4815
J. Phys. Chem. 1992,96,4815-4820
Solid-State %o NMR in Tetrahedral Clusters JCr8me Hirschinger,' Pierre Granger, Institut de Chimie, BP 296/R8, UMR 50 CNRS-Bruker-UniversitE Louis Pasteur, 67008 Strasbourg CEdex. France
and Jacky RosC Laboratoire de Chimie de Coordination, URA 416 CNRS, UniversitC Louis Pasteur, 4 rue Blaise Pascal, 67070 Strasbourg CEdex, France (Received: December 18, 1991; In Final Form: February I I, 1992)
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A series of tetrahedral mixed-metal clusters A[MCO,(CO),~]with M = Fe, Ru and A = NEt4, H has been studied by solid-state s9C0NMR at two different magnetic field strengths (4.7 and 7.1 T). Due to the large value of the quadrupole coupling constant, only the central transition is detected experimentally. We have developed a moment analysis of the central transition powder pattern that permits a direct evaluation of the quadrupole coupling constant as well as the isotropic part and anisotropy of the chemical shift tensor without having to assume coincident chemical shift (CS) and electric field gradient (EFG) tensors. Using the method of the moments to obtain starting guess parameter sets, both the magnitudes and relative orientation of the CS and EFG tensors have been determined by nonlinear least-squares fitting of the line shapes at the two magnetic field strengths. The CS tensor is almost completely axially symmetric for the whole series of clusters. While the substitution of the apical metallic atom M does not affect the CS anisotropy,all the clusters can clearly be distinguished by their quadrupole coupling constant. The EFG tensor is found to be significantly less symmetric than the CS tensor. Furthermore, the line-shape fits demonstrate that the CS and EFG tensors are noncoincident and that their relative orientation mainly depends on the nature of the counterion A. The solid-state isotropic chemical shifts and quadrupole coupling constants are consistent with s9C0 NMR data previously obtained in solution.
I. Introduction With the increasing availability of commercial high-field multinuclear N M R spectrometers, it has become possible to observe N M R powder spectra of several half-integer spin quadruof the periodic Among these, the polar nuclei (I > s9C0nucleus (I = 7/2) has a rather large quadrupole moment Q. This is probably the reason why only polycrystalline complexes with a high molecular symmetry have been studied by s9c0NMR? Generally, the spectra of quadrupolar nuclei depend on both the magnitudes and relative orientation of the electric field gradient (EFG) and chemical shift (CS) t e n s o r ~ . ~ . ~When - ~ the site symmetry at the s9Conucleus of interest is high, the principal axes of the EFG and CS tensors must be coincident? Obviously, this is likely to be the case for symmetric octahedral cobalt com~lexes.2.~On the other hand, when the site symmetry is low, the principal axes of the EFG and CS tensors may be noncoincid e r ~ t . ~This . ~ , ~fact that was first observed in 51VN M R spectra of a single crystal of V2058 has recently been confirmed in a sV N M R study of polycrystalline V205.9 Furthermore, the noncoincidence of the EFG and CS tensors has also been detected by 87Rband 133CsN M R in Therefore, the N M R interaction parameters may also be accurately determined in systems where the lack of long-range order precludes the application of standard diffraction methods. In the present article, the tetrahedral mixed-metal clusters [NEt4][MCo3(CO),2] ( M = Fe (la), M = Ru (lb)) and H M C O ~ ( C O )(M , ~ = Fe ( 2 4 , M = Ru (2b)) are studied by solid-state s9C0NMR. As a consequence of the relatively low site symmetry at the 59C0sites, the EFG and C S tensors are expected to be noncoincident. Moreover, these heteronuclear cobalt containiig clusters are of structural and catalytic interest.IO As we have recently shown in the liquid state,"J2 s9C0NMR may be a very powerful method that will improve our knowledge about metallosite selectivity in cluster reactions. 11. Theory
A. Orientation-Dependent Transition Frequencies. The total spin Hamiltonian His written as a sum over the different couplings A
iltonians HA(A # Z) can then be treated as perturbations of the Zeeman coupling. In the presence of quadrupolar (A = Q) and anisotropic chemical shift interactions (A = CS) and considering that IHQI >> IHcsl, the effective Hamiltonian in the rotating frame is writtenI3J4
Herr = 4)+
x
(1)
(2)
(3) (4)
R ~ - ~ R R I z [ ~+ I ( 1) Z - 21zz - 11) (5) where CQ =
4I(2I - 1)h
wo = YBO the isotropic chemical shift. The quantities Rki (i = 0, f l , f 2 ) are irreducible spherical tensor components corresponding to the interaction A = Q, CS.IsJ6 The resonance frequency in the rotating frame of an arbitrary single quantum transition is thus uo is
wm,m-l
= (mlHerrlm) - ( m - 1 I H d m - 1)
(6)
By substitution of eq 2 into eq 6 we readily obtain wm,m-l
= w$&
+ w:g:)l
+ Wmm-1 Q(2)
(7)
with W$$l
w$fl
= (2m - ~ ) C Q ~ I / ~ R $ $
(8)
2 = - C Q ~ ( R ~ - I R ~ [+ ~ I1)( I- 24m(m - 1) - 91 + WO
H =E H ~
+ %2)
with
R!$,R$[2I(I+
In a strong magnetic field Bo,the Zeeman interaction H z of the nuclear spin I with Bo is predominant. The internal Ham0022-3654/92/2096-48 15%03.00/0 0 1992 American Chemical Society
1) - 6m(m - 1) - 31) (9)
4816 The Journal of Physical Chemistry, Vol. 96, No. 12, 1992
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Equation 8 implies that the or “central” transition (m = is not influenced to first order by the quadrupolar interaction. This is the reason why it is possible to observe the central transition NMR signal even if the quadrupolar interaction is very large. On the other hand, eq 10 shows that, for the shift interaction, the single quantum transition frequencies are equal for all Im) * Im-1) transitions. Restricting the problem to the relevant central transition, we obtain
4cQ2 = -[I(I
+ 1 ) - ’/,](-2RX’RR + R$qRR)
(12)
WO
At this point, we assume that molecular dynamics are not contributing to the observed N M R signal. The Rii in the laboratory frame L are then expressed as a function of the spherical components pii in the principal axes system P of the interaction tensor according toI5J6
Hirschinger et al. wCs = wou0
+ ~ ~ 6 ~ ~ [ Pe)~ -( !/zqcs c o ssinZe cos 2$0]
(21)
where P~(COS e) = ~ ( cosz 3 e- 1) The quadrupole coupling constant Qm = 2qQ/h is independent of I. It is remarked that the Euler angle yiLis no longer present because the magnetic field Bo is a symmetry axis for the spin system. 6’ and 9 simply are the spherical polar angles that define the orientation of Bo in the P system. Moreover, since we have dropped the superscript A, eqs 20 and 21 implicitly assume that the EFG and CS tensors are coincident. However, for crystallographic sites of low symmetry, the EFG and CS tensors may not have the same orientation, i.e., fl& # fig, It is then necessary to take into account the relative orientation of the two coupling tensors. The second-order quadrupolar interaction which involves products of spherical tensor components (eq 5 ) leads to a complicated orientation dependence (eq 20). Thus, as in refs 3 , 4, and 6, we express the shift interaction in the EFG tensor principal axes system. This is easily done by further expansion into Wigner rotation matrices Z
where the Df)are the second-rank Wigner rotation matrix elements.” aiLare the Euler angles (aiL,siL,Ai,) which give the orientation of the considered interaction tensor in the laboratory frame. The irreducible principal components p$i are expressed as a function of the Cartesian components A? of the coupling tensor ( i = 1-3),
RY$ = C p$sDb2’(fl,) iJ=-2
= pi-1 = 0
(15)
(16) = Pi-2 = - j / Z ‘ l X h = !h(A: - A$) The principal axes 1, 2, and 3 are labeled according to the following c o n ~ e n t i o n ~ ~ J ~ IAi - AXI 1 IA: - AAl 1 IAi - A X / (17) P$2
where AX
1/3(Af+ A$ + A:)
(18)
A Xis the isotropic or scalar part of the coupling tensor (Acs = uo and AQ = 0). The A? ( i = 1-3) are the principal axes com-
ponents of the EFG tensor. In the following, instead of using the principal values At, a coupling tensor will be conveniently characterized by its isotropicpart AX,its anisotropy 6X (eq 14) which is a measure of the strength of the interaction, and its asymmetry parameter qx,which is a measure of the deviation of the interaction tensor from axial symmetry, = (A$ - A ? ) / 6 ,
(05 5 1 ) (19) Substituting eqs 13-19 into eqs 10 and 12 and considering that (&, &) (Ee) and 6 , = eq, we derive the same result as Baugher et al. W?/$1/2 = - ( W Q ~ / ~ W O ) [ I (+I 1 ) - 3/41 x qX
[ ~ ( pcos4 )
e + B ( P ) cos2 e + c(p)] (20)
(22)
where $2, relates the orientation of the CS tensor to the EFG principal axes system which is itself related to the laboratory frame by QQL. Inserting eq 22 into eq 10 and writing ,lf = (a,8, y) and ( ~ Q L PQL) , = ($0, 0) gives us wcs = 1 wouo woScs P2(cos 8) - -qcs sin2 ,8 cos 2a Pz(cos 0) 2 sin 28 sin 28 [ 3 + qcs cos 2a] cos (y + $0) + 4 1 ;qcs sin 0 sin 28 sin 2a sin (y cp)
1
(1
+
pi1
Dg’(flQL)
+ +
L.
1 sin2 0 - -qcs(cos2 4
6 + 1 ) cos 2a
cos 0 sin2 e sin 2a sin 2(y + pCs 1
$0)
)
(23)
In a powder, all orientations of the principal axes system are equally probable, and the NMR spectrum is obtained by averaging eq 7 over all elements of solid angle. The powder pattern will then be given by
where dfl = 217 sin e dB d q and g(w) is a normalized Gaussian broadening function of standard deviation Aw accounting for unresolved dipolar interaction, relaxation effects, inhomogeneous magnetic fields, etc. The powder average over 8 and 9 is done numerically. B. Moment Analysis. Instead of directly computing I(w), one can undertake the simpler task of constructing a theory of independent moments of this line shape, these being experimentally defined byI5J9
where wQ =
and
3e2qQ =- WCC 21(2Z- 1)h Z(2Z- 1 )
M , = ~ ~ m w n Z (dw w)
(n = 0, 1, 2, ...)
(25)
Mo is equal to the area of the spectrum (Mo = 1) while the first moment M I is the center of gravity of the line shape. For theoretical calculations, a more useful definition of the moments of the central transition line shape is given by
In the following, using eq 26, we will express the first and second moments of the line shape as a function of the interaction parameters.
The Journal of Physical Chemistry, Vol. 96, No. 12, 1992 4817
Solid-state 59C0NMR in Tetrahedral Clusters First Moment M1.In this case, the contributions of the chemical shift and the quadrupolar interactions can be separated
MI
=ws+@
(27)
bf)(Q) dQ = 0
(28)
Since we
Equations 10 and 13 (Q =
QpL)
(37) Using eq 35, the integrals involving the products of four Wigner rotation matrices are expressed as a sum over Clebsch-Gordan coefficient~l~
or eq 22 (0= QQL) imply that
This is the well-known result stating that only the isotropic part of a given interaction contributes to M Ito first order. To evaluate the quadrupolar contribution, we use the well-known orthonormality relations”
where 6 is the Kronecker symbol. Using eqs 12, 13, and 30, we readily obtain
30 = -(@)’ (39) 7 It is convenient to define the second moment (MJc0 about the center of gravity of the line shape. It is straightforward to derive an expression similar to eq 32 (M2)CG
=
(MS>CG
+ ( W s ) C G + (@)CG
(40)
M S- (es)’
(41)
with (KS>CG (Mp”’)CG
This is the result originally found by Behrens and SchnabeLzo It shows that the second-order quadrupolar interaction causes NMR line shifts that must be taken into account for the determination of the chemical shiftpla especially at low magnetic fields. Second Moment M2. According to eq 26, M2 is defined as the sum of three terms
Mz = My
+ Mp“S + M#
(32)
As for the calculation of MI, eqs 10, 13, or 22,28, and 30 @vel5
The cross-term
is written (34)
The calculation of this term requires the integration of products of three Wigner rotation matrices. This is easily performed using the relationk7
bhz)*(Q) DU)(Q) D$(Q) dQ =
8u2 -6k+m,iC(222;mk) C(222;Oj) (35) 5 where the C(jJd3;mlm2)are Clebsch-Gordan coefficients. For a given orientation of the CS tensor relative to the EFG tensor specified by Qm = (cu,@,r), eqs 10, 12, 13,22, 29,31,34, and 35 lead to the expression
(M#)CG
=
=
w s- 2 w s @
= M# - (@I2
=
(42)
23
?(@I2
(43)
Note that eq 43 has previously been obtained by Freude et a1.22 At this point, it is interesting to investigate how to extract relevant parameters from the line-shape moments. From eqs 27 and 32, we immediately see that no interaction parameter can be deduced from the measurement of M I and ( M 2 ) C G at a single magnetic field. A possible solution would be to apply sample reorientation in order to average the chemical shift anisotropy (MAS,I3VAS23 experiments) or both the CS and EFG anisotropies (DAS,24 DORZ5experiments). However, in order to avoid the overlap of spinning side bands, these experiments require the spinning rate to be higher than the homogeneous line broadening and comparable to the inhomogeneous line broadening due to the secondorder quadrupolar interaction. Obviously, this cannot be achieved for the clusters presented in this study (see below). In any case, note that the relative orientation of the principal axes of the interaction tensors is lost upon applying MAS, DAS, or DOR techniques. An alternative method consists of measuring M Iand (MJm at two different magnetic field strengths. As will be shown below, this permits the determination of u,, as well as limiting values for WQ and with no need for assuming coincident CS and EFG tensors. Indeed, from the experimental lineshapes at two different magnetic fields, it is possible to calculate using eq 25 the first moment M l ( w o ) and the second moment *‘(w0) at the corresponding Larmor frequencies w,,’ and w$. Equations 27,29, and 31 then lead to a system of two linear equations from which we can extract the isotropic part of the CS tensor W,,’Ml
tro
=
(w,,’) (w,,’)2
- OO”Ml(W{) - (Wgll)Z
(44)
and the quadrupolar parameter
i+ 1
WQ2
cos 2a cos 27 ( 1
+ cos2 8) - sin 2cu sin 27 cos 6 (36)
The last term of eq 32 is the quadrupolar contribution
1
1 3
-r)Q2
=
3 0 [ W { k f 1 (W() -~ ( [Z(I + 1 ) - 3 / 4 ] ( ( W , , ’ / W / )
k f (lw { ) ]
- (W$/W())
(45) Since qQ2 I1, the ratio of the highest to the lowest possible value of Iw I (and lQccl) is 1.15, thus, giving a good estimation of IWQ~ (an% IQccl). Furthermore, using the fact that is independent of wo, this complicated cross-term (eq 36) can be eliminated by forming the difference of the second moments
Hirschinger et al.
4818 The Journal of Physical Chemistry, Vol. 96, No. 12, 1992
TABLE I: Moment Analvsis for the Series of Mixed-Metal Clustersa (ppm) IQccl, MHz
16,--l, ppm
la
lb
-2698 20.5, 23.7 590, 681
-2693 17.9, 20.7 1097, 1266
2a -2832 12.4, 14.3 1036, 1196
2b -3016 13.7, 15.8 723, 835
For lQccl and IScsl, the upper and lower limits given correspond to q = 0 and q = 1, respectively (see section IIB). bRelative to K,Co(Ca
N)6 in H20. l a M-Fe l b M-RU
21 M = Fe Zb M = RU
Figure 1. Schematic of the mixed-metal clusters.
obtained at the two different magnetic field strengths. Moreover, since @ is proportional to ( I @ ) ~ , it is determined by eq 45. Therefore, we get the following chemical shift parameter
then be neglected.28 Indeed, the effective radio-frequency field intensity is equal to ( I + 1/2)w1.14 The spectral width was 1.667 or 2.5 MHz. The delay time r2 was carefully adjusted in order to obtain a sampled data point at the peak of the echo. After left shifting, the signal is Fourier transformed with no line broadening starting at the top of echo to yield an undistorted NMR spectrum.
IV. Results and Discussion Figure 2 shows the experimental spectra obtained at the two available field strengths for the series of tetrahedral mixed metal clusters. Interestingly, considerable changes are detected in the lineshapes not only going from the anionic (la, lb) to the hydrure (h, 2b) compounds but also by substitution of the apical metallic atom. Moreover, the line shapes vary with the magnetic field As for the quadrupolar parameter (eq 4 9 , we obtain limits on strength, demonstrating that the magnitude of the second-order the absolute value of the chemical shift anisotropy 16csl having quadrupolar and first-order chemical shift interactions are comthe same ratio (1.15). Since the homogeneous broadening is well accounted for by an even function (g(w)), it does not affect M I , parable. Therefore, it is expected that both the CS and EFG tensors as well as their relative orientation may be determined so that eqs 44 and 45 are always valid. Similarly, eq 46 can be by analysis of the NMR lineshapes of Figure 2.3*6As described applied in the presence of strong homogeneous broadening provided in section 11, there are, in principle, nine parameters to determine: that it is essentially independent of wo, as it must be. up 6-, q-, Qcc, qQ, a,8, y, and Aw. This problem is not trivial. 111. Experimental Section First of all, a good starting guess data set must be found before iterative powder pattern fitting may be applied. As previously Samples. The tetrahedral clusters (Figure 1) [NEt,] [FeCo,( C O ) l ~ l(la), HFeCo3(C0)12 [NEt41[ R ~ C O , ( C O ) ~ ~ I proposed,6.18this could be achieved by determination of the locations of the lineshape singularities or ‘critical frequencies” (lb),27and H R u C O ~ ( C O )(2b)” , ~ were prepared using the pub(divergences and shoulders), preferably at several magnetic field lished procedures. strengths. Unfortunately, these frequencies are generally not NMR. 59C0NMR measurements were carried out on Bruker related in a simple manner to the interaction parameters,18esMSL-300 (Bo= 7.1 T, w0/27r = 71.213 MHz) and CXP-200 (Bo pecially when the EFG and CS tensors are noncoincident.6 = 4.7 T, w0/27r = 47.468 MHz) spectrometers. The spectra were Moreover, homogeneous line broadening causes the singularities obtained by using a standard broad-band Bruker probe. Due to (particularly the shoulders) to be less sharp or not distinguishable. the large magnitude of both the chemical shift anisotropy and the On the other hand, as shown in section 11, three important paquadrupolar coupling constant (see below), the free induction rameters (uo,16al, IQccl) can directly be evaluated from the first decay (fid) following a single radio-frequency pulse is very rapid and second moments of the line shapes (MI, (Mz)CG)at two (a few microseconds), so a significant part of the signal is lost different magnetic field strengths. Since the line shape does not during the receiver dead time. In such case, the powder pattern depend on the sign of Qcc (see section 11), it cannot (and need of the outer lines ( m m - 1 transitions with m # 1/2) submitted not) be determined. The sign of 6cs and the remaining parameters to first-order quadrupolar interaction (section 11) is so broad that (v-, qQ, CY,j3,y) may subsequently be estimated from the positions only the central transition signal is observable. Since the radioof the line-shape singularities. In our case, it is also useful to frequency field intensity wIis much smaller than IwQI (see section consider the symmetry at the cobalt site in the clusters. UnforIV), the central transition can be considered as that of a ‘fictive tunately, to our knowledge, there are no X-ray structure deterspin l/? (pure selective excitation).14 Therefore, the dead time minations of the clusters studied here. However, according to problem has been overcome by applying the well-known two-pulse X-ray studies on analogue c o m p o ~ n d s , 2we ~ ~may ~ ~ assume that Hahn echo sequence to the fictive spin central transition3 a mirror plane forces one axis of each tensor to lie perpendicular O1s - ~I-02s-~2-acquire-Do to it and hence to coincide with each other (Figure 1). In these conditions, a and y can be set to 0, ?r or f*/2 and only the Euler where Bls, BE are the pulse angles (the subscript s referring to angle j3 has to be adjusted. We have then to fit the experimental selective pulses) and r I ,T~ are the delay times. T~ was fixed to spectra with only seven parameters. A reasonable value for the 20 ps. The recycle time Do was in the range 300-500 ms. The last parameter, the line-broadening Aw which essentially affects nonselective 77/2 pulse length was determined for a liquid sample the sharpness of the singularities, is readily obtained by visual of K3[Co(CN6)] salt to be 6.5 ps. The corresponding selective comparison of the simulation with the experiment. This is the 7r/2 pulse length in the solid state would be 6.5/(1+ = 1.6 general strategy we have used in order to get good starting pap . I 4 We checked experimentally that this value is consistent with rameters for iterative powder pattern fitting (see below). the variation of the echo intensity as a function of the pulse The results of the moment analysis applied to the experimental durations. The 16-step phase cycling proposed by Rance and lineshapes of Figure 2 are reported in Table I. First, they clearly Byrd28causing destructive interference of the fid tails but coadc o n f m that, for all the clusters, the quadrupole coupling constant dition of the spin-echo signals was employed. Note that, in our Qcc is higher than 10 MHz, Le., l w ~ l / 2 7 r> 714 kHz. Therefore, case, phase cycling is not very useful since the fid has decayed we effectively have w1/27r = 38.5 kHz
