13318
J. Phys. Chem. 1993,97, 13318-13325
Theory of the SCOTCH Experiment for Reactions Involving Radical Pairs Petra J. W. Pouwels and Robert Kaptein’ Bijvoet Center for Biomolecular Research, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands Received: August 4, 1993; In Final Form: September 28, 1993’
SCOTCH (spin coherence transfer in chemical reactions) is a 2D N M R experiment in which cross peaks connect the chemical shifts of nuclei in starting compound and reaction product. If a reaction proceeds via a radical pair, the intensity of the cross peaks may be reduced due to dephasing processes occurring in the paramagnetic intermediate. This reduction is different for transverse and longitudinal nuclear spin magnetization. First, an expression for the amount of transferred magnetization and thus for the 2D cross peak intensity is derived in terms of the strength of the electron-nuclear hyperfine interaction and the lifetime of the radical pair; the influence of electron and nuclear spin relaxation is discussed as well. Then, the nuclear spin magnetization that is transferred to product is expressed as function of the reencounter probability and of the expectation value of finding the radical pair in a singlet electronic state. Two models that describe the reencounter probability are considered: an exponential model and a diffusion model. The resulting theoretical expressions are compared with experimental results of the SCOTCH experiment performed on the photochemical reaction of methyl terf-butyl ketone with tetrachloromethane. 1. Introduction
The SCOTCH experiment (spin coherence transfer in chemical reactions) is a useful tool for studying photochemical reactions in solution.1 It is a 2D NMR experiment involving pulse sequences that combine rf and light pulses and is based on transfer of either transverse or longitudinal nuclear spin magnetization during a reaction. The resulting 2D spectrum contains cross peaks that connect the frequencies of corresponding nuclei in starting compound and product. The intensity of these cross peaks will generally decrease if the reaction proceeds via a paramagnetic intermediate (e.g., a radical pair). In that case the hyperfine coupling of the unpaired electron with the nucleus induces a field in the z direction, in addition to the external magnetic field. As a result transversenuclearspin magnetizationin the intermediate performs a precession in the rotating frame of the corresponding diamagnetic molecule, either clockwise or anticlockwise, dependent on the orientation of the electron spin along the z axis. Opposite precession directions and a spread in lifetime of the paramagnetic intermediate cause cancellation and dephasing of nuclear spin coherence, respectively. An expression for the ratio of transverse and longitudinal nuclear spin magnetization has previously been derived from the modified Bloch equations for a one-sided first-order reaction, neglecting relaxation effects.* Here a complete derivation is presented, and the influenceof electron and nuclear spin relaxation is discussed. In this classical approach, any reaction of a radical pair, recombination and escape, is regarded as a unimolecular process. This is a correct description for a radical escaped from the radical pair that is scavenged by solvent molecules, a reaction that follows a pseudo-first-order rate law. However, the behavior of the radical pair itself is more appropriately described by a model that allows radicals to diffuse apart, reencounter again, and possibly react with each other. Therefore, we also derived a general expression for the transverse and longitudinal nuclear spin magnetization that is transferred from radical pair to product. This is determined by the probability that the radicals react with eachother and by the magnetizationthat is present in the radicals at the time of reaction. Equivalently, it depends on the spinindependent encounter probability f i t ) and on the expectation
* To whom correspondence should be addressed. @Abstractpublished in Advance ACS Absrracrs, November 15, 1993. 0022-3654/93/2097-133 18004.00/0
value of magnetizationin a singlet radical pair. This expectation value is derived from a density matrix calculation, and it is demonstrated that the probability of finding a radical pair in a singlet electronic state depends to some extent on the orientation of the nuclear spin. The exact form of the functionfit) depends on the model chosen: in the literature several models have been presented that describe the geminaterecombinationprocess, either with or without taking spin effects into account.”16 A large number of experimentalstudies, including a quantitative CIDNP (chemically induced dynamic nuclear polarization) study,14Js have shown that these models are qualitatively correct. Here we apply both an exponential model, which is shown to be equivalent to the approach based on the Bloch equations, and a diffusion model. The latter model results in an expression that contains physical parameters like diffusion coefficients, the original distance between the radicals, the reaction distance and the jump length of a diffusion step. Both models are compared with experimental results of the SCOTCH experiment performed on the photochemical reaction of methyl tert-butyl ketone with tetrachloromethane. 2. Theory
The evolution of nuclear spin magnetization in a radical pair R is discussed on the basis of a general photochemical reaction: hu
A+R-.B where A and B are the diamagnetic starting compound and product, respectively. Absorption of a photon and formation of the radical pair is supposed to be an instantaneous process that does not interfere with the electron spin orientation or the nuclear spin magnetization. The system is therefore considered to consist of two components: at time t = 0 the radical pair R is formed that evolves into the diamagnetic product B. For simplicity we assume that only one of the radicals of the radical pair contains a proton or a set of equivalent protons. (2.1) Magnetization in Radical Pair and Product Described with the Bloch E q ~ r t i o ~A.derivation from the modified Bloch equations is made assuming that the reaction from radical pair to product follows a (pseudo-) first-order rate law, characterized by rate constant k. In this approach R denotes only the radical containing the proton, because the radical without protons has 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 50, 1993 13319
Theory of the SCOTCH Experiment
etTjRP3 -
t
k
.-
MRZ
lk
(3) The Bloch equations in eq 2 likewise lead to an expression for the final amount of longitudinal magnetization in the product, M.:
MBZ
MB+ Figure 1. Reaction scheme for transverse and longitudinal nuclear spin magnetization, M+ and M,,respectively, transferred from radical R to product B with a first-order rate constant k. Indicated is the precession of transverse magnetization in the field of the unpaired electron with frequency ffo,the sign dependingon the orientation of the electron spin. Longitudinal electron spin relaxation is described with kl. = Tle-', longitudinaland transverse nuclear spin relaxation with k l =~ T11t-l and k 2 ~= Tz1t-1, respectively.
no influence on the nuclear spin magnetization. This gives a correct description for a radical that is scavenged by molecules present in large excess. But it leads to only an approximate expression for recombination reactions of a geminate pair, and at a later stage the radical pair character will be treated more explicitly. The reaction scheme is presented in Figure 1. For the description of transverse magnetization (M+ = Mx iMy)in the radical, a distinction is made between radicals with electron spin a and 8, because the orientation of the electron spin determines the precession direction of the nuclear spins. If described in the rotating frame of the diamagnetic product B, the precession frequency of the nuclear spin in the radical is only determined by the hyperfine coupling constant (a possible difference in the nuclear Zeeman terms of radical and product is negligibly small) and equals w = a / 2 h (in angular frequency units). Transitions between the twoelectron spinorientationsoccur with rateconstant kl,, related to the longitudinal electron spin relaxation time: kl, = TI,-'. Magnetization in the radical decays by nuclear spin relaxation processes: k2R = T z R -describes ~ transverse relaxation, while longitudinal relaxation is denoted by klR = TIR-I. Relaxation in the diamagnetic product is neglected, because it has no influence on the time scale of a radical reaction. The modified Bloch equations for the system are17J8
+
Equations l a and 1b were solved under the initial condition of equally populated electron spin states: MR,+(O) = Mw+(O) = &(0)/2, where MR(O)stands for the nuclear spin magnetization in the radical at the time of its origin. Substituting the resulting expressions for M b + ( t )andMw+(t) intoeq lcgivesanexpression for Me+(?),the transverse magnetization in the product formed up to time t . We are interested in the final amount of transverse magnetization in the product, called Mxy. Because all radicals have disappeared at the moment that the magnetization is detected, this corresponds to the value of Me+(t) for t is infinity:
(4)
Consequently, the ratio of transverse and longitudinal magnetization can be expressed as
If the nuclear spin relaxation times are long compared to the lifetime of the radical, T ~ and R T ~ >> R kl, this ratio simplies to
If, in addition, the longitudinal electron spin relaxation time is also long on the timescale of the reaction (Ti,>> kl)no transitions between the electron spin states a and j3 occur during the lifetime of the radical. In this case eq 6 reduces to the simple formula derived previous1y:z
(7) A comparison of eqs 6 and 7 shows that the electron spin relaxation process counteracts the decay of transverse magnetization: the value of Mxy/Mzin eq 6 is higher than that in eq 7. This can also be understood qualitatively from thevector model. Because the electron spin states are equally populated initially, half of the nuclear spins rotates with frequency w, while the other half rotates in opposite direction with frequency -a. Starting, for instance, with magnetization along the x axis, the y components cancel and only the x component survives. In the absence of electron spin relaxation the resulting vector oscillates along the x axis as cos(wt) during the lifetime of the radical. If, however, transitions between the electron spin states a and j3 occur very frequently during the lifetime of the radical, the precession of a nuclear spin is no longer exclusively in one direction but changes now and then from w to Y and vice versa. As a consequence, the resulting vector along the x axis oscillates at a slower rate. Thus, the dephasing of nuclear spin magnetization becomes less effective if the electron spin relaxation time is short. Nuclear spin relaxation processes can both counteract and contribute to the decay of transverse magnetization, as becomes clear from comparing eqs 5 and 6. If the transverse and longitudinal relaxation times are equal, this results in an apparent increase of the reaction rate k, thereby enhancing the fraction Mxy/Mz. If, on the other hand, the transverse relaxation time T ~ isRconsiderably shorter than the longitudinal relaxation time T ~ Rmagnetization , in the transverse plane will relax faster than magnetization along the z axis, resulting in a lower value for
1
M x y Mz
(2.2) Geminate Recombinationin the Light of the High-Field Radical Pair Theory. Until now, recombination and dissociation processes of a radical pair were considered as unimolecular reactions that could be described with a first-order rate constant. We will now give a description of the geminate pair analogous to the high-field radical pair theory of the CIDNP effect.I9,20 A geminate radical pair is formed from one or more precursor molecules in an electronic state that is either singlet or triplet initially. The radicals start to diffuse, and during this diffusion singlet-triplet transitions are induced by a difference in the
Pouwels and Kaptein
13320 The Journal of Physical Chemistry, Vol. 97, No. 50, 1993
electron Zeeman interaction of the radicals and by the electronnuclear hyperfine interaction. A recombination product may be formed if the radicals meet each other (reach a specific reaction distance) and if at the same time their unpaired electrons have the appropriate electronic configuration (i.e., the singlet state usually). Pairs not reacting at the first encounter again diffuse apart and may recombine during a following reencounter. A fractionof the radicals escapes thegeminate pair and never meets its original partner again but either forms random pairs with other radicals (so-called F pairs) or is scavenged by other molecules. Thus, geminate recombination depends both on the probability of an encounter,fit), on the probability of finding a radical pair in a singlet electronic state, ( S ( t ) ) ,and on the steric configurationof the radicals during an encounter, described with a factor A. The probability P(t) that a geminate pair has recombined at time t is proportional to the product of fit) and ( S ( t ) ) ,integrated from 0 to t:
P ( t ) = XJat) ( S ( t ) )dt
(8)
This expression takes into account only the single encounter that is described by the function At). The total probability of recombination is obtained by combining the contributions of all individual encounters, which is treated in the part following eq 26. We are especially interested in the magnetization that is present in the pair at the moment of recombination. This quantity is expressed similarly to the function P ( t ) but now containing the expectation value of nuclear spin magnetization in a singlet radical pair. The total amount of magnetization in the product can be written
characters successively indicate the electron spins of radicals 1 and 2 and the nuclear spin of radical 1. In this basis the Hamiltonian is diagonal:
%=
Y + O 0 Y 0 0 0 0
First, weconsider singlet-born radical pairs both with transverse and with longitudinal nuclear spin magnetization, initially described by the density matrices ps1+(0) and psl,(O). These matrices are formed by the direct product of the singlet electron and the transverse or longitudinalnuclear spin matrix, respectively. In contrast to CIDNP, where population differences in Boltzmann equilibrium are neglected, in the SCOTCH experiment the description of the evolution of nuclear spin magnetization in a radical pair is applied to the excess populations. It is only the population differencein Boltzmann equilibriumat the beginning of a pulse sequence that determines the finally detected signal, as is normally the case in NMR experiments. Thus, for the excess magnetization we have
0 1 0 - 1
-1 2
We make the usual assumption that the functionsfit) and (SI+( t ) ) or (SI,(t)) can be treated independently. Two models for a spin-independent encounter probability fit) are discussed in section 2.2.2. First, the amount of magnetization present in a singlet radical pair is expressed in terms of the nuclear spin orientation and of the electronic parameters of the radical pair (gvalues, hyperfine coupling constantand initial spin multiplicity). 2.2.1. Expectation Value of Magnetization in a Singlet Radical Pair. The Bloch equations show that transverse nuclear spin magnetization is influenced by the electron spin, but effects of the nuclear spin on the electronspins are not taken into account. In this section the mutual interaction between the spins is discussed, using a density matrix description. We consider the simplest case of a radical pair consisting of two radicals, with one nucleus coupled to the unpaired electron of radical 1. The spin Hamiltonian for this system in the high-field approximation is
(10) The exchange interaction J(r) depends on the distance between the two radicals and is therefore the only mechanism that couples the encounter probabilityflt) and the probability of finding the radical pair in a singlet configuration. To treat these probabilities independently, J(r) is further neglected, which is justifiable because the interaction is active only if the distance between the radicals is very small. In the high-field approximationtransitions are possible only between S and 7‘0 states. Because the energy difference between these states and the other triplet states is too large, S and TOare not coupled with T+ and T-, which implies that the CYUand Bj3 electron spin states can be neglected. We use the reduced basis (aBa, a/?@,Baa, @a/3),where the three
0 0 0 -Y-
O 0 -Y+ 0
1
0 - 1 0 - 1 0 - 1 0 1 0 1 0
(12a)
1 0
0 - 1 0 - 1 0 1 (12b) 2 - 1 0 1 0 0 1 0 - 1 Under the influence of the Hamiltonian these density matrices evolve according to the transformation21.22
-1
= exp[-i%t]p(O) exp[i%t] (13) Because the Hamiltonian is diagonal in the chosen basis the exponential matrices are diagonal as well, containing the terms exp[-iy+t], exp[-iy-t] and so on. The expectation value of magnetization in a singlet radical pair as a function of time is expressed as the trace of the density matrix p ( t ) multiplied by SI+ or SIz,21,22 the latter matrices containing the transverse or longitudinalnuclear spin operator I+or I, and the singlet electron spin operator S. The following expressions can be derived for the expectation value of magnetization in a singlet radical pair, dependent on whether the nuclear spins have been labeled with transverse or longitudinal magnetization,given a singlet electronic state initially: p(t)
(Sl+(t))s= Tr[p,,(t)SI+]
= ‘/2[cos(wt) + cos(At)] (14a)
( S I z ( t ) ) S= Tr[pS,z(t)SIzl = L/2[l+
/2
cos(@ ‘/2
+ wit) +
cos(@ - 4 t ) l
1 + cos(ot) cos(At)] (14b) The observation that these expectation values are not equal has some resemblance with the CIDNP experiment in which the singlet character of a radical pair depends on whether the nuclear spin is a or 8. Apparently, in the SCOTCH experiment the
=
The Journal of Physical Chemistry, Vol. 97, No. 50, 1993
Theory of the SCOTCH Experiment singlet character depends on whether the radical pair contains transverseor longitudinalnuclear spin magnetization. However, whereas in a CIDNP experiment the bulk of the radical pairs is considered, the singlet character as denoted in eq 14 applies only to the small amount of radical pairs that is investigated in the SCOTCH experiment, corresponding to the excess nuclear spin population. It must be noted that eqs 14a and 14b are identical if w or A are equal to zero. If the hyperfine interaction is absent, there is no dephasing or cancellationof transversemagnetization. The only effect that causes an oscillation of singlet character is the singlet-triplet mixing as a result of a difference in g values. Because this mechanism is not coupled with the nuclear spin, the orientation of this spin does not influence the amount of magnetization in a singlet radical pair. The orientation of the nuclear spin also has no effect if the radicals of the geminate pair have the same g value. However, in that case the hyperfine interaction still contributes to a decrease of transverse magnetization. Apparently, if the singlet-triplet mixing is caused only by the hyperfine interaction, its effect on longitudinal magnetization exactly matches the combined effect of singlet-triplet mixing and dephasing/cancellationon transverse magnetization. For comparison with the classical description, section 2.1, the influence of the hyperfine interaction on dephasing and cancellation processes can be expressed neglecting singlet-triplet transitions. This leads to the expectationvalues of the transverse and longitudinal nuclear spin operatorsirrespective of the electron spin state, Z+ and Zz:
It is clear that these expressions are equivalent to the classical description. If a radical pair is labeled with transverse nuclear spin magnetization, the amount of magnetization oscillates as cos(wt), as a result of opposite precession frequencies (see reasoning following eq 7), whereas it is constant if the radical pair is labeled with longitudinal magnetization. If the radical pair is initially formed in a triplet state, expressions similar to eq 14 can be derived for the expectation values of magnetization in a singlet radical pair, including both the effect of singlet-triplet mixing and of dephasing and cancellation:
(SI+(?)), = '/,[cos(wt) - cos(At)]
(16a)
(SZ,(t)),= '/'[1 -cos(wt) cos(At)]
(16b)
Finally, the density matrix approach is used for determining the total amount of geminate products, as expressed in eq 8. In this case the bulk of the nuclear spins, and not only the excess magnetization, is taken into account. The expectation value of finding a singlet-born radical pair in a singlet electronic state, irrespective of the nuclear spin state, can then be written as
( S ( t ) ) ,= '/'[1
+ cos(wt) cos(At)]
(17) This expression has the same time dependence as eq 14b, but eq 17 applies to the bulk of the nuclear spins rather than the excess population investigated in the SCOTCH experiment. 2.2.2. Encounter Probabilities According to the Exponential and the Diffusion Model. The probability that radicals from a geminate pair will encounter each other within the time interval ( t , t dt) is given byflt) dt. This distribution function has been described in many different ways. A simple model used in the early days of the radical pair theory is the so-called exponential model.23.24 In this model the radicals forming a geminate pair stay together during their lifetime and then either recombine or separate. Reencounters are neglected, which implies that radicals
+
13321
once separated never meet again. This behavior corresponds with an encounter probability that decays exponentially with rate constant k:
f , ( t ) = k exp[-kt]
(18) Alternatively, diffusion models have been developed that describe the reencounter statistics of particles without spin. In the random flight model according to Noyes the particles stay at a specific position for a short time and then move by a small diffusion step of length u . 3 ~Two ~ partners come into contact and can recombine if the distance between them is equal to or smaller than the reaction distance d , which approximately equals the sum of the van der Waals radii of the partners. If they do not react while in contact, they diffuse apart again and might react during a following reencounter. If both the jump length u and the positional lifetime approach zero, this model passes into the continuous diffusion model. Although in this model the time of an encounter is infinitely short, the recombination probability still has a nonzero value because the number of reencounters tends to infinity. We use the same approach as Salikhov et al.13 and Vollenweider and Fischer14J5and distinguish between the first encounter and reencounters by applying the functionsfo( t ) and f i ( r ) , respectively. At the moment of birth of a radical pair, the radicals usually are not in direct contact but are separated by a distance ro > d . If, for example, a radical pair is formed by photochemical dissociation, the excess energy (the differencebetween the energy of the photon and that needed for bond-breaking) is dissipated by giving an impulse to both fragments, which thereby reach an initial distance r0.25 For this case, the distribution function that expresses the probability of the first encounter, f o ( t ) , has been solved analytically from the continuous diffusion equation:'
In this expression the parameter D denotes the sum of the diffusion coefficients of the two radicals. The total recombination probability as a result of the first encounter is defined po and is equal to J"$o(t) dt = d/ro. Note that the function fo(t) is applicable only if the initial separationro is larger than the reaction distance d . During an encounter the partners are separated by the reaction distance d . If they do not react, this is their initial position for the description of a following reencounter which cannot be described byfo(t) because ro = d . In this case the encounter probability is taken from the random flight model. The distribution functionfi(t) is applicableto reencountersand to the first encounter if the geminate pair is generated at the reaction
m = (27/8r)'/'(1 - p ) ' ( d / ~ ) ' ~ ' / ' (20) Here T is the average time interval between two subsequent diffusion steps, which in the limit of continuous diffusion amounts to u2/6D. The parameter p is the probability of at least one reencounter (which also follows from J"$l(t) dt = p ) and is related to the ratio of reaction distance and jump length. If d / u > 1, p can in a very good approximation be written as3
2.2.3. Magnetization Transfer in the Exponential and the Diffusion Model. The amount of transverse and longitudinal nuclear spin magnetization that is transferred to product will now be expressed as the product of the encounter probability and the expectation value of magnetization in a singlet radical pair.
Pouwels and Kaptein
13322 The Journal of Physical Chemistry, Vol. 97, No. 50, 1993 a
encounter and for one of the following encounters. Substitution offo(t), eq 19, into eqs 23 and 22, respectively, gives the following expressions for the ratio of transverse and longitudinal magnetization as a result of the first encounter:
:/
1 1
I'll O V 0
,
where ,
0.25
.
,
.
0.5
I
0.75
'
,
'
1
!
1.25
'
I
1.5
t(1095.1) A
Figwe2. Influenccofsinglet-triplet mixingontheratioM,/M,described with the exponential model, as a function of reaction rate constant k,eq 24. The curves correspond to different values of A: (a) A = 0 rad s-l, (b) A = 5 X IO7 rad s-1, (c) A = 2 X 108 rad s-l, (d) A = 5.36 X lo9 rad s-l. Curved is identical to the graph belonging to eq 7, which denotes the ratio M /M,in the absence of singlet-triplet mixing. For all curves w = 2 X Igrrad s-1.
A similar expression is obtained for the ratio Mxy/Mzin the case of an encounter with the initial condition ro = d, which is described withfi(t) of eq 20:
For a singlet-born radical pair eq 14 can be substituted into eq 9, resulting in the following expression for the ratio Mxy/Mz: where To discuss the influence of singlet-triplet transitions, this ratio is also calculated for the situation in which singlet-triplet mixing is neglected, indicated by an asterisk, by substituting eq 15:
The encounter models presented in section 2.2.2 are substituted and discussed below, starting with the exponential model. The description based on the Bloch equations does not contain the influenceof singlet-triplet transitions, which corresponds with the assumption that a singlet-born radical pair remains in the singlet state. Substitution off0(r), eq 18, into eq 23 results in exactly the same expression for (Mxy/Mz)*as was derived in eq 7 , which shows the equivalence of the Bloch equations and the exponential model. If singlet-triplet transitions are taken into account, eq 22, application of the exponential model yields
.-+
M*., --
(k2 A')-'
-J
Mz
k-'
-
I
=--3(1 -P)
3d 2 D ) l i 2 2 P The ratios Mxy/Mzgiven in eqs 25 and 26 denote the effect of a single encounter, either the first or one of the following. A complete description is obtained if the contributions of all individual encounters are added together. We have done this by calculating the longitudinal and transverse magnetization components separately. This is illustrated for Mxy*,the transverse magnetization neglecting S - TOtransitions, using eqs 9a and 1 Sa. The first encounter contributes an amount MR(O)AJ"afo(t) cos(wt) dt to the transverse magnetization in the recombination product. Of the magnetization in the radical pairs that did not react during the first encounter, M~(0)(1-A)Jmdo(r) cos(wt) dt, a fraction AJmafi(r) cos(ot) dt is transferred to product during the second encounter, and so on. This gives the total amount of transverse magnetization:
X(4)
+ ( k 2+ w2'I-'
+ ( 2 [ k 2+ (A + u]'])-' + ( 2 [ k 2+ {A - o]~])-'
(24) In Figure 2 the ratio Mxy/Mzcalculated with eqs 7 and 24 is displayed for a few values of A as a function of reaction rate constant k. If A is larger than w , there is no difference between Mxy/Mzand (Mxy/Mz)*,curve d, but if A is on the order of w , it is clear that the effect of singlet-triplet mixing partly compensates the decrease of transverse magnetization. For a singlet-born radical pair transitions to the triplet state reduce the amount of magnetization in the product, because of a smaller chance of recombination and consequently a longer lifetime of the radical pair. But, as can be concluded from the increase of Mxy/Mzif singlet-triplet mixing is taken into account, the effect on longitudinal magnetization is apparently larger than on transverse magnetization. Now the magnetization transferred to product is described according to thediffusion model. First wediscuss the contribution of a single encounter on the ratio Mxy/Mz,both for the first
(27) A similar reasoning yields the expression for longitudinal magnetization in the product:
(28) MR(o)l - (1 - A)p The final ratio between transverse and longitudinalmagnetization in the product as a result of all encounters, neglecting S - TO transitions, is now expressed as
It can be recognized that XO(O) denotes the contribution of the
Theory of the SCOTCH Experiment
:: cn3-c-y-m3 ?3
+
hv
m,
,343
y3 .C-cH3
The Journal of Physical Chemistry, Vol. 97, No. 50, 1993 13323 S + "3
0
II cH3-c-cI
f
1
,343
c ~ l p and e
scavenging
L
I
3
4
3
Figure 3. Reaction scheme of the photochemical reaction of methyl ferfbutyl ketone with tetrachloromethane.
first encounter,while reencounters are represented by the fraction on the right-hand side. Now we derive similar expressions taking singlet-triplet mixing into account. At the first encounter a fraction A of the singlet radical pairs reacts to product. A fraction (1 - A) of the singlet encounters and all triplet encounters do not recombine and start the process of diffusion and singlet-triplet mixing again. This means that for the description of the following reencounters the radical pairs should be considered partly singlet-bornand partly triplet-born. The amount of transverse and longitudinal magnetization in the product is given by infinite geometric series, resulting in
with aof
1
= /zpo(xo(w)
a, = "x(4
* xo (A)) *X ( W
* 1/2(x0(A + w ) + xo(A a, = l/2P11 * '/21x(A + + x(A -
Bof
= '/ZPO11
0)
0)H
0)H
We followed the same method to calculate the total yield of geminate products G, formed from an initial number of radical pairs Ro. Using eqs 8 and 17,this resulted in
3, Results
The experimental arrangement for studying photochemical reactions with NMR consists of a Bruker AM 360 spectrometer and a Lambda Physik EMG 101 excimer laser operating at 308 nm and has been described in detail previously.2 The SCOTCH experiment was applied to the photochemical reaction of methyl tert-butyl ketone in tetrachloromethane. This reaction proceeds via a singlet precursor (seeFigure 3): the excited methyl tert-butyl ketone reacts with tetrachloromethane under formationof acetyl chlorideand a singlet radical pair that consists of the tert-butyl and the trichloromethyl radical.26 Three types of product can be recognized in this reaction: acetyl chloride 1 that is formed without passing through a radical intermediate, the geminaterecombination products 2-4, and the escape product terr-butyl chloride 5. The reaction was studied with both the transverse and the longitudinal version of the experiment. The 2D NMR spectra that are obtained with the basic pulse sequences of the SCOTCH experiment' contain off-diagonal cross peaks at position (WA,WB), resulting from transfer of either transverse or longitudinal
magnetization, and diagonal peaks at (UA,WA) and (WB,OB), belonging to nonconverted starting material A and accumulated product B, respectively. In a SCOTCH spectrum of the reaction recorded in this way, the small cross peaks lying close to the diagonal could not be detected because of very strong diagonal intensity. Therefore, use was made of pulse sequences in which diagonal peaks are s~ppressed.2~In the resulting spectra the cross peaks are well resolved.28 The intensity of the cross peaks is proportional to the magnetization that has been transferred to product, either M,or M,.Comparison of the two spectra provides the experimental ratio M,,/M,. For a quantitative comparison the cross peak of acetyl chloride 1 has been used as an internal standard: during the formation of this compound the magnetization is not influenced, because there is no interaction with an unpaired electron, which implies that in both types of spectrum the intensity of this peak is proportionalto the number of radical pairs formed. The relative amounts of transverse and longitudinal magnetization in the geminate products have been calculated from the intensities of the four cross peaks belonging to products 2-4, measured in a large number of experiments. The resulting value for M , , / M , was found to be 0.74 f 0.1 1.2 The cross peak belonging to the escape product tert-butyl chloride 5 could be detected only in the longitudinal magnetization spectrum. Its absenceinthe transverse magnetization spectrum has been explained by a complete dephasing of coherence during the lifetime of the escaped tertbutyl radical. It has been confirmed by time-resolved CIDNP experiments2 that the free tert-butyl radical is scavenged by the solvent tetrachloromethanewith a pseudo-first-orderrate constant k = 2.2 X 106 s-I corresponding to a lifetime of 0.45ps. The yield of geminate products was determinedfrom 1 D NMR spectra recorded after photolysis. The number of radical pairs formed, Ro, was obtained from the intensity of the acetyl chloride signal, and the amount of geminate products, G, was derived from the intensities of the methyl groups of 2 and 4. This method resulted in a value for G/& of 64 f 7%. 4. Discussion The experimental results are now compared with the theoretical expressions according to the exponentialand the diffusion model, using known or estimated values for the various parameters. The magnetic parameters for the geminate pair are as follows: the gvalues are 2.0098 and 2.0026for the trichloromethyl29 and the tert-butyl radical20 respectively, which means that A as defined in eq 1 1 amounts to 5.36 X 109 rad S-I in a BO field of 8.46 T. The hyperfine coupling constant of the protons in the latter radical is 2.27 mT,JOcorresponding to a / h = 4.0 X 108 rad s-1 and o = 2.0 X lo8rad s-l. A value of 100 ps for T ~ has R been estimated experimentally2l and T ~ will R be of the same order of magnitude. For this type of organic radicals Tle also amounts to a few microseconds. This implies that relaxation can be neglected on the time scale of recombination processes of the geminate pair. Thevalue 3.7 X 1t9m2s-l wasestimatedfor themutualdiffusion coefficient D.32-34 An average value of van der Waals radii and hydrodynamic radii resulted in a reaction distance d = 5.5 A.34 The value for the initial distance ro will be discussed below. (4.1) Exponential Model. The experimental ratio Mxy/M,of 0.74f 0.11 was substituted in the theoretical expressionobtained from the exponential model, eq 7. For the values of A and w as given above, this equation is identical to eq 24, as also follows from Figure 2. It appears that in this model, for the radical pair studied, the influence of singlet-triplet transitions is negligible. The decrease of transverse magnetization is completelydetermined by cancellation and dephasing processes as a result of the hyperfine interaction. The resulting reaction rate constant k amounts to (3.4f 1.0) X 108 s-1. Application of the exponential model thus leads to an average lifetime of the geminate pair of 2.9 1 .O ns. (4.2)Diffusion Model. First, the effect of the first encounter will be discussed. If singlet-triplet transitions are neglected, the
*
13324 The Journal of Physical Chemistry, Vol. 97, No. 50, 1993
t M.v Y 0.8-
, 0
0.2
0.4
0.6
0.8
A
1
+
Figure 4. Influence of singlet-triplet mixing and the value of p on the ratio M,/M, described with the diffusion model, as a function of the recombination probability during a singlet encounter, X. Curve a and b eq 29, in absence of singlet-triplet mixing, curve c and d: eq 30, with singlet-triplet mixin for p = 0.5 (a and c) and p = 0.9 (b and d). For all cuwcs ro = l Z . O X , d = 5.5 A, D = 3.7 X 10-9 m* s-1, A = 5.36 X lo9 rad s-l and w = 2 X 108 rad s-l.
contributionofthe first encounter to the ratio (Mv/Mz)* amounts to X O ( W ) , eq 25a. The argument of the exponential and cosine function, XO(W), is proportional to the difference between the initial separation and the reaction distance, (ro- 4,which implies that the choice of ro highly influences the value of XO(O). This is readily understood, because the initial distance ro determines the time between the formation of a radical pair and the first encounter of the radicals. A large distance corresponds with a relatively long time and consequently, a large decrease of transverse magnetization. From an estimation of the excess energy, an additional separation between the radicals was calculated to amount to 1.5 A, resulting in ro = 7.0 A.25.33935 This method for calculating the initial distance is reliable if a radical pair is formed by the splitting of one molecule, but it might be inaccurate if the radical pair is formed in another way. The exact mechanism of formation of the tert-butyl-trichloromethyl radical pair is not known, and it is possible that the acetyl chloride molecule is positioned between the radicals of the radical pair during its formation. The initial separation is then enlarged by at least 5.2 A (the diameter of the acetyl ~hloride3~), resulting in ro > 12 A. Substitution of the values of w, A, d, and D as given above results in (Mxy/MZ)o*= 0.959, 0.894, and 0.766 for ro = 8, 12, and 20 A, respectively, which shows the large effect of the choice of ro. If singlet-triplet transitions are introduced,the contribution of the first encounter to Mxy/Mzbecomes smaller, as follows from substitutionin eq 25b: instead of the above-mentionedvalues for (Mxy/MJ0*,the ratio (Mxy/MZ)ois calculated to be 0.977, 0.928, and 0.786 for ro = 8, 12, and 20 A. As already appeared from application of the exponential model, Figure 2, the ratio Mxy/Mzincreases if singlet-triplet mixing is taken into account. The effect of reencounters depends on parameters like the diffusion step u and the steric factor A. In Figure 4 the ratios (Mxy/Mz)*,q 29, and Mx,./Mz,eq 30, are plotted as a function of A, for two values of p (which is correlated with the step-size u) and for ro = 12A. If thesteric factor A equals 1, each encounter in the singlet state results in recombination, and the value for Mxy/Mzis completely determined by the contribution of the first singlet encounter. For lower values of A the total amount of reencounters increases, which corresponds with a longer average time before the radicals recombine and consequently a decrease of Mxy/Mz. It follows from this figure that the length of the diffusion step u, for u I d or p 1 0.5 as is usually assumed, has no strong influence on the ratio Mxy/Mz. Although the experimental value for MxyIMzof 0.74 & 0.1 1 suggests that ro is about 20 A, we consider such a large value for the initial distance not very realistic. A choice for ro of 12 A
Pouwels and Kaptein corresponds just within the error limits with the experimental ratio, Figure 4. This initial distance would be in agreement with a mechanism of radical pair formation in which the acetyl chloride molecule is positioned between the radicals of the pair in the primary step. Another way of estimating the initial distance is the yield of geminate products, which is connected with ro as shown in eq 3 1. The experimental ratio G/& = 64 f 7% would suggest an initial distance of 8.0 A. This clearly disagrees with the results from the SCOTCH experiment. However, the yield of geminate products is possibly overestimated as a consequence of further reactions with acetyl chloride (resulting in some small unidentified signals in the methyl region of the NMR spectrum recorded afterward), which decreasethe amount of acetyl chloride and thereby increase GI&. Generally, it can be concluded that for a singlet-born radical pair the ratio Mxy/M,is mainly determined by the first encounter and by the effect of singlet-triplet mixing. For a triplet-born radical pair, reencounters become much more important, because only a very small amount of radical pairs will react during the first encounter. The same conclusion could be drawn from a quantitative CIDNP study:l4J5 the choice of ro hardly influences the polarization or the yield of geminate products originating from a triplet-born radical pair, whereas it strongly determines thesevalues for a singlet-born radical pair. Toexplain the intensity of the polarizations and geminate yields for a singlet-bornradical pair, it appeared necessary to assume either a large initial distance (ro = U)I4 or a low recombination reactivity A,15 which corresponds with our results. 5. Conclusion The ratio of transverse and longitudinal nuclear spin magnetization transferred to a product has been expressed in terms of a function f i t ) that describes the encounter probability of the radicals. It has been shown that this ratio decreases due to dephasing and cancellation processes as a result of the electronnuclear hyperfine interaction during the lifetime of a radical pair. For a singlet-born radical pair this decrease is counteracted by singlet-triplet transitions. This has been explained by thedistinct effects that transverse and longitudinal nuclear spins have on the amount of singlet character. If the behavior of a radical pair is described with a diffusion model, it has been shown that for a singlet-born radical pair the first encounter of the radicals has a large influence on the ratio Mxy/Mz,while the reencounters only have a (small) effect if the recombination chance per singlet encounter, A, is very small. Because the first encounter is so important, the initial distance between the radicals largely determines the total decrease of transverse magnetization. For the photochemical reaction of methyl tert-butyl ketone with tetrachloromethane the experimental ratio Mxy/Mzhas been compared with the theoretical expressions. If the encounter probability is described with the exponential model, an average lifetime of the geminate radical pair of 2.9 ns is obtained. From a comparison with the diffusion model it appeared that the initial distance between the radicals is rather large, which suggests that during the formation of the geminate radical pair the acetyl chloride molecule is inserted between the trichloromethyl and rert-butyl radicals. Ackwwledgment. This work was supported by the Netherlands Foundation for Chemical Research (SON)with financial aid from the Netherlands Organization for Scientific Research (NWO). References and Note (1) Kemmink, J.; Vuiater, 0. W.; Boelens, R.;Dijkstra, K.: Kaptein, R. I. Am. Chem. Soc. 108, 5631. (2) Pouwels, P.J. W.; Kaptein, R. Z . Phys. Chem., in press. (3) Noyea, R. M.J . Chem. Phys. 1954, 22, 1349.
Theory of the SCOTCH Experiment (4) Noycs, R. M. J. Am. Chem. Soc. 1956,78,5486. ( 5 ) Adrian, F. J. J. Chem. Phys. 1970,53,3374; 1971,54, 3912. (6) Kaptein, R. J. Am. Chem. Sa.1!372,9I, 6251. (7) Deutch, J. M. J . Chem. Phys. 1972,56,6076. (8) Evans, G. T.; Fleming 111, P. D.; Lawler, R. G. J. Chem. Phys. 1973, 58, 207 1. (9) Salikhw, K. M.; Sarvarov, F. S.;Sagdeev, R. Z.; Molin, Yu. N. Kine?. Katal. 1975, 16, 279. (10) Freed, J. H.; Pcdersen, J. B. Ado. Magn.Reson. 1976, 8, 1. (1 1) Pcdersen, J. B. J. Chem. Phys. 1977,67,4097. (12) Muus, L. T.; Atkins, P. W.; McLauchlan, K. A.; Pcdersen, J. B., Eds. Chemically Induced Magnetic Polarization; D. Reidel: Dordrecht, 1977. (13) Salikhov, K. M.; Molin, Yu. N.; Sagdeev, R. Z.; Buchachenko, A.
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