J. Phys. Chem. 1992,96, 561-566 Based upon a previous detailed analysis for several diatomic molecules, this overage for the nitrogen 2p, character could have been anticipated.' With FACM, the nitrogen anisotropy is attributed solely to local 2p character while there are clearly anisotropic contributions from orbitals centered on boron, especially the large amount of spin density in the boron 2p, orbitals directed toward nitrogen. Even though there is an f 3 dependence, this nonlocal contribution will not be negligible since the B-N bond distance is small.
Conclusion The BNB radical has been identified by matrix isolation ESR spectroscopyas a vapor species above laser-vaporized solid boron nitride. The same experimental ESR results were obtained in neon, argon, and krypton matrices. Supporting evidence that BNB is a direct vapor species rather than a recombination reaction product formed during condensation is the fact that it was observed in a krypton matrix. The heavier rare gases under these experimental conditions are known to hinder or prevent such deposition or recombination reactions such as B BN BNB or B2 N BNB. ESR and theoretical calculations indicate that the electronic ground state of BNB is linear X %"+. The large boron isotropic hyperfine interaction and the properties of the g tensor eliminate the possibility of a ground 211 state. A detailed analysis of the orbital containing the unpaired electron in BNB shows most of the spin density located in boron sp, orbitals. A test of the free atom comparison method (FACM) revealed that this approach yields orbital characters similar to those obtained directly from a calculated CI wave function that closely reproduces the observed hyperfine parameters, namely Ai, and AdiY The geometric de-
+
-
+
-
561
pendence of the calculated A values was conducted as a function of bond length and bond angle in BNB. Chemical vapor deposition (CVD) of the type that is typically used for thin film formation applications has apparently not employed the direct pulsed laser vaporization of a refractory material such as boron nitride. A homogeneous gas-phase method for forming BN films involves the heating of NH,(g) and B2H6(g).29 Film formation by the laser decomposition of a gas-phase compound in contact with a substrate surface has been studied in detail.30 An extensive review of laser-assisted deposition of thin films from gas-phase and surface-adsorbed molecules has been presented by Herman.31
Acknowledgment. Project support from the National Science Foundation (L.B.K., CHE-9019511) and the donors of The Petroleum Research Fund, administered by the American Chemical Society, is gratefully acknowledged. Undergraduate students were supported by an NSF-REU grant and a Duke Endowment grant to Furman University. A Du Pont College Science Grant to Furman's chemistry department provided valuable equipment support. The authors are indebted to Professor E. R. Davidson for the use of his MELD program for calculating nuclear hyperfine interactions and to Dr. David Feller for helpful discussions concerning various features of these programs. We are also indebted to Dr. Michael Morse at the University of Utah for kindly supplying the TOF-MS data on laser-vaporized boron nitride. (29) Adams, A. C.; Capio, D. C. J. Electrochem. SOC.1980, 127, 399. (30) Copley, S. M. J . Appl. Phys. 1988, 64, 2064. (31) Herman, I. P. Chem. Rev. 1989, 89, 1323.
Nuclear Magnetic Relaxation in Cyclopropenone M. T. Cbenon,*Vt C. Coupry,+and L. G. Werbelow*.t LASIR, CNRS, 94320 Thiais, France, and Laboratoire des Mgthodes Spectroscopiques, Centre St. JZrame, URA CNRS 773, Case 541, 13397 Marseille, Cedex 13, France (Received: June 28, 1991)
The nuclear spin relaxation characteristics of the oleftnic carbon in cyclopropenonewere investigated over a range of temperatures. When relaxation-induced multispin order is monitored rather than the decay of athermal one-spin order, molecular detail otherwise obscured is rendered accessible. Known dipoledipole interaction anstants provide the basic description of molecular dynamics. With knowledge of the relevant molecular dynamics, it was possible to determine the orientation and antisymmetry of the carbon chemical shielding tensor.
Introduction Nuclear multispin relaxation is rapidly gaining deserved recognition as an important physicochemical probe which details the anisotropy of interaction, the modulation of interaction, and, most importantly, the temporal correlation between interactions. Although certain features of multispin relaxation are seeded in the works of Abragam, Solomon, and Lurqat' dating back to the 1950s, it was Redfield's development of a perturbative treatment of spin relaxation2 that provided direction for further exploration in this field. With a sound theoretical basis for describing the spin relaxation process, Hubbard, Blicharski, Shimizu, Schneider, McConnell, Freed, and others investigated 'crosscorrelation", 'anomalous relaxation", 'differential relaxation", and other embryonic concepts during the 1 9 6 0 ~ .The ~ arrival of the 1970s saw operator descriptions of the relaxation experiment being popularized by P ~ p e r . ~ Using these pioneering studies as a foundation, it has been established that temporal correlations between various time-deLASIR.
* Laboratoire des Methodes Spectroscopiques.
pendent spin interactions play a central role in multispin relaxation and are responsible for interconversions between various forms of spin order.5 Of course, during the 1980s, operator (multispin) descriptions of the multipulse 2D NMR experiment also became (1) Abragam, A.; Pound, R. N. Phys. Rev. 1953, 92, 943. Solomon, I. Phys. Rev. 1955,99, 559. LurGat, F. C. R. Hebd. Seances Acad. Sci. 1955, 240, 2402. (2) Redfield, A. G. Adu. Magn. Reson. 1965, I , 1 . (3) For listings of older literature in this field, see: Werbelow, L. G.; Grant, D. M. Adu. Magn. Reson. 1977, 9, 189. Werbelow, L. G.; Grant, D. M. J . Magn. Reson. 1915, 20, 554. (4) Pyper. N. C. Mol. Phys. 1971, 21, 1; 1972, 22, 433. (5) For listings of newer literature in this field see: Canet, D. Prog. NMR Specrrosc. 1989,21,237. Hartzell, C. J.; Stein, P. C.; Lynch, T. J.; Werbelow, L. G.; Earl, W. L. J . Am. Chem. SOC.1989,111, 5114. Also see: Elbayed, K.; Canet, D. Mol. Phys. 1989,68, 1033. Chang, W. T.; Wang, P. L.; Duh, D. M.; Hwang, L. P. J. Phys. Chem. 1990,94, 1343. Konrat, R.; Sterk, H. J . Phys. Chem. 1990.94, 1291. Foucat, L.; Chenon, M. T.; Werbelow, L. G. J . Phys. Chem. 1990,94,6663. Bull, T. E. J . Magn. Reson. 1988,80,480. Bull, T. E. J . Chem. Phys. 1990,93, 6824. Decatur, J. D.; Farrar, T. C. J . Phys. Chem. 1990, 94,7395. Krishnan, V. V.; Kumar, A. J . Magn. Reson. 1 9 9 1 , 92, 293. Kontaxis, G.; Muller, N.; Sterk, H. J . Magn. Reson. 1991, 92, 332. Fuson, M. M.; Anderson, D. J.; Liu, F.; Grant, D. M. Macromolecules 1991, 24, 2594.
0022-365419212096-561$03.00/00 1992 American Chemical Society
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562 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
popular! Indeed, quantitative descriptions of relaxation and 2D NMR experiments such as COSY look suspiciously similar, and numerous groups, most notably the Lausanne group, are capitalizing on this fundamental relationship.’ This brief overview sets in context the present study in which it is illustrated show multispin relaxation can povide unique detail of the submicroscopic world. It is argued that it is possible to study informative interactions in the presence of other, more dominant interactions. It is demonstrated that one may utilize interference effects to discriminate between two interactions with similar field, frequency, and temperature dependencies. In particular, it is suggested that multispin relaxation will prove indispensable in detailed studies of the chemical shift tensor. Cyclopropenone, a relatively novel chemical species, provides the simple, yet illustrative, spin system selected for this investigation.
Experimental Section The actual sample studied was a 30% by volume solution of cyclopropenone in hexadeuteroacetone. The cyclopropenone (natural abundance 13C)was synthesized in this laboratory using procedures proposed by Breslow and Ryan.8 The acetone-d6 was obtained from Spectrometrie, Spin et Techniques. Dissolved oxygen was removed by successive freezepumpthaw cycles. To suppress the diffusion of resonant nuclei out of the probe coil, a 9-mm-0.d. microcell was used. Given the thermal instability of cyclopropenone, between experiments the microcell was stored at -78 OC. Spectra were obtained on a Brucker-AM 300 spectrometer. To avoid spinning artifacts, all spectra were measured on nonspinning samples. The FID consisted of 8K data points, and the spectral width was 1 kHz. The various proton and carbon pulse lengths were calibrated before each experiment. A A / 2 pulse for 13Cwas effected in 8.2-9.6 ps depending upon the temperature of the sample. The proton 17 pulse was calibrated by proven method^.^ At maximum decoupling power, the proton A pulse duration is approximately 40 ps and, once again, depends upon temperature. The inversion of the magnetization was always within 1% of the theoretical value. The time between any two acquisitions was at least 7 times greater than the T I of 13C. The response characteristics of the carbon magnetization of the I3C spin grouping were observed subsequent to two different perturbations: (1) inversion of the carbon magnetization (with and without proton decoupling) or (2) inversion of the proton magnetization. The number of acquisitions required to achieve acceptable SIN varied from 1 (proton-decoupled experiment) to 64 (proton inversion, carbon recovery experiment). The proton inversion, carbon recovery experiment was performed only at two temperatures, -80 and -60 OC, whereas the other experiments were performed at -90, -80, -60, and -30 OC. Given the small magnitude of the effects observed, every effort was made to eliminate all potential sources of error. For example, every third acquisition recorded the equilibrium magnetization. Although quite time-consuming, this provides a very accurate and extremely reproducible measure of the equilibrium magnetization. Each of the four peaks of the 13Cmultiplet has been normalized, independently, by comparison with this equilibrium value.
Theoretical Section The theoretical framework used for subsequent analysis considers a three-spin system where spin I is an ethylenic I3C, spin ~~
(6) Sorensen, 0. W.; Eich, G. W.; Levitt, M. H.; Bodenhausen, G.; Ernst, R. R. Prog. NMR Specrrosc. 1983, 16, 163. Wang, P. K.; Slichter, C. P. Bull. Magn. Reson. 1986, 8, 3. Goldman, M. Quantum Description of High Resolution NMR in Liquids; Oxford University Press: Oxford, U.K., 1989. (7) Exemplary references include: Wimperis, S.;Bodenhausen, G. Mol. Phys. 1989,615,897. DiBari, L.; Kowalewski, J.; Bodenhausen, G. J . Chem. Phys. 1990, 93,7698. Burghardt, I.; Dibari, L.;Bonvin, A.; Bodenhausen, G. J. Mogn. Reson. 1990,86,652. Brunschweiler, R.;Griesinger, C.; Ernst, R. R. J. Am. Chem. Soc. 1989, I l l , 8034. Oschkinat, H.; Limat, D.; Emsley, L.; Bodenhausen, G. J. Magn. Reson. 1989.81, 13. Dalvit, C.; Bodenhausen, G. Adu. Mogn. Reson. 1990, 14, 1, and references cited therein. (8) Breslow, R.; Ryan, G. J . Am. Chem. Soc. 1972, 94, 4787. (9) Bax, A. J. Magn. Reson. 1983, 52, 76.
NOE.154
/
NOE.1.62
I
-l,o -I+
’
4,o
6,O
5,O
loo0 KIT
Figure 1. Plot of the proton-decoupled, 13Clongitudinal relaxation rate, R I ( l / T l or pc), vs reciprocal temperature for a 30% v/v cyclopropenone/acetone-d, solution.
S is the directly bonded proton (H), and spin S’ is the distant proton (H’). It is anticipated that spin S is relaxed by C-H and H-H’ dipolar and proton chemical shift anisotropy interactions. Spin Z is relaxed by the C-H dipolar interaction and tensor/vector 13Cshift anisotropies/asymmetries. Except for the fact that spin S‘ provides possible interferences with spins Z and S, the explicit relaxation characteristics of spin S’ prove unimportant in this study and are not detailed. If nuclear spin relaxation is effected by these relaxation pathways, the relevant longitudinal relaxation characteristics of the I3CHH’ spin grouping can be summarized by an abbreviated set of conversion rates:
:I
In extreme narrowing, the familiar self-relaxation and cross-relaxation rates are defined as pc = 10JcH/3 + (4JAc+ 4J6c)and u = 5JcH/3, respectively. The various spectral densities appearing in eq 1 are defined such that in extreme narrowing with isotropic, diffusional reorientation (correlation time T ~ ) JcH , = (3/10)(yHyCh/rCH3)2Tc,JAn = (1/30)(1 + q2/3)(unA~n)2Tc, Kn = (1/20)(yCyHhu,Aun/rcH3)~ ( Q , q ) r c , J6c = ( 1 / 2 ) ( 6 u ) 2 a c 2 7 c , and K m n p = ( 3 / 2 0 ) ( y ~ y m y , , h 2 / r n m 3 r ,cos2 ~ ) ( 30 - 1 ) ~ ~The . chemical shift antisymmetry is defined as ( 6 ~ = ) ~( 1 / 4 ) { ( u x-y u,,)~ + (uxzu2J* + (uY2- u,)*}. The angular function associated with the dipolar cross-correlation spectral densities (Kmn) is the familiar correlation factor relevant for isotropic motions.lg For nonaxially symmetric shift tensors, the analogous correlation factor for the shift anisotropy-dipolar cross correlation,f(Q,q), cannot be written so simply. For isotropic reorientation, this factor, (3 cos2 0 - 1 - 7 sin20 cos 24), positions the dipolar vector in the principal axis system of the shift tensor. The dipolar coupling between spins I and S’ is assumed too weak to contribute a significant cross relaxation or dipolarshift anisotropy cross-correlation term (KH,). It must be recognized that eq 1 provides a convenient pictorialization for further discussion. This description is purposely schematic in nature. An asterisk should not be interpreted as a (10) Hubbard, P. S. Phys. Reu. 1969, 180, 319.
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 563
Nuclear Magnetic Relaxation in Cyclopropenone
2,2
-
Protoninversion carbon recovery
2.01
,
+
*
*
I
.
.
+
.
1
'
ad
'60
PPM
*
LINE A LINE B LINEC LINED
e
158
Figure 2. I3CSpectra of cyclopropenone's ethylenic carbon for a series of recovery times subsequent to inversion of the proton magnetization (30% v/v cyclopropenone/acetone-d6solution; T = -80 "C).
zero. Only those couplings pertinent to future discussions are shown. To define the explicit time evolution behavior in this system, additional spin variables must be introduced along with appropriate couplings. However, rather than attempting to fit a heptaexponential recovery, a more qualitative, instructive approach utilizing initial evolution rates will be employed.
Results and Discussion Since I3C spin-rotation relaxation would mask many of the novel features that are of interest in this study, the relaxation characteristics of cyclopropenone were studied over a range of temperatures. Figure 1 provides a plot of RI(l/Tl or p ~ )measured , from the conventional decoupled inversion recovery experiment, as a function of 1/ T. Corresponding values for the observed NOE are indicated for each temperature. From this plot it is apparent that in the temperature range between -60 and -80 OC spin rotation is ineffective and extreme narrowing obtains. The data obtained at -80 OC are used in subsequent discussion. The nondipolar component of relaxation demonstrates quadratic field dependence which justifies the identification of p c with 10JcH/3 4(JAc + J6,-). Furthermore, numerous studies with elevated proton concentrations have shown that intermolecular dipolar interactions do not effectively relax the I3C spina5 Experimental 13Cspectra that illustrate the magnitude and form of the observed multiplet relaxation are shown in Figure 2. The isotropic chemical shifts of the ethylenic and carbonyl carbons are 158.77 and 155.76 ppm, respectively. These spectra were obtained for a series of delay times following inversion of the proton magnetization. The ethylenic carbon appears as a doublet of doublets. The splitting due to the directly bonded proton is 219.9 Hz and the long-range two-bond coupling is -6.0 Hz. These couplings are temperature dependent and vary linearly with respect to 1 / T ' J = (-7.2 + 220K/T) Hz, I J = (215.6 825K/T) Hz. Since the integrated intensities of these four components are equal, the apparent peak heights differ slightly because the intrinsic T2sof the various components are not identical."J' A residual differential broadening on the order of 1-276 exists between the various components. This factor has been taken into account in the subsequent analysis. The observed differential broadenings are in accordance with theory which predicts that the high-field components are broadened relative to the low-field components. Figure 3 demonstrates the time evolution of individual line intensities within the carbon multiplet. Initially, the inner lines (lines B and C) evolve differentially from the outer lines (lines A and D), whereas later in the evolution the high-field components (lines C and D) evolve separately from the low-field components (lines A and B). Since the observed pulse does not scramble the population^,'^ the intensity of each multiplet component, Is$,',
+
+
(11) Mackor, E. L.;McLean, C. Prog. N M R Spectrosc. 1967, 3, 129. (12) Werbelow, L. G.; Marshall, A. G. Chem. Phys. Lett. 1973, 22, 568. (13) Schaublin, S.; Hohener, A.; Ernst, R. R.J . Mugn. Reson. 1974,13, 196.
8'
1.21
,
I
1
2
tlsec Figure 3. Individual line intensities of the peaks shown in Figure 2 plotted as a function of recovery time. Lines A-D are associated with multiplet components of decreasing frequency (Le., line A is the lowest field component).
-40 0
2
1 VSeC
+
Figure 4. Plot of the high-field components (lines C D) minus the low-field components (lines A + B). Data are taken from Figures 2 and 3.
can be identified with a unique combination of spin operators, 41++ = (1,) + (21$,) + (213;) + (413$,'), 41,- = ( 1 2 ) + ( 2 4 4 ) - ( 21$; ) - ( 4 1 3 3 ; ) 41-t = ( 41 - ( 213,) + (21s;) - (41$&'), and 4 L - = ( I , )- (213,) - (213;) + (41$3;). For a positive signed scalar coupling, the high-field component is associated with the transition where the projection of the passive spin is The summed intensity of all four lines provides the value of ( Z , ( t ) ) , and from the initial behavior of this parameter, it is determined that u = 0.23 f 0.01 s-l. However, the most interesting evolutions involve multiplet differences. In Figure 4, the intensity of the high-field components minus the intensity of the low-field components is plotted as a function of time following the inversion of the proton magnetization. As can be seen from the operator identifications given above, this difference samples the evolution of 21JZ(t). F ject to the boundary conditions imposed by proton inversion, (Z,(O)) = Z,(eq), (S,(O)) = -S,(eq), (S,,(O)) = -Si(eq), the initial evolution of 21$,(t) is described as (2Z3,(t))/(Z2(eq)) = -8KH. (YH/YC)t. Since this initial evolution is positive, the interference term, KH, is negative. To deduce the sign of KH,it has been assumed that the onebond 13C-H scalar coupling is positive. From the initial slope of this curve, it is determined that KH = -0.00075 f 0.00012 It is recognized that this coupling is quite small and the larger coupling, Kc, assumes dominance after a brief induction period during which cross relaxation drives ( I , ) away from equilibrium. The quantification of this term is remarkable when one considers that the effective time constant for this channel of magnetization transfer is on the order of 5 min, whereas TI(l3C) is on the order of 2 s! 9
564 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
8 Proton Inverslon
Chenon et al.
- carbon recovery
V '
0
1
v
3
2
0
VSeC
Figure 5. Plot of the inner lines (lines B + C) minus the outer lines (lines + D). Data are taken from Figures 2 and 3.
A
TABLE I: Relevant Structural Parameters-Cyclopropeaoae"' RcH = 109.7 pm RcH, = 228.9 pm RHHl= 309.8 pm
C-H-H' angle = 35.09'; P2(35.09") = 0.504 C-H'-H angle = 16.00"; P2(16.00') = 0.886 H-C-H' angle = 128.91'; P2(128.9lo)= 0.092
Figure 5 plots the sum of the inner two lines minus the intensity of the outermost components. There is only one explanation for this behavior. Either the C-H or C-H' dipolar interaction must be temporally correlated with the H-H' dipolar interaction. It is seen from this figure that the initial growth of threespin order, given by the expression (41JJ9(t))/(Zz(eq)) = 16(KCHH, KcHtH)t is 0.18 f 0.01. Assuming that the gas-phase structure (see Table I) and the solution structure of cyclopropenone do not differ greatly, the strengths of the three dipolar interactions are 22.9 (C-H), 4.05 (H-H'), and 2.52 kHz (C-H'). In principle, the three-spin order observed can be attributed to either of the two interferences involving the H-H' dipolar interaction. Comparison of the interaction strengths suggests that, in practice, &HHt dominates KcHtH. However, relevant correlation factors must be considered. For isotropic reorientations the correlation factors for these two terms are 0.50 (KcHH,)and 0.88 (KcH'H). Thus, the tentative values KCHH, = 0.0095 f 0.0007 s-I and KCHHi/KCHw 5 can be assigned. Because of the molecular topology (cf. Table I), the values of KcHH, and KcHrH must be positive. Therefore, the identification of (4ZJJ9) with (lines B + C) minus (lines A D) is correct and the one-bond and two-bond W - H scalar couplings are opposite in sign.I2 Since. one-bond W - H couplings are invariably positive, the two-bond I3C-H coupling in cyclopropenone is negative. When the difference in intensity in the high-field components (lines C - D) plus the difference in intensity of the low-field components (lines A - B) is plotted as a function of time, no trend is observed. As previously assumed, the interference, KHi, is too small to produce an observable effect in these experiments. In addition to the proton hard pulse preparation, a conventional I3C proton-coupled inversion recovery curve was obtained. As shown in eq 1, this study makes possible the observation of additional interferences-e.g., a plot of the high-field components minus the low-field components yields a value for the interference, Kc. Using the data summarized in Figure 6, the following parameters were determined with good precision: pc = 0.58 f 0.04 s-' and Kc = -0.012 f 0.001 s-I. The value for pc compares favorably with the more accurate value, 0.585 f 0.007 s-l, determined from the protondecoupled inversion recovery experiment. The value of the dipolar interference, KHCH,,is below the observable threshold of these experiments-no noticeable three-spin order is generated from simple inversion of the carbon magnetization. Three factors make the measurement of KHCH, difficult:
+
+
2
1
3
VSeC
Figure 6. Plot of the high-field components (lines C + D) minus the low-field components (lines A + B) subsequent to inversion of the 13C magnetization (30% v/v cyclopropenone/acetone-d6solution; T = -80 'C).
(1) the C-H' distance renders KHC+9 times smaller than JcH; (2) when compared with the proton inversion experiments, the transient Overhauser effect is suppressed; and finally, (3) the intrinsic correlation parameter, Pz(cos QHcw),is close to zero (cf. Table I). At -80 OC, the value of the NOE is 1.62 f 0.06. Therefore, u 0.24 f 0.01 s-' and JCH= o.145 f 0.01 S-'. This value of u agrees with the value determined from the proton inversion, carbon recovery experiment described earlier. The value for 4(JAc PC)= pc - 2u = 0.10 f 0.03 s-l. Hastily assuming that pc - 2u = 4JAc,the apparent shift anisotropy (in parts per million) = (1 + q 2 / 3 ) . can be determined from the ratio, JAC/JCH (75Au/68700)*. This suggests an effective shift anisotropy on the order of 400 ppm. However, effects of motional anisotropy and possible contributions of the antisymmetric component have been ignored. Since a 50 ppm shift antisymmetry effects the same degree of relaxation as a 200 ppm shift anisotropy,15neglect of the antisymmetric component can lead to serious error. Once the various spectral densities have been determined, the associated interaction constants can be deduced. In the limit of isotropic reorientation, this is a trivial task. At the other extreme, if both the shift tensor and diffusion tensor lack cylindrical symmetry, extraction of the various parameters is tedious or impossible. However, for the specific system under consideration, certain simplifications are applicable. For example, theoretical calculations on cyclopropene-likemolecules and other similar systems reveal that the most shielded component of the olefinic carbon chemical shift is perpendicular to the molecular plane.I6 For the purposes of this work, this will be labeled the i axis. The x and y axes, which lie in the molecular plane, satisfy the inequality a,, 1 u,,? 1 u,. Furthermore, since the olefinic carbons are charactenzed by C, local symmetry, only one component of the pseudo vector associated with the antisymmetric components of the shift tensor is nonzero." This component, (uxy- uyx)/2 = 8u, is perpendicular to the molecular plane. Likewise, diagonalization of the diffusion tensor for a planar molecule is realized by an axis perpendicular to the molecular plane and two axes in the plane.I8 Because of the additional symmetry elements characterizing cyclopropenone, one of these planar axes is collinear with the C=O bond and the other collinear with the H-H' vector. In the present study, motion will be
+
(14) Benson, R.C.; Flygare, W. H.; Oda, M.; Breslow, R. J . Am. Chem. SOC.1973, 95, 2772.
(IS) Blicharski, J. S. Z.Nufurforsch. 1972, A27, 1456.
(16) Facelli, J. C.; Orendt, A.; Grant, D. M.; Michl, J. Chem. Phys. Len. 1984,112,147. Hansen, A. E.; Bouman, T.D.J . Chem. Phys. 1989,91,3552, and references cited therein. (17) Robert, J. B.; Wiesenfeld, L. Phys. Rep. 1982, 86, 363. (18) Huntress, W. T. Adu. Mugn. Reson. 1970, 4, 1.
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 565
Nuclear Magnetic Relaxation in Cyclopropenone modeled as quasi symmetric top diffusional rotation with diffusion constants Dzz= DIland (Dxx Dyy)= 20,. This approximation relaxes the unrealistic requirement of isotropic motion yet retains expressions that are tractable. Furthermore, if motion about one axis is rapid annpared with other motions, the motional asymmetry (Dyy- Dxx)/Dzzis relatively small and the symmetric top approximation is valid even if Dyyand D, are quite different. Subject to the placement of various tensors for planar molecules mentioned above, the z axis of the shift tensor is parallel or perpendicular to the principal (z)axis of the diffusion tensor. The most appropriate placement is deduced from the spectral density ratio, &HH,/JcH. Expimentally, this ratio, based on the tentative value of KcHHt = 0.0095 s-I, equals 0.065 f 0.010 and can be identified with the term ( ~ H / ~ c ) ( ~ c H / ~ H = 0H. 1, )7’7~~ .Thus, the value of the correlation factor, x, is 0.37 f 0.05, which is significantly less than the isotropic value of 0.50. Defining the correlation time, 7, = (60, + n2(DII- D1))-l, and the motional parameter, F = Dll/Dl = 2Dzz/(Dxx+ Dyy) shows’ that ifthe z axis of the shift tensor and the unique axis of the diffusion tensor are collinear x = ( r o 7 2 ) / ( 7 0 3 ~ =) (E 2)/(5 + 5), whereas x = 6 ~ ~ / ( 3 7871 ~ T ~ =) (2t2 + 115 5)/(t2 221 13) ifthe H-H’ vector and the unique axis of the diffusion tensor are collinear. In either case, the minimum value of x approaches 0.40 and occurs when D,, 0. The angle a defines the angle between the dipole moment and the x axis of the chemical shift tensor. In contrast to simple ethylenic linkages, theoretical evidence2’ suggests a nonzero value for a in cyclopropenone. A most crucial spectral density ratio in this analysis is Kc/JcH
KC/ JCH = ( 1/ 3) (mCAurCH3/YCYH h )x
(4)
+
where the correlation parameter x = ((-27~2~1~ sin 2 a ) ~ (-3 ~ + 7 COS 2a)72)/(471 + 272) = ((-3 7 cos 2a)(5 [) - 472l/* sin k ( l + 2[))/18(1 + F). Using a value x(5 = 2.5) = -(5/14)(1 - (7/3)(cos 2a - (9/2) sin Za)), the chemical shift anisotropy and the spectral density, J*c, are given by the expressions
+
+
(19) Vold, R. L.; Vold, R. R.; Canet. D. J. Chem. Phys. 1977, 66, 1202. Vold, R. L.; Vold, R. R. Prog. N M R Spectrosc. 1978, 12, 19. (20) Spiess, H . W . NMR: Basic Prin. Prog. 1978, IS, 55. (21) Orendt, A. M.; Facelli, J. C.; Beeler, A. J.; Reuter, K.; Horton, W. J.; Cutts, P.; Grant, D. M.; Michl, J. J . Am. Chem. Soc. 1988, 110, 3386.
Au(1 - (v/3)(cos 2a - (9/2) sin 2a)) = (-2560)(Kc/JcH) = 210 f 50 ppm (5) and
JAc = (0.070 f 0.01 5)JCHf(v,a) (6) wheref(v,a) = (1 2/5)v cos 2a (3/7)$)(-1 + (v/3)(cos 2a - (9/2) sin 2c~))-~.[By comparison with other ethylenic moieties, it is expected that the most “unique” axis of the chemical shift tensor lies in the molecular plane, and it could be argued that a more appropriate definition of Au and q should be used for this analysis. Indeed, there are certain advantages for the choice where A d = uxx- (uyy uzz)/2and 7’ = 3(uyy- uz,)/2Ad. With this choice, the presumed principal axis of the shift tensor lies in the molecular plane forming an angle a with the molecular dipole. This angle is identical to the angle a defined in eqs 5 and 6. Obviously, if Au is positive, A d is negative. Of course, the physics are invariant with any change in coordinates and (uz2 uy; uxx2 - uxxuyy - uxxuzz - u 1 = (1/2)(bXX - uyy)2 + (%x - %A2 + ( u - uzz)2)can be id%ked with either (Au)z(l + q2/3) or (Ad’ (1 7”/3). Alternatively, defining (uzz- uyy)/(uyy - uxx) = 5; Au/Ad = -(1 20/(2 + 0 , =~3/(1 + 2 0 and 4 = 3f/(2 l).Using this alternative representation, eq 4 can be rewritten by simply replacing Au with A d and x with x’ = (3(21/2sin 2 4 1 - v’/3))71 (3 sin2 a + $(cos2 a 1 ) ) ~ ) / ( 4 7+~2 ~ ~ ) . ] Unfortunately, three variables appear in eqs 5 and 6. Obviously, without additional information, it is impossible to deduce unique values for Au(Ad), ~ ( v ’ ) a, , or JAc. However, it is interesting to examine plausible sets of parameters compatible with the observed data. For small values of v, the function f(q,a) is close to unity, whereas for larger values of v, f(7,O)increases without bound. Indeed, in the limit where v 3 and a 0, the shift tensor is axially symmetric, with the unique axis and the molecular dipole being collinear. The angle between this axis and the C-H dipolar vector is approximately 126’, and hence the cross-correlationterm, Kc, vanishes regardless of the degree of motional anisotropy! In this limit, a residual value of KCprovides a very demanding test of the symmetric top model. However, it does not appear that this limit obtains because a relatively large value for & is observed. This can be rationalized only if Au is surprisingly large, v is vani&ingly small, or a is nonzero. All available evidence indicates a nonzero value for a. Assuming very modest restrictions on the bounds of Au and 1, 100 ppm < Au < 200 ppm and 1 < < 2, it is found that a = 7’ 2’. For cyclopropene, the calculated value of a is 18.4-20.5°.22 Allowing for large variabilities in a,Au, and 7, eq 6 predicts JAc= 0.010 f 0.002 s-l. In turn, this suggests that the value for J6c, 0.015 f 0.003s-l, is nonzero and may be up to 2 times larger than JAc. Antisymmetric shift tensors were unobserved curiosities until a recent, exacting study convincingly demonstrated the existence of these terms.23 In this cited study, separation of the chemical shift anisotropy into symmetric and antisymmetric components was effected from Tl/T2 ratios. Although straightforward in principle, spin relaxation must be dominated completely shift anisotropy interactions for this method to be useful. In the present study where anisotropicshift interactions are a mere peruturbation, a more general approach utilizing the rotational differences of the rank one (antisymmetric) and the rank two (symmetric) interaction has been implemented. Combining eqs 2, 3, and 6 yields (uxy- uyx)/2 = 6u = 260(J6c/J~H)’/2 = f(80 f 40 ppm). The theoretical value for the antisymmetry (6u) in cyclopropene is on the order of 1 0 0 ppm.16 One final comment on a unique ratio of spectral densities is in order. Since the principal axes of the proton and 13Cchemical shift tensors can be chosen as collinear and perpendicular to the C-H internuclear vector, it may appear that the ratio, Kc/KH is
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(22) Zilm, K.; Conlin, R. T.; Grant, D. M.; Michl, J. J . Am. Chem. Soc. 1980, 102,6612.
(23) Anet, F. A. L.;OLeary, D.;Wade, C. G.; Johnson, R. D. Chem. Phys. Lett. 1990, 171, 401, and references cited therein.
J . Phys. Chem. 1992,96, 566-571
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well-defined. However, this is somewhat illusionary unless both shift tensors are axially symmetric. In the present case, Kc/KH = (Yc/YH)(A~c/AQH)(~ - r)c(COS 2% - (9/2) sin 2ac)/(3 ~ ~ ( 2ffH ~ 0 (9/2) s sin 2 4 = 16. If the ratio of angular terms is not considered, the observed value of KH implies a proton shift anisotropy of approximately 3 ppm.
Conclusion A relaxation study of a 30% v/v solution of cyclopropenone in acetone-d6 a t -80 O C has been described. For this system, R l d i ~ l a r / R I t= o t 0.82. a 1 At first glance, this suggests that the information available from a relaxation study will be limited. However, it has been demonstrated that if relaxation-induced multispin order is monitored, the dipolar interaction serves to amplify rather than obscure weaker interactions. Using well-defined interaction constants, dipolar interferences provided information about the relevant molecular dynamics. With the basic motional features understood, additional interaction constants could be determined. It was possible to deduce the orientation of the shift tensor (the least shielded component is rotated 7 O f 2 O from the perpendicular to the olefinic linkage and lies in the molecular plane) and the sign of the two-bond C-H coupling constant (-6.3 H z a t -30 OC) and to assess the relative importance of the proton and 13C chemical shift anisotropies.
Although it did not prove possible to isolate individually A u and r), it is apparent how these methods could be implemented in conjunction with complementary techniques. It has also been argued that relaxation-induced multispin order can be used to isolate the symmetric from the antisymmetric components of the shift tensor (the present study suggests a shift antisymmetry of 80 f 40 ppm). Although nuclear spin relaxation via the anisotropic chemical shift is recognized as important, relatively few, thorough investigations of this mechanism have appeared in the literature. In most studies, a t least two, and often three, of the parameters, r), 60, or D,, - (Dxx D,,,,)/2,are assumed to be zero for no other reason than convenience. However, as the present study clearly indicates, this can be quite dangerous. It is expected that multispin relaxation will prove essential for further developments in this area.
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Acknowledgment. We thank Madame Nicole Ratovelomanana for her aid in the synthesis of the cyclopropenone. Note Added in Proof: A recent theoretical calculation (Facelli, J., private communication) suggests that the shielding tensor of cyclopropenone contrasts markedly with other olefinic Carbons and is characterized by a large anisotropy, Au 250 ppm, and a small asymmetry, r) 0.2. The calculated values for 6u and a! are 20 ppm and 14O, respectively.
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Fluorescence Anisotropy of Reversible Interacting Fiuorophores in Solutions. A Theoretical Study Z. LimpouchovQ and K. ProchPzka* Department of Physical and Macromolecular Chemistry, Charles University, Prague, Albertov 2030, 128 40 Prague 2, Czechoslovakia (Received: July 10, 1991; In Final Form: September 16, 1991)
A model for the time-resolved fluorescence anisotropy is presented for a dilute solution containing symmetric-top molecules where rotational and translational diffusion take place simultaneously with a reversible complex formation. The description of the molecular behavior is based on a rotational diffusion model and the Smoluchowski theory of diffusioncontrolledreactions. A system of differential equations for orientational probability densities was solved both numerically for the time-dependent rate constant of the complex formation and analytically under the simplifying assumption of the time-independentrate constant of the complex formation. Parametric studies of fluorescence, difference, and anisotropy decays were performed for typical values of parameters describing reorientational motion of molecules and kinetics of complex formation and dissociation.
Introduction For nmiy years, the reversible kinetics of intermolecular excited complex formation and dissociation as well as other possible reactions of an excited fluorophore has been attracting attention of a number of researches.'-* Formation of complex in a solution is a diffusion-controlled reaction between molecules which undergo simultaneous random translation and rotation. With the exception of a few recent theories of diffusion-controlled reactions do not take into account the orientational factor and consider only the influence of the translational diffusion. Their experimental verification is than based only on the time-resolved fluorescence decay measurements. Fluorescence anisotropy measurements with excited complexes have been reported recently;'*15 however, these experiments did not study kinetics of excited complex formation and dissociation in solutions. The aim of this paper is to include kinetics of excited complex formation and dissociation into a rotational diffusion model for the fluorescence anisotropy in an isotropic solution. The timeresolved polarization spectrofluorimetry may be used to test the derived expressions. Mathematical treatment is presented for *Towhom correspondence should be addressed.
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exciplexes, however, the results are suitable for other reversible interacting fluorophore systems, e.g., for excited complexes, which do not exhibit fluorescence or for excimers under the condition (1) Birks, J. B. Phorophysics of Aromatic Molecules; Wiley: London, 1970. (2) Weixelbaumer, W. D.; Burbaumer, J.; Kauffmann, H. F. J. Chem. Phys. 1985,83, 19980. (3) Lee, S.;Karplus, M. J . Chem. Phys. 1987, 86, 1883. (4) Andre, J. C.; Baros, F.; Winnik, M. A. J . Phys. Chem. 1990,94,2942. ( 5 ) Hauser, M.; Wagenblast, G. In Time-Resoloed Fluorescence Spectroscopy in Biochemistry and Biology; Cundall, R. B., Dale, R. E., Eds.; Plenum: New York, 1983. (6) Sienicki, K.; Winnik, M. A . J . Chem. Phys. 1987, 87, 2766. (7) Martinho, J. M. G.; Winnik, M. A. J . Phys. Chem. 1987, 91, 3640. (8) Vogelsang, J.; Hauser, M. J . Phys. Chem. 1990, 94, 7488. (9) Baldo, M.; Grassi, A.; Raudino, A. J . Chem. Phys. 1989, 91, 4658. (10) e n a b l e , T.; Cranston. D. H.; Soutar, I. Eur. Polym. J. 1989, 25,221. (1 1) Yliperttula, M.; Lemmetyinen, H.; Mikkola, J.; Virtanen, J.; Kinnunen, P. K. J. Chem. Phys. Lett. 1988, 152, 61. (12) Stegemeyer, H.; Hasse, J.; Laarhoven, W. Chem. Phys. Lett. 1988, 137, 516. (13) Fraser, I. M.; MacCallum, J. R. Eur. Polym. J . 1988, 23, 171. (14) Gardette, J.; Phillips, D. Polym. Commun. 1984, 25, 366. (15) MacCallum, J. R. Eur. Polym. J . 1981, 17, 953.
0 1992 American Chemical Society
