Quadrupolar Isotope-Correlation Spectroscopy in Solid-State NMR

Quadrupolar solid-state NMR carries a wealth of structural information, including insights about chemical environments arising through the determination of local coupling parameters. Current methods can successfully resolve these parameters for individual sites using sample-spinning methods techniques applicable to quadrupolar I ≥ 1 nuclei, provided second-order central transition broadenings do not exceed by much the spinning rate. For large quadrupolar coupling (CQ) values, however, static acquisitions are often preferable, leading to challenges in extracting local structural information. This study explores the use of two-dimensional QUadrupolar Isotope Correlation SpectroscopY (QUICSY) experiments as a means to increase the NMR spectral resolution and enrich the characterization of quadrupolar NMR patterns under static conditions. QUICSY seeks to correlate the solid-state NMR powder line shapes for two quadrupolar isotopes belonging to the same element via a 2D experiment. In general, two isotopes of the same element will have different nuclear quadrupole moments, gyromagnetic ratios, and spin numbers but essentially identical chemical environments. The possibility then arises of obtaining sharp “ridges” in these 2D correlations, even in static samples showing large quadrupolar effects, which lead to second-order line shapes that are several kilohertz wide. Moreover, pairs of quadrupolar isotopes are recurrent in the periodic table and include important elements such as 35,37Cl, 69,71Ga, 79,81Br, and 85,87Rb. The potential of this approach is explored theoretically and experimentally on two rubidium-containing salts: RbClO4 and Rb2SO4. We find that each compound gives rise to distinctive 2D QUICSY line shapes, depending on the quadrupolar and chemical shift anisotropy (CSA) parameters of its sites. These experimental line shapes show good agreement with analytically derived 2D spectra relying on literature values of the quadrupolar and CSA tensors of these compounds. The approach underlined here paves the way toward better characterization of wideline NMR spectra of quadrupolar nuclei possessing different nuclear isotopes.

S2 scaled CQ was used. (a-b) Quadrupolar and isotropic chemical shift only, with (a) = 0 and (b) = 1. In both cases a narrow ridge is preserved. (c-e) Addition of a collinear CSA tensor with a shielding anisotropy of =-13.8ppm. (c-e) Effects of adding and . When these are zero, a narrow correlation is still maintained, albeit with a hook-like shape. Lack of axial symmetry for both tensors, even if still collinear, results in some loss in resolution. This is only exacerbated when the tensors are noncoincident, as shown in (f-h) with a relative orientation of the CSA and quadrupolar tensors of = 90 , =30̊ and =60. Figure S2. 85 Rb CT line shape and intensity for Rb2SO4 as a function of spin-lock length (0-70 ms), collected using an RF amplitude of ca. 26 kHz at three different transmitter frequency offsets: (a) -35 kHz (b) -10 kHz and (c) 18 kHz, marked by the yellow arrows. The sequence is depicted in figure 4b-c, and the echo acquisition time used was 0.64 ms, with a dead time of 37 s before and after each  pulse. Figure S3. 87 Rb CT line shape and intensity as a function of spin-lock length (0-70 ms) using an RF amplitude of ca. 57 kHz for RbClO4 (a) and Rb2SO4 (b). The transmitter offset frequency is marked by the yellow arrows. The sequence is depicted in figure 4b-c, and the echo acquisition time used was 1.28 ms for both RbClO4 and Rb2SO4, with a dead time of 7-8 s before and after each  pulse.

S3
In addition to a conventional CP, this study also considered a variation of CP where the spin lock pulse on the 85 Rb channel was replaced by WURST pulse (WCP), followed by a WURST-CPMG (WCPMG) acquisition block 4 ( Figure S4a-b). The purpose of this variation was to ensure broadband polarization transfer from the 85 Rb channel. This sequence is similar in nature to BRAIN-CP-WCPMG 5 , but involves the reversal of the WURST pulse and spin lock. Admittedly, this method still requires a spin-lock on 87 Rb. In order to incorporate a t1-encoding for the 2D experiment, it is necessary to add an excitation block ( Figure S4b), which is expected to lower the SNR of this experiment by a factor of ~2. The results of a 1D CP-CPMG experiment including both (±1) SQCs on Rb2SO4, were almost identical in SNR and line shape to those of a 1D WCP-WCPMG experiment ( Figure   S4c-d). For the constant-time experiment at t1=0 (maximal signal) the SNR dropped by a factor of 2-3 for the WCP-WCPMG experiment, whereas it remained similar for the CP experiment (due to the lack of flip-back pulse). Due to this difference in signal, we preferred to acquire 2D CP-CPMG experiments at 3 different offsets and sum them.
However, the WCP-WCPMG sequence could be of use for wider patterns, where the excitation block in Fig. S4b, could easily be converted to swept-pulses.

Figure S6
Experimental QUICSY spectra of RbClO4 with whole-echo acquisition. (a) Same raw data as in Figure 6a, but processing only the echo pathway (positive t1s) using 12 t1 increments and phasing the data (sw1=50 kHz). (b) Same parameters and processing as Figure 6a, but with ca. 1/10 th less transients. The data is presented in magnitude. The SNR is lower, but most of the information is preserved (13.5 hr acquisition time).
The frequency-stepped 2D QUICSY acquisitions on Rb2SO4 resulted in small intensity jumps following the summation of the spectra. These jumps occurred in the points of overlap between the ranges covered by the spectral width in the indirect dimension (sw1). For Figure S7 and figure 6, the overlap occurs outside the range [-50 30] kHz due to setting sw1=150 kHz. In order to reduce experimental time, it is possible to set sw1=100 kHz or even lower. Then, some of the overlap will occur in this range, resulting in small intensity jumps marked by arrows ( Figure S8). However, these can be easily corrected in the projection along F1, and are barely noticeable in the 2D contour. Another way to reduce experimental time, is to acquire a slightly asymmetric echo in the t1 domain. In this case, the dispersive component might not be zero, but the effect on the 2D contour is barely visible ( Figure S8). Whole echo acquisition was carried out, with a CPMG echo acquisition time TE=320 s (7 s dead time before and after each  pulse, sw=100 kHz) and a total of 49 t1 increments constituting a symmetric t1-echo (sw1=150 kHz). Notice (by the noise) that the 2D contours and projections of (a)-(c) are not presented in the same intensity scale. The 2Ds in (a) and (c) are presented with 20 contour levels, whereas the 2D in (b) is presented with 30 contour levels.

Figure S8
2D QUICSY spectra of Rb2SO4 acquired at almost the same conditions as Figure S7.
The main difference was setting sw1=100 kHz and a total of 22 t1 increments (not a fully symmetric t1-echo). The data 2D is presented in magnitude.

Figure S9
Difference spectra arising between the experimental data and analytical calculations based on the parameters depicted in figure 6. Both subtracted 2D datasets were normalized to a maximum of 1 before taking their difference, and then subtracted over the same range. Since the experimental data is in magnitude, the analytical calculated was also subtracted in magnitude mode.    (5), was calculated from the Δ there given based on the relation = 2 3 Δ . 4 b The notation Δ used in this publication refers to the used in our study. We converted from Hz to ppm. In these expressions VLAB1 is the orientation-dependent term of a single quadrupolar tensor in the lab frame as written in equation A.11-A.12 of the appendix (VLAB1= 2,−1 2,1 + 1 2 2,−2 2,2 ). VLAB2 is the orientation dependence of a second magnetically inequivalent quadrupolar tensor with an additional transformation as specified in A.23. DLAB1 is the orientation-dependent term of a chemical shift tensor in the presence of a quadrupolar tensor as specified in A.20-A.21 (DLAB1=√ 2 3 20 ). DLAB2 is the orientation dependence of a second magnetically inequivalent CSA tensor with an additional transformation as specified in A.24. S11

Appendix: Tensors, orientations, and transformations considered in this work
As a number of different parametric definitions regarding coupling parameters and rotations of Hamiltonians have been given in the solids NMR literature, we define in a short Appendix the conventions used in this work.
All interactions were represented by second-rank Cartesian tensors 11 . The quadrupolar interaction was described by: and the chemical shift interaction by: In the principal axis system (PAS) each tensor will be characterized by its three principal The chemical shift tensor is described (in ppm) by the following three parameters 13,15 : Typically, one needs to transform the Hamiltonian from its PAS to various frames, 12,11 including the laboratory frame where the secular approximation is taken. If spin anisotropies are described in terms of irreducible tensors involving products of spatial and S12 spin spherical tensors , , then the quadrupolar interaction involve second rank components: Second-order perturbation theory is then necessary to calculate the central-transition frequency of half-integer spins, which is 12 : (2) = − For the chemical shift first-order perturbation theory suffices, leading to 41,12 : which is Equation (3) in the main text. In the present study, the orientation-dependence of the chemical shift was calculated by first transforming this tensor from its PAS into the PAS of the quadrupolar interaction, and then onto the lab frame. This transformation is given in Cartesian components by: Here , the rotation matrix from the PAS of the CSA tensor to the PAS of the EFG tensor, is described by the Euler angles , , : = ( , , ), and is as described above. The elements of the Cartesian tensor can then be used to calculate the chemical shift frequency according to: as expected.

S14
The simulations in the main text called for accounting for a second, chemically equivalent but magnetically inequivalent site, related to one another by a crystal-imposed transformation. The anisotropies of this second site can be accounted for by an additional set of transformations taking the second nucleus' interactions into the quadrupolar PAS of the first nucleus -and proceeding thereafter as above. Denoting for simplicity the second site by a subindex 2, the transformation of its Cartesian EFG tensor from its PAS to the lab frame can be described as: Here 2 = 1 , the tensor of site 1, yet the PAS orientations themselves are not aligned.

A.25
Here DCM is the direction cosine matrix describing the quadrupolar tensor in the crystalframe (e.g., Table 2