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Molecular and structural information from variable-angle spinning NMR dipolar spectra of 13C-14N systems
JOURNAL
OF MAGNETIC
RESONANCE
w,522-534
( 1990)
Molecular and Structural Information from Variable-Angle Spinning NMR Dipolar Spectra of 13C-14NSystems ZHEHONG GAN AND DAVID M. GRANT* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Received April 6, 1990 A sample-spinning solid-state NMR study of a spin-1 nucleus coupled to a quadrupolar nucleus is presented. Using a simple approach, the quadrupolar effect is expanded in terms of irreducible spherical tensor components up to I = 4. Information on the electric field gradient tensor orientation, dipolar coupling, and chemical-shielding tensor for the ‘%“‘N system can be obtained experimentally from the magic-angle and the off-magic-angle sample-spinning spectra. Satisfactory results are obtained by comparing the simulations with the experimental spectra of tetramethylpyrazine, dimethylglyoxime, and triethylenediamine. 0 1990 Academic Press. Inc.
Cross polarization of dilute spins from proton magnetization combined with magicangle spinning (CP/MAS) constitutes the main technique for obtaining high-resolution NMR spectra of organic solids ( I, 2). An interesting feature observed in MAS spectra of a carbon directly bonded to a nitrogen is the splitting of the carbon line into an asymmetric doublet while other carbon lines continue to remain narrow. In a 13C14N coupled spin pair, the quantization direction of the nitrogen spin is not strictly along the external magnetic field since the quadrupolar interaction of the nitrogen spin has a magnitude comparable to that of the Zeeman interaction. Failure to achieve the alignment of the quadrupolar nuclear spin with the external magnetic field can make a difference in the spectra of a spin coupled to a quadrupolar nucleus, and this quadrupolar effect is manifested in the fine structure lineshape of the MAS spectra. Spectral features, such as the separation of the asymmetric doublet, are directly related to the dipolar coupling and to the magnitude and the orientation of the electric field gradient (EFG) tensor of the nitrogen in the molecule. Vanderhart et al. (3) have explained this quadrupolar effect theoretically by calculating the eigenstates of the quadrupolar nucleus under the presence of both the Zeeman and the quadrupolar interaction for each orientation of the molecule with respect to the external magnetic field. In the past, several studies (4-7) have been initiated to obtain information on the quadrupolar interactions and on the dipolar coupling from the MAS spectral lineshape of carbons coupled to nitrogens. Although the MAS technique gives relatively high-resolution spectra, information on the chemical-shift anisotropy (CSA) cannot be recovered from such spectra. Spinning at a variable angle other than the magic angle has been used to obtain CSA * To whom correspondence should be addressed. 0022-2364190 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights ofreproduction in any form reserved.
522
VARIABLE-ANGLE
SPINNING
DIPOLAR
SPECTRA
523
information from the powder pattern which is scaled by a factor of ( 3 cos’& - 1 )/ 2 (8). With the proper selection of the spinning angle, variable-angle spinning can therefore improve the resolution by separating the overlapping lineshapes from different carbon spins. For a coupled ’ ‘C- 14N system in the variable-angle spinning experiment, the spectra are also scaled except for the part due to the quadrupolar effect term. Thus, the quadrupolar effect under different spinning angles may be studied as a way to retrieve CSA and dipolar data from spectral lineshapes. In this paper, we present a simple analysis of the quadrupolar effect when the quadrupolar interaction is small compared to the Zeeman interaction. The first-order nonsecular terms in the rotating frame give an analytical solution for the transition frequencies in both the stationary and the sample-spinning cases. Experimentally, tetramethylpyrazine and dimethylglyoxime were studied to capitalize on the large separation of the lines from the carbon spin coupled to the nitrogen and on the magnitudes of these quadrupolar interactions as determined by other techniques. Triethylenediamine provides an interesting case for study since the molecular motion at room temperature alters the lineshape and can be analyzed with this technique. THEORY
The Hamiltonian of a nuclear spin system in terms of the irreducible spherical tensor operators is given by i% = c c x
I
;
(-l)“R;-,T;m,
111
m=-I
where X represents the type of interaction. For the 13C(I)- 14N( S) system, X includes Z, and Zc, the Zeeman interactions for the carbon and the nitrogen spins (on the order of 10 MHz) ; Q, the quadrupolar interaction of the nitrogen spin (on the order of MHz); D, the dipolar interaction between the carbon and the nitrogen spins; and CS, the chemical shielding of the carbon spins (the later two terms are both on the order of kHz). The rotational property of the irreducible spherical tensor operators, R,, , and the construction of the bases spin operator, T,,,, , of the irreducible representation have been given by Haeberlen (9). The same convention as that of Haeberlen is used in this work. The rotating frame of both the carbon and nitrogen spins at their Larmor frequencies is characterized by the rotation operator
and the Hamiltonian
is given by
&f R = RtiR-’
+ @R-j dt
[21
524
GAN
AND
GRANT
For most r3C- 14N cases, the Zeeman term of the nitrogen spin in the Hamiltonian is large compared to the quadrupolar interaction. The time dependence of the Hamiltonian in Eq. [ 21 can be removed by applying average Hamiltonian theory ( 10). The average Hamiltonian includes the terms which do not commute with the Zeeman Hamiltonian. These terms contribute to the spectral frequency in higher orders. The portion of the average Hamiltonian which contains the carbon spin operator and which commutes with the Zeeman Hamiltonian is given to first order by SC = #NV + SC’), (31 where
The zeroth-order term, S(“, includes the secular terms of the chemical shielding and the dipolar interaction. The .X’ (I) term gives the first-order quadrupolar effect. The results obtained with Eq. [ 3] are equivalent to those obtained using first-order perturbation theory to calculate the eigenstates of the nitrogen spin under both the quadrupolar interaction and the Zeeman interaction ( 7). It is well known that average Hamiltonian theory has a close relationship to perturbation theory. For example, the secular approximation is equivalent to the zeroth-order perturbation of the eigenstates ( 10). In Eq. [ 3 1, RF:, , the irreducible tensor components in the lab frame, are related to those in the molecular frame by
where the Dht,,,( (Y, 6, y ) are the Wigner rotation matrix elements. In our case, the principal-axis frame of the dipolar coupling tensor is well defined in the molecule and is selected to be the molecular frame. From Eqs. [ 31 and [ 41, the term characterizing the quadrupolar effect can be expanded as s%(l) =
C /=0,2,4
;: D!,&O, /9, y)A1,ZZ(3St m=-I
- S2>,
[51
where the nonzero A,, elements are 1 rforfo- ’ 5q.J
Am = - -
1 ~~~~ --$‘rQ.
7oN
20
20,
DQ A40 = - l2 r20r20; 35WN
A 211
=..--
A 4+1
=
1 DL? r20r2+1; 14WN 2,k D Q r20r2&1; 35WN
-
A2&2
= G
AJr2 = g
I
DQ r20r2*2;
r%%+,.
[61
VARIABLE-ANGLE
SPINNING
DIPOLAR
525
SPECTRA
Explicit expressions are given for D2mjm( (Y, /3, y) in the book by Haeberlen (9) and for D4ti(0, ,6, y) in the Appendix. In the molecular frame, the components rim are related to the dipolar coupling constant and quadrupolar coupling tensor by rfo=-
f 60 VZX
Q-
y20 - 8S(2S - 1)
Y2q1 + = - ,,~~~~
{ 3 cos2pD - 1 - 7) sin2PDcos 27,)
1) {+3sin2PD+9
&,+ = - ,;;;fl,
sin 2PDsin 27~ + 2i?7sin BDsin 2y,J
sin2PD + q( 1 + cos2~D)cos 2~~
{-3
* 2irj cos PDsin 2yD},
[ 71 where D is the dipolar coupling constant, X is the quadrupolar coupling constant, and n is the asymmetry of the EFG tensor, which take the familiar form D
=
YcYd
x
4a*r3
-
e*Qq h
’
’
9YY -
t=
4xX
9 Zi
s
The quantities qxx, qyy, and qzz are the principal values of the EFG tensor with qz2 = eq; CQ, /LID,and 70 are the Euler angles for the rotation from the molecular frame to the EFG principal frame; and POand yD are the polar angles of the dipolar vector in the EFG tensor principal-axis frame. Equation [ 51 gives, in the stationary case, the solution for the quadrupolar effect for a certain orientation of the crystallite with respect to the external magnetic field. In the sample-spinning case, the Wigner rotation matrix from the lab frame to the molecular frame can be expressed as a product of two corresponding Wigner rotation matrices from the lab frame to the rotor frame (0, &, w&) and from the rotor frame to the molecular frame ( yo, 0, 4). Thus, the expansion in Eq. [ 5 ] can be rewritten as A?(‘) =
2
2
1=0,2,4
Dh~o(0, f?,, wrt + yo)Dfnmr(O, 8, 4)&1,(3S;
m,m ‘=-
- S2).
[8]
1
The time-independent terms (m’ = 0) give the quadrupolar effect on the spectral frequency in the fast-spinning case,. The spectral frequency including the chemicalshielding, the dipolar coupling, and the first-order quadrupolar effect for a nitrogen spin state is given by wte~
4)
=
-{
CT i: is0 wc -I- P2(cos 8,)
\J2/3 Dh(O,
0, c$)rEJ
m=-2
- DP2(cos 19,)(3 cos28 - l)m}
-{Jm +
(Ao
+
P2tcos
f-4)
;; m=-2
D2dt0,
4
4b42m
4
+ P4(cos~,)
2
D4&(0, B, I#J)A~,,,}{S(S+
1) - 3m2}.
[91
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CAN AND GRANT
The parameters AI, are given in Eq. [ 61, and Y$2 are the spherical components of the chemical-shielding tensor in the molecular frame. The spectral lineshape for a certain spinning angle can be obtained by a powder average of the transition frequencies for the polar angle (0, 4) over a unit sphere. Figure la shows the simulated lineshape for 13C- 14N MAS spectra using the firstorder quadrupolar effect from Eq. [ 51 and the method of numerical diagonalization. Equation [ 91 explains the features of the spectral lineshape in the simulations. At the magic angle, only quadupolar effect terms depend on the orientation of the molecule (0, 4). The spectral lines for the nitrogen spin states 1+ 1 ) and ) - 1 ) give the same quadrupolar effect and this results in an asymmetric doublet in the MAS spectrum with intensity ratio 2: 1. The height ratio of the asymmetric doublet is 4: 1 because of the difference in the factor (S + 1 )S - 3m2 for m = 0 and m = + 1 (S = 1). Restricted by the spectral resolution, the peak height ratio may be reduced by line broadening. It has been reported that differential line broadening is observed for the transitions corresponding to the different nitrogen spin states in a single crystal. This variation has been ascribed to different proton decoupling efficiencies (II ). Thus, it may be possible to observe a peak height ratio less than 2: 1. Figure 1 also contains the MAS calculated lineshapes using the numerical diagonalization method to compare
I
I
-300
Hz
0
1
300
Hz
FIG. 1. The simulations of the MAS spectral lineshape for a carbon coupled to a quadrupolar nucleus which has a symmetric EFG tensor and its symmetric axis along the dipolar vector. (a) Simulated hneshape obtained using Eq. [ 91 with D * X = 1200 Hz. 1.O MHz. (b) Simulated lineshape obtained by diagonalizing the Hamiltonian of the nitrogen spin to calculate the dipolar coupling energy with D = 1200 Hz and x = 1.0 MHz. (c) Same as b but with D = 400 Hz and X = 3.0 MHz. The comparison of(b) and (c) with (a) shows the higher-order quadrupolar effect.
VARIABLE-ANGLE
SPINNING
DIPOLAR
SPECTRA
527
with the lineshape calculated using the first-order approximation. In Figures lb and lc, different quadrupolar coupling constants are used but the product of the dipolar coupling constant and the quadrupolar coupling constant remains the same as that used in Fig. 1a. The lineshapes for the nitrogen spin states m = - 1 and m = 1 in c are no longer the same as those in a, due to the higher-order quadrupolar effect. When spectral line broadening is present, the higher-order quadrupolar effect can still be neglected even for a ratio X/UN as large as 0.42, the one used in the simulation of Fig. lc. The most directly observed feature of MAS spectra for 13C-14N is the separation of the doublet which is related to the A0 term in eq. [ 61: A=%{(3
cos’fl, - 1) - 9 sin2/YDcos 27,).
[lOI
The doublet splitting provides both structural and molecular information if the other parameters have been determined. For example, amino acids and cyanide compounds have EFG tensors nearly axially symmetric with the principal axes closely aligned along their dipolar vectors. In this instance, a large and well-resolved doublet will give X from D or vice versa ( 7). Most aromatic or olefinic nitrogen compounds have very asymmetric EFG tensors and the orientation of the EFG tensor in the molecular frame varies considerably. Thus, the orientation of the EFG tensor, unavailable from NQR data, can be measured provided the absolute value of the quadupolar coupling constant and the asymmetry of the EFG tensor are known. From Eq. [ 51, it can be seen that each transition for a nitrogen spin state has the same lineshape while the position, width, and height are determined by the factor S( S + 1) - 3m*. A spin coupled to a quadrupolar nucleus S = $ gives a symmetric doublet in the MAS spectrum since the factors S( S + 1) - 3m 2 for m = *i and m = +$ have the same absolute value but a different sign. In the case of variable-angle spinning, the CSA and dipolar coupling dominate the spectral lineshape. The quadrupolar effect distorts the lineshape not only by changing the position of the breakpoint but also by changing the whole lineshape due to the presence of the 1 = 4 terms. As the sample is spun at the angle 70.12” (or 35.56”), P4(cos 0,) = 0. The contribution from the 1 = 4 terms is averaged to zero and the quadrupolar effect shifts only the breakpoints. If the quadrupolar parameters, particularly the orientation of the EFG tensor, are obtained from the MAS spectra, the chemical-shielding and dipolar coupling information can be retrieved from the variableangle spinning spectra. EXPERIMENTAL
SECTION
Cross polarization with variable-angle spinning (VAS) experiments were performed on a Bruker CXP-100 instrument where the 13C Larmor frequency is 25.152 MHz and the 14N Larmor frequency is 7.29 MHz. A homebuilt VAS probe with a spinning speed of 4 kHz was used for the measurements. The spinning angle was adjusted by measuring the scaling factor in the off-magic-angle spinning spectrum of the aromatic carbon lineshape of hexamethylbenzene. The dimethylglyoxime and triethylenediamine were purchased from Fluka and the tetramethylpyrazine was obtained from Aldrich.
528
GAN AND GRANT
The numerical simulations were performed on a VAX 11/750 computer using the POWDER program (12). RESULTS
AND DISCUSSION
Tetramefhylpyrazine. The tetramethylpyrazine molecule is planar and the four aromatic carbons are equivalent due to molecular symmetry. The quadrupolar coupling constant and the asymmetry of the EFG tensor have been measured to be 4.67 MHz and 0.45 at 77 K (13, 14). From molecular symmetry, the three principal axes of the EFG tensor are restricted to being along the N-N axis, perpendicular to the molecular plane, and in the molecular plane perpendicular to the N-N axis. There are a total of six possible assignments of the three principal axes to the three symmetric axes. Using the dipolar coupling constant estimated from the X-ray bond length (IS) and the NQR data, the lineshapes of all six assignments of principal axes were simulated. The simulated MAS lineshape in Fig. 2a gives the assignment of the principal axes of the EFG tensor in the molecule with qZZalong the N-N axis and with qYYperpendicular to the molecular plane. Other assignments of the principal axes are clearly excluded since the simulated lineshapes are distinguishably different from the experimental spectra. As discussed in the previous section, the quadrupolar effect would reflect the
I
,
I
-2 kliz
0
2 kHz
FIG. 2. “C NMR spectra of the aromatic carbon of tetramethylpyrazine spinning at the magic angle (a) and 8, = 70.12” (b) with their best-fit simulations.
VARIABLE-ANGLE
SPINNING
DIPOLAR
SPECTRA
529
MAS lineshape if the sign of the quadrupolar coupling were changed. The nonsymmetric experimental lineshape in Fig. 2a gives a negative sign for the quadrupolar coupling constant of tetramethylpyrazine. Figure 2 also presents the off-magic-angle spinning spectrum of tetramethylpyrazine at & = 70.12” and its best-fit simulation using the orientation of the EFG tensor determined from the MAS spectrum. The principal values of the chemical-shielding tensor in the best-fit simulation are a, = 243 ppm, a,+, = 160 ppm, and u,, = 45 ppm. The orientations of the principal axes are: bzz is perpendicular to the aromatic ring, bYYis close to the tangent of the ring and forms an angle of 32” from the C-N bond, and a, is in the plane of the ring and is generally oriented radially. The dipolar coupling constant is measured to be 890 Hz, corresponding to a bond length of 1.38 A. This compares with an X-ray bond length of 1.334 A (15). Dimethylglyoxime. Although the molecules of dimethylglyoxime are planar and the two olefinic carbons are equivalent, the nitrogen spins experience less symmetry than in the case of tetramethylpyrazine. Without the aid of symmetry to assign the principal axes of the EFG tensor, the orientation has to be determined by fitting the MAS spectra of the compound. The quadrupolar coupling constant and the asymmetry of the EFG tensor were measured to be 4.97 MHz and 0.68 at room temperature by NQR (13, 16) and one of the principal axes is perpendicular to the molecular plane due to the planar symmetry. Thus, the angle between the principal axis and the dipolar vector and the assignment of the principal axis along the direction perpendicular to the molecular plane describe the orientation of the EFG tensor in the molecule. Using a quadrupolar coupling constant of 4.97 MHz and an asymmetry of 0.68, the MAS lineshapes with different angles between one principal axis and the dipolar vector were simulated with three assignments of the principal axis to the perpendicular direction. Figure 3a presents the MAS spectral lineshape of the olefinic carbon of dimethylglyoxime and its bestfit calculated lineshape. The best-fit simulation gives the orientation of the EFG tensor with the qzzaxis at a 34” angle to the dipolar vector and with the qY,,axis perpendicular to the molecular plane. The sign of the quadupolar coupling constant is determined to be negative. The spectral lineshape, particularly the separation of the asymmetric doublet, is very sensitive to the angle between the qzz axis and the dipolar vector, which ensures the accuracy of the orientation determination. Figure 3b contains the off-magic-angle spinning (& = 70.12” ) spectrum of dimethylglyoxime and the best-fit simulation using the EFG tensor orientation obtained from the MAS spectral lineshape. The best-fit simulation gives a dipolar coupling constant of D = 930 Hz and the three principal values of the chemical-shielding tensor of a, = 26 1 ppm, uYY= 147 ppm, and uzz = 60 ppm. The orientation of the chemicalshielding tensor obtained from the best-fit simulation specifies that the u,, axis be perpendicular to the molecular plane and u,,~forms a 12” angle to the dipolar vector. Because of the cylindrical symmetry around the dipolar axis, there are two possible orientations for the chemical-shielding tensor and the EFG tensor even though the magnitudes of these angles are determined. From the result by Hsieh et al. (16), the orientation is determined to be the one shown in Fig. 5. The C-N bond length from the dipolar coupling constant is 1.33 A and is about 4% longer than the bond length of 1.27 A obtained from X-ray measurements ( 17).
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CAN AND GRANT
-1.5
kHz
0
FIG, 3. 13CNMR spectra of the olefinic carbon of dimethylglyoxime 8, = 70.12” (b) with their best-fit simulations.
1.5 kHz
spinning at the magic angle (a) and
Triethylenediamine. The molecule of triethylenediamine has a C3” symmetry axis along the N-N direction. All six carbons are equivalent in one molecule. The angle between the C-N bond and the N-N axis is 70.16” and the C-N bond length is 1.46 A, which corresponds to a dipolar coupling constant of 700 Hz ( 18). The EFG tensor is symmetric due to the CjV symmetry and the quadrupolar coupling constant has been measured to be 4.92 MHz at a temperature of 77 K by NQR (13, 19). The motion of the molecule in the solid state has been studied by the relaxation of i3C and 14N at temperatures from 355 to 408 K (20). It was found that both the motion around the C3” axis and the reorientation of the axis were occurring in this temperature range but that the reorientation of the CjV axis was more hindered than the motion around this axis. The MAS spectrum of triethylenediamine at room temperature shown in Fig. 4a gives a well-resolved doublet with intensity ratio of 2: 1. The simulated MAS lineshape obtained by assuming that the molecule is static is similar to the lineshape calculated for a molecule undergoing rapid rotation around the CsV symmetry axis. Due to the spectral resolution, the experimental results of these two cases are sufficiently similar that it is impossible to distinguish between them. As discussed in the previous section,
VARIABLE-ANGLE
SPINNING
r
-1 KHz
DIPOLAR
531
SPECTRA
I
0
IKHz
FIG. 4. (a) The MAS spectrum of triethylenediamine at room temperature compared with the best-fit simulation assuming that the molecule has rapid axial motion around the Csv axis (bottom trace) and the simulation assuming no molecular motion (top trace). (b) Same as a but with a spinning angle of 70.12”.
the most visible feature of the asymmetric doublet due to the quadrupolar effect is the separation of the doublet, which is determined in Eq. [lo]. In the static case, the formula in Eq. [lo] gives a geometrical factor of (3 cos2a! - 1)/2 to the quadrupolar effect term, where CYis the angle between the dipolar vector and the N-N axis. Assuming the molecule undergoes rapid motion around the CsV symmetry axis, the motion averages the C-N dipolar coupling to the N-N direction by a factor of (3 cos2a - 1)/2, which is the same as the geometrical factor in the static case. Therefore, a similar spectral lineshape may be expected for these two cases. In off-magic-angle spinning spectra, the lineshape with molecular motion is much narrower. In Fig. 4b, the spectral lineshapes with and without molecular motion are simulated and compared with the experimental spectra. Since the zeroth-order dipolar coupling becomes dominant in the off-magic-angle spinning spectral lineshape, it is obvious that the molecule has rapid axial motion and that the reorientational motion of the symmetry axis is either prohibited or at least slower than the inverse of the width of the lineshape at room temperature. The best-fit simulation with molecular motion gives D = -220 Hz, the quadrupolar coupling constant has a negative sign, and the motionally averaged chemical-shielding tensor is u,, = 57 ppm and CT~= 43 ppm. Both the dipolar coupling tensor and the chemical-shielding tensor averaged by the molecular motion are axially symmetric and have their symmetry axis along the N-N direction. The effective dipolar coupling constant, -228 Hz, obtained from the bond length in the molecular structure measured by X-ray, suggests that there is no
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GAN AND GRANT
significant librational motion of the symmetric axis at room temperature which would reduce the dipolar coupling constant by a factor of sin 28iib, where Biibis the mean angle of the librational motion (21). The orientations of the EFG tensor and chemical-shielding tensor for tetramethylpyrazine and dimethylglyoxime are described in Fig. 5. CONCLUSIONS
The quadrupolar effect on NMR dipolar spectra of spin-4 nuclei coupled to a quadupolar nucleus has been solved analytically to a first-order approximation. The frequency shift due to the quadrupolar effect can be expanded in terms of Wigner matrix elements up to I = 4. Under sample spinning, these terms are scaled by factors of P2( cos 0,) and P4(cos (3,). In the magic-angle spinning case, the 1 = 4 terms in the quadrupolar effect give an asymmetric doublet in the high-resolution spectra. The fine structure lineshape gives critical information on the EFG tensor, particularly the sign and the orientation of the tensor, which is unavailable from NQR studies of a powder sample. With the knowledge of the EFG tensor and its orientation in the molecule, both the dipolar coupling constant and the chemical-shielding tensor of the carbon coupled to the nitrogen can be obtained from the off-magic-angle spinning spectra. Also, off-magic-angle spinning improves both resolution and sensitivity compared to those in the stationary sample experiment.
'\H FIG. 5. The orientations of 14N EFG tensor and “C chemical-shielding tensor for the molecules of (a) tetramethylpyrazine and (b) dimethylglyoxime.
VARIABLE-ANGLE
SPINNING
DIPOLAR
533
SPECTRA
In summary, the quadrupolar effect in sample-spinning dipolar NMR spectra, treated both theoretically and experimentally, provides structural and molecular information such as the orientation of the EFG tensor. The dipolar coupling constant and the chemical-shielding tensor along with its orientation can be obtained by variable angle sample-spinning experiments for the carbon coupled to a nitrogen nucleus. APPENDIX
The Wigner matrix elements with one zero projection are related to the spherical harmonics by
XlJt~,8,Y)= t-1)”
d
&
r,(P,71,
which gives the expressions, for 1 = 4, D&(0,8,
4) = $ (35 cos40 - 30 cos28 + 3)
D&(0,8,
4) = 4ti (7 cos30 - 3 cos8)ei6
D&(0,
0, 4) = 8fi
sin20(7 cos28 - l)e2’+
D&(0,
Ei 8, 4) = 4
sin36 cos 8e3’@
D&(0,
8, C#J)=
d
35 . i-g sm40e4i4
ACKNOWLEDGMENT This work GM08521-29.
was supported
by a NIH
research
grant
from
the Institute
of General
Medical
Sciences,
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. Y.
J. A. D. A. J. A. A. N. U.
SCHAEFER AND E. 0. STUESKAL, .I. Am. Chem. Sot. 98, 1031 ( 1976). PINES, M. G. GIBBY, AND J. S. WAUGH, J. Chem. Phys. 59,569 ( 1973). L. VANDERHART, G. S. GUTOWSKY, AND T. C. FARRAR, J. Am. Chem. Sot. 89,5056 NAITO, S. GANAPATHY, AND C. A. MCDOWELL, J. Magn. Reson. 48, 367 ( 1982). G. HEXEM, M. H. FREY, AND S. J. OPELLA, J. Chem. Phys. 77,3847 ( 1982). NAITO, S. GANAPATHY, AND C. A. MCDOWELL, J. Chem. Phys. 77, 5393 ( 1981). C. OLIVIERI, L. FRYDMAN, AND L. E. DIAZ, J. Magn. Reson. 75, 50 ( 1987). K. SETHI, D. M. GRANT, AND R. J. PUGMIRE, J. Magn. Reson. 71,476 (1987). HAEBERLEN, “High Resolution NMR Spectroscopy Selective Averaging,” Academic
( 1967).
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GAN
10. M. II.
K.
12. D. 13. G. 14. L. 15. D. 16. Y. 17. L. 18. G. 19. P. 20. R. 21. K.
AND
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York, 1976. (Page 10 of this reference should reed pZr2 = (R, - R,,,)/2 = -11612. The negative sign has been omitted.) M. PINTAR, “NMR Basic Principles and Progress,” Vol. 13, “Introductory Essays,” Springer-Verlag, Berlin/Heidelberg/New York, 1976. TAKEGOSHI, A. NAITO, AND C. A. MCDOWELL, Magn. Reson. Chem. 24, 544 ( 1986). W. ALDERMAN, M. S. SOLUM, AND D. M. GRANT, J. Chem. Phys. 84,3717 ( 1986). K. SEMIN, T. A. BABUSHKINA, AND G. G. YAKOBSON, “Nuclear Quadrupole Resonance in Chemistry.” Wiley, New York, 1975. GUIBE AND E. LUCKEN, Mol. Phys. 10,273 (1966). T. CROMER, J. Phys. Chem. 61,254 (1957). N. HSIEH, P. S. IRELAND, AND T. L. BROWN, J. Magn. Reson. 21,445 ( 1976). L. MERRITT JR. AND E. LANTERMAN, Acta Crystallgr. 5, 8 11 ( 1952). S. WEISS, A. S. PARKES, E. R. NIXON, AND R. E. HUGHES, J. Chem. Phys. 41, 3759 ( 1964). HAIGH AND L. GUIBE, Compt. Rend. 261, 2328 (1965). E. WASYLISHEN, B. A. PETTITT, AND R. Y. DONG, J. Chem. Sot. Faraday Trans. 2 76,57 1 ( 1980 ). W. ZILM AND D. M. GRANT, J. Am. Chem. Sot. 103,2913 ( 198 1).
OF MAGNETIC
RESONANCE
w,522-534
( 1990)
Molecular and Structural Information from Variable-Angle Spinning NMR Dipolar Spectra of 13C-14NSystems ZHEHONG GAN AND DAVID M. GRANT* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Received April 6, 1990 A sample-spinning solid-state NMR study of a spin-1 nucleus coupled to a quadrupolar nucleus is presented. Using a simple approach, the quadrupolar effect is expanded in terms of irreducible spherical tensor components up to I = 4. Information on the electric field gradient tensor orientation, dipolar coupling, and chemical-shielding tensor for the ‘%“‘N system can be obtained experimentally from the magic-angle and the off-magic-angle sample-spinning spectra. Satisfactory results are obtained by comparing the simulations with the experimental spectra of tetramethylpyrazine, dimethylglyoxime, and triethylenediamine. 0 1990 Academic Press. Inc.
Cross polarization of dilute spins from proton magnetization combined with magicangle spinning (CP/MAS) constitutes the main technique for obtaining high-resolution NMR spectra of organic solids ( I, 2). An interesting feature observed in MAS spectra of a carbon directly bonded to a nitrogen is the splitting of the carbon line into an asymmetric doublet while other carbon lines continue to remain narrow. In a 13C14N coupled spin pair, the quantization direction of the nitrogen spin is not strictly along the external magnetic field since the quadrupolar interaction of the nitrogen spin has a magnitude comparable to that of the Zeeman interaction. Failure to achieve the alignment of the quadrupolar nuclear spin with the external magnetic field can make a difference in the spectra of a spin coupled to a quadrupolar nucleus, and this quadrupolar effect is manifested in the fine structure lineshape of the MAS spectra. Spectral features, such as the separation of the asymmetric doublet, are directly related to the dipolar coupling and to the magnitude and the orientation of the electric field gradient (EFG) tensor of the nitrogen in the molecule. Vanderhart et al. (3) have explained this quadrupolar effect theoretically by calculating the eigenstates of the quadrupolar nucleus under the presence of both the Zeeman and the quadrupolar interaction for each orientation of the molecule with respect to the external magnetic field. In the past, several studies (4-7) have been initiated to obtain information on the quadrupolar interactions and on the dipolar coupling from the MAS spectral lineshape of carbons coupled to nitrogens. Although the MAS technique gives relatively high-resolution spectra, information on the chemical-shift anisotropy (CSA) cannot be recovered from such spectra. Spinning at a variable angle other than the magic angle has been used to obtain CSA * To whom correspondence should be addressed. 0022-2364190 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights ofreproduction in any form reserved.
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VARIABLE-ANGLE
SPINNING
DIPOLAR
SPECTRA
523
information from the powder pattern which is scaled by a factor of ( 3 cos’& - 1 )/ 2 (8). With the proper selection of the spinning angle, variable-angle spinning can therefore improve the resolution by separating the overlapping lineshapes from different carbon spins. For a coupled ’ ‘C- 14N system in the variable-angle spinning experiment, the spectra are also scaled except for the part due to the quadrupolar effect term. Thus, the quadrupolar effect under different spinning angles may be studied as a way to retrieve CSA and dipolar data from spectral lineshapes. In this paper, we present a simple analysis of the quadrupolar effect when the quadrupolar interaction is small compared to the Zeeman interaction. The first-order nonsecular terms in the rotating frame give an analytical solution for the transition frequencies in both the stationary and the sample-spinning cases. Experimentally, tetramethylpyrazine and dimethylglyoxime were studied to capitalize on the large separation of the lines from the carbon spin coupled to the nitrogen and on the magnitudes of these quadrupolar interactions as determined by other techniques. Triethylenediamine provides an interesting case for study since the molecular motion at room temperature alters the lineshape and can be analyzed with this technique. THEORY
The Hamiltonian of a nuclear spin system in terms of the irreducible spherical tensor operators is given by i% = c c x
I
;
(-l)“R;-,T;m,
111
m=-I
where X represents the type of interaction. For the 13C(I)- 14N( S) system, X includes Z, and Zc, the Zeeman interactions for the carbon and the nitrogen spins (on the order of 10 MHz) ; Q, the quadrupolar interaction of the nitrogen spin (on the order of MHz); D, the dipolar interaction between the carbon and the nitrogen spins; and CS, the chemical shielding of the carbon spins (the later two terms are both on the order of kHz). The rotational property of the irreducible spherical tensor operators, R,, , and the construction of the bases spin operator, T,,,, , of the irreducible representation have been given by Haeberlen (9). The same convention as that of Haeberlen is used in this work. The rotating frame of both the carbon and nitrogen spins at their Larmor frequencies is characterized by the rotation operator
and the Hamiltonian
is given by
&f R = RtiR-’
+ @R-j dt
[21
524
GAN
AND
GRANT
For most r3C- 14N cases, the Zeeman term of the nitrogen spin in the Hamiltonian is large compared to the quadrupolar interaction. The time dependence of the Hamiltonian in Eq. [ 21 can be removed by applying average Hamiltonian theory ( 10). The average Hamiltonian includes the terms which do not commute with the Zeeman Hamiltonian. These terms contribute to the spectral frequency in higher orders. The portion of the average Hamiltonian which contains the carbon spin operator and which commutes with the Zeeman Hamiltonian is given to first order by SC = #NV + SC’), (31 where
The zeroth-order term, S(“, includes the secular terms of the chemical shielding and the dipolar interaction. The .X’ (I) term gives the first-order quadrupolar effect. The results obtained with Eq. [ 3] are equivalent to those obtained using first-order perturbation theory to calculate the eigenstates of the nitrogen spin under both the quadrupolar interaction and the Zeeman interaction ( 7). It is well known that average Hamiltonian theory has a close relationship to perturbation theory. For example, the secular approximation is equivalent to the zeroth-order perturbation of the eigenstates ( 10). In Eq. [ 3 1, RF:, , the irreducible tensor components in the lab frame, are related to those in the molecular frame by
where the Dht,,,( (Y, 6, y ) are the Wigner rotation matrix elements. In our case, the principal-axis frame of the dipolar coupling tensor is well defined in the molecule and is selected to be the molecular frame. From Eqs. [ 31 and [ 41, the term characterizing the quadrupolar effect can be expanded as s%(l) =
C /=0,2,4
;: D!,&O, /9, y)A1,ZZ(3St m=-I
- S2>,
[51
where the nonzero A,, elements are 1 rforfo- ’ 5q.J
Am = - -
1 ~~~~ --$‘rQ.
7oN
20
20,
DQ A40 = - l2 r20r20; 35WN
A 211
=..--
A 4+1
=
1 DL? r20r2+1; 14WN 2,k D Q r20r2&1; 35WN
-
A2&2
= G
AJr2 = g
I
DQ r20r2*2;
r%%+,.
[61
VARIABLE-ANGLE
SPINNING
DIPOLAR
525
SPECTRA
Explicit expressions are given for D2mjm( (Y, /3, y) in the book by Haeberlen (9) and for D4ti(0, ,6, y) in the Appendix. In the molecular frame, the components rim are related to the dipolar coupling constant and quadrupolar coupling tensor by rfo=-
f 60 VZX
Q-
y20 - 8S(2S - 1)
Y2q1 + = - ,,~~~~
{ 3 cos2pD - 1 - 7) sin2PDcos 27,)
1) {+3sin2PD+9
&,+ = - ,;;;fl,
sin 2PDsin 27~ + 2i?7sin BDsin 2y,J
sin2PD + q( 1 + cos2~D)cos 2~~
{-3
* 2irj cos PDsin 2yD},
[ 71 where D is the dipolar coupling constant, X is the quadrupolar coupling constant, and n is the asymmetry of the EFG tensor, which take the familiar form D
=
YcYd
x
4a*r3
-
e*Qq h
’
’
9YY -
t=
4xX
9 Zi
s
The quantities qxx, qyy, and qzz are the principal values of the EFG tensor with qz2 = eq; CQ, /LID,and 70 are the Euler angles for the rotation from the molecular frame to the EFG principal frame; and POand yD are the polar angles of the dipolar vector in the EFG tensor principal-axis frame. Equation [ 51 gives, in the stationary case, the solution for the quadrupolar effect for a certain orientation of the crystallite with respect to the external magnetic field. In the sample-spinning case, the Wigner rotation matrix from the lab frame to the molecular frame can be expressed as a product of two corresponding Wigner rotation matrices from the lab frame to the rotor frame (0, &, w&) and from the rotor frame to the molecular frame ( yo, 0, 4). Thus, the expansion in Eq. [ 5 ] can be rewritten as A?(‘) =
2
2
1=0,2,4
Dh~o(0, f?,, wrt + yo)Dfnmr(O, 8, 4)&1,(3S;
m,m ‘=-
- S2).
[8]
1
The time-independent terms (m’ = 0) give the quadrupolar effect on the spectral frequency in the fast-spinning case,. The spectral frequency including the chemicalshielding, the dipolar coupling, and the first-order quadrupolar effect for a nitrogen spin state is given by wte~
4)
=
-{
CT i: is0 wc -I- P2(cos 8,)
\J2/3 Dh(O,
0, c$)rEJ
m=-2
- DP2(cos 19,)(3 cos28 - l)m}
-{Jm +
(Ao
+
P2tcos
f-4)
;; m=-2
D2dt0,
4
4b42m
4
+ P4(cos~,)
2
D4&(0, B, I#J)A~,,,}{S(S+
1) - 3m2}.
[91
526
CAN AND GRANT
The parameters AI, are given in Eq. [ 61, and Y$2 are the spherical components of the chemical-shielding tensor in the molecular frame. The spectral lineshape for a certain spinning angle can be obtained by a powder average of the transition frequencies for the polar angle (0, 4) over a unit sphere. Figure la shows the simulated lineshape for 13C- 14N MAS spectra using the firstorder quadrupolar effect from Eq. [ 51 and the method of numerical diagonalization. Equation [ 91 explains the features of the spectral lineshape in the simulations. At the magic angle, only quadupolar effect terms depend on the orientation of the molecule (0, 4). The spectral lines for the nitrogen spin states 1+ 1 ) and ) - 1 ) give the same quadrupolar effect and this results in an asymmetric doublet in the MAS spectrum with intensity ratio 2: 1. The height ratio of the asymmetric doublet is 4: 1 because of the difference in the factor (S + 1 )S - 3m2 for m = 0 and m = + 1 (S = 1). Restricted by the spectral resolution, the peak height ratio may be reduced by line broadening. It has been reported that differential line broadening is observed for the transitions corresponding to the different nitrogen spin states in a single crystal. This variation has been ascribed to different proton decoupling efficiencies (II ). Thus, it may be possible to observe a peak height ratio less than 2: 1. Figure 1 also contains the MAS calculated lineshapes using the numerical diagonalization method to compare
I
I
-300
Hz
0
1
300
Hz
FIG. 1. The simulations of the MAS spectral lineshape for a carbon coupled to a quadrupolar nucleus which has a symmetric EFG tensor and its symmetric axis along the dipolar vector. (a) Simulated hneshape obtained using Eq. [ 91 with D * X = 1200 Hz. 1.O MHz. (b) Simulated lineshape obtained by diagonalizing the Hamiltonian of the nitrogen spin to calculate the dipolar coupling energy with D = 1200 Hz and x = 1.0 MHz. (c) Same as b but with D = 400 Hz and X = 3.0 MHz. The comparison of(b) and (c) with (a) shows the higher-order quadrupolar effect.
VARIABLE-ANGLE
SPINNING
DIPOLAR
SPECTRA
527
with the lineshape calculated using the first-order approximation. In Figures lb and lc, different quadrupolar coupling constants are used but the product of the dipolar coupling constant and the quadrupolar coupling constant remains the same as that used in Fig. 1a. The lineshapes for the nitrogen spin states m = - 1 and m = 1 in c are no longer the same as those in a, due to the higher-order quadrupolar effect. When spectral line broadening is present, the higher-order quadrupolar effect can still be neglected even for a ratio X/UN as large as 0.42, the one used in the simulation of Fig. lc. The most directly observed feature of MAS spectra for 13C-14N is the separation of the doublet which is related to the A0 term in eq. [ 61: A=%{(3
cos’fl, - 1) - 9 sin2/YDcos 27,).
[lOI
The doublet splitting provides both structural and molecular information if the other parameters have been determined. For example, amino acids and cyanide compounds have EFG tensors nearly axially symmetric with the principal axes closely aligned along their dipolar vectors. In this instance, a large and well-resolved doublet will give X from D or vice versa ( 7). Most aromatic or olefinic nitrogen compounds have very asymmetric EFG tensors and the orientation of the EFG tensor in the molecular frame varies considerably. Thus, the orientation of the EFG tensor, unavailable from NQR data, can be measured provided the absolute value of the quadupolar coupling constant and the asymmetry of the EFG tensor are known. From Eq. [ 51, it can be seen that each transition for a nitrogen spin state has the same lineshape while the position, width, and height are determined by the factor S( S + 1) - 3m*. A spin coupled to a quadrupolar nucleus S = $ gives a symmetric doublet in the MAS spectrum since the factors S( S + 1) - 3m 2 for m = *i and m = +$ have the same absolute value but a different sign. In the case of variable-angle spinning, the CSA and dipolar coupling dominate the spectral lineshape. The quadrupolar effect distorts the lineshape not only by changing the position of the breakpoint but also by changing the whole lineshape due to the presence of the 1 = 4 terms. As the sample is spun at the angle 70.12” (or 35.56”), P4(cos 0,) = 0. The contribution from the 1 = 4 terms is averaged to zero and the quadrupolar effect shifts only the breakpoints. If the quadrupolar parameters, particularly the orientation of the EFG tensor, are obtained from the MAS spectra, the chemical-shielding and dipolar coupling information can be retrieved from the variableangle spinning spectra. EXPERIMENTAL
SECTION
Cross polarization with variable-angle spinning (VAS) experiments were performed on a Bruker CXP-100 instrument where the 13C Larmor frequency is 25.152 MHz and the 14N Larmor frequency is 7.29 MHz. A homebuilt VAS probe with a spinning speed of 4 kHz was used for the measurements. The spinning angle was adjusted by measuring the scaling factor in the off-magic-angle spinning spectrum of the aromatic carbon lineshape of hexamethylbenzene. The dimethylglyoxime and triethylenediamine were purchased from Fluka and the tetramethylpyrazine was obtained from Aldrich.
528
GAN AND GRANT
The numerical simulations were performed on a VAX 11/750 computer using the POWDER program (12). RESULTS
AND DISCUSSION
Tetramefhylpyrazine. The tetramethylpyrazine molecule is planar and the four aromatic carbons are equivalent due to molecular symmetry. The quadrupolar coupling constant and the asymmetry of the EFG tensor have been measured to be 4.67 MHz and 0.45 at 77 K (13, 14). From molecular symmetry, the three principal axes of the EFG tensor are restricted to being along the N-N axis, perpendicular to the molecular plane, and in the molecular plane perpendicular to the N-N axis. There are a total of six possible assignments of the three principal axes to the three symmetric axes. Using the dipolar coupling constant estimated from the X-ray bond length (IS) and the NQR data, the lineshapes of all six assignments of principal axes were simulated. The simulated MAS lineshape in Fig. 2a gives the assignment of the principal axes of the EFG tensor in the molecule with qZZalong the N-N axis and with qYYperpendicular to the molecular plane. Other assignments of the principal axes are clearly excluded since the simulated lineshapes are distinguishably different from the experimental spectra. As discussed in the previous section, the quadrupolar effect would reflect the
I
,
I
-2 kliz
0
2 kHz
FIG. 2. “C NMR spectra of the aromatic carbon of tetramethylpyrazine spinning at the magic angle (a) and 8, = 70.12” (b) with their best-fit simulations.
VARIABLE-ANGLE
SPINNING
DIPOLAR
SPECTRA
529
MAS lineshape if the sign of the quadrupolar coupling were changed. The nonsymmetric experimental lineshape in Fig. 2a gives a negative sign for the quadrupolar coupling constant of tetramethylpyrazine. Figure 2 also presents the off-magic-angle spinning spectrum of tetramethylpyrazine at & = 70.12” and its best-fit simulation using the orientation of the EFG tensor determined from the MAS spectrum. The principal values of the chemical-shielding tensor in the best-fit simulation are a, = 243 ppm, a,+, = 160 ppm, and u,, = 45 ppm. The orientations of the principal axes are: bzz is perpendicular to the aromatic ring, bYYis close to the tangent of the ring and forms an angle of 32” from the C-N bond, and a, is in the plane of the ring and is generally oriented radially. The dipolar coupling constant is measured to be 890 Hz, corresponding to a bond length of 1.38 A. This compares with an X-ray bond length of 1.334 A (15). Dimethylglyoxime. Although the molecules of dimethylglyoxime are planar and the two olefinic carbons are equivalent, the nitrogen spins experience less symmetry than in the case of tetramethylpyrazine. Without the aid of symmetry to assign the principal axes of the EFG tensor, the orientation has to be determined by fitting the MAS spectra of the compound. The quadrupolar coupling constant and the asymmetry of the EFG tensor were measured to be 4.97 MHz and 0.68 at room temperature by NQR (13, 16) and one of the principal axes is perpendicular to the molecular plane due to the planar symmetry. Thus, the angle between the principal axis and the dipolar vector and the assignment of the principal axis along the direction perpendicular to the molecular plane describe the orientation of the EFG tensor in the molecule. Using a quadrupolar coupling constant of 4.97 MHz and an asymmetry of 0.68, the MAS lineshapes with different angles between one principal axis and the dipolar vector were simulated with three assignments of the principal axis to the perpendicular direction. Figure 3a presents the MAS spectral lineshape of the olefinic carbon of dimethylglyoxime and its bestfit calculated lineshape. The best-fit simulation gives the orientation of the EFG tensor with the qzzaxis at a 34” angle to the dipolar vector and with the qY,,axis perpendicular to the molecular plane. The sign of the quadupolar coupling constant is determined to be negative. The spectral lineshape, particularly the separation of the asymmetric doublet, is very sensitive to the angle between the qzz axis and the dipolar vector, which ensures the accuracy of the orientation determination. Figure 3b contains the off-magic-angle spinning (& = 70.12” ) spectrum of dimethylglyoxime and the best-fit simulation using the EFG tensor orientation obtained from the MAS spectral lineshape. The best-fit simulation gives a dipolar coupling constant of D = 930 Hz and the three principal values of the chemical-shielding tensor of a, = 26 1 ppm, uYY= 147 ppm, and uzz = 60 ppm. The orientation of the chemicalshielding tensor obtained from the best-fit simulation specifies that the u,, axis be perpendicular to the molecular plane and u,,~forms a 12” angle to the dipolar vector. Because of the cylindrical symmetry around the dipolar axis, there are two possible orientations for the chemical-shielding tensor and the EFG tensor even though the magnitudes of these angles are determined. From the result by Hsieh et al. (16), the orientation is determined to be the one shown in Fig. 5. The C-N bond length from the dipolar coupling constant is 1.33 A and is about 4% longer than the bond length of 1.27 A obtained from X-ray measurements ( 17).
530
CAN AND GRANT
-1.5
kHz
0
FIG, 3. 13CNMR spectra of the olefinic carbon of dimethylglyoxime 8, = 70.12” (b) with their best-fit simulations.
1.5 kHz
spinning at the magic angle (a) and
Triethylenediamine. The molecule of triethylenediamine has a C3” symmetry axis along the N-N direction. All six carbons are equivalent in one molecule. The angle between the C-N bond and the N-N axis is 70.16” and the C-N bond length is 1.46 A, which corresponds to a dipolar coupling constant of 700 Hz ( 18). The EFG tensor is symmetric due to the CjV symmetry and the quadrupolar coupling constant has been measured to be 4.92 MHz at a temperature of 77 K by NQR (13, 19). The motion of the molecule in the solid state has been studied by the relaxation of i3C and 14N at temperatures from 355 to 408 K (20). It was found that both the motion around the C3” axis and the reorientation of the axis were occurring in this temperature range but that the reorientation of the CjV axis was more hindered than the motion around this axis. The MAS spectrum of triethylenediamine at room temperature shown in Fig. 4a gives a well-resolved doublet with intensity ratio of 2: 1. The simulated MAS lineshape obtained by assuming that the molecule is static is similar to the lineshape calculated for a molecule undergoing rapid rotation around the CsV symmetry axis. Due to the spectral resolution, the experimental results of these two cases are sufficiently similar that it is impossible to distinguish between them. As discussed in the previous section,
VARIABLE-ANGLE
SPINNING
r
-1 KHz
DIPOLAR
531
SPECTRA
I
0
IKHz
FIG. 4. (a) The MAS spectrum of triethylenediamine at room temperature compared with the best-fit simulation assuming that the molecule has rapid axial motion around the Csv axis (bottom trace) and the simulation assuming no molecular motion (top trace). (b) Same as a but with a spinning angle of 70.12”.
the most visible feature of the asymmetric doublet due to the quadrupolar effect is the separation of the doublet, which is determined in Eq. [lo]. In the static case, the formula in Eq. [lo] gives a geometrical factor of (3 cos2a! - 1)/2 to the quadrupolar effect term, where CYis the angle between the dipolar vector and the N-N axis. Assuming the molecule undergoes rapid motion around the CsV symmetry axis, the motion averages the C-N dipolar coupling to the N-N direction by a factor of (3 cos2a - 1)/2, which is the same as the geometrical factor in the static case. Therefore, a similar spectral lineshape may be expected for these two cases. In off-magic-angle spinning spectra, the lineshape with molecular motion is much narrower. In Fig. 4b, the spectral lineshapes with and without molecular motion are simulated and compared with the experimental spectra. Since the zeroth-order dipolar coupling becomes dominant in the off-magic-angle spinning spectral lineshape, it is obvious that the molecule has rapid axial motion and that the reorientational motion of the symmetry axis is either prohibited or at least slower than the inverse of the width of the lineshape at room temperature. The best-fit simulation with molecular motion gives D = -220 Hz, the quadrupolar coupling constant has a negative sign, and the motionally averaged chemical-shielding tensor is u,, = 57 ppm and CT~= 43 ppm. Both the dipolar coupling tensor and the chemical-shielding tensor averaged by the molecular motion are axially symmetric and have their symmetry axis along the N-N direction. The effective dipolar coupling constant, -228 Hz, obtained from the bond length in the molecular structure measured by X-ray, suggests that there is no
532
GAN AND GRANT
significant librational motion of the symmetric axis at room temperature which would reduce the dipolar coupling constant by a factor of sin 28iib, where Biibis the mean angle of the librational motion (21). The orientations of the EFG tensor and chemical-shielding tensor for tetramethylpyrazine and dimethylglyoxime are described in Fig. 5. CONCLUSIONS
The quadrupolar effect on NMR dipolar spectra of spin-4 nuclei coupled to a quadupolar nucleus has been solved analytically to a first-order approximation. The frequency shift due to the quadrupolar effect can be expanded in terms of Wigner matrix elements up to I = 4. Under sample spinning, these terms are scaled by factors of P2( cos 0,) and P4(cos (3,). In the magic-angle spinning case, the 1 = 4 terms in the quadrupolar effect give an asymmetric doublet in the high-resolution spectra. The fine structure lineshape gives critical information on the EFG tensor, particularly the sign and the orientation of the tensor, which is unavailable from NQR studies of a powder sample. With the knowledge of the EFG tensor and its orientation in the molecule, both the dipolar coupling constant and the chemical-shielding tensor of the carbon coupled to the nitrogen can be obtained from the off-magic-angle spinning spectra. Also, off-magic-angle spinning improves both resolution and sensitivity compared to those in the stationary sample experiment.
'\H FIG. 5. The orientations of 14N EFG tensor and “C chemical-shielding tensor for the molecules of (a) tetramethylpyrazine and (b) dimethylglyoxime.
VARIABLE-ANGLE
SPINNING
DIPOLAR
533
SPECTRA
In summary, the quadrupolar effect in sample-spinning dipolar NMR spectra, treated both theoretically and experimentally, provides structural and molecular information such as the orientation of the EFG tensor. The dipolar coupling constant and the chemical-shielding tensor along with its orientation can be obtained by variable angle sample-spinning experiments for the carbon coupled to a nitrogen nucleus. APPENDIX
The Wigner matrix elements with one zero projection are related to the spherical harmonics by
XlJt~,8,Y)= t-1)”
d
&
r,(P,71,
which gives the expressions, for 1 = 4, D&(0,8,
4) = $ (35 cos40 - 30 cos28 + 3)
D&(0,8,
4) = 4ti (7 cos30 - 3 cos8)ei6
D&(0,
0, 4) = 8fi
sin20(7 cos28 - l)e2’+
D&(0,
Ei 8, 4) = 4
sin36 cos 8e3’@
D&(0,
8, C#J)=
d
35 . i-g sm40e4i4
ACKNOWLEDGMENT This work GM08521-29.
was supported
by a NIH
research
grant
from
the Institute
of General
Medical
Sciences,
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. Y.
J. A. D. A. J. A. A. N. U.
SCHAEFER AND E. 0. STUESKAL, .I. Am. Chem. Sot. 98, 1031 ( 1976). PINES, M. G. GIBBY, AND J. S. WAUGH, J. Chem. Phys. 59,569 ( 1973). L. VANDERHART, G. S. GUTOWSKY, AND T. C. FARRAR, J. Am. Chem. Sot. 89,5056 NAITO, S. GANAPATHY, AND C. A. MCDOWELL, J. Magn. Reson. 48, 367 ( 1982). G. HEXEM, M. H. FREY, AND S. J. OPELLA, J. Chem. Phys. 77,3847 ( 1982). NAITO, S. GANAPATHY, AND C. A. MCDOWELL, J. Chem. Phys. 77, 5393 ( 1981). C. OLIVIERI, L. FRYDMAN, AND L. E. DIAZ, J. Magn. Reson. 75, 50 ( 1987). K. SETHI, D. M. GRANT, AND R. J. PUGMIRE, J. Magn. Reson. 71,476 (1987). HAEBERLEN, “High Resolution NMR Spectroscopy Selective Averaging,” Academic
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10. M. II.
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12. D. 13. G. 14. L. 15. D. 16. Y. 17. L. 18. G. 19. P. 20. R. 21. K.
AND
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York, 1976. (Page 10 of this reference should reed pZr2 = (R, - R,,,)/2 = -11612. The negative sign has been omitted.) M. PINTAR, “NMR Basic Principles and Progress,” Vol. 13, “Introductory Essays,” Springer-Verlag, Berlin/Heidelberg/New York, 1976. TAKEGOSHI, A. NAITO, AND C. A. MCDOWELL, Magn. Reson. Chem. 24, 544 ( 1986). W. ALDERMAN, M. S. SOLUM, AND D. M. GRANT, J. Chem. Phys. 84,3717 ( 1986). K. SEMIN, T. A. BABUSHKINA, AND G. G. YAKOBSON, “Nuclear Quadrupole Resonance in Chemistry.” Wiley, New York, 1975. GUIBE AND E. LUCKEN, Mol. Phys. 10,273 (1966). T. CROMER, J. Phys. Chem. 61,254 (1957). N. HSIEH, P. S. IRELAND, AND T. L. BROWN, J. Magn. Reson. 21,445 ( 1976). L. MERRITT JR. AND E. LANTERMAN, Acta Crystallgr. 5, 8 11 ( 1952). S. WEISS, A. S. PARKES, E. R. NIXON, AND R. E. HUGHES, J. Chem. Phys. 41, 3759 ( 1964). HAIGH AND L. GUIBE, Compt. Rend. 261, 2328 (1965). E. WASYLISHEN, B. A. PETTITT, AND R. Y. DONG, J. Chem. Sot. Faraday Trans. 2 76,57 1 ( 1980 ). W. ZILM AND D. M. GRANT, J. Am. Chem. Sot. 103,2913 ( 198 1).