Correlating quadrupolar nuclear spins: a multiple-quantum NMR approach

19 November 1999

Chemical Physics Letters 313 Ž1999. 763–770 www.elsevier.nlrlocatercplett

Correlating quadrupolar nuclear spins: a multiple-quantum NMR approach M.J. Duer ) , A.J. Painter Department of Chemistry, UniÕersity of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK Received 19 August 1999; in final form 7 September 1999

Abstract An NMR experiment is presented which allows the relative orientation of nuclear quadrupole and dipole coupling tensors to be determined. The experiment uses a 4 I quantum filter to select dipolar-coupled spin I pairs, and the relative tensor orientations are determined by lineshape analysis. One of the key features of this experiment is that because it selects dipolar-coupled spin pairs, even in a multi-spin system, it leads to data which are straightforward to interpret. The same approach is used to perform a two-dimensional correlation experiment showing which spins are close in space. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction The drive to use solid-state NMR for quantitative structural studies is ever increasing. Already, structural information has been obtained on solid proteins not amenable to conventional diffraction techniques; complete structure determinations analogous to those in solution-state NMR are a realistic possibility for the near future. The key to the development of this area is the invention of solid-state NMR techniques to measure dipole–dipole couplings between nuclei. The dipole– dipole coupling depends on the inverse cube of the internuclear separation as well as the angular distribution of the nuclei with respect to the applied ) Corresponding author. Fax: q44-1223-336362; e-mail: [email protected]

magnetic field. Many ingenious methods for measuring dipolar couplings between spin y 1r2 nuclei now exist and continue to be developed, including spin–echo double resonance ŽSEDOR. w1–3x and rotational echo double resonance ŽREDOR. w4,5x for heteronuclear couplings; rotational resonance w6x, dipolar recovery at the magic angle ŽDRAMA. w7x and related methods for homonuclear couplings. Perhaps the most useful methods for structure determination in complex solids are those based on a double-quantum filter w8x. By arranging the radiofrequency pulse sequence to select the double-quantum transition a a ™ bb between two coupled spin y 1r2 nuclei, only signals due to dipolar-coupled spin pairs appear in the final signal at the end of the pulse sequence. Thus such sequences may be used successfully on complex solids where there are many coupled spins. Other techniques mentioned, such as

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 0 4 3 - X

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M.J. Duer, A.J. Painterr Chemical Physics Letters 313 (1999) 763–770

rotational resonance suffer from the drawback that they require relatively isolated spin y 1r2 pairs in the sample under study if useful quantitative information is to be obtained. Despite the recent successes for measuring dipolar couplings between spin y 1r2 nuclei, there has been comparatively little work on measuring dipolar couplings between quadrupolar spins. This is in large part due to the added complexities that the quadrupole coupling presents in these systems. Indeed, the presence of quadrupole coupling prevents any of the aforementioned techniques being successfully applied to dipolar couplings between quadrupolar nuclei with anything other than very small quadrupole couplings. The few studies which have appeared in the literature have utilised quadrupolar spins with very small quadrupole moments or have been at a site of cubic symmetry, where quadrupole coupling is not an issue w9–17x. Nevertheless, quadrupolar nuclei represent ; 75% of NMR-active nuclei and in many materials of interest, the only NMR-active nuclei are quadrupolar. Thus methods need to be found by which dipolar couplings between quadrupolar nuclei can be determined. We have recently described an experiment w18x based on the successful MQMAS experiment to study dipolar couplings in homonuclear quadrupolar spin systems. As with many experiments mentioned previously, the data from this experiment can be difficult to interpret unambiguously if there are several dipolar-coupled spins rather than isolated spin pairs. In this Letter, we describe an experiment which allows the determination of the relative orientations of the quadrupole coupling and dipole coupling tensors. Since the dipole coupling tensor is always axial with its unique axis lying along the internuclear axis between the coupled spins, information on the relative orientations of the dipolar and quadrupolar tensors effectively fixes the orientation of the quadrupole tensor relative to the molecular geometry. This latter information is of particular use now that workers in the area are using comparisons of calculated Žby ab initio means. and experimental quadrupole coupling tensors to reveal information on the electronic as well as molecular structure of materials. One of the key features of this experiment is that it selects dipolar-coupled spin pairs in an analogous manner to the double-quantum-filtered experiment for spin y

1r2 nuclei, and thus leads to data which are straightforward to interpret. The new experiment is applied here to spin y 3r2 systems but is equally applicable to other half-integer spin systems. The experiment is shown schematically in Fig. 1. An initial radiofrequency Žrf. pulse Žor sequence of pulses. creates "4I-quantum coherence Žamongst others. in the system of nuclei with spin I. For spin y 3r2, this is six-quantum Ž6Q. coherence associated with the transition Žy3r2, y3r2. l Žq3r2, q3r2. between two coupled spins. A twelve-step phase cycle of this initial pulse Žor sequence of pulses. is used to select only "6Q coherence from all the other coherences generated in this step. After a short period, the "6Q coherence is converted to single-quantum ŽSQ. coherence associated with the central transition q1r2 l y1r2 for each spin and a free-induction decay ŽFID. recorded. The 6Q filter ensures that only signals from spins suffering significant dipolar coupling appear in the final spectrum. However, the final spectrum contains more information than simply a list of dipolar-coupled spins. The spectrum from this 6Q-filtered experiment will consist of a powder pattern arising from the q1r2 l y1r2 transition. That is, the frequency of this transition for a given spin depends on the orientation of the quadrupole coupling tensor associated with that spin relative to the applied magnetic field. In a sample with many crystallites Ža so-called powder sample. all possible orientations of the

Fig. 1. Schematic representation of the 6Q-filtered pulse sequence and corresponding coherence transfer diagram.

M.J. Duer, A.J. Painterr Chemical Physics Letters 313 (1999) 763–770

quadrupole coupling tensor are present resulting in a broad powder pattern. The powder pattern may be envisaged as consisting of many overlapping sharp lines, one line arising from each different quadrupole tensor orientation, or equivalently, from each different crystallite orientation. The overall shape of the powder pattern comes from the relative intensities of the individual lines it is composed of. In the 6Qfiltered experiment, the intensity of each line is proportional to the amount of 6Q coherence generated in the initial step of the pulse sequence as it is only this coherence which carries on through the pulse sequence to generate the signal at the end. In turn, the amount of 6Q coherence generated depends on: Ž1. the strength of the quadrupole coupling associated with each of the two coupled spins and Ž2. the strength of the dipole–dipole coupling between the two spins. Both the strength of the quadrupole coupling and the strength of the dipole coupling depend on the orientations of their respective tensors with respect to the applied magnetic field. In other words, crystallites in the sample with different orientations will contribute different intensities to the final powder pattern, those intensities being characteristic of the orientations of the quadrupole and dipole coupling tensors in each crystallite. The overall powder pattern resulting from the 6Q-filtered experiment thus depends on the relative orientations of the dipole and quadrupole coupling tensors for the two coupled spins. The entire experiment is carried out under magicangle spinning ŽMAS., primarily to reduce the width of the powder pattern produced by the experiment and so improve resolution. MAS has the effect of averaging second-rank terms in the hamiltonian describing the spin system to zero, providing that the rate of spinning is fast compared with the anisotropy associated with the second-rank terms. In spin y 1r2 systems where dipolar couplings are to be measured, if MAS of moderate speed or greater is used, then so must pulse sequences which act to prevent the averaging of the dipolar coupling between the spins Žsee, e.g., Ref. w7x.. For quadrupolar spins, the net second-rank term that MAS acts on is the sum of the second-rank quadrupole coupling term plus the dipolar coupling term. This net second-rank term has an anisotropy associated with it which depends on the anisotropy associated with the quadrupole term and

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the dipolar coupling constant. The anisotropy of the quadrupole term at least, is large Žfor quadrupole couplings of the order of MHz. and thus high spinning speeds are required to completely average out the net second-rank term in the spin hamiltonian. A simple example serves to illustrate the point: consider a single spin under the influence of two interactions, each described by second-rank tensors, which for simplicity we shall consider to be axial and coparallel. Suppose the anisotropy of each interaction is 5 kHz. If either one of the interactions acted alone, then under MAS of 5 kHz, the interaction would be largely averaged and in the spectrum of the spin system, only a few small spinning sidebands would be seen. If however, the two interactions act together, the spin system is affected by a net interaction with a net anisotropy of 10 kHz. Now spinning at 5 kHz would do little to average the net interaction, and the spectrum would contain several spinning sidebands whose intensities equally represent the strengths of both interactions. In other words, neither interaction is ‘spun out’ despite the spinning speed being sufficient to largely ‘spin out’ each interaction alone. Accordingly, for quadrupolar systems with large quadrupole coupling, multiple-quantum coherences between spins can be excited without the need to apply pulse sequences to prevent the dipolar coupling being averaged to zero by the MAS, providing spinning speeds less than the net secondrank anisotropy are used. This condition is met if the spinning speed is such that spinning sidebands appear in the normal one-dimensional, single-quantum spectrum. Such spinning speeds generally are quite sufficient to provide good resolution in the final spectrum as the dipolar couplings are generally weak compared to even the second-order quadrupole coupling, so that homogeneous linebroadening in the spectra of half-integer quadrupolar nuclei is generally insignificant. Only in strongly dipolar-coupled spin systems are very high spinning speeds required for the sake of good resolution. Finally, the 6Q-filtered approach is used in a two-dimensional correlation experiment analogous to the double-quantum experiments described previously for spin y 1r2 nuclei w8x. Such an experiment allows a qualitative assessment of which spins are close in space to be read off the two-dimensional experiment and as such, allows the experimenter

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invaluable insight into structural detail without the need for complex simulations.

The hamiltonian governing a two-spin quadrupolar system Žneglecting chemical shift anisotropy for clarity. in the rotating frame is:

2. Theory

Hs

We assume that the eigenstates in the quadrupolar spin system are largely governed by the Zeeman and quadrupolar terms acting on the spins, and that the dipolar coupling is small relative to the quadrupole coupling. For quadrupole couplings of the order of MHz, this is certainly the case. Thus in periods of free precession Žno changes in coherence order., we consider the spin system to be acting only under the chemical shift and quadrupolar interactions Žin the rotating frame.. In coherence transfer steps Žforced by radiofrequency pulses. involving coherences on one spin, the quadrupole coupling and interaction with the radiofrequency field are expected to dominate. The dipolar coupling between spins is only important in coherence transfer steps generating coherences involving more than one spin, as the amplitude of these coherences would otherwise be zero in the absence of dipolar coupling between the spins involved. The generation of six-quantum coherence between two spin y 3r2 nuclei depends only on the initial eigenstate of the spin system before the radiofrequency pulse Žlargely determined by the chemical shift and quadrupole coupling as explained above., the radiofrequency pulse and the dipolar coupling between the two spins. It does not depend upon other dipolar couplings in the spin system, even if those two spins are part of a much larger coupled spin system, except through the weak dependence of the initial eigenstate on these other couplings. Given this, we can then approximate the six-quantum spectrum for a multi-spiny 3r2 system as the sum of spectra arising from all possible inequivalent spin pairs. Thus, rather than having to consider a basis set of 4 N functions for an N-spin system, we simply consider the system as a set of discrete two-spin systems, with a corresponding basis set of 4 2 for each inequivalent spin pair. By making this approximation we are excluding the possibility that sixquantum coherence can arise through correlations between more than two spins. The amplitudes of such coherences are expected to be low however, so this neglect is unlikely to be serious. We are presently performing calculations to investigate this further.

Ý  HCS Ž i . q HQŽ 1 . Ž i . q HQŽ 2 . Ž i . 4 is1, 2

q H DŽ1. q Hrf ,

Ž 1.

where the isotropic chemical shift term is HCS s yD v CS Iz ,

Ž 2.

where D v CS is the isotropic chemical shift. The first- and second-order quadrupole terms are given by HQŽ1. s v Q Ž 3 Iz2 y I Ž I q 1 . . 1

=

2

Ý

2

C2 n Ž h .

Ý Ý

nsy1 isy2 lsy2

=D 22 n , i Ž c Q , u Q , f Q . Di2l Ž a , b , g . 2 =D 10 Ž v R t , u R , 0. ,

HQŽ2. s

v Q2

2

2 v0

Iz

2

Ý Ý

Ž 3.

2k

2k

Ý

Ý

A2 k Ž I, Iz .

ks0 nsy2 isy2 k lsy2 k

=B22nk

Ž h . D 22 nk , i Ž c Q , u Q , cQ . =Di2l k Ž a , b , g . D l20k Ž v R t , u R , 0 . ,

Ž 4.

where the quadrupole coupling constant v Q is given by

vQ s

e 2 qQ 4 I Ž 2 I y 1.

,

Ž 5.

the parameters C2 nŽh . by h C0 Ž h . s 1 , C " 2 s '6 , and the operators A2 k Ž I 2 , I z . are given by

Ž 6.

A0 s I Ž I q 1 . y 3 Iz2 , A2 s 8 I Ž I q 1 . y 12 Iz2 y 3 , 4

A s 18 I Ž I q 1 .

Ž 7.

y 34Iz2 y 5 .

The parameters B22kn Žh . are given in Ref. w19x. The first-order dipolar term in Eq. Ž1. is given by HDŽ1. s D Ž 3 I1 z I2 z y I1 P I2 . 2

=

2

Ý Ý

D 02 i Ž c D , u D , f D .

lsy2 isy2

=Di2l Ž a , b , g . D l20 Ž v R t , u R , 0 . ,

Ž 8.

M.J. Duer, A.J. Painterr Chemical Physics Letters 313 (1999) 763–770

where D is the dipolar coupling constant between the two spins: Ds

m0

g2

4p

r3

ž /

"y1 ,

Ž 9.

where r is the internuclear distance. In Eqs. Ž3., Ž4. and Ž8. for the various spin interactions, the Euler angles, cA , uA , fA ; A s Q, D for quadrupolar and dipolar interactions respectively describe a common reference frame in a given crystallite with respect to the interaction principal axis frame ŽPAF.; the Euler angles a , b , g describe a frame fixed in the sample rotor with respect to the common crystallite reference frame and finally, Ž v R t, u R , 0. describe the laboratory frame in which the applied field is along z with respect to the rotor-fixed frame. The term describing the interaction of radiofrequency pulses with the spin system in Eq. Ž1., Hrf is given by: Hrf s yv 1 Ž I1 x q I2 x . cos frf q Ž I1 y q I2 y . sin frf 4 ,

Ž 10 . where v 1 is the amplitude of the pulse and frf is its phase. Eq. Ž1. is used to calculate the spin density matrix of the two-spin system during the radiofrequency pulses for the pulse sequence shown in Fig. 1 Žsee, e.g., Ref. w20x.. The rotor is assumed to be stationary during any rf pulses, providing that the rotor period is long compared with the pulse period Žrotor period ) 10 = pulse length.. During periods of free precession, only isotropic chemical shift and quadrupole interactions are considered. The rate of spinning is considered to be fast relative to the quadrupolar anisot-ropy during these periods, corresponding to

Fig. 2. Simulations of the 6Q-filtered lineshapes arising from the experiments shown in Fig. 1 for different relative orientations of the quadrupole and dipole tensors for a spiny3r2 pair: Ža. quadrupole tensors on spins 1 and 2 coparallel; and Žb. quadrupole tensor on spin 2 perpendicular to that on spin 1 Ži.e. related to that on spin 1 by the Euler angles Ž08, 908, 08... The normal central transition lineshape that would arise from a conventional singlepulse, single-quantum experiment is shown at the bottom for comparison. In all simulations, the quadrupole coupling constant was 2.6 MHz and the quadrupole asymmetry, 0.6, and spectral frequency 105.78 MHz. Details of the calculation can be found in the text.

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zeroing of the subscript l in Eqs. Ž3., Ž4. and Ž8.. This is clearly an approximation, but results in little change to the powder lineshape, but simply neglects the spinning sidebands in the spectrum. A FID is calculated at the end of the pulse sequence by following the evolution of density matrix elements corresponding to the central transitions on each spin. To produce the final lineshape, FIDs from over 800 000

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crystallite orientations Žas given by the Euler angles Ž a , b , g . are summed and the result Fourier transformed to produce a frequency spectrum.

3. Experimental The 23 Na 6Q-filtered MAS spectrum of Na 2 SO4 was recorded on a Chemagnetics CMX400 NMR spectrometer operating at 105.78 MHz, using the pulse sequence in Fig. 1. The rf power was 106 kHz throughout. The transfer pulse for the 6Q ™ single quantum step was optimised to be 4.2 ms and the delay between scans was 15 s. Excitation pulse lengths are given in the appropriate figures. The sample was spun at the magic angle at 5 kHz, stabilised to "5 Hz by standard Chemagnetics equipment. The two-dimensional 6Q-correlation spectrum of sodium zirconate ŽNa 2 ZrO 3 . used rf power of 99 kHz, excitation pulse of 5.8 ms, transfer pulse of 8 ms, and 2 s delay between scans. The delay Ž t 1 . between the 6Q excitation and 6Q ™ SQ transfer step Žsee Fig. 1. was incremented in 5 ms steps corresponding to an f 1 spectral width of 200 kHz. The spectral width in f 2 was 50 kHz. 50 t 1 slices were collected with 3168 repetitions in each, giving a total acquisition time of 88 h. The sample was spun at 5 kHz throughout with the spinning speed stabilised as above. The coherence pathway 0 ™ q6 ™ y1 was selected by appropriate phase cycling. The two-dimensional dataset was processed by Fourier transforming in each dimension and then forming a magnitude spectrum. Sodium sulfate and sodium zirconate were both obtained from Aldrich Chemical and used without further purification.

sors as expected. Further simulations show that the effect of decreasing the size of the dipolar coupling constant is simply to decrease the intensity of the final signal, with no effect on the lineshape, again as would be predicted qualitatively. 6Q experiments were performed on 23 Na Ž I s 3r2. in Na 2 SO4 . The results for different excitation pulse lengths are shown in Fig. 3a. The crystal structure of this compound shows each 23 Na to have two near 23 Na neighbours at a ˚ Other Na sites are 3.6 A˚ or further distance of 3.2 A.

4. Results and discussion Fig. 2 shows simulations of the central transition lineshapes expected for different relative orientations of the quadrupole and dipole coupling tensors for two coupled spin y 3r2, with the same quadrupole coupling constants and asymmetry. There are clearly marked differences in lineshape accompanying the changes in relative orientation of the interaction ten-

Fig. 3. Ža. Experimental 6Q-filtered lineshapes for 23 Na in Na 2 SO4 for different excitation pulse lengths Žas given in the diagram.. Note that the broad features either side of each powder pattern are spinning sidebands. Žb. Simulations of the 6Q-filtered lineshapes for 23 Na in Na 2 SO4 . The quadrupole parameters used in the simulations are the same as those for Fig. 2. Further details in text.

M.J. Duer, A.J. Painterr Chemical Physics Letters 313 (1999) 763–770

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Ž08, 23.48, 08. with respect to the dipole coupling tensor, i.e. with respect to the 23 Na– 23 Na axis. The agreement with the experimental lineshapes is excellent for all excitation pulse lengths. A variation of 38 in the Euler angles produces a significantly worse fit to experiment. This orientation of the quadrupole tensor places its principal axes parallel to the crystallographic axes, which indeed is reasonable, given the coordination geometry around the 23 Na spins shown in Fig. 4. The ultimate aim of this experiment is of course to perform two-dimensional correlation experiments using the 6Q filter. We have performed such an experiment on sodium zirconate ŽNa 2 ZrO 3 ., which a previous NMR study shows to have three distinct

Fig. 4. Ža. The crystal structure of Na 2 SO4 . Žb. Definitions of the angles used in the simulations of Fig. 3b. The orientation of the quadrupole tensors shown is illustrative only.

away and are ignored in our simulations. The coordination geometry about each 23 Na is identical but neighbouring 23 Na have different orientations of their respective coordination spheres, being related by a 1808 rotation about the b axis ŽFig. 4a.. It is reasonable to expect that the orientation of the quadrupole coupling tensor, whatever it may be, of neighbouring 23 Na spins are related in a like manner, as shown in Fig. 4. Thus, there is only one variable in our simulations of the 6Q 23 Na spectra for this system: the relative orientation of the dipolar and quadrupolar coupling tensors for one of the two 23 Na spin pairs considered in the calculation Žangle a in Fig. 4b.. Fig. 3b shows simulations of the 23 Na Na 2 SO4 6Q-filtered lineshapes for the quadrupole coupling tensor 1 in Fig. 4b oriented by the Euler angles

Fig. 5. The two-dimensional 6Q 23 Na correlation spectrum for Na 2 ZrO 3 . Above is shown the normal one-dimensional 23 Na spectrum, along with its decomposition into three simulated components as proposed in Ref. w21x. The labels NaŽ1. –NaŽ3. and A–E refer to the labelling scheme used in the text.

M.J. Duer, A.J. Painterr Chemical Physics Letters 313 (1999) 763–770

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Table 1 Shortest internuclear distances in Na 2 ZrO 3 Na sites

Shortest internuclear distance between Na sites Žnm.

NaŽ1. –NaŽ1. NaŽ1. –NaŽ2. NaŽ1. –NaŽ3. NaŽ2. –NaŽ2. NaŽ2. –NaŽ3. NaŽ3. –NaŽ3.

0.307 0.322 0.310 0.563 0.370 0.563

Na sites w21x. The two-dimensional 23 Na spectrum obtained is shown in Fig. 5 along with the normal one-dimensional 23 Na spectrum, showing the three components determined in the previous NMR study w21x. Spins which are close in space will give rise to 6Q coherences, which in the two-dimensional spectrum will give rise to cross-peaks between the spins’ signals in f 2 , the normal single-quantum dimension, and the corresponding 6Q signal in f 1. The closest internuclear distances between different spins, determined from X-ray diffraction data w21x are listed in Table 1. From these data it is clear that we should expect to see 6Q signals corresponding to interactions between NaŽ1. –NaŽ1., NaŽ1. –NaŽ2. and NaŽ1. –NaŽ3. Žinternuclear distances all of the order 0.31–0.32 nm., but not between NaŽ2. –NaŽ2., NaŽ3. –NaŽ3. or NaŽ2. –NaŽ3.. The 23 Na resonances from sodium zirconate overlap in the normal single-quantum spectrum of the central transitions. The lineshape from NaŽ1. is in the central region of the overlapping. NaŽ1. is therefore responsible for the intensity in the two-dimensional spectrum ŽFig. 5. labelled A–C; there are clearly at least two separate signals in this region. The intensity in the region labelled D correlates in f 2 with the region of the spectrum associated with NaŽ2., and region E with NaŽ3.. The overlapping regions B ŽNaŽ1.. and D ŽNaŽ2.. correspond to similar 6Q frequencies in f 1 and must therefore both arise from the same Na–Na interaction, namely that between NaŽ1. and NaŽ2.. Regions A ŽNaŽ1.. and E ŽNaŽ3.. also have similar 6Q frequencies and must therefore each arise from the dipolar interaction between NaŽ1. and NaŽ3.. Finally, the region C ŽNaŽ1.. in the two-dimensional spectrum has no other corre-

sponding signal at a similar 6Q frequency in f 1 , and must therefore arise from an autocorrelation between NaŽ1. spins. In summary, the 6Q correlation spectrum for Na 2 ZrO 3 contains all the 6Q signals expected Žand none that were not. given the crystal structure of the compound and previous assignment of the 23 Na NMR spectrum, and with relative intensities that do indeed reflect the various internuclear distances. The strength of this experiment is that no calculations or spectral simulations are required; the structural information is ‘read’ directly off the experimental spectrum.

23

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