Quantitative structure parameters from the NMR spectroscopy of quadrupolar nuclei

Double-rotation NMR

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 [37].

Fig. 1:

The DOR double-rotor system is shown in (a) and the geometry of the rotor is shown in (b). The sample is packed within the smaller inner rotor and is spun simultaneously about two angles.

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 [38]. This has an important impact on the properties of the dipolar coupling Hamiltonian under MAS conditions. The full dipolar Hamiltonian (Ĥ_DD, eq. 2), given below, contains terms where the spins are aligned with the magnetic field (I_z and S_z) 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  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 (arccos1/3 ≈ 54.74 °). 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 (R_eff=R_DD – ΔJ/3). If ΔJ is small, such as with most light elements, R_eff is nearly equal to R_DD. 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) [43].

Fig. 2:

The ¹¹B DOR NMR spectra of B-bromocatecholborane and bis(catecholato)diboron are shown in (a) and (b), respectively, demonstrating the effects of RDC (a) and homonuclear J coupling (b) on the resonances. Adapted with permission from reference [42].

To date, this, and a similar approach using MQMAS [34], 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 [46]. 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 [47], a curious 1:2:2:2:1 multiplet was observed [42], 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 [42] 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 [44].

Dipolar recoupling

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 [48], 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 [49]. 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) [50] and rotational echo adiabatic passage double resonance (REAPDOR) [51] have recently been largely superseded by the more efficient, and easier to analyze, rotary resonance echo saturation pulse double resonance (RESPDOR) [52] and low amplitude/low alpha REDOR (LA-REDOR) [53], as well as its phase-modulated version [30]. These experiments have recently been reviewed in detail by Goldbourt [54], 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) [55] 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 [56], in which ²⁷Al-¹³C recoupling is applied, as well as to a vanadium complex [57], 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.

Fig. 3:

²⁷Al-¹³C S-RESPDOR curves measured for aluminium lactate. The shaded area corresponds to the theoretical recoupling curves’ upper and lower error bounds. The carbon atom associated to each curve is circled on the structures. Reproduced with permission from reference [56].

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 [58]. 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 [61], and dipolar recoupling (applied using HMQC [62, 63], HSQC [64], INEPT [65], and PRESTO [66]) 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 [65] 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 [72]. 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 [73].

Qualitative proximity information may be obtained by using this recoupling sequence to measure double-quantum-single-quantum (DQ-SQ) homonuclear correlation spectra [72], however, unlike spin-1/2 nuclei, the build-up rate of the DQ coherences may not be used to determine distances [33]. 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 [33]. 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 [74]. 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.

Fig. 4:

Left column: normalized RMS deviations between experimental ²³Na CT DQ spinning sideband amplitudes and summed spin-pair simulations in Na₂SO₄ plotted against the maximum Na–Na internuclear distance included in the summation. Blue triangles and black squares correspond to the results obtained in the absence and presence of the chemical shift anisotropy in the simulations, respectively. Right column: dashed sticks depict the experimental CT DQ sideband amplitudes. Solid sticks represent the sum of numerical simulations obtained by including all Na–Na distances up to the R-value marked in the left column. Reproduced with permission from reference [74].

J-resolved experiments

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 [75]. 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 [61].

Unlike the spin-1/2 case, some attempts at performing 2D J-resolved NMR experiments on pairs of quadrupoles had, unfortunately, been unsuccessful [77]. 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 [78]. 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 [79].

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 [44]. 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 [80], 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.

Fig. 5:

¹¹B DQF-J-resolved NMR data for bis(pinacholato)diboron (a), bis(catecholato)diboron (b) and bis(catecholato)diboron coordinated to the iMes carbene (c). Breaking the symmetry of the molecule in (c) reduces the splitting by a factor of 3. The signals from both the three- and four-coordinate boron bonding partners is clearly seen in the two-dimensional spectrum in (c). Adapted with permission from reference [79].

We have applied this methodology towards the characterization of β-boration reagents [80]. 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 [80]. 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 [83]; 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 [84], 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) [85]. 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 [86]. 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.

Fig. 6:

Molecular structures of the singly-, doubly-, and triply-bound diboron compounds studied are shown along with the correlation of their reduced ¹¹B-¹¹B J coupling constants to the reduced ¹³C-¹³C J coupling constants of simple analogous organic compounds. Adapted with permission from reference [85].

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 [87]. 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 [88]. We have then attempted to perform J-resolved experiments under static conditions using only a small fraction of the full powder pattern [89]; digallium compounds were chosen for the task. Modifications of the experiment were however needed such as the removal of the double-quantum filter [79] to avoid lineshape distortions and the echo was shifted in order to obtain purely absorptive 2D lineshapes [90]. 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 [89]; 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) [26] 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) [89] 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.