J . Phys. Chem. 1991, 95,6404-6405
6404
Anlsotroplc Nuclear Magnetic Shleldlng In Cbo P. W. Fowler,**+P. Lazzeretti,t M. Malagoli,* and R. Zanasi* Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, U.K., and Dipartimento di Chimica, Universitd di Modena, Via G. Campi 185, Modena I41 - 100, Italy (Received: April 17, 1991; In Final Form: June 26, 1991) Extrapolation from ab initio coupled Hartree-Fock calculations is used to estimate the anisotropic I3C nuclear shielding tensor for each site in Cw The principal components of the symmetric shielding tensor are 179, 10, and -51 ppm. The derived chemical shifts have the same pattern as those deduced from solidstate NMR measurements, and their mean differs from the experimental shift by less than 4 ppm. The large diamagnetic component is associated with a near-radical local axis and the paramagnetic component with the normal to the local mirror plane.
The most striking evidence for the icosahedral symmetry of the Cm molecule is its I3C NMR spectrum. In solution'*2and room-temperaturepowder spectra? a single peak is found at 143 ppm, demonstrating the equivalence of all 60 carbon nuclei. However, though equivalent, the nuclei lie at sites of relatively low symmetry and so have asymmetric nuclear shielding tensors. Line-shape analysis on samples of fullerite powder cooled to 77 K gives the principal components of the chemical shift tensor at 220, 186, and 40 ~ p m In . ~this Letter we compare these values with those obtained from ab initio calculation and use the theoretical results to describe the orientation of the shielding tensor in the local axis frame. The observed chemical shifts can be converted to absolute shieldings by calibration against a substance for which both are known. In ref 5 we used values for benzene to derive a conversion equation: absolute shielding = 185.7 ppm - chemical shift. The quoted components for C, are therefore equivalent to shieldings of -34,0, and 146 ppm. The discrepancy of 6 ppm between their average and the absolute shielding of C, in solution (43 ppms) gives an estimate of the likely error in the lineshape analysis. Note that one component of the "shielding" tensor is actually a paramagnetic deshielding and that the tensor has one very large and diamagnetic component. Previously, we calculated electric and magnetic properties of C, by the ab initio coupled HartretFock method using the SYSMO program with a sequence of basis sets of increasing size and flexibility. Results for energy, polarizability, magnetizability, and mean nuclear electric and magnetic shieldings in STO-3G, STO-3G*, 6-31G, and 6-31G* basis sets were reported in ref 4, along with computational details. Full icosahedral symmetry was utilized throughout. Even with the 900 CGTOs of the 6-31G* basis, the computed properties are likely to be still some way from the Hartree-Fock limit, and because of the relatively heavy cost of response calculations, there is little prospect of further basis improvement in the near future. In ref 4 we developed an extrapolation procedure for estimating the basis set limit and applied it to the magnetizability' and mean nuclear shieldings of Cm Here the same technique is used to estimate theoretical nuclear shielding components from the ab initio results. The site symmetry of each carbon in C, is the mirror group C,,since the nucleus lies in one of the 15 reflection planes of Zh but on no other nontrivial symmetry element. When the origin of vector potential lies in the same plane (e.g., at the nucleus or at the molecular center), the nuclear shielding tensor u may have up to five nonzero elements
-
u =
(:;: :;i3) 0
(1)
0
where 1 and 2 are orthogonal directions within the plane and 3 'Universitv of Exeter. *Universiti di Modena.
is normal to it. This matrix is not symmetric, because the paramagnetic contribution to u is asymmetric, but may be diagonalized to yield three distinct principal components. The eigenvalues of u are usually real but can become complex for sufficiently large asymmetry. The symmetric tensor d = 1 / 2 ( u + 5 ) has three real eigenvalues, one of which coincides with u33 and two that sum to the trace of the (12) block of u. Hansen and Bouman6 have analyzed the spectroscopic consequencesof asymmetry in u. Only the eigenvalues of d play a role in the observed anisotropies, since the full shielding tensor always appears in company with its adjoint in expressions for the resonance frequencies. The shielding tensors in the four basis sets are listed in Table I. All are calculated with the origin of vector potential at the nucleus of interest. In finite-basis CHF calculations the total shielding shows a spurious dependence on this choice of gauge, and different results would be obtained if the origin were taken for example at the molecular center of mass. Previous experience with the mean shieldingSsuggests that the nuclear gauge gives more accurate results for Cso,probably because the basis functions are themselves centered on atoms. The chosen carbon nucleus has coordinates (1/2rh,1/2#(rb+ 2rp),0)in the Boyle-Parker axis system' where e.g. xyz is a C3 axis, x , y , and z are C2axes, and xy is a mirror plane. r is the pentagonal (-single) bond length, rhis the hexagonal (-iouble) + 1). In our bond length, and # is the golden ratio 1/2(d/5 calculations we used the optimized STO-3G values* of rp = 1.465 and rh = 1.376 A. Experimental estimates of rpand rhhave since become a~ailable:~ rp = 1.45 and rh = 1.40 A with uncertainties of *0.015 A. Table I shows that the tensor components vary strongly with basis. This is an expected consequence of the form of u. The total shielding consists of a sum of large and opposing diamagnetic and paramagnetic contributions; the positive diamagnetic term is an expectation value and is stable against basis improvement (the mean, 8, is 1080 1 ppm for all basis sets used heres); the paramagnetic term has negative diagonal elements and as a second-order property is poorly represented in minimal and split-valence sets. In the present case, its average grows in magnitude as the basis is e~tended,~ the cancellation in u becomes more complete, and the total shielding decreases with improve-
*
(1) Taylor, R.; Hare,J. P.; AWul-Sala, A. K.; Kroto, H. W. J. Chem. Sor., Chem. Commun. 1990. 1423. ( 2 ) Bethune, D. S.;Meijer, G.; Tang, W. C.; Rosen. H. J. Chem. Phys. Lctr. 1990, 174, 219. (3) Yannoni, C . S; Johnson, R. D.; Meijer, 0.; Bethune, D. S.;Salem. J. R. J . Phys. Chem. 1991, 95,9. ( 4 ) Fowler,P. W.; Lazzeretti, P.; Zanasi, R. Chem. Phys. k r t . 1990, 165,
..
70
( 5 ) Fowler. P. W.; Lazzerctti. P.; Malagoli, - M.; Zanasi. R. Chcm. Phys. Leri.1991, 179, 174. (6) Hansen. A. E.: Bouman. T. D. J . Chem. Phys. 1989. 91. 3552. (7) k y l e , L. L.; Parker, Y..Mol. Phys. 1980, 99, 95. (8) Schulman. J. M.;Disch, R. L.; Miller, M. A.; Peck,R. C . Chem. Phys. Len. 1981, 141, 45. ( 9 ) Yannoni, C . S.;Bernier, P. P.; Bethune, D. S.;Meijer, G.;Salem, J. R. J . Am. Chem.Soc. 1991, 113, 3190.
0022-365419112095-6404302.5010 0 1991 American Chemical Societv
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6405
Letters
TABLE I: Coupled hrtree-Fact Calclculrtiorrr of Carbon Nuclear Sbieiding Tensors in Co (ppm)"
basis (P,P) STO-3G 90.1 6-31G 176.4 STO-3G' 186.4 6-31G' 243.9
extrap 360
9
689.95 -1 1.67 0 480.90 -1.67 0 464.64 -2.32 0 315.00 5.50 0 36.242 17.956 0 0.999 1 0.9880
25.09 626.95 0 53.51 483.71 0 44.9 1 453.45 0 7 1.77 357.09 0 103.545 153.536 0 0.9293 0.9978
diag (a')
diag (a)
U
0 0 653.57 0 0 418.92 0 0 423.03 0 0 247.19
0
684.90
690.66 632.00
482.31
626.24 653.51
+ 9.351
653.57 508.26
482.3 1-9.351 459.05
456.35 418.92
+ 8.541
418.92 48 1.06
459.05 - 8.541'
437.03 423.03
364.99
423.03 380.04
307.10
292.05 247.19
22.097
0 -5 1.246
247.19 10.449
167.68 1
179.329
-5 1.246
-5 1.246
0.9936 a Details of basis sets are given in ref 4. (P,P)is the TRK sum rule (2). 9 is the Bravais coefficient of the linear regression of each shielding component on (P,P).
ments in the basis. The eigenvalues of u are also strongly basis dependent, with complex values in two cases (where u12u21is negative and larger in magnitude than 1/4(u11- u ~ ~ ) * The ). eigenvalues of the symmetric part of u show a more reasonable behavior, falling monotonically with basis size. To make an educated guess at the results to be expected from future calculations on larger basis sets, we consider the Thomas-Reiche-Kuhn sum rule in the momentum formalism. In a complete basis the quantity
is equal to the total number of electrons.I0 In (2) Pa is the a component of the total electronic linear momentum and {fin,En] is the complete set of eigenfunctions and eigenvalues of the unperturbed molecule. (P,P)is a ''diagonal- perturbation sum and hence obeys a Hylleraas variation principle;" an approximate calculation with a sufficiently accurate zeroth-order function fi0 will underestimate (P,P). Since the physical magnetic properties depend on similar integrals, it seems reasonable to take (PIP)as a guide to basis quality. In the basis sets used for Cbo a CHF calculation of (P,P) recovers 25.0% (STO-3G), 49.0% (6-31G), 51.8% (STO-3G*), and 67.8% (6-31G*) of the full complement of 360 electrons. The calculated average shielding grows monotonically with (P,P),and simple linear regression on the mean nuclear shieldingss gives B = 46.2 ppm with a Bravais coefficient of 0.9995. In view of the measured value of B = 43 ppm, this is encouraging. Extrapolation of each component uil in the same way yields the tensor shown at the foot of Table I, with Bravais coefficients in the range 0.929-0.999. The extrapolated u* matrix has three real eigenvalues: one large and diamagnetic, one small and diamagnetic, and one paramagnetic. The comparison with experimental values is extrapolated: 176, 10, -51 experimental: 146, 0, -34 i.e. an average discrepancy of -20 ppm per component. The extrapolated eigenvalues of u itself are 168,22, and -51. Given the crude extrapolation and the difficulty of ab initio calculation on a molecule of this size, the level of agreement with experiment is satisfying. It confirms the qualitative pattern of u and allows us to assign directions to the measured principal components. The paramagnetic eigenvalue of both symmetric and full tensors is uj3; Le., it refers to the normal to the local mirror plane. Principal directions of the (12) block are more problematic since this (IO) Lazzeretti, P.;Zanasi, R. Phys. Rev. 1985, ,432, 2607. ( I I ) Hylleraas, E.A.; Undheim, R. Phys. Rev. A 1930,65, 759.
- ...
... e,
F i i 1. Axes of the I3Cshielding tensor in CW The plane of the figure is a mirror plane for the icosahedral group. In the molecule-fixed frame (MFF) with origin at the molecular center of mass the nucleus C has coordinates (0.6880,3.4836,0) A. The unit vectors e, and e define the local frame parallel to the MFF, e, is the radial vector, and e,i! and e! arc the (nonorthogonal) right eigenvectors of u. The orthogonal vectors # and et are the eigenvectors of the symmetric part of u. The dotted curve is the surface of the sphere through the 60 nuclear positions. The sub script 1 is associated with the largest, and 2 with the middle, eigenvalue of the respective tensors. In the local frame er = (0.6188,0.7855), e:
= (-0.9908,0.1354), e; = (0.3908,0.9205), and e! = (-0.9205,0.3908).
asymmetric 2 X 2 block has nonorthogonal eigenvectors. The directions of the right eigenvectors, i.e., those defined for each total shielding eigenvalue X by the equation (3)
are shown in Figure 1. If, however, we take the symmetric part of u (which is the measurable), we find eigenvalues close (*I0 ppm) to those of the total tensor, and the large diamagnetic component then refers to an axis lying in the mirror plane at an angle of -11.8' to the radius vector (see Figure 1). This is broadly compatible with the picture of Cboas a molecule with a surface r system, with diamagnetic circulation induced within the faces of the truncated icosahedron by a magnetic field at right angles to them. Whether this constitutes a "ring current" is difficult to say, since the pattern of anisotropy is cOmmon to several types of ?r system-isolated, conjugated, and aromatic. For example, 13Cin ethylene has experimental shielding components12 of -83 and +82 ppm (in-plane) and +178 ppm (out-of-plane) and benzene" has an average in-plane shielding of 6 f 10 ppm and an out-of-plane component of 186 f 10 ppm. It would be premature to take the "C NMR spectrum as decisive proof of aromaticity in Cbo. (12) Appleman, B. R.;Dailey, B. P. Adu. Magn. Reson. 1974, 7, 231. (13) Zilm, K.W.;Conlin. R.T.; Grant, D. M.; Michl, J. J . Am. Chem. Soc. 1980, 102, 6672.
