The shape and information content of high-field solid-state proton NMR spectra of methyl groups

ARTICLE IN PRESS

Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

The shape and information content of high-field solid-state proton NMR spectra of methyl groups Peter Gutsche,a Monika Rinsdorf,b Herbert Zimmermann,b Heike Schmitt,b and Ulrich Haeberlenb,
b

a SAP AG, Neurottstr. 16, 69190 Walldorf, Germany Max-Planck-Institut fu¨r Medizinische Forschung, Jahnstrasse 29, 69120 Heidelberg, Germany

Received April 30, 2003

Abstract The possible variation of the lineshape of the high-field 1 H spectrum of methyl groups is explored by simulation and experiment. The spectrum of an isolated methyl group depends, apart from the orientation of the applied field B0 relative to the C3 -axis of the group, on its rotational tunnel frequency nt and on its stochastic reorientation rate k: For quantitative analyses, the directional mobility of the C3 -axis must also be taken into account. A distinct but frequently occurring case arises when the methyl groups come as pairs of magnetically equivalent close neighbours. For the experiments, single crystals of four compounds I–IV were grown that were isotopically substituted such that they contained protons only in the methyl positions. The crystal symmetry of all compounds I–IV allowed us to record spectra with all methyl groups being orientationally and otherwise equivalent. I; acetonitrile in deuterated hydroquinone, represents the case of a well-isolated methyl group with a ‘‘high’’ tunnel frequency nt : Its spectrum is (almost) independent of the temperature T: In II; monomethyl malonic acid, nt is comparable in size with the strength of the intramolecular dipolar H–H interaction. All seven theoretically expected lines in the 1 H spectrum are clearly resolved in the spectra of II: nt can be inferred with an uncertainty of only 7300 Hz: nt ðTÞ is found to possess a (flat) maximum near 40 K: Compound III; l-alanine, allows the study of the case of a methyl group with an extremely low, although nonzero tunnel frequency (nt E3 kHzÞ while IV; dimethylglyoxime, represents the case of a close pair of equivalent methyl groups. Its spectrum reflects intriguing structural implications. r 2003 Elsevier Inc. All rights reserved.

1. Introduction Our subject is an old one; it is almost as old as NMR itself. Already in 1950, Andrew and Bersohn (AB) calculated the single crystal and powder sample proton NMR spectrum of a motionless as well as of a frequently reorienting methyl group [1]. They compared their results with powder spectra from 1,1,1-trichloroethane (90 K; supposedly motionless methyl group) and acetonitrile (93 K; frequently reorienting methyl group) that were recorded the year before by Gutowsky and Pake [2]. The reason why we return to this subject is that almost all the previous methyl group solid-state proton NMR experiments were carried out on powder samples [3–5] for which much of the wealth of


Corresponding author. Fax: +49-06221-486-351. E-mail address: [email protected] (U. Haeberlen). 0926-2040/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0926-2040(03)00061-4

information contained in principle in the spectra is buried under the ‘‘powder distribution’’. Methyl group spectra recorded from single crystals are scarce in the literature and the few examples we know of [6–8] by no means exhaust the full range of possible different cases. The methyl group spectra are governed by three, in practice, however, by only two dynamic parameters (see below), namely the (effective) rotational tunnel frequency nt and the rate k of stochastic reorientations of the group about its approximate C3 -axis. Both nt and k depend on the temperature T and, depending on the circumstances, may be accessed by analysing the spectra. Here, we present proton NMR spectra recorded at nL ¼ 270 MHz of single crystals of four compounds I–IV that were selected such that each highlights a particular special case. These compounds are I acetonitrile, N  C–CH3 ; trapped in the inclusion compound hydroquinone (perdeuterated), II monomethyl malonic acid, CH3 DCðCOODÞ2 ; III l-alanine,

ARTICLE IN PRESS 228

P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

CH3 DCðCOO ÞNDþ and IV dimethylglyoxime, 3 ðCH3 CNODÞ2 : As indicated, all these compounds were isotopically labelled such that protons are (nominally) present exclusively in the methyl group(s). Either the crystal symmetry or the special choice of the orientation of the sample crystal relative to the applied field B0 always made sure that the observed spectrum is that of a set of equivalent methyl groups. Acetonitrile in perdeuterated hydroquinone is special in two respects. First, the methyl protons are exceptionally well isolated from other nuclei with strong magnetic moments. This feature of I leads to an exceptionally good resolution of the structure of the solid-state proton NMR spectra. Second, for T ¼ 0 K up to the decomposition temperature of I (well above room temperature) either nt or k; or both nt and k; are considerably larger than the strength b of the intra-group dipolar coupling of the protons. The consequence of this feature is that the proton NMR spectrum is virtually independent of T: Methylmalonic acid, by contrast, highlights the situation for which nt ðT ) 0Þ is comparable in size with b: Consequently, the spectral shape strongly depends on T and enables us to measure the temperature dependence of nt with unprecedented precision. l-Alanine represents the case where nt ðT ) 0Þ is exceptionally low, although nonzero. Dimethylglyoxime is a representative of the rather frequently occurring case where two magnetically equivalent methyl groups related by an inversion centre of the crystal form a near-neighbour pair with strong intermolecular dipolar couplings. The structure information contained in the spectra of especially compounds I and IV reveals that molecular vibrations and librations affect the spectra in a significant way. Finally, we cannot resist showing a wide line proton NMR spectrum of a single crystal of aspirin (acetylsalicylic acid). The richness of the resolved structure and the strong asymmetry of this spectrum contradict in a striking manner the wisdom of classic NMR textbooks according to which wide line solid-state proton NMR spectra ought to possess a gaussian shape and thus to be symmetric and structureless. This wisdom was, e.g., one of the starting points of the renowned Anderson–Weiss theory [9,10].

2. Review of theoretical background Understanding die 1 H spectrum of an isolated methyl group progressed in four major steps. The first was carried out, as indicated above, by AB. The Hamiltonian that they considered for the immobile methyl group consisted, apart from the Zeeman term, of the secular 23 31 part HD;sec ¼ HD;M¼0 ¼ H12 D;0 þ HD;0 þ HD;0 of the dipolar Hamiltonian HD of three immobile spin-1/2 nuclei located at the corners of a general triangle. For

the rapidly reorienting methyl group they took the time average of HD;sec ; /HD;sec ðtÞS: Of real interest is only the case where the nuclei are located at the corners of an equilateral triangle. It is then convenient to also introduce in addition to the quantum number I (having values 3/2 and 1/2) of the total spin and that of its z-component, M; the symmetry labels G ¼ A; Ea and Eb : A spin function of symmetry A; Ea and Eb ; respectively, is an eigenfunction of the cyclic spin permutation operator Ps with eigenvalue 1; e ¼ expð2pi=3Þ and e
; respectively. Using these labels we have Eb Ea HD;sec ¼ HA 20 þ H20 þ H20

ð1Þ

with HA 20 HE20a HE20b

rffiffiffi 3 3 cos2 b  1 A T20 ; ¼ b 2 2 rffiffiffi 3 3 2 2if Ea ¼ b sin be T20 ; 2 4 ¼ ðHE20a Þ


ð2Þ

and G 12 23 31 ¼ uT20 þ vT20 þ wT20 T20

ð3Þ

with u; v; w ¼ ð1; 1; 1Þ; ð1; e
; eÞ; ð1; e; e
Þ for G¼A; Ea ; Eb qffiffi m0 2 kl 3 and T20 ¼ 23 fIzk Izl  14 ðIþk Il þ Ik Iþl Þg: b ¼ 8p 2 gH _=rHH is a measure of the strength of the intra-group dipolar coupling. For a ‘‘standard’’ methyl group with tetra( implying hedral H–C–H bond angles and rCH ¼ 1:093 A ( we have b ¼ proton–proton distances rHH ¼ 1:779 A; bs ¼ 21:3 kHz: The angle b is the angular distance of the applied field B0 from the C3 -axis of the methyl group, f is the azimuth of B0 in the plane of the protons. Because it will turn out that the spectra are independent of f it is not necessary to specify a reference direction for f: The most important feature of the matrix of HD;sec in the jGIMS basis is that matrix elements of the type /12 32 A j HE20a j Eb 12 12S are nonzero for ba0: This is Ea because the operator T20 produces in the ket the labels G ¼ A; I1 ¼ ð2 þ 12Þ ¼ 52 (not possible in a three-spin-1/2 system) then I2 ¼ ð2 þ 12  1Þ ¼ ð2  12Þ ¼ 32 (bottom of the angular momentum addition ladder), and finally M ¼ 12: The labels A; I2 and M match those of the bra, hence, the matrix element can be, and actually is nonzero. The consequence is that HD;sec mixes the jEa;b 12 12S and jA 32 12S states. On the other hand, any G matrix element of the type /12 12 G0 jT20 jG0012 12 S must vanish because I2 does not match I in the bra, regardless of G0 ; G and G00 : A closer look at the matrix elements reveals that all dipolar matrix elements formed with the linear combination jE 12 12S :¼ p1ffiffi2 fe2if jEb 12 12S  2if 11 e jEa 2 2Sg vanish. This implies that only its orthogonal companion function jE 012 12S mixes with the jA 32 12S state. Something analogous is true, of course, of the E

ARTICLE IN PRESS P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

(a)

AB, 1950, k=0 HZ + H D

HZ M

229

A, I=3/2

I=1/2

3/2 Ea

A Eb

A/E’

E

-1/2

A/E’

E

-3/2

A

1/2

β =90˚

(b)

AB, 1950, k>>b HZ + H D A

(c)

Ea

Eb

Ea

Eb

AC, 1968, νt >>b , any k HZ + H D + H t A

β =90˚

h νt

Fig. 1. Energy level diagrams and spectra of isolated methyl groups. (a) Immobile methyl group. E and E 0 are l.c. of Ea and Eb ; only E 0 mixes with the corresponding A; M ¼ 712 state. (b) Rapidly reorienting methyl group. The time average of HD;sec contains only the HA 20 term which cannot mix A; Ea and Eb states. (c) Methyl group with ‘‘large’’ tunnel frequency nt : The degeneracy of the A; M ¼ 712 states with the corresponding Ea;b states is lifted, no mixing occurs and the spectrum is indistinguishable from that of a rapidly reorienting methyl group. By showing DM ¼ 71 transitions with different line styles that are repeated beneath corresponding lines in the spectra, an effort is made to show which line arises from which transition.

and E 0 states with M ¼ 12: Hence, the level diagram and the spectrum appears as shown in Fig. 1(a). For a rapidly reorienting methyl group, f jumps 2p randomly between f0 ; f0 þ 2p 3 and f0  3 ; the average 2p of expð2ifÞ; expð2iðf þ 3 ÞÞ and expð2iðf  2p 3 ÞÞ is zero, hence the time average /HD;sec ðtÞS contains only the A T20 term which cannot mix the E a;b and A states. Therefore, the level diagram and spectrum of the rapidly reorienting methyl group appears as shown in Fig. 1(b). The intermediate case of a methyl group group that reorients with an arbitrary rate k can be treated by standard classical methods [11]. The results are contained in what we are going to present below. Although already in their 1950 paper, AB mention the possibility of methyl group tunneling (spelled with a single l) it took 18 more years until Apaydin and Clough (AC) found an ingenious way to include tunnelling (spelled with two l’s) into the calculation of the methyl group proton NMR spectrum [12]. The crucial step was to express the spatial part HR of the Hamiltonian of the methyl group, considered to be a rigid uniaxial rotor in a

threefold potential V ðfÞ; by an approximate equivalent spin Hamiltonian Ht : What is neglected in the approximation is minuscule and the form of Ht is ð0Þ extremely simple: Ht ¼ 13 hnt ðPs þ P1 s Þ: The replacement of HR with Ht is made possible by restricting attention to only the three lowest states jA0 S; jEa0 S and jEb0 S of HR ; the index 0 indicating the ground torsional level of the rotor. The jEa S and jEb S states of each torsional level n are degenerate (they form a Kramers ð0Þ doublet) so that only the energy difference hnt between a;b jA0 S and jE0 S enters Ht : The restriction just mentioned obviously makes sense at low temperatures where in an ensemble of methyl groups at thermal equilibrium the overwhelming majority of the groups will occupy the lowest torsional level. However, as thermal fluctuations mix the jAn S and the jEna;b S states separately to a large extent, the concept of AC also works at intermediate temperatures with the only ð0Þ difference that nt be replaced with the temperatureðnÞ dependent thermal average /nt Sth ¼: nt ðTÞ; to be called the effective tunnel frequency. After replacing

ARTICLE IN PRESS 230

P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

HR with Ht ; the calculation of the 1 H spectrum exactly follows the procedure of AB for the immobile methyl group, and the result is again a symmetric seven-line spectrum with, however, modified positions and intensities of the lines, see Table 1 of [12]. Two features of the result are worth mentioning. (i) For nt comparable in size with b; nt can be inferred from the 1 H spectrum with remarkable precision. This will be exploited in Section 4. (ii) If, on the other hand, nt =bb1; i.e., if the tunnel frequency is large, there is no more mixing between the Ea;b and the A states with equal M and the 1 H spectrum is indistinguishable from that of a rapidly reorienting methyl group, cf. Fig. 1(c). In this respect, the 1 H spectrum differs in a striking manner from the 2 H and the 13 C spectrum of, respectively, a CD3 and a 13 CH3 group, see [13,14]. The origin of the difference is the following: in the 1 H case, nonzero dipolar matrix elements between any pair of jE a;b IMS states do not exist as explained above because all these states come with I ¼ 1=2: By contrast, corresponding matrix elements of the secular quadrupolar Hamiltonian HQ;sec (also coming with a spin tensor operator T20 ) of a CD3 ; and of HD;sec of a 13 CH3 group do not vanish because the jE a;b IMS states in these larger spin systems come with total spin quantum numbers I41=2: This circumstance makes the above-invoked angular momentum addition ladder longer and allows a match of the labels in the bra and the ket. So far, the development led to an understanding of the 1 H spectra of immobile, jumpwise reorienting, and tunnelling methyl groups. For quite some time, people had difficulties accepting the idea that a methyl group may undergo stochastic reorienting jumps and simultaneously be in a delocalized state that is implied by tunnelling. Guided by intuition, Clough wrote down in 1976 a Liouvillian (note, it could not be a Hamiltonian!) that is adequate to this situation [15, Eq. (16)]. Expressed in the form of a spin-density-matrix equation and using our notation it reads dR=dt ¼  i_1 ½ðHZ þ HD;sec þ Ht Þ; R 1 þ kðPs RP1 s þ Ps RPs  2RÞ:

ð4Þ

k is the rate of stochastic jumps of the methyl group rigid rotor. This, although heuristic, combination of tunnelling (Ht in Eq. (4)) with stochastic reorientation (last term in Eq. (4)) constitutes the third large step in our understanding of the 1 H spectra of methyl groups. The fourth and final step was carried out by Szyman´ski who put Eq. (4) on a sound quantum mechanical basis [16]. In so doing he discovered that Eq. (4) must be supplemented by a term that has no classical analogue and therefore cannot be guessed intuitively. Likewise, he showed that a proper description of the dynamics of the methyl group requires, in addition to the tunnel frequency nt ; the introduction of

two rate constants kK and kt : Szyman´ski’s extension of Eq. (4) reads dR=dt ¼  i_1 ½ðHZ þ HD;sec þ Ht Þ; R 1 þ kK =3fPs RP1 s þ Ps RPs  2Rg þ ðkt  kK Þ=2fURU  Rg;

ð5Þ

where U ¼ 1=3f1  2ðPs þ P1 s Þg is a unitary, selfinverse matrix. kK relates to transitions between the Kramers doublet jEa S and jEb S while kt relates to transitions between the jAS and the jE a;b S states separated by the (tunnel) energy hnt : Clearly, if kt ¼ kK ; and setting kK =3 ¼ k; Eq. (5) becomes identical to Eq. (4). By supplementing the Hamiltonian in Eq. (5) by the quadrupolar term HQ ; Eq. (5) also applies to CD3 groups. The relevance of the nonclassical term ðkt  kK Þ=2fURU  Rg and the necessary distinction between kK and kt have been demonstrated by Szyman´ski through an extremely careful iterative lineshape analysis of quadrupolar echo 2 H spectra recorded in our lab by Olejniczak from a single-crystal sample of acetylsalicyclic acid CD3 (aspirin) [17]. The effects induced by this term on the 1 H spectra of protonated methyl groups are, however, too small to be observed presently. In our simulations of 1 H spectra we therefore neglect this term and effectively use Eq. (4). It follows from the analysis outlined above that 1 H spectra of (isolated) methyl groups depend on the three quantities b; nt and k: The distance rHH of the protons and, hence, b should not vary significantly from one compound to another. For b ¼ 0 ; i.e., for B0 parallel to the C3 -axis of the methyl group, Eqs. (2) state that E H20a;b ¼ 0; which implies that the spectrum is independent of both nt and k: It is a 1:2:1 triplet with a splitting of the outer pair of lines of 3b (=63:9 kHz for a ‘‘standard’’ methyl group). For ‘‘large’’ k (4108 s1 ; say) and/or ‘‘large’’ nt (4107 Hz; say) the spectrum is, as explained above, a 1:2:1 triplet for all values of b (except for b ¼ bm ¼ 54:7 ; where it is a singlet). For b ¼ 90 ; the splitting of the outer pair is 3=2  b: If neither k nor nt is ‘‘large’’, the most interesting situation for the spectroscopist arises if b ¼ 90 : In Fig. 2(a) we show how the spectrum develops for this value of b if k ¼ 0 and nt increases stepwise. It should be clear from this figure that nt can be inferred with great precision from such spectra provided that nt is larger than the width of an individual spectral component and smaller than an upper limit of roughly 100 kHz; set by the detectability of the outer satellites and thus depending on the sensitivity of the spectrometer. However, as the comparison of the nt ¼ 70 kHz and 10 MHz spectra in Fig. 2(a) shows, the splitting of the inner pair of satellites also carries information about the tunnel frequency. This source of information allows to access nt up to about 400 kHz; albeit with reduced accuracy and the measurement requires a precise knowledge of b:

ARTICLE IN PRESS P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

νt

k/ s−1

k / s−1

0 kHz

3 . 10

3

3 . 10

3

10 kHz

3 . 10

4

3 . 10

4

30 kHz

3 . 10

5

3 . 10

5

70 kHz

3 . 10

6

3 . 10

6

10 MHz

3 . 10

7

3 . 10

7

(a)

(b)

k=0

νt =70 kHz

(c)

231

νt =30 kHz



Fig. 2. Simulations of methyl group spectra for b ¼ 90 and different combinations of the dynamic parameters k and nt : The simulations are based on Eq. (4). The Cþþ program library G was used. The spectral range shown is 175 kHz.

Fig. 2(b) and (c) show how the spectrum develops if the tunnel frequency is fixed at 70 and 30 kHz; respectively, and k increases from 3  103 (‘‘slow’’) to 3  107 s1 (‘‘very fast’’). The interesting feature of Fig. 2(b) is the broadening of the three inner spectral components, in particular that of the central line. As Fig. 3 shows, this broadening is largest when k ¼ 2nt : It may be observed experimentally up to about nt ¼ 5 MHz: Tunnel frequencies larger than that classify as ‘‘large’’ and cannot be quantified by analysing the highfield 1 H spectrum. Note that the series of spectra in Fig. 2(b) and (c) do not reflect typical temperature dependences of methyl group spectra because along with k; the tunnel frequency nt will vary with temperature as well, see Section 4.

3. Experimental 3.1. NMR All spectra to be reported were recorded with the homemade 270 MHz multiple pulse spectrometer that in the past we have used intensively to study proton chemical shift tensors in single crystals [18]. Two modifications were found necessary for the present project. (i) A new probe had to be designed and constructed that could be cooled down to about 5 K in an Oxford Instruments helium flow cryostat. To keep the unavoidable background signal as low as possible, care was taken that no material possibly containing protons was near the sample coil. (ii) While for the former multiple pulse work a Transient Recorder (Datalab, model 922) with 8 bit dynamic range in the detection channel had proven adequate, it turned out

that for the detection of weak tunnel satellites next to much larger resonances (see Fig. 2(a), simulated spectrum for nt ¼ 70 kHz and, below, Fig. 6) this was no longer the case. Note that spin–lattice relaxation times sometimes exceeding several hours made signal accumulation and the resulting increase of the dynamic range virtually impossible. Therefore, we replaced the Transient Recorder with a homemade VME IC board containing at its input a commercial 10 MHz; 12 bit ADC. 3.2. Sample preparation I; space group P3; the molecular C3 -axes of all three sets of inequivalent acetonitrile molecules coincide with trigonal crystal axes [19]. Perdeuterated hydroquinone was prepared following the procedure described in [20]. Single crystals of I of suitable size (hexagons with diameter and height of approximately 4 and 2 mm; respectively) were grown from a saturated solution of hydroquinone-d6 in acetonitrile (Aldrich). The initial temperature was 40 C; the cooling rate 0:2 =h: Because the trigonal axis is perpendicular to the hexagon, the crystals could easily be orientated by inspection. An NMR sample crystal with the trigonal axis perpendicular to the axis of the NMR sample tube (5 mm diameter) was prepared. The crystal was held in place by a glass rod tightly fitting into the sample tube and ending in a fork, the width of which was adapted to the height of the crystal. The sample tube can be rotated in the NMR probe about an axis that is perpendicular to the applied field B0 : This sample thus allowed us to record spectra for any angle b: II; space group P1% [21], was prepared by repeated exchange of the commercial product of natural isotopic

ARTICLE IN PRESS P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

232

4 0.5 MHz 1 MHz 1.5 MHz 2 MHz 5 MHz 10 MHz

3.5

broadening of central line / kHz

3

2.5

2

1.5

1

0.5

0 4 10

10

5

10 6

10

7 −1

10 8

10

9

k/ s

Fig. 3. Broadening of the central line of the methyl group triplet as a function of k for various values of nt : For a given value of nt ; the broadening is a maximum when k ¼ 2nt : For nt exceeding 10 MHz; the broadening becomes negligibly small.

composition (Aldrich) with D2 O (99.7%) at room temperature. It is a very special property of methylmalonic acid, which it shares with malonic acid itself, that in D2 O the ready exchange of the hydrogens of the carboxyl groups is inevitably accompanied by an exchange of the hydrogen(s) attached to the central carbon. As usual, the methyl group hydrogens do not exchange. Crystals were grown from a solution of II in ether. Unfortunately, all crystals of suitable size turned out to be multiple twins. We found out, however, that the twins shared common ab-planes with the c
directions alternating up and down. This lucky circumstance implied that there is a direction d> which is perpendicular to the C3 -axes of the methyl groups in both twins. We determined d> in a suitable specimen (approx. size 4  3:5  2 mm3 ) and using glass rods with appropriately ground end faces, we fixed it in an NMR sample tube such that d> was perpendicular to the tube axis. Rotating the sample tube into the appropriate position, the methyl groups of both twins indeed yielded identical b ¼ 90 spectra, see Fig. 6, below. III; space group P21 21 21 ; Z ¼ 4; [22], contains four differently oriented methyl groups. They become equivalent if B0 jj a; b or c: III was prepared from commercially obtained l-alanine-a-d1 (Dr. Glaser AG, Basel, Switzerland) by repeated exchange in D2 O (99.7%). Crystals were grown in the course of several weeks from a saturated solution of l-alanine-d4 in D2 O at room temperature in a desiccator at about 10 torr vs. P2 O5 : This procedure was chosen to prevent protons from atmospheric moisture from being incorporated into the crystals. An NMR sample was prepared by

fixing in the way described above a specimen of size 3  2  1:5 mm3 in a sample tube with the crystal b-axis parallel to the tube axis. In the spectrometer, this choice allows us to reach the orientations B0 jj a and B0 jj c: IV; space group P1% ; Z ¼ 2; [23] was obtained from the compound with natural isotopic composition (Aldrich) by repeated exchange with EtOD. Crystals were grown in a closed flask from a saturated solution of IV in EtOD at 40 C and cooling at a rate of 0:2 C=h: The ab-planes (c
in the reciprocal lattice) of the harvested crystals turned out to be prominent natural growth planes. Inside the sample tube, we cautiously squeezed the c
and c
planes of the crystal specimen selected for the NMR work between glass rods with endfaces ground obliquely in such a way that the C3 -axes of the methyl groups were aligned perpendicular to the sample tube axis (64 was calculated to be the appropriate angle between the normal of the endface and the axis of the rod). By rotating the sample tube in the NMR probe, this procedure allowed us to access any angle 0pbp90 :

4. Results and discussion 4.1. Acetonitrile in perdeuterated hydroquinone, I In Fig. 4 we show the 1 H spectrum of I recorded at T ¼ 216 K with the trigonal axis of the crystal aligned parallel to B0 ; thus b ¼ 0 for all methyl groups. The spectrum is the expected triplet and its resolution is indeed exceptionally good. The central line is not quite symmetric, it has broad feet and its intensity is larger

ARTICLE IN PRESS P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

233

T/K

217

170

115

58.4 kHz 76

1

Fig. 4. H spectrum of I; that is acetonitrile, N  C2CH3 ; trapped in the clathrate of perdeuterated hydroquinone, DO2C6 D4 2OD: B0 is parallel to the trigonal axis of the crystal which implies b ¼ 0 ; T ¼ 216 K: Note the unusual resolution of the expected triplet.

33

than that of the sum of the satellites. These observations are easily explained by a slightly chemically shifted background signal from (highly diluted) protons in the hydroquinone matrix. Note that there are six times as many hydrogen sites in the matrix than in the trapped acetonitrile molecules so that even a small imperfection of the deuteration procedure leads to a noticeable background signal. The splitting of the satellites is Dn ¼ 58:4 kHz; i.e., significantly less than 3bs : This result reflects librations of the C2C  N axes of the acetonitrile molecules in their cages which lead to partial averaging of HD;sec : From the ratio Dn=ð3bs Þ we estimate the root-mean-square amplitude of the librations to be 14 at T ¼ 216 K: This is a plausible value. As pointed out in Section 2, the b ¼ 0 orientation of the crystal is not useful for studying the tunnelling and stochastic dynamics of the methyl group. Therefore, we turn now to the b ¼ 90 orientation, see Fig. 5. A quick glance at this series of spectra leads to the conclusion that there is no temperature dependence of the spectral shape, hence at all temperatures either kðTÞ or nt ðTÞ; or both, are ‘‘large’’. This is true roughly. However, a closer inspection of the spectra reveals that they are not entirely independent of T: First, the splitting of the satellites increases from 29:2 kHz at 217 K to 30:8 kHz at 6 K: This again reflects librations of the molecular axes which decrease in amplitude with falling T: However, even at 6 K the observed splitting is smaller by about 1 kHz than that of a ‘‘standard’’ methyl group with immobile C3 -axis. The best explanation for the deficit are zero-point librations with a root-mean-square amplitude of about 6 : We should mention that evidence of librations of the acetonitrile axes in the cages of hydroquinone was also seen in the deuteron NMR work on the isotopically complementary version of I (deuter-

14

6 200 kHz Fig. 5. H spectrum of I for b ¼ 90 when T is increased from 6 to 217 K implying a decrease of nt from more than 10 MHz down to zero, and an increase of k from less than 10 s1 to more than 109 s1 : See text for subtle variations of the satellite splitting and of the width of the central line. 1

ated acetonitrile, natural isotopic composition of hydroquinone) [24]. Second, the width d (FWHH) of the central component of the spectra in Fig. 5 increases from 7:15 kHz at TX76 K to 8:8 kHz at 14 K and then drops again to 8:2 kHz at 6 K: According to the last paragraph of Section 2 this suggests that in the temperature regime from, say, T ¼ 14 to 35 K neither nt ðTÞ nor kðTÞ are sufficiently large to keep the system in the limiting regime of fast reorientation rate or ‘‘large’’ tunnel frequency. This is quite plausible in the light of Detken’s deuteron NMR results: the –CD3 groups in two of the three inequivalent sites possess low-temperature limiting values of nt of 53 kHz; while for the third nt ðT ) 0Þ ¼ 29 kHz [24]. Assuming a purely threefold potential V ðfÞ hindering the rotation of the methyl group rotor, these numbers translate into potential heights of 75:8 and 80:6 meV; respectively, which for –CH3 groups in turn imply nt ðT ) 0Þ ¼ 24 and 18 MHz; respectively. Both these values are ‘‘large’’ on the scale discussed in

ARTICLE IN PRESS 234

P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

Section 2, but nt ðTÞ decreases with rising T and eventually goes to zero, for the –CD3 group, see Fig. 5 of [24]. For –CD3 ; the descent of nt ðTÞ starts at around 30 K; for –CH3 the onset of the descent is expected at a somewhat lower temperature. The reorientation rate k is of the order of 1012 s1 at room temperature (see Fig. 7 of [24]) and decreases with falling T to reach 103 s1 somewhere around 30 K: Therefore, our observation that the central line of the 1 H NMR triplet is broader around this temperature than at both high and very low temperatures is perfectly consistent with the deuteron NMR results. We emphasize, however, that deuteron NMR on I with deuterated acetonitrile gives much more information about the system, in particular about the potential V ðfÞ; than can be accessed through 1 H NMR. This will not be true for compound II; to which we turn now. 4.2. Monomethyl malonic acid, CH3 DCðCOODÞ2 ; II Compound II in its natural isotopic composition has repeatedly played a key role in the development of methyl group spectroscopy. Applying ESR to the free radical CH3 CðCOOHÞ2 ; produced by g-irradiating II; Clough and Hill demonstrated in 1974 that nt depends on T [25]. In 1982, Clough and McDonald were able to detect and to correctly interpret very weak tunnel satellites in high-field NMR spectra of a powder sample of II [26]. They reported nt ðT ) 0Þ ¼ ð6872Þ kHz and found within experimental accuracy that nt ðTÞ is constant up to 60 K whereupon it decreases, see Fig. 4 of [26]. Three years later, Clough’s group again used II to introduce dipole–dipole-driven NMR [27]. The value obtained for nt ðT ) 0Þ was ð7275Þ kHz: We will now demonstrate that the use of a single-crystal sample of the isotopically manipulated version of II allows to observe clearly both pairs of tunnel satellites predicted by theory, and to access nt ðTÞ with a precision that exceeds that of the older experiments by a full order of magnitude. The left column of Fig. 6 shows a selection of the spectra recorded from II: The sample described in Section 2 was rotated in such a position that b ¼ 90 for both its twins. The sharpness of the resonances in Fig. 6 confirms that the orientation procedure worked as intended. As the asymmetry of the central part of the To85 K spectra reveals, we again have a (slight) problem with a background signal. For T490 K; this signal could be suppressed on the basis of its long T1 compared with that of the methyl groups of II: This was no longer possible for To85 K where T1 of the methyl group becomes longer than that of the background signal and eventually grows into the range of hours and even days.1 Actually, the low-temperature spectra in 1

We wonder how Clough’s team overcame this problem!

simulation

νt / kHz

T/K

k/s-1 6

>5 . 10

250

0

104

0

92

8

85

15

4 4 . 10

70

42

1000

60

60

200

39

73.5

<30

17

71.9

0

5 6 . 10

5

1.3 . 10

250 kHz

Fig. 6. Left column, 1 H spectrum of II; CH3 CDðCOODÞ2 ; for b ¼ 90 when T is increased from 17 to 250 K: Note, in particular, the tunnel satellites in the low-temperature spectra. Right column, simulations of l.h. counterparts with nt and k as fitting parameters.

Fig. 6 were obtained by warming the sample up to about 80 K; where the spins still come to thermal equilibrium in a reasonable time, and then cooling it down to the desired temperature where, after an appropriate waiting time to allow establishment of a uniform lattice temperature over the sample volume, the rf pulse exciting the FID was fired. No signal accumulation was applied. The 250 K spectrum in Fig. 6 is the expected triplet of a rapidly reorienting methyl group. The central component has only 7% more intensity than the satellites taken together. This gives credit to the successful suppression of the background signal by T1 filtering. The splitting of the satellites is Dn ¼ 30:3 kHz; i.e., 5% smaller than expected from a ‘‘standard’’ methyl group, whose C3 -axis is fixed in space. The deficit is again ascribed to vibrational averaging. By contrast to the spectra of I; those of II undergo a dramatic change on lowering T: For Tp70 K; in particular, the expected ‘‘tunnel satellites’’ become a prominent feature of the spectra. On the r.h.s. of Fig. 6 we show simulations based on Eq. (4), i.e., with nt and k as adjustable parameters. For 100 K4T460 K; and for To85 K; respectively, the

ARTICLE IN PRESS P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

80 (b)

νt / kHz

60

(a)

40 20 0 0

20

40

60

80

T/K Fig. 7. The temperature dependence of nt for II: Note the maximum of nt ðTÞ near 40 K: Curve (a) is a one-parameter fit to the data according to Eq. (6); it does not reproduce the maximum. Curve (b) is a fourparameter fit according to Eq. (7).

spectral shape depends sensitively on k and nt ; respectively. Actually, for To70 K; the tunnel frequency nt can be inferred from the spectra with an uncertainty not larger than 70:3 kHz: Also note that the very sharpness of the tunnel satellites in Fig. 6 gives evidence of a remarkable uniformity of nt over the sample volume. In Fig. 7 we present a plot of nt ðTÞ together with two attempts (a) and (b) to account for these data. The new feature of our results is that on increasing T we first observe a definite increase of the tunnel frequency (ntmax ¼ nt ð40 KÞ ¼ ð73:270:3Þ kHz), before it starts to drop as observed by Clough and McDonald. The value of nt ðT ) 0Þ is ð71:970:3Þ kHz: From this result we deduce, assuming a purely threefold potential V ðfÞ ¼ 1 2 V3 f1  cosð3fÞg; that the potential height is V3 ¼ 158 meV: The highest temperature at which we can still detect tunnelling is 92 K: There, nt is about 8 kHz: The temperature dependence of nt may be compared with predictions from theory. The simplest approach is the coherent averaging model whose underlying idea is that methyl groups undergo frequent (on the NMR scale!), symmetry ðA; Ea ; Eb Þ preserving transitions up and down the ladder of librational states. The effective tunnel frequency nt ðTÞ is then the Boltzmann-weighted ðnÞ average of the tunnel frequencies nt of all librational 2 levels n having energies En [28]: PN ðnÞ n¼0 nt expfEn =kB Tg nt ðTÞ ¼ P : ð6Þ N n¼0 expfEn =kB Tg ðnÞ

Note that jnt j increases with rising n but because sign ðnÞ nt ¼ ð1Þn ; the effective tunnel frequency nt ðTÞ drops ðnÞ with rising T: The En and nt ; and hence nt ðTÞ; can be calculated numerically once the potential V ðfÞ is given. For V ðfÞ purely threefold and V3 ¼ 158 meV; the result for nt ðTÞ is shown by curve (a) in Fig. 7. In view of the fact that Eq. (6) relies on only one adjustable parameter 2

Actually, Allen considered only the lowest two librational levels of the methyl group rotor. Press pointed out that all levels should be taken into account [28].

235

ðV3 Þ; the match of curve (a) to the experimental data is quite impressive. However, the initial increase of nt ðTÞ is missed and this deficiency cannot be remedied by modifying V ðfÞ; e.g., by including a sixfold term 1 2 V6 f1  cosð6fÞg: Tunnel frequencies increasing initially with increasing temperature have been observed before by inelastic neutron scattering. The best-studied example is probably methyliodide, CH3 I; see Fig. 7 of [29]. The initial increase of nt may be traced back to a special feature of the coupling of the methyl group to (low-frequency) phonons of the bath. Namely, it needs to be modelled by two terms Hc and Hs proportional to, for each phonon mode l; xl cos 3f and xl sin 3f; xl being the displacement of oscillator l: Hc couples the librational ground state n ¼ 0 with n-even excited states which all possess ðnÞ positive tunnel splittings hnt ; increasing with n: Obviously, this coupling leads to an increase of nt : By contrast, Hs couples the ground state with n-odd ðnÞ excited states having negative tunnel splittings hnt : It thus will depend on the relative size of the coupling constants gcl and gsl coming along with each cosine and sine term, whether or not an initial increase of nt ðTÞ will occur [30]. Guided by the insight that Hc and Hs play separate roles, and simulating the phonons by a dispersionless Einstein mode, Prager et al. proposed to represent nt ðTÞ as the following superposition of two contributions [29]:  1  expðE c =kB TÞ nt ðTÞ ¼ nt ð0Þ 1  ð1 þ Ac Þ expðE c =kB TÞ   As expðE s =kB TÞ : ð7Þ The indices c and s of the prefactors Ac ; As and of the activation energies E c ; E s refer to Hc and Hs ; respectively. By adjusting Ac ; As ; E c and E s in Eq. (7) we obtained curve (b) in Fig. 7. The virtually perfect fit to the data comes as little surprise because of the large number (four) of available fitting parameters. The bestfit parameter values are E c ¼ 11:2 meV; E s ¼ 23:0 meV; Ac ¼ 1:7 and As ¼ 37: A look at the structure of Eq. (8) reveals that it is mainly the result E c oE s which is responsible for the initial increase of nt ðTÞ: Similar relations between E c and E s ; and between Ac and As were reported for methyliodide [29]. For 60 KtTt120 K; spectra as shown in Fig. 6 also allow us to access the stochastic reorientation rate kðTÞ: When nt ðTÞ has dropped to 12 nt ðT ) 0Þ; which occurs at around 76 K; kðTÞ has grown to 3:3  103 s1 : In Fig. 8(a) we show a plot of k vs. 1=T as obtained from lineshape analyses. These data may be compared with results from measurements of T1 which we have also carried out, see Fig. 8(b) [31]. Without going into details we just note that for b ¼ 90 ; unlike for ba90 ; the spin relaxation of the methyl protons in II proceeds

ARTICLE IN PRESS P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

236

200 150

k / s-1

10 10 10

8

10

6

10

4

10

2

100

75

T/K

50

from relaxation Ea = 124 meV

10 0 5

10

(a)

100

T1 / s

10

300

200

15 1000 K / T 150

20

T/K

100

4.3. l-Alanine, CH3 CDðCOO ÞNDþ 3 ; III Ea = 124 meV

From Detken’s 13 C NMR experiments on CH3 CHðCOO ÞNDþ 3 ; it is known that the lowtemperature limiting tunnel frequency in l-alanine is ‘‘low’’. He found nt ðT ) 0Þp4 kHz [14]. The same conclusion can be drawn indirectly from measurements (via deuteron NMR lineshape analyses) of kðTÞ of the CD3 groups which one of us (P.G.) has carried out on a single crystal of CD3 CHðCOO ÞNHþ 3 (similar measurements on a powder sample of III were reported by Griffin et al. [36]). The reorientation rate k is thermally activated; from the activation energy a height V3 of the hindering potential V ðfÞ of about 220 meV may be inferred from which the value nt ðT ) 0ÞE1 kHz may be estimated. We thus expect—and shall find indeed—that the temperature dependence of the 1 H spectra of III displays essentially the classical transition from a fast reorienting to an eventually static methyl group. The question is whether tunnelling in the regime of 1?5 kHz still leads to observable effects and thus possibly allows one to distinguish a low-frequency tunnelling from a truly static methyl group. As mentioned in Section 3, the four differently oriented sets of methyl groups in III become equivalent for B0 parallel to any of the primitive crystal axes. When B0 is parallel to a; b and c; respectively, b is 36:9 ; 55:6 and 78:4 ; respectively. In terms of spectral resolution, the most promising situation is obtained for B0 jjc: We chose this crystal orientation for all experiments to be discussed here. In III; there are close-neighbour pairs of methyl groups with a distance rCC of the methyl carbons ( We shall see that meaningful simulations of only 3:68 A: of spectra must explicitly take into account the intergroup dipolar interactions. The left column of Fig. 9 shows how the 1 H spectrum of III develops when the temperature is lowered. The transition from the fast to the slow reorientation regime occurs between 220 and 125 K: Above 220 K and below 125 K the lineshape is independent of T: Note that the spectral resolution is considerably poorer 13

1

0.1 0.01 4

(b)

T1 results with those from the lineshape analyses, see the thick full line in Fig. 8(a). Three points are noteworthy. (i) The relaxation and lineshape data complement each other seamlessly. (ii) The stochastic reorientation rate kðTÞ follows the Arrhenius relation over more than seven orders of magnitude. (iii) Only when kðTÞ has dropped below 300 s1 does the Arrhenius-type descent of k give way to a definitely slower one. This kind of behaviour has been observed before in other systems with the switchover from the Arrhenius type to a slowed-down decrease of kðTÞ occurring at similar values of k as found here for II [34,35]. It can be rationalized in terms of a particle moving in a doublewell potential and being coupled to a thermal bath [34].

6 1000 K / T

8

10

Fig. 8. (a) The temperature dependence of the rate k of stochastic reorientational methyl group jumps in II inferred from lineshape analyses. (b) Temperature dependence of T1 in II: The full curve is a fit of Eq. (8) to the data assuming an Arrhenius behaviour of tc : Translating tc ðTÞ into kðTÞ gives the thick full line in part (a). Note how well the relaxation and lineshape data complement each other.

monoexponentially at all temperatures. This observation implies that II represents an example for Symmetry Restricted Spin Diffusion (SRSD) according to the classification of Emid/Wind. This is a remarkable result because the spectrum consists of a well-resolved triplet for which case Emid/Wind anticipated Limited Spin Diffusion (LSD) [32]. For SRSD and b ¼ 90 ; we obtain [31]   9 1 1 T11 ¼ p2 b2 tc þ : ð8Þ 4 1 þ o2L t2c 1 þ 4o2L t2c Note that unlike in the BPP formula both terms in the curly brackets have equal weights. In deriving Eq. (8) we set nt ¼ 0: This is justified because the data in Fig. 8(b) extend only to Tmin ¼ 110 K where nt ðTÞ has already dropped to zero, see Fig. 7. The full curve in Fig. 8(b) is a fit of Eq. (8) to the data and assuming tc ¼ tco expðEa =kB TÞ: As can be seen, the fit is excellent over the whole range where data have been collected. The best-fit parameters are tco ¼ 4:7  1013 s and Ea ¼ ð12471Þ meV: Using the familiar relation k ¼ 1=ð3tc Þ [33] we can translate the correlation time tc into the stochastic reorientation rate k and can thus compare the

ARTICLE IN PRESS P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

T/K 223 νt = 0, k = 10 9 s -1 170 165

150 145 142 135

124 νt = 0, k = 0

250 kHz 1

Fig. 9. Evolution with T of H spectrum of III; l-alanine-d4 ; CH3 CDðCOO ÞNDþ 3 with B0 along the crystal c-axis, thus b ¼ 78:4 : In an effort to reduce the background contamination of the spectra as much as possible, the nonzero signal from an NMR tube without sample crystal, but otherwise equal to that containing the crystal, was subtracted from the original signal. This procedure caused the glitches that can be seen, in particular, in the centres of the 165 and 170 K spectra. R.H.S.: simulations for rapidly reorienting and immobile methyl groups, respectively, with the inter-group pair interaction explicitly taken into account.

of simulations calculated under various assumptions. In part (b) the model is that of a lone methyl group of proper orientation ðb ¼ 78:4 Þ whose individual resonances, shown in part (a), are gaussian broadened and scaled such that the satellites match those in the experimental spectrum with regard to splitting, width and height. To match the splitting, b ¼ 20:7 kHz must be chosen, which appears quite satisfactory. The problem is obviously the centre part of the simulated spectrum which is off its experimental counterpart with regard to both its shape and height. We stress that there is no choice of nt a0 which would reduce or remove the discrepancy. In part (c) the model is extended and now includes the nearest-neighbour methyl group. Both groups are considered to be static and all inter-group (secular) dipolar interactions are taken into account explicitly. The required structural information is taken from [22]. In part (d) the 32 individual lines in (c) are again broadened and scaled such that the satellites match those in the experimental spectrum. The match of the centre parts of simulated and experimental spectra is now much better than in simulation (b). However, the simulation is somewhat too low in the centre. This can readily be remedied by assuming nt ¼ 3 kHz; see part (e). How sensitively the spectral shape depends on the value of nt is demonstrated in part (f), where nt ¼ 5 kHz is assumed and where the height of the centre in the simulation significantly exceeds that of the experiment. We are thus led to conclude that nt is nonzero and definitely is smaller than 5 kHz: The most likely value is 3 kHz: To the best of our knowledge, this is the smallest nonzero value of the tunnel frequency of any methyl group reported so far.

H3 C

NOD C–C

4.4. Dimethylglyoxime,

DON

than for II; cf. Fig. 6, compare in particular the limiting high-temperature spectra of II and III: The main reason for the difference is the strong inter-group dipolar interaction in III; a minor reason is the somewhat less favourable angle b (78:4 for III vs. 90 for II). On the r.h.s. of Fig. 9 we show spectral simulations for the limiting cases of fast and slow reorientation rates, assuming nt ¼ 0 and adopting gaussian filtering such that the satellites in the experimental spectra are reproduced optimally. The inter-group dipolar interaction between closest pairs of methyl groups is taken into account explicitly. The fast-reorientation simulation compares satisfactorily with the high-temperature experimental spectrum, however, between the k ¼ 0; nt ¼ 0 simulation and the low-temperature experimental spectrum there are substantial differences. Therefore, we compare in Fig. 10 the 124 K spectrum with a number

237

; IV

CH 3

All methyl groups in IV are magnetically equivalent. The peculiarity is that they come in close pairs. The distance of the midpoints of the methyl proton triangles ( [23]. It causes no problem to in the pair is ‘‘only’’ 2:92 A orient the sample crystal such that b ¼ 0 : This orientation is the most favourable one to demonstrate the influence of the dipolar pair interaction HD inter on the shape of the spectrum. In part (a) of Fig. 11 we show the 1 H spectrum of IV; recorded at T ¼ 100 K; for b ¼ 0 : This spectrum is to be compared with that of I for b ¼ 0 ; see Fig. 4. The striking contrast results from the strong inter-group interactions in IV while I represents an extreme case of a well-isolated methyl group. The interesting features of the spectrum in Fig. 11(a) are (i) the satellite splitting which is Dnsat ¼ 71:8 kHz; i.e., it is significantly larger than 3bs : Note that this result cannot be due to a misorientation of the sample crystal because any ba0 would lead to a reduction of Dnsat : (ii) The

ARTICLE IN PRESS 238

P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

(a)

(c)

(e)

(b)

(d)

(f)

Fig. 10. Attempts to simulate the 124 K spectrum of III reproduced as noisy trace in (b), (d), (e) and (f). See text for description of (a)–(f). The spectral width is 160 kHz:

(a)

(c)

(e)

(b)

(d)

(f)

Fig. 11. (a) spectrum of IV; dimethylglyoxime, ðCH3 CNODÞ2 ; for b ¼ 0 and T ¼ 100 K: (b)–(c) Various attempts to fit the experimental spectrum. The total spectral width is 200 kHz. See text for details.

centre component of the expected triplet is clearly split while the satellites are not. This pattern was also observed by Raber and Mehring [6], and later by Vuorima¨ki and Punkkinen [8] in the 19 F spectrum of CF3 COOAg; and by the latter authors in the 1 H spectra of CH3 COONa  3D2 O and CH3 COOLi  2D2 O [8]. (iii) The satellites appear to be ‘‘nicely’’ bell shaped. This observation will become relevant when we try to simulate the spectrum in Fig. 11(a). Judging from T1 measurements which one of us (P.G.) has carried out, the reorientation rate k of the methyl groups in IV is about 109 s1 at T ¼ 100 K: While for b ¼ 0 this is irrelevant for the intra-group dipolar interactions, it means that HD inter ðtÞ can be viewed as being in the fast motional limit where the dipolar interactions /HD ij 0 ðtÞS of all proton pairs i and j 0 ; the prime distinguishing the two groups of the pair, are equal to each other and equal to * + rffiffiffi 2 0 0 1  3 cos Y m 3 ij ij 0 2 g _ /HD T ij 0 ðtÞS ¼ 2 20 4p H 2r3ij 0 with Yij 0 ¼ -ðB0 ; rij 0 Þ: Vuorima¨ki and Punkkinen succeeded in analytically calculating the spectrum of a pair

of equivalent methyl groups (either protonated or fluorinated) in this limit [8]. We find it equally convenient to simulate it numerically using as before the Cþþ library G: The simulation that results for a pair of standard rapidly reorienting methyl groups with HD ij 0 ðtÞ calculated according to the X-ray structure of IV [23] is shown in Fig. 11(b). The points to be noted are, first, that the central component in the simulated spectrum is rather flat but not split and, second, that the splitting of the satellites (62:1 kHz) is significantly smaller than in the experimental spectrum (Dnsat ¼ 71:8 kHz). Parts (c)–(f) of Fig. 11 demonstrate how modifications of the strengths of the inter- and intra-group interactions affect the shape of the simulated spectrum. In (c)–(e) /HD inter ðtÞS is increased by a trial factor cinter of 1.2, 1.5 and 1.35, respectively. The choice cinter ¼ 1:35 produces the correct splitting of the centre component of the spectrum, while for cinter ¼ 1:2 it is definitely too small and for cinter ¼ 1:5 it is too large. To correctly reproduce the splitting of the centre component as well as that of the satellites it is necessary to increase in the simulation not only the inter- but also the intra-group dipolar coupling. This we do by introducing a second trial factor, cintra : Simulation (f)

ARTICLE IN PRESS P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

in Fig. 11 was calculated with cintra ¼ 1:1 and cinter ¼ 1:35: Both kinds of splittings of the experimental spectrum are well matched by this choice, but the shape of the simulated spectrum is still not perfect: the satellites are somewhat too large and, in contrast to the experiment, they are not ‘‘nicely’’ bell shaped but display shoulders on their outer wings. It is not obvious to us how the simulations can be extended to correctly reproduce the shape of the satellites. In an even more pronounced manner the shoulders just mentioned also show up in the simulated spectrum for CF3 COOAg shown in Fig. 2(d) of [8], while the satellites in the corresponding experimental spectrum are again nicely bell shaped. The authors of [8], however, do not pay any attention to this contrast. Is there a way to rationalize cintra 41 and cinter 41? We think that the answer is YES with regard to cinter : librations of the molecules, even if purely harmonic, have a definite tendency to increase cinter because what counts is not /rij 0 ðtÞS3 ; but /rij ðtÞ3 S which puts stronger weight on those librational displacements that bring the two groups of the pair into closer proximity. In view of the well-known fact that the amplitude of librations of (small) molecules is a sizeable fraction of the time-average intermolecular distance, the size of cinter found here is quite plausible. The situation with cintra is more difficult. The need to choose for IV b4bs in order to obtain the correct satellite splitting can hardly be traced back to intragroup vibrations which are much higher in frequency and smaller in amplitude than molecular librations. And there is no reason why the methyl group in IV should behave differently in this respect than those in I; II and III where we always found bobs : Translating b ¼ 1:1bs in other than ‘‘standard’’ H–H distances rHH ( instead of would mean that in IV rHH ¼ 1:712 A ( 1:780 A: Such a modification might arise from shorter than standard C–H distances rCH ; or smaller than tetrahedral H–C–H bond angles. The modifications that would be required to account for the observed value of ( Dns on the basis of this proposition are rCH ¼ 1:048 A ( if tetrahedral bond angles are instead of 1.09 A preserved, or -ðHCHÞ ¼ 103:5 instead of 109:5 if the standard C–H bond length is retained. Whether such substantial modifications of the geometry of the methyl group in IV are realistic is at least doubtful. However, at least there is a way to verify or falsify this proposition by independent means, namely by neutron diffraction. Actually, an n-diffraction study of IV was reported as early as 1961 [37]; however, the accuracy achieved ( is not (uncertainty of C–H bond length 70:065 A) sufficient to resolve the present issue. Therefore, we propose to the n-diffraction community to refine with present state-of-the-art precision the low-temperature structure of IV and in particular that of its methyl group.

239

5. Concluding remarks We have shown, both by experiment and simulation, that the shape of the 1 H spectrum of a methyl group may greatly vary depending on (i) the angle b subtended by the C3 -axis of the group and B0 while the azimuth of B0 in the plane of the protons is irrelevant; (ii) the rate k of stochastic reorientational jumps and, simultaneously, the frequency nt of coherent tunnelling; (iii) the extent to which the methyl groups can be considered as being isolated entities, isolated pairs or being subject to many-body dipolar interactions without prominent pair interactions. Depending on the strength of the inter-group interactions, tunnelling is reflected by the spectra provided that nt exceeds, say, 2 kHz: The accuracy of the measurement of nt depends on whether tunnel satellites can be observed or whether features (splitting, broadening) of the inner part of the spectrum must be analysed. When circumstances are favourable, tunnel frequencies can be measured with an uncertainty not larger than 7300 Hz: The structure information contained in the shape of the spectrum is by no means trivial. It tells about librations of the directions of the methyl group C3 -axes and, in the case of close pairs of methyl groups, about librations of their centres of mass. Quantitative conclusions drawn from the shape of the spectra in terms of structure and librations must, however, still be met with caution. This remark may sound surprising because after all the methyl group is one of the simplest molecular entities we can imagine, one whose geometry we hardly suspect to vary from one compound to the next and yet, as we have seen above, the well-defined spectral splittings we observe cannot be interpreted easily and without a certain degree of uncertainty. All the spectra we have reported so far comply basically with the expectation of the traditional NMR spectroscopist that a high-field, high-temperature wide line solid-state 1 H NMR spectrum ought to possess mirror symmetry. In Fig. 12 we show an example which strikingly is at odds with this expectation. It is from a single crystal of aspirin, acetylsalicylic acid C6 D4 ðCOOHÞOCOCH3 ; whose aromatic hydrogen sites were deuterated. The applied field B0 was in a general but otherwise unknown direction relative to the monoclinic crystal with Z ¼ 2 molecules in the unit cell. The temperature was 294 K; hence k was in the fast motion limit. The interpretation of the spectrum is as follows: the outermost pair of lines with a splitting of 48 kHz are the satellites of the triplet from one of the methyl groups in the unit cell. The other methyl group also gives rise to a triplet, the left satellite of which is the second line from the left. The right satellite contributes to the second,

ARTICLE IN PRESS 240

P. Gutsche et al. / Solid State Nuclear Magnetic Resonance 25 (2004) 227–240

complex spin–lattice relaxation phenomena of methyl groups [32].

References

28 kHz 48 kHz Fig. 12. Room temperature spectrum of single crystal of aspirin, C6 D4 ðOCOCH3 ÞðCOOHÞ; for a general orientation of B0 : It demonstrates how richly structured and how asymmetric a high-field wide line 1 H spectrum can be. See text for interpretation.

large line from the right. Another, somewhat larger contribution to that line arises from the chemically shifted, dipolar split doublet of the carboxyl protons. These protons are engaged in hydrogen bonds and inversion-symmetry related such bonds contain pairs of strongly coupled protons. As a matter of fact, that contribution must arise from the carboxyl protons of both molecules in the unit cell whose resonances coincide accidentally. The partner of the doublet is visible as the shoulder on the left side of the large central resonance stemming from the central resonances of the methyl triplets. The chemical shift of the carboxyl protons relative to that of the methyl protons is 16 ppm which is a plausible value in view of the large chemical shift anisotropies typical of protons in hydrogen bonds. The spectrum in Fig. 12 implies that 1 H wide line spectra of organic solids by no means need necessarily be homogeneous in the sense that, due to a tight network of dipolar interactions, each proton site of the sample contributes more or less equally to each part of the spectrum. Rather, the spectrum can readily be decomposed, as is typical in liquid state NMR, into subspectra of small units (methyl groups, pairs of hydrogen bonds) the protons of which are tightly coupled among themselves and only much weaker to the remaining protons in the sample. Note that this observation also applies to powder samples, a fact that has largely been overlooked so far but that is relevant, e.g., for the

[1] E.R. Andrew, R. Bersohn, J. Chem. Phys. 18 (1950) 159. [2] H.S. Gutowsky, G.E. Pake, J. Chem. Phys. 18 (1950) 162. [3] C. Mottley, T.B. Cobb, C.S. Johnson Jr., J. Chem. Phys. 55 (1971) 5823. [4] C.S. Johnson Jr., Carolyn Mottley, Chem. Phys. Lett. 22 (1973) 430. [5] J.A. Ripmeester, S.K. Garg, D.W. Davidson, J. Chem. Phys. 67 (1977) 2275. [6] M. Mehring, H. Raber, J. Chem. Phys. 59 (1973) 1116. [7] A.H. Vuorima¨ki, M. Punkkinen, E.E. Ylinen, J. Phys. C.: Solid State Phys. 20 (1987) L749. [8] A.H. Vuorima¨ki, M. Punkkinen, J. Phys.: Condens. Matter 2 (1990) 993. [9] P.W. Anderson, P.R. Weiss, Rev. Mod. Phys. 25 (1953) 269. [10] P.W. Anderson, P.R. Weiss, J. Phys. Soc. Japan 9 (1954) 316. [11] T.B. Cobb, C.S. Johnson Jr., J. Chem. Phys. 52 (1970) 6224. [12] F. Apaydin, S. Clough, J. Phys. C: Solid State Phys. 1 (1968) 932. [13] Z.T. Lalowicz, Ulrike Werner, W. Mu¨ller-Warmuth, Z. Naturforsch. 43a (1988) 219. [14] A. Detken, H. Zimmermann, U. Haeberlen, Mol. Phys. 96 (1999) 927. [15] S. Clough, NMR Basic Principles Progress 13 (1976) 113. [16] S. Szyman´ski, J. Chem. Phys. 111 (1999) 288. [17] S. Szyman´ski, Z. Olejniczak, A. Detken, U. Haeberlen, J. Magn. Reson. 148 (2001) 277. [18] R. Prigl, U. Haeberlen, Adv. Magn. Optical Reson. 19 (1996) 1. [19] T.C.W. Mak, K. Lee, Acta Crystallogr. B 34 (1978) 3631. [20] H. Zimmermann, Liquid Cryst. 4 (1989) 591. [21] J.L. Derissen, Acta Crystallogr. B 26 (1970) 901. [22] H.J. Simpson, R.E. Marsh, Acta Crystallogr. B 29 (1966) 550. [23] C.L. Raston, B.W. Skelton, A.H. White, Aust. J. Chem. 33 (1980) 1519. [24] A. Detken, P. Schiebel, M.R. Johnson, H. Zimmermann, U. Haeberlen, Chem. Phys. 238 (1998) 301. [25] S. Clough, J.R. Hill, J. Phys. C: Solid State Phys. 7 (1974) L20. [26] S. Clough, P.J. McDonald, J. Phys. C: Solid State Phys. 15 (1982) L1039. [27] S. Clough, A.J. Horsewill, P.J. McDonald, F.O. Zelaya, Phys. Rev. Lett. 55 (1985) 1794. [28] P.S. Allen, J. Phys. C: Solid State Phys. 7 (1974) L22; W. Press, Springer Tracts Mod. Phys. 92 (1981) 95. [29] M. Prager, J. Stanislawski, W. Ha¨usler, J. Chem. Phys. 86 (1987) 2563. [30] A. Wu¨rger, Z. Phys. B Condensed Matter 76 (1989) 65. [31] P. Gutsche, Ph.D. Thesis, University of Heidelberg, 1999. [32] S. Emid, R.J. Baarda, J. Smidt, R.A. Wind, Physica B 93 (1978) 327. [33] D.A. Torchia, A. Szabo, J. Magn. Reson. 49 (1982) 107. [34] A. Detken, H. Zimmermann, J. Chem. Phys. 109 (1998) 6791. [35] T. Schmidt, Heike Schmitt, U. Haeberlen, Z. Olejniczak, Z.T. Lalowicz, J. Chem. Phys. 117 (2002) 9818. [36] K. Beshah, E.T. Olejniczak, R.G. Griffin, J. Chem. Phys. 86 (1987) 4730. [37] W.C. Hamilton, Acta Crystallogr. 14 (1961) 95.