Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

each connected to five other pentagons through a connection at each apex. The connection could formally be given a carbon-carbon double bond. The bonds in each pentagon are formally single, emphasising its alkene nature, conjugation between adjacent double bonds taking a secondary role. To illustrate the possible aromatic

would be to have extended conjugation over the whole ring, although hindered by the curvature imposed by the spherical structure. Whatever view is taken both indicate that they should be good candidates for treatment by resonance theory [15].

for a relation of the type

where a and b are adjustable constants, centres j, k and l are adjacent to centre i and where the modulus sign is necessary to allow for negative spin densities. A simpler, and more easily manipulated, relation is The equation has been plotted on Fig. 4 to show how well it correlates the data. The oscillation

This relationship has been found in spite of the fact that the spherical cage of

has been severely distorted in the region of the defect which could have mitigated

It is also possible to try and rationalise the variation in bond length of over the surface using resonance theory. This is rather more difficult than the similar calculation made above for To do this we assume that the muon is attached to atom 1 (Fig. 1) and the unpaired electron is localised at position 2. Certainly this structure will dominate. Two other adjacent atoms are selected to form a bond and the number of classical structures which can be drawn from this configuration is enumerated and the number is a measure of the double bond character. These have been directly plotted (Fig. 5) against the optimised bond lengths calculated for Biaxial error bars are used to include a range of values which would otherwise have overlapped. A number of points on the graph have been indicated for

450 T. A. CLAXTON

further comment. These points refer to bonds which are the immediate vicinity of the defect and conceivably still liable to the large distortion.

Nevertheless the unexpectedly long bonds, (7-11,8-12), are well predicted as are the very short bonds (3-7,4-8,9-11,10-12). All of these bonds are part of the hexagonal carbon atom structure. On the other hand the bonds of the neighbouring pentagons, 3-5,5-6,etc., are certainly out of place although still qualitatively in the right order. Presumably these bonds have absorbed most of the residual distortion of the defect for geometrical reasons.

Apart from these all other bond lengths correlate well with the corresponding number

associated by Dewar and Schmeising [17] to a measure of the normal single bond

predicted by resonance theory, the bond lengths being either slightly longer than a typical double bond or slightly shorter than a single bond. This would seem to rule out significant conjugation as expected in an arene. The alkene properties of

since there are 5 different types of carbon atom leading to eight different bond lengths. These have been obtained from a calculation on fully optimising the geometry using the ROHF method and an STO-3G basis set (see Table 4, the bond lengths are

within in ref.[12|).

The corresponding numbers of classical structures for each bond are also given. The column marked in Table 4 is where n is the number of classical structures for the bond in question, no = 14196, the number of classical structures for the typical bond 31-40, and the total number of classical structures for

We are, without justification, assuming that no classical structures are unimportant to the bond order and are subtracted from all classical structures. The ratios of the residuals are taken as a measure of the mobile bond order [16]. This fixes the typical bond 31-40 to be single. The bond order for the aromatic bond is then 1.5 (the same as benzene in resonance theory). At present this is no more than a useful correlation. The almost identical numbers of classical structures for the ajacent typical bonds 21-22 and 21-31 strongly indicates that these form an aromatic type system because the symmetry of the molecule makes these typical bonds form complete hexagonal arrangements. The arene properties are also demonstrated from the almost identical bond lengths which arise even though the bond environments of 21-22 and 21-31 are different. These aromatic rings are apparently not directly conjugated since the bond which connects them, typically 31–40, is the longest bond with the smallest number of classical structures. The other single bonds have similar immediate environments. Their bond lengths are typical for single bonds connecting carbon atoms [17]. One

1The point is emphasised from STO-3G RHF calculations on butadiene. The optimised central

which have the same bond environment (pentagon/hexagon). It was therefore with some confidence that the valence-bond type structure in Fig. 2a was presented as representing the structure which describes the chemistry of If muonation occurs more readily at alkene bonds the expected sites are typically 1, 6 and 11, that is, all type a structures (Fig. 2b). Type b and c structures involve bonds with arene character.

4.2. AB INITIO CALCULATIONS

Our interest is two-fold, we wish to know whether the defect is firstly structurally localised and secondly electronically localised. Our second interest extends to know

adduct did not depend on the full structure but corresponded to a localised ‘defect’, both structurally and electronically.

The calculations reported here are for a fully optimised geometry. The difference between the calculation using the fully optimised geometry and that which allowed only six of the carbon atoms to relax from the underlying structure is not

only a localised distortion.

The ROHF calculations (Table 2) show a clear progression of decreasing stability the closer the muon is placed to the equator of This is shown by whichshould be less than -.5 Hartrees for a stable adduct. The most stable structures are of type a,

(compare Figs. 2a and 2b). The corresponding UHF calculations (Table 3) do not show the same behaviour although many of the differences can be associated with the problems expected from large values of in other words, the exceptionally large spin contamination makes these UHF calculations particularly uncertain. Perhaps this has arisen due to the restrictions imposed by the partial geometry optimisation procedure. In any case the results will not be discussed further.

Apart from type which is only slowly convergent to the optimised geometry, the other centres are well described by the ROHF method. Polyhedral views of the three type a structures are shown in Fig. 6. These all illustrate the change of hybridisation at the point of muonium attachment and at the adjacent carbon atom where the unpaired electron is effectively localised as expected from addition to an alkene. The

and c defects (Fig. 7) are quite different. The expected hybridisation change to

is clearly present for the atom bonded to muonium, but other significant distortions are not obvious. This is consistent with the prediction from resonance theory (Fig. 8) that the unpaired electron for these structures is delocalised over a large number of centres.

This seems to imply that the associated distortion of the cage is only large for one atom (not two as for the type a structures) and therefore possibly better accomodated by the defects defined in Table 4. It must be concluded that either structures b and c are chemically unrealistic or there is a subtle, but significant, stabilisation provided by a complete geometry optimisation.

In Fig. 8 the preferred sites for the unpaired electron are estimated from the numbers of resonance structures counted when the electron is fixed at that site. The structural differences between Figs. 6 and 7 can now be understood. In addition the similarity

Overall Tables 2 and 3 suggest that only type a has a firm foundation. If we accept that the other structures may be possible only types and show smaller muon spin density. Experimentally [4] three sites for muonium attachment have been detected with muon hyperfine coupling constants 278.2MHz, 342.8MHz and 364.6MHz with amplitudes 17.1%, 13.1% and 11.2% respectively. The smallest coupling constant of the three is the strongest. The most obvious explanation is that only type a structures have been observed, type with the smallest of the type a coupling constants, has twice the number of sites in the molecule than the other type a structures. This would fit both the amplitude and coupling constants results qualitatively. Since structures

of decreasing hyperfine coupling constant. It is not adequate to suggest that the number of sites of each should be proportional to the amplitude in this case since type a structures are ‘alkene’ and type b structures are ‘arene’. However since alkene sites are expected to form adducts more readily than arene sites [18] the predicted amplitudes are qualitatively incorrect.

cessful in reproducing the trends obtained from ab initio calculations. Not only has this helped the intepretation of the results but has contributed to testing the relia-

(type a) alkene structures. It does seem unlikely that the type c structure will be observed. Although partial geometry optimisations seem satisfactory for the type a defects, they are not for the other structures and full geometry optimisations seem to be necessary. It is possible that only type a structures are chemically feasible. Further work is necessary to confirm these conjectures but it is clear that the interpretation of the experimental data in terms of isomers of is correct.

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