J . Phys. Chem. 1990, 94, 1291-1293
1291
Relaxation and Molecular Dynamics. Overall Movement and Internal Rotation of Methyl Groups in N,N-Dimethylformamide 13C NMR
R. Konrat and H. Sterk* Institute f o r Organic Chemistry, Karl- Franzens- University, Heinrichstrasse 28, Graz, Austria (Received: March 15. 1989; In Final Form: July 27, 1989)
To describe. the overall movement of NJ-dimethylformamide (DMF) the magnetization modes formalism proposed by Werbelow and Grant has been used. Thereby, the carbon of the aldehyde group has been considered as an AX spin system with an additional coupling between dipolar and chemical shift anisotropy (CSA) contributions. The rotation of the methyl groups has been described by a formalism applicable to molecules with an anisotropic overall motion and additional internal mobility. The methyl groups were treated as AX3 spin systems. The rotation of the trans-methyl group was found to be faster than that of cis-methyl group.
Introduction There are numerous recent papers dealing with the spin-lattice relaxation times of methyl groups.'" Some of them only qualitatively discussed the rotation of the methyl group. A theoretical study of the nuclear spin-lattice relaxation in molecules with an additional internal motion has been made by Woessner et al.,' who treated axially symmetric ellipsoid molecules. Levine et aI.* have reported a different formalism for describing the internal rotation in an axially symmetric molecule. On the other hand, Wallach et aL9 have studied the anisotropic molecular motion in liquid DMF using a quadrupolar relaxation mechanism. In this work dipolar and chemical shift anisotropy (CSA) interactions are under consideration. To describe the relaxation behavior of the methyl groups the magnetization modes formalism for an AX3 spin system has been used. To take into account the additional internal rotation of the two methyl groups, a recently developed formalism has been applied. The description of the carbon located at the aldehyde group was achieved by using the well-known magnetization modes formalism proposed by Werbelow and Grant.lo An AX spin system with additional coupling between dipolar and CSA contribution has been used. The applicability of this formalism has already been shown by Koenigsberger and Sterk." Experimental Section DMF of commercial origin was distilled before use. The sample was completely degassed by bubbling nitrogen through it. I3C NMR measurements were recorded on a Varian XL 200 spectrometer and on a Bruker MSL 300 spectrometer. Spin-lattice relaxation times were determined by the conventional inversionrecovery experiment. Also a IH inversion with consecutive I3C detection has been performed. The pulse delays for both experiments were set at greater than 10 times the longest TI to be measured. The error ranges in the T I measurements did not exceed 5%. All measurements have been done at room temperature. As will be presented in the theoretical section, the total 13C magnetization as well as the intensity difference of the I3C (1) Kuhlmann, K. E.; Grant, D. M. J. Chem. Phys. 1971, 55, 2998. (2) Schmidt, C. F.; Chan, S . I. J. Magn. Reson. 1971, 5, 151. (3) Werbelow, L. G . ;Marshall, A. G. J . Am. Chem. Soc. 1973,95, 5132. (4) Leipert, T. K.; Noggle, J. N. J . Magn. Reson. 1974, 13, 158. (5) Buchner, W. J . Magn. Reson. 1973, 12, 82. (6) Levy, G . C.; Nelson, G. L. J . Am. Chem. Sor. 1972, 94, 4897. (7) Woessner, D. E.; Snowden, B. S.; Meyer, G. H. J. Chem. Phys. 1969, 50, 719. (8) Levine, Z. K.; Birdsall, N. J. M.; Lee, A. G.; Metcalfe, J. C.; Partington, P.; Roberts, G.C. K. J . Chem. Phys. 1974, 60, 2890. (9) Wallach, D.; Huntress, W. T. Jr. J. Chem. Phys. 1969, 50, 1219. (IO) Werbelow, L. G.; Grant, D. M. Adv. Magn. Reson. 1977, 9, 189. (1 1) Kocnigsberger, E.; Sterk, H. J . Chem. Phys. 1985, 83, 2723.
0022-3654/90/2094-1291$02.50/0
doublet components have been measured. For the case of the methyl groups only the total I3C magnetization has been measured. They are presented graphically in Figures 2-5. A nonlinear least-squares procedure has been developed to fit the diffusion constants, random field contributions, and the CSA term to the measured time evolution of the magnetization.
Theory As stated above, we have adopted the model of a 13C-IH system relaxed by dipolar interactions with an additional CSA interaction at the 13C nucleus. To account for relaxation by spin rotation and intermolecular dipolar interactions, random field terms are employed. In the following, we use the magnetization modes formalism proposed by Werbelow and Grant.lo A detailed description is given elsewhere;" here only a short summary will be presented. Normal modes are defined as traces over the products of the deviation spin density operator with (in trace form) orthonormal irreducible tensor operators representing, at least in part, observables of the spin system. In the case of an AX spin system there are four normal modes, of which three can be detected, corresponding to the total A magnetization (mode l ) , the total X magnetization (mode 2), and the intensity difference within each doublet (mode 3). Mode 4 is zero and accounts for the invariance of the trace and need not be considered further. The spectral densities p,FSA, and psA (the notation of the excellent review of Werbelow12 is used) can be calculated by using standard assuming throughout exteme narrowing. For the description of the relaxation behavior of the methyl groups also the normal modes formalism has been used. In this case an AX3 spin system is present (denoting the carbon by A and the protons by X). A detailed description is given by Werbelow.I2 Mode 1 corresponds to the total A magnetization and is the only one being considered. For the calculation of the spectral densities the well-known expressions for the asymmetric top of Vold16 have been used. For the description of the rotation of the methyl groups a formalism has been applied which is given in detail elsewhere." In the fit procedure first the random field contributions have been treated as parameters. Second, attention has been drawn to the fact that the spectral density functions depend in a nonlinear fashion on the anisotropy of the rotational diffusion tensor, the interatomic distances, and the orientation of the internuclear vector (or the principal axes of the CSA tensor) with respect to the main (12) Werbelow, L. G.; Grant, D. M. Adv. Magn. Reson. 1977,9, 189-299. (13) Bain, A. D.; Lynden-Bell, R. M.; Mol. Phys. 1975, 30, 325. (14) Werbelow, L. G.; Marshall, A. G. Mol. Phys. 1974, 28, 113. (15) Fuson, M. M.; Prestegard, J. H. J . Chem. Phys. 1982, 76, 1539. (16) Vold, R. L.; Vold, R. R.Prog. Nurl. Magn. Reson. Spectrosc. 1978, 12, 79. (17) Konrat, R.; Sterk, H. Chem. Phys. Lett. 1989, 159, 137.
0 1990 American Chemical Society
1292 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 TABLE I: Diffusion Constants, Rwdom Field Contributions, and Chemical Shift Anisotropy (out of the AX System) 200 MHz 300 MHz a" b" a" bb 5.74 5.75 6.13 D,, 5.79 40.00 36.00 47.00 DYY 36.00 377.00 375.00 346.00 4, 397.00 CSA 170 169 170 170
!c JH
0.003
0.003
0.003
Konrat and Sterk TABLE IV: Diffusion Constants for the Internal Rotation (DR, s-') of the Methyl Groups" E-Me Z-Me DR 1.5 x IO" 6.8 X IO'*
" Calculations based on an asymmetric overall motion and internal rotation
0.003
" a corresponds to mode 1; b corresponds to mode 3. The diffusion constants are given in IO9 s-'; the CSA term is given in ppm. The error ranges for the fitted diffusion constants are 10%.
TABLE 11: Swctral Densities Calculated from the Values of Table I 200 MHz 300 MHz 0.016 16 JD 0.01609 0.00 1 76 JCSA 0.000 82 KD-CSA -0.003 30 -0.004 80 TABLE III: Diffusion Constants for the Methyl Groups and Autocorrelation Function as Well as Random Field Contributionsd E-Me Z-Me E-Me Z-Me D,, 9.95 5.05 JAx 0.0054 0.0015 Dyy 74.80 504.00 Jc 0.0257 0.0220 D,, 759.00 18260.00 "Calculation based on an asymmetric overall motion exclusively. The diffusion constants are given in IO9 s-l. The error ranges are about 10%.
axes of the rotational diffusion tensor. In calculating these spectral density functions, needed in the relaxation matrix, the main axes of the inertial tensor of the molecule rather than the main axes of the rotational diffusion tensor have been ~sed.'*9*~ The inertial tensor has been calculated by using standard bond lengths and assuming a planar molecule. The principal axis of the shielding tensor has been chosen along the CO bond.20 JD and FsA are the spectral density functions for the dipolar and CSA contributions and F A means the interference or crms term between the dipolar and the CSA interaction. They are calculated according to Vold and Vold.I6 Results In order to get a description of the overall motion, the relaxation behavior of the aldehyde carbon has been measured. Due to the fact that in a nonlinear least-squares fit procedure no unique set of parameters can be evaluated, a second TI experiment at 300 MHz has been done. The fitted values are given in Table I. As one can see, the values for the chemical shift anisotropy are exactly the same and agree very well with the ones reported in the literature.20 To expand the data set, mode 3 has also been considered. The results of this fit procedure corroborate the former values for the diffusion constants and the chemical shift anisotropy (Table I). In Table I1 the dipolar and CSA contributions at 200 and 300 MHz are given. One can see that the CSA term and moreover the coupling between dipolar and CSA interaction can by no means be neglected and, as expected, gain in significance the higher the applied field becomes. To describe the dynamic behavior of the methyl groups, both of them have been treated as an AX3 spin system. The obtained results are given in Table 111. The (Z)-methyl group shows significantly higher diffusion constants than the @)-methyl group and the overall motion. This is due to the fact that rotation takes place, which has not yet been considered. In order to do so, an extended formalism17has been used. Thereby, the autocorrelation (18) Sterk, H.; Kalcher, J.; Kollenz, G.; Waldenberger, H. Z . 2.Naturforsch. 1979, 34A, 375. (19) Sterk, H.; Maier, E. Ado. Mol. Relax. Interact. Processes 1982, 23,
247. (20) Fyfe, C. E. Solid State N M R for Chemists: C . F. C. Press: Guelph, Ontario. Canada, NIH 629.
Figure 1. The direction of the principal diffusion axes in DMF and the direction of the molecular dipole moment ( p = 3.91 D). The x diffusion axis is perpendicular to the plane. The direction of the chemical shift anisotropy is assumed to lie along the CO bond and in the plane.
function JAX (coming out of the fit procedure and given in Table 111) has to be described in terms of the diffusion constants for the overall motion and for the internal rotation. The diffusion constants for the overall motion were set equal to those determined from analysis of the AX system and the diffusion constant for the internal rotation has been fitted onto the spectral density function JAx. The results in Table IV show that the rotation of the (Z)-methyl group is about 10 times faster than the rotation of the (E)-methyl group, which is in good agreement to the values obtained by Nakanishi and Yamamoto.21 Another fact which can be seen from Table I11 is that the random field contributionmainly determined by spin rotation-cannot be neglected, especially in the case of the (Z)-methyl carbon. This leads to a lower NOE factor for the (Z)-methyl carbon compared to the ( E ) methyl carbon, which is also in good agreement with the results of Nakanishi and Yamamoto.21 Discussion The diffusion equation is an appropriate equation of motion in the liquid in the limit that the molecular reorientation is very much slower than in the free gas or that the decay of the rotational correlation function for the liquid is very much slower than the decay for the gas. If the diffusion equation, or exponential decay, is assumed, then the characteristic decay time for the diffusional rotor tensor correlation function is 7 , = (6l4-I. The correlation time for a free spherical rotor is given by T,(fr) = ( 2 ~ / 9 ) ( 1 , / k T ) ~ ' * where the Ziare the principal moments of inertia. An appropriate test to determine the validity of the assumption of a diffusion equation for the motion in the liquid is, therefore,
xi = 7,/7,(fr) The values of Xi in DMF are X , = 26.0, X, = 4.2, X,= 1.2. They are in good agreement to those given by W a l l a ~ h .These ~ values seem to indicate that the diffusion model is a reasonable first approximation to the rotational motion of DMF, though inertial effects probably do affect the motion to some extent. What, in addition, can be seen out of these values is that the reorientation about the z diffusion axis is not hindered at all, or is hindered only to a very small extent. The reorientation about the x diffusion axis is the one which is hindered most, due to an effective dipolar interaction (consider the direction of the dipole moment in Figure 1). This is quite similar to the conclusions drawn by Wallach and H ~ n t r e s s . ~ (21) Nakanishi, H.; Yamamoto, 0. Chem Phys. Lett. 1975, 35, 407.
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1293 0
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Figure 2. Magnetization mode 1 of the AX system. I3C inversion recovery (lower trace) and 'H inversion with consecutive "C detection (upper trace). The curves represent the fits (200 MHz).
Figure 4. Magnetization mode 1 of the AX3 system for the cis-methyl group. "C inversion recovery (lower trace) and IH inversion with consecutive I3C detection (upper trace). The curves represent the fits (200
MHz).
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Figure 3. Magnetization mode 3 of the AX system. 13C inversion recovery. The curve represents the fit (200 MHz).
But there is also a striking difference between their results and ours, because in their work it was assumed that the diffusion constant for the internal rotation was the same for both methyl groups, whereas we have found that the methyl groups rotates with different velocities, in agreement with Nakanishi and Yamamoto.*' The question is, what is this difference about, because it is quite unexpected. From a simple steric hindrance point of view, the rotation of the (2)-methyl group is considered to be more restricted (because of the carbonyl oxygen). A possible explanation might be the fact that DMF forms linear intermolecular 1,4 van der Waals interactions, which are more stable than cyclic ones. In this case only the (,?)-methyl group is involved and therefore exhibits lower rotation rates compared to the (2)-methyl group. Summary
To sum up, the anisotropic overall movement of DMF has been determined by using the magnetization modes formalism. CSA
7 TAU tsl
.
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Figure 5. Magnetization mode 1 of the AX, system for the transmethyl group. 13Cinversion recovery (lower trace) and 'H inversion with consecutive "C detection (upper trace). The curves represent the fits (200
MHz). autocorrelation and D-CSA crosscorrelation terms have been used to determine the three diffusion constants for the overall movement. To get an additional proof for the accuracy of the revealed values the methyl groups have been treated as an AX3 spin system. The diffusion constants, which are revealed in the first step, thereby neglecting the internal rotation, are too great. They came out in the right amount if this internal rotation is included in the calculation of the spectral densities. It revealed that the two methyl groups rotate with different velocities, which is in good agreement with the existing literature. Registry No. N,N-Dimethylformamide,68-12-2.
