J‐recoupling patterns arising from two chemically equivalent nuclear spins inmagic‐angle spinning spectra of solidsGang Wu and Roderick E. Wasylishen Citation: The Journal of Chemical Physics 98, 6138 (1993); doi: 10.1063/1.464852 View online: http://dx.doi.org/10.1063/1.464852 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/98/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dipolar truncation in magic-angle spinning NMR recoupling experiments J. Chem. Phys. 130, 114506 (2009); 10.1063/1.3089370 Recoupling of chemical shift anisotropies in solid-state NMR under high-speed magic-angle spinning andin uniformly 13 C -labeled systems J. Chem. Phys. 118, 8378 (2003); 10.1063/1.1565109 Synchronous helical pulse sequences in magic-angle spinning nuclear magnetic resonance: Doublequantum recoupling of multiple-spin systems J. Chem. Phys. 112, 8539 (2000); 10.1063/1.481458 Magic‐angle spinning nuclear magnetic resonance spectra of second‐order two‐spin systems in the solidstate J. Chem. Phys. 99, 6321 (1993); 10.1063/1.465870 Calculation of magic‐angle spinning nuclear magnetic resonance spectra of paramagnetic solids J. Chem. Phys. 89, 4600 (1988); 10.1063/1.454800

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J-recoupling patterns arising from two chemically equivalent nuclear spins in magic-angle spinning spectra of solids

Gang Wu and Roderick E. Wasylishena) Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4J3

(Received 1 December 1992; accepted 4 January 1993)

A general description of magic-angle spinning (MAS) nuclear magnetic resonance (NMR) spectra arising from a pair of chemically equivalent nuclear spins is presented in terms . of average Hamiltonian theory (AHT). In general, the MAS NMR spectra of such a spm­pair exhibit a spinning frequency dependent four-line pattern from which it is possible to extract the indirect spin-spin coupling constant, J, involving the "equivalent" spin pair. Explicit expressions for the spinning frequency dependence of the four-line pattern have been derived using AHT. In principle, correction terms to any order can be included; however, consideration of correction terms up to and including third order appear to be sufficient to interpret the most important features characteristic of J-recoupled spectra involving chemically equivalent spin pairs. The average Hamiltonian theory predicts three different general types of recoupling patterns. The type of recoupling pattern observed for a particular chemically equivalent spin pair is predicted to depend on the relative magnitudes of the indirect homonuclear coupling constant, J, the direct homonuclear dipolar coupling constant, R, the magnitude of the instantaneous chemical shift difference between the "equivalent" spins in frequency units, and the MAS spinning frequency. All reported examples of spinning frequency dependent MAS NMR spectra arising from a pair of chemically equivalent spins can be understood using the theoretical expressions derived here. As an example, we interpret the unusual J-recoupling pattern observed in 31p MAS NMR spectra of Hg(PPh3h(N03h The recoupling pattern is unusual in that 2J(p,p) is given by the separation of alternate lines in the four-line pattern. Similar unusual J-recoupling patterns were first reported by Eichele, Wu, and Wasylishen.

I. INTRODUCTION

In the solid state, nuclear magnetic resonance (NMR) spectra of magnetically dilute spino! nuclei such as J3C, 15N, and 31p, generally exhibit relatively sharp peaks under conditions of magic-angle spinning (MAS) , cross­polarization (CP), and high-power abundant spin decou­pling. l -6 The implementation of these techniques in NMR studies of solids has become routine, making it possible for chemists to study numerous diverse systems in the solid state.7- JO

If two nuclei with the same magnetogyric ratio are coupled to one another, they constitute a homonuclear spino! pair. If the spins are dipolar coupled to <.me another and have different isotropic chemical shifts, w~o, interest­ing line-broadening effects and cross-relaxation behavior are observed in MAS NMR spectra of the spin-pair when wf matches a multiple of the sample spinning frequency, WR, i.e., when w~=n WR, where n is an integer. Such MAS NMR spectra are said to be obtained under rotational res­onance conditions. More than 25 years ago, Andrew et al. 11,12 first recognized enhanced line broadening and cross-relaxation in 31p MAS NMR spectra of solid PCl5 [PClt PCI6"] obtained under conditions of rotational res­onance. Recently, considerable attention has been devoted to investigations of MAS NMR spectra arising from homo­nuclear spino! pairs under rotational resonance condi-

a) Author to whom all correspondence should be addressed.

tions. 13-24 For example, Griffin and co-workers have re­cently demonstrated that it is possible to obtain the distance between two weakly dipolar-coupled homonuclear spins by analyzing the time evolution of differences in their

. Z . . 1519 respective eeman magnetizatIOn. ' Several theoretical treatments have been developed in

order to describe MAS NMR line shapes of a homonuclear spino! pair under rotational resonance conditions. Raleigh et al. 15 performed numerical simulations to reproduce the observed MAS line shapes of J3C doubly labeled zinc ace­tate. Gan and Grant20 proposed a pseudospin model to analyze the MAS spectra under on- and off-rotational res­onance conditions. Levitt et al. 21 presented the results of an extensive study of MAS NMR spectra of homonuclear spin pairs. Recently, Schmidt and Vega,23 and Nakai and Mc­Dowe1l24 have independently applied Floquet theory to provide a general description of MAS spectra.

If two nuclear spins are crystallographically equivalent in the solid state, we will also define them as being chem­ically equivalent since they have identical isotropic chemi­cal shifts. In the solid state the chemical shift is orientation dependent and is characterized by a second-rank tensor, namely the chemical shift tensor. The chemical shift tensor components of two chemically equivalent nuclei are related by the rotation-reflection symmetry operations which re­late their spatial positions. That is, the chemical shift ten­sors of two chemically equivalent nuclei may have different orientations with respect to the crystallographic axis sys­tem. In such cases, the two nuclei have different chemical

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G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra 6139

shifts at certain crystallite orientations with respect to the applied magnetic field. If two nuclei are related by a center of inversion, their respective chemical shift tensors are co­incident and their chemical shifts will be identical at any orientation of the crystallite in the applied magnetic field. Such spin pairs are designated as being magnetically equiv­alent.

MAS NMR spectra arising from a pair of chemically equivalent nuclear spins which are not related by a center of inversion have been noted to exhibit some interesting spectral features. Since the isotropic chemical shift differ­ence vanishes for a pair of chemically equivalent spins, it represents a special case, i.e., an "n = 0 rotational reso­nance".21 In a classic paper, Maricq and Waugh25 analyzed the MAS NMR spectra arising from a pair of chemically equivalent 13C nuclei in 13C doubly labeled diammonium oxalate, (NH4)2(0213C-13C02)' using average Hamil­tonian theory. By considering the average Hamiltonian and its first order correction term, they showed that the direct dipolar interaction between the two 13C nuclei causes sig­nificant line broadening in MAS NMR spectra and further­more, such broadening is dependent on the sample spin­ning frequency. More recently, Hayashi and Hayamizu26

and Kubo and McDowe1l27 independently reported that the 31p CP/MAS NMR line shapes of Na4PP7' lOH20 are sensitive to sample spinning frequency. In particular, a single peak was observed at rapid and slow spinning fre­quencies while a doublet was observed at intermediate spinning frequencies. The splitting of the doublet was de­pendent on the sample spinning frequency in this interme­diate regime. Based on the Floquet formalism, Kubo and McDoweU27 have presented theoretical expressions which successfully describe the spinning frequency dependent spJittings observed in the 31 P CP IMAS spectra of Na4P207' lOH20. Similar spinning frequency dependent phenomena involving two chemically equivalent 31p nuclei in transition metal phosphine compounds have also been reported by Lindner et al 28 and Power and Wasylishen.29

In each of the aforementioned examples the "equivalent" spins are only dipolar coupled.

Another interesting phenomenon arises when two chemically equivalent spin-4 nuclei are J coupled. As is well known, the J coupling constant between two chemi­cally equivalent spins in a two-spin system is an unobserv­able in isotropic fluids. In the solid state, however, the absence of an inversion center between two chemically equivalent spins makes it possible for the J interaction be­tween them to be reintroduced into MAS NMR spectra, leading to J recoupling. Although the J-recoupling phe­nomenon between two chemically equivalent spins in MAS NMR spectra was predicted in the early pioneering paper of Maricq and Waugh,25 it was not observed experimen­tally until the recent careful work of McDowell and co­workers.3(}-32 These authors noticed that the 31p MAS spectra of I ,2-bis( 2,4,6-tri-tert-butylphenyl) -diphosphene (TBPDP) exhibit a general four-line pattern. They dem­onstrated that such J-recoupled spectra can be understood by considering the average Hamiltonian and its first-order correction term. Using this average Hamiltonian theory

they predicted that the J coupling constant can be mea­sured directly from splittings in a J-recoupled spectrum. They concluded that the J-recoupling effect between two chemically equivalent spins is enhanced by differences in the orientations of their respective chemical shift tensors and by the strength of the direct homonuclear dipolar cou­pling between them.

Recently, Eichele et al. 33 observed that the 31p J-recoupled spectra of solid Cd(PPh3h(N03h exhibit un­usual structures; i.e., the 2 Je l p,3Ip) coupling constant is given by the separation between alternate lines in the four­line pattern, in contrast to the J-recoupled spectra of TBPDP. They also noted that the observed unusual J-recoupled spectra cannot be predicted using average Hamiltonian and its first-order correction term.

In this paper, we present a general description of the MAS NMR spectra arising from a pair of chemically equivalent nuclear spins in terms of average Hamiltonian theory. Such MAS NMR spectra are generally dependent on the sample spinning frequency. Explicit expressions for the spinning frequency dependence have been derived for correction terms of any order. We report J-recoupled 31p MAS NMR spectra of Hg(PPh3)2(N03}z, which exhibit unusual spinning frequency dependent splitting patterns similar to those observed by Eichele et al. 33 It is shown that by considering correction terms of the average Hamil­tonian to third order, the theory is capable of reproducing the unusual spinning frequency dependent J-recoupling pattern observed for Hg(PPh3h(N03}z. The magnitUde of 2Jelp,3lp) obtained from an analysis of the J-recoupling pattern is confirmed by two-dimensional (2D) 31p CPI MAS J-resolved NMR experiments.

II. THEORY

A. The average Hamiltonian and its high-order correction terms

In the solid state, the nuclear spin Hamiltonian for a pair of spin-4 nuclei can be written as

(1)

where 7r' z is the Zeeman interaction, 7r' cs is the chemical shielding interaction, 7r'D is the homonuclear dipole­dipole interaction, and 7r'J is the indirect spin-spin (J) interaction. For a polycrystalline sample, each of the later three interactions is dependent on the orientation of the crystallite in the applied magnetic field. Under magic-angle spinning conditions, the nuclear spin Hamiltonian becomes time dependent,

7r'(t) = -WI (t)Ilz- w2(t)I2z+ [2Wd(t) +wJlhhz

+H WJ-Wd(t) 1 (/1 +Jz- +II-Jz+), (2)

where Wi(t) Ci= I and 2) and wit) are NMR frequencies arising from chemical shielding and the direct dipolar in­teractions, respectively; WJ is the isotropic indirect spin­spin coupling constant in angular frequency units. Explicit expressions for the orientation dependence and time depen­dence of these factors can be found in the literature.7.25 In this paper we follow the convention of Kubo and McDow-

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6140 G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra

ell.27 For a pair of chemically equivalent spin-~ nuclei, CUi(t) U= 1 and 2) and CUd(t) can be written as

cui=cuiso+ C'{i cos ( r+CURt) +Sfi sine r+CURt)

+Cii cos(2r+2cuRt) +S2i sin(2r+2cuRt), (3)

CUd=c{ cOS(r+CURt) +s1 sin(r+cuRt)

+C1 cos(2r+ 2cuRt) +~ sin(2r+ 2cuRt), (4)

where CUiso is the isotropic chemical shift for both nuclei and CUR is the MAS sample spinning frequency. The coef­ficients c!. and S~ (m=1, 2; A=a, d) are functions con­taining the principal components and their orientations. Expressions for these coefficients can be found in Ref. 27.

From Eq. (2), it is clear that the nuclear spin Hamil­tonian in the MAS NMR experiments is periodic. To treat such a time-dependent periodic Hamiltonian, Waugh and co-workers have developed a successful method known as average Hamiltonian theory (AHT).25,34-36 According to AHT, at t=NtR (tR is the rotor period and N is an integer variable) the spin system can be described by an effective Hamiltonian, Eq. (5):

,w'elf=,w'(O) +,w'(I) +,w'(2) + ..• ,

where

(5)

(6)

(7)

(8)

etc. The zeroth term, ,w'(O), is called the average Hamil­tonian; ,w'(I) and ,w'(2) are the first and second-order cor­rection terms, respectively. Substituting Eq. (2) into Eqs. (6) and (7), one can readily obtain the average Hamil­tonian and its first-order correction term:25

,w'(O) = -CUiso (/'z+ 12z ) +CU.fl, .12 , (9)

(I). (/,+12--1,-12+) ,w' = -IK, (CUR) 2 (10)

where

The superscript !.:J. denotes the difference between corre­sponding coefficients, for example, q. = Cj' - Cj2. After some lengthy but straightforward algebra, one can also obtain the second-order correction term as

(12)

where

G2(CUR) =9L1 (7q.C{C1-12q.s1s1+ 12S~C{sf

+ 12S~s1c{-4S~s1C1-6~C{C1+4~s1s1

-6cts1~ -3~~~ -12S~cts1

+6S~S1C1+3S~C1~) =4, CUR

F 2(CUR) = 4L~ (7c{q.~-12c{S~S~+ 12s1cts~

+ 12s1s~ct-4S1s~~-6C1q.~

+4C1s~s~ - 6C1s~s~ - 3C1s~s~

(13)

d,.,A A d A,.,A A,.,d d 12 -12S2L,IS, +6S2S, L,2+ 3S2 L,:iS2) =--z. (14)

CUR

For a pair of chemically equivalent nuclear spins, it can be shown that the 2n and (2n + 1 )th order correction terms to the average Hamiltonian have the following forms:

(15)

(/,+12- -1,-12+) q;p(2n+') - -·K ( ) dl - I 2n+' CUR ----2---' (16)

where

(17)

and

(18)

The coefficients g2m 12n, and k 2n +, are functions of the principal components of the two chemical shift tensors, the direct dipolar interaction and its orientation with respect to the chemical shift tensors. Although the explicit expres­sions for these coefficients may be complicated, it is useful to examine their magnitudes qualitatively. One can show that

and

k cc1:2n+'·R 2n+' u , n=0,1,2,3, ... (20)

where {j is a quantity which depends on the difference be­tween the two chemical shift tensors and R is the direct dipolar coupling constant. Therefore, the effective Hamil­tonian can be written as

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G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra 6141

co co

=~O)+ L K(2n)+ L K(2n+1)

n=1 n=O

(21)

where

(22)

(23)

(24)

Using the product basis functions 11)= laa), 12)= lap), 13) = I pa), and 14) = I Pp), the effective Hamiltonian is usually expressed in a matrix form

WJ -WiSO+"4 0 0 0

WJ G wJ-F iK 0 --+- -- - 0

4 2 2 2 K etf=

wJ-F iK wJ G 0 --+- -"4-"2 0

2 2

0 0 0 wJ

Wiso+"4

(25)

Following steps analogous to those used in solving the "AR-spin problem" in solution NMR, the aforementioned matrix can be diagonalized, so that the four transition fre­quencies and their relative intensities are obtained; Eqs. (26)-(29):

(26)

(27)

(28)

(29)

where

(30)

One of the important implications ofEqs. (26)-(30) is that under conditions of magic-angle spinning, the NMR spectrum of two chemically equivalent nuclear spins can give rise to a general four~line pattern for a powder sample. In contrast, only a single peak would be observed for an isotropic liquid sample. Analysis of the solid-state NMR spectrum yields the J coupling constant between two chemically equivalent spins. Equally important is the fact that the appearance of the four-line pattern is dependent on the sample spinning frequency, wR' A plot of the peak positions in the four-line pattern as a function of sample spinning frequency is referred to as a J-recoupling pattern in this paper. The J-coupling constant, wJ' can be directly measured once the type of J-recoupling pattern is identified (vide infra).

Several common features of J-recoupled spectra are worth mentioning at this stage. First, the chemical shift tensors of the two chemically equivalent nuclear spins must be noncoincident in order to observe J recoupling. Secondly, direct dipolar coupling between the two homo­nuclear spins is essential to recouple the J interaction. Of course, if the anisotropy in the J tensor is large, it is also possible, in principle, to observe J recoupling even in the absence of the direct dipolar interaction. However, we as­sume in this paper that the anisotropy in the J tensor is negligible. Third, at the high spinning frequency limit, the two central peaks in the J-recoupled spectra collapse to the isotropic chemical shift, Wiso' while the two outer peaks vanish at Wiso ± W J'

It is also worth noting that since the MAS experiments for a pair of chemically equivalent spins satisfy the condi­tion for a "n=O rotational resonance" at any spinning fre­quency, the four transitions derived in this paper [Eqs. (26)-(30)] are similar to the general eight-line patterns derived by Levitt et al. 21 under rotational resonance using Floquet theory. Since the isotropic chemical shift differ­ence vanishes for a pair of chemically equivalent spins, the two central levels in the general four-level system become degenerate. A resonant splitting of them leads to only four observable single-quantum transitions between the virtual states and the two outer levels. Interestingly, the even­order correction terms introduce a shift for the state 12) and 13), i.e., G(11z-12z)!2=GI~23). This shift is directly analogous to the Bloch-Siegert shift37 as noted by Levitt et al. 21 Moreover, the even-order correction terms intro­duce an additional component, F, which could reduce the apparent J-coupling constant. It will be shown in the fol­lowing section that this additional term is responsible for the variations in different types of J-recoupIing patterns.

It is important to emphasize that the effective Hamil­tonian given by Eq. (5) can characterize the behavior of spin systems only at t=NtR, which corresponds to a spec­trum obtained with data sampling synchronized with the rotor period. In ordinary MAS NMR experiments, how­ever, the effective Hamiltonian can be used to describe the total MAS NMR line shape, which is a spectrum summed over all spinning sidebands.

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6142 G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra

spll1ningtrequency

FIG. 1. Schematic diagram illustrating the MAS frequency dependence of the mixing term D when different order correction terms are consid­ered.

B. Three different types of J-recoupling patterns In MAS NMR spectra

It is interesting to note that Eqs. (26)-(30) are similar to those which result from solving the AD problem in so­lution NMR studies, except that the isotropic chemical shift difference vanishes in the solid-state systems consid­ered here, and the mixing term, D, has a different form. As is well known in solution NMR, the ratio of the mixing term, D, to the J-coupling constant, illJ, determines the general appearance of the AD spectrum. It will be shown in this section that different magnitudes of the mixing term, D, compared to illJ can result in three basic types of J-recoupling patterns. However, before we discuss these, it is useful to examine the magnitude of D as a function of sample spinning frequency.

It is readily seen in Eq. (30) that if only the average Hamiltonian, JY<0), is considered, Le., K=G=F=O, the D term is independent of sample spinning frequency and al­ways equals illJ. When the first-order correction term, JY'(l), is added, D becomes greater than illJ, Le., (illJ+Kr) 1I2>wJ' As the spinning frequency increases, D decreases monotonically until it reaches WJ at the high spinning limit. When the second-order correction term, JY'(2), is further included, the D term is given by

D={G~(WR) + [illJ-F2 (illR) ]2+K~(illR)}'/2

~ 12/2 112

= [-i-+ (WJ-~) +~] . (31) ill R W R ill R

Based on Eq. (31), it becomes possible for the D term to be smaller than W J in a range of spinning frequencies because of the introduction of the F2 term. Figure 1 shows sche­matically the magnitude of D as a function of sample spin­ning frequency when different correction terms in the ef­fective Hamiltonian are considered. Note that the curve in Fig. 1 illustrates the effect of including the second-order correction term is just one possible case, Le., for a partic­ular set of k" g2, 12' and WJ values. The curve containing the second-order correction can also be similar to that ob­tained by considering only the first-order correction term, provided the second-order correction term is not large

c.>iso+ c.>J c.>iso c.>iso- c.>J

(a) ~

c.>J j! :1

c.>1 c.>2 c.>3 c.>" ~

c.>iso + c.>J c.>iso c.>iso- c.> J

(b)

~

i! c.>1 c.>3 c.>2

:~ c.>" ~

c.>iso + c.>J c.>. 180 c.>iso- c.>J

(c)

i jl

c.>J :1 Po

c.>1 c.>2 c.>3 c.>" !Xl

line position

FIG. 2. Schematic diagram illustrating each of the three types of J-recoupling patterns.

enough to make the D term smaller than WJ. However, it is the curve shown that causes the unusual appearance of the J-recoupling patterns, which will be discussed later.

Based on Eqs. (26)-(30), there are, in general, three possible types of J-recoupling patterns depending on the magnitude of the mixing term, D, compared with WJ:

(I) D>wJ,

(II) D<wJ and

(III) D>wJ at slow spinning frequencies while D < W J at high spinning frequencies.

The spinning frequency dependence of these three dis­tinctive types of J-recoupling patterns are sketched in Fig. 2.

In the type I J-recoupling pattern, where D> W J, the four-line pattern is analogous to the AD spectrum predicted

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G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra 6143

in solution NMR. We refer to this type of pattern as a usual J-recoupling pattern. In usual J-recoupling patterns, the J-coupling constant is always given by the separation of the two adjacent outer peaks in the four-line pattern inde­pendent of illR' As shown in Fig. 2(a), the separation be­tween /1}I and /1}2 (as well as that between /1}3 and /1}4) equals /1}J' so that these transitions are always parallel to each other as a function of /1}R' At the high spinning limit, the two central peaks collapse and the intensities of the outer peaks vanish, leading to a single peak at the position of the isotropic chemical shift. At slower spinning frequencies, the central peak splits with two non vanishing outer peaks, leading to a four-line spectrum. As the sample spinning frequency decreases, the low and high frequency parts of the four-line pattern move apart from each other with the two outer lines becoming more intense. Note that as the sample spinning frequency decreases, the separation be­tween the two outer peaks increases. The type I J-recoupling pattern was first observed in the 3Ip CP/MAS spectra of TBPDP by McDowell and co_workers.3

O-32

In the type II J-recoupling pattern, where D</1}J' the two central lines in the four-line pattern interchange their relative positions. Consequently, the J coupling constant is given by the separation between alternate peaks, i.e., be­tween /1}1 and /1}2 or /1}3 and /1}4' rather than by the two outer splittings in the usual J-recoupling patterns (type I). Since this situation can never be observed in solution NMR stud­ies, we denote this type of J-recoupling pattern as unusual. At the high spinning limit, a single peak is also observed at the position of the isotropic chemical shift. At slower spin­ning frequencies, a general four-line pattern appears. In contrast to the situation in the usual J-recoupling pattern (type I), however, the separation between the two outer peaks in the unusual J-recoupling pattern decreases as the sample spinning frequency decreases. The first observation of such an unusual J-recoupling pattern has recently been reported by Eichele et al. 33 They observed spinning fre­quency dependent splittings in the 3Ip CP/MAS spectra of Cd(PPh3h(N03h

In the type III J-recoupling pattern, D-/1}J changes sign at a specific spinning frequency, and the resultant pat­tern can be thought of as a combination of the type I and II patterns. At the rapid spinning extreme, similar to the cases in type I and II patterns, the MAS spectrum consists of a single isotropic peak centered at the isotropic chemical shift position. As the spinning frequency decreases, but still lies in the high frequency regime, the four-line pattern ex­hibits the same behavior as that in the unusual J-recoupling pattern, i.e., the J coupling constant is given by the separation between alternate peaks and the separa­tion between the two outer peaks decreases as the spinning frequency decreases. However, after the spinning fre­quency reaches a certain value, at which the D term reaches its minimum, the separation between the two outer peaks starts to increase as the spinning frequency de­creases. As the sample spinning frequency is further re­duced, the two central peaks merge giving rise to a three­line pattern and then cross as the MAS frequency enters the slow spinning regime. In this regime the J coupling

C/).

~ 1.0

~ ~ ~

~ 0.5 :g ...... ~

Spinning Frequency

FIG. 3. Schematic diagram illustrating the MAS frequency dependence of the relative intensity within J-recoupled spectra.

constant is given by the separation between the two adja­cent outer peaks in the four-line pattern, i.e., a usual J-recoupling pattern. The spinning frequency dependence of the complete pattern looks rather complicated. To date this type of J-recoupling pattern has not been observed. Based on Eqs. (19), (20), (22)-(24), and (26)-(30), it is apparent that this type of recoupling pattern will only be observed in extreme situations (vide infra).

From Eqs. (26)-(30) it is clear that not only do the peak positions in a J-recoupled spectrum change as a func­tion of sample spinning frequency, but also the relative intensities of the four lines. Since the J-recoupled spectrum is always symmetric, as indicated by Eqs. (26)-(30), it is only necessary to discuss the relative intensities of the peaks within the four-line patterns. Here we define the intensity ratio of the outer peaks to the inner peaks by the parameter S. Since the two central peaks always have equal intensity, a switch in their relative positions in the different types of J-recoupled spectra does not change S. Hence, it is unnecessary to make a distinction between the different types of J-recoupling pattern when we discuss the relative intensities of the peaks. Because D approaches /1}J at the high spinning frequency limit, it leads to a vanishing S. Under such a condition, the J coupling constant will not be observable. As the sample spinning frequency decreases, S increases, i.e., the outer lines become more intense at slower spinning frequencies. Thus a four-line pattern re­sults and the J coupling constant is reintroduced. A sche­matic plot of S vs the sample spinning frequency is shown in Fig. 3.

III. EXPERIMENT

All 3Ip NMR spectra were recorded on a Bruker MSL-200 (Bo=4.70 T) and a Bruker AMX-400 (Bo=9.40 T) operating at 3Ip NMR frequencies of 81.03 and 162.0 MHz, respectively. The white crystalline sample was packed into zirconium oxide rotors, 7 mm o.d. and 4 mm o.d., for the low and high field experiments, respectively. Cross polarization under the Hartmann-Hahn match con­dition and high power proton decoupling were employed for acquisition of all solid-state 3Ip NMR spectra. Typical lH 90° pulses were 4.5-5.0 /-LS for the low field experiments

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6144 G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra

(a)

(b)

(c)

(d)

(e)

(f)

,,, ........ ,,.,,, .. 1 ... ,,,,,, •• 11''''''10''',,11,,'''' .. ,,1 .. ,,,,,,,,,,,,,, •• ,

400 200 0 -200 -400

Hz

FIG. 4. The 31p CP/MAS spectra of Hg(PPh3h(N03h obtained at 4.70 T with MAS frequencies of (a) 4514, (b) 3831, (e) 2206, (d) 1845, (e) 1600, and (f) 1020 Hz.

and 3.5-4.0 f-Ls for the high field experiments. Contact times of 3-5 ms were used for experiments at both fields. In the MAS experiments, the sample spinning frequency ranged from 1.0 to 5.0 kHz at 4.70 T and 1.5 to 9.0 kHz at 9.40 T.

The solid-state 31p 2D J-resolved experiments were performed at 4.70 T under conditions of CP/MAS. The tl increments were synchronized with the rotor, i.e., tl

=2nT R' where T R is the rotor period. A simple eight-step phase cycling38 was used. All two-dimensional (2D) spec­tra were acquired in the magnitude mode. All 31p NMR spectra were referenced with respect to 85% H3P04(aq) by setting the NMR peak of solid NH4H2P04 to 0.81 ppm.

IV. RESULTS AND DISCUSSION

A. One-dimensional NMR spectra; unusual J­recoupling patterns

The 31p CP/MAS spectra of Hg(PPh3h(N03h as a function of the sample spinning frequency at 4.70 Tare shown in Fig. 4. Note that each spectrum is the total MAS NMR spectrum, i.e., the intensities of all the spinning side­bands have been added. At sample spinning frequencies greater than 3.8 kHz, the 31p CP/MAS spectrum exhibits a single isotropic peak centered at 40.32 ppm. X-ray dif-

fraction experiments indicate that the crystals of Hg(PPh3)2(N03h are monoclinic with four molecules in the unit cell belonging to the space group Cl/c. 39 Each molecule has a crystallographic C2 axis relating the two phosphine ligands and the two nitrate anions, thus the two phosphorus atoms are crystallographically and, hence, chemically equivalent. The observation of a single peak in the 31p CP/MAS spectra at high spinning frequencies con­firms the chemical equivalence of the two triphenylphos­phine ligands. As the spinning frequency decreases, the isotropic peak becomes broader and splits. Further de­crease in the MAS frequency causes the intensity of the outer peaks to grow. Simultaneously their separations de­crease. In general, a four-line pattern was observed over the normal range of sample spinning frequencies, i.e., 1.0-4.0 kHz. Moreover, it is clear that this four-line pattern belongs to the unusual J-recoupling pattern (type II) as defined in Sec. II. From the separation of Ull and Ul2 as well as that of Ul3 and Ul4' the value of 2Je1p,3Ip) in Hg(PPh3h(N03)2 was determined to be 250± 10 Hz. This value for 2Je1p,3Ip) is in excellent agreement with the corresponding value found for a related compound, Hg(PMe3h(N03h, 250 Hz.40

The 31p CP/MAS spectra of Hg(PPh3h(N03h ob­tained at a higher magnetic field, 9.40 T (shown in Fig. 5), exhibit essentially the same behavior as those observed at 4.70 T, except that the peaks are broader. For example, at spinning frequencies greater than 7.0 kHz, the 31p NMR spectrum exhibits a single peak centered at 40.32 ppm, again confirming that the two 31p nuclei are chemically equivalent. At slower spinning frequencies, a four-line pat­tern is also observed.

Equations (26)-(30) predict that for a given system, the J-recoupling patterns obtained at different applied magnetic fields should be coincident, provided that the G term in Eq. (30) is negligible and that spectra obtained at a reduced sample spinning frequency, UlR/UlO' are com­pared; Ulo is the Larmor frequency. In Fig. 6 the line posi­tions in the 31p CP/MAS spectra obtained at 4.70 Tare plotted against the actual spinning frequency while for the corresponding line positions at 9.40 T, the actual spinning frequencies were scaled by a factor of i. The agreement between the two J-recoupling patterns is excellent. From the pattern in Fig. 6, it is also apparent that the alternate lines are always parallel to each other, the separation of which equals the J coupling constant between the two chemically equivalent phosphorus nuclei in Hg(PPh3h(N03h Again, the value of 250± 10 Hz for 2Je1p,3Ip) is also confirmed from the 31p CP/MAS spec­tra obtained at 9.40 T.

Based on Eqs. (26)-(30), it is clear that the J-recoupling patterns are determined only by the mixing term, D, since the J coupling constant is measured directly from the J-recoupling patterns. Therefore, instead of show­ing all four lines in the J-recoupling patterns, only the observed and calculated D terms are compared; see Fig. 7. The four line positions can be easily obtained using Eqs. (26)-(30). The effective Hamiltonian includes J¥'(O),

J¥'(1), J¥'(2), and J¥'(3). The agreement between the ob-

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G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra 6145

(a)

(b)

(c)

(d)

(e)

",,' ' ..... ! III ' '',. ' ...... II. II" .. " ...... !

~oo 200 -200

Hz

FIG. 5. The 31p CP/MAS spectra ofHg(PPh3h(N03)2 obtained at 9.40 T with MAS frequencies of (a) 7000, (b) 5200, (c) 4000, (d) 3400, and (e) 2000 Hz.

served and calculated D terms is good in the spinning fre­quency range of 1.5-4.0 kHz, indicating that the effective Hamiltonian inclUding high-order correction terms up to Jlr<3) is sufficient to describe the observed unusual J-recoupling patterns between two chemically equivalent spins in this spinning frequency range. At frequencies be­low 1.5 kHz the experimental data deviate from the theo­retical curve, implying that higher-order correction terms become important for such slow frequencies.

The parameters used in the simulation are k l/27T=3.2 ±0.8X lOS (Hz2), i2/27T=5.5±0.8X 108 (Hz3

), g2/27T =0.0 (Hz3), and k 3/27T=-3.5±0.8XlOll (Hz4 ). These parameters are independent of the applied magnetic field when the MAS frequencies measured at 9.40 T are scaled by a factor on. It is readily seen from Eqs. (19) and (20) that the coefficients g2n are small if the difference between the two chemical shift tensors is larger than the direct dipolar coupling. From the ratio of i2lkl' the magnitude of the difference between the two chemical shift tensors can be estimated, i.e., S=i2Ikl=1.7 kHz. Based on a P-P separation of 4.474 A obtained from the x-ray diffraction study of Hg(PPh3)2(N03}z,39 the direct dipolar coupling constant was calculated to be 220 Hz, which is indeed much smaller than S. Moreover, our observation that the J-recoupling patterns obtained at two different applied

0 0 0

,.--... 4000 N 0 0

:c '--" • • >-u c 3000 • • Q)

0 0 0 0 ::J 0- • • • • Q) l-

LL S .., ~ ·0

Ol 2000 • o. 10 • C 0 0 0 0

• • • • c 0 0 0 0

c 0 0 0 0 0 0 0 0

0.. 0 0 0 0

(/) 1000

O+-~-.--~'-~--r-~-r~--r-~~ 300 200 100 0 -, 00 -200 -300

Line Position (Hz)

FIG. 6. The J-recoupling patterns observed at 4.70 T (open circles) and 9.40 T (closed circles). The spinning frequency for data obtained at 9.40 T is scaled by a factor of i.

magnetic fields become coincident provides further evi­dence that the g2n terms are negligible, as we have noted previously. In principle, based on these calculated coeffi­cients, the relative orientations of the two phosphorus chemical shift tensors can be deduced using the structural data obtained from the x-ray diffraction study and the three principle components of the phosphorus chemical shift tensor obtained from the NMR spectrum of a static powder sample. The detailed analysis is, however, reserved for a later publication.41

0.35

0.30

0.25

~

0.20 N :r: .::£ "-""'

0 0.15

0.10

0 0.05

0.00 1000 1500 2000 2500 3000 3500 4000

Spinning Frequency (Hz)

FIG. 7. Calculated (continuous curve) and observed D terms in the J-recoup\ed spectra of Hg(PPh3h(N03)2 at 4.70 T (open circles) and 9.40 T (closed circles) as a function of MAS frequency.

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6146 G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra

1.0

0.8 V>

:i-. iii 0.6 c

q) +'

.EO q) 0.4 > ~ q)

et:: 0.2

O.O...jL---.----r-----r-------,r-----''I'''------, 1000 1500 2000 2500 3000 3500

Spinning Frequency (Hz)

4000

FIG. 8. Calculated (continuous curve) and observed relative intensity within the J-recoupled spectra of Hg(PPh3)2(N03lz at 4.70 T (open circles) and 9.40 T (closed circles) as a function of MAS frequency.

B. Relative intensities and line shapes in the J-recoupled spectra

As noted in Sec. II, the relative intensity within the J-recoupled spectrum is also sensitive to the sample spin­ning frequency. This is obvious in the experimental spectra shown in Figs. 4 and 5. Using k" 12> g2' and k3 values obtained from the aforementioned line-position simulation, the relative intensities of the outer and inner peaks, S, were also calculated using Eqs. (26)-(30). From Fig. 8, it is clear that the agreement is good in a spinning frequency range of 1.5-4.0 kHz, indicating that the effective Hamil­tonian induding the average Hamiltonian and its first three correction terms can reproduce the main characteristics of the observed J-recoupled spectra, i.e., both the peak posi­tions and their relative intensities. It is also obvious that at spinning frequencies below 1.5 kHz, the agreement be­tween calculated and observed relative intensities deterio­rates. The inclusion of higher order correction terms would be necessary to reproduce the relative intensities of peaks observed at such slow MAS frequencies.

It is of interest to compare the peak positions and the relative intensities of peaks in the J-recoupled spectra with AB spectra obtained in solution NMR studies. Consider the usual J-recoupled spectrum (type I) as an example. For such systems, it is only necessary to consider the K term to first order because the magnitudes of the G and F terms are much smaller relative to the K term. At the high spinning frequency limit, an A2 spectrum is observed. As the spinning frequency decreases, a typical AB quartet re­sults. As the spinning frequency continues to decrease, the low and high frequency parts move apart with the outer peaks being more intense, i.e., approaching an AX spec­trum. It is known that such changes in AB spectra are associated with an increase in the chemical shift difference between the two strongly J-coupled spins. The appearance of the J-recoupled spectrum can be readily understood by considering the similarity of the mixing term, D, and the corresponding expression in solution NMR, [wJ+ (WA

- W B) 2] 1/2. Since the K term in Eq. (30) is analogous to the chemical shift difference term in solution cases, we can think of the K term as an effective chemical shift difference. Because this effective chemical shift difference is inversely proportional to the sample spinning frequency, WR' it van­ishes at high spinning frequencies, leading to an A2 spec­trum. At slower spinning frequencies, typical "AB quar­tets" are observed. The slower the sample spinning frequency, the larger the effective chemical shift difference and therefore the less AB character in the resultant J-recoupled spectra.

It was found in the experimental spectra that although the peak positions and relative intensities are sensitive to the sample spinning frequency, the linewidths of the indi­vidual peaks are approximately independent of the sample spinning frequency. This observation indicates that inho­mogeneous line broadening due to the orientation depen­dence of various order correction terms is small, compared with the naturallinewidth in the 31p CP/MAS spectra of Hg(PPh3)2(N03h This also provides proof that our ap­proximation of using peak positions and peak intensities instead of the complete line shapes in the simulations is valid for the present case.

If we neglect the detailed line shape of the individual peaks in J-recoupled spectra and use the square root of the second moment of such a four-line pattern, (M2 ) 1/2, to describe the "width" of the observed J-recoupled spec­trum, then we have

(M2) 1/2= [(K I +K3)2+G~+F~] 1/2

= (.;+ ~+ l~i2klk3 +~) 1/2:::::~. (32) wR wR wR WR

Equation (32) predicts a linear relationship between (M2) 1/2 and 1/wR' provided that high order terms are neg­ligible. In Fig. 9, the observed (M2 ) 1/2 was plotted against the sample spinning frequency. Indeed, a linear relation­ship was found for high spinning frequencies with the ini­tial slope being equal to k l •

In their study of 13C MAS spectra of (NH4h(0213C-13C02), Maricq and Waugh25 found the 1/wR dependence of the linewidth at high sample spinning frequencies. Re­cently, Challoner et al. 3o also observed that the linewidth in the J-recoupled spectra of TBPDP increases as the sam­ple spinning frequency is decreased. Levitt et al. 21 distin­guished between linewidths due to a resonant splitting and inhomogeneous broadening. They demonstrated a different spinning frequency dependence for the 13C MAS NMR linewidths of dimethylsulphone- 13C2 and 1,2-13C2-glucose. For Hg(PPh3}z(N03h we find that the "width" of the J-recoupled spectra exhibit a 1/ W R dependence at high spinning frequencies. Since the chemically equivalent 31p spin pair in Hg(PPh3h(N03h satisfies an "n=O rota­tional resonance" condition, it is clear that our case is associated with a resonant splitting, as defined by Levitt et al. 21 In fact, the inhomogeneous broadenings are negli­gible in our case.

In more general cases, as in the J-recoupled spectra of TBPDP reported by Challoner et al.,30 inhomogeneous

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G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra 6147

300

slope = kl/21t ,-.,

200

::a 0 0 0

'-'

• li

til 100

o+-----~----~----._----~----._--__, 0.00 0.20 0.40 0.60 0.80 1.00 1.20

FIG. 9. Spinning frequency dependence of (M2) 1/2 in J-recoupled spectra of Hg(PPh3)2(N03 )2 at 4.70 T (open circles) and 9.40 T (closed cir­cles). The initial slope is given by k/21r, 3.2X 105 (Hz2

).

broadening can be large especially in the slow spinning frequency region. For simplicity, we consider only the av­erage Hamiltonian, jy(O) , and its first-order correction term, ~ I), in the following discussion. Theoretical J-recoupled spectra (type I) as a function of sample spin­ning frequency are shown in Fig. 10. Since the first-order correction coefficient, kh depends on crystallite orientation in the applied magnetic field, for a powder sample it results in inhomogeneous broadening for individual peaks in J-recoupled spectra. It is seen in Fig. 10 that, in addition to changes in the line position, the line shapes of individual peaks show significant broadening at slow spinning fre­quencies.

C. Two-dimensional J-resolved spectra

In order to obtain an independent measurement of the two-bond phosphorus-phosphorus J-coupling constant in Hg(PPh3)2(N03h, 20 31p homonuclear J-resolved exper­iments were performed at 4.70 T under MAS conditions. In the 20 experiments, the tl increments were synchro­nized with the rotor.38

,42 The homonuclear J-coupling con­stant is given by the splitting along the II dimension in the 20 J-resolved spectrum. The contour plot of a homonu­clear 31p J-resolved spectrum of Hg(PPh3h(N03h ob­tained at a spinning frequency of 1.80 kHz is shown in Fig. 11. From the 2D spectra, 2Je lp,3Ip) = 250 ± 15 Hz, confirming our analysis of the 10 spectra.

It is also apparent from Fig. 11 that the four-line pat­tern in the isotropic region is different from that in the first-order low frequency spinning sideband. In particular, the two outer lines in the isotropic part are too weak to be seen while the two central lines in the first low frequency spinning sideband are less intense than the two outer lines. Such a difference in relative intensity within the four-line pattern also manifests itself in the 2D J-resolved spectrum. As we have pointed out earlier in this paper, the effective Hamiltonian derived from the AHT cannot describe fea-

(a)

(b)

0.5 o kHz

-0.5

FIG. 10. Theoretical J-recoupled spectra for a pair of chemically equiv­alent spins at MAS frequencies of (a) 4000, (b) 3000, (c) 2000, and (d) 1000 Hz. The central feature in the spectra shown in (b)-(d) has been cut off for clarity. The parameters used in the simulations are .5 11 = 150, .522 =50, and .5)3=0.0 ppm; the .533 components of the two chemical shift tensors are coincident; the two 822 components make an angle of 65°; the internuclear vector bisects the angle made by the two .522 components; the dipolar coupling constant R/27T=400 Hz; U)J/27T=200 Hz; Larmor fre­quency, 81.03 MHz.

tures in the spinning sidebands of MAS NMR spectra. Hence, although the total MAS NMR spectrum (summed over all spinning sidebands) can be accurately predicted using the effective Hamiltonian in Eq. (5), the detailed structure of individual spinning sideband cannot be under­stood using the effective Hamiltonian derived in this paper. A complete description of spinning sideband structure in the MAS NMR spectra can be obtained using the Floquet theory as demonstrated by Schmidt and Vega23 and Nakai and McDowell. 24,43

V. CONCLUSIONS

We have shown that for a pair of chemically equivalent spins, the solid-state MAS NMR spectra exhibit a general four-line pattern. The J coupling constant, which cannot be observed in solution-state NMR spectra of such a spin pair, can generally be determined directly from the MAS frequency dependent NMR spectra of solid samples. Using average Hamiltonian theory, explicit expressions for the spinning frequency dependence have been derived to any order correction terms. All observed spinning frequency dependent phenomena involving two chemically equivalent

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6148 G. Wu and R. E. Wasylishen: J-recoupling in MAS NMR spectra

(a) (b)

I I 42 38

f2 ppm

\") 00 C>o

I 20

@.::)

I 16

200

-200

Hz

FIG. 11. (a) Isotropic and (b) first-order low frequency spinning side­band regions of the 1D 31p CP/MAS and 2D 31p CP/MAS J-resolved spectra ofHg(PPh3)2(N03)2 at 4.70 T. The spinning frequency was 1.80 kHz.

nuclear spins can be understood on this theoretical basis. The theory predicts three general types of J-recoupling patterns. Although the type I and II J-recoupling patterns have been observed experimentally, the type III pattern has not yet been reported.

When the J coupling constant involving two chemi­cally equivalent spins is small relative to the direct dipolar coupling constant between them, the resultant J-recoupled spectrum usually exhibits a type I pattern. If the J coupling constant is comparable to the dipolar coupling constant, a type II J-recoupling pattern is possible. Only in systems where the J coupling constant is large and comparable to the slowest spinning frequencies, may the type III J-recoupling pattern be observed.

The theory derived here was successfully applied to analyze the unusual J-recoupling patterns (type II) ob­served in 31p CP/MAS spectra of Hg(PPh3h(N03h. We have demonstrated that the unusual J-recoupling pattern observed in this system cannot be understood with the average Hamiltonian and its first order correction term; instead, higher correction terms must be included. By con­sidering the average Hamiltonian and its first three correc­tion terms, the J-recoupled spectra of Hg(PPh3)2(N03)2 observed at 4.70 and 9.40 T were well reproduced for sam­ple spinning frequencies in the range of 1.5-4.0 kHz and 3.0-8.0 kHz, respectively. It is also clear that higher order correction terms are necessary for reproducing the J-recoupled spectra at very slow sample spinning frequen­cies. Analysis of the J-recoupling patterns obtained at 4.70 and 9.40 T yields 2Je1p,31 p) = 250 ± 10 Hz. This value was confirmed by obtaining the 2D 31p J-resolved spec­trum. Analysis of the J-recoupling pattern arising from a chemically equivalent spin pair in the solid state potentially

permits one to deduce information concerning the relative orientation of their respective chemical shift tensors.

In this study we have demonstrated that it is important to first identify the type of J-recoupled spectrum by obtain­ing MAS NMR spectra at a wide range of sample spinning frequencies. This will assure one of obtaining the correct J-coupling constant. We anticipate that J-recoupling phe­nomena can be of general importance in a wide variety of systems containing chemically equivalent spin pairs.

ACKNOWLEDGMENTS

We wish to thank Mr. Michael D. Lumsden for pro­viding a sample of Hg(PPh3}z(N03}z, Dr. Bill Power for obtaining some preliminary 31p CP/MAS spectra, and Dr. Klaus Eichele for his interest and suggestions. This work was financially supported by the Natural Sciences and En­gineering Research Council of Canada (NSERC) in the form of operating and equipment grants (R.E.W.).

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