Journal of Magnetic Resonance 242 (2014) 23–32

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Journal of Magnetic Resonance

journal homepage: www.elsevier .com/locate / jmr

Theoretical study of homonuclear J coupling between quadrupolarspins: Single-crystal, DOR, and J-resolved NMR

http://dx.doi.org/10.1016/j.jmr.2014.01.0151090-7807/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Fax: +1 613 562 5170.E-mail address: dbryce@uottawa.ca (D.L. Bryce).

Frédéric A. Perras, David L. Bryce ⇑Department of Chemistry and Centre for Catalysis Research and Innovation, University of Ottawa, 10 Marie Curie Private, Ottawa, Ontario K1N 6N5, Canada

a r t i c l e i n f o

Article history:Received 23 October 2013Revised 12 January 2014Available online 8 February 2014

Keywords:Numerical simulationsJ-resolvedDORQuadrupolar nucleiJ couplingMagnetic equivalence

a b s t r a c t

The theory describing homonuclear indirect nuclear spin–spin coupling (J) interactions between pairs ofquadrupolar nuclei is outlined and supported by numerical calculations. The expected first-order multi-plets for pairs of magnetically equivalent (A2), chemically equivalent (AA0), and non-equivalent (AX)quadrupolar nuclei are given. The various spectral changeovers from one first-order multiplet to anotherare investigated with numerical simulations using the SIMPSON program and the various thresholdsdefining each situation are given. The effects of chemical equivalence, as well as quadrupolar coupling,chemical shift differences, and dipolar coupling on double-rotation (DOR) and J-resolved NMR experi-ments for measuring homonuclear J coupling constants are investigated. The simulated J coupling mul-tiplets under DOR conditions largely resemble the ideal multiplets predicted for single crystals, and acharacteristic multiplet is expected for each of the A2, AA0 , and AX cases. The simulations demonstratethat it should be straightforward to distinguish between magnetic inequivalence and equivalence usingJ-resolved NMR, as was speculated previously. Additionally, it is shown that the second-order quadrupo-lar-dipolar cross-term does not affect the splittings in J-resolved experiments. Overall, the homonuclear J-resolved experiment for half-integer quadrupolar nuclei is demonstrated to be robust with respect to theeffects of first- and second-order quadrupolar coupling, dipolar coupling, and chemical shift differences.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Since its discovery over 60 years ago [1], indirect nuclear spin–spin (J) coupling has been used to gain insight into chemical struc-ture [2–6]. Unlike direct dipolar coupling, which is a through-spaceinteraction, the J coupling interaction is mediated by the electronsthat form bonds [7,8]. As J coupling is sensitive to bonding interac-tions, it has been used to probe the nature of hydrogen bonds [9–11], CH–p interactions [12], as well as van der Waals’ interactions[13]. Although the first direct observation of J coupling involved aquadrupolar nucleus (spin S > 1/2), 121Sb [14], the vast majority ofthe applications of J coupling have focussed solely on spin-1/2 nu-clei [15]. This is because the quadrupolar interaction leads to rapidrelaxation in solution, which often severely broadens the lines andobscures the fine structure of the resonances. In solids, anisotropicline broadening is present which cannot be completely removed bymagic angle spinning (MAS) and thus also obscures fine structurein conventional NMR experiments.

A great deal of progress has however been made in recent yearsfor studying J coupling involving quadrupolar spins. Heteronuclear

J coupling information for both spin-1/2-quadrupole [16–22] andquadrupole–quadrupole [23,24] spin pairs has been obtained withthe use of multidimensional heteronuclear correlation experi-ments. Precise J coupling constants have also been measured forspin-1/2-quadrupole spin pairs with the use of central transition(CT) selective J-resolved experiments [25,26]. There are very fewreports of the measurement of J coupling involving two quadrupo-lar nuclei in condensed phases. These mostly originate from themeasurement of solution NMR spectra of highly symmetric mole-cules [27–32] and residual dipolar/J coupling multiplets in MAS[33], double-rotation (DOR) [34,35], and multiple quantum MAS(MQMAS) [36,37] NMR spectra of powders.

Recently, we have shown that J coupling can be measured insolids for pairs of magnetically equivalent quadrupolar nuclei[35]. The use of high-resolution DOR NMR, where the sample isspun simultaneously about two angles, the magic angle of 54.74�and an angle of 30.56� [38], was however necessary in order to re-solve the unusual multiplets. This led us to develop homonuclear2D MAS J-resolved experiments [39,40] that can be used to mea-sure highly accurate J coupling constants for pairs of quadrupolarnuclei [41]. These methods show promise for studying bondinginteractions between homonuclear quadrupolar nuclei such asthose found in homometallic metal–metal bonds [42,43], diboron

24 F.A. Perras, D.L. Bryce / Journal of Magnetic Resonance 242 (2014) 23–32

reagents [44,45], and borate materials [46], as well as oxygen–oxy-gen coupling in materials [47] and biological systems [48,49].

In solids, nuclei are said to be chemically equivalent (AA0) whenthey are related by a symmetry operation. The nuclei then have thesame tensor magnitudes and give the same powder NMR spectra. Ifthese nuclei are related by an inversion center, or if there is a C3

rotation axis, or higher, along the internuclear vector, the nucleiadditionally share the same tensor orientations and thus resonateat the same frequencies for all crystallite orientations [50]. Thesespin pairs are termed magnetically equivalent (A2) and J couplingdoes not affect the NMR spectra of A2 spin-1/2 spin-pairs. If thecoupled spins are not related by any symmetry operation then theyare deemed inequivalent (AX). All of these situations are summa-rized in Table 1. Intermediate conditions also exist if the differencein chemical shifts is of the same order of magnitude as the J cou-pling; these are known as AB spin systems.

Our previous work has elucidated the expected multiplets inDOR [35] and J-resolved [41] experiments for both magneticallyequivalent and non-equivalent spin pairs; however, it remains un-clear what can be expected in less ideal situations. A particularlyimportant case is that of chemically equivalent yet magneticallyinequivalent spins pairs which are more common than magneti-cally equivalent spin pairs. The effects of J coupling in MAS and J-resolved spectra of chemically equivalent spin-1/2 pairs have beeninvestigated where J-recoupling effects are introduced [51–53]. Itwould also be important to determine the effects of the chemicalshift difference (in AB and AX spin pairs) and the size of the quad-rupolar and dipolar interactions if this method is to be used to helpsolve chemical problems.

This work explores the effects of various NMR parameters on the Jcoupling multiplets for pairs of quadrupoles by following the variousAB-type spectral changeovers with the use of numerical density ma-trix calculations of single-crystal NMR experiments using the SIMP-SON program [54]. Simulations of DOR and J-resolved experimentsare also performed in order to evaluate the effects that these AB-typespectral changeovers have on the results of available NMR experi-ments for measuring homonuclear J coupling. The effects of chemi-cal equivalence on DOR and J-resolved data are also explored.

2. Computational details

All spectral simulations were performed using the SIMPSONprogram, version 3.1.0, with the exception of the DOR simulationswhich used version 1.1.2 [54]. Calculations of the idealized multi-plets were performed using a single crystallite orientation and thequadrupolar interaction was treated using first-order perturbationtheory. By using only first-order quadrupolar effects, the second-order quadrupole broadening is eliminated and the equivalenceof the central transition is imposed; this is relevant for fast-spin-ning DOR NMR experiments, vide infra. Second-order quadrupoleshifts could, in principle, lift the magnetic equivalence of the CTsignals for both nuclei in an orientation-dependent way for singlecrystals, but removing these effects better represents spinningexperiments. With this approach, it is possible to isolate each indi-vidual effect on the multiplets that are, in principle, simulta-neously present in an NMR experiment. All calculations were setto detect only the central transition.

Simulations of double-quantum filtered J-resolved experimentsand DOR experiments were performed for a pair of 11B nuclei at9.4 T. Typically the quadrupolar coupling constant (CQ) was set to3 MHz and the asymmetry parameter (g) to 0.5. Unlike the sin-gle-crystal simulations, the second-order quadrupolar interactionwas included in these simulations.

Simulations of the J-resolved experiments were performedusing 10 kHz MAS spinning, 30 and 60 ls CT-selective 90� and

180� pulses, respectively, and 232 crystallite orientations accord-ing to the ZCW scheme [55–57]. The J-DQF pulse sequence previ-ously published [41] was employed with a total double-quantumfiltration time of 40 rotor cycles. 256 data points were acquiredwith a 1 ms t1 increment. The data were apodized using 15 Hz ofGaussian broadening and processed in magnitude mode. This pulsesequence uses a refocused-INADEQUATE block [58,59] to removethe non-modulated signals from spin pairs where both nuclei arenot in central states [41]. Double-quantum filtering is also usefulfor overcoming the low natural abundances of different isotopes[60,61] as well as mapping out the distributions of J coupling con-stants [62,63].

The DOR simulations were typically performed unsynchronizedusing a 2 kHz outer rotor spinning frequency and an 8 kHz innerrotor spinning frequency. A 20 kHz spectral window was usedand 1024 data points were acquired and apodized using 60 Hz ofLorentzian broadening. All DOR simulations used 66 crystallite ori-entations and the REPULSION scheme [64]; simulations using agreater number of orientations showed only negligible differences.

3. Results and discussion

3.1. Idealized spin systems

The isotropic J coupling Hamiltonian is given by [65]:

HJ ¼ JisoS1zS2z þ12

JisoðS1þS2� þ S1�S2þÞ ð1Þ

where the first term is secular, does not affect the wavefunctions,and simply returns the product of the magnetic quantum numbers,m, of both nuclei and the second term mixes the eigenstates whichdiffer in m by 1. The full mixing which is induced by the second partof the J coupling Hamiltonian may only occur if the eigenstates inquestion are degenerate [66]. For a quadrupolar nucleus, due tothe presence of the quadrupolar interaction, typically only theeigenstates with the various permutations of the same m quantumnumbers can be degenerate. For a pair of quadrupolar nuclei, theeigenstates can then be written as [35]:

1ffiffiffi2p m1;m2j i þ m2;m1j ið Þ

1ffiffiffi2p m1;m2j i � m2;m1j ið Þ

ð2Þ

where there are perfectly symmetric and perfectly antisymmetriceigenstates. These, however, only correspond to the eigenstatesin situations where the nuclei are magnetically equivalent. The en-ergy levels need to be strictly degenerate at all crystallite orienta-tions for the mixing to consistently occur. The coupled nucleimust then have the same chemical shift and electric field gradienttensor magnitudes and orientations. This corresponds to the spinsystems that have been observed with DOR NMR [35] and the mag-netically equivalent spin pairs observed using J-resolved NMR [41].

Typically, only the CT (m = 1/2 to �1/2 transition) is observedexperimentally and only the CT can be easily manipulated usingradio-frequency pulses. This is due to its much narrower spectralline shape since it is unaffected by the first-order quadrupolarinteraction. We will then only concern ourselves with the CTmagnetization.

For simplicity, the CT multiplets can be thought of as a series ofdoublets due to the coupling of the central states to each of thepairs of |m| states of the other nucleus. The transitions where bothnuclei are in central states (i.e., m = 1/2 or �1/2) are condensedinto a single, amplified, doublet with a splitting of(2S + 3)(2S � 1)J/4 [41] and originate only from the fully symmetricstates. The antisymmetric state does not lead to any central

Table 1Definitions of the various spin systems discussed in the text, typical compounds which would have the appropriate symmetry, and the various equivalencies of the differenttransitions of quadrupolar nuclei.

Spin system Label Notes Example compound Equivalence of thedifferent transitions ofa quadrupolar nucleus

CT:ST CT:CT ST:ST

Central-satellitemagneticequivalence

CS-A2 The nuclei have identical tensor magnitudesand orientations and no quadrupolarcoupling

This situation occurs in crystals with cubic symmetry Ex.23Na nuclei in NaCl(s)

U U U

Magneticequivalence

A2 The nuclei have identical tensor magnitudesand orientations

The spins need to be related by an inversion center or liealong a C3 or higher rotation axis Ex. 35Cl nuclei in Cl2(g)

X U U

Chemicalequivalence

AA0 The nuclei have identical tensor magnitudesbut differing tensor orientations

The spins need to be related by a reflection, rotation,translation, or glide symmetry operation Ex. 17O nuclei inH2O2

X Ua X

Non-equivalence AX The nuclei have generally different tensorialinteractions

The spins are not related by any symmetry operation Ex. 17

nuclei in ROOR’X X X

a Note that the central transition resonances are not strictly magnetically equivalent in this case when there is large second-order quadrupolar broadening. This broadeningis however removed by DOR.

F.A. Perras, D.L. Bryce / Journal of Magnetic Resonance 242 (2014) 23–32 25

transitions; however, unlike spin-1/2 nuclei [67], the central anti-symmetric state of quadrupolar spin pairs cannot be used to storehyperpolarized magnetization for long periods of time as it canundergo satellite transitions to other antisymmetric states withhigher m values.

The doublet originating from the coupling between the centralstates and states with m values of 3/2 and �3/2 is split into twoseparate doublets with half the intensity each. There are two pairsof resonances in this case instead since the symmetric and anti-symmetric states differ in energy because of the second term inthe J coupling Hamiltonian. The coupling in the symmetric statesleads to an amplified splitting of (S2 + S + 9/4)J and thus alwaysleads to the doublet of largest splitting within the A2 multiplets.The splitting of the resonance of the antisymmetric states is of(2S + 5)(2S � 3)J/4 and is thus smaller than that of the otherdoublet.

The raising and lowering operators in the second term of the Jcoupling Hamiltonian do not affect the energy of the states withm values greater than 3/2, for the CT. The splittings of the doubletsdue to coupling to states with higher angular momentum are thusof 2|m|J, where |m| is the magnetic quantum number of the nucleuswhich is not in a central state. These splittings correspond to theusual, unamplified, splittings that are observed in AX spin systems,vide infra.

The idealized CT multiplets which are expected for pairs ofmagnetically equivalent quadrupolar nuclei are shown in Fig. 1for nuclei with spin of 3/2, 5/2, 7/2, and 9/2. For example, for a pairof magnetically equivalent spin-3/2 nuclei, a 1:2:2:2:1 pentet withequal spacings of 3/2J is expected. These are first-order multipletsand do not originate from a J recoupling interaction [51] or a cross-term interaction [68]. We note that the intensity we had previouslyreported [35] for the symmetry-amplified doublet for the centralstates is doubled here since we had not considered that the anti-symmetric state does not lead to a central NMR signal. In singlecrystals, these multiplets will be largely affected by direct dipolarcoupling; however, under spinning conditions, the dipolar cou-pling would be averaged, leaving only the effects of the J coupling.

In AX spin systems, the eigenstates take the familiar Zeemanproduct form, m1;m2j i, and the CT NMR spectra are composed of2S + 1 uniformly spaced lines of equal intensity separated by theJ coupling constant, as is usual for heteronuclear J coupling. Thiscorresponds to a superposition of S + 1/2 doublets with line split-tings of 2|m|J, where m is the magnetic quantum number of thecoupled nucleus.

In the special case where the quadrupolar interaction is zero,or is motionally averaged to zero, as in solution, the satellite

transitions are no longer separated from the CT by a first-orderquadrupolar splitting and all transitions are degenerate. The eigen-states with the same

Pm value are also degenerate and are, in

principle, allowed to mix. In that case, which we will refer to as‘central-satellite magnetic equivalence’ (see Table 1), the J couplingno longer affects the NMR spectrum and a single resonance isexpected. This is what is observed for magnetically equivalentspin-1/2 nuclei. For quadrupolar nuclei in the solid state, thisspecial situation can, however, only occur in cubic structureswhere the EFG is zero at the nuclei.

In realistic cases, it can be understood that there is a partialbreaking of central-satellite magnetic equivalence which is in-duced by the first-order quadrupolar interaction. The central tran-sition of each nucleus, however, remains equivalent with that ofthe other nucleus, and so do the satellite transitions of the two nu-clei, in spin systems which are magnetically equivalent accordingto the definition of Waugh [50]. It is important however to under-stand the thresholds for which the spins can be safely assumed tofall into this regime of magnetic equivalence. We have performednumerical simulations of these multiplets for magnetically equiva-lent spin pairs as a function of the first-order quadrupolar splitting(DmQ) which leads to the partial breaking of central-satellite mag-netic equivalence. The calculations were performed using a singlecrystal with an axially symmetric EFG tensor whose largest compo-nent is oriented along the magnetic field such that DmQ is given by:

DmQ ¼3CQ

2Sð2S� 1Þ ð3Þ

and represents the frequency separation between the different sin-gle-quantum transition resonances.

The simulations for a pair of spin-3/2 nuclei are shown in Fig. 2where it can be seen that the singlet obtained under strict condi-tions of central-satellite magnetic equivalence (DmQ = 0) rapidlysplits into a multiplet with as many as nine resonances as the DmQ:Jratio is increased. Two of these resonances disappear at a DmQ:3Jratio of 1–2 whereas another two converge to form the resonancesof the 1:2:2:2:1 pentet for magnetically equivalent spin-3/2 nuclei(top trace of Fig. 2). Although five resonances are clearly seen witha DmQ/6J ratio as low as 5, a properly spaced first-order multiplet isonly observed when this ratio is larger than 10.

These simulations show that the first-order A2 multipletsshould nearly always be observed in experimentally relevant casessince a DmQ as small as 6 kHz is large enough to break the central-satellite magnetic equivalence for a J coupling constant of 100 Hz,which is of the expected size for second- and third-row elementswith moderate nuclear magnetogyric ratios. Similar simulations

Fig. 1. Simulated ideal, CT-only, first-order J coupling multiplets for the AX, AA0 , and A2 spin systems of two homonuclear J coupled quadrupolar nuclei with spin quantumnumbers of (a) 3/2, (b) 5/2, (c) 7/2, and (d) 9/2 under single-crystal NMR conditions. These multiplets assume a significant first-order quadrupolar interaction such that theCTs are not equivalent with the STs for both nuclei.

Fig. 2. Simulated single-crystal CT-only J coupling multiplets for a pair ofmagnetically equivalent spin-3/2 nuclei are shown as a function of the ratio ofthe first-order quadrupolar splitting (DmQ) to the maximum J splitting (6J). For mostrealistic experimental cases, the ratio will exceed 100 and a simple 1:2:2:2:1 pentetis expected. The bottom trace corresponds to the CS-A2 case whereas the top trace isthe A2 case. Negative resonances in some of the spectra arise from the ST which areslightly mixed with the CT and thus appear in the spectra calculated with the ‘Inc’detection operator.

26 F.A. Perras, D.L. Bryce / Journal of Magnetic Resonance 242 (2014) 23–32

for spin pairs with spin quantum numbers of 5/2, 7/2, and 9/2 areshown in the Supplementary Material. Since the value of the quad-rupolar splitting is reduced for higher spin nuclei (see Eq. (3)) and

the maximum J splittings are also greater (see Fig. 1), somewhatlarger CQ values are necessary to ensure that first-order A2 multi-plets (in the absence of central-satellite equivalence) are observed.Generally, if the DmQ is over 100 times the size of the maximumspectral J splitting, then first-order A2 multiplets will be observed.

Some resonances with negative intensities are calculated insome of the spectra with small quadrupolar coupling constants.These are ST resonances which are not completely filtered out bythe ‘Inc’ detection operator due to state mixing. Calculations usingthe ‘Inp’ detection operator feature only positive resonances butthe multiplets are severely complicated by the overlap of the mul-tiplets from the CT and the STs.

If the spins are chemically equivalent but not magneticallyequivalent (i.e., AA0), the satellite transition signals from the twocoupled nuclei will appear at different resonance frequencies andthus the satellite transitions behave as if they were non-equivalent.The CT, however, is unaffected by the first-order quadrupolar inter-action and will only lose its magnetic equivalence if there are largeenough anisotropic second-order quadrupolar effects; note thatthese are magnetic field dependent and are eliminated by DOR.The splitting for the central states can be expected to remain sym-metry-amplified due to the much smaller (or absent) anisotropicsecond-order quadrupolar effects, whereas the splittings observedfor the coupling of a central state to the satellite states will take theform Jm1m2. These multiplets for AA0 spin systems are also shownin Fig. 1 along with the multiplets for the A2 and AX spin systems.

Describing the changeover from the A2 spin system to the AA0

spin system is somewhat more complicated than those from thecentral-satellite magnetic equivalence spin systems discussed ear-lier, as it depends on the size of the first-order quadrupolar inter-action as well as the J coupling and the relative orientation of theEFG tensors. In principle it is the ratio of the difference in quadru-polar splittings (DDmQ) between the two nuclei with different ten-sor orientations to the maximum J splitting which dictates the

Fig. 3. Simulated single-crystal CT-only J coupling multiplets for a pair ofchemically equivalent (AA0) spin-3/2 nuclei are shown as a function of the ratioof the difference in quadrupolar splittings (DDmQ), induced by different tensororientations, to the maximum J splitting (3J). In many realistic experimental cases,the ratio will exceed 10 and a simple splitting of 3J is expected. The bottom tracerepresents the A2 case.

F.A. Perras, D.L. Bryce / Journal of Magnetic Resonance 242 (2014) 23–32 27

form of the multiplets. This parameter is calculated as follows inthe case of axially symmetric EFG tensors, where b is the angle be-tween the principal component of the EFG tensor and the magneticfield:

DDmQ ¼3CQ

4Sð2S� 1Þ � Dð3 cos2 b� 1Þ: ð4Þ

The changeover from the A2 to the AA0 case is illustrated in Fig. 3for a spin-3/2 pair. It is seen that the central and outer resonancesof the A2 pentet, which originate from the coupling to the satellitestates, are split and eventually converge to form a doublet with asplitting of 3J. This splitting is coincidentally the same splittingas the symmetry-amplified splitting of the central states and thusa doublet is observed. Similar simulations for the larger spin quan-tum numbers are shown in the Supplementary Material where, ingeneral, the AA0 multiplet is expected when the ratio of DDmQ tothe maximum J splitting is larger than 100. It is important to notethat the second-order quadrupolar interaction was omitted in this

Fig. 4. Simulated single-crystal CT-only J coupling multiplets for a pair of non-equivalchemical shifts (Dd) to the maximum J splitting (3J). In (a) the case for zero quadrupolar inquadrupolar interaction.

case to impose the equivalence of the central states. The simula-tions are therefore relevant to methods which remove the sec-ond-order quadrupolar broadening such as DOR (vide infra).Under single-crystal conditions, the spectra would typically alsobe affected by the different chemical shifts of the two sites at dif-ferent crystallite orientations as well as dipolar coupling.

Using numerical simulations it is also possible to look at the AXto A2 changeover as was done for spin-1/2 nuclei [69]. Such simula-tions are shown for spin-3/2 nuclei in Fig. 4 for limiting cases wherethe quadrupolar interaction is either zero or non-negligible. It canbe seen that for both cases, simple quartets are observed in the AXlimit when the ratio of the chemical shift to the maximum J splittingis above 100. As the difference in chemical shifts decreases, thefamiliar ‘roofing’ effect is observed in each of the doublets; however,the central doublet is much more strongly affected. For the casewhen CQ is zero, the resonances then converge to form a single res-onance (Fig. 4(a)), as is observed for spin-1/2 nuclei. When the quad-rupolar interaction is non-negligible, however, the various doubletsare shifted with respect to each other as the chemical shift differ-ence is reduced. The central doublet splitting then becomes ampli-fied to 3J at a Dd:3J ratio of around 1. As was observed in the AA0

case, the peaks attributed to the satellite states converge later thanthe central states onto the 0 and 6J splittings of the A2 multiplet.Simulations for the higher spin quantum numbers are given in theSupplementary Material and show a similar behavior.

Clearly, the possible forms of the multiplets attributed to J cou-pling in homonuclear spin pairs of quadrupolar nuclei are more di-verse than those for their spin-1/2 counterpart but these idealmultiplets can only be measured in special cases, such as spinningsingle-crystals, single-crystals with specific orientations, and per-haps in partially aligned media. In powdered solids, the second-or-der quadrupolar interaction often obscures all fine structure in theNMR spectra under static and MAS conditions. Homonuclear J cou-pling has; however, recently been observed in DOR NMR experi-ments [35] as well as J-resolved experiments [41]. The followingsections explore the impact of chemical equivalence and otherNMR interactions on the spectra obtained from these experiments.

3.2. Double-rotation (DOR) NMR

In double-rotation NMR, the sample is spun about two anglessimultaneously to average the second-order quadrupolarinteraction to its isotropic part [38]. The only broadening that

ent spin-3/2 nuclei (AX) are shown as a function of the ratio of the difference interaction is shown whereas the multiplets in (b) are for nuclei with a non-negligible

Fig. 6. Simulated DOR NMR spectra for a pair of magnetically equivalent (A2) 11Bspins are shown as a function of the quadrupolar coupling constant (CQ). Theapparent differences in chemical shift originate from the second-order quadrupoleshift. The inner and outer rotor spin rates were of 8 and 2 kHz, respectively.

28 F.A. Perras, D.L. Bryce / Journal of Magnetic Resonance 242 (2014) 23–32

remains is that from the spin–spin coupling interactions[34,35,70,71] and thus the homonuclear J coupling multiplets canbe observed in powders.

The individual components of the multiplets however have dif-ferent dipolar coupling strengths and will be affected differently bythe dipolar interaction. Simulations were performed for an ideal-ized diboron spin pair, as was mentioned in the computational sec-tion, with varying dipolar coupling strength; the typical dipolarcoupling constant (RDD) for a diboron spin pair ranges from 2.3 to2.7 kHz. The dipolar coupling tensor was oriented perpendicularto the largest principal components of the EFG tensors, as inbis(catecholato) diboron [35]. It can be seen in Fig. 5 that dipolarcoupling affects the intensities of the various components of themultiplet but does not affect the positions of the resonances. Itshould then still be possible to accurately measure J coupling con-stants using DOR NMR even when dipolar coupling is present.Additionally, with proper modeling, it would also be possible to ex-tract the relative orientations of the EFG and dipolar coupling ten-sors in favorable cases.

The DOR multiplets are only visible for A2 spin pairs if the quad-rupolar interaction is non-zero, as the symmetry of the spin statesneeds to be partially broken. It can be seen in Fig. 6 that these mul-tiplets are ideal when the quadrupolar interaction is larger than100 kHz, which is usually the case for any four-coordinate orthree-coordinate boron nucleus in a diboron compound, for exam-ple. Numerical simulations may also be used to simulate the DORNMR spectra in cases when a distorted multiplet is observed be-cause of a small quadrupolar interaction.

Simulations were also performed as the orientation betweenthe EFG tensors was modified, to model a pair of chemically equiv-alent but magnetically inequivalent 11B spins (see Fig. 7). Clearly,when the angle between the largest principal tensor components(b) is equal to zero, the 1:2:2:2:1 pentet is observed since the spinsare magnetically equivalent. This characteristic multiplet is com-pletely lost when the EFG tensor orientations differ by as little as0.25�! With larger deviations, only a single doublet with a splittingof 3J is observed, along with the spinning sidebands. As was men-tioned in the previous section, this is caused by the magneticequivalence of the CTs but inequivalence of the satellite transitions

Fig. 5. Simulated 11B DOR NMR spectra for a pair of magnetically equivalent (A2)spins are shown as a function of the dipolar coupling constant (RDD). Typical RDD

values for a diboron spin pair are of the order of 2.3–2.7 kHz. The inner and outerrotor spin rates were of 8 and 2 kHz, respectively, with the exception of the toptrace. The rotation frequencies were 10 times larger in the top trace to demonstratethe more complete averaging of the dipolar interaction at higher spinningfrequencies.

due to their much larger first-order broadening. Under DOR condi-tions, all CTs resonate at the same frequencies and can thus be con-sidered magnetically equivalent. The first-order quadrupolarbroadening of the STs is however much larger and the averagingis incomplete at moderate DOR frequencies. This is analogous tothe case of chemically-equivalent spin-1/2 nuclei under MAS con-ditions [51]. Under high-frequency MAS, the spins behave asthough they are magnetically equivalent.

The multiplets that are calculated for DOR NMR spectra areidentical to those from the single-crystal simulations and thus,DOR NMR spectra can be fit using a simple idealized multiplet.First-order multiplets should be observed using DOR NMR innearly all cases and the J splitting should only disappear for cubicsalts where the quadrupolar interaction is exactly zero.

Fig. 7. Simulated DOR NMR spectra for a pair of chemically equivalent (AA0) 11Bspins are shown as a function of the relative orientation of the boron EFG tensors.The doublet with a splitting of 3J which is expected for an AA0 pair is clearlyobserved when there is even a minute difference in the EFG tensor orientations(�0.5�). The inner and outer rotor spin rates were of 8 and 2 kHz, respectively. Thebottom trace is equivalent to the case of magnetic equivalence (A2).

Fig. 8. The indirect dimension of a 2D DQF-J-resolved experiment is shown as afunction of the difference in chemical shifts between two spin-3/2 nuclei. CQ is thesame for both nuclei. In the bottom spectrum, the spins are magnetically equivalent(A2) and the observed splitting is 3J whereas when the chemical shift difference islarge, the splitting is of J, as is expected for an AX spin pair. Experimentally, mostrealistic cases will correspond to the top or bottom traces. The simulationparameters are given in the computational details section.

F.A. Perras, D.L. Bryce / Journal of Magnetic Resonance 242 (2014) 23–32 29

3.3. J-resolved NMR

J-resolved spectroscopy [39,40] provides an approach to mea-sure precise J coupling constants for pairs of quadrupolar nucleiwith the use of a two-dimensional experiment [41]. This experi-ment can typically be performed with MAS and does not requireany other specialized equipment.

The typical J-resolved experiment for quadrupolar nuclei [41]features a double-quantum filter as in the refocused INADEQUATEexperiment [58,59] with four rotor synchronized delays whoseoptimal durations are of 1/(4J) where J is either the J coupling con-stant, or the amplified splitting in the magnetically equivalent case.All the pulses are CT-selective and so only the transitions betweenthe central states will be detected. The signal is detected using asimple Hahn-echo for which the echo delay is increased to formthe indirect dimension of the 2D experiment. During this period,the second-order quadrupolar broadening and chemical shifts arerefocused but the homonuclear J coupling between the centralstates is unaffected and the signal will be modulated [40]. A simpledoublet is then observed for both the A2 and AX spin systems;however, the splittings are of (2S + 3)(2S � 1)J/4 and J in each case,respectively [41]. An open question concerns whether chemicalequivalence will give an amplified splitting, as in DOR, or if a split-ting of J will be observed. It is also important to investigate the ef-fects of similar chemical shifts and dipolar coupling on theobserved multiplets.

In Fig. 8, simulations of the indirect dimension of a double-quantum filtered J-resolved experiment are shown as the chemicalshift difference is increased. It can be seen that a simple doubletwith a splitting of 3J is observed in the magnetically equivalentcase, which then increases as the chemical shift difference in-creases while a doublet with a splitting of J begins to appear. Inthese intermediate cases, the splitting of the outer doublet maybe unreliable but the inner doublet would yield the correct J cou-pling constant. These simulations show that fairly large chemicalshift differences are necessary in order to observe only a simpledoublet with a splitting of J. In realistic cases, however, the differ-ences in quadrupolar coupling and tensor orientations between thetwo coupled nuclei would further break the magnetic equivalenceand a single doublet with a splitting of J would be observed.

The effect of including dipolar coupling simply reduces theintensity of the doublet whereas the splitting remains the same(see Supplementary Material). The intensity loss can be regainedby increasing the spinning frequency in order to better averagethe dipolar coupling. Similarly, if the quadrupolar coupling con-stant is smaller than 500 times the J coupling constant, a significantdecrease in the intensity of the doublet is observed (see Supple-mentary Material).

If the relative orientation of the EFG tensors is altered to gen-erate a chemically equivalent yet magnetically inequivalent spinpair (Fig. 9), the outer doublet with a splitting of 3J decreasesin intensity whereas the doublet with a splitting of J increasesin intensity. The inner doublet is dominant when the differencebetween the tensor orientations is greater than 5� for a typicaldiboron spin pair. Interestingly, unlike what was seen for DORNMR, vide supra, the splitting of the central states does notremain amplified for a chemically equivalent spin pair under J-re-solved MAS conditions. This is caused by the second-order quad-rupolar broadening which separates the CT signals of the twonuclei to different portions of the MAS powder patterns. Underinfinitely fast MAS conditions, the resonance frequencies for thetwo spins would still differ and they would remain inequivalent.If a J-resolved experiment were performed under DORconditions, or at a higher magnetic field strength so that the sec-ond-order quadrupolar interaction is absent, a splitting of 3Jwould be observed.

3.4. Second-order quadrupolar-dipolar cross-term

It is well known that the quadrupolar interaction that affectsquadrupolar nuclei is often quite large compared to all other inter-nal NMR interactions. As it also has a molecule-fixed quantizationaxis, instead of a quantization axis aligned with the magnetic field,it does not commute with the dipolar and chemical shift anisot-ropy interactions. There can then be significant cross-terms be-tween these different interactions that can affect the NMRspectra [68].

The second-order quadrupolar-shielding cross-term does notaffect the symmetric transitions and would thus not affect the CTof a quadrupolar nucleus, but the second-order quadrupolar-dipo-lar cross-term does affect the CT NMR spectrum [68]. The second-order dipolar-quadrupolar cross-term, also commonly referred toas residual dipolar coupling [72], has been observed in MQMAS[36,37,73,74], STMAS [68], and DOR [34,35] NMR spectroscopy ofquadrupolar nuclei and leads to the formation of characteristicmultiplets.

In the heteronuclear case, which also applies to AX homonu-clear spin systems, the second-order quadrupolar-dipolarcross-term shifts the resonances according to the spin state ofthe coupled nucleus as follows [68,36]:

mSðmIÞ¼3RDDCI

Q

20mI0

!IðIþ1Þ�3m2

I

Ið2I�1Þ

� �3cos2 b�1þgI

Q sin2 bcos2a� �

: ð5Þ

In the expression above, RDD is the dipolar coupling constant, CIQ

and mI0 are the quadrupolar coupling constant and Larmor fre-

quency of the coupled nucleus, I is the spin quantum number ofthat nucleus, mI is the magnetic quantum number of its spin state,gI

Q is its quadrupolar asymmetry parameter, and b and a are polarangles relating the orientations of the EFG tensor of I to the dipolarvector.

Fig. 9. The indirect dimension of a 2D DQF-J-resolved experiment is shown as afunction of the difference in EFG tensor orientations between two chemicallyequivalent (AA0) spin-3/2 nuclei. In the bottom spectrum, the spins are magneticallyequivalent (A2) and the observed splitting is 3J whereas when the difference intensor orientations is large, the splitting is J. The simulation parameters are given inthe computational details section.

30 F.A. Perras, D.L. Bryce / Journal of Magnetic Resonance 242 (2014) 23–32

It can be seen from the above expression that the frequencyshifts caused by the second-order quadrupolar-dipolar cross-term(mS) only depend on the square of the magnetic quantum numberof the coupled nucleus. This is due to the second-order nature ofthis interaction which will always depend on the absolute magni-tude of m but not its sign, much like the second-order quadrupolarinteraction which is identical for both satellite transitions. Thisinteraction would then affect the multiplets that are observed inDOR NMR by shifting the different doublets which form the J cou-pling multiplet (vide supra) with respect to each other. The splittingwithin each individual doublet would however be unaffected sincethe sign of m does not affect the second-order quadrupolar-dipolarshift. In a J-resolved experiment, the splitting would then alwaysbe equal to the J coupling constant, or the amplified splitting, evenin the presence of a strong quadrupolar interaction. Since the sec-ond-order quadrupolar-dipolar cross-term would simply shift bothcomponents of the doublet by the same amount, it would be refo-cused by the Hahn echo and the correct splitting would be ob-tained. This is demonstrated using the density matrix formalismin the Supplementary Material. To verify this experimentally, J-re-solved experiments were performed on bis(catecholato)diboron atan ultra-high magnetic field of 21.1 T (see Supplementary Mate-rial) which provided the same splittings as the data acquired at9.4 T, [41] indicating that residual dipolar coupling does not affectthe splitting at lower field. Interestingly, this conclusion also ap-plies to heteronuclear J-resolved experiments [25,26] for whichthe effects of residual dipolar coupling would be eliminated inthe indirect dimension. This means that the effects of isotropic Jcoupling and dipolar coupling can be separated conveniently usingeither a heteronuclear or homonuclear J-resolved experiment. Inthis respect, the J-resolved experiment does have another

advantage over DOR NMR, since the DOR multiplets may becomedistorted by the dipolar-quadrupolar cross-term when the quadru-polar interaction is sufficiently large, but the J-resolved splittingsremain unaffected. The resolution in a J-resolved experiment isalso superior since simpler multiplets are measured and the distri-bution of chemical shifts, the magnetic field inhomogeneity, andthe second-order dipolar-quadrupolar cross-term are refocused,thus leading to much sharper resonances.

4. Conclusions

We have outlined the theory for the homonuclear J couplinginteractions between quadrupolar nuclei. The J coupling multipletsthat are expected under various conditions for the cases of inequiv-alent, chemically equivalent, and magnetically equivalent spinpairs have been described. It has also been shown that the first-or-der multiplets that are observed for magnetically equivalent quad-rupolar spin pairs originate from a partial symmetry breakingcaused by the first-order quadrupolar interaction. Using numericalsimulations, we have explored the changeovers between each ofthe four possible first-order situations (AX, AA0, A2, and central-sa-tellite magnetically equivalent A2) and provide thresholds for eachof the situations. Generally, the ideal first-order multiplet is ob-served when the difference in frequency caused by first-orderquadrupolar splittings or chemical shifts is 100 times greater thanthe maximum spectral splitting due to J.

We have also used numerical calculations to simulate DOR andJ-resolved MAS NMR spectra of homonuclear J coupled spin pairs.We found that the DOR NMR spectra closely match those whichwere calculated for single crystals and that dipolar and quadrupo-lar interactions have little effect on the resulting spectra underrealistic experimental conditions. The J-resolved MAS NMR spectrashow that it should be possible to distinguish between magneticand chemical equivalence in a spin pair since an unamplified split-ting would be obtained even for small differences in the EFG tensororientations of the two nuclei. For example, the doublet with asplitting of J would dominate when the difference in EFG tensororientations is greater than 10� for a typical diboron spin pair.The presence of dipolar and small quadrupolar interactions wouldpartially deplete the signal whereas the second-order quadrupolar-dipolar cross-term would simply shift the doublet from zerofrequency without affecting the splitting. This is also relevant toheteronuclear J-resolved experiments for which the effects of thesecond-order quadrupolar-dipolar cross-term would be elimi-nated. Chemical shift differences would also significantly affectthe J-resolved spectra; however, these would usually be accompa-nied by differences in EFG tensor magnitudes and orientations.

Our simulations show that J-resolved NMR spectroscopy is a ro-bust tool for studying the bonding in homonuclear spin systemsinvolving quadrupolar nuclei. The J splittings are largely indepen-dent of the chemical shift differences, first- and second-orderquadrupolar interactions, as well as dipolar interactions.

Acknowledgments

We are grateful to Dr. Tom Daff, Mr. Peter Boyd, and Dr. VictorTerskikh for technical support. D.L.B. thanks the Natural Sciencesand Engineering Research Council of Canada (NSERC) for fundingand F.A.P. thanks NSERC for a graduate scholarship. Access to the900 MHz NMR spectrometer was provided by the National Ultra-high-Field NMR Facility for Solids (Ottawa, Canada), a national re-search facility funded by the Canada Foundation for Innovation, theOntario Innovation Trust, Recherche Québec, the National ResearchCouncil Canada, and Bruker Biospin and managed by the University

F.A. Perras, D.L. Bryce / Journal of Magnetic Resonance 242 (2014) 23–32 31

of Ottawa (nmr900.ca). NSERC is acknowledged for a MajorResources Support grant.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jmr.2014.01.015.

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