Deuteron NMR on single crystals. EFGs, signs of quadrupole coupling constants, and assignment of σ tensors in malonic acid

JOURNAL

OF MAGNETIC

RESONANCE

47,

221-239 (1982)

Deuteron NMR on Single Crystals. EFGs, Signs of Quadrupole Coupling Constants, and Assignment of g Tensors in Malonic Acid CARMEN MILLER, Department

of Molecular

S. IDZIAK,* N. PI~LEWSKI,* Physics,

Max-Planck-Institute 6900 Heidelberg,

for Germany

AND U. HAEBERLEN Medical

Research,

Jahnstrasse

29,

Received July 2, 1981 The high-field NMR of two dipolar-coupled deuterons experiencing different EFGs is investigated theoretically and shown to be solvable analytically. The determination of the signs of quadrupole coupling constants from the spectral appearance is discussed. Room-temperature FT deuteron spectra of deuterated crystals of malonic acid were recorded at 54.7 MHz. The dipolar fine structure resulting from the coupling of the deuterons of the CD2 group could be resolved. From the rotation patterns of the line splittings, the EFGs at the sites of the four inequivalent deuterons are determined and compared with the previous work of Derbyshire et al. The results confirm the conclusion, drawn in a previous paper on the proton NMR of malonic acid, that the methylene deuteron internuclear vector and the normal of the C-C-C backbone of the molecule subtend an angle of about 6’. The signs of the quadrupole coupling constants of the methylene deuterons are shown to be positive. The correlation of chemical shifts with quadrupole splittings allowed to assign directly the chemical shift tensors measured previously by multiple-pulse proton NMR to the methylene hydrogen sites. 1. INTRODUCTION

In the high-temperature regime NMR spectra governed by nuclear quadrupoleelectric field gradient (EFG) interactions are invariant against a change of sign of the quadrupole coupling constant QCC = e*qQ/h, where eQ and eq = V,, are, respectively, the nuclear quadrupole moment and the largest component of the EFG tensor at the site of the nucleus in question. In most measurements of EFG tensors the sign of V,, remains, therefore, undetermined. Although there is a little doubt (I) that for a deuteron bound to a carbon Vzz > 0, and hence QCC > 0 since eQ > 0 (2) it is desirable to check this expectation by experiment. An opportunity to do this arises when two deuterons 1 and 2 are coupled by dipolar interactions and when the difference of their quadrupolar splittings Au, and Au2 happens to be comparable to or smaller than the dipolar splitting D. In such a case the dipolar fine structure of a resonance line is not symmetric with respect to its center of gravity. From the asymmetry of the dipolar structure of the spectra the relative sign of the dipolar and quadrupolar interactions can be inferred. Since the * On leave from the Institute of Molecular Physics of the Polish Academy of Sciences, Poznan, Poland. 221 0022-2364/82/050227-13$02.00/O Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved

228

MijLLER

ET AL.

sign of the dipolar interactions can be determined from the orientation of the crystal in the applied field B,, the absolute sign of the QCC follows. Equal quadrupolar splittings of two coupled deuterons may be the result of crystal symmetry, e.g., when deuterons 1 and 2 are related by an inversion center (3). Another phenomenon which leads to Au, = +Av2 is rapid site exchange, e.g., of the deuterons of “flipping” water molecules in hydrates (4, 5) or of the deuterons in methyl groups (7). In all these cases only the limiting case of exactly equal quadrupolar splittings can be studied. In the case of two coupled deuterons with Au, = +Av2 the dipolar fine structure consists of 1:3:2 asymmetric triplets (4). In the other limiting case, /Au, - Avzl B D, the spectra consist of four symmetric 1: 1: 1 triplets centered at the positions of the resonances of the uncoupled deuterons 1 and 2. To our knowledge intermediate cases have not been studied so far. They may be observed if the degeneracy of the quadrupolar splittings is not the consequence of natural phenomena (symmetry, motion) but is produced artificially by the experimenter through aligning a crystal in the applied field in such a way that Au, x Avz. By changing the orientation of the crystal in small steps it is possible to scan all situations from IAvr - Au,/ % D to Au, - Au2 = 0 and to observe the changeover of the dipolar structure of the spectra from the one to the other characteristic limiting case. In a reinvestigation of the EFGs at the sites of the deuterons in deuterated malonic acid CD2(COOD)2 we observed for some orientations of the crystals spectra with structure arising from the dipolar interactions of the methylene deuterons. The observation of such a structure is not mentioned in the classic paper of Derbyshire et al. on the EFGs in malonic acid (8). With a properly oriented crystal we could observe the changeover of the dipolar structure of the spectra from the limiting case of symmetric 1:l: 1 triplets to the 1:3:2 asymmetric triplets. From the latter the sign of I’,, at the sites of the methylene deuterons could be determined to be positive. In Section II we treat briefly the NMR of two spin-l nuclei in a large magnetic field coupled by dipolar interactions. We point out that not only the case I’$$ = V$$, treated by Chiba (4b), but also the case I$& = --I’& leads to mixing of the states of the uncoupled nuclei and to characteristic features in the spectra. In Section III we describe the determination of the sign of I’,, at the sites of the methylene deuterons in deuterated malonic acid. In Section IV we compare the EFGs which we obtained in malonic acid with Derbyshire’s results and draw some conclusions about the positions of the deuterons in the crystal lattice. Section V is about deuteron chemical shifts. It is pointed out that the correlation of chemical shifts with quadrupole splittings allows a straightforward assignment of chemical shift tensors which is not possible in multiple-pulse determinations of proton chemical shift tensors. II. THEORY

The system we consider consists of two coupled spin-l magnetic field &. It is described by the Hamiltonian

nuclei in an external

DEUTERON x

=

NMR

221

+

x22

OF +

MALONIC

ZQl

+

229

ACID

xQ2

+

x01.2

[II

,

represent, respectively, the Zeeman and where 2~1, zz2; zQ19 xQ2, and ~D,.z quadrupolar interactions of spins 1 and 2, and their dipolar coupling. Our experiments are done at a Larmor frequency of 54.7 MHz. This is large enough to justify restriction to the so-called secular parts of ‘%?Q and XD. The various terms in Eq. [I] are then given by 22

=

‘%@Q

= XQ, +

2?Dl.2

=

XZl

+

zz2

=

all,,

+

%Q,

= A,(3&

dz2

- 1:) + A,(3&

- (1/4)(&G

~[~Zl~ZZ

>

- c),

[la1

+ C&)1,

with, for i = 1, 2

A,

e2Qiqi I 7(3 =

COS*

Bi - 1 + ?Jisin2 0i COS 2’Pi),

(1 - 3 cos2 &), where 19,~= X(Bo, r12), r12 = ]r12]. Bi, ‘Pi are the polar angles of Be in the principal axes system of the ith EFG tensor. The remaining symbols have their usual meaning. Chiba has diagonalyzed this Hamiltonian for two limiting cases, ]A,] - IA21 %-D, a, = u2, and A, = AZ, al = a2 and has calculated the resulting spectra. In the absence of relative chemical shifts of nuclei 1 and 2, i.e., al = u2, the inherent symmetry of the Hamiltonian allows the problem to be easily solved analytically for arbitrary values of A,, AZ, and D. This can be seen as follows. Choose as a base the set of spin functions ]m,m& where ml and m2 are the eigenvalues of Z,, and Zz2,respectively; ml,* = - 1, 0, 1. Z, = Z,, + Zr2 commutes with &“, the eigenfunctions of A? may, therefore, be classified by A4 = ml + m2, the eigenvalue of Z,. Note that Xz + &%?Q (but not &” = Xz + &“Q + zD) also commutes with Z$, + c2: t(zZ

+

ZQ),

(El

+

c2)]

=

0,

[21

The quantum number associated with c, + c2 we call 2’. It classifies the three eigenfunctions of Xz + AYo with A4 = 0 into a doublet with 2’ = 2 and energy (A, + AZ) and a singlet with 2’ = 0 and energy -2(A, + A2). From the two degenerate states (1, - 1) and (-1, 1) we form linear combinations %o,z2=2 = cp;, = $

(]I, -1)

+ I-1, 1))

PaI

and %=O,Z2=2

=

‘pi%

= &

(11,

-1) - I-1, 1)).

Values (Pi2 and ‘P$ are, respectively, symmetric and antisymmetric interchange of nuclei 1 and 2. Value ‘PO0= 10, 0) is symmetric.

[3bl with respect to If we now add

230

MtiLLER

ET AL.

TABLE THE

HAMILTONIAN

[I]

M = 52

I+1 2 I)

I+1 + I)

f2la

M=l

IlO)

101)

110)

IOl)

a t A, - 2A> -D/2

-D/2 a - 2A,

M=O

II - 1)

1%

I1 - 1) W l-11)

A, + A2 - D -D/2 0

-WI

‘All

t

A,

t

MATRIX

1

IN THE Imlms)

REPRESENTATION”

A2 + D M=

t

-I

IO- 1) A>

I-10)

IO-

1)

-a - ZA, -Df2

1-q

t

A2

-D/2 --(1 t A,

2A2

l-11)

-D/2

+ Ad

-D/2

0 -D/2 A, + A2 - D

matrix elements MI shown are ZCTO.

ZD, which is also symmetric with respect to interchange of nuclei 1 and 2, to Zz + X’o, group theory tells that (P& can be mixed by flD to neither Pt2 nor ‘PO0 = (P,,,=o;.~=o.This means (Pt2 must be an eigenfunction of A? = Zz + Zo + XD This is easily verified by directly evaluating XPo2. We conclude that &“D can mix only the states and 11, 0) IO, 1>7 I-1,0) 62

and

IO, -I>,

and

(doe.

The diagonalization of A? requires, therefore, only the solution of quadratic equations. Table 1 gives the A? matrix, Table 2 the eigenvalues and eigenfunctions of A?. The states 11, 0) and 10, 1), and j-1, 0) and IO, -1) are mixed significantly by xD only when A, - A2 x D (see Fig. la), whereas P& and P, are mixed significantly when A, + A2 = D (see Fig. 1b). Note that A, and AZ may have opposite signs because of the angular factors involved, even if the corresponding QCCs have equal signs. The three different situations JA,J - lAzl $ D, A1 - AZ x D, and A, + A2 FZ D lead to different characteristic features in the dipolar structure of the NMR spectra; see Figs. 2a-g. Only the right halves of the complete spectra (which are necessarily symmetric about VJ are shown. The spectra for (AlI - lAzl & D and for Al = A2 = A, sign A = f sign D (Figs. 2b, c) and the possibility to derive from the latter two the relative signs of A and D have been discussed by Chiba (4) and will not be further commented on. Three spectra for Al + A2 N D, IAll, iA21 $ D are shown in Figs. 2e-g. In Fig. 2e, a value of l/3 was chosen for the parameter S = (A, + A,)/D, because in this case complete mixing of Pt2 and PO0occurs; see Fig. lb. The lines at the center of the spectrum (remember, we are talking of only the right half of the total spectrum) have weak intensities. The opposite is true if S = - l/3. This feature can be used as a fingerprint of the relative signs of D and Al and A2: Suppose that by rotating a crystal about a certain axis we pass continuously from the situation IAll > IAll to IAll < IA?/. By comparing the spectrum obtained for exact degeneracy, IAll = lAzl with the spectra shown in

DEUTERON

NMR OF MALONIC

231

ACID

Figs. 2c, d, and f we will be able to tell whether we are dealing with the case A, = A2 or with Al = -A*. Suppose we are dealing with the latter. If, on rotating the crystal, we first encounter a spectrum of type 2c, then, after passing through exact degeneracy, a spectrum of type 2g and if IA,( and IA21vary in the following sequence: first IAll > )Azl, then (AlI = \A*[, finally IAll < IAl1 (this we can tell from the “rotation pattern”) then we may conclude that sign A, = sign D. If we observe the spectra 2e, f, and g in the reverse order and all other things are equal, we may conclude sign D = sign A2 (#sign A,). It is thus possible to exploit the spectral features in both cases, A, - A2 N D and A, + A2 = D for determining the relative signs of D and A,, AZ. For IAll - IA21 b 2101 two triplets are obtained. The center line of the inner triplet is, depending on whether sign (A, + A?) = sign D or sign (A, + AZ) = -sign D, larger or smaller than the outer two lines of this triplet; see Figs. 2a, b. The opposite is true of the outer triplet. Only for (A,[ - IAJ % D are 1: 1: 1 triplets obtained.

TABLE THE

EIGENVALUES

2

AND EIGENFUNCTIONS

OF THE HAMILTONIAN

Eigenvalues

Eigenfunctions

E, = 2a + A, + A2 + D

*I = Ill)

E2 = a - $A,

+ A,)

- y

$2 = 410) - Pm)

E, = a - $A,

+ AZ) + y

$3 = PIlO) + 401)

E4 = - ;(A,

+ AZ) - f D - ;

ES = - ;(A,

+ A2) - ; D + f

$5 = 69& t y’pw

Eg = A, + A2 - D

$6 = (pa02

E7 = -a

- $A,

+ AZ) + y

J/l = al0 - 1) + P/-IO)

E8 = -a

- :(A,

+ AZ)

$8 = 010 - 1) - al-IO)

Eg = -2a U = [9(A,

a = [-3(A,

’ =[-3(A,

- t

+ A, -t A2 + D - A*)*

[l]

+ D2]“*

ti9 = I-1 - I) W = [9(A,

D - AZ) + U2 + @I”’ -3(A, - AZ) + U - AZ) + C? + D2]“*

= [-3(A, b=

[-3(A,

+ A$

+ A,)

+ 3D* - 6D(A,

+ AZ)]“’

D(2)“’ + D + IV* + 2d]“2

-3(A, + AZ) + D + W + AI) + D + W’* + 2D2]1’2

232

MijLLER

ET AL.

R=/A,-A,)/D

-

S = (A, + A,)/D FIG. I. The mixing + Az)/D, respectively. III.

coefficients

DETERMINATION METHYLENE

(Y, fl (top)

and

OF THE SIGNS DEUTERONS

y,

-

6 (bottom)

vs R = (A,

OF THE QCCs AT IN MALONIC

THE ACID

- A,)/D

SITES

and

S = (A,

OF THE

FT deuteron spectra from perdeuterated single crystals of malonic acid were recorded at 54.7 MHz at room temperature using the apparatus described in Ref. (9). Figure 3 shows the angular variation of the deuteron resonances for rotating

FIG. 2. Stick spectra for two coupled spin-l nuclei fqr various values of A,, AZ, and D leading characteristic spectra. Note that only for [A,/ - IA21 @ IDI 1:l:l triplets are obtained.

to

DEUTERON

NMR OF MALONIC

ACID

233

h E ; c '50

0

-50

-100

---01 @orbox,il

-

~--Dzlcaboxyfl

- -04

D3lmethylenei lmethylene I

FIG. 3. Angular variation of the deuteron resonance positions for rotating a crystal of malonic acid about its c axis. The labeling of the deuterons is as in (I 1).

a crystal about its c axis. The rotation axis is perpendicular to b. Such patterns were also recorded from crystals which were rotated about their a* and b* axes, respectively. At ambient temperature T, of the deuterons in malonic acid varies, depending on the orientation of the crystals, between 0.5 and 3 sec. The accumulation of not more than 10 FIDs already gave excellent S/N, therefore, the spectrometer time required for collecting the data was not more than a few hours. From the rotation patterns of the line splittings we determined by standard fit procedures the EFGs at the sites of the four inequivalent deuterons in malonic acid. The EFGs are easily assigned by comparing the directions of the principal axes of the measured EFG tensors with the C-D and O-D directions in the crystal. This implies that we can tell which of the traces in Fig. 3 belongs to which deuteron. The line splittings of the methylene deuterons D4 and D3, respectively, pass through maxima near gC= 15 and Cp= 123”. There, the line splittings are within 5% of the absolute maximum of the line splittings. It follows that sign AD~,P=w = sign QCCD3 and sign AD4,v=150= sign QCCD4 . The traces of D3 and D4 cross near Cp= 70 and V = 160”. Near the crossing at (0 = 160” the spectra exhibit pronounced, well-resolved fine structure attributable to the dipolar coupling of the methylene deuterons D3 and D4 In this crossing region Bc, is almost along the vector r,, j oming ’ D3 and D4; hence the dipolar coupling strength D is near its maximum value. This region, therefore, appears to be very favorable for observing the changes in the dipolar fine structure of the spectra on approaching degeneracy, IAD = IAD4(. Unfortunately, however, the line from the carboxylic deuteron D, happens to overlap with the methylene resonances just in the crossing region and spoils the most interesting spectra. By computer

234

MijLLER

ET AL.

FIG. 4. Deuteron FT spectra of a single crystal of perdeuterated malonic acid from the crossing region of the methylene resonances. Only the right halves of the spectra are shown. The two single lines at the right-hand side of the spectra are from the carboxyl deuterons. The crystal is rotated about the (3i6) axis of the reciprocal lattice. A+’ indicates the increments of the rotation angle; rL = 54.7 MHz.

simulation of rotation patterns we searched, therefore, for a more suitable rotation axis in the neighborhood of c. The 3i6 direction in the reciprocal lattice turned out to be acceptable. Accordingly a sample crystal was prepared with this direction along the rotation axis. Figure 4 shows a series of spectra from the crossing region of the methylene resonances from this crystal. The lines from the carboxyl deuterons are well separated from the methylene resonances. The spectrum shown in Fig. 4e corresponds most closely to the case of complete degeneracy, J&1 = IAD4). A barely visible shoulder on the r.h.s. of the large component is all that remains from the small component of the 2:3:1 triplet in Fig. 2d. A rotation of the crystal by only kO.2” results in the spectra 4d and f which have a completely different appearance than spectrum 4e. Despite the fact that the resolution of the dipolar structure is rather poor in the spectra shown in Fig. 4 (D is only about 0.62(&,,/) the characteristics of the case sign A n3 = sign AD.,, sign (Am/D) = -1 are borne out in remarkably fine details. In spectrum 4a, (AD3 - &,)/ID1 is as large as about nine,

DEUTERON

NMR OF MALONIC

ACID

235

nevertheless the center line of the outer triplet is clearly larger than its counterpart from the inner triplet, just as expected from the stick spectra shown in Fig. 2. IDI = 0.62)0,,,) implies D < 0, hence, sign AD3 = sign AD4 = fl for all spectra shown in Fig. 4. The signs of An3 and An4 do not change between the orientations corresponding to the spectra 4a-g and the orientations which give the maximum splittings of the resonances of D3 and D4 (this we can infer from our computer simulations of rotation patterns); hence we may conclude that the quadrupole coupling constants of the methylene deuterons D3 and D4 in malonic acid are positive. It would, of course, be interesting to look for orientations (0, cb) of B,, relative to a crystal fixed reference frame for which AD3 = -AD4 and to see whether we can observe spectra displaying the characteristic features of the case JAmI - IAD4/ G D, sign A,, = -sign AD4, cf. Figs. 2e-g. A continuous set of such (0, ‘P) values does exist in malonic acid. We calculated it numerically at intervals of ACP= 10”. For each such (0, P) pair we also calculated the dipolar splitting D. It turns out that IDI is always <0.46)0,,,~. Unfortunately, for most of the orientations for which A = -AD4 we get interference with the lines from the carboxyl deuterons. This is?rue of all the AD3 = -AD4 orientations which exist for the crystals which we had already oriented. It turns out that there are no AD3 = -AD4 orientations for which there is no interference between the carboxyl and methylene resonances and for which IDI > 0.3~0,,,~. In view of the considerable width of the fine-structure component lines a splitting IDI f 0.3~0,,,) was considered to be too small to warrant the effort of orienting another crystal. We point out that avoiding the interference problem by selective deuteration of only the methylene group is not possible in the case of malonic acid for chemical reasons (10). It is, however, conceivable that spectra of the type of Figs. 2e-g can be observed from CD2 groups in other compounds or from (nonflipping) DzO molecules. IV. REDETERMINATION

OF THE EFGs AT THE HYDROGEN

SITES IN MALONIC

ACID

The EFG tensors derived from the rotation patterns of the deuteron line splittings are shown in Table 3 together with those of Derbyshire et al. (8). The most important point to note is that there is excellent overall agreement between this work and Ref. (8). Only a few minor differences are worth mentioning. In (8) a significant difference between the QCCs of the two CD2 deuterons is reported. Such a difference is not supported by our data. The principal directions e, found in this and the earlier work coincide very closely. The major aspect of this point is that it gives credit to the accuracy of crystal orienting in both works. Larger differences show up in the directions of e, and e,. This is not surprising in view of the small asymmetry factors n which leave the e, and e, directions less well defined than the e, directions. It seems that the x and y axes given in (8) for site 3 are interchanged with respect to ours. This, again, is not surprising since the error limits of q3 in (8) are consistent with q, = 0, which leaves the x and y axes of the corresponding EFG tensor completely undefined. To Derbyshire’s discussion of the EFGs we would like to add two remarks. First, if we assume, as is widely done in the literature, that e, coincides with the C-D (or O-D) bond direction and if we interpret the equality of the QCCs at sites 3 and 4 as providing evidence that the C-D, and C-D4 bonds have equal

236

MijLLER

DEUTERON

EFG

ET AL.

TABLE

3

TENSORS

IN MALONIC

Quadrupole coupling constant (kHz)

Asymmetry factor

Deuteron site’

This workd

Ref. (8)

This workd

1

179.1

179.9

0.112

0.125

* 0.2

f 0.2

2 0.002

If: 0.002

180.9

181.9

0.098

0.103

f 0.2

+ 0.2

k 0.002

f 0.002

165.0

165.0

0.024

0.010

+ 0.2

+ 2.0

* 0.002

* 0.003

165.2

168.1

0.037

0.038

+ 0.2

2 0.02

?I 0.002

+- 0.002

2

3

4

ACID

Polar angles of principal direction@ in SO-system’ This work

Ref. (8)

Ref. (8)

e, ey e,

37.0 102.4 124.2

353.3 66.4 327.8

40.7 109.8 123.8

7.64 72.7 328.7

e, ey e,

116.6 43.9 121.9

308.5 7.1 56.7

112.5 40.5 121.7

312.5 13.6 57.1

e, ey e,

85.8 35.5 54.8

55.9 151.8 323.0

36.6 88.4 53.4

148.5 56.3 325.1

e, ey e,

90.1 34.3 55.7

324.0 234.1 53.9

72.5 39.8 55.7

312.0 199.7 54.5

u Sites 1 and 2 are the carboxyl, sites 3 and 4 the CD2 sites. The labeling of the deuterons is as in Ref. (II). The assignment of the EFG tensors to the various deuterons is based on a comparison of the e, directions with bond directions. * Our convention is 1V,,l * IV,,] 2 IVyyI. ‘The standard-orthonormal (SO-) system is defined as follows: x/la, zIIc*, y in a-b plane of the crystal. d These error limits follow from the assumption that the error limits of the measurements of all principal values of the QC tensor are +0.2 kHz.

lengths, we can determine the direction of the vector rDD which connects D3 and Dq. In a previous paper on the proton NMR in malonic acid (II) we determined the direction and the length of the corresponding vector rH,. The surprising point was that rHH subtends an angle of (6 +- 2)” with the normal to the plane spanned by the C1-Co-C2 backbone of the molecule. This result is now confirmed by the deuteron EFG measurements: the directions of rDD and rHH differ by only 1.5”, which is well inside the combined error limits of the respective measurements. Neglecting isotope effects of the geometry of the methylene group we may now calculate the C-H bond length, using rHH = ( 1.76 + 0.02) A from the proton data and <(D3C,,D4) = *(e,(3), e,(4)) = 108.0” from the deuteron data. The result is rceH = 1.09 A. A ssuming that e,(3) and e,(4) deviate by not more than lo from the respective bond direction--this is a reasonable assumption-and taking the uncertainty of lrHHl into account, the error limits of Ire-HI are t-O.025 A. Second, Looyestijn et al. (12) have distilled from experimental data an expression which relates in linear (and almost linear) O-D * * -0 hydrogen bonds the O-D distance to the QCC of the deuteron,

DEUTERON

NMR

OF MALONIC

a

237

ACID

QCWHzl = Rb-D

b

with a = 533 kHz k’ and b = 327 kHz. By solving Eq. [4] for Rome and inserting the measured values for the QCCs of deuterons D, and D2 we get Ro-D = 1.01(6) A. The differences obtained for D, and D2 are not significant. The lengths of the hydrogen bonds formed by D, and D2 are, respectively, 2.68 and 2.71 A. The value of the O-D distance derived from the deuteron QCCs fits well into the diagram of O-D vs 0-D. * -0 distances prepared by Olovsson et al. (13). V. .CHEMICAL

SHIFT

Figure 5 shows two deuteron spectra where chemical shift differences are obvious from displacements of the centers of gravity of different line pairs. The displacements are only detectable because the S/N ratio is so high in the spectra. They remain smaller than the linewidths. With regard to measuring chemical shift tensors deuteron NMR on single crystals is thus inferior to proton multiple-pulse NMR where the chemical shift differences amount to many linewidths (1 I ). There is, however, one aspect that makes the observation of deuteron chemical shifts very attractive. It is the fact that the chemical shifts which can be observed are correlated with quadrupolar interactions. This has implications with regard to assigning chemical shifts. In proton NMR the assignment of chemical shift tensors 0 is often a difficult and sometimes a not unambiguously solvable problem (I I, 14). On the

..:.t-71 .. .,..,.

.

-. . .. . .. -:

FIG. indicate

5. Two centers

. . :.4

:

.

. . :-T..

. ;....-.

. . . . . . .a’ .

;i:’

,‘. / . : . :,.

deuteron spectra of malonic acid showing chemical shifts. The of line pairs. Note that the shifts are smaller than the linewidths.

vertical

lines

near

vL

238

MijLLER

ET

AL.

other hand, the assignment of deuteron EFG tensors is usually straightforward because of the close parallelism of the unique axis of the EFG tensor and the direction of the chemical bond involved. Because this is so the assignment of deuteron chemical shifts is also straightforward. Deuteron NMR can thus be used to resolve ambiguities in the assignment of Q tensors which remain in proton multiplepulse studies. An example in case is malonic acid. In (II) we have determined all the proton u tensors in this compound. The size of the shielding anisotropy allowed unambiguously to tell the carboxyl from the methylene u tensors. The directions of the unique shielding axes al!owed, in addition, to assign with certainty the two carboxyld tensors to “their” sites. On the other hand, no feature of the methylene u tensors gave a hint about their correct assignment. For both possible alternatives the principal shielding directions match the local (near) symmetry equally well (or poorly). One implies that the C-H bond direction is approximately the most-, the other that it is approximately the least-shielded direction. In order to solve this assignment problem experimentally, i.e., without invoking intuition or chemical experience, the chemical shifts must be correlated with other, well-understood properties of the nuclei involved. In (11) we tried to correlate the chemical shifts with dipolar interactions by tracing very small differences of linewidths in the experimental spectra back to differences of dipolar interactions. This led us to propose that the assignment with the most-shielded axes along the bond direction is the correct one. Another approach to correlate dipolar interactions with chemical shifts is 2D spectroscopy (15). We have carried out such an experiment (16). The resolution along the axis which displays dipolar couplings was good enough to distinguish carboxyl and methylene sites. It was, however, not good enough to differentiate between the two methylene sites. A definite solution of the assignment problem became possible with deuteron NMR. The most informative spectra in the present context are those with B,, along either the C-D, or the CD, bond direction. These cases we encounter in the rotation pattern shown in Fig. 3atcPx 13and’P N 123”. The line pairs with the largest splittings correspond to those deuterons with B0 parallel to the bond. By carefully measuring the centers of gravity of all the line pairs in the corresponding spectra it could be established beyond any doubt that the center of gravity of the line pair with the largest splitting is shifted upfield with respect to the centers of gravity of all the other pairs. As expected from the proton data, the shift is considerably smaller than the linewidth, on the other hand, it is about an order of magnitude larger than the second-order quadrupole shift which, at vLarmor= 54.7 MHz, does not exceed l/2 ppm. This finding confirms the assignment of the proton u tensors proposed in (I). The bond direction of a methylene hydrogen in malonic acid is (approximately) the mostshielded direction. ACKNOWLEDGMENTS We would like to thank H. Zimmermann for synthesizing and growing the deuterated malonic acid. The idea that we could check the proposed assignment of the proton chemical by deuteron NMR came up in discussion with A. M. Achlama. N. PiHlewski and S. Idziak hospitality and stipends from the Max-Planck-Gesellschaft.

crystals of shift tensors acknowledge

DEUTERON

NMR OF MALONIC

239

ACID

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