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The creation of off-diagonal elements in chemically induced dynamic nuclear polarization experiments
ChemicalPhysics 14 (19761285793 0 North-Holland Publishing Company
THE CREATION OF OFF-DIAGONAL
ELEMENaS IN
CHEMICALLY INDUCED DYNAMIC NUCLEAR POLARIZATION
EXPERIMENTS
S. SCtiUBLJN, A. WOKAUN and R.R. ERNST Lnbomrorimn fiir physikafisclre Clrernic, Eidgetkische TeclrnisclzcHoclrschule, 8006 Ziirich. Swif~erla~ld Received 22 December 1975 The nuclear density matrix created in putsed CIDNP experiments contains off-diagonal elements whenever the resulting nuclear spin system is strongly coupled. These off-diagonal elements, which connect spin states with the ume magnetic quan. turn number, are due to the mhing of nuclear state functions during the process of product formation. The observation of these elements by means of a two-pulse experiment is described.
1. Introduction Radical reactions in solution are known to provide,
under certain conditions, enhanced nuclear polarization of the diamagnetic reaction products [l] _The occurrence of bath emission and absorption lines in an NMR spectrum is evidence for the existence OFa non-equilibrium state which may be of the first or the second kind * [2] . The current theories of chemically inhuced dynamic nuclear polarization [3-S] are confmed to the computation of populations of the nuclear spin states, i.e., they assume the creation of a non-equilibrium state of the first kind with a diagonal density matrix. Off-diagonal elements are usually disregarded as they can not be observed in a steady state experiment. The possibility to create off-diagonal elements by an experiment with pulsed chemical excitation has recently been predicted on theoretical grounds [2]. The purpose of this paper is to report experimental evidence to support these predictions. In addition, the earlier theoretical treatment [2] is extended by including dispersive parts of the resulting line shapes. It is found that off-diagonal elements connecting nuclear spin states with equal magnetic number can OG *
A non-equilibrium state of the fust kid is characterized by a density matrix diagonal in the eigenbase of the hamiltonian, i.e., the state can completely be described by population numbers In conlrast, a non-equiliirium state of the second kind is described by a density matrix containing off-diagonal elements in the eigenbase.
cur exclusively. A prerequisite is the existence of strong scalar or dipolar coupling among the nuclear spins in the diamagnetic reaction product. The off-diagonal elements are evidence for a non-adiabatic development of the product formation. For the experimental detection of the created offdiagonal elements, it is necessary to utilize pulse experiments with pulsed chemical excitation and with pulsed radio frequency fields to generate observable transverse magnetization. The experimental technique is equivalent to the detection of a general non-equilibrium state of the second kind [2,6] and is intimately
related to
the techniques of two-dimensional spectroscopy [7,8]. In section 2, the conditions for the creation of offdiagonal elements are discussed and the formalism for their computation is developped. Section 3 describes the method of detection of non-equilibtium states of the second kind with particular emphasis on CIDNP. For the experimental verification, the most simple nontrivial system, an AB system, was selected. The theoretical expressions for the signal line shapes are derived in section 4 for tlGs special case and the experimental results are described in section 5.
2. Off-diagonal elements in CIDNP experiments
The chemically induced dynamic nuclear polarization is generally assumed to be caused by the nuclearspin dependent singlet-triplet mixing during the dif-
fusional motion of an electronic spin pair and preferred recombinktion from the singlet state. To decide upon possibilities For the occurrence of off-diagonal elements, the process of polarization will be reexamined briefly. The special case of radical recombination will be discussed only. This situation corresponds to the practical example described in section 5. The consideration of escape products would lead qualitatively to exactly the same results.
The polarization process can be represented schematically by the following diagram: nuclear density % operator of educt optical excitation I initia! excited state pi sudden separation 1 Ps@)
diffusion and time evolution
initial radical pair state
I 2&r) projection onto singlet state
pair state before recombination
singlet pair state
pair recombination h
PJTr) trace aver election spin states, average over ‘or
,
diamagnetic reaction product
tor, do not change, i.e., the sudden approximation is valid, and all nuclear states are equally populated in the
initial excited state, represented by the electron spin nuclear spin density operator pi. It is diagonal in the product base formed by the electronic singlet and trip let functions, s, t _L, f. and t 1, and the nuclear spin functions Qi = cu,/3of nucleus i: 1y. I
(17zI,...,772J=s
Q-1
(y,.
3J$J
(y
3G1
(ml,..
.’
nuclear density operator of reaction product. The discussion will be limited to the high field case. For all but extremely low temperatures, it is possible to approximate the nuclear density operator of the educt, uO, by a multipk of the unity operator. The very small thermal population differences of the nuclear spin levels are completely immaterial; they wilI affect the final CIDNP intensities only by one part in 106. During the very fast optical excitation process (
Q,
. ..4.,>
= t-1
$1
02
.
=$I
6,
0,
. . o,,,
#1
o2
. . . o,,.
9,,, (0
I...,
q,> .,m
n
)=
r1
The separation of the excited state pi into a radical pair, described by p (0), occurs in the time of one diffusional step (-10-E s). This prao%s can again be de. scribed by the sudden approximation and one obtains for the radical pair density operator Pr&O> = Pi.
(2)
The change of base from the bound state to the radical pair state involves exclusively the electronic base functions. The nuclear base functions remain unaffected as long as the high field approximation applies and the hyperfme interaction can be treated in the weak coupling approximation. The interacticns between the nuclei are always much weaker than the hypertine interactions and can be completely neglected for the radical pair state. The eigenbase of the radical pair is given by the functions
. . o,,, .‘- ’ --I”) = I32c, @1_ J, 1,2,_1,2(ml,. . . .m,> = 4 9, 9, . . .9,, 9 l/2, l/2(%
4 -I,& &$. I ~(0)
P,,)
d,
Q&_&Q,
-. .3,,)
(3)
= @ 01 $2 “. o,lY
. >m,J = PP 9, 4r
#rl.
p&O) is no longer diagonal in this eigenbase. The offdiagonal elements connect exclusively states differing in the electronic part of the base functions. The nuclear spin states are still equally populated and no off-diagonal elements involving different nuclear spin states occur. During the diffusional motion of the radical pair, the off-diagonal elements of the density operator p,(t) will oscillate with frequencies which depend on the corresponding nucIear spin state. This evolution is responsible for the nuclear-spin-selective singlet-triplet mixing which leads, finally, to nuclear polarization of the reaction product. The nuclear spin states influence the time evo-
S. Scltiublin er al./Of(-diagonalelemenrsin CIDNP lution of the electronic state functions but they are themselves not directly involved in the evolution and remain mvariant (for the high field approximation). No off.diagonal elements of nuclear spin states occur. Following the accepted theories of CIDNP [j-5], it is assumed that the radical recombination proceeds exclusively from the singlet state of the radical pair during a reencounter. This may formally be expressed by the action of a projection operator PS which projects the spin state onto the singlet subspace. The projected density operator is then given by [9] sPr,(Q = t F, P&I
+ P,,(Q~J.
(4)
The radical recombination is again assumed to occur in a sufficiently short time to permii the application of the sudden approximation and cne obtains for the density operator of the diamagnetic reaction product P&J
= %&).
(G
The nuclear density operator of the reaction product, u(O), is finally obtained by calculating the trace over the electron spin states and by averaging over the distribution of the recombination times or. o(O) is still diagonal in the product base of the nuclear spin functions. For weak coupling among the nuclear spins, this base is identical with the eigenbase of the hamiltonian. On the other hand, for strong coupling, the eigenfunctions are linear combinations of product functions. The density matrix in the eigenbase of the hamiltonian, oe(@), is obtained from the density matrix in the product base, oP(O), by means of an orthogonal transformation T, oe(0) = 2-l UP(O)T.
(6)
~~(0) will, in general, contain off-diagonal elements which connect states with the same total magnetic quantum number. It is, therefore, exclusively the nuclear spin-spin coupling which is responsible for the occurrence of offdiagonal elements in the nuclear dertiity matrix. The generation of these elements occurs in the very last step, during the recombination of the radical pair. This phenomenon is not restricted to cases with a scalar spinspin coupling, but it does occur also for dipolar coupling and in the presence of radio frequency perturbations [lo] whenever the eigenbase of the nuclear hamiltonian does not coincide with the nuclear product functions. The prediction of the occurrence of off-diagonal eiements is strongly dependent on the sudden approxima-
287
tion used for the recombination of the radical pair. The conditions for the validity of the sudden approximation are stated in textbooks on quantum mechanics [l l] . It can be estimated that the !ime To during which the char.ge of the hamiltonian takes place should fulfti the condition To < l/A%,I
(7)
where (4si,J2
= (,11121&
(nIRln~2.
8i is the hamiltonian in rad/s averaged over the time T,, and In) is an eigenfunction of the system before the change. For the occurrence of off-diagonal elements between nuciear spin eigenfunctions, the internuclear scalar coupling terms in the hamiItonian Rare relevant. The following e.
with mi = -mi.
(9
The summation runs over pairs of nuclei with opposite magnetic quantum numbers in state II, only. As the coupling constants are in the order of a few hertz, and Tc is of the order of one diffusional step (-lo-l2 s), condition (7) is perfectly fulfilled. For low field experiments, an additional origin of off-diagonal elements exists. Here, the nuclear states are mixed by the hyperfiie interaction and they participate actively in the evolution of p,.&t) of the radical pair. This will be described further at another place. Finally, it should be mentioned that a related experiment has recently been performed by Fischer and Laroff [lo]. The chemically induced nuclear polarization was investigated in the presence of a radio frequency field. From the observed Torrey oscillations, it can be concluded that the produced polarization is built up along the static field. This suggests that during the diffusion period of the radical pair the rf field can be completely neglected, as the effects of the hypertine interaction are much stronger. After the recombination step, however, the interaction with the rf field becomes important causing a sudden change of the eigenbase of the hami!. tonian. The transformation of the density matrix into the new eigenbase (quantization along the effective field in the rotating frame) leads again to off-diagonal elements.
3. Detection’of non-equilibrium states of the second kind
The non-equilibrium state c(O), generated in a pulsed CIDNP experiment, is a particular exampIe of a nonequilibrium state of the second kind which occurs in various types of experiments [2,6]. A non-equilibrium state of the second kind may involve matrix elements which produl:e observable transverse magnetization, as well as ma!rix elements which are not directly observable, for example matrix elements connecting states with magnetic quantum numbers differing by 0 or 52, *3,.... The general observation scheme for a state of this kind consists of a set of experiments in which the system is allowed to develop freely during an evolution period of length rI. The evolution period is followed by an rf pulse after which the transverse magnetization is detected as is indicated in fig. 1 [!3] Due to the mixing effec? of the rf pulse, the transverse magnetization components will obtain also contributions from off-diagonal elements of ue(tl) with AM= 0 and AM = +_2, + 3, . . . . To trace out the development during the evolution period, tl is varied systematically from experiment to experiment. The mixing effect of the rf pulse is strongly dependent on the flip angle Q, and the relative intensities will become functions of a. This has been extensively described earlier [2,6]. The time development of the off-diagonal elements during the evolution period is given by
as there are no degenerate transitions. On the other hand, the time dependen= of the diagonal elements is governed by the master equation for populations [IZ] and will be non-exponential for coupled spin systems. The rf pulse with flip angle (I at time t = tr transforms the density operator into u(tI ,O) = exp(--iFya)
oftl)
exp(tiF,,a).
(10)
The free induction decay WJt, ,r2) is then recorded as a function of the second time parameter t2 t - 1,. The contribution of transition fjk) to the observed signal is calculated to be q
(13) Taking into account that the matrix representation of iFY in the eigenbase of !he hamiltonian is real, one obtams for the amplitudesAikQl) and D,X(tI)
where
Relaxation of the off-diagonal elements, measured by the relaxation time Tzrs, is purely exponential as long
Fig. 1. Pulsed C!DNP experiment with delay I, between lightand rL-pulse.
The diagonal elements of the hermitian matrix ae(t,) are real and, therefore, contribute to Alk(rl) only. This is put into evidence by writing
1~ [~"(tl),,l
The contribution of transition Gk) to the spectrum is finally obtained by means of a cosine Fourier transformation of eq. (13),
= oYO>,~ sid~,,~~)ed--ltITZrs).
It is then seen that the amplitude Aik(tl) of the absorption mode signal consists of terms showing a cosine dependence on I~, and of contributicns from the populations u”(c~),, which exhibit only a slow variation due to longtudinal relaxation. On the other hand, the an;plitude Djk(ll> of the dispersive signal consists of oscillating terms only, their dependence on I~ being given by sine-functions. The only non-vanishing off-diagonal elements ~~(0)~~in pulsed CIDNP experiments connect states with the wme magnetic quantum number. The oscillation frequencies wsr are, therefore, very low and of the order of the chemical shift differences and of nuclear coupling constants. This makes them easily observable.
4. The strongly coupled AB system with the result
For a non-equilibrium state of the first kind, Dik(ll) = 0, and the phase is in pure absorption over the entire spectrum. In contrast, D,%(rl) # 0 for a non-equilibrium state of the second kind, and the resulting shape is a superposition of an absorptive and a dispersive signal, different for each transition in the spectrum. This effect has recently also been observed in Fourier double resonance experiments [6,13]. The same effect occurs in repetitive Fourier experiments due to interference of the transverse magnetization of successive experiments [2] ; this has also been demonstrated by computer simulations [ 141 . In the case of pulsed CIDNP experiments discussed in section 2, the density matrix ~~(0) is real. Thus, rea! and imaginary parts of the off-diagonal elements IJ~(L~)~~, given by eq. (9) and used in eq. (1 S), are obtained in the form
The most simple nontrivial example of a CIDNP product with a non-diagonal initial density matrix u’(O) is the strongly coupled AB system with the resonance frequencies R, and R, and the coupling constant 2n.r. The initial density matrix in the product base can be described by four population numbers P,, ,P4:
(1%
The density matrix in the eigenbase, ue(tl), immediately before the rf pulse is then determined by the base transformation given in eq. (6) with
T=
tan(20) = 2lrJ/(SI,-a,),
(20)
and by the time evolution described by eq. (9). Neglecting relaxation, one finds
290
S. Sctiublir:
et aL/Of~dia~orral
where D = cos0, b = sinU, wz3 = JIn,-n,)2
t (2i#.
(22)
The neglect of relaxation is well justified for the present application, since the maximum value of I~ used in the experiments was 120 ms < Tzrr. Evaluation of eq. (15) yields, finally, the signal amphtudes+(rI) and Dik(tI):
elenrents in CIDNP
constant part (which depends on the populations and is independent of pi due to the neglect of relaxation) and an oscillating term proportionalto co~(w~~ri), while the dispersive amplitudes D,.Jfl) are purely OScillatory and show a sine.dependence on t1. A single oscillation frequency w23 occurs in this case as there is only one pair of AM= 0 off.diagonal elements in the AB system. The amplitudes of the oscillating terms are proportional to the density matrix eIements e(O),, = o(Q~2 = aW3 - P&.
(241
A,&)
= Cz4 +A ~os(o+~rJ
A&)
= C,, + B cos(o~~$J>
D,,(t,)
=D,,(t,)
= -W2-P3>
sin(+3t1),
The non-oscillating terms Cjk result from differences in the popuiations of the nuclear spin levels (classical CIDNP effect). Usually, they c0nstitute.a mayor contribution to the observed line intensities while the oscillatory parts are comparatively small. Due to the superposition of an absorptive and dispersive contribution, the resulting line shape is asymmetric with a positive and a negative extremum. The maximum of the absolute value is given by
&(fJ
= D&i)
= +W2-P3)
sI+J*$J>
l~~k~(r17~Z,opt)I = $T2jk[l’jk(ri)l
+(‘i)
= Ci2 -A cos(o~~$~ ),
A,,($)
= Cl, -B
cosGJ& (23)
with the abbreviations D = SCab(o’-b2),
+ &ia(rl)2 s = sirl(a/?-),
c = cos(a/2),
A = D(2? f 4s’ob - l)(P+‘~), B = D(2.s2 - 4s*& - l)(+P$,
CIZ = SC@fb)2 [&,
lS;,)(rl
‘9,@
)I=iT2jk[lAjk(f~)l
‘~lDj~(‘~)lID~~(‘~)I’j~(~~)lI,
s*+E)P,] ,
Cl 3 = $cfo-b)2 [GP, - s2P4
(25)
If the relation lD.,(r,)l+djk(tljl is fulfilled, the square root may 6 e expanded to give
- s2P4
- (u2 - ,I - E) P2 - (b2 1
+D;#$] .
06)
showing that the change in peak height aused by the superposition of a weak dispersive signal onto a strong absorption line is rather small.
-(62-s2-~P,-(n2-s2+E)P3], C*4 = SC@+b)2 [s2P1 - c+4 +(a* -s2-E)P2 CM =x&b)*
+(b2-s2tE)p3],
[s*P, - c2P4
t (6” - sz - E) fz + (a2 - s2+E)P,], E = ?.s2 ob(a2- b2).
The four transitions of the Al3 spectrum exhibit the characteristic behaviour discussed at the end of section 3. The absorptive amplitudesdjk(rl) consist each of a
5. Experiment To verify the theoretical results of section 4, the photochemical decomposition of di-p-chlorobenzoylperoxide-d4 * [i 51 was chosen. This molecule yields a suitable recombination product. The reaction scheme, fig. 2, was fomrulated in accordance with the results of Blank and Fischer [16] on the decomposition of aromatic peroxides. The product molecule of interest is p-chlorophenyl-p-chIorobenzoatedd (II, fig. 2), where I
Synthesized by Merck, Shop & Dohme.
S. Scltiublin
et al./Of~diagor~i eIemenrs in CIDNP
291
.
(24)
w (12) (341
,,,;I-
II
Fig. 2. Reactions in the phctochemicll decomposition of di-pChlorobcnzoSfl-peroKide-d4 (I) in a mixture of CC14 and p diosane_dB. Asterisks denote polarized nuclei.
only the two strongly coupled protons on the phenolic moiety are polarized [ 161. The reaction of escaping radicals with the solvent Ccl, yields p-dichlorobenzened2 (III, fig. 2). A 0.14 M solution of di-p-chlorobenzoyl-peroxided4 in a 50% (v/v) mixture of Ccl, and p.dioxane-dg was investigated. The latter solvent provided the signal for the internal heteronuclear lock. A low pressure xenon flash tube, with an average flash duration of 2.5 ms and and energy of 500 joule, was used to initiate the photochemical reaction. The experiment described in fig. 1 was completely computer-controlled, and for each value of the time delay tl, 100 free induction decays were coadded. A flip angle (Y= 45” was used. A series of experiments was performed where rI was varied in steps of 10 ms, from 7.5 to 117.5 ms. Details of the apparatus and methods used will be described at another place. A typical 90 MHz proton spectrum obtained in this manner (fig. 3) shows the four lines of an AB system, labelled (34), (12), (24), (13), and an additional strong emission line in the center. The AB spectrum is characterized by the parameters J= 8.7 Hz, (5ZA-RB)/2n = 20.5 Hz, with an oscillation frequency ~~~/27r= 22.3 Hz [eq. (22)] . It is assigned to the polarized ester II (fig. 2). The emission line is due to p.dichlorobenzened2 (III, fig. 2). It was verified that reactions of escaping radicals with the second solvent, p.dioxanedg, are not observable in the presence of 50% Ccl,.
Fig. 3. CIDNP signals from an benzoyl-pcrosided4 in a 50% diosaneds. 50 FIDs, obtained fg. 1, were accumulated; I, q were used.
0.14 M solution of di-p-chlora(v/v) misture of CCL, and pwith the experiment shown in 57.5 ms and a flip an@ u = 45’
To compensate for small homogeneity changes and for longterm drifts of the spectrometer, the peak heights of the AB spectrum were normalized by the height of the (nomoscillatingj emission signal of p-dichloroben-
L
___.A
._I_
.
4
.2. .l
M
r
1 -I_
20
4n
80
I, e.0
ti --* 100
tj (ml)
Fig. 4. Measured and fitted peak heights (normal&d by the height of the emission line) of the polarized AB system of Fchlorophenyl-p-chlorobenzoated~ (II, fg. 2) for differen,; time deIays II berwee‘n light- and rf-pulse.
S. Schiubli~l et ol/Offdiogoml
292
.I01
Fig. 5. Calculated flip anple deper.j,:nce of the osci!lation amplitudes A and B defined in eq. (23). The same plramcters and the WTUZS&C as in fii. 4 were used.
elements in UDNP
in the case of strong nuclear coupling within the reaction product. They connect states with equal magnetic quantum number, and are due to the fact that product formation occurs in a non-adiabatic manner. In an experiment with pulsed chemical excitation and with pulse Fourier detection, these off-diagonal elements wiLlcontribute to the transverse magnetization. Consequently, line intensities show an oscillatory dependence on the time between the chemical excitation and the observing pulse. A single oscillation frequency was observed in the case of the strongly coupled two spin system, in agreement with the theoretical treatment. In more complicated spin systems, the number2 of oscillation frequencies quickly increases. For a system of N strongly coupled, non-eqtiivalent spins i, for example, it is given
by zene$. The results are plotted in fig. 4. The solid curves were obtained by adjusting the populations P,, __,P, [eq. (19)] in a leas! squares fit to the measured peak heights [eqs. (23), (25)]. The oscillation is given by a slightly distorted cosine-function due to the contribution of dispersion-type signals [eq. (26)]. In fig. 5, the theoretical oscillation amplitudes A and B, defined in eq. (23), are plotted against :he flip angle a; the curves were calculated using the parameters of the measured example. The amplitudes show a pronounced dependence on the flip angle;_4 is zero for u= 2 arcTin { [~(LI + b)]-l}. whereas B vanishes foror=2arcsin{[~(u-b)l-1).Forol=90”,A=-B, which means that the osci!latory contribution to the absorptive part of the signal has the same absolute amplitude for all four transitions for this particular flip angle. Similar experiments were performed on solutions of di-(2,3,4-trimethyl-bemoyl)-peroxide * in the same solvent mixture. The photodecomposition leads in this case to at least two polarized reaction products with strongly overlapping AB spectra in the aromatic region. They all exhibit the described oscillation phenomenon.
6. Conclusions In radicai reactions leading to enhanced nuclear po&i&ion (CIDNP), offdiagonal elements are produced l
Synthes~kd in our laboratory.
z
=
io I
1 2N _ 2” 2
.
(27)
N
Six ze;o quantum ;ransition frequencies are obtained for three coupled nuclei, while already 27 frequencies are expected for the case of four strongly coupled spins. These oscillations can be observed exclusively in pulsed UDNP experiments. In experiments with cop tinuous
chemical
excitation,
the off-diagonal
elements
They will partially be averaged and their influence on the spectrum is unobservably small for all practical purposes. are produced
over an extended
period
of time.
This research was supported by the Swiss National Science Foundation. Some help by Mr. J. Keller in the numerical data evaluation is acknowledged.
References [I] J. Bargon, H. Fischer and I-I. Johmn, 2. Naturforsch. 22a (1967) 1551. [2] S. Schiublin, A. HGhener and RR Ernst. J. Magn. Res. 13 (1974) 196. j3] R. Kaptein and L.J. Ooskrhoff, C&m. Phyr Lett. 4 (1969)
195,214; R. Kaptein, J. Amer. Chem. Sot. 94 (1972) 6251. [4] G-L. Gloss and L.E. Gloss, I. Amer. Chem. Sac. 9i (1969) 4549,455Q;
S. Schjublin
[S] [6] [7] [a] [9]
er al/Off-dbgoml
G.L. Gloss, I. Amer. Chem. SOE. 91 (1969) 4552; G.L. Clossand A.D. Trifunac, J. Amer. Chem. Sot 91 (1969) 4554; J. Amer. Chem. SOL 92 (1970) 2183,2186. F.J. Adrian, J. Chem. Phys. 54 (1970) 3912. R.R. Ernst, W.P. Aue, E. Bxtholdi, A. Hohener and S. Sch;iubiin, Pure AppL Chem. 37 (1974) 47. R.R. Ernst, Chiti 29 (1975) 179. W.P. Aue, E. Bartholdi and R.R. Ernst, J. Chcm. Phys. 64 (1976) 2229. M. Tomkiewicz, A. Croen and M. Cocivera, J. Chem. Phys. 56 (1972) 5850.
elmmnis in CIDNP
293
[IO] H. Fischer andC.P. Lxoff, Chcm. Piiys. 3 (1974) 217. [ll] See, for example, A. Messiah,QuailturnMechanics, VoL II (North-Holland, Amsterdam, 1969) p. 739. [12] A. Abragam, The Principles of Nuclear Magnetism(Clarcndon Press, Oxford, 1961) p. 274. [13] N.R. Krishna,J. Chem. Phys. 53 (1975) 4329. !14] P. h&kin and J.P. Jesson, J. Msg. Res. 18 (1975) 411. [lj] J.L Charlton and I. Bargon. Chem. Phys. Lett. 8 (1971) 442. (16) B. Blank and H. Fischer, Helv. Chim. Acta 54 (1971) 905.
THE CREATION OF OFF-DIAGONAL
ELEMENaS IN
CHEMICALLY INDUCED DYNAMIC NUCLEAR POLARIZATION
EXPERIMENTS
S. SCtiUBLJN, A. WOKAUN and R.R. ERNST Lnbomrorimn fiir physikafisclre Clrernic, Eidgetkische TeclrnisclzcHoclrschule, 8006 Ziirich. Swif~erla~ld Received 22 December 1975 The nuclear density matrix created in putsed CIDNP experiments contains off-diagonal elements whenever the resulting nuclear spin system is strongly coupled. These off-diagonal elements, which connect spin states with the ume magnetic quan. turn number, are due to the mhing of nuclear state functions during the process of product formation. The observation of these elements by means of a two-pulse experiment is described.
1. Introduction Radical reactions in solution are known to provide,
under certain conditions, enhanced nuclear polarization of the diamagnetic reaction products [l] _The occurrence of bath emission and absorption lines in an NMR spectrum is evidence for the existence OFa non-equilibrium state which may be of the first or the second kind * [2] . The current theories of chemically inhuced dynamic nuclear polarization [3-S] are confmed to the computation of populations of the nuclear spin states, i.e., they assume the creation of a non-equilibrium state of the first kind with a diagonal density matrix. Off-diagonal elements are usually disregarded as they can not be observed in a steady state experiment. The possibility to create off-diagonal elements by an experiment with pulsed chemical excitation has recently been predicted on theoretical grounds [2]. The purpose of this paper is to report experimental evidence to support these predictions. In addition, the earlier theoretical treatment [2] is extended by including dispersive parts of the resulting line shapes. It is found that off-diagonal elements connecting nuclear spin states with equal magnetic number can OG *
A non-equilibrium state of the fust kid is characterized by a density matrix diagonal in the eigenbase of the hamiltonian, i.e., the state can completely be described by population numbers In conlrast, a non-equiliirium state of the second kind is described by a density matrix containing off-diagonal elements in the eigenbase.
cur exclusively. A prerequisite is the existence of strong scalar or dipolar coupling among the nuclear spins in the diamagnetic reaction product. The off-diagonal elements are evidence for a non-adiabatic development of the product formation. For the experimental detection of the created offdiagonal elements, it is necessary to utilize pulse experiments with pulsed chemical excitation and with pulsed radio frequency fields to generate observable transverse magnetization. The experimental technique is equivalent to the detection of a general non-equilibrium state of the second kind [2,6] and is intimately
related to
the techniques of two-dimensional spectroscopy [7,8]. In section 2, the conditions for the creation of offdiagonal elements are discussed and the formalism for their computation is developped. Section 3 describes the method of detection of non-equilibtium states of the second kind with particular emphasis on CIDNP. For the experimental verification, the most simple nontrivial system, an AB system, was selected. The theoretical expressions for the signal line shapes are derived in section 4 for tlGs special case and the experimental results are described in section 5.
2. Off-diagonal elements in CIDNP experiments
The chemically induced dynamic nuclear polarization is generally assumed to be caused by the nuclearspin dependent singlet-triplet mixing during the dif-
fusional motion of an electronic spin pair and preferred recombinktion from the singlet state. To decide upon possibilities For the occurrence of off-diagonal elements, the process of polarization will be reexamined briefly. The special case of radical recombination will be discussed only. This situation corresponds to the practical example described in section 5. The consideration of escape products would lead qualitatively to exactly the same results.
The polarization process can be represented schematically by the following diagram: nuclear density % operator of educt optical excitation I initia! excited state pi sudden separation 1 Ps@)
diffusion and time evolution
initial radical pair state
I 2&r) projection onto singlet state
pair state before recombination
singlet pair state
pair recombination h
PJTr) trace aver election spin states, average over ‘or
,
diamagnetic reaction product
tor, do not change, i.e., the sudden approximation is valid, and all nuclear states are equally populated in the
initial excited state, represented by the electron spin nuclear spin density operator pi. It is diagonal in the product base formed by the electronic singlet and trip let functions, s, t _L, f. and t 1, and the nuclear spin functions Qi = cu,/3of nucleus i: 1y. I
(17zI,...,772J=s
Q-1
(y,.
3J$J
(y
3G1
(ml,..
.’
nuclear density operator of reaction product. The discussion will be limited to the high field case. For all but extremely low temperatures, it is possible to approximate the nuclear density operator of the educt, uO, by a multipk of the unity operator. The very small thermal population differences of the nuclear spin levels are completely immaterial; they wilI affect the final CIDNP intensities only by one part in 106. During the very fast optical excitation process (
Q,
. ..4.,>
= t-1
$1
02
.
=$I
6,
0,
. . o,,,
#1
o2
. . . o,,.
9,,, (0
I...,
q,> .,m
n
)=
r1
The separation of the excited state pi into a radical pair, described by p (0), occurs in the time of one diffusional step (-10-E s). This prao%s can again be de. scribed by the sudden approximation and one obtains for the radical pair density operator Pr&O> = Pi.
(2)
The change of base from the bound state to the radical pair state involves exclusively the electronic base functions. The nuclear base functions remain unaffected as long as the high field approximation applies and the hyperfme interaction can be treated in the weak coupling approximation. The interacticns between the nuclei are always much weaker than the hypertine interactions and can be completely neglected for the radical pair state. The eigenbase of the radical pair is given by the functions
. . o,,, .‘- ’ --I”) = I32c, @1_ J, 1,2,_1,2(ml,. . . .m,> = 4 9, 9, . . .9,, 9 l/2, l/2(%
4 -I,& &$. I ~(0)
P,,)
d,
Q&_&Q,
-. .3,,)
(3)
= @ 01 $2 “. o,lY
. >m,J = PP 9, 4r
#rl.
p&O) is no longer diagonal in this eigenbase. The offdiagonal elements connect exclusively states differing in the electronic part of the base functions. The nuclear spin states are still equally populated and no off-diagonal elements involving different nuclear spin states occur. During the diffusional motion of the radical pair, the off-diagonal elements of the density operator p,(t) will oscillate with frequencies which depend on the corresponding nucIear spin state. This evolution is responsible for the nuclear-spin-selective singlet-triplet mixing which leads, finally, to nuclear polarization of the reaction product. The nuclear spin states influence the time evo-
S. Scltiublin er al./Of(-diagonalelemenrsin CIDNP lution of the electronic state functions but they are themselves not directly involved in the evolution and remain mvariant (for the high field approximation). No off.diagonal elements of nuclear spin states occur. Following the accepted theories of CIDNP [j-5], it is assumed that the radical recombination proceeds exclusively from the singlet state of the radical pair during a reencounter. This may formally be expressed by the action of a projection operator PS which projects the spin state onto the singlet subspace. The projected density operator is then given by [9] sPr,(Q = t F, P&I
+ P,,(Q~J.
(4)
The radical recombination is again assumed to occur in a sufficiently short time to permii the application of the sudden approximation and cne obtains for the density operator of the diamagnetic reaction product P&J
= %&).
(G
The nuclear density operator of the reaction product, u(O), is finally obtained by calculating the trace over the electron spin states and by averaging over the distribution of the recombination times or. o(O) is still diagonal in the product base of the nuclear spin functions. For weak coupling among the nuclear spins, this base is identical with the eigenbase of the hamiltonian. On the other hand, for strong coupling, the eigenfunctions are linear combinations of product functions. The density matrix in the eigenbase of the hamiltonian, oe(@), is obtained from the density matrix in the product base, oP(O), by means of an orthogonal transformation T, oe(0) = 2-l UP(O)T.
(6)
~~(0) will, in general, contain off-diagonal elements which connect states with the same total magnetic quantum number. It is, therefore, exclusively the nuclear spin-spin coupling which is responsible for the occurrence of offdiagonal elements in the nuclear dertiity matrix. The generation of these elements occurs in the very last step, during the recombination of the radical pair. This phenomenon is not restricted to cases with a scalar spinspin coupling, but it does occur also for dipolar coupling and in the presence of radio frequency perturbations [lo] whenever the eigenbase of the nuclear hamiltonian does not coincide with the nuclear product functions. The prediction of the occurrence of off-diagonal eiements is strongly dependent on the sudden approxima-
287
tion used for the recombination of the radical pair. The conditions for the validity of the sudden approximation are stated in textbooks on quantum mechanics [l l] . It can be estimated that the !ime To during which the char.ge of the hamiltonian takes place should fulfti the condition To < l/A%,I
(7)
where (4si,J2
= (,11121&
(nIRln~2.
8i is the hamiltonian in rad/s averaged over the time T,, and In) is an eigenfunction of the system before the change. For the occurrence of off-diagonal elements between nuciear spin eigenfunctions, the internuclear scalar coupling terms in the hamiItonian Rare relevant. The following e.
with mi = -mi.
(9
The summation runs over pairs of nuclei with opposite magnetic quantum numbers in state II, only. As the coupling constants are in the order of a few hertz, and Tc is of the order of one diffusional step (-lo-l2 s), condition (7) is perfectly fulfilled. For low field experiments, an additional origin of off-diagonal elements exists. Here, the nuclear states are mixed by the hyperfiie interaction and they participate actively in the evolution of p,.&t) of the radical pair. This will be described further at another place. Finally, it should be mentioned that a related experiment has recently been performed by Fischer and Laroff [lo]. The chemically induced nuclear polarization was investigated in the presence of a radio frequency field. From the observed Torrey oscillations, it can be concluded that the produced polarization is built up along the static field. This suggests that during the diffusion period of the radical pair the rf field can be completely neglected, as the effects of the hypertine interaction are much stronger. After the recombination step, however, the interaction with the rf field becomes important causing a sudden change of the eigenbase of the hami!. tonian. The transformation of the density matrix into the new eigenbase (quantization along the effective field in the rotating frame) leads again to off-diagonal elements.
3. Detection’of non-equilibrium states of the second kind
The non-equilibrium state c(O), generated in a pulsed CIDNP experiment, is a particular exampIe of a nonequilibrium state of the second kind which occurs in various types of experiments [2,6]. A non-equilibrium state of the second kind may involve matrix elements which produl:e observable transverse magnetization, as well as ma!rix elements which are not directly observable, for example matrix elements connecting states with magnetic quantum numbers differing by 0 or 52, *3,.... The general observation scheme for a state of this kind consists of a set of experiments in which the system is allowed to develop freely during an evolution period of length rI. The evolution period is followed by an rf pulse after which the transverse magnetization is detected as is indicated in fig. 1 [!3] Due to the mixing effec? of the rf pulse, the transverse magnetization components will obtain also contributions from off-diagonal elements of ue(tl) with AM= 0 and AM = +_2, + 3, . . . . To trace out the development during the evolution period, tl is varied systematically from experiment to experiment. The mixing effect of the rf pulse is strongly dependent on the flip angle Q, and the relative intensities will become functions of a. This has been extensively described earlier [2,6]. The time development of the off-diagonal elements during the evolution period is given by
as there are no degenerate transitions. On the other hand, the time dependen= of the diagonal elements is governed by the master equation for populations [IZ] and will be non-exponential for coupled spin systems. The rf pulse with flip angle (I at time t = tr transforms the density operator into u(tI ,O) = exp(--iFya)
oftl)
exp(tiF,,a).
(10)
The free induction decay WJt, ,r2) is then recorded as a function of the second time parameter t2 t - 1,. The contribution of transition fjk) to the observed signal is calculated to be q
(13) Taking into account that the matrix representation of iFY in the eigenbase of !he hamiltonian is real, one obtams for the amplitudesAikQl) and D,X(tI)
where
Relaxation of the off-diagonal elements, measured by the relaxation time Tzrs, is purely exponential as long
Fig. 1. Pulsed C!DNP experiment with delay I, between lightand rL-pulse.
The diagonal elements of the hermitian matrix ae(t,) are real and, therefore, contribute to Alk(rl) only. This is put into evidence by writing
1~ [~"(tl),,l
The contribution of transition Gk) to the spectrum is finally obtained by means of a cosine Fourier transformation of eq. (13),
= oYO>,~ sid~,,~~)ed--ltITZrs).
It is then seen that the amplitude Aik(tl) of the absorption mode signal consists of terms showing a cosine dependence on I~, and of contributicns from the populations u”(c~),, which exhibit only a slow variation due to longtudinal relaxation. On the other hand, the an;plitude Djk(ll> of the dispersive signal consists of oscillating terms only, their dependence on I~ being given by sine-functions. The only non-vanishing off-diagonal elements ~~(0)~~in pulsed CIDNP experiments connect states with the wme magnetic quantum number. The oscillation frequencies wsr are, therefore, very low and of the order of the chemical shift differences and of nuclear coupling constants. This makes them easily observable.
4. The strongly coupled AB system with the result
For a non-equilibrium state of the first kind, Dik(ll) = 0, and the phase is in pure absorption over the entire spectrum. In contrast, D,%(rl) # 0 for a non-equilibrium state of the second kind, and the resulting shape is a superposition of an absorptive and a dispersive signal, different for each transition in the spectrum. This effect has recently also been observed in Fourier double resonance experiments [6,13]. The same effect occurs in repetitive Fourier experiments due to interference of the transverse magnetization of successive experiments [2] ; this has also been demonstrated by computer simulations [ 141 . In the case of pulsed CIDNP experiments discussed in section 2, the density matrix ~~(0) is real. Thus, rea! and imaginary parts of the off-diagonal elements IJ~(L~)~~, given by eq. (9) and used in eq. (1 S), are obtained in the form
The most simple nontrivial example of a CIDNP product with a non-diagonal initial density matrix u’(O) is the strongly coupled AB system with the resonance frequencies R, and R, and the coupling constant 2n.r. The initial density matrix in the product base can be described by four population numbers P,, ,P4:
(1%
The density matrix in the eigenbase, ue(tl), immediately before the rf pulse is then determined by the base transformation given in eq. (6) with
T=
tan(20) = 2lrJ/(SI,-a,),
(20)
and by the time evolution described by eq. (9). Neglecting relaxation, one finds
290
S. Sctiublir:
et aL/Of~dia~orral
where D = cos0, b = sinU, wz3 = JIn,-n,)2
t (2i#.
(22)
The neglect of relaxation is well justified for the present application, since the maximum value of I~ used in the experiments was 120 ms < Tzrr. Evaluation of eq. (15) yields, finally, the signal amphtudes+(rI) and Dik(tI):
elenrents in CIDNP
constant part (which depends on the populations and is independent of pi due to the neglect of relaxation) and an oscillating term proportionalto co~(w~~ri), while the dispersive amplitudes D,.Jfl) are purely OScillatory and show a sine.dependence on t1. A single oscillation frequency w23 occurs in this case as there is only one pair of AM= 0 off.diagonal elements in the AB system. The amplitudes of the oscillating terms are proportional to the density matrix eIements e(O),, = o(Q~2 = aW3 - P&.
(241
A,&)
= Cz4 +A ~os(o+~rJ
A&)
= C,, + B cos(o~~$J>
D,,(t,)
=D,,(t,)
= -W2-P3>
sin(+3t1),
The non-oscillating terms Cjk result from differences in the popuiations of the nuclear spin levels (classical CIDNP effect). Usually, they c0nstitute.a mayor contribution to the observed line intensities while the oscillatory parts are comparatively small. Due to the superposition of an absorptive and dispersive contribution, the resulting line shape is asymmetric with a positive and a negative extremum. The maximum of the absolute value is given by
&(fJ
= D&i)
= +W2-P3)
sI+J*$J>
l~~k~(r17~Z,opt)I = $T2jk[l’jk(ri)l
+(‘i)
= Ci2 -A cos(o~~$~ ),
A,,($)
= Cl, -B
cosGJ& (23)
with the abbreviations D = SCab(o’-b2),
+ &ia(rl)2 s = sirl(a/?-),
c = cos(a/2),
A = D(2? f 4s’ob - l)(P+‘~), B = D(2.s2 - 4s*& - l)(+P$,
CIZ = SC@fb)2 [&,
lS;,)(rl
‘9,@
)I=iT2jk[lAjk(f~)l
‘~lDj~(‘~)lID~~(‘~)I’j~(~~)lI,
s*+E)P,] ,
Cl 3 = $cfo-b)2 [GP, - s2P4
(25)
If the relation lD.,(r,)l+djk(tljl is fulfilled, the square root may 6 e expanded to give
- s2P4
- (u2 - ,I - E) P2 - (b2 1
+D;#$] .
06)
showing that the change in peak height aused by the superposition of a weak dispersive signal onto a strong absorption line is rather small.
-(62-s2-~P,-(n2-s2+E)P3], C*4 = SC@+b)2 [s2P1 - c+4 +(a* -s2-E)P2 CM =x&b)*
+(b2-s2tE)p3],
[s*P, - c2P4
t (6” - sz - E) fz + (a2 - s2+E)P,], E = ?.s2 ob(a2- b2).
The four transitions of the Al3 spectrum exhibit the characteristic behaviour discussed at the end of section 3. The absorptive amplitudesdjk(rl) consist each of a
5. Experiment To verify the theoretical results of section 4, the photochemical decomposition of di-p-chlorobenzoylperoxide-d4 * [i 51 was chosen. This molecule yields a suitable recombination product. The reaction scheme, fig. 2, was fomrulated in accordance with the results of Blank and Fischer [16] on the decomposition of aromatic peroxides. The product molecule of interest is p-chlorophenyl-p-chIorobenzoatedd (II, fig. 2), where I
Synthesized by Merck, Shop & Dohme.
S. Scltiublin
et al./Of~diagor~i eIemenrs in CIDNP
291
.
(24)
w (12) (341
,,,;I-
II
Fig. 2. Reactions in the phctochemicll decomposition of di-pChlorobcnzoSfl-peroKide-d4 (I) in a mixture of CC14 and p diosane_dB. Asterisks denote polarized nuclei.
only the two strongly coupled protons on the phenolic moiety are polarized [ 161. The reaction of escaping radicals with the solvent Ccl, yields p-dichlorobenzened2 (III, fig. 2). A 0.14 M solution of di-p-chlorobenzoyl-peroxided4 in a 50% (v/v) mixture of Ccl, and p.dioxane-dg was investigated. The latter solvent provided the signal for the internal heteronuclear lock. A low pressure xenon flash tube, with an average flash duration of 2.5 ms and and energy of 500 joule, was used to initiate the photochemical reaction. The experiment described in fig. 1 was completely computer-controlled, and for each value of the time delay tl, 100 free induction decays were coadded. A flip angle (Y= 45” was used. A series of experiments was performed where rI was varied in steps of 10 ms, from 7.5 to 117.5 ms. Details of the apparatus and methods used will be described at another place. A typical 90 MHz proton spectrum obtained in this manner (fig. 3) shows the four lines of an AB system, labelled (34), (12), (24), (13), and an additional strong emission line in the center. The AB spectrum is characterized by the parameters J= 8.7 Hz, (5ZA-RB)/2n = 20.5 Hz, with an oscillation frequency ~~~/27r= 22.3 Hz [eq. (22)] . It is assigned to the polarized ester II (fig. 2). The emission line is due to p.dichlorobenzened2 (III, fig. 2). It was verified that reactions of escaping radicals with the second solvent, p.dioxanedg, are not observable in the presence of 50% Ccl,.
Fig. 3. CIDNP signals from an benzoyl-pcrosided4 in a 50% diosaneds. 50 FIDs, obtained fg. 1, were accumulated; I, q were used.
0.14 M solution of di-p-chlora(v/v) misture of CCL, and pwith the experiment shown in 57.5 ms and a flip an@ u = 45’
To compensate for small homogeneity changes and for longterm drifts of the spectrometer, the peak heights of the AB spectrum were normalized by the height of the (nomoscillatingj emission signal of p-dichloroben-
L
___.A
._I_
.
4
.2. .l
M
r
1 -I_
20
4n
80
I, e.0
ti --* 100
tj (ml)
Fig. 4. Measured and fitted peak heights (normal&d by the height of the emission line) of the polarized AB system of Fchlorophenyl-p-chlorobenzoated~ (II, fg. 2) for differen,; time deIays II berwee‘n light- and rf-pulse.
S. Schiubli~l et ol/Offdiogoml
292
.I01
Fig. 5. Calculated flip anple deper.j,:nce of the osci!lation amplitudes A and B defined in eq. (23). The same plramcters and the WTUZS&C as in fii. 4 were used.
elements in UDNP
in the case of strong nuclear coupling within the reaction product. They connect states with equal magnetic quantum number, and are due to the fact that product formation occurs in a non-adiabatic manner. In an experiment with pulsed chemical excitation and with pulse Fourier detection, these off-diagonal elements wiLlcontribute to the transverse magnetization. Consequently, line intensities show an oscillatory dependence on the time between the chemical excitation and the observing pulse. A single oscillation frequency was observed in the case of the strongly coupled two spin system, in agreement with the theoretical treatment. In more complicated spin systems, the number2 of oscillation frequencies quickly increases. For a system of N strongly coupled, non-eqtiivalent spins i, for example, it is given
by zene$. The results are plotted in fig. 4. The solid curves were obtained by adjusting the populations P,, __,P, [eq. (19)] in a leas! squares fit to the measured peak heights [eqs. (23), (25)]. The oscillation is given by a slightly distorted cosine-function due to the contribution of dispersion-type signals [eq. (26)]. In fig. 5, the theoretical oscillation amplitudes A and B, defined in eq. (23), are plotted against :he flip angle a; the curves were calculated using the parameters of the measured example. The amplitudes show a pronounced dependence on the flip angle;_4 is zero for u= 2 arcTin { [~(LI + b)]-l}. whereas B vanishes foror=2arcsin{[~(u-b)l-1).Forol=90”,A=-B, which means that the osci!latory contribution to the absorptive part of the signal has the same absolute amplitude for all four transitions for this particular flip angle. Similar experiments were performed on solutions of di-(2,3,4-trimethyl-bemoyl)-peroxide * in the same solvent mixture. The photodecomposition leads in this case to at least two polarized reaction products with strongly overlapping AB spectra in the aromatic region. They all exhibit the described oscillation phenomenon.
6. Conclusions In radicai reactions leading to enhanced nuclear po&i&ion (CIDNP), offdiagonal elements are produced l
Synthes~kd in our laboratory.
z
=
io I
1 2N _ 2” 2
.
(27)
N
Six ze;o quantum ;ransition frequencies are obtained for three coupled nuclei, while already 27 frequencies are expected for the case of four strongly coupled spins. These oscillations can be observed exclusively in pulsed UDNP experiments. In experiments with cop tinuous
chemical
excitation,
the off-diagonal
elements
They will partially be averaged and their influence on the spectrum is unobservably small for all practical purposes. are produced
over an extended
period
of time.
This research was supported by the Swiss National Science Foundation. Some help by Mr. J. Keller in the numerical data evaluation is acknowledged.
References [I] J. Bargon, H. Fischer and I-I. Johmn, 2. Naturforsch. 22a (1967) 1551. [2] S. Schiublin, A. HGhener and RR Ernst. J. Magn. Res. 13 (1974) 196. j3] R. Kaptein and L.J. Ooskrhoff, C&m. Phyr Lett. 4 (1969)
195,214; R. Kaptein, J. Amer. Chem. Sot. 94 (1972) 6251. [4] G-L. Gloss and L.E. Gloss, I. Amer. Chem. Sac. 9i (1969) 4549,455Q;
S. Schjublin
[S] [6] [7] [a] [9]
er al/Off-dbgoml
G.L. Gloss, I. Amer. Chem. SOE. 91 (1969) 4552; G.L. Clossand A.D. Trifunac, J. Amer. Chem. Sot 91 (1969) 4554; J. Amer. Chem. SOL 92 (1970) 2183,2186. F.J. Adrian, J. Chem. Phys. 54 (1970) 3912. R.R. Ernst, W.P. Aue, E. Bxtholdi, A. Hohener and S. Sch;iubiin, Pure AppL Chem. 37 (1974) 47. R.R. Ernst, Chiti 29 (1975) 179. W.P. Aue, E. Bartholdi and R.R. Ernst, J. Chcm. Phys. 64 (1976) 2229. M. Tomkiewicz, A. Croen and M. Cocivera, J. Chem. Phys. 56 (1972) 5850.
elmmnis in CIDNP
293
[IO] H. Fischer andC.P. Lxoff, Chcm. Piiys. 3 (1974) 217. [ll] See, for example, A. Messiah,QuailturnMechanics, VoL II (North-Holland, Amsterdam, 1969) p. 739. [12] A. Abragam, The Principles of Nuclear Magnetism(Clarcndon Press, Oxford, 1961) p. 274. [13] N.R. Krishna,J. Chem. Phys. 53 (1975) 4329. !14] P. h&kin and J.P. Jesson, J. Msg. Res. 18 (1975) 411. [lj] J.L Charlton and I. Bargon. Chem. Phys. Lett. 8 (1971) 442. (16) B. Blank and H. Fischer, Helv. Chim. Acta 54 (1971) 905.