Truncated Dipolar Recoupling in Solid-State Nuclear Magnetic Resonance

Transcription

1 Truncated Dipolar Recoupling in Solid-State Nuclear Magnetic Resonance Ildefonso Marin-Montesinos a, Giulia Mollica b, Marina Carravetta a, Axel Gansmüller a, Giuseppe Pileio a, Matthias Bechmann c, Angelika Sebald c, and Malcolm H. Levitt a a Chemistry Department, Southampton University, SO17 1BJ, UK, b Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Pisa, Italy c Department of Physics, University of Dortmund, D Dortmund, Germany mhl@soton.ac.uk Tel: Fax: IMM-TDR, version 5.0 (for submission to CPL) 1

2 Abstract We describe a solid-state NMR concept for the estimation of individual spinspin couplings in strongly-coupled homonuclear spin systems. A radiofrequency pulse sequence, synchronized with the magic-angle sample rotation, recouples zero-quantum dipolar interactions as well as a frequency-dispersing interaction such as the chemical shift anisotropy. The combination of these two recoupled interactions causes the spin system to behave in an approximately weaklycoupled fashion. Individual spin-spin couplings may then be disentangled by using frequency-selective radiofrequency pulses. Theoretical results and numerical simulations are compared with experimental data for the 13 C nuclei in [ 2 H 7, 13 C 3, 15 N]-L-alanine. 2

3 1 Introduction Solid-state nuclear magnetic resonance (NMR) is useful for obtaining atomicscale structural information, especially in materials that lack long-range molecular order, such as many polymers, biomolecules, and network solids (1-3). Many of these achievements rely on dipolar recoupling methods which allow the estimation of internuclear dipole-dipole couplings in the presence of magicangle sample spinning (MAS), which provides chemical site resolution and good sensitivity (4-12). However, most quantitative recoupling experiments require approximately isolated pairs of nuclear spins. In some cases, it is possible to engineer the necessary spin-pair distributions by isotopic labelling, but this is always expensive and laborious. The use of multiply-labelled samples would be more general and cost-effective. Unfortunately, it is difficult to interpret the data from multiply-labelled samples, due to the many-body spin dynamics (13). Typically, the dynamics of multiple-spin systems are dominated by strong, short-range couplings, with the influence of long-range couplings being sharply attenuated (14). This property is unfavourable for structural studies since long-range couplings usually have more structural significance than short-range couplings. There are several solid-state NMR experiments that do permit the estimation of individual long-range couplings in the presence of short-range couplings, but these methods suffer from strong limitations. For example, selective RE- DOR experiments (15) are restricted to heteronuclear spin systems. Rotational 3

4 resonance techniques (14,16-20) allow the activation of individual spin-spin couplings in a homonuclear multiple-spin system, but are restricted to certain ranges of the isotropic chemical shift differences. In this communication we describe a new recoupling concept, called truncated dipolar recoupling (TDR), which holds promise for selective internuclear distance estimations in a wide variety of homonuclear multiple-spin systems. We support the validity of this concept by numerical simulations and experiments on systems of three coupled 13 C nuclei. We indicate some of the difficulties we have experienced with implementing TDR in practice. 2 Theory 2.1 Weak coupling Consider a homonuclear system of many coupled spins-1/2 in high magnetic field. The nuclear spin Hamiltonian is given by H = j H j + j<k H jk [1] where H j represents the individual spin-field interactions and H jk represents the spin-spin interactions. In general, the coupling Hamiltonians of connected spin pairs do not commute: [ Hjk, H kl ] 0 [2] Non-commutativity makes it very difficult to disentangle the effect of an individual spin-spin coupling, especially if the coupling of interest is weaker than 4

5 the others. In solution NMR, the term strong coupling is used to describe this complicated situation. In solution NMR, the spin Hamiltonian in Eq.[1] is often simplified by omitting terms which are off-diagonal in the Zeeman basis (21). This weak-coupling approximation is valid if the Larmor frequency differences between coupled spins greatly exceed the off-diagonal parts of the relevant coupling Hamiltonians (proportional to the J-couplings in the case of isotropic liquids). The truncated coupling terms H 0 jk commute both with each other and with the spin-field terms H j. This very convenient property is the basis of most solution NMR methodology (21,22). In particular, the spin echoes induced by frequency-selective rf pulses may be used to estimate small spin-spin couplings in the presence of larger ones (23). The weak-coupling condition also applies to some cases in broad-line solidstate NMR. For example, in static solids, isotropic and anisotropic chemical shifts create a frequency dispersion which sometimes leads to weak coupling for the spin-spin interactions. A similar effect is created by second-order quadrupolar interactions in the NMR of half-integer quadrupolar nuclei (24). For spins- 1/2, rapid sample rotation about an axis which is not at the magic-angle with respect to the field generates scaled dipolar couplings which are often truncated by the isotropic and anisotropic chemical shifts (25-27). 5

6 2.2 Truncated Recoupling In magic-angle-spinning (MAS) NMR, the rapid rotation of the sample averages out the chemical shift anisotropies and dipole-dipole couplings, giving rise to good resolution and sensitivity. The informative dipole-dipole coupling information may be retrieved by applying recoupling pulse sequences (4-12). However, the recoupled dipole-dipole interactions do not commute, leading to strong coupling in the case of more than two coupled spins. The truncated dipolar recoupling (TDR) concept involves two essential features: (i) zero-quantum recoupling of homonuclear dipole-dipole interactions (9); (ii) simultaneous recoupling of spin interactions which provide a frequency dispersion. The average Hamiltonian under a TDR pulse sequence may be written H = H 0 j + j<k(h 0 jk + H ± jk) [3] j where H 0 jk and H ± jk represent the parts of the recoupled dipole-dipole interaction which are diagonal and off-diagonal in the Zeeman basis, and H 0 j is the frequency-dispersing interaction of spins S j. The frequency-dispersing interactions may involve chemical or paramagnetic shifts, quadrupolar couplings, or heteronuclear couplings. In the work discussed here, isotropic and anisotropic chemical shifts are used. The average Hamiltonian terms may be written: H j = ω 0 js jz H 0 jk = ω 0 jk2s jz S kz H ± jk = ω ± 1 ( jk S + j 2 S k + S j S+ k ) [4] 6

7 with ω 0 jk = ω DD jk + πj jk ω ± jk = ω DD jk + 2πJ jk [5] The recoupled dipole-dipole interaction for the pair of spins {S j, S k } is denoted ω DD jk. If the weak-coupling condition is satisfied for all spin pairs: ω 0 j ω 0 k >> ω ± jk [6] then the coupling terms may be truncated, leading to a weakly-coupled spin system. The condition in Eq.[6] is satisfied if the differences in recoupled chemical shift frequencies (including both isotropic and anisotropic contributions) are large compared to the recoupled spin-spin couplings. The validity of this approximation depends on the spin system parameters, the pulse sequence used, and the molecular orientation. Some cases are explored below. 3 Implementation 3.1 Symmetry-Based Recoupling Symmetry may be used to guide the design of recoupling sequences (7-11). In the case that anisotropic chemical shifts or heteronuclear couplings are used for the frequency dispersion, the following selection properties are needed to implement TDR: (i) recoupling of terms with space-spin quantum numbers 7

8 (l, m, λ, µ) = (2, ±1, 2, 0) or (2, ±2, 2, 0) to ensure zero-quantum recoupling of the dipole-dipole interactions; (ii) recoupling of terms with space-spin quantum numbers (l, m, λ, µ) = (2, ±1, 1, 0) or (2, ±2, 1, 0) to ensure zero-quantum recoupling of the chemical shift anisotropy or heteronuclear dipolar couplings; (iii) finite scaling factors for the symmetry-allowed terms (for notation, see ref.10). Symmetry analysis (7-11) reveals several appropriate symmetries, including C3 1 3, which indicates three radiofrequency cycles synchronized with three complete sample rotations, with a phase shift of 2π/3 between consecutive cycles. The symmetry-allowed first-order terms have the following sets of spacespin quantum numbers: (l, m, λ, µ) = (2, ±2, 1, 0), (2, ±1, 1, 0), (2, ±2, 2, 0), (2, ±1, 2, 0), (0, 0, 1, 0) and (0, 0, 0, 0). The first two terms correspond to CSA or heteronuclear DD interactions, the next two correspond to homonuclear DD interactions, and the last two terms correspond to isotropic chemical shifts and J-couplings. The results discussed here were obtained with a C3 1 3 sequence of the following form: C3 1 3 = C 0 C 2π/3 C 4π/3 [7] where each windowed cycle is given by C φ = [τ r /4] (π/2) φ (2π) φ+π (3π/2) φ (π/2) φ+π (2π) φ (3π/2) φ+π [τ r /4] [8] }{{} τ r/2 A rectangular pulse of flip angle β and phase φ is denoted (β) φ, and a window without rf irradiation of duration τ is denoted [τ]. As indicated, the duration 8

9 of the pulse cluster is τ r /2, where a rotor period is defined τ r = 2π/ω r, and ω r denotes the angular spinning frequency. This requires an rf field providing a nutation frequency ω nut = 8ω r. The C3 1 3 sequence has a duration of 3τ r, and is sketched in Fig.1a. This incarnation of C3 1 3 has a scaling factor of Fig.1 and 0 for the (l, m, λ, µ) = (2, ±1, 2, 0) and (2, ±2, 2, 0) dipole-dipole terms, and scaling factors of 0, and 0.50 for the (2, ±2, 1, 0), (2, ±1, 1, 0), and (0, 0, 1, 0) chemical shift terms, respectively. Since the average dipole-dipole coupling Hamiltonian contains only m = ±1 terms, its sign may be reversed by shifting the starting time point t 0 of the C3 1 3 sequence by one half of a rotor-period: ω DD jk (t 0 ) = ω DD jk (t 0 + τ r /2) [9] 3.2 Chemical Shift Refocussing The C3 1 3 sequences generate, under the weak coupling approximation, a spin propagator which may be expressed a product of all recoupled chemical shift interactions as well as recoupled spin-spin interactions. The recoupled chemical shift interactions may be removed by the insertion of two strong π pulses, as shown in Fig.1b. Since the recoupled chemical shift terms are removed by this procedure, their precise form and orientation-dependence is unimportant. They should simply be large enough to impose weak coupling on the recoupled dipolar interactions. 9

10 3.3 Selection of Individual Internuclear Couplings The TDR pulse sequences may be combined with frequency-selective pulses in order to isolate the effect of individual couplings. One possible scheme is shown in Fig.1c, which only involves a single radiofrequency channel. This pulse sequence includes two shift-refocussed TDR sequences each of duration τ/2, and two selective rf pulses with a Gaussian amplitude modulation (23). In ideal circumstances, each selective pulse rotates resonant spins through an angle of π/2 while leaving off-resonant spins unperturbed. In the discussion below, the selected spins are called S j, while non-selected spins are denoted S k. The first Gaussian pulse has zero phase, while the second Gaussian pulse (drawn in black) has a phase φ j which is varied between 0 and 2π in a series of experiments. Both Gaussian pulses have a duration equal to an integer number of rotor periods. The intervals of 3τ r /4 bracketing the selective pulses ensure that the starting time points of the two TDR sequences are one-half of a rotor period out of phase with each other. According to Eq.[9], the recoupled dipoledipole Hamiltonians are opposite in sign for the two TDR sequences. First ignore J-couplings, and consider the case where the two Gaussian pulses are out of phase (φ j = π). The rotations induced by the two Gaussian pulses cancel, so that the selective pulses may be ignored, to a first approximation. The first π/2 pulse converts longitudinal magnetization into transverse magnetization which evolves freely during the first quarter rotor period before encountering the TDR sequence. The transverse magnetization dephases under 10

11 the recoupled spin-spin interactions over the first TDR interval τ/2. However, since the TDR dipole-dipole Hamiltonian for the second sequence is opposite in sign to the first, all of these antiphase terms refocus by the end of pulse sequence. The arrangement of τ r /4 and 3τ r /4 intervals ensures that the chemical shift anisotropy leads to no net rotation over the free-evolution intervals. The second π/2 pulse converts the transverse magnetization into longitudinal magnetization, and the last π/2 pulse, which is phase-cycled to suppress spurious signals, selectively detects this longitudinal magnetization. If J-couplings and relaxation are neglected, the NMR signal in the case φ j = π is therefore the same as that generated by a single π/2 pulse. Now consider an experiment in which the phase of the second Gaussian pulse is zero (φ j = 0). In this case, the selected spins S j undergo a π rotation, which prevents the refocussing of S j S k dipole-dipole couplings. The transverse magnetization of non-selected spins S k is therefore reduced according to the strength of the couplings ω DD jk to the selected spins S j, multiplied by the total recoupling interval τ. The NMR signals from spins S k which are not resonant with the Gaussian pulses increases as φ j is increased from 0 to π, and then decreases again as φ j is increased from π to 2π. The depth of this phase-dependence may be used to assess the magnitude of the corresponding coupling. The protocol operates in constant-time mode and hence is rather insensitive to relaxation interference (13,28). If the NMR spectrum is well-resolved, many internuclear distances r jk may be estimated in a single set of experiments. 11

12 It will be shown elsewhere that for idealized weak coupling and perfectly selective pulses, the powder-average NMR signal of observed spins S k depends on the rf phase of the selective pulse φ j and the recoupling interval τ according to s k (τ, φ j ) s k (τ, π) = sin 2 ( φ j 2 ) cos(πj jkτ) + cos 2 ( φ j )π 4 J 1/4( 4 κ 2120b jk τ)j 1/4 ( 4 κ 2120b jk τ) [10] where the scaling factor κ 2120 is equal to for the case considered here. The symbols J ±1/4 are quarter-integer-order Bessel functions (29). The dipole-dipole coupling constant b jk is inversely proportional to the cube of the internuclear distance: b jk = µ 0 γ 2 4π rjk 3 [11] Four theoretical curves described by Eq.[10] are shown in Fig.2 (solid lines). The Fig.2 depth of the modulation curves is deeper for the short internuclear distances (left column) than for the longer internuclear distances (right column). The results of 3-spin SIMPSON numerical simulations (30) are shown by the dashed lines in Fig.2. All simulations were performed using parameters (31) appropriate to the 13 C nuclei of 13 C 3 -L-alanine, at a magnetic field of 9.39 T and a spinning frequency of ω r /2π = khz. The powder average was calculated using 256 pairs of Euler angles, distributed according to the REPULSION 12

13 scheme (32), with the third Euler angle stepped through a full revolution in 20 steps. The dashed black lines show the results of SIMPSON simulations using idealized selective rotations of zero duration on the selected spins, instead of the Gaussian pulses. The dashed grey lines show realistic simulations, in which each selective Gaussian pulse was simulated explicitly, using an overall duration of ms and an amplitude modulation function proportional to exp{ at 2 } where a = s 2, as used in the experimental tests (see below). The 3-spin simulations using idealized selective rotations (dashed black lines) agree rather well with the analytical curves in all cases. This supports the validity of the TDR concept. Note that 13 C 3 -L-alanine is a relatively challenging case for testing the truncated dipolar recoupling, since the isotropic and anisotropic shifts of two of the 13 C sites are relatively small. The realistic simulations including the Gaussian pulses (dashed grey lines) continue to agree well with the analytical formulae when the 13 CD 3 site is observed while applying the selective pulses to the 13 CD or 13 CO sites (upper row). The agreement is less good when the 13 CO site is observed (lower row). This may reflect the difficulty in achieving a clean discrimination of the 13 CD and 13 CD 3 sites by a selective rf pulse. 13

14 4 Experimental Results During our preliminary experimental trials on 13 C-labelled organic solids, we discovered that the performance of TDR sequences is strongly effected by heteronuclear 13 C- 1 H interactions. This is not too surprising since the heteronuclear interactions have the same transformation properties under 13 C spin rotations as the chemical shift anisotropies, which the C3 1 3 sequences are designed to recouple. In the future it should be possible to control the heteronuclear interference by suitable proton irradiation schemes. Nevertheless, in this first report we avoid the issue of proton interference by studying the 13 C spin systems in [ 2 H 7, 13 C 3, 15 N]-L-alanine, in which all protons are replaced by deuterons, to a level of around 98%. The sample was prepared by mixing [2,3-2 H 4, 13 C 3, 15 N]-Lalanine with 2,3-2 H 4 -L-alanine in a ratio of 1:10, and recrystallizing several times from 2 H 2 O. The isotopically-labelled alanines were purchased from Cambridge Isotope Laboratories. Experiments were performed using the pulse sequence in Fig.1c at a 13 C Larmor frequency of MHz and a spinning frequency of khz. A 4 mm magic-angle-spinning rotor was used with the sample at ambient temperature. In all cases the total recoupling interval was τ = ms. Each Gaussian pulse had a duration of ms. An interval of 120 seconds was left between successive transients, to allow recovery of the 13 C magnetization. The rf carrier was positioned in the centre of the 13 C spectrum, with the selective rf field temporarily shifted to the desired frequency by using a phase ramp. Fig.3 14

15 Fig.3 shows experimental signal amplitudes for selection of the 13 CD 3 site (top row) and selection of the 13 CO site (bottom row). The experimental amplitudes are compared with the analytical formula of Eq.[10] (solid lines) and accurate numerical simulations (broken grey lines). These curves have been adjusted to take into account the natural abundance 13 C signals from the 2,3-2 H 4 - L-alanine matrix, which comprises about 10% of each observed peak and which is not modulated by the phase φ j. For this reason the solid and broken lines do not coincide precisely with the corresponding curves in Fig.2. Agreement between the experimental measurements and the accurate numerical simulations is acceptable in all cases. However, the agreement with the analytical curves is less good, especially when the 13 CO site is observed. As shown in Fig.2, these discrepancies are mainly associated with imperfect performance of the selective pulses. 5 Conclusions These results show that truncated dipolar recoupling is a promising concept for the estimation of selected internuclear couplings in strongly-coupled solid-state NMR spin systems, but also that several obstacles remain. In the case of 13 C 3 -L-alanine, simulations show that the TDR concept is reasonably valid. Most of the observed discrepancies between analytical theory and numerical simulations are due to imperfect performance of the selective pulses. 15

16 Improvements in this regard may be anticipated, for example by using optimal control theory to design modulation schemes with a better performance (33). Numerical simulations agree reasonably well with experimental measurements on [ 2 H 7, 13 C 3, 15 N]-L-alanine at short recoupling times. However, unpublished results at longer recoupling times show larger discrepancies which we tentatively attribute to rapid 2 H relaxation. This is under investigation. In addition, we have found it difficult to apply these pulse sequences to protonated organic solids and biomaterials due to strong interference from heteronuclear interactions. More effective heteronuclear decoupling methods are required in this context. In summary, TDR has considerable potential for selective spin-spin coupling measurements in multiple-spin systems in solid-state NMR. In principle, the idea is not restricted to nuclei with large chemical shift interactions. Even strongly-interacting abundant nuclei such as protons may be amenable to TDR if the external magnetic field is large enough, or if other anisotropic interactions such as paramagnetic shifts or heteronuclear dipole-dipole couplings are used to provide the necessary frequency dispersion. 6 Acknowledgements This research was supported by the EPSRC (UK). We thank Ole G. Johannessen for experimental support. 16

17 References 1. F. Castellani, B. van Rossum, A. Diehl, M. Schubert, K. Rehbein and H. Oschkinat, Nature 420 (2002) A. T. Petkova, Y. Ishii, J. J. Balbach, O. N. Antzutkin, R. Leapman, F. Delaglio and R. Tycko, Proc. Natl. Acad. Sci. USA 99 (2002) D. H. Brouwer, R. J. Darton, R. E. Morris and M. H. Levitt, J. Am. Chem. Soc. 127 (2005) R. Tycko and G. Dabbagh, Chem. Phys. Lett. 173 (1990) S. Dusold and A. Sebald, Annual Reports NMR Spectrosc. 41 (2000) D. D. Laws, H.-M. L. Bitter and A. Jerschow, Angew. Chem. Int. Ed. 41 (2002) M. H. Levitt, in Encyclopedia of Nuclear Magnetic Resonance: Supplementary Volume., Ed. D. M. Grant and R. K. Harris, (Wiley, Chichester, UK, 2002). 8. M. Carravetta, M. Edén, X. Zhao, A. Brinkmann and M. H. Levitt, Chem. Phys. Lett. 321 (2000) A. Brinkmann, J. Schmedt auf der Günne and M. H. Levitt, J. Magn. Reson. 56 (2002) A. Brinkmann and M. H. Levitt, J. Chem. Phys. 115 (2001) A. Brinkmann and M. Edén, J. Chem. Phys. 120 (2004) M. Bak and N. C. Nielsen, J. Chem. Phys. 106 (1997) J. Schmedt auf der Günne, J. Magn. Reson. 180 (2006) V. Ladizhansky and R. G. Griffin, J. Am. Chem. Soc. 126 (2004) C. P. Jaroniec, B. A. Tounge, J. Herzfeld and R. G. Griffin, J. Am. Chem. Soc. 123 (2001) M. H. Levitt, D. P. Raleigh, F. Creuzet and R. G. Griffin, J. Chem. Phys. 92 (1990)

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19 Figure Captions Fig.1. (a) C3 1 3 pulse sequence, in which three radio-frequency cycles with overall phases {0, 2π/3, 4π/3} are applied in sequence. Each cycle has a duration of one rotor period. The sketch shows the windowed cycles specified in Eq.[8]. (b) Chemical shift refocussing is implemented by placing strong π pulses at 1/4 and 3/4 of the total sequence duration. (c) The two TDR sequences bracket a combination of two weak rf pulses and one strong π pulse, which has phase φ = 0. The modulated weak pulses act as a selective π/2 rotations on spins in site j and have a duration equal to an integer number of rotor periods. The phase of the first selective π/2 pulse is zero, while the phase φ j of the second selective pulse (shown in black) is varied from 0 to 2π in a series of experiments. The phase of the last pulse is cycled in four steps together with the digitizer to suppress signals which do not pass through longitudinal magnetization before the final π/2 pulse. 19

20 Fig.2. Theoretical modulation curves using Eq.[10] (solid lines), and numerically simulated modulation curves using idealized selective pulses (thin dashed lines) or realistic selective pulses (dashed gray lines) for pairs of 13 C nuclei in 13 C 3 -L-alanine at a recoupling time of τ = ms and a spinning frequency ω r /2π = khz. All curves are normalized to have a value of 1 at φ j = π. (a) The 13 CD site is observed with selective pulses applied to the 13 CD 3 site (r jk = nm); (b) The 13 CD 3 site is observed with selective pulses applied to the 13 CO site (r jk = nm); (c) The 13 CO site is observed with selective pulses applied to the 13 CD site (r jk = nm); (d) The 13 CO site is observed with selective pulses applied to the 13 CD 3 site (r jk = nm). Fig.3. Theoretical modulation curves using Eq.[10] (solid lines), numerically simulated modulation curves (dashed gray lines), and experimental measurements (filled circles) for a 1:10 solid solution of [2,3-2 H 4, 13 C 3, 15 N]-L-alanine in 2,3-2 H 4 -L-alanine. The theoretical and simulated curves have been adjusted to take into account the natural abundance 13 C signals from 2,3-2 H 4 -L-alanine. All curves are normalized to have a value of 1 at φ j = π. (a) The 13 CD site is observed with selective pulses applied to the 13 CD 3 site (r jk = nm); (b) The 13 CD 3 site is observed with selective pulses applied to the 13 CO site (r jk = nm); (c) The 13 CO site is observed with selective pulses applied to the 13 CD site (r jk = nm); (d) The 13 CO site is observed with selective pulses applied to the 13 CD 3 site (r jk = nm). 20

21 a C3 3 1 τ r τ r τ r C 0 C 2π/3 C 4π/3 b π π c τ /2 τ /2 (π/2) 0 (π) 0 (π/2) 0 (π/2) φ π/2 π/2 τ r /4 3τ r /4 3τ r /4 τ r /4 Marin-Montesinos et al. Fig.1. 21

22 a b c d φ j / φ j / Marin-Montesinos et al. Fig.2. 22

23 a b c d φ j / φ j / Marin-Montesinos et al. Fig.3. 23