Nuclear magnetic resonance (NMR) spectroscopy is one of the most important characterization tools in chemistry, however, 3/4 of the NMR active nuclei are underutilized due to their quadrupolar nature. This short review centers on the development of methods that use solid-state NMR of quadrupolar nuclei for obtaining quantitative structural information. Namely, techniques using dipolar recoupling as well as the resolution afforded by double-rotation are presented for the measurement of spin–spin coupling between quadrupoles, enabling the measurement of internuclear distances and connectivities. Two-dimensional J-resolved-type experiments are then presented for the measurement of dipolar and J coupling, between spin-1/2 and quadrupolar nuclei as well as in pairs of quadrupolar nuclei. Select examples utilizing these techniques for the extraction of structural information are given. Techniques are then described that enable the fine refinement of crystalline structures using solely the electric field gradient tensor, measured using NMR, as a constraint. These approaches enable the solution of crystal structures, from polycrystalline compounds, that are of comparable quality to those solved using single-crystal diffraction.
Nuclear magnetic resonance (NMR) spectroscopy has, alongside X-ray diffraction (XRD), become the most important structure determination tool that is available to chemists. Most commonly used in analyzing liquid solutions, for example, for the determination of the molecular structure of organic compounds or the three dimensional structure of proteins, NMR is increasingly being applied to solve the structure of solids. The use of solid-state NMR (SSNMR) for the determination of crystalline structures, a feat more commonly associated to XRD, has been gaining importance [1, 2]. IUPAC has in fact recently appointed the first commission on NMR crystallography at the International Union of Crystallography’s General meeting in August 2014 in Montreal . Although XRD will likely always dominate the field, SSNMR does have several advantages for studying the structure of solids, when compared to XRD. For example, polycrystalline samples can be analyzed by SSNMR, large single crystals are not needed, and non-crystalline samples can be studied just as easily. In fact, many of the species of current interest to chemists, such as heterogeneous catalysts , mesoporous materials , polymers , nanoparticles , powdered pharmaceuticals , and glasses  cannot be studied using single-crystal diffraction techniques and SSNMR can be used as an alternative. There is, however, no clear path towards the determination of the crystal structure of an arbitrary compound from NMR data.
Several approaches to NMR crystallography include, for example, structure prediction software to aid in the determination of a crystalline structure [10, 11]. Computationally-predicted chemical shift tensors have also been used as restraints for crystal structure refinements akin to the Rietveld approach used in powder XRD [12–14]. Most promising, however, is the use of the measurements of dipolar coupling between various nuclei. The dipolar coupling constant (RDD) depends only on the internuclear distance between the atoms and thus provides very strong experimental structure restraints (eq. 1) [15–17]. In fact, if the dipolar coupling between all atoms in a system could be measured, the structure solution would be fairly direct, and this has been applied to some zeolites [18, 19] and aluminophosphates [20, 21], however, in general, this approach is far from trivial.
In eq. 1 μ0 is the permittivity of free space, γ1 and γ2 are the magnetogyric ratios of the coupled nuclei, ћ is the reduced Planck constant, and r1,2 is the internuclear distance.
Aside from sensitivity, one of the largest hurdles for the application of NMR crystallography, specifically for materials chemistry, is the predominance of quadrupolar nuclei across the periodic table [22–24]. Just under 3/4 of all the NMR active nuclei are quadrupolar; that is, they have spin quantum numbers greater than 1/2. Quadrupolar nuclei have a non-spherical charge distribution and thus possess a nuclear electric quadrupole moment that couples to the electric field gradient (EFG) at the nucleus generated by the electrons and nuclei in a compound. This quadrupolar coupling often causes overwhelming anisotropic broadening of the NMR resonances thus obscuring all information from the weaker chemical shift and spin–spin coupling interactions. Clearly, the extension of NMR crystallography to quadrupolar nuclei is highly desired but different approaches from those proposed for spin-1/2 nuclei are necessary. This article will summarize some approaches to obtaining quantitative structural parameters from the NMR spectroscopy of quadrupolar nuclei, with particular emphasis on the results from my doctoral work.
Measuring spin–spin coupling
As was mentioned, the most straightforward approach for solving solid structures from SSNMR is through the measurements of spin–spin coupling. Different nuclei can couple either through the electrons participating in bonding interactions (J coupling) or directly through space (dipolar coupling). Both interactions yield extremely valuable information that can be used to both map the bonds between the atoms in a compound as well as obtain distance restraints that can elucidate the packing order and conformations of the species . Although the measurement of spin–spin coupling between spin-1/2 nuclei and even between a quadrupolar nucleus and a spin-1/2 nucleus is a well-developed field [26–30], measuring spin–spin coupling in pairs of quadrupolar nuclei is far from trivial [31–34]. This is the case since the dominating quadrupolar interaction is orders of magnitude stronger than the spin–spin coupling interactions (several MHz vs. a few Hz to kHz). The work presented in the following sections will highlight some approaches using dipolar recoupling, double-rotation (DOR), and J-resolved NMR to the measurement of spin–spin coupling involving quadrupolar nuclei.
The largest spin–spin coupling constants involving spin-1/2 nuclei can be easily measured from the resulting spectral splittings in the one-dimensional magic-angle-spinning (MAS) SSNMR spectra. As the second-order quadrupolar broadening is not averaged by MAS, these splittings typically cannot be observed in 1D NMR spectra for quadrupolar nuclei. The second-order quadrupolar interaction can, however, be averaged by spinning the sample about two angles simultaneously using a specialized NMR probe (see Fig. 1), a technique known as double-rotation (DOR) [35, 36]. The resolution provided by DOR can, in principle, match that which is obtained from the MAS NMR of spin-1/2 nuclei; yielding a simple way to probe spin–spin coupling between quadrupolar nuclei .
Residual dipolar coupling
The quadrupolar interaction has the interesting effect of ‘tilting’ the quantization axis of equilibrium magnetization away from the axis of the large applied magnetic field . This has an important impact on the properties of the dipolar coupling Hamiltonian under MAS conditions. The full dipolar Hamiltonian (ĤDD, eq. 2), given below, contains terms where the spins are aligned with the magnetic field (Iz and Sz) as well as perpendicular to it (I+, I–, S+, and S–). For spin-1/2 nuclei, which are aligned perfectly with the magnetic field, only the Â term has significance, and due to this term’s 3cos2θ-1 dependence (θ corresponds to the angle between the internuclear vector and the applied magnetic field), dipolar coupling is averaged under MAS conditions. The magic angle, by definition, is the spinning angle that makes this term average to zero The higher-order terms containing magnetization oriented perpendicularly with respect to the magnetic field do not share this orientational dependence and are thus not removed by MAS. The MAS, and DOR, NMR spectra of quadrupolar nuclei are then affected by the dipolar coupling between quadrupolar spins; this is known as residual dipolar coupling (RDC) [39, 40]. In principle, the measurement of internuclear distances between quadrupolar nuclei is then as simple as acquiring and analyzing a DOR NMR spectrum.
We have demonstrated that it is indeed possible to measure dipolar and J coupling between quadrupoles for numerous spin pairs and compounds [41, 42]; see Fig. 2a for the 11B DOR NMR spectrum of B-bromocatecholborane showing J and dipolar coupling between 11B and 79/81Br, all of which have spins of 3/2. Note that the anisotropic part of the J coupling tensor (ΔJ) cannot be distinguished from dipolar coupling and only an effective dipolar coupling constant is measured (Reff=RDD – ΔJ/3). If ΔJ is small, such as with most light elements, Reff is nearly equal to RDD. Otherwise, as is the case here, internuclear distances must be measured independently from diffraction experiments in order to allow the determination of ΔJ. The DOR NMR experiment on B-bromocatecholborane enabled the first measurement of ΔJ between quadrupolar nuclei. In order to analyze this data, additional software needed to be developed since the quadrupolar interaction needs to be treated exactly due to its immense size. The quadrupolar interaction at both nuclei was treated using a diagonalization approach and the resulting ‘tilting’ of the quantization axes are used to calculate the eigenvalues of the dipolar coupling Hamiltonian, which is averaged over a rotation period. This software also led to the development of the QUadrupolar Exact SofTware (QUEST) .
To date, this, and a similar approach using MQMAS , is the only approach enabling the determination of either J or dipolar coupling between pairs of quadrupolar nuclei in the case where one spin’s quadrupolar interaction is too strong to enable the detection of that isotope by NMR.
Homonuclear J coupling
In the case of homonuclear spin systems, an additional complication occurs due to the mixing of the eigenstates of the two coupled spins [44, 45]. This mixing leads to the common high-order multiplets in solution NMR spectra of spin-1/2 nuclei, such as ‘AB’ and ‘A2’ spin systems . A well-known feature of this mixing is that if two spin-1/2 nuclei are in an identical chemical environment, and share the same coupling strengths to all other nuclei in the molecule, then the J coupling will not affect the NMR spectrum. This phenomenon, which is taught to all undergraduate chemistry students, was often thought to be a general feature in the NMR spectroscopy of all nuclei.
In an effort to measure RDC from a homonuclear dipolar coupled spin pair in bis(catecholato)diboron (11B) using DOR NMR, something that had been already accomplished using MQMAS , a curious 1:2:2:2:1 multiplet was observed , see Fig. 2b. This spectral feature could only be explained by reevaluating the NMR wavefunction of the system. Since the molecule has inversion symmetry, both boron atoms are magnetically equivalent, in other words, they form an A2 spin system. All of the degenerate eigenstates then need to be mixed to form the corresponding symmetric and antisymmetric eigenstates. This treatment predicts a 1:2:2:2:1 pentet, as observed experimentally, where the splittings between the resonances are, curiously, equal to 3/2 J. The spectral splittings found in the NMR multiplet are then magnified by the high symmetry of the molecule. Significantly, this also proves that, in the case of quadrupolar nuclei, J coupling can be measured in pairs of magnetically equivalent spins.
Similarly, an A2 multiplet was measured in dimanganese decacarbonyl  for which there is a metal–metal bond connecting two manganese atoms. 55Mn has a spin of 5/2 and a 1:2:3:3:2:1 hextet with unequal line spacings was observed, as was predicted by the theory. Numerically exact calculations also predict the same fine structure as the simple state mixing model; however, these may also be applied in the case of tightly coupled AB spin systems .
In cases when the dipolar coupling is too weak to directly affect the multiplicities of the lines, a dipolar recoupling approach may be used. It is beneficial to maintain the resolution afforded by MAS as this allows the distinction of the resonances from various nuclei and the selective measurement of pairwise distances. MAS, however, also has the effect of largely averaging the dipolar interaction which holds the distance information, leaving only the minute effects of RDC. The dipolar coupling can, fortunately, be selectively reintroduced by interrupting the MAS averaging by the application of rotor synchronized RF pulses. For example, in the rotational-echo double-resonance (REDOR) scheme , two inversion pulses are applied to one nucleus every rotor cycle to interrupt the averaging of the dipolar coupling while a second nucleus is observed. As a result of the recoupling, the signal intensity of the observed nucleus is attenuated and this attenuation is modulated by the strength of the dipolar interaction. The REDOR curve can be fit using the internuclear distance as the sole variable.
The applicability of the REDOR experiment to quadrupolar nuclei is, however, very limited to the case when the nuclei have very small quadrupolar coupling. In most situations, the quadrupolar nucleus cannot be properly inverted by a RF pulse and the REDOR experiment becomes less efficient . For this reason an arsenal of recoupling schemes applicable to quadrupolar nuclei have been developed which rely on the scrambling of the states of a quadrupolar nucleus when a long, low-power, RF pulse is applied. The original sequences based on this principle: the transfer of population in double resonance (TRAPDOR)  and rotational echo adiabatic passage double resonance (REAPDOR)  have recently been largely superseded by the more efficient, and easier to analyze, rotary resonance echo saturation pulse double resonance (RESPDOR)  and low amplitude/low alpha REDOR (LA-REDOR) , as well as its phase-modulated version . These experiments have recently been reviewed in detail by Goldbourt , the exact specifics of these experiments are beyond the scope of this article. Note that these experiments are generally only applicable to the case of a spin-1/2 nucleus that is coupled to a quadrupolar spin since complex recoupling schemes need to be applied to the spin-1/2 nucleus.
Pourpoint and co-workers have recently demonstrated the utility of the symmetry-based-RESPDOR (S-RESPDOR)  experiment in mapping out the structures of metal complexes. For example, the rigorous measurement of the internuclear distances between the 13C spins of a ligand, whose resonances are generally well-resolved in an MAS spectrum, and the central metal atom can be used not only as a means of resonance assignment but also to determine the three-dimensional conformation of a complex. They have applied this methodology to the NMR crystallography of a series of aluminium complexes , in which 27Al-13C recoupling is applied, as well as to a vanadium complex , in which 51V-13C recoupling is performed. Example recoupling curves are shown in Fig. 3 below for the case of aluminium lactate. The precise distances to the aluminium center can be measured for all three of the different carbon sites, providing strong structural constraints of use in NMR crystallography.
Li and co-workers have recently applied the LA-REDOR experiment in order to measure the nitrogen-vanadium distance in a series of oxovanadium complexes and determine the order of the bond connecting the atoms . The knowledge of this bond order is important in the determination of the geometrical position of the nitrogen atom of the ligand with respect to the terminal oxo group. By measuring the 15N-51V internuclear distance they were able to determine that the bond length was consistent with it being a single bond, in agreement with the expected molecular structure of the species.
Proximities between spin-1/2 and quadrupolar nuclei can also be probed with the use of heteronuclear correlation experiments. Numerous schemes have been presented to this means using cross-polarization [59, 60], INEPT , and dipolar recoupling (applied using HMQC [62, 63], HSQC , INEPT , and PRESTO ) as a means of transferring polarization from one spin to the other. Dipolar recoupling-based experiments appear to be the most efficient as the number of pulses applied to the quadrupolar nucleus can be minimized and the dipolar coupling is generally stronger than the J coupling. These techniques have been extensively reviewed [26, 67, 68]. The most useful of these experiments, for NMR crystallographic purposes, are those which also make use of the resolution available from MQMAS and STMAS experiments [65, 69–71]. In these cases, high resolution can be obtained along both the spin-1/2 and quadrupolar nucleus’ dimensions enabling the identification of a greater number of cross-peaks.
Taulelle and co-workers have notably applied the MQ-D-R-INEPT experiment  for the acquisition of high-resolution 27Al-31P correlation spectra of aluminophosphate (AlPO) samples. Instead of determining precise conformations via the determination of a large number of internuclear distances, they used through-space correlation experiments in order to determine the appropriate building blocks that composed the AlPO [20, 21]. The knowledge of these building blocks can then be used to accelerate ab initio crystal structure determination from powder diffraction, an approach they have termed NMR-driven crystallography. Without the knowledge afforded by the HETCOR experiments, the solution of the crystal structures of some of these species would be beyond our current computational means. Clearly, this is a very powerful approach to NMR crystallography that can only be improved upon the inclusion of a larger number of intra and inter-molecular correlations, ideally for all pairs of nuclei in a sample.
The recoupling of dipolar interactions between quadrupolar nuclei is, unfortunately, much more challenging than in the case of a spin-1/2 and a quadrupolar spin. In pairs of quadrupolar nuclei, the recoupling pulse sequence needs to be applied to the quadrupolar isotope which leads to a tremendous loss in sensitivity from the difficulty in performing precise manipulations of a quadrupolar nucleus . In principle, the central transition (CT, m=1/2 to –1/2 transition) may be manipulated selectively using low-power pulses, however, this limits the recoupled signal to 1/(S+1/2) and leads to offset dependencies due to the use of low RF power. The later issue has been greatly improved with the development of the BR212 supercycled symmetry-based R212 sequence, which is simply composed of a CT-selective inversion pulse every rotor period .
Qualitative proximity information may be obtained by using this recoupling sequence to measure double-quantum-single-quantum (DQ-SQ) homonuclear correlation spectra , however, unlike spin-1/2 nuclei, the build-up rate of the DQ coherences may not be used to determine distances . Brinkmann and co-workers have, however, recently demonstrated that accurate internuclear distances may be obtained by measuring the double-quantum sidebands of a DQ-SQ correlation experiment on quadrupolar nuclei . The intensities of these sidebands depend nearly exclusively on the dipolar coupling constant and can be used to measure accurate internuclear distances in isolated spin pairs. Brinkmann further expanded this technique to general solids containing non-isolated spin pairs and demonstrated that the double-quantum sideband pattern may be simulated directly from a given crystal structure . Via homonuclear distance measurements between quadrupolar isotopes, a crystal structure may in principle be validated, or refined, on the basis of the spread of internuclear distances that are observed. The 23Na DQ sidebands acquired for sodium sulfate at three different recoupling times are shown in Fig. 4, below. It can be seen that the experimental DQ sidebands are very well reproduced using only the crystal structure as input. The use of longer recoupling times increases the radius within which the dipolar interactions need to be considered, thus allowing the study of the crystal packing.
Although J coupling to a spin-1/2 nucleus may be measured by MAS NMR and, similarly, J coupling to quadrupoles can be measured by DOR NMR, the most accurate J coupling constants are hardly ever measured by 1D NMR spectroscopy. 2D J-resolved experiments typically provide a much higher resolution of the J coupling over 1D NMR since the line width is solely determined by the spin–spin relaxation time constant: T2 . J-resolved NMR experiments would also no longer need high resolution in the directly directed NMR signal since the multiplet is detected indirectly by its modulation of an echo signal. This is highly desired as conventional NMR hardware could be used to perform such experiments instead of an expensive, specialized, DOR probe, for example.
In a 2D J-resolved experiment, a spin is excited using a 90° pulse and is allowed to freely evolve for a period of time of t1. An inversion pulse is, however, applied in the middle of the evolution period to the nuclei whose J coupling is desired. This refocuses the chemical shifts, but leaves the J coupling untouched. The signal amplitude is then modulated by the J coupling frequency and a Fourier transform yields the J coupling multiplets free of chemical shift effects with a higher resolution. J-resolved experiments have been performed for measuring precise J coupling between spin-1/2 and quadrupolar isotopes [61, 76]. CT-selective pulses need to be applied to the quadrupolar nucleus meaning that only 1/(S+1/2) of the signal is J modulated. This has been greatly applied towards the acquisition of HETCOR spectra between spin-1/2 and quadrupolar nuclei, most notably between 31P and 27Al, using the INEPT polarization transfer approach .
Unlike the spin-1/2 case, some attempts at performing 2D J-resolved NMR experiments on pairs of quadrupoles had, unfortunately, been unsuccessful . From our previous theoretical treatment of the 1D DOR NMR multiplets it became clear why conventional J-resolved experiments fail when applied to quadrupolar nuclei. Unlike spin-1/2 nuclei, all of the transitions cannot be simultaneously manipulated by radio frequency pulses due to the shear breadth of the lineshapes . Instead, only the CT can be manipulated with any degree of accuracy. Therefore, only spin pairs where both spins were simultaneously in their 1/2 or –1/2 states would modulate the NMR signal. The remaining signal, which is not modulated, forms a strong peak at zero frequency in the center of the spectrum. We have then developed specialized MAS J-resolved pulse sequences including a double-quantum filter to remove this unwanted non-modulated signal .
Aside from being generally accessible to all chemists having MAS NMR hardware, these experiments also provide a much improved resolution and ease of interpretation over the DOR NMR experiments. Only a single doublet is observed for every spin pair and the effects of second-order quadrupolar coupling and RDC are completely refocused and do not affect the end result . Additionally, we observed that the splitting is amplified, by a factor of (2S+3)(2S – 1)/4, corresponding to 3, 8, 15, and 24 for nuclei with spins of 3/2, 5/2, 7/2, and 9/2, respectively in A2 spin systems. This very interesting feature has several applications. Namely, it can be used as a very stringent test of the molecular symmetry of a compound; where symmetric molecules would feature amplified splittings, and it can be used to magnify small coupling constants in order to facilitate the measurement of small J coupling across multiple intervening bonds, for example. In fact, we have used this effect to measure the small 11B–11B two-bond J coupling constant in solid 9-BBN , the important hydroboration reagent. We have proven this amplification effect of the J splittings by breaking the symmetry of bis(catecholato)diboron via a reaction with an N-heterocyclic carbene. As is shown in Fig. 5, upon coordination, the J splitting is reduced by a factor of three.
We have applied this methodology towards the characterization of β-boration reagents . The β-boration reaction is an important method towards the production of organic compounds featuring B–C bonds for use in cross-coupling reactions [81, 82]. Generally, nucleophilic boron moieties are obtained by the activation of diboron compounds either by using a metal catalyst, or by generating a sp2-sp3 diboron compound such that the B–B bond is weakened and polarized. We have shown using a localized molecular orbital approach that the experimentally measured J(11B, 11B) coupling constants are strongly correlated to the strength, length, and s-character of the boron–boron bond . The fact that the strength of the boron–boron bond, a strong indicator of the success of a reagent, can be experimentally determined using a sensitive SSNMR experiment prior to performing extensive synthetic work, supports the application of this methodology as a rapid screening method.
Most recently, there has been some controversy regarding the nature of the boron–boron bond in Braunschweig and co-workers’ diboryne compound ; the only stable species to have been synthesized featuring a boron–boron triple bond. Given the success of our method at characterizing the electronic character of boron–boron single bonds, we have decided to investigate this compound using 11B double-quantum-filtered J-resolved NMR. By studying this compound, as well as two diborenes and a diboracumulene , we were able to observe a strong correlation between the J(11B,11B) coupling constants in these compounds and the J(13C, 13C) coupling constants in analogous organic compounds (see Fig. 6) . Additionally, the J(11B, 11B) coupling constant is in quantitative agreement with the predicted value based on the J(13C,13C) coupling in acetylene, when considering the fact that boron has five electrons as opposed to carbon’s six . Since the bonding electrons are mainly responsible for the J coupling, this provides strong experimental evidence, as opposed to computational evidence, in support of the classification of the boron–boron bond in the diboryne as a triple bond.
The MAS double-quantum-filtered J-resolved experiments work remarkably well for 11B and other quadrupolar nuclei with weak quadrupolar coupling; however, they are not applicable to the vast majority of quadrupolar nuclei whose lineshapes are far broader than the achievable MAS spinning speeds . In fact, for many quadrupolar isotopes, it is necessary to acquire the NMR spectra piecewise in non-spinning samples since the full breadth of the spectra cannot be excited at a single transmitter offset . We have then attempted to perform J-resolved experiments under static conditions using only a small fraction of the full powder pattern ; digallium compounds were chosen for the task. Modifications of the experiment were however needed such as the removal of the double-quantum filter  to avoid lineshape distortions and the echo was shifted in order to obtain purely absorptive 2D lineshapes . Using this experiment not only is it possible to measure the J coupling in ultra-wideline NMR situations, but the effective dipolar coupling between the gallium atoms could also be measured ; dipolar coupling being related to the internuclear distance. Additionally, it was discovered that, under static conditions, the sense of the powder patterns is reversed if the spins are magnetically equivalent; a fact that allowed us to solve the 3D structure of a digallium compound whose crystal structure had not yet been determined.
The suite of methods presented in this section enables the measurement of J and dipolar coupling between heteronuclear spin pairs involving a spin-1/2 nucleus (dipolar recoupling and J-resolved)  or pairs of quadrupolar nuclei where one spin has strong quadrupolar coupling (DOR) [41, 42] as well as between homonuclear spins having either weak (dipolar recoupling, DOR, and J-resolved) [42, 44, 45, 74, 79, 80, 85] or strong (J-resolved)  quadrupolar coupling. This enables chemists to probe internuclear distances, connectivities, and electronic structure in nearly any arrangement of quadrupolar nuclei; an important step towards ab initio crystal structure solution from solid-state NMR.
EFG-based NMR crystallographic structure refinement
Although the clearest structural information is obtained by measuring spin–spin coupling, in many situations the coupling may be too weak or the quadrupolar interaction may simply be too strong. In those cases only the quadrupolar coupling can be measured with a significant degree of accuracy, permitting the measurement of the EFG tensor. Luckily, although the EFGs cannot be translated directly into structure, it is an electrostatic property that can be quickly calculated from any electronic-structure method  or even a simplistic point charge model , albeit with lessened accuracy. This is in contrast to the chemical shift tensors, that have been used in structure refinement protocols [12–14], which are relatively expensive to calculate . Therefore, assuming that the EFG tensor components can be predicted with a high degree of accuracy, a model structure can be refined directly against the EFGs in order to yield a higher quality structure.
This approach to crystal structure refinement was first implemented by Widdifield and co-workers who had noticed that the 79/81Br quadrupolar coupling constant in MgBr2, calculated using the projector-augmented-wave (PAW) density functional theory (DFT) method , was in disagreement with the experimentally determined value . This was in contrast to other bromide salts whose EFG tensor components were well reproduced by PAW DFT. The crystal structure of MgBr2 is highly symmetric and featured only one variable coordinate: the ‘c’ coordinate of the bromine atoms, which was originally assumed to being 0.25. They noticed that the variation of this coordinate by as little as 0.04 Å would vary the CQ by as much as 11 MHz (i.e. 50% its experimental value). By optimizing this coordinate they could then both improve the experimental agreement of the quadrupolar coupling as well as the DFT-calculated energy of the crystal, thus showing the EFG tensor’s remarkable sensitivity to structure and its utility as a fine refinement parameter. A plot showing the variability of the 81Br CQ value as a function of the ‘c’ coordinate for MgBr2 is shown in Fig. 7.
Wang and co-workers applied a similar methodology for the refinement of the niobium coordination geometry in triple-layered perovskites . Instead of using DFT calculations to predict the EFG tensor components, an electrostatic model was used enabling the solution of closed form equations relating the quadrupolar coupling parameters to the coordination geometry of the niobium.
We have decided to generalize this NMR crystallographic approach such that it can be applied to any material. Like Widdifield, we chose to use the PAW DFT method for the prediction of the EFG tensor components since, as shown in Fig. 8, it reproduces experimental EFG tensor components with a staggering degree of accuracy for many elements . Using PAW DFT calculations and experimentally-determined EFG tensor components, the quality of a given model structure can be quantified using the following χ2 parameter (eq. 3), which tends towards zero in the case of perfect agreement with experiment .
In the above equation, S corresponds to an atom in the unit cell, Vii are the EFG tensor components with a standard deviation of σ, and E and Eopt are the DFT-computed energies of the model and the lowest energy structure. β is an experimentally optimized scaling factor and α represents the constant by which the PAW DFT calculations overestimate the EFG tensor components.
This method was initially applied, and cross-validated, using the 11B, 17O, 23Na, and 27Al SSNMR data of sodium aluminoborate (Na2Al2B2O7) . Na2Al2B2O7 is a non-linear optical material produced using a high temperature synthesis [99, 100]. This procedure precludes the possibility of growing high quality single crystals thus rendering the task of solving a high resolution crystal structure extremely challenging. Our approach, however, does not share the same stringent sample requirements as XRD and we were able to solve a high resolution crystal structure for this material .
We have then sought to improve this method by combining it with dipolar coupling measurements, as mentioned in the previous section. For this we have studied sodium pyrophosphate samples in which it is possible to perform 23Na spin diffusion 2D correlation experiments under DOR conditions , enabling the qualitative measurement of 23Na-23Na dipolar coupling . This necessitated the expansion of the rate matrix model for spin diffusion to quadrupolar nuclei. Although the dipolar coupling could not be used in fine structural refinements, due to its low accuracy, it was useful for the crucial prior step of resonance assignment .
Lastly, we have applied our EFG-based NMR crystallographic structure refinement method towards the solution of the crystal structure of the near-zero thermal expansion material: ZrMgMo3O12 . In this case we needed to solve the structure ab initio as no structural model existed. Accurate unit cell dimensions and possible space groups could be determined from powder XRD data which enabled the ab initio solution of a model structure using a Monte Carlo-type crystal structure solution program . This XRD structure was not, however, chemically reasonable; even after the application of a Rietveld refinement. Using 25Mg, 91Zr, and 95Mo SSNMR data we were able to obtain a high quality crystal structure for this compound (see Fig. 9). 17O SSNMR data was later acquired and used for cross-validation. As seen in Fig. 9, the 17O SSNMR data is best reproduced using the NMR crystallographic structure, even when compared to a DFT-optimized structure, further showing the importance of incorporating experimental data into the structure solution process.
The quadrupolar exact software (QUEST)
The techniques presented in the previous sections, or any other approaches aimed at extracting structural information from an NMR spectrum, require the accurate measurement of the EFG tensor. This can at times be complicated by the fact that, as an electrostatic interaction, the quadrupolar interaction can be quite large, even when compared to the Zeeman interaction. In those situations when the quadrupolar interaction is of similar magnitude to the Zeeman interaction the so called high-field approximation is no longer valid [104, 105]. Simulations treating the quadrupolar interaction as a small perturbation to the Zeeman interaction then fail at reproducing experiment and yield wrong numbers when used to fit experimental data. In principle, only a truly general, exact, treatment of the quadrupolar interaction can be trusted; however, no such user friendly software was available to the scientific community.
Given the similarity between this simulation problem and the process involved in simulating RDC multiplets [41, 42], we were able to adapt the previous code and write the QUadrupolar Exact SofTware (QUEST) . QUEST is a graphical program that can simulate exact NMR spectra for any quadrupolar nucleus at any applied magnetic field strength (including zero field). The program diagonalizes the full Zeeman-quadrupolar Hamiltonian and uses the eigenvalues to calculate the resonance frequencies and the eigenvectors to calculate the transition probabilities. The user interface for QUEST is pictured in Fig. 10. Using efficient interpolation schemes, exact NMR spectra can be calculated in fractions of seconds, thus disposing of the need for perturbation theory entirely.
The importance of using an exact treatment of the quadrupolar interaction was demonstrated by measuring, for the first time, the 35Cl NMR spectra for some organic, covalently-bound, chlorine sites . We showed that the asymmetry parameter of the EFG tensor and the chemical shift of 35Cl are strong indicators of the chlorine’s chemical environment. Perturbation theory, however, leads to errors in the measured chemical shifts and quadrupolar coupling constants on the order of 600 ppm and 700 kHz, respectively. The differences in the spectra simulated using QUEST and software using second-order perturbation theory are shown in Fig. 11.
We have shown that tremendous structural detail can be extracted from the sometimes overwhelming NMR spectra of quadrupolar nuclei. Using DOR it is possible to resolve the fine structure of the NMR lines that is usually obscured by quadrupolar coupling. In heteronuclear spin systems, RDC can be measured enabling the determination of dipolar coupling, and internuclear distances, as well as J coupling. In the case of homonuclear spin systems, DOR data revealed that, surprisingly, the J splittings are not eliminated in the case of magnetically equivalent pairs of quadrupolar nuclei, as they are for spin-1/2 nuclei. This discovery, lead to a greater understanding of the spin–spin coupling between quadrupolar spins and the development of J-resolved NMR pulse sequences that can be used measure J and dipolar coupling between quadrupolar nuclei with a high degree of accuracy. These experiments are very easy to implement and analyze, yielding a single doublet for each spin pair in the sample. Interestingly, the splitting of the doublet is amplified when the spins are magnetically equivalent, yielding additional structural information and enabling the measurement of smaller J coupling constants.
Much progress has also been done in the field of dipolar recoupling to quadrupolar nuclei. This has permitted the development of advanced NMR crystallographic approaches that take advantage of this dipolar coupling for studying the local coordination of metal complexes, the solution of crystal structures from correlation experiments involving quadrupolar nuclei, as well as probing the crystal packing.
It was also shown that the quadrupolar coupling itself may be used as a structural restraint in crystal structure refinements, enabling the solution of high resolution crystal structures in polycrystalline samples. This approach has been applied to a number of systems including a non-linear optical material and a near-zero thermal expansion material. Notably, this method seems to yield higher quality crystal structures than DFT-only based crystal structure refinements.
Lastly, we have written a user friendly and graphical program for the simulation of exact NMR spectra of quadrupolar nuclei: QUEST. QUEST has been notably used in order to extract chemical information from 35Cl SSNMR spectra of organic molecules. Others have used QUEST as well to analyze 185/187Re , 75As , and 33S  SSNMR spectra, for example.
The utilization of quadrupolar nuclei for NMR crystallography is still a very young and underdeveloped field of research. New NMR crystallographic approaches beyond the selected examples presented here are expected to appear in the coming years, furthering the applicability of the NMR spectroscopy of quadrupolar nuclei towards the solution of structural problems and the determination of quantitative structural parameters.
A collection of peer-reviewed articles by the winners of the 2015 IUPAC-SOLVAY International Award for Young Chemists.
I would firstly like to thank IUPAC for providing me with the opportunity of writing this short review. NSERC is acknowledged for a graduate scholarship. Current support is from a Spedding fellowship funded by the LDRD program. Most importantly, I would like to thank my thesis advisor Prof. David L. Bryce for his guidance and support throughout my graduate work. Prof. Bryce is also kindly thanked for his useful comments regarding this article.
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