may give rise to radical pairs. In the case of highly reactive pairs (/1 close to l), it may be possible for the sign of [pT - (1 - i)p,] to be positive (and therefore lead to CIDNP consistent with a triplet precursor) even though /I,~ > 3/j,.. that is. most of the pairs ari
3.4. Drpcvdencr
of Sign
of Enhuncment
on the S -
7;) Mi.uing
Cotfficiewt,
6,
The quantity 6, which is defined in equation (23) is responsible for the nuclear-spin dependent mixing of the singlet and triplet states through its linear dependence on the quantum numbers m, and m,, for nuclei on the two radical fragments. The actual intensity of an NMR transition in a product will then depend on the difference 4, - $,, and therefore on sin2 s,(t, - t,) - sin’ s,(t, - tl), for two nuclear spin states which differ by one unit in the sums of the nuclear spin quantum numbers. To illustrate how the intensities depend on the quantum numbers, on hyperfine splittings and on w1 - o2 we will consider the commonly-occurring case of an NMR transition in a product molecule which to a first approximation Hips only the spin of one set of equivalent nuclei, A. These nuclei may also be coupled to another set of equivalent nuclei, B, on the same or the opposite radical. The transition is thus from spin state II,,. r~,)ll,,. ~1,~)to II,,. ~1, - I ))I,,. ~rl,,).where I, and I,, arc in general irreducible components of the coupled angular momcuL;1 of equivalent A and B
CHEMICALLY
nuclei. In a normal NMR
INDUCED
spectrum this transition
DYNAMIC
NUCLEAR
has a first-order
169
POLARIZATION
absorption
intensity
given by’79’ I(IA, MA3IS, q’
IA3 mA -
I. I,? md x
81A81,(1A
+
mA)(h
-
‘?4
+
I)
(26)
where g,, and gr, ‘are degeneracy factors. For n equivalent spin l/2 nuclei these are gr = n!(n - 2k + l)/k!(n - k + l)! k=
0,1,2 ,..., 0, L2,.
n/2
..,(n
-
I = (1/2)n - k n even
1)/2
(27)
nodd.
The line position, v, in frequency units, for transition mA to mA - 1, is given to secondorder by”” v = VA- JABmB+ (J~,/2~dUB - ‘%)tlB +
“*B
+
1)
+
(JiB/6AB)mAmB
(28)
whereh,, = (vA - vB)is the resonance frequency difference and J,, is the nuclear spin-spin splitting constant. In the first-order limit where JAB 4 a,, the last two terms in equation (28) are very small and the line positions for A transitions depend only on vA, JAB and the quantum number for the nuclei B. The last term, however, shows that when the nuclear spin coupling is not weak, transitions with different values of mA will have different line positions. In the usual field-swept NMR experiment, of course, a transition with a large value of v will occur at low field and vice versa. Substitution of the appropriate expressions for 6, in equation (24) for the transition above yields an expression for the difference in reaction efficiencies which depends only on mAand mg, i.e. is independent of I, and I,. Intensity cc fj m.&m”(r)- &, - 1.m”(T) = + AfJ . -‘,=[pT
- (I - /.)ss] (N f
M + P)
(29)
where N = sin(A,/2)r sin(o, - tozh cos(&m,)r
cos AA(mA- l/2)r
M = sin(A,/2)r sin(A,m,)r cos[(w, - w2) + A,(m,
-
1,‘2)]r
P = sin(A,/2)7 sin A,(m, - 1/2)r COS(W,- Wl).r cos(A,m,)r
(30a) (30b) (3Oc)
and T = t2 - t, is the time between the first and second encounter of a pair. The first ) sign is positive if the product is of the combination type, C, and negative if it is of the scavenging type, S. The second f. sign is positive if the B nuclei are the same radical as A and minus if it is on the other radical. Each of the three terms in parentheses in equation (29), which are defined in equations (30), give rise to CIDNP effects of a different type. In genera1 all three kinds of effects will occur simultaneously, but under certain limiting conditions each term alone may be responsible for the observed effects. These effects are discussed below. It is well to point out, however, that all three terms vanish if A is zero. This, therefore, emphasizes the requirement that enhanced lines arise only from transitions involving nuclei which are coupled to the unpaired electron in one of the radicals. 3.4.1. NET ENHANCEMENT. If we consider the case where there are no other nuclei coupled to A, in effect I, = mB = 0, and we see that M vanishes. Furthermore, all transitions of A occur at the same frequency regardless of mA. Finally, since mA - l/2 = I, - l/2,
170
R.G.LAWLER
I, - 312,. . . , -(I, - 3/2), -(I, - l/2) the sum over all mA of the sin A,(m, - 1/2)r term in P vanishes. Similar conclusions are also reached even when B spins are present, provided both that A, is zero and the spectrum is sufficiently first-order that the last term in equation (2X) is negligible compared with the NMR linewidth. Under either of these conditions CIDNP arises solely from the term N, which is non-zero only if wi # 02, i.e. only if the g-factors of the two radicals are different. When N is averaged over all m, one is still left with a net enhancement of the transitions due to A. The spectra shown in Figs. 2 and 3 are both of this type. In the first case the enhancement is generated in the radical pair consisting of the phenyl radical, C,H, . , and the benzoyloxy radical, C,H,CO, . , in which the g-factor of the former radical is less than that of the latter. (*O)The net enhancement in Fig. 3 arises in the CH,. . Ccl, radical pair.“’
3.4.2. MULTIPLET EFFECT.In the case where o, = o2 and the spectra are also first order the contributions from N and P vanish and leave M as the only possible source of CIDNP. M is non-vanishing, however, only if both A and B are coupled to one of the unpaired electrons in the radicals. The enhancement obtained under these conditions is such that a line in a multiplet with quantum number mB has an intensity opposite in sign to that for the line on the other side of the multiplet with -mB. This kind of enhancement has come to be known as the multiplet efSect. It is shown in the spectra in Figs. 4 and 5 where the enhancement is generated in pairs of hydrocarbon radicals which have almost identical g-factors (see Schemes IX and X). 3.4.3. HIGHER ORDER EFFECTS.(i) Second-order CIDNP. It is possible to envisage a case in which B may be coupled to A in a product but where o, = w2 and A, = 0. This happens if the radical pair involves identical radicals and the nucleus B is either in a part of the radical inaccessible to the unpaired electron, e.g. the CH, group in a radical of the type CHJH,CX,., where X has no nuclear spin, or is transferred to the radical from a diamagnetic molecule in a scavenging reaction. 2H,R.
=
H,RH,
In either case both N and M vanish. Since AA is not zero, however, P may still make a non-zero contribution to the spectrum provided that the last term in equation (28) exceeds the line width so that transitions with different values of mA appear at resolvably different positions. The signs of enhancement will be independent of the sign of A, and will be opposite for the oppositely-signed value of /nA - l/2. Each first-order line for a given lug will therefore exhibit a “multiplet effect”. So far the only clear cut example of this rather striking effect has been found by Kaptetn “2.‘x’ in the NMR spectra of the second-order components of the methyl group doublet of 2-cyanopropane formed in Scheme XII, which is of the second type mentioned above. Scheme XII :
(CH,),C(CN)N,(CN)C(CH,), -
2(CH,),(CN)C. 5
(CH,),(CN)CH
(ii) intensity borrowing. The expression given in equation (26) for the intensity of a transition arising from A is strictly valid only in the limit where J,, 4 6,, so that transitions are essentially pure A or pure B transitions. We have already mentioned the line shifts which may arise when this condition breaks down and may expose strongly-enhanced transitions of opposite sign which otherwise cancel. The observation of these second-order
CHEMICALLY
INDUCED DYNAMIC NUCLEAR POLARIZATION
171
effects, however, still requires that the nuclei whose spins flip in the transition be coupled to the unpaired electron on one of the radicals. It is also possible to have an apparent enhancement of nuclei which are not coupled to an unpaired electron, provided that they become strongly coupled to another set of nuclei which are strongly enhanced. Under these conditions what appear to be pure A transitions, for example, are really a mixture of A and B transitions and may therefore reflect the enhancement of B. If B is very strongly enhanced, A intensities may be affected even though the amount of intensity borrowing in the normal spectrum is rather small. A rather nice example of this effect is given by Lehnig and Fischer@i’ in the spectra of chlorobenzene and para-dichlorobenzene formed from the phenyl and paru-chlorophenyl radicals respectively. In these radicals the ortho protons are more strongly coupled to the unpaired electron than are the metu protons. In the non-first-order spectra of these products, however, the lines nominally due to transitions of the metu protons appear in emission, probably because of intensity borrowed from the strong emission lines arising from the ortho protons. 3.5. Rules for Predicting the Sign of Enhancement Equations (26H30) contain, in principle, all of the information required to calculate the intensity of first-order CIDNP NMR lines in an A,B, spin system provided that one knows how to average over a distribution of re-encounter times, z. Even in the absence of such averaging, however, it is possible to predict the sign of enhancement of such a transition by considering the way in which the two + signs in equation (29) and the signs of the parameters A,, A,, o1 - o2 and m, (which in turn requires knowing the sign of J,,) affect the expression for the intensity. The effects of the f signs are trivial and given by definition. Furthermore, only the sine factors in equations (30) will have an effect on the overall sign of the intensity, provided that the distribution of re-encounter times is a monotonically decreasing function of time. Rules based on these considerations for determining whether net enhancement will be enhanced absorption (A) or emission (E) and whether the multiplet effect will be EA or AE have been recently provided in tabular form (14)and in rather convenient formulae.‘*.“’ The formulae given by Kaptein “‘)for the net enhancement, r,,, and multiplet effect, r,, are net enhancement: r”, = F(SI
- SzM*
+=A
-=E
multiplet effect:
rm, = wWBJAB~~~~
+ =
EA
- = AE
where p, E and g’ABare signature symbols given by the following rules: p=
+ when the radical pair is formed from a triplet (t) or by an encounter of free radicals with uncorrelated spins (u). i - when the pair is formed from a singlet (s) precursor. + for combination
E=
products
( - for products of scavenging. + if nuclei A and B belong to the same radical uAB
=
-
if nuclei A and B belong to different radicals.
172
R.G.LAWLER
As an illustration of these rules we consider the spectrum of n-butyliodide shown in Fig. 5. The nuclear spin system is the RCHy’CH\*‘I formed as a scavenging product of the radical RCH,CH, . group. Both the top and the bottom spectra are of the pure multiplet effect type. It is now well known (82) that for alkyl radicals of the above kind the hyperfine coupling constants of the protons c(and fi to the unpaired electron are negative and positive respectively. We therefore have A, = -, A, = +, E = +, J,, = +, cAB = + from which we conclude that the multiplet effect should be AE if the radical pair precursor were singlet and EA if it arose from encounters of free radicals with uncorrelated spins. (The possibility of a triplet radical precursor is highly unlikely for a thermally-induced reaction such as this.) The vicinal proton coupling constant in RCH,CH,I is also known to be positive.‘83’ From the chemical standpoint the spectra in Fig. 5 lead us to the rather remarkable conclusion that the addition of an alkyne to the solution of an organolithium compound and alkyliodide specifically intercepts those radical pairs formed in the initial reaction between the two reagents. It has been proposed that this occurs because alkynes may form complexes with alkyllithium compounds and therefore be held near the first formed singlet-state radical pair.‘44’ The above rules may be considered absolute as long as (a) only two sets of equivalent nuclei are involved, (b) the radical-pair precursor fits neatly into one of the categories s, t or U, i.e. is not formed in an unequal mixture of s and t, (c) the effects are either pure multiplet effect (gl = g2) or pure net enhancement (J,, or A, = O), (d) the spectra are first order. Although the rules undoubtedly still sometimes hold even when these restrictions are relaxed, their range of validity has not yet been explored sufficiently to guarantee that misinterpretation will not result. When the rules are not applicable the signs of enhancement: the higher order effects discussed above and others as well, may be taken into account by numerical solution of the expressions for NMR intensities and frequencies, suitably modified to include changes in populations, in a manMer analogous to that now routinely applied to ordinary spectra. (84) Before discussing computer simulation, however, we must consider different schemes for averaging equation (24) over a distribution of re-encounter times, t.
3.6. Quantitative
Calculation
of CIDNP
Intensities
3.6.1. PROBABILITY OF DIFFUSIVE RE-ENCOUNTER OF PAIRS WITH CORRELATED SPINS.In
ELECTRON
Section 2.2.2 the opinion was expressed that at present there is considerable doubt about the proper time dependence of the function f (t)dt which describes the probability of a re-encounter of a pair taking place between t and t + dt. One can at least make a reasonable guess, however, that it will have the following properties: (a) After some short time after a pair separates f (t) will decrease monotonically with increasing time, (b) the functional dependence will be approximately t-“, (c) each degree of freedom for a random walk separation of the pair will make a contribution of l/2 to the power qc4’) (d) the number of degrees of freedom will lie somewhere between 3, for the three relative translational modes for the pair of radicals, and 9, which would include the addition of three rotational degrees of freedom for each radical, and therefore that n will lie between 3/2 and 9/2. Furthermore, it is probably valid to. define a characteristic “encounter lifetime”, zD, which is the time required for one diffusive step in the random walk, and before which ,f’(t) is small. We can also define the probability, p = j,“,,f’(t)dc that a pair
CHEMICALLY INDUCED DYNAMIC NUCLEAR POLARIZATION
will undergo a re-encounter approximation forf(t)
sometime between Q, and infinity. This gives us the following o
f(t) = 0 f(t)
173
= (n -
5D ‘< t
l)z;;‘pt_”
(31) 312 d
n d
912.
The parameters n, p and Q, are, unfortunately, all highly subjective gnd depend on how we define an encounter. In general an “encounter” will require that the pair be in close enough proximity and have the r.ight orientation of the two radicals for there to be a probability i that they react before separating. If the rate of rotation of the fragments is much faster than the reciprocal of the encounter lifetime, TV),the rotational degrees of freedom will not count in the random walk and one should find approximately
a t-3’2
dependence for f( t).
For a three-dimensional
random walk TVmay be approximately zc,
g
defined as
2 WD,,,
where dz is a mean square “encounter diameter” and Dre, is the diffusion coefficient for relative translational displacement of the two radical fragments.‘85’ For small molecules in relatively nonviscous liquids Dre, is approximately 5 x lo-’ cm’ set- ‘. Therefore an encounter diameter of 5 A would give a value of.T, = lo- ’ 1 sec. To calculate the observable value of 4 we must integrate the expression in equation (24) over all intervals T = t, - t,, using an appropriate form forf(z).’ 4 = IT
(32)
Wf(z)dz.
0
The integration is simplified if we recognize that typical values of 6, are 5 x lo8 rad set- ‘, while most scavenging and relaxation processes probably take place with characteristic lifetimes, ~~~longer than lo-’ sec. Under these conditions the sine-squared term in equation (24) undergoes many oscillations before scavenging takes place. It is therefore permissible to ignore the exponential in the integral. In the case of inverse 3/2 power re-encounter probability the averaging of equation (24) yields 4: = &[l
+ p(1 - ;1)] + L7t*(p/2)z#2[p. - (1 - Q&J+
= 1 - 4;.
(33)
The conclusion 4: = 1 - & is unaltered by the averaging, provided that scavenging product is formed before relaxation in the radical takes place. It should be pointed out that the dependence of the reaction probability on the square root of the mixing coefficient, 6,, is a peculiarity of the inverse 3/2 power dependence assumed forf(t). For example, if the unlikely assumption is made that the pairs disappear predominantly via the exponential decay in equation (24) the expression for I#$ analogous to equation (33) varies as the square of fi,,.(16,21*28,34) A ch aracteristic of the early stages of evolution of the radical-pair model was theinclusion of a non-zero exchange interaction J, for D pairs. Cl8,3s) This may also lead to a 6: dependence for J larger than the hyperfine splittings. It now appears, however, that virtually all known high-field CIDNP spectra may be adequately explained by a S$ dependence, (*) although the predicted intensities are in many cases quite similar for the two different functional forms of 6,. 3.6.2. COMPUTER SIMULATION OF CIDNP SPECTRA.The singlet-triplet mixing coefficient, 6,,, is a function of nuclear spin quantum numbers in a representation, x,,, consisting of
174
R. G. LAWLER
products of functions for individual spin Hamiltonian for a diamagnetic C. will be of the form
sets of equivalent nuclei [equation (12b)]. The nuclear product molecule, for example,.a combination product,
HC = 1 Vilf + C Jijlilj I
where vi coupling. generally states xn.
(34)
i
is the Larmor frequency of the nucleus i and Jji is the nuclear spiny spin In the case of non-first-order spectra. however, the spin functions z,, will not be eigenfunctions of H, and the transitions in C will involve mixtures of the If we write the eigenfunctions, xy, of H, as linear combinations of xn we have
The intensity,
I,.,, of a transition
between
two eigenstates
1’and p of Hc is given by (36)
where the resonance frequency of the spectrometer, v,, , satisfies the conditions v0 = v, - vp. The diagonal density matrix elements in the xy representation are, however, related to those in the x,, representation by
where U,, and (U-I),, are the elements of the unitary matrix which transforms xn into xy, as in equation (39, and its inverse respectively. The intensity of the transition is therefore related to the reaction efficiencies 4,. e.g. as given in equation (33), by
The transition probabilities (x,,lC I&,)’ and the matrix U are both calculated in the course of the usual numerical simulation of normal NMR spectra.@5’ These programs may be relatively easily modified for simulating high field CIDNP spectra by providing for weighting of the level populations as in equation (38) before the intensities are calculated. including magnetic equivalence The programs NMRIT”” and LAOCOON,‘57’ factoring, Cl*,l 9, have now been modified to perform such calculations. Consequently simulations are now beginning to appear in the literature as a standard part of reporting CIDNP results (5’,60,81,8’) An example of a simulated high-field CIDNP spectrum calculated by using equations (33) and (38) is given in Fig. 6 which shows the spectrum of the n-propyl sidechain of n-propylbenzene formed from the coupling of ethyl and benzyl radicals produced during the reaction of ethyllithium with benzylchloride.(88) Scheme XIII : C,H,CH,Cl
+ LiCH,CH,
----+
C,H,CH,
. . CH,CH,
-
C,H,CH,CH,CH,
A rather reliable computer simulation can be made in this case because all of the hyperfine splittings in the free radicals are known from ESR measurements(89) and the NMR spectrum of the product has been analysed completely to give chemical shifts and spin-spin splittings of the protons. (90,9 ‘) The small deviations which appear between the simulated and experimental spectra are probably due to a combination of effects such as variations of Tl from line to line, cross-relaxation between enhanced transitions. partial breakdown in the
CHEMICALLY +cH2
FIG. 6. Experimental n-propylbenzene
INDUCED
Cl t Li CeCH,
(upper) and simulated formed in the coupling
-
DYNAMIC
NUCLEAR
POLARIZATION
175
#GHz CHz CH,
(lower) CIDNP ‘H NMR spectrum (60 MHz) from reaction between benzylchloride and ethyllithium.
assumptions involved in deriving equation (33) and, perhaps most significantly, the fact that lines in different parts of the spectrum were of necessity scanned at different times during the chemical reaction. In fact, since all of these effects undoubtedly do occur to some extent in all CIDNP spectra, the agreement between calculated and observed spectra in this and other cases is quite remarkable.
4. Dynamic
Effects
4.1. Ej>cts of’Rela.uation
OH CIDNP
Intensities
(a) Tl the same for all lines in the spectrum. In Section 2.3 we discussed qualitatively the time dependence of the CIDNP signal. In the special case where all lines in a spectrum have the same T,, as they might for example if all nuclei relaxed by interaction with an added paramagnetic substance, the relative intensities of lines are unaffected by relaxation. If the overall rate constant for the chemical reaction, k,, is much smaller than T;’ the observed intensity is simply reduced by the factor k,T, as shown in equation (8b). If the two rate constants are more nearly of the same magnitude, however, the expressions for intensity as a function of time become more complicated. Buchachenko and coworkers”” ““ha ve derived extensive formulae for CIDNP intensities including relaxation.
176
K. G. LAWLER
Although some of their conclusions are independent of the mechanism by which the enhancement is produced, they have not incorporated the radical pair model into their discussions and their studies to date are of rather limited utility. Although it is possible, in principle, to analyze time-dependent CIDNP spectra to obtain both Ti and k, it is doubtful if either the precision, convenience or generality of the method can compare with the more conventional methods of obtaining these quantities. (b) A slowly relaxing nucleus coupled to a set of rapidly relaxing m&i. It has become generally well established that the relaxation time of a proton bound to a carbon atom containing no other magnetic nuclei (a methyne proton) is often an order of magnitude longer than if the carbon contains one or more additional protons which may relax by interaction. (lol) This means, for example, that for a the intramolecular dipole-dipole CH,CH or CH,CH group the nuclear spin levels which differ only in the spin state of the CH, or CH, protons will reach thermal equilibrium more rapidly than those which differ only in the spin of the methyne proton. The redistribution of populations for an AX, spin system exhibiting a multiplet effect and/or net enhancement and showing such selective relaxation of the X spins is illustrated in Fig. 7. The effects are as follows: (i) a pure multiplet effect disappears for both sets of nuclei and approximately with the relaxation time of the faster relaxing set, (ii) net enhancement also disappears for the fast relaxing set but is retained in the slower set, (iii) a spectrum exhibiting both multiplet and net effects relaxes with the faster relaxation time to give no enhancement of the fast relaxing nuclei and to destroy the multiplet effect components of the enhancement of the After
relaxation
of x
-“A
-“A
-“A +nA
-“A
+5
+“A
+nA
I
I
I
I
I
I
FIG. 7. Etfect on CIDNP intensities of fast relaxation of two nuclei. X2. coupled to a slow relaxing nucleus, A. Symbols for contributions to level populations given at the right of the energy levels to net effects, are: 11”= common to all levels z N,,,/8; m = multiplet effect: nA, )1Xare contributions assumed negative for A and positive for X. Pertinent quantum numbers for each level are given at left.
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
177
slow
set. The net enhancement of the slow set remains, however. Consequently, spectra corresponding to case (iii) appear with exaggerated net effects. Attempts to simulate such spectra when all the necessary g-factors and hyperfine splittings are known from ESR would therefore be expected to predict a larger amount of multiplet effect than observed. An example of such a failure of simulation is shown in Fig. 8 which gives a comparison of the expected and observed CIDNP spectrum of the methyne proton in C,H,C(CH,),H produced in the reaction”87’ in Scheme XIV.
FIG. 8. Disagreement (60 MHz) of methyne
between observed (A) and simulated (B) CIDNP proton NMR spectra proton septet in C,H,CH(CH,), brought about by destruction of multiplet effect by rapid relaxation of CH3 protons.
Scheme XIV : C,H,C(CH,),CHO
hv
C,H,C(CH&
. . HCO -
C,H,C(CH,),H
+ CO
In this example both the hyperfine splittings and g-factors necessary to simulate the spectrum of the product are known and lead to a simulation with an appreciable multiplet effect component in contrast to the experimental results. In cases where all of the hyperfine splittings and both g-factors are not known the above effect can lead to estimated g-factor differences which are too large.‘8’
178
R. G. LAWLER
In cases where only partial equilibration of the levels of the faster relaxing set of nuclei takes place, distortion (but not complete removal) of the multiplet effect is observed.‘“” Both steady state and transient relaxation effects in this intermediate region have been elegantly demonstrated recently by Muller and C10ss.‘~~’
4.2. Conrpc~titiori hetwwrl Scrrt’rn~~ir~yard Rr~bsrrtion of’ N uclcar Spins in Free Radicals The sequence of events by which enhancement shown in Scheme XV.
in a scavenging
Schcmc XV :
x”’
%
product,
S, is generated
is
T,
R-E-D-E-D-S-S, TO
- ?I
b
'c&, D
0
The times t, to t, are those discussed in Section 2.4.2 and it is assumed that both scavenging and relaxation are first-order kinetic processes describable by characteristic lifetimes. Using equations (25) and (8b) and the analogous relationship for the scavenging product, the observed population difference (S, - S,), for a transition between states n and tn of product S containing a set of nuclei which may also end up in’ a combination product under steady-state conditions is
6, - %a)= (UK =
-(cl
‘(4: - &,)T,,,,T, ‘/bi ’ + T&-1 + T&J - ctJ(Ts,,JG,,,)~S %G l + 4- l + ~i&?J.
(39a)
Wb)
In the special case where the rate of the scavenging reaction leading to S is much faster than either nuclear relaxation in the diffusively-separated pair, D, or competing reactions which give rise to other scavenging products, S’, corresponding transitions in S and the combination product, C, have relative intensities which are affected only by the relaxation times in the two kinds of diamagnetic products. If T, and Tc are the same, the CIDNP spectra of S and C will have opposite but equal intensities. Although these limiting cases are sometimes observed (see for example, Fig. 3) there are at least two reports of differences of intensities of CIDNP lines in scavenging and combination products which have been attributed to loss of intensity during the scavenging step through competition between the scavenging rate and the rate of relaxation in the intermediate radicals.(4h*57,58) Comparison was made of the intensity of net enhancement of protons in the products R-R’ and R-R formed according to Scheme XVI. Scheme XV 1: R-X-_-‘-R..
--
R’-R./.R’%rR
Aqzo1~-~’
-_R-_R
kd As=0
(see reference 46) R = . CHCl,, R’ = . CH,COCH, (see reference R = .CHZC,Hs, R’ = .CH(p-BrC,H,),
57)
In both cases the “scavenging” process was recombination of R. with a radical produced from another encounter, i.e. r; ’ = k,(R.). Under these conditions net enhancement is generated only in the non-identical pair and singlet-triplet mixing during the diffusive encounter between identical radicals, necessary to produce R-R products, may be ignored. Use of the simplified scheme above, which ignores other competing paths for loss of R.,
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
179
and estimation of k,(R.), leads to reasonable estimates of the order of 10e4 set for the proton relaxation time in benzyl and substituted methyl radicals. Relaxation has also been invoked by C10ss’~~) to explain variations in the relative CIDNP intensities of the ortho and aldehyde protons of p-chlorobenzaldehyde serving both as a scavenger and as a scavenging product .in the reaction sequence shown in Scheme XVII. Scheme XVII : p-CIC,H,CH,OH
+ “(I?-BrC,H,),C=O
&ClC~H,CH*OH-
-
,,_(.,$ 1(.Hot p-ClC,H,CH*O
.(p-BrC,H4)ZCOH
+ p-ClC6H4i3HOH
It was found that at high concentrations of added benzaldehyde the ratio of the aldehyde to ortho proton intensity in the benzaldehyde product approached that calculated for the radical-pair model as in equation (33). At lower concentrations, however, the (starred) aldehyde proton (which was benzylic in the radical) was of anomalously low relative intensity. This was attributed to competing nuclear relaxation in the hydroxybenzyl radical which is more effective for the benzylic than for the ring protons owing to the higher unpaired electron density at the former position. If TD for the benzylic proton was assumed to be 10m4 set the concentration dependence of the relative intensities was best fit using a value of 8 x lo4 M-’ set- ’ for the rate constant, kt,, for transfer of a hydrogen atom from the hydroxybenzyl radical to the corresponding aldehyde. This value is, however, very sensitive both to the estimate of TD and to possible changes in relative relaxation times of the two kinds of protons in the diamagnetic product brought about by changes in the medium. It has been found.““’ for example. that the addition of as little as 0.1 M molecular iodine as a scavenger during the decomposition of benzoyl peroxide causes up to a sixfold, but non-uniform, decrease in relaxation time in the various types of protons in reaction products.
4.3. scuucmjng
R~~~~ctims which cornprte with Rr-crJc~our7tc~r.s qJ’Rudictr1 Pairs
In the derivation of the nuclear-spin dependent combination efficiency given in equation (25) of Section 2.4.3 it was assumed that a D pair formed after the initialization step retains its identity during the time t, - t, between the first and second encounters of the pair. Consequently the nuclear-spin dependent mixing coefficient, 6,, remains constant between encounters. It is possible, however, to have a rapid reaction, or other rate process, which competes with the re-encounter frequency, changing the identity of one partner, but not destroying the pair. In such a case the new pair may undergo a re-encounter to form a new combination product, but one in which the unchanged radical partner has felt the influence of two mixing coefficients, 6, and Sk, and may “remember” its partner from the original encounter. The life history of such a pair is shown schematically in Scheme XVIII. g 7s’ Scheme XVIII : R-E-D
-D’--E’-C’
‘0
‘I t-t,+
‘1
‘2
180
R. G. LAWLER
Five such processes
are illustrated
below with reactions
where CIDNP
has been observed:
Scheme XIX : A. Fragmentation’102’ -
C,HsCO,O,CC,H,
-“Oi b C,H, . .CH,CH,
. CH,CH,
C,H,CO,
L C,H,CH,CH,
I C,H,CO,CH,CH, B. Rearrangement’io3’
(CH,),CHCIH,C(CH,), + LiC(CH,),
-
(CH,),CHCJH, . WH3)3
+ CH2=C(CH3)2
-
.CH2CH2CH=CH2.C(CH3),
(CH,),CHCH,Br CH3CH2CH=CH2 C. Conformation
change with exchange
of hyperfine
splittings”
2,
(A’ )
(A)
H\c-ci /H ,,’
‘R
D. “Pair substitution”‘102’ C,H,C02.
.C,H,
“‘HZ’H’K6H,C02.
E. Racemization
. CH,CH,
---+
C,H,CO,CH,CH,
+ C,H,I
1 C,H,CO,C,H, of pairs generated
from optically
active precursors(io4)
R
The expression analogous to equation (22~) for the probability of finding the pair D’ in the singlet state at time t, if it was formed suddenly for D at t; is (Appendix A.111) PF’~ n, n(t 1t’) = 2;’ If the mixing coefficients dependent recombination
r-“‘S((l
- 1.)~~ + [pr - (1 - A)pJ sin2[&t
- (6; - &)t’]).
(40)
for D and D’ are different the probability of an electron-spinoccurring at t, therefore depends not only on the time interval
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
181
POLARIZATION
t = t, - t, between encounters, but also on the difference t’ = t; - t, between the time the first encounter ended and the time D changed to D’. The latter time interval is, however, governed by the rate constant, k, for the pair reaction process. Thus the probability of reaction occurring between t’ and t’ + dt’ will be k exp( - kt’) dt’. The probability of combination product C’ in spin state n forming at the end of the second encounter then becomes, by analogy with equations (3 1) and 32),
42’ =
l’k
m ‘p&. ss0 0
(t,t’) e-““f(t)
dr’ dt.
(41)
If the rate constant k is less than both 6, and 6, -6: (typically lo8 set- ‘) but greater than the rate constant rs 1 for processes which scavenge both D and D’, the integral in equation (41), evaluated with the three-dimensional random walk expression for_/“(t)given in equation (3 l), becomes f#$’ = (X/2) pq
7c1’2 CO -4~s
+
PTIk”’ + CPT-(I - 4psl W/W, - 4Jl x
(42)
Thus the enhancement in the product C’ depends, as expected, on the mixing coefficients in both D and D’ pairs. The functional dependence is, however, more complicated than a simple weighted average of the mixing coefficients for the two kinds of pairs. This may lead to some rather striking intensity effects which deviate markedly from a simple superposition of net enhancement and multiplet effects even for first-order spectra in simple spin systems. Figure 9 shows the calculated and observed CH, quartet of ethylbenzene formed from benzoylpropionyl peroxide in the 2-step fragmentation process in Example A. In this particular example the difference in mixing coefficients, 6, - Sk,is nuclear-spin independent and reflects only the g-factor difference of the first pair. The rate constant k in equation (42) serves only as a scale factor for the intensity of each line in the spectrum of C’ and therefore does not affect the relative intensities of lines. It is in principle, however, possible to obtain a value for k by comparing the intensities of lines from C’ and those in a product C formed by combination of pairs which did not undergo the reaction but nevertheless underwent a reactive diffusive re-encounter. In Example A the product C is ethyl benzoate formed from the initially formed benzoyloxy-ethyl radical pair. Any such analysis of the relative intensities will inevitably, however, suffer from lack of information about the relative values of the parameters 1, p and zn for the D and D’ pairs. Nevertheless it is often very difficult to obtain even reasonable.estimates for rate constants in the range 10’ to lo9 see-’ which would be detected by this phenomenon. For example, the best estimate”05’ for the rate constant for decarboxylation of benzoyloxy radical under the conditions in Example A is about lo5 set- ‘. The mere appearance of a “memory effect” in ethyl benzene shown in Fig. 9 is good evidence that this estimate is at least two orders of magnitude too low. Examples B and C are in the same category as A, but have not yet been studied in
182
R. G. LAWLER
METHYLENE
GROUP
OBSERVED
OF ETHYL
BENZENE CALCULATED
FK. 9. Comparison of observed (A) and calculated (B) CIDNP proton NMR spectrum of -CH,quartet in ethylbenzene formed from decomposing C,H,CO,O,CCH,CH,.
(60 MHz)
sufficient detail to permit quantitative comparison between the experimental and calculated spectra. The “pair substitution” process given in Example D is invoked to explain cases in which combination products have been observed containing molecular fragments which were not in the original pair but which, by application of the rules in Section 3.5, must have been formed from singlet born rather than uncorrelated radical pairs. Pair substitution has also been invoked to explain the change in sign of the enhancement of HCCl, obtained during thermal decomposition of isobutyryl peroxide with high concentrations of added BrCCl,. Cl06) Kaptein (i2) has attempted to estimate the rate constant for the reaction Scheme XX: 2(CH,),CH.
Br’TA+ (CH,),CHBr
+ (CH,),CH.
.CCl,
-
HCCI,
by comparing the experimental dependence of the enhancement factor on BrCCl, concentration with that calculated using a model for the competition between diffusion and reaction similar to the above. Example E is included because of the observation of CIDNP from rearrangement reactions in which only partial racemization of an initially optically-active radical-pair precursor occurs. Cl04) The “rate constant” for the racemization process in this case, however, is simply the correlation time for rotational motion of the asymmetric radical. Since this is typically IO- ’ ’ set, rotation should be completely randomized before re-encounter of pairs which have lived long enough to develop appreciable nuclear spin dependent S-T mixing. This in turn leads to the prediction of equally intense CIDNP from the two
CHEMICALLYINDUCEDDYNAMICNUCLEARPOLARIZATION
183
enantiomers. This would, of course, be observable only if one could devise a suitably unsymmetrical environment, e.g. by the use of asymmetric paramagnetic shift reagents, to simultaneously observe the NMR spectrum of both species. Such an experiment has not yet been reported, but would provide a good opportunity for confirming the rather long times necessary for the production of CIDNP.
5. Variation of CIDNP Intensities with Magmetic Field 5.1. High Field .Eficts In one of the first reports of CIDNP by Bargon and Fischer,‘36’ it was stated that the intensity of the emission line from benzene formed from decomposing benzoyl peroxide depended on the strength of the magnetic field in which the measurements were carried out. At that time the field effect was interpreted to arise from the frequency dependence of relaxation times in the free phenyl radical and was taken as evidence in favour of the Overhauser mechanism for CIDNP. The fields used in this experiment, however, were only 10, 13 and 23 kilogauss because both the reaction and the NMR spectrum were run in the same field. In later work(81,107-1 lo) it h as been found that nuclear T1 in reaction products are sufficiently long that the reaction may be carried out in a variable field and the sample rapidly transferred to a spectrometer operating at constant field and still retain an appreciable portion of the CIDNP generated in the external field. This has greatlyincreased the ease with which the field dependence of CIDNP may be studied. 5.1.1.CHARACTERISTICSOFHIGHFIELDEFFECTS. Theabovefieldeffectonbenzeneemission and the results of subsequent investigations may now also be understood when viewed from the vantage point of the radical-pair model. In fields sufficiently high that the singlet state of the pair mixes only with the T, triplet state the variations of CIDNP intensities with field depends on the following qualitative factors: (a) Variations with magnetic field of enhancement generated in radical pairs occurs only if the radical pairs have different g-factors, i.e. if there is net enhancement. (b) The thermal polarization, p”, used to calculate the enhancement factor increases linearly with the field [equation (3)]. Therefore the enhancement factors for CIDNP spectra, such as the benzene emission spectrum above, in which the reaction and spectra are run in the same field, have this additional source of field dependence. In particular, even if the radical pair interactions produce no field dependent CIDNP, e.g. for identical radical partners (Ag = 0), the enhancement factor calculated by comparing the “normal” and CIDNP intensities at the same field will decrrasr with increasing field. (c) Since nuclear relaxation times and intensity borrowing may be field dependent, enhancement factors must be suitably corrected for these added sources of intensity variation with field.
5.12. EXAMPLL OF FIELD IXPI.SIWW~-.IN Tw: HIGH FIELD REGION: BENZENE FROM BENZOYL PI:KOXIIX.The
field dcpendcncc of the CIDNP spectrum of benzene formed from benzoyl
184
R. G. LAWLER
peroxide according to Scheme XXI has been studied extensively,by Fischer and Lehnig’lo8’ and by Slonim et al.” lo) Scheme XXI :
(C,H, CO,),
-
C6H5 CO,.
Since the ortho protons in the phenyl radical are most strongly coupled to the unpaired electron, the contribution of the other three protons to the enhanced spectrum may be ignored as a first-approximation. The net intensity of the tw.o transitions mortho= + 1 + 0 and 0 + - 1 of the two equivalent ortho protons as given by equation (33) then becomes 1 cc (4+1 - 4-i) K C(91 - S~W’~O
+ A? - C(s1 - SJWBO
- ‘a+
(43)
where A is the hyperfine coupling of the ortho protons and radical 1 is the phenyl radical. The intensity in equation (43) passes through a maximum which depends on both A and (gi - gz) and falls off at both high and low fields according to the. asymptotic relationships IAl
(gi - g&?, $ A(“high” field)
III @z
(gI - g&j
I11oc I91 - g$
e A(“low” field) By
I91 - gPf
= 2JA(gauss)l lgl - gJ_l.
IAl+
’ Wc)
In the present case g1 = 2.002,““’ g2 = 2.010,‘1’2’~ A = + 17 gauss.““’ According to equation (44~) ,r should occur at approximately 4000 gauss, in almost exact agreement with the experimental observation. (i lo) The functional dependence on field predicted by equations (44) in both the “high” and “low” field limits also agrees well with the results obtained by running the reaction in the variable field and observing in a 60 or 100 MHz spectrometer. In experiments where the spectrometer field is also varied, however, the enhancement factor is obtained by dividing equations (44a) and (44b) by B,. This leads to the prediction that the enhancement factor, measured, for example, by determining the amount of added unenhanced product needed to cancel a CIDNP emission signal,” l 3, should be independent of Ho when (gl - g2)Bo 6 A and fall off as BG” in the “high” field region. The latter field dependence predicts enhancement factors for benzene in the early work of Bargon and Fischerc3@ m the ratios 1: 2.4: 3.9 at 100,56.4 and 40 MHz respectively. These are in reasonable agreement with the observed ratios 1:2.6:6.8. 5.2. Low Field EfSects In the field region below approximately 1000 gauss the strength of the coupling of the unpaired electrons to the applied magnetic field becomes comparable in magnitude to the electron-nuclear hyperfine coupling. Under these circumstances the singlet electron-spin state of the radical pair mixes with all three of the triplet states and the mathematical simplification leading to equation (22) no longer occurs (see Appendix AIV.). Much of the t Assuming that the g-factors of C,H,CO, and D02CCH=CHC0,. are similar.
CHEMICALLYINDUCED
DYNAMICNUCLEARPOLARIZATION
18.5
qualitative understanding of CIDNP leading to the simple high field rules given in Section 3.5 is lost and it is necessary to resort to computer-assisted numerical calculation of nuclearspin state populations. (1g,3L,86)In the simple case of a radical pair with only one set of equivalent nuclei on one radical fragment and no nuclei on the other an analytical expression for &,&t) is m fact obtainable’86’ which is valid for all fields. It is, however, too complicated to be very useful in obtaining physical insight into the problem. As in the high field case, the shape of the field dependence in the majority of examples of low field CIDNP observed to date may apparently be simulated adequately using only hyperfine splittings and g-factors. Differences in g-factors, however, become unimportant in low fields. The parameters A, p and z,, which describe the reactivity and diffusive properties of radical pairs serve only as scaling factors in low field as well as in high. 5.2.1. SOMECHARACTERISTICS OF LOW FIELD CIDNP. The only practical way at present of determining the field dependence of CIDNP over a wide range of magnetic fields is to observe the spectrum at a constant high field, i.e. in a conventional NMR spectrometer, and run the reaction in a variable field. In order for the observed NMR spectrum to reflect the population differences built up in the auxiliary held, however, two conditions must be met: (a) The polarization must persist during the transfer of the sample between the two magnets. A sufficient condition for this is that the spin-lattice relaxation time must not be too much shorter than the time required for transfer. (b) The relative populations of the nuclear spin levels must not be changed during the time that the system spends in passing through the intermediate field between the variable magnet and the spectrometer. Requirement (a) has limited the observation of low field CIDNP to protons with 7” of several seconds or more. These have so far included benzene, CHCl, and the methyne protons in (CH,),CHI and C,H,CHClCH,CH,. Attempts to observe low field CIDNP from protons in methyl or methyiene groups have been generahy unsuccessful except when the enhancements are very large. There are three more important and subtle factors which determine as in (b) whether the process of transferring a sample between two magnets alters the relative populations of the nuclear spin states. Two of these are common to all spectra and are discussed below. The third pertains only to spectra exhibiting spin-spin multiplets and is discussed in the next section. (i) The relative enhancements of different lines observed in the spectrometer will reproduce accurately those in the variable field only if the relative Tl in the variable, intermediate and spectrometer fields are the same. This is a good approximation in nonspin-coupled systems for the usual relaxation mechanisms with correlation times of lo- lo to 10-l’ set because 7” are field independent except at very high fields.. A possible exception may, however, occur at very low fields if relaxation is dominated by a scalar relaxation process, such as chemical exchange or spin-spin splitting modulated by quadrupole relaxation, with a long characteristic time. (I 14, In multiplet spectra the troublesome effects of coupled relaxation discussed in Section 4.1 also complicate the interpretation of variable field CIDNP. (ii) In order for the populations in the external field to correlate directly with those for the sample in the spectrometer it is necessary that the nuclear spin system remain quantized along the instantaneous static field as the sample is transferred. This occurs
186
R. G. LAWLER
only if the length of time spent at any given field during the transfer is long compared to the Larmor frequency of the nuclei in the instantaneous field.““) In other words the adiabatic condition dB,/dt
=+yB;
(45)
must be satisfied. Since the lowest field usually experienced by the sample is of the order of the Earth’s field (0.5 gauss) in which the Larmor frequency for protons is about 2 kHz, the condition in equation (45) will be met if the time spent in this field region is longer than about 1 msec. This is clearly not a serious restriction considering that at least 1 set is required to transfer a sample by hand or in a practical flow system. The adiabatic condition also implies that the instantaneous field is followed by the system regardless of the field direction. Consequently, simply determining whether the observed CIDNP changes
Br
FIG. IO. Field dependence of CIDNP proton NMR signal from HCCI, formed in reaction between BrCCI, and t-butylmagnesium bromide. Solid line is computer simulation using known q-factor for ‘Ccl, andg-factor and proton hyperfine sphtting of (CH,),C. but ignoring Cl hyperfine splitting in Ccl,. Dashed line is simulation with Ag = 0.
with the relative orientation of the external and spectrometer way of verifying that the transfer is indeed adiabatic.
magnetic fields provides a
5.2.2. EXAMPLE OF LOW FIELD CIDNP FROM CHCl,. The arguments in (i) and (ii) above make it appear reasonable that the field dependence of a CIDNP line arising from a single set of equivalent nuclei should be insensitive to the rate of transfer of the sample between fields as long as the time required for transfer, and therefore the-degree of relaxation of the sample, is the same at all fields. An example of the agreement between the calculated and observed field dependence of the proton CIDNP line from HCCl, generated in the reaction in Scheme XXII is shown in Fig. 10.
CHEMICALLYINDUCEDDYNAMICNUCLEARPOLARIZATION
187
Scheme XXII : (CH,),CMgBr
+ BrCCl, -
(CH,),C:
. Ccl,
-
HCCl, + (CH,),C==CH,
This spectrum illustrates the features characteristic of low field CIDNP of a single NMR line arising from a set of equivalent nuclei: (a) The signal goes to zero, as expected, when the magnetic field in which the reaction is carried out, B,, is zero. (b) The enhancement changes sign in the region below about 100 gauss where B, is comparable to the nuclear hyperfine splitting. This change in sign does not occur, however, unless more than two equivalent protons are in the radical pair. (c) The sign of the enhancement at all fields depends on the multiplicity of the radical pair precursor and on whether the product molecule is formed in a combination or scavenging step (the parameters p and i respectively in the simple rules for high field CIDNP given in Section 3.5). (d) The low field enhancement is independent of the g-factor difference in the radical pair, and net enhancement is obtained in low field even when Ag = 0. The dotted line in Fig. 10 indicates the fall off of intensity expected if Ag were zero for the radical pair in Scheme XXII. (e) As the high field limit is approached the enhancement increases linearly with &, just as predicted in equation (44b). 5.2.3. Low FIELD MULTIPLET SPECTRA. A more restrictive effect than either (i) or (ii) above arises when a spin-coupled product molecule containing two or more nuclei with different chemical shifts is transferred through a region of magnetic field which varies rapidly with distance. Even though the spins remain individually quantized along the instantaneous field as required in (ii), the passage from a low field region where the nuclear coupling constant, J ABt is greater than the difference in Larmor frequency, (vA - v,), into a high field region where JAB < (vA - vB) will, if carried out slowly enough, lead to adiabatic transfer of the population of the lowest low field energy level to the lowest high field energy level with the same value of Z; + I;. This second kind of adiabaticity is more restrictive than the kind pointed out in (ii). It requires that the length of time spent in the “intermediate coupling” field region where J,, w (vA - ve) be longer than the reciprocal of JAB. For values of J,, of a few hertz this requires that passage through the region around 1000 gauss be accomplished in a few tenths of a second or longer. For smaller values of J,, the time for adiabatic transfer must be proportionately longer. The above effect will produce changes in the observed .spectrum only if the CIDNP populations of some levels with the same value of (I:, + 1;) = I’ are different from others. This condition is met when the field in which the reaction is run is much smaller than any of the electron-nuclear couplings in the radical. Under these conditions the Hamiltonian for a radical pair containing two magnetic nuclei A and B on one radical fragment and none on the other is simply H, = A,AS,‘I,
+ A,,&‘I,.
(46)
From the properties of coupled angular momenta it is easily shown”‘@ that this form of HD commutes with both the square of the total angular momentum of the system, F = S, + S, + IA + I, = S + 1. and with its component, F’. along the weak, static magnetic field. H, generally does not commute, however, with S2, S’, 1’ or I’. Consequently,
188
R. G. LAWLER
the singlet-triplet mixing of the electrons induced by H, and responsible for CIDNP must be accompanied by a simultaneous change in I in such a way that F remains constant. Therefore in products formed in very low fields the nuclear spin states with the same value of I’ but different values of Z will be affected differently by singlet-triplet mixing and will therefore have different populations.
b”e - b$,Q,,
I
I
I
I
FIG. 11. Energy levels, relative populations (indicated by widths of levels), and high field spectra for adiabatic (A)and sudden (B) transfer of two coupled protons from low to high field. It is assumed that the 1 = 1 states are preferentially populated in low field and that JAB > 0.
The above effect is illustrated by the energy level diagram in Fig. 11 for the case of a radical containing two non-equivalent protons for which I = 0 or 1 in zero field. It has been assumed that the I = 1 state of the product molecule was populated by the chemical reaction to a greater extent than the Z = 0 state. Furthermore, at zero field all three of the Z = 1 states must have the same population. Adiabatic passage from low to high field (Fig. 11~) produces equal and opposite enhancement only of the outer two lines of the fourline multiplet of the AB spin system in the product while sudden passage (Fig 11~) would produce enhancement of all four lines. Figure 12 shows the NMR spectrum obtained formed from thermal decomposiexperimentally (l 09)for 1-bromo-4-chloro-benzene-2,3-d, tion (Scheme XXIII) of 4-chloro-dibenzoylperoxide-2,3-d,.
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
189
POLARIZATION
Scheme XXIII :
The experimental to high field.
result is thus in complete
agreement
with adiabatic
passage
from low
A
FIG. 12. Proton
NMR spectrum at 60 MHz of p-bromochlorobenzene-2,3-d, after decomposition of the peroxide and (B) several minutes
taken immediately later.“04’
(A)
The appearance of spin-spin multiplets in AX, proton spectra of products formed in zero field and observed in high field was first discussed by Glarum.” ‘) He has pointed out that (a) the 12+ 1 lines normally exhibited by the A proton should reduce to n lines, (b) the highest field line of the low field multiplet and the lowest field line of the high field multiplet vanish, and (c) the remaming lines should have approximately the binomial intensities expected for coupling to n +_ 1 protons. The spectrum of the two protons in Fig. 12 obviously obeys this rule. An additional example is given in Fig. 13 which shows the
190
R. G. LAWLER
multiplet obtained in Scheme XXIV.
from the methyne
proton
of 2-iodopropane
enhanced
by the reaction”
07)
Scheme XXIV: I 2i-C,H,I
ZR-
2j-CsH,.
3
___ 2i-CaH;
i-c&i+
CH;_C_CH’; I!I*
when the reaction is run in the spectrometer (Fig. 13~) and in nearly zero field (Fig. 13~). The EA multiplet effect observed in the high field reaction thus goes over in low field to a net effect with only six of the expected seven lines of the methyne proton multiplet.
I
Me-b-Me
J
5
I
4 mm
FIG. 13. NMR spectrum of the methyne proton of 2-iodopropane benzoyl peroxide when reaction is run in a 60 MHz spectrometer
The above “n and breaks down are present, e.g. in from the benzylic xxv.
in the presence of decomposing (A) and in the Earth’s field (B).
1 multiplet effect” in zero field was derived only for AX,, spin systems either in the A,X, case or when more than two sets of coupled protons the AM,X, case. This is illustrated in Fig. 14 by the three-line multiplet proton in C,H,CHClCH,CH, formed in the reaction in Scheme
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
191
Scheme XXV : C,H,CHCl,
+ LiCH,CH,
-
C,H,CHCl.
.CH,CH,
-
C,H,-CH*ClCHyCH:
Since the 3-line multiplet arises only from the AM, subsystem of the six-proton sidechain, the n - 1 rule would predict enhancement only of the two lowest field lines. The observed spectrum shows, however, that the presence of the methyl group apparently affects the correlation of low-field and high-field energy levels in such a way that all three lines appear enhanced. The high field simulated and obscrvcd spectra are shown for comparison.
Cs t-I5 CHClCHzCH3
Obs.
-&em.BR= 14.1OOG
CdC. +H
-CHF
5-o
4-o
-a-l3 1.
I
3-o
20
10
....,
0
FIG. 14. Calculated and observed proton NMR spectra at 60 MHz from I-chloro-I-phenylpropane produced in the reaction ofr,r-dichlorotoluene with ethyllithium. The upper spectrum was obtained when the reagents were mixed in the NMR probe and the lower spectrum of the methyne proton (d = 4.6) was recorded after the reagents were mixed outside of the probe and transferred to it after the reaction was complete. Spin-lattice relaxation of the CH, and CH, protons destroyed the low field enhancement for these groups before the spectrum could be scanned.
In the above case it is, nevertheless, possible to simulate both the low-field CIDNP population scheme and the effects of adiabatic passage by using a digital computer both to solve the complete equations (Appendix A) for the populations and to calculate spin energy levels in the product as a function of field. This approach has been discussed in detail recently by Kaptein and den Hollander’19’ and has also been employed by Evans@‘j’ to simulate the above spectrum (Fig. 14).
R. G. LAWLER
192
5.2.4. QUALITATIVE RULES FOR LOW FIELD CIDNP. It would clearly be desirable to have a set of qualitative rules relating the observed sign of enhancement of low field lines to the initial electron spin multiplicity of the radical pair, signs of hyperfine splittings, etc., in the same way as the high field rules discussed earlier. Kaptein has, in fact, proposed a modification of the high field rule for multiplet effects l-k
= ~AAARJABflTAH + gives EA
- gives AE
that may be applied to the zero field enhancement of two sets of coupled protons if the symbol EA is taken to mean emission for the lowest field multiplet and enhanced absorption for the high field multiplet. (i7) Application of the rule to the low field spectra presented here for Schemes XXIII-XXV is presented below. .-. /I D
Br
Cl
\
H
4,
II,
J hll
c \I,
_
+
+
+
+
1.
+
_ ._
+ _
+ +
+ _
E
D
/
u-
Zero field enhanceInlent of proton A
6
(Fig. 12)
H
CH,CHAICH; C,H,CHAC‘IC‘H~CH,
(Fig. 131
+
(Fig. 14)
t _
Even though all three of these examples agree with the prediction of the rule, the rule itself has apparently not yet been proven for zero field in the same sense as was done in Section 3.4 for its high field counterpart. Until this is accomplished and its range of applicability has been explored further, the rule should be used cautiously in treating any of the six parameters (usually p, the parameter for electron-spin multiplicity) as unknowns to be evaluated from the observed sign of the enhancement.
6. Effects arising from Electron-Electron
Interaction in Radical Pairs
Until now we have been concerned with CIDNP effects which were explainable in terms of magnetic coupling parameters of isolated free radicals. In particular, neither the high or low field effects presented so far have required explicit consideration of the exchange interaction between the two unpaired electrons which is turned on in an encounter pair: H, = -J(l/2
+ 2s, .S,).
(19)
We will now discuss three kinds of effects which seem to come from radical pairs in which this interaction must be taken into account. In general the exchange integral, J, for a pair of radicals is a rapidly-decreasing function of the distance between the two species. (70,7’) Figure 15 shows a schematic representation of the way the energy of separation, 25, between the singlet and triplet states of a two-electron bond varies with distance. The blown-up region of the diagram shows the energy levels when the pair is in a magnetic field, contains one proton and
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
193
FIG. 15. Energy levels for two unpaired electrons and one I = I/2 nucleus in a magnetic field. The upper diagram shows the variation of singlet and triplet state energies with distance. The lower diagram is an expanded view of the region of fixed r where J - u)‘,.
has a value of J between 0 and the order of the electron Larmor frequency 0,. In the case where J = 0 the So and r, states are degenerate and the pair behaves as already described at length in our previous discussion of high field CIDNP. As long as o, is large compared to both J and the strength of the singlet-triplet mixing (of the order of the hyperfine splitting, lo* rad set-‘), the T+ and T_ states are effectively isolated from the other two states. When o, -c A, i.e. in low fields, all four states mix as we have seen, even when J = 0. In the case where 25 _ o,, however, the S, and T_ levels become nearly degenerate and S, mixes more strongly with T_ than with To or T+. The difference between S, - To and S, - T_ mixing is that the selection rule for the former is Am, = 0, Am, = 0, i.e. no spins flip, while the latter is accompanied by simultaneous electron and nuclear spin flips (Am, = f 1, Am, = f 1). This is produced by the off-diagonal part of the hyperfine interaction in equation (21). (These terms were, of course, included in the low field calculations already discussed, but in that case J = 0 and they produced essentially equal mixing of S, with T_ and T+ .) As a consequence of this selective exchange of angular momentum between the electron and nuclear spin systems the total numbers of spin up and spin down nuclei in the sample changes and a true nuclear polarization results. The result is similar
194
R. G. LAWLER
to that obtained in the Overhauser effect where combined electron and nuclear spin flips transfer thermal polarization from the electron-spin system to the nuclei.” ’ 7, 6.1. CID N P,from lntramolrcular
Biradicals
There is at present only one rather special example of the above model in which J remains roughly constant and of the order of magnitude of w, for a sufficiently long time for nuclear spin induced S, - T mixing to occur. This arises during the photolysis of compounds capable of forming “intramolecular radical pairs”, or biradicals, in which the portions of the molecules containing the two unpaired electrons are never able to diffuse very far apart. Gloss has found, for example, that irradiation of an alkyl-o-benzoyl-benzoate (Scheme XXVI) at high field in the probe of the spectrometer Scheme XXVI : G-k co
hv CO&H,
0
W’5
=d-b CH,CH(CH,),
HO ,
*COH CO2 CHZ CHZ i: (CH&--
I
1; 0
CO2
) / P Cyclization
’ /
C,H, \I
HCOH 0,
CH,
CH-C(CH,),
1: o-’
produces an intense CIDNP spectrum which exhibits emission for all of the lines of the cyclization and disproportionation products formed from the intermediate biradical.‘38’ In Fig. 16 is presented a qualitative energy level scheme which shows how emission arises from T_ - S, mixing in a biradical born in the triplet state. It is significant that the rate of S - T mixing varies as the absolute value of the mixing coefficient, H_ +,s_, and the population difference between + and - nuclear spin states is, unlike S, - TO mixing, independent of the sign of the hyperfine splitting A. Consequently, all NMR lines in biradical products will occur in emission provided that (a) the biradical is produced initially in the triplet state and (b) the triplet state of the biradical lies above the singlet. Condition (a) is, in addition, probably a necessary condition for even seeing CIDNP via biradical intermediates because singlet biradicals are likely to cyclize or disproportionate before sufficient time has elapsed for the necessary nuclear spin flips to take place. If the triplet energy were lower than the singlet [i.e. J > 0 in equation (19)], as is, in fact, usual for excited electronic S and T states, S, would mix most strongly with the T+ state and enhanced absorption of all lines would result. This has, however, so far not been observed. Kaptein has recently shown (’ ’ 8, that S, - T_ mixing may produce CIDNP of carbon- 13 nuclei, again completely in emission, during the photolysis of cyclic ketones. Scheme XXVII :
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
19.5
It is interesting that in this example the starting ketone also exhibits CIDNP, thus demonstrating the significance of the biradical cyclization as an effective pathGay for inter-system crossing of the photo-excited triplet to the ground singlet state.
hv
(triplet excitation)
/
FIG. 16. Schematic representation of the origin of a proton emission line in both scavenging and combination products when S,, - T- mixing dominates in a triplet-born radical pair or biradical.
S, - T_ mixing has also been invoked by C10ss’~~’ to explain obtained by Berson rt L/I.‘39’ from the dimers of two intermediate ground states.
6.2. Low Field CIDNPfiom
S -
the emission spectrum biradicals with triplet
Ti Mixing
our discussion of high field CIDNP it was pointed out that the enhancement expected for a combination product is opposite in sign to that of a scavenging product derived from the same radical pair. In the limiting case where both products were the same the contributions of each would cancel and no enhancement would be observed. Another way of stating this result is that every nuclear spin state of the reactant would have a unit probability of ending up in the product, so that no nuclear level population differences would arise. Garst rt al.‘56) have reported what is apparently an example of just such an effect in the formation of propane from the reaction of 2-iodopropane with naphthalene radical ion, N I, in the presence of a weakly acidic proton donor, RH. In
196
R. G. LAWLER
Scheme XXVIII : ICH(CH,),
J%
I- + N + . CH(CH,),
-&+ NL .CH(CH,), e”CO”“tel
3 2
3
H,C(CH,),
Virtually all of the 2-iodopropane is converted to propane via the 2-propylanion. As expected, therefore, no CIDNP is observed from the propane protons in high field because of the cancellation of the effects of the combination and scavenging steps.” ’ 9, Remarkably, however, rather intense emission lines are observed from both sets of protons in the product when the reaction is carried out in a magnetic field of approximately 100 gauss. The observed effect is reminiscent of that previously discussed for triplet-born intramolecular biradicals. The emission lines are consistent with the postulated triplet-like diffusive encounter pair and dominant S, - T- mixing. The illustration in Fig. 16 shows that under these conditions a proton would appear as emission in both combination and scavenging products. The actual physical situation in the above low field case is, of course, a good deal more complicated than that for the intramolecular biradicals. In low field o, and the hyperfine splittings are of the same order of magnitude and all four of the electron spin levels mix, but with some preference for S, - T- mixing. Furthermore, since a T- level may mix with any SOlevel in which the nuclear spin state differs by one unit of angular momentum, ihe mathematical problem generally involves mixing of more than two levels at a time. Adriancz6’ has attempted to treat the problem of low field S, - T+ mixing for a one-proton radical pair quantitatively, but was forded to consider mixing of only two levels at a time, a result which is strictly valid only for very short radical-pair lifetimes. His treatment of the problem does, however, employ the physically realistic assumption that H, is turned on only during an encounter. The computer simulations of Garst et ~1.~~~)for a one-proton pair confirm the qualitative idea that S, - T_ mixing requires a non-zero exchange interaction, but treats J unrealistically as a constant throughout the lifetime of a radical pair. It should also be emphasized that the low field CIDNP from S, - T_ mixing will generally be superimposed on that arising from pairs in which there is no exchange interaction. Although it is tempting to treat these two contributions to low field CIDNP as separable, there seems at present little theoretical justification for doing so. In limiting cases, such as Garst’s or that of intramolecular biradicals, where only the results of S, - Tt mixing are observed, one may, however, write a qualitative rule for the net enhancement expected from a set of protons: r;, = pj
+ gives A
- gives E.
The exchange integral, J, has the following property, J=
+ when triplet state lies below singlet r -when
singlet below triplet (usual case)
and p, as before, is positive when the radical pair is triplet-born or results from a diffusive encounter and negative if it is singlet born. Note that the sign of enhancement is the same for combination and scavenging products and is independent of the sign of the hyperfine splitting, for reasons already mentioned.
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
197
6.3. Chemically Induced Dynamic Electron Polarization (CIDEP) In 1963 Fessenden and Schuler,‘89’ m . the report of their classic investigation of the ESR spectra of transient radicals formed during radiolysis of hydrocarbons, showed a spectrum of hydrogen atoms in which the low field line of the doublet was in emission. No explanation was offered at that time for this remarkable observation and it was largely ignored until the advent of CIDNP. Since that time several more examples ofemission and enhanced absorption in ESR have been reported. These are summarized in Table 1. In all of these cases the effect is observed only while the sample is being rather violently disturbed, almost always by radiolysis or photolysis, and the lifetimes oft the radicals themselves are short. It therefore seems proper to call it chemically induced dynamic electron polarization (CIDEP). Fischerc3’ was the first to point out a possible connection between CIDEP and CIDNP, but Kaptein and Oosterhoff (l 5, first provided a satisfactory explanation in terms of the Radical-Pair Theory. Subsequently, Glarum’120’ and Fischer’27’ offered a modified explanation based on considerations of level crossing and Atkins et al.“21) extended somewhat the ideas of Kaptein and Oosterhoff. As in the case of CIDNP, however, it was Adrian”?2’ who recently supplied the most physically-realistic explanation of CIDEP by incorporating the proper diffusive characteristics of radical pairs into the model. The essential feature of his model is that CIDEP arises only in a radical which has undergone at least one unreactive re-encounter with its partner, but one in which the exchange interaction, J, is not zero. The sequence of events leading to CIDEP in a radical Ry in nuclear spin state n is shown in Scheme XXIX. Scheme XXIX : (D
(D)
R:--X-RR,k,_R;..R,-R;./.R,-R:.
CD) .R,-
(JZO)
'0
‘I
:. 0
Te
(E)
‘2
N, .
+
\ Ts
R,
R; X
‘3
This is identical to the scenario in Scheme VII which was employed in the derivation of the equations (22H24) for the reaction probabilities for high field CIDNP. Since virtually all CIDEP so far reported has been obtained in fields of about 3OtMgauss we need again consider only S - To mixing. The experimental observable in CIDEP is the expectation value of the z-component of the spin ofelectron 1 when the nuclear spins are in state n. We assume that the measurement takes place at time t, after the radical has escaped from the re-encounter which began at t,. That is, (S;(r3L
= Tr p%)&
= &~,0n(t3)
+ &n,m(t3)1 = Re ~~,On(h)~
(47)
It is easily shown, (l 22, however that PE,0n(r3) = PE,0n(t2)e-i(H~-H~O)(f3--fZ) = pkeo,(tz) e-2iJTE
(48)
where J is the average exchange interaction during the encounter and rE is the time during which it operates. The off-diagonal matrix element P:,~.(!~) may, however, be evaluated (Appendix A.1) in exactly the same way as the diagonal elements were evaluated in
a
/‘--\
OH
in butane
HO,CCH(OI’I)C.H(OH)CO,H
peroxide
+ I”,, HzOr
of alcohols.
of 2chlorobenzaldehyde in liquid paraffin of solutions of K. Rb and Cs in 1.2-dimethoxyethane of benLophenone at 4K
of aqueous
Photolysis
Photolysis Photolysis Photolysis
of di-r-butyl
Photolysis
solutions
of production
of bcnrene
(L’H,)CHOH
+ CH,CH~CO~H
of aqueous
solutions
of cyclopentanc
of aqueous
Ti(II1) + HL02
Photolysis
Radiolysis
Pulsed radiolysis
Electron radiolysis of hydrocarbons Pulsed radiolysis of H,O and DZO Steady electron irradiation of aqueous acids and esters
Method
of CIDEP
observations
t E.4 refers to hyperfine lines on low field side of ccntre in emission. high field in enhanced absorption I,” R. W. F-I.SSLNIXNand R. H. SCHLLLK, J. C‘hrrr~.Phy.t. 39, 2 147 (1963). lb) B. SMALLER, E. C. AVERY and J. R. REMKO, ibid. 55, 2414 (1971). (‘I E. EIBEN and R. W. FESSENDEN,J. PhJa. Chrm. 75, 1186(197 I). N) P. NETA, R. W. FESSENDENand R. H. SCHULER, ibid. 75, 1654 (1971). (<) B. SMALLER,J. R. REMKO and E. C. AVERY, J. Che,n. Phys. 48,5174 (1968). If) R. LIVINGSTONand H. ZELDES, J. Amer. Chrk. Sot. 88, 4333 (1966). (a) H. PAUL and H. FISCHER, Z. Naturforsch. 25a, 443 (1970). (hi P. J. KRUSIC and J. K. KOCHI, J. Amu. Chu. Sot. 90, 7155 (1968). “i R. LIVINGSTONand H. ZELDES, J. Chem. Phys. 53, 1406 (1970). tJ) P. W. ATKINS, I. C. BUCHANAN, R. C. CURD, K. A. M~LAUCHLAN and A. F. SIMPSON. Chum. Comm. 513 (1970) “I S. H. GLARUM and J. H. MARSHALL, J. Chrrn. Phys. 52, 5555 (1970). “I W. S. VCEMANand J. H. VAN IXR WAALS. Chew Phys. Lrttw$, 7, 65 (1970).
2-CICeH&HOH Solvated electron H. (trapped in solid benzophenone)
!.H(oH)
HOzCCH(OH)
CH&HCOO-
.CH,CH,COO-
(CH,),COH
\\_,
I
H
0 .
H., D. H.. D. H.
Radical
TAI~L~ I. Summary
E E
t
IL l (tentative, detected by monitoring benrophenonc phosphorescence intensity as a function of magnetic tield)
.
E
E,4 (very weak)
EA
EA (very weak)
E 4 (weak)
E :I
EA EA EA
Type of enhancement
J k I
h
d
e
a b cd
Ref.
F
p’ E
F
x
z X
CHEMICALLY
199
INDUCED DYNAMIC NUCLEAR POLARIZATION
the presentation of high field CIDNP. Substitution and averaging as before over a distribution of times between the first and secdnd diffusive encounters yields equation (49). (S;),
= 2p(7rr,)+(sin
[(i - Il)p, - pT]f$
252,) Z;’
16.1”
(49)
n
Z, is the total number of nuclear spin states for radical 1, radical 2 is assumed to have no magnetic nuclei, (sin 252,) is an average over all encounter pairs, and all other quantities have the same meaning as in equation (33) for CIDNP. The observed value, ST,, for the electron magnetization will also be influenced by relaxation and scavenging processes. In fact, if the processes producing (S;), occur much more rapidly than either relaxation or scavenging (i.e. t, < T,, us) we have dS;, ~ = -&J’s;, dt
- T&S;,
- s;;p)
(50)
where si;p = P,o[l - &(2 P, is the probability reacting and
that
radicals
1 and
- ;*)I (>-rrs = p;p@r-*‘rS 2 will escape
from
their
(5la) encounters
without
P,” = z, 1 (gB B,/2kT)
(51b)
is the thermal electron polarization for nuclear spin state II. If we let S;,(O) be approximately (S;),, the time-dependent solution to equation (50) is S;,(t) = pfp,e-f/Ts The ESR enhancement
+ [(Sl),
- p,op,] ,-(rs-‘+T,‘)f.
(52)
V:(t), then becomes
factor Vi(t) =
s;,w-
%B)
sf;p(t)
=
[%
-
1]e-tlT1c
(53)
The above transient decay of enhancement has in fact been observed by Smaller et al.,” 23) Atkins et .1.“24’ and Glarum and Marshall. (12’) This is certainly the most sensitive way to detect CIDEP, and the only one where large enhancements have been seen, since the effects disappear by relaxation with r,, - 10m5 to lop6 sec. Under steady-state conditions, such as employed in the majority of other reports of CIDEP, V!(t) must be integrated over a distiibution of radical lifetimes using a weighting function ZS ’ exp( - t/r,) dt. This yields v: = 7;’
[T;' +
T,']-'
If zs B T,,, as may often be the case, the steady state enhancement fraction of the maximum transient enhancement. 6.3.1. QUALITATIVE ESR
line to equation
gyric ratio requires
RULES FOR CIDEP.
In order
to relate
(49) we must realize that the negative that I ;SR cc ( - s&J.
(54)
v:(o).
may be only a small
the observed
intensity
sign of the electron
of an
magneto-
(55)
200
R. G. LAWLER
We also recall that for a radical pair with only one set of.equivalent 6” = i[(g1 - g2)pc1Bo
nuclei on radical 1
+ Am].
Furthermore, if A is positive, the ESR hyperfine lines with positive values of m occur on the low field side of the spectrum. Referring to equation (49) we may therefore derive the following qualitative rules for CIDEP: Net enhancement (occurs only when (gr - gz) 9 Am) ri, = ,&(g, - $,)
+ gives A
- gives E
where the signs of p, J and (gr - gJ are determined as described previously. As an example we consider the radical 2-ClC,H,CHOH reported by Atkins et al.‘124’ and most likely formed according to Scheme XXX. Scheme XXX : 2-ClC,H,COH
--%
0 I
2-ClC,H,-C-H
(triplet)
“ky’-H )
OH
t
2-ClC,H,CH(OH).alkyl Yl ’
-
I
2-Cl,H,C-H
92
If J < 0 during the re-encounter of the pair, then we see that ri, = + - + and emission is predicted (and observed). Multiplet efSect (for the special case of one set of nuclei and Ag = 0) rz, = PJA
+ gives EA
- gives AE.
The simplest example of a pure multiplet CIDEP effect is the EA spectrum observed for atomic hydrogen and deuterium under radiolysis conditions-(see Table 1). The hyperfine splitting for both atoms is positive. If we assume that J < 0 for any pair involving these atoms, then the observed EA effect requires that ,LL be positive, i.e. that H. is either formed from a dissociated triplet excited state or, as seems more likely, undergoes a diffusive encounter with another radical with a similar g-factor. Unfortunately, CIDE,P spectra do not usually fit cleanly into either of the two limiting categories above. In fact, equation (49) is much less successful’12” in predicting quantitatively the relative intensities of lines within an ESR spectrum than the corresponding equation (33) is in predicting CIDNP intensities. Part of the problem undoubtedly lies in the greater importance of relaxation during observation of CIDEP. It is also possible, however, that the approximate physical model behind equation (49) is not a completely adequate description of the phenomenon. (241,243)Additional and better,-albeit labo.riously obtained, data appear to be needed in the field of CIDEP even more than in CIDNP.
7. Guide to the Literature of CIDNP In the 6 years since the discovery of CIDNP more than 200 publications dealing directly with the phenomenon have appeared. We have so far quoted those references which have
CHEMICALLY INDUCED DYNAMIC NUCLEAR POLARIZATION
201
been most concerned with establishing the mechanisms producing CIDNP and illustrating its scope and utility. In the hope of making this wealth of information more readily available to the uninitiated, we present below the complete list of CIDNP and CIDEP publications up to December 1972. These are organized into six broad categories and a number of sub-categories. The order of listing within a given category is the approximate order in which the papers were listed as received by the journals. 1. REVIEWSANDCONFERENCEREPORTS References: 3, 96, 125, 141, 9, 14, 126, 142, 127, 10, 136, 13, 8, 99, 117, 7, 240, 246. 2. TYPESOFREAGENTSANDREACTIONSPRODUCING
CIDNP
A. Peroxides a. Benzoyl peroxide References: 1, 128, 129, 92, 130, 131, 107, 132, 81, 95, 133, 134, 135, 46, 76, 109, 102, 137, 138, 139, 140, 219. b. Other acyl peroxides References: 41, 92, 130, 132, 94, 143, 144, 97, 145, 146, 9, 106, 42, 147, 148, 149, 238. c. Other peroxides References: 150, 151, 152. B. Nitrogen-eliminating Compounds a. Azoalkanes References: 128, 74, 153, 154, 55, 57; 155, 156, 245. b. Diazonium salts References: 157, 158, 159, 160. c. Other nitrogen-eliminating compounds References: 162, 163, 134, 164, 165, 166, 167, 168. C. Organometallic Compounds a. Alkyllithiums References: 2, 169, 170, 4, 5, 171, 172, 173, 107, 174, 77, 44, 227. b. Grignard reagents References: 175, 176, 177. c. Organomercurials References : 178, 179. d. Radical anions References: 56, 119, 180. e. Alkylsodiums Reference : 181. f. Alkyllead Reference : 182 D. Rearrangements (R-X-Y R.X-Y. X-Y-R) References: 183, 184, 185, 186, 187, 188, 189, 190, 191, 104, 192, 193, 194, 195, 196, 197, 198, 199,200,201,202, 203,204,205.
202
R. G. LAWLkR
E. Divalent Carbon Species, RR’C References: 73, 74, 153, 206, 57, 167, 207, 208, 209, 210, 250 F. Photochemical Reactions (photo-CIDNP) a. Carbonyl compounds References: 211, 45, 212, 213, 58, 75, 76, 214, 215, 33, 60, 216, 217, 78, 218, 38, 220, 221, 61, 34, 222, 87, 251, 254, 242. b. Other compounds References: 73, 206, 57, 167, 2 IO, 223, 255, 257. 3. THEORY References: 36, 37, 173, 93, 20, 74, 15, 16, 29, 30, 27, 21, 24, 28, 25, 22, 32, 17, 26, 119, 23, 109, 108, 35, 61, 225, 14. 117. 100, 226. 18. 19, 252, 253. 4. Low FIELD EFFECTS References:
107, 56, 109, 110, 108, 119, 31, Il. 19. 244.
5. CIDNP FROMNUCLEIOTHERTHAN‘H A. 19F References:
128, 227,209, 180, 248, 256.
B. 13C References: 228, 54, 229,230, 12, 118, 247, c. 3’P References : 23 1, 249. D. 2H. References : 12, 238. E. 15N Reference : 230. 6. CHEMICALLY INDUCED
DYNAMICELECTRONPOLARIZATION
A. Experimental References: 89, 123, 232, 124, 120, 233, 234, 235, 236, 258, 259. B. Theoretical References: 15, 120,27, 121, 122, 108, 237, 241,243.
8. Conclusions
In the preceding pages an attempt has been made to present a unified version of the theoretical basis which has recently been developed for CIDNP as well as discussing the phenomena which it is capable of explaining. Although the theory has certainly been
CHEMICALLY
INDUCED DYNAMIC NUCLEAR POLARIZATION
203
successful in qualitative terms, it remains to be seen whether it will stand up to more quantitative tests. A major stumbling block in this endeavour is sure to be the adequately accounting for relaxation effects. Conversely, almost no use has been made of CIDNP as a way of preparing a nuclear spin system with non-equilibrium populations for studying relaxation effects. With a careful choice of the free radical precursor of a product of interest, however, it should be possible to drive specific sets of energy levels to an extent not attainable with radiofrequency fields in ordinary multiple resonance methods. There is the added advantage that CIDNP samples are usually automatically deoxygenated by the radical reaction. This eliminates the often appreciable intermolecular relaxation effects of dissolved oxygen. Multiple resonance on samples exhibiting CIDNP may, nevertheless, be employed both as spin of spectra and as a way of probing the CIDNP enhancedecoupling (54) for simplification ment of specific transitions when single resonance spectra are severely perturbed by relaxation effects.‘61’ Multiple quantum transitions in samples exhibiting CIDNP do not appear to have been reported, but their intensities should be predictable by the radical-pair model provided that only the populations of the nuclear spin levels are perturbed by the chemical reaction. It can be expected that the amount of CIDNP data for nuclei other than protons will increase markedly in the near future. Hyperfine splittings for the less abundant nuclei in reactive radicals should thus be easier to obtain by NMR, via CIDNP, than by ESR since the latter generally requires isotopically enriched samples. The wide availability of pulsed Fourier transform spectrometers should make ’ 3C CIDNP especially attractive. Proton CIDNP spectra may also be obtained up to two orders of magnitude more rapidly using pulsed methods. This will make it possible to accurately follow the course of even very rapid CIDNP-producing reactions. One can also expect increased use of on-line data processing by small computers in applications such as automatically subtracting unwanted, unenhanced lines from a spectrum. In short, the development of a satisfactory theoretical basis for CIDNP now makes it clear that the phenomenon belongs to both NMR spectroscopy and free radical chemistry. It is sure to find further valuable applications in both fields.
9. Acknowledgements It is a pleasure to acknowledge the co-operation of many of the research groups working in the field of CIDNP for providing preprints or otherwise indicating new results prior to publication. Most of the experimental examples used as illustrations were good-naturedly provided by the students at Brown University. Financial support was received from the Alfred P. Sloan Foundation and the National Science Foundation. I owe a special debt of gratitude, however, to four people: my colleague Harold R. Ward of Brown University with whom I have had the good fortune to collaborate for some 5 years; Glenn T. Evans also of Brown University and Jan A. den Hollander of the University of Leiden who patiently and constructively participated as I groped my way toward an understanding of the theoretical basis of CIDNP; and, finally, Professor L. J. Oosterhoff for his thoughtfulness and generosity during my stay in the Department of Theoretical Organic Chemistry of the University of Leiden where most of this review was written late in 1971.
204
R. G. LAWLER
10. Appendices A.I. Solution of the Master Equation (17b).for the Radical Pair Density Matrix between First and Second Encounters In the time region t, < t < t,, pD obeys the equation pD = i[pD, HD] - z; ‘pD. By a simple transformation,
however,
(A.11
we can write (A.2a)
p”(t) = P(t) exp( - t/T,) where by substitution it can be shown that the matrix representation as pD, obeys the equation
P(t), which is defined
in the same
P(t) = i[P(t), HD].
(A.2b)
Since HD, given by equation (21), is time independent, a representation which diagonalizes HD is
the solution
of equation
(A.2b) in
P&(t) = Pzl(0) exp[ - i(EF - Ey)t]
where EF and Ey are two eigenvalues of HD (in rad set-‘). serves to define a transformation matrix, B, such that
(A.3)
The matrix
of these eigenvalues
ED = B-‘HDB
(A.4)
where the elements of HD are given in the EJ, representation energy’matrix. The matrices P*(t) and P(t) then obey the relationship
and
ED is the diagonal
P(t) = BP*(t)B- ‘. By combining of pD
equations
t%(t) = Cexp( -
WI 1 C C 1 k
I
s
where cukl = EF - EF. The matrix elements tion are Diagonal elements: H:.sn = H&,, H’&+, Off-diagonal
~mk&1~Ps(0)41K~
element,
exp( - iok, t)
1
of HD from equation
&,(t),
(‘4.6)
(2 1) in the l,x,, representa-
(A.7a)
= 0
= +(1/2)c(w,
+ 02) + cc Alams + c (1 b
~d%)l.
(A.7b)
elements: HZ”,,, = (1/2)W,
- %) + (c Al,m, a
HP,,, = i8-“2i”iAi[Ii(Ii
HP,,, wher.
(A3
(A.5), (A.3) and (A.2a) we have for an arbitrary
he nuclear
- c
A,bmb)l
=
&I
(A.7c)
b
+ i) - mi(mi + 1)]“2
= 8-1’LA,[Zi(Zi + 1) - mi(mi + l)]“’
spin state x_+ = I.. . Ii, ‘Hi + 1,. . . ) corresponds
(A.7d) (A.7e)
to a change of one unit
CHEMICALLY
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
205
in m, for the ith set of equivalent nuclei and I is + 1 or - 1 depending on whether the ith nuclei are on fragments 1 or 2 respectively. A.11. High Field Solutions In the special case where l(wi + oJl B Aili the diagonal elements of the Tilxn states are much larger than any of their off-diagonal elements and they are only weakly coupled to the T, and S states. To a first approximation, then
ED,,= H%,t, B *n,Lbl = B&W
= Sk,,,o,n,.
(A.8a) (A.8b)
Therefore by substitution in equation (A.6) P%,,,,, = &,&O)
exp( -t/Q.
(A.9)
The states r,x, and Sx,, on the other hand, are strongly coupled by the off-diagonal element 6, since their diagonal elements vanish. We therefore find by solving equation (A.4) using the required 2 x 2 matrices that (A.lOa)
=
B-‘.
Substitution into equation (A.6) using the elements of B and ED in equations (A.lO) and the initial conditions in equation (22b) gives the equation (22~) for p:,,,(t). A.111. Memory Effect in a New Pair, ID’,formed from D Equation (40) for pf&,,(tz, t;) for a new pair which was formed at time t; after the first encounter is obtained by first solving equation (A.6) for p$Jt;), &,,,(t;) and &,,$t;), using these in another equation of the same form as (A.6) for the matrix element &,,,sn(tJ of the new pair formed at t;, and employing the new values + Sk for the eigenvalues of ED’.
A.IV. Low Field Populations In the case where l(wr + 02)) < Aili all four electron spin states are mixed by HD and numerical methods must be employed to solve equations (A.4) and (A.6) for &&t). When several kinds of nuclei are involved a further complication arises because elements of pD may build up which are diagonal in s but off-diagonal in the nuclear spin states, i.e. n # n’. In this situation the description of the nuclear spin system in terms only of “populations” breaks down and the elements of the form &,, must also be included in equation (37) to properly simulate low field CIDNP.‘31’
206
R. G. LAWLER
11. References I. 2. 3. 4. 5. 6. 7. 8. 9. IO. Il. 12. 13.
14. 15. 16.
17.
18. 19. 20.
21. 22. 23. 24. 25. 26. 27. 28. 29.
30. 31. 32. 33.
34. 35.
36. 37. 38. 39. 40. 41.
42. 43. 44. 45. 46. 47. 48. 49.
50. 51. 52. 53. 54. 55.
56. 57.
J. BAHC;ON. H. FIS(.HI.K and U. JOHNSEN.Z. Ntrtl!rfi>r.sch. 22a, I55I (19671. H. R. WAHI) and R. G. LAwLtK, J. A~rlrr. C~PIII. Sot. 89, 551X (1967). H. FISCHER and J. BARGON, Accounts Chem. Rrs. 2, 1 IO (1969). H. R. WARD, R. G. LAWLER and R. A. COOPER, J. Amer. Chem. Sot. 91,746 (1969). A. R. LEPLEY and R. L. LANDAU, J. Ajnrr. Chem. Sot. 91,748 (1969). P. LIVANT, Unpublished results, Brown University (1970). H. R. WARD, in Free Radicals, Vol. 1, Ed. J. K. KOCHI, Wiley-Interscience, New York (1972). G. L. CLOSS, Spec. Lrct. XXIIIrd Int. Congr. Purr Appl. Chem. 4, 19 (1971). H. R. WARD, Accounts Chrm. Rrs. 5, 18 (1972). Ind. Chim B&J 36, 1051-89 (1971); P. BAEKELMANSand G. J. MARTENS, I&. J. Chrrn. Kinetics3, 375(1971). S. GLARUM, in Chrmically Induced Magnrcic Polarization, Ed. G. L. CLOSS and A. R. LEPLEY, Wiley, New York (1973). R. KAPTEIN, Doctoral Dissertation, University of Leiden (1971). H. FISCHER, Fortschr. drr Chum. For&. 24, 1 (1971). R. G. LAWLER, Accounts Chrm. Rrs. 5,25 (1972). R. KAPTEIN and L. J. OOSTERHOFF, Chrm. Phys. Lvtters 4, 195 (1969). R. KAPTEIN and L. J. OOSTERHOFF,Chem. Phys. Letters 4,214 (1969). R. KAPTEIN, Chrm. Cornm. 732 (1971). R. KAPT~IN, J. Amer. Chem. Sot. 94, 6251 (1972). R. KAPTEIN and J. A. DEN HOLLANDEK. J. Amer. Chem. Sot. 94,6269 (1972). G. L. CLOSS. J. Amer. Chrm. Sot. 91, 4552 (1969). G. L. CLOSS and A. D. TRIFUNAC, J. Amer. Chem. Sot. 92, 2183 (1970). ’ G. L. CLOSS. J. Amer. Chern. Sot. 93, 1546 (1971). K. MOLLEK, Chem. Commun. 45 (1972). F. J. ADRIAN. J. Chrm. Phys. 53, 3374 (1970). F. J. AIIRIAN. J. C’heln. Phys. 54, 3912 (1971). F. J. ADRIAN, Chrm. Phys. Letters IO, 70 (1971). H. FISCHER, Chrm. Phys. Letters 4, 61 I (1970). H. FISTHER, Z. Naturforsch. 25a, 1957 (1970). F. GFKHART and G. OSTERMAN. Trtrtrhedron Letfers 4705 (1969). F. GERHART, Tptrahrdron Letters 5061 (1969). J. 1. MORRIS, R. C. MORRISON, D. W. SMITH and J. F. GAKST, J. Amer. C~L’III.Sot. 94, 2406 (1972). M. TOMKIEWI~Z and M. COCIVERA, Chem. Phq’s. Letters 8, 595 (1971). M. TOMKI~WICZ, A. GROEN and M. COCIVEKA, Chew. Phys. Letters IO, 39 (1971). M. TOMKIEWICZ, A. GROEN and M. COCIVERA, J. Chrm. Phys. 56,585O (1972). J. M. DEUTCH, J. Chem. Phys. 56, 6076 (1972). J. BARGON and H. FISCHER, Z. Naturjorsch. 22a, 1556 (1967). R. G. LAWLER, J. Amer. Chrm. Sot. 89, 5518 (1967). G. L. CLOSS, Ind. Chim. Brig. 36, 1064 (1971). J. A. BEKSON, R. J. BCSHRY, J. M. MCBRIDE and M. TRAMELLING, J. Atner. Chrrn. Sot. 93, 1544 (1971). See, for example, Frrr Radicals, Ed. J. K. KOCHI, Wiley-Interscience, New York (1972). R. KAPTEIN. Chern. Phys. Letters 2, 261 (1968). R. A. COOPER, R. G. LAWLER and H. R. WARD. J. Amer. Chem. Sot. 94, 545 (1972). S. ROSENF~LU, Ph.D. Dissertation, Brown University (1972). R. A. COOPER. R. G. LAWLCR and H. R. WARD. in Chumicully Induced Magnetic Polarization, Ed. G. L. CLOSS and A. R. LEPLEY, Wiley, New York (1973). G. L. CLOSS and L. E. CLOSS. J. Amer. Chew. Sot. 91, 4550 (1969). M. LEHNIG and H. FISCHER, 2. Naturforsch. 25a, 1963 (1970). A. M. NORTH, Collision Theory of Chemicul Reuctions in Liquids, Methuen, London (1964). S. CHANI~RASEKHAR,Rrc. Mod. Phys. 15, 1 (1943). R. M. NOYI s. J. Chvm. Phys. 22, 1349 (1954). E. RABINOWITZ, Trans. Faraday Sot. 33, 1225 (1937). R. M. N0vr.s. Progr. Reaction Kinetics I, 131 (1961). C. A. EMIEISand P. L. FEHDEK, J. Amer. Chem. Sot. 92,2246 (1970). N. K. AILAWADI and B. J. BERNE, J. Chem. Phys. 54, 3569 (1971). E. LIPI’MAA, T. PEHK, A. L. BUCHA~HENKO and S. V. RYKOV, Chew. Phy.s. Lettws 5, 521 (1970). H. IWAHLKA and M. IWAML~KA,Tetrahedron Letters 3723 (1970). J. G. GAKST. R. H. Cox. J. T. BARBAS, R. D. ROBERTS, J. I. MORRIS and R. C. MORRISON. J. Amer. Chem. Sot. 92, 5761 (1970). G. L. CLOSS and A. D. TRIFUNAC. J. Amrr. Chrm. Sot. 92, 7227 (1970).
CHEMICALLY
58. G. 59. 60. 61. 62. 63. 64.
65. 66. 67. 6X. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
101. 102. 103. 104. 105. IOh. 107. 108. 109. I IO.
11I.
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
207
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160. 161. 162.
163. 164. 165.
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2650
(1970).
Ser. Khim.
CHEMICALLY 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177.
178.
INDUCED
DYNAMIC
NUCLEAR
POLARIZATION
209
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SSSR, Ser. Khim 454 (1971). 179. 180. 181. 182. 183. 184. 185.
186. 187.
188. 189.
190. 19 1.
192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224.
225. 226. 227.
228.
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