J . Phys. Chem. 1988, 92, 4859-4863
4859
From the above analysis it seems that new theories of bonding of electron correlation is unavoidable, e.g., C U C ~ HNiC2H4,21 ~,~~ interpretation of metal-ligand interaction are actually highly and NiC2H2;22for MCO examples and discussion, M = Rh, Pd, Pt, Sc, Ni, Cu, Fe, see ref 47c; also C U H ~ ~The . ~ligand ~ ~is , ~ ~desirable. Perhaps they will be more elaborate than the simple and intuitive model proposed by Dewar,' and at this point one quite unperturbed, and the complex (when possible) does not hold faces a problem quite usual in science: a simple theory is more Dewar's model. A quantitative, and even qualitative in some cases, easily handled than a complicated one, but sometimes, that relative description of these systems often requires to go further than advantage may be a real disadvantage if one does not obtain a valence space ~ o r r e l a t i o n . ~ ~ reliable picture or lead to misleading conclusions with regard to (2) Metals in excited states (with promotion of u electrons): delocalization of a electrons into the a* MO's of the ligand takes interpretations of bonding. In order to be generalized such a new place. However, the u charge transfer does not occur, e.g., theory should take into account the electron correlation effects. C U C ~ H For ~ . ~MCO ~ examples see ref 47d and 48. Recent attempts toward that direction appear to be quite promi~ing.5~ Positively Charged Metals. The main interaction is of electrostatic nature, and hence the SCF solution does not suffer from Acknowledgment. We are indebted to the "Laboratoire de large quantitative errors. However, to improve the description Physique Quantique l'Universit6 Paul Sabatier", Toulouse, France, of these systems, a limited C I is not enough (selecting only the for making available to us the FSHONDCWIFSI computer programs. more important configurations) since it overestimates the S C F Pseudopotential and basis set for the silver atom were kindly trend. Therefore, a large scale CI is necessary to ensure the proper provided by Dr. J. C. Barthelat. The present work has been correlation of the ionic forms. Contribution of the two-way dosupported by the CAICYT Project No. 714/84. nor-acceptor model to the bonding is negligible, e.g., (CuC2H2+), (AgC2H2+) (in the present study), NiC0+,47dNiH20+,47dand Registry No. C2H2, 74-86-2; (CuC2H2)+, 45326-64-5; (AgC2H2)+, cuco+.4* 114885-45-9;CU', 17493-86-6;Ag+, 14701-21-4. ~~
(49) Sauer, J.; Haberlandt, H.; Pacchioni, G. J. Phys. Chem. 1986, 90, 3051.
(50)Schipper, P. E. J . Phys. Chem. 1986, 90, 2351.
'H and 13C Spin-Lattice Relaxation in Gaseous Benzene Michael M. Folkendt, Boris E. Weiss-Lopez, and Nancy S. True* Department of Chemistry, The University of California, Davis, California 95616 (Received: September 3, 1987; In Final Form: February I , 1988)
The nuclear spin-lattice relaxation time, T I ,measured for benzene protons at densities between 0.81 and 54.4 mol/m3 (15 and 980 Torr) at 381 K exhibits a characteristic nonlinear density dependence. Analysis of the density-dependent T I data yields a spin-rotation coupling constant, CeR,of (182.6 (0.4)( Hz and an angular momentum reorientation cross section, u, of 131 (1) A*. The 13C spin-lattice relaxation time of singly labeled I3C benzene is a linear function of density over the density range 1.07-75.12 mol/m3 (20-1330 Torr). "C TI values are shorter than 'H T I values by a factor of ca. 100 at comparable densities. The nuclear Overhauser enhancement factor, 7, is 0.0 f 0.02 at densities between 11 and 85.3 mol/m3 (200 and 1500 Torr), demonstrating that dipole-dipole relaxation is relatively inefficient in this region. The spin-rotation coupling constant, Ceff,for I3C nuclei in benzene is estimated to be 11602 (68)l Hz.
Introduction This study reports 'H and I3C relaxation times and I3C nuclear Overhauser enhancement factors for gaseous benzene that demonstrate that spin-rotation interactions provide the dominant relaxation mechanism for both nuclei at moderate densities. Although relaxation rates have been measured and mechanisms determined for the IH and nuclei of benzene in condensed phases, little is known about the nuclear spin-lattice relaxation processes of this molecule in the gas phase. Dipolar interactions provide the dominant mechanism for spin-lattice relaxation of both the 'H and 13Cnuclei of benzene in the liquid phase. Other relaxation mechanisms have been found to be much less efficient. Near the critical point, there is a suggestion of a spin-rotation contribution to relaxation of the ring protons.lV2 Ring carbons with attached protons relax mainly through dipole-dipole intera c t i o n ~ . The ~ nuclear Overhauser enhancement factor, 7, is 1.60 at 3 11 K for I3C in degassed liquid b e n ~ e n e . ~In the gas phase, (1) Green, D. K.; Powles, J. G. Proc. Phys. SOC.1965,85, 87-102. (2) Smith, R. E. Ph.D. Dissertation, 1966, Texas A&M University. (3) Alger, T. D.; Grant, D. M. J. Phys. Chem. 1971, 75, 2538-2539. (4) Levy, G. C.; Cargiolli, J. D.; Anet, F. A. L. J . A m . Chem. SOC.1973, 95, 1527-1535.
0022-3654/88/2092-4859$01 S O / O
spin-rotation interactions provide the dominant spin-lattice relaxation mechanism for protons in most molecules studied to date.5 It was suggested, on the basis of temperature-dependent relaxation measurements and theoretical considerations, that there is a 12% contribution to relaxation from dipole-dipole interactions in methane gas? Proton T Idata for several symmetric top molecules have been analyzed by assuming all the observed relaxation can I3Cspin-lattice be attributed to the spin-rotation mechani~m.~-'~ relaxation in gases has not been widely studied. A recent study of C O in various buffer gases has attributed nuclear spin relaxation to spin-rotation interactions.1° Gas-phase relaxation studies of I3C nuclei with attached protons have not been reported. The present study was undertaken to determine the relative importance of spin-rotation and dipoledipole relaxation for both ( 5 ) Bloom, M. In?. Reu. Sci.: Phys. Chem., Ser. One 1972, 4, 1-42. (6) Bloom, M.; Bridges, F.; Hardy, W. N. Can. J . Phys. 1967, 45,
3533-3554. (7) Lemaire, C.; Armstrong, R. L. J . Chem. Phys. 1984,81, 1626-1631. (8) Dong, R. Y.; Bloom, M. Can. J . Phys. 1969, 48, 793-804. (9) Pandey, L.; Lalita Sarker, K. Chem. Phys. Left. 1978, 58, 375-378. (10) Jameson, C. K.; Jameson, A. K.; Bochi, K. J . Chem. Phys. 1986,85,
697-700.
0 1988 American Chemical Society
4860
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
Folkendt et al.
the ' H and 13C nuclei in benzene gas.
Experimental Section Benzene samples used for 'H T I measurements were prepared from Fischer Scientific spectroscopic grade benzene. For 13C T1 measurements, 13C-enrichedbenzene (MSD, 9 1.6% singly labeled) was used. A mass spectroscopic analysis demonstrated that 5.1% of the sample was doubly labeled. There were traces of cyclohexyl fluoride and singly labeled (I3C)toluene in the sample. To facilitate handling of the small amount of labeled sample, we diluted some of this material in unlabeled benzene to a 6% concentration and used it in higher density measurements. All materials were thoroughly degassed by using the freeze-pumpthaw method prior to preparation of N M R samples. Sample cells used for the gas-phase T , measurements were constructed from Wilmad 12-mm insert tubing (10.95-mm 0.d.). Sample tubes were 17 mm long and had flat termini at both ends to confine the gas samples to the active volume region in our probes. A 3-mm Pyrex stem blown onto the top allowed attachment to a vacuum line. Benzene gas samples with pressures below 75 Torr (ambient vapor pressure) were made by filling the vacuum system to the desired pressure, sealing the sample tube with a glass torch, and promptly immersing the sample in liquid nitrogen to minimize sample decomposition at the hot sealing site. An estimated '% density uncertainty from this preparation technique. The low vapor pressure of benzene at room temperature necessitated quantitative transfer of material from preiiously calibrated glass- bulbs into calibrated sample cells to prepare samples covering a useful density range. These samples were frozen in liquid N2 before sealing. Pressures of samples prepared by this method have an estimated 3% uncertainty. In all cases, pressures were measured with a Baratron capacitance manometer with a 1000-Torr head (fO.O1-Torr accuracy), and all samples were sealed as close to the top as possible to minimize contributions to the measured T I values from diffusion of molecules from the active volume into the tube stem. Sample cells were positioned inside a standard 12-mm Wilmad N M R tube that had a small indentation 5 mm from the bottom to allow accurate positioning of the sample within the active volume. Sample pressures cited in the present paper refer to the sample pressures at the temperature at which they were prepared. For subsequent data reduction and analysis, these pressures were converted to densities by solving the van der Waals equation",'2 to correct for nonideal behavior. The van der Waals parameters, a = 1.881 02 Pa m6 mol-2 and b = 1.192 62 X m3 mol-', were calculated from reported critical parameter^.'^ It is interesting to note, however, that up to a 4%error in the pressure resulted in a mear 0.5% error in the resulting coupling constant. All NMR measurements were made with a GE NT-300 spectrometer with proton observation at 300.068 MHz and I3C observation at 75.092 MHz. For ' H and I3C measurements, 12-mm proton and 12-mm broad-band probes (Cryomagnet Systems, Inc.) were used, respectively. All T , measurements were made at 381 K. Samples were spun at a rate of ca. 22 Hz. Spectrometer field drift is negligible on the time scale of the T1 measurements reported here, and a field lock was not employed. The digital resolution in proton and I3C spectra was at least 0.7 and 0.6 Hz/point, respectively. The gas-phase spin-lattice relaxation times were determined by the inversion-recovery method14 using a composite pulse s e q ~ e n c e . ' ~Typically, for proton T1 measurements, a single transient was acquired for each delay time except for pressures below 100 Torr, where eight transients were acquired and averaged to produce spectra with signal/noise ratios in excess of 75/ 1. To ensure that system instability did not cause ( 1 1 ) Microsoft MuMATH Symbolic Mathematics Package, Version 9014.412, 1983, The Soft Warehouse. (12) MathCAD, Version 2.01, 1987, Mathsoft., Inc. (13) Kudchadker, A. P.; Alani, G. H.; Zwolinski, B. J. J. Chem. Reu. 1968, 68, 659-7 3 5 , (14) Martin, M. L.;Martin, G. J.; Delpuech, J.-J. Practical N M R Spectroscopy; Heyden: London, 1980. ( 1 5 ) Freeman, R.; Kempsell, S . P.; Levitt, M. H. J . Magn. Reson. 1980, 38, 453-419.
-
50 mrec.
100
Fipre 1. Inversion-recovery experiment using a 992-Torr sample of 6% 13C-enrichedbenzene,
errors in T l measurements using single transients, we performed additional experiments using multiple transients. There was no significant difference in the results obtained. 13C T I measurements were obtained with continuous proton decoupling. To test if sample heating from continuous proton decoupling was occurring, we measured 13C T1values for several samples under conditions of continuous proton decoupling and also under completely undecoupled conditions with the variable-temperature unit set at the same temperature. No significant differences in T I values were observed. For benzene at the temperatures and pressures used in the present experiments, data described below demonstrate that there is no significant contribution to I3C spin-relaxation from dipolar interactions. Therefore, any differences in sample temperatures between decoupled and undecoupled experiments would be simply related to differences in observed T , values. Benzene 13C T , values increase by ca. 1% for each 1 "C increase in temperature in the pressure and temperature regions studied. The lack of measurable differences in T1values under decoupled and undecoupled conditions demonstrates that continuous decoupling did not result in significant sample heating. To obtain acceptable signal/noise ratios, we acquired at least 400 transients for each delay time at pressures above 100 Torr and at least 1000 transients at pressures below 1000 Torr. Delay times between transient acquisitions were at least 7 times the determined T I to allow reestablishment of the Boltzmann distribution between the spin states. At least 12 delay times were used in each T , determination. Figure 1 illustrates typical data obtained for a I3C T I determination for a sample containing 54.4mol/m3 (992 Torr) of 6% 13C-enriched benzene gas. Tl was obtained from a nonlinear least-squares fit of the time-dependent intensity data. This procedure eliminates systematic errors in T I determination from the delay times used in the experiment. The delay times chosen for each T Idetermination were a compromize between T I sensititiy and signal-to-noise considerations. Nuclear Overhauser effect (NOE) measurements were made for the 13Cnuclei in benzene gas at four densities between 7 and 81 mol/m3. These measurements are complicated by the low concentrations of the probe nuclei and also by the short I3C T1 values observed. For compensation for possible changes in the spectrometer long term stability, NOE enhanced and control transients were acquired alternately in blocks of 8000 by using the pulse sequence shown in Figure 2. For the NOE enhanced measurements, on resonance continuous proton decoupling was applied for 10 T , values prior to sample acquisition switching to
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 4861
Spin-Lattice Relaxation in Gaseous Benzene NOE ENHANCED
05
CONTROL End
6locC B K
DECOUPLER
-51, 0 10 I
TRANSMITTER
"
,
20 CENSY
,
30 (TO
,
,
,
LO
5G
6C
M3)
Figure 4. Difference between the experimental proton T , values of benzene gas and those calculated from the two-parameter fit described in the text. RECEIVER
h -
h F.1.D.
F.I.D.
1 sec.
1 sec.
Figure 2. Experimental pulse sequence used to determine NOE factors for "C in benzene gas.
04
C
10
23
30
40
50
6C
3 E h S I N (rno /M3)
Figure 3. Density dependence of the proton T I for benzene gas at 381 K. The solid line is calculated from a two-parameter fit of the data as described in the text.
broad-band decoupling during transient acquisition. Control measurements were obtained with continuous proton decoupling with the decoupler frequency 2000 Hz off resonance to compensate for possible sensitivity changes in the 13C receiver channel as a result of decoupling while ensuring that no NOE enhancement could occur. For the control spectrum the broad-band decoupler was off during acquisition so that N O E could not build during the acquisition time. In all cases sufficient transients were obtained and averaged to produce signal/noise ratios in excess of 200, with at least 15 frequency points over the resonance. To measure the areas of the resonance, we transferred the Fourier transforms of the sums of the collected transients to a Zenith 2-200 microcomputer and simulated with Lorentzian line shapes using the program MASTER.FIT,~~ which utilizes a Fletcher and Powell minimization technique.
Results and Discussion IH Spin-Lattice Relaxation. The gas-phase nuclear spin-lattice relaxation time, T1, of benzene protons was measured in samples containing the naturally occurring I3C/l2C isotopic ratio at 36 densities between 0.8 and 54.4 mol/m3 (1 5 and 980 Torr) at 38 1 K. This temperature was chosen to maximize the experimentally accessible density range without exceeding instrumental temperature capabilities. Below 1 mol/m3, acquisition times are prohibitively long due to rapid polarization decay rates and associated line broadening, poor signal/noise, and the long delay times required to measure the long Tl values that occur at densities below the TI minimum. Figure 3 displays density-dependent T, values for benzene protons at 381 K. The increased scatter in (16) Badilla, I.; Weiss, B. E., copyright 1987.
the higher density data is attributed to sample preparation techniques. The following analysis of the relaxation data obtained for the protons in benzene gas is based on the assumption that the only important relaxation mechanism is spin-rotation. Proton dipole-dipole relaxation is estimated to be inefficient in light of NOE measurements described below, which show that relaxation from dipole-dipole (dd) interactions is insignificant for the 13C nuclei in benzene at the densities studied. We estimate that for adjacent protons is ca. 7 and 65 s at 7 and 81 mol/m3, respectively, using the approximate upper limit correlation times, T ~calculated , from the NOE enhancement factor described below and a dipole-dipole coupling contant, Cdd, of 3.3 x io9 s - ~ ,which is consistent with the smallest proton-proton distance (Le., 2.48 A). Relaxation from chemical shift anisotropy (csa) is also estimated to be insignificant. In the extreme narrowing region the relative contribution from this mechanism can be calculated from the relationship" where Au is the anisotropy and WI is the nuclear Larmor angular , impact apvelocity. The correlation time, T ~ in ~the~ sudden proximation18 is equal to l/z, where z is the collision frequency. Assumming a chemical shift anisotropy of 10 ppm, Tl,c:lis 21 s for a correlation time of 1 X s and 21 X lo3 s for a correlation time of 1 x s. The solid line in Figure 3 corresponds to a single relaxation time approximation (SRTA) fit of the density-dependent proton TI data to the equation Ti = A p + B / p (2) where p is the density in mol/m3 and A and B are curve-fitting parameters corresponding to the quantities ( a / 4 ~ ~ C , , 2 ) ( w-, a J ) / p m i nand ( c u / 4 ~ ~ C ~ &-( qWJ)P,,,~,,, respectively where a = h2/2ZokTand Io is the moment of inertia, 1.475 X g cm2, nuclear and molecular Larmor frequencies are 300.068 MHz and 14.6 kHz,22respectively, and pmin is the density at which T, is a minimum.19 Values of A and B obtained from the density-dependent TI data are 0.1941 (3) s m3/mol and 5.42 (3) s mol/m3, respectively. The constants derived from these quantities are Ceff = 1182.6 (0.4)l Hz and pmin = 5.28 (2) mol/m3. Figure 4 displays the difference, observed - calculated, for the 36 measured TI values. There is a suggestion of a systematic error in fitting the observed relaxation data to the function shown in eq 1. At densities below 11 mol/m3 the majority of the measured TI values are longer than those calculated from the best values of the curvefitting parameters. At the higher densities, in the range 27-54 mol/m3, the majority of the measured T , values are shorter than (1 7) Abragam, A. Principles of Magnetic Resonance; Oxford University Press: New York, 1961. (18) Chen, F. M.; Snider, R. F. J . Chem. Phys. 1967, 46, 3937-3940. (19) Tamagawa, K.; Iijima, T.; Kimura, M. J . Mol. Struct. 1976, 30, 243-253. (20) Stoicheff, B. Can. J . Phys. 1954, 32, 339-346. (21) Shoemaker, R. L.; Flygare, W. H. J . Chem. Phys. 1969, 51,
2988-2991.
(22) Courtney, J. A,; Armstrong, R. L. Can. J . Phys. 1972,50, 1252-1261.
4862
- ..
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
.
Folkendt et al.
the calculated values. This observation is similar to previous results obtained for N H 3 and HC1 gas.' This suggests that a multiple relaxation time approximation (MRTA) should provide a better model for the observed data.' A fit of the observed benzene proton TI data to the function 1 / ~ ,= (4*2/.)[(~,ffZro)/(r02
+ w2)][1+ [(ro2 - 3w2) x (@ro)2/(ro2 + w 2 ) 2 ~(3) ~
where rois l / r o , the reciprocal of the average correlation time, and @ is the dispersion parameter defined as Ar/ro, which characterizes the range of effective correlation times in the system, yields the following results: Ceff= 1180 (3)l Hz, ro= 21 ( 5 ) MHz atm-I, and @ = 0.3 (0.2). These parameters have larger uncertainties than those obtained in the two-parameter fit but a systematic error in the fitting procedure is not observed. Despite the approximations made in modeling the density dependence of T I for the protons of benzene, the spin-rotation coupling constant is reasonably well determined. There are no direct measurements of this parameter from analysis of rotational hyperfine structure. On the basis of molecular beam magnetic resonance results, it was estimated that C,, for benzene is much less than 1 ~ H z A . contribution ~ ~ to the overall relaxation from the spin-rotation interaction was suggested to explain the magnitude and temperature dependence of proton spin-lattice relaxation in liquid benzene, and a coupling constant of 410 H z was suggested.2 A larger value of 540 H z was estimated to explain the anomalous decrease in the T , values of liquid benzene observed near the critical temperature.' The spin-rotation coupling constant for benzene protons obtained in the present study may also be compared to recent theoretical chemical shift calculations. The nuclear shielding constants of benzene were calculated by using coupled Hartree-Fock perturbation theory.24 For the protons in benzene, the paramagnetic shielding components were calculated as up(xx)= -57.45 ppm, u0(yy) = -175.16 ppm, and up(zz) = -221.75 ppm for a coordinate system with its origin at a proton. The principal components of the paramagnetic shielding tensor are directly related to those of the spin-rotation tensor.25 For benzene, the theoretical shielding constants are consistent with principal ' H spin-rotation tensor elements, C(xx) = 5258 Hz, Cby) = 4651 Hz, and C ( z z ) = 492 Hz, by using the geometry reported in ref 19. For benzene, which is a symmetric top, these coupling constants are consistent with an average spin-rotation coupling constantz6 of 464 Hz, slightly more than twice the experimental value obtained in the present work. The cross section for angular momentum reorientation for benzene can be determined from the density at which T I is a = 1, where T~ = minimum. At this density the quantity (ucN/ V)-I and u is the cross section, c is the average speed of the colliding molecules, and N / V is the gas density. The minimum in the T I versus density curve occurs at a gas density of 5.28 (2) mol/m3 at 381 K, and the corresponding value of the cross section is 130 (1) A2. This result can be compared to cross sections determined from transport measurements and other spin-rotation cross sections. Lennard-Jones parameters for benzene, determined from viscosity measurements, are consistent with a cross section, u8, of 87.25 i42.27The geometrical cross section, us, of benzene is ca. 78 A2. The angular momentum reorientation cross section is larger than ug by a factor of ca. 1.3. The angular momentum reorientation cross section, u8, of "3, 105 A', is also significantly larger than the geometric cross section, u8, of 60 A2.' I3CSpin-Lattice Relaxation. I3Cspin-lattice relaxation times of singly I3C-enrichedbenzene were measured at densities between ~~~~~~
(23) Ramsey, N. F. Sci. Prog. (New Haven) 1963, 59-78. (24) Lazzeretti, P.; Zanasi, R. J . Chem. Phys. 1981, 75, 5019-5027. (25) Flygare, W. H. Molecular Structure and Dynamics; Prentice-Hall: Englewood Cliffs, NJ, 1978. (26) (a) See eq 4 of the following: Weiss-Lopez, B. E.; Winegar, E. D.; True, N. S. J . Phys. Chem., in press. (b) Armstrong, R. L.; Courtney, J. A. Can. J . Phys. 1972,50, 1262-1272. (c) Dong, R. Y.; Bloom, M. Can. J . Phys. 1970, 48, 793-804. (27) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954.
-
2y
0
3 :51
J L
n ; - zz
_I
e
"
i
d
_
I
_ i
I
.,"
Figure 5. Density dependence of the "C Ti for singly "C labeled benzene gas at 381 K.
1.0 and 80 mol/m3 (16 and 1325 Torr) at 381 K and are shown in Figure 5 . In this density range, I3C relaxation in benzene is faster than 'H relaxation by a factor of ca. 100. The densitydependent I3C relaxation times are a linear function of gas density in this region with a slope dTl/dp of 2.53 (0.05) ms m3/mol. Poor signal/noise limited T I measurements to densities above 1.O mol/m3, and a characterization of the T I minimum for I3C was not experimentally feasible. A consideration of reasonable estimates and experimental measurements of the relative contributions to the observed spinlattice relaxation times of the I3C nuclei in benzene gas from all possible mechanisms leads to the conclusion that only spin-rotation interactions are important at the temperatures and densities where measurements were made. The observed NOE enhancement factor, 7, is 0 with a ca. 2% uncertainty at densities between 7 and 81 mol/m3 at 381 K. This result is consistent with additional T I measurements which were performed to determine if proton decoupling affected observed T I values. Six T I values were measured at densities between 17 and 66 mol/m3 with continuous proton decoupling, and an additional set of measurements were made on the same samples without proton decoupling. There was no significant difference between the corresponding measurements. These results confirm that dipole-dipole interactions do not provide a significant source of spin-lattice relaxation for the I3C nuclei of benzene gas. An estimate of the contribution to I3C spin-lattice relaxation from shielding anisotropy can be made by using eq 1. The shielding anisotropy, Au, for I3C nuclei in benzene has been determined from solid NMR spectra to be 180 (2) ppm.z8 With the angular momentum reorientation cross section of 130 A2 determined from the proton measurements described above, TI,,, for I3Cnuclei in benzene is calculated to be 15.6 s at I moI/m3 and 1175 s at 80.7 mol/m3. Although experimental limitations precluded a characterization of the T I minimum and a direct determination of the spin-rotation coupling constant for the I3C nuclei of benzene, the slope in the extreme narrowing region, coupled with the results obtained for the protons, described above, allows an estimate of this parameter to be made. In the extreme narrowing region the spin-lattice relaxation time is
( T 1 , J ' = (4*2/.)CeffZ7,
(4)
Assuming that the angular momentum correlation times are identical for 'H and I3C nuclei at a given density
With the ratio of the slopes in the extreme narrowing region, (dT,/dp)(lH) = 0.194 s m3/mol and (dTI/dp)(l3C) = 2.53 (0.05) ms m3/mol, and the spin-rotation coupling constant, Cefi2,of 3.335 value of 2.56 (0.014) X lo4 Hz2 determined for the protons, a Ceff2 (0.01) kHz2 (Cer = 11602 (68)l Hz) is obtained for the I3Cnuclei. The larger value of CeffZfor I3C nuclei compared to IH nuclei in benzene is consistent with the molecule's geometry. Our estimated (28) Gibby, M. G.; Pines, A.; Waugh, J. S. Chem. Phys. Lett. 1972, 16, 296-299.
J . Phys. Chem. 1988, 92, 4863-4868 spin-rotation coupling constant for 13C in singly labeled benzene may also be compared to theoretical and experimental chemical shift data. Calculated principal axis components of the paramagnetic shielding tensor, up(xx) = -446.57 ppm, .,(yy) = -351.56 ppm, and u (zz) = -307.27 ppm, evaluated at a carbon nucleus in b e n ~ e n e ~ ~consistent ~ a r e with a C eof~ 1712 Hz, which is close to our experimental value. The estimated Cenfor 13C can be used to estimate the density at which Tl is a minimum for I3C nuclei in benzene by using the relationship
A = (./4r2)(1
/Ced)(WI/Prnin)
(6)
where A is the slope in the extreme narrowing region and the other parameters are defined above.22 At 381 K, the calculated pmin is 1.32 mol/m3 (24.08 Torr). Although data were obtained at densities as low as 1.0 mol/m3 in the present study, the data were not of sufficient quantity or quality to detect a departure from linear density dependence for the lower density data points. The lack of appreciable dipole-dipole spin-lattice relaxation for the I3C nuclei is an interesting result of this study. In the extreme narrowing region, Tl,dd is given by (Tl.dd)-' = CddTO
(7)
where Cdd is the dipole-dipole coupling constant and T~ is the molecular reorientation correlation time. Tlgrcan be calculated from eq 4. The dipole-dipole coupling constant, Cdd, for a 13C nucleus and attached proton is 2.00 X 1Olo s-*, assuming an effective bond length of 1.101 A2.14 The quantity (4r2/a)Cef:
4863
is 1.38 X 10" s - ~by using the spin-rotation coupling constant estimated above. A N O E enhancement factor, 7, of 0.14 is expected at densities well above the T1minimum, by assuming that the geometric and angular momentum reorientation correlation times, TO and 75,are identical. Experimentally is substantially longer than Tl,srand does not contribute significantly to the relaxation process. The uncertainty in the experimentally determined NOE factor, 7,of 2% allows an estimate of the lower limit for Tl,dd to be made. For 7 of 0.02, the corresponding lower limit T1,dd is 3 s at 10.8 mo1/m3 and 20 s at 80.7 moi/m3. The estimated lower limit Tl,dd values are consistent with molecular framework reorientation correlation times of 1.7 X lo-" s and 2.5 X lo-'* s at 10.8 and 80.7 mol/m3, respectively. The angular momentum reorientation correlation times determined above for the benzene protons are 2.68 X 10-'0 and 3.41 X lo-" s, respectively, at 10.8 and 80.7 mol/m3.
Conclusion Spin-rotation interactions provide the dominant spin-lattice relaxation mechanism for both protons and I3C nuclei of gaseous benzene. 13Crelaxation is facilitated by the larger spin-rotation coupling constant, and (13C)benzene TI values in the gas phase are ca. 1000 times shorter than in the liquid phase. Acknowledgment. We are pleased to acknowledge support from the National Science Foundation (CHE-83-511698-PYI and CHE-85-03074) and the Alfred P. Sloan Foundation for support of this research. Registry No. Benzene, 71-43-2.
Circular Dichroism Band Shapes for Helical Polymers David A. Rabenold Biochemistry and Biophysics Department, Iowa State University, Ames, Iowa 5001 1 (Received: October 26, 1987; In Final Form: March 14, 1988)
Three approaches are discussed for handling band shapes in circular dichroism (CD) and absorption spectra calculations for helical polymers. The effect of interaction between chromophores upon polymer transition band shapes is explored by employing Gaussian band shapes for isolated chromophore transitions in the classical coupled oscillator scheme. The results are compared to those obtained from using Lorentzian band shapes and to those obtained by using 6 function line shapes that are replaced by Gaussians after interactions are taken into account. The three approaches yield very different calculated AT* absorption and circular dichroism spectra for an a-helix. Skewing of band shapes derived from input Gaussians is analyzed by employing input bands that are composite bands of several Lorentzians and mimic the input Gaussians. Skewing is shown to result from coupling of different vibronic components. Distortion of the CD helix band is discussed by using an approximate formulation based on periodic boundary conditions that includes end effect corrections. The classical scheme is also related to the so-called matrix method.
Introduction Calculations of the circular dichroism (CD) and absorption spectra of helical polypeptides have been based on a variety of formulations. Use of the so-called matrix method yields polymer rotational and oscillator strengths and polymer transition frequencies to which band shapes are assigned.'+ On the other hand, a classical coupled oscillator scheme, with Lorentzian band shapes, all of the same half-peak width, for the oscillators yields the same Lorentzian band shapes for the normal modess-' In another
method,8 a time-dependent Hartree schemeg-" is used that is equivalent to the decorrelation approximation,12each normal mode of which obtains a &function line shape that to do calculations is replaced, e.g., by a Gaussian. The latter two methodss,8 can yield identical polymer rotational and dipole strengths and polymer transition frequencies. All of the above methods yield polymer transition frequencies that are shifted from unperturbed values
(1) Woody,R. W.J . Chem. Phys. '1968, 49, 4797. (2) Pysh, E.S.J. Chem. Phys. 1970, 52, 4723. (3) Ronish, E.W.;Krimm, S.Biopolymers 1974, 13, 1635. (4) Madison, V.;Schellman, J. Biopolymers 1972, 1 1 , 1041. (5) Applcquist, J.; Sundberg, K. R.; Olson, M. L.; Weiss, L. C. J. Chem. Phys. 1979, 70, 1240; 1979, 71, 2330.
I70 ._.
0022-3654/88/2092-4863$01.50/0
(6) Applequist, J. J. Chem. Phys. 1979, 71, 1983;1979, 71, 4324;1979, 71,4332;1980, 73, 3521. (7) Applequist, J. Biopolymers 1981, 20, 387;1981, 20, 2311; 1982, 21,
.
(8) Rabenold, D. A,; Rhodes, W. J. Chem. Phys. 1986, 90,2561 (9) McLachlan, A. D.;Ball, M. A. Mol. Phys. 1964, 8, 581. (10)Harris, R. A. J . Chem. Phys. 1965, 43,959. (11)Rabenold, D.A. J. Chem. Phys. 1982, 77,4265. (12)Rhodes, W.;Chase, M. Reu. Mod. Phys. 1967, 39, 348.
0 1988 American Chemical Society
