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Казанский национальный исследовательский технологический университет
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Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

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436

A. RAHMOUNI AND C. BARBIER

In order to compare the electronic properties of the (001), (010) and (100) faces three

clusters have been selected each one modelling one specific face of the

crystal, these

clusters contain the same number of oxygen and molybdenum atoms :

and

for

the (001), (010) and (100) faces respectively.

 

 

The total energy values (Table 6) show that the (010) face is more stable than the (001) and (100) faces, in agreement with the results of Firment[21], also our calculations show that the (001) face is more stable than the (100) one.

FROM CLUSTER TO INFINITE SOLID

437

Since the type of oxygen vacancy depends on the (001) or (100) faces, the unsaturated molybdenum charge is highest in the negative charge of unsaturated oxygen is higher than in the coordinatively saturated atoms. Analysis of the Mo-O overlap population shows that the bond is stronger on the (100) face whereas the other Mo-O bonds are stronger on the (001) face.

438

A. RAHMOUNI AND C. BARBIER

4. Conclusion

The similarity of the results obtained for finite clusters and the infinite slab allows to conclude in favour of the validity of the cluster model of adequate size (6 or 8 molybdenum atoms). In addition to the chemisorption of organic molecules on solid surfaces which is generally considered as a localized phenomenon, the interaction between molybdenum oxide and an adsorbate can also be represented by a local complex formed by a finite cluster and the adsorbed molecule. Indeed, the study of the evolution of the electronic properties as a function of the cluster size shows that, for a cluster containing 6 or 8 molybdenum atoms, most of the electronic properties converge towards limit values. This convergence is sensitive to the direction of the cluster growth. On the other hand, the electronic properties of the (001), (010) and (100) faces are not identical, the type of surface atoms being different ; these results allow to predict that the characteristics of the chemisorption step will depend on the particular face on which it takes place.

Acknowledgement

The authors gratefully acknowledge technical assistance of the "Centre d'Informatique Scientifique et Médical" of the University Claude Bernard Lyon I.

References

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2.A. B. Anderson, Y. Kim, D. W. Ewing, R. K. Grasselli and M. Tenhover, Surf. Sci., 134, 237 (1983).

3.J. B. Moffat, J. Mol. Catal. 260, 385 (1984).

4.D. Masure, P. Chaquin, C. Louis, M. Che and M. Fournier, J. Catal. 119, 415 (1989).

5.M. Fournier, C. Louis, M. Che, P. Chaquin and D. Masure, J. Catal. 119, 400

(1989).

6.J. Sylvestre, J. Am. Chem. Soc. 109, 594 (1987).

7.M.-H. Whangbo and L. F. Schneemeyer, Inorg. Chem. 25, 2424 (1986).

8.M.-H. Whangbo and E. Canadell, J. Am. Chem. Soc. 110, 358 (1988).

9.M.-H. Whangbo, E. Canadell and C. Schlenker, J. Am. Chem. Soc. 109, 6308 (1987).

10.E. Canadell and M.-H. Whangbo, Inorg. Chem. 27, 228 (1988).

11.

E. Canadell, M.-H. Whangbo, C. Schlenker and C. Escribie-Filippini,

 

Inorg. Chem. 28, 1466 (1989).

12.M.-H. Whangbo, R. Hoffmann and R. B. Woodward. Proc. R. Soc. London,

A366, 23 (1979).

13.J. Howell, A. Rossi, D. Wallace, K. Haraki and R. Hoffmann, QCPE, 4, 344

(1984).

14.M.-H. Whangbo , M. Evain, T. Hugbanks, M. Kertesz, S. Wijeysekera, C. Wilker, C. Zheng and R. Hoffmann, QCPE, 9, 591 (1989).

15.R. Hoffmann , J. Am. Chem. Soc. 112, 50 (1990).

16.L. Kihlborg, Arkiv Kemi, 21, 357 (1963).

17.G. Pacchioni, P. S. Bagus, M. R. Philpott and C. J. Nelin, Int. J. Quant. Chem.

XXXVIII, 675 (1990).

FROM CLUSTER TO INFINITE SOLID

439

18.V. D. Sutula , A. P. Zeif, B. I. Popov and P. I. Vadash, React. Kinet. Catal. Letters ,9, 79 (1978).

19.A. B. El Awad, E. A. Hassan, A. A. Said and K. M. Abd El Salaam,

Monatschete für Chemie, 120, 199 (1989).

20.R. Hoffmann, J. Angew, Chem. Int. Ed. Engl. 26, 846 (1987).

21.L. E. Firment and A. Ferretti, Surf. Sci. 129, 155 (1983).

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Ab initio Calculations on Muonium Adducts of Fullerenes

T.A. CLAXTON

Department of Chemistry, University of Leicester, Leicester LE1 7RH, United

Kingdom

1.Introduction

This paper is concerned with the structures of the simplest possible adducts of the and fullerenes, namely the monohydrides, and These open shell species or radicals may be considered as the product of the addition of one atom of hydrogen or one of its isotopes, among which we include specifically the light pseudoisotope of hydrogen known as muonium, Although has been observed [1], the stimulus for these calculations arose from the experiments on muon

implantation in solid

and

and higher fullerenes are distinguished from other allotropes of carbon, diamond and graphite, in that they exist as discrete molecules. The spherical or ellipsoidal nature of the monotropes opens up the possibility of intriguing new areas of chemistry. Here we are only interested in the hydrogen (or muonium) adducts, although this study has important implications to the very vigorous and extensive research in fullerene chemistry.

Two types of species have been detected in the

spectrum of

One shows an

unreacted or meta-stable muonium state which may well correspond to an ‘internal’

state, muonium is trapped inside the cage:

in the current notation [2].

This may be compared with ‘normal’ muonium (Mu´) in diamond and many other elemental and compound semi-conductors, where the trapping site is in one of the

cavities of tetrahedral symmetry.

This state of

is not discussed here, but

it does exhibit all the characteristics expected of the ‘internal’ chemistry of

The ‘anomalous’ muonium state,

observed in semi-conductors and generally

accepted to arise from muonium being trapped within one of the chemical bonds of the crystal, is unknown in molecules [5,6]. The constraints of the crystal lattice are necessary for the bond-centred state to be stable.

The other muonium adduct of

has very similar

hyperfine coupling con-

stants, 326MHz [2], to the the addition compound of muonium and ethene,

(329MHz)[7]. This is strongly indicative of a similar local structure and formation mechanism, represented formally in process (1). Muonium attacks and reduces one the short C-C bonds (common to two hexagons), bonds which have formally double bond character. It bonds to and saturates one of the carbon atoms leaving major spin density on the other. The rotational degree of freedom present in the ethyl radical is absent in caged structures and in this respect is simpler theoretically [8]. Nevertheless

441

Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 441–456.

© 1996 Kluwer Academic Publishers. Printed in the Netherlands.

442 T. A. CLAXTON

the similarity in the coupling constants argues against substantial delocalisation of the unpaired electron.

These observations are supported by the calculations of Estreicher et al. [9], using the the PRDDO method and density functional theory, the semi-empirical calculations of Percival and Wlodek [10] and ab initio ROHF calculations [11]. They confirm that

the most stable state of

results from the muon bonding itself to one of the

carbon atoms from outside the cage.

 

Here this work is continued and

extended to

where a considerable amount of

experimental work is currently in progress. The observation [4] of three electronmuon hyperfine coupling constants in not unexpected since there are five chemically distinct sites for muon to attack. The lower symmetry of makes the molecule much more interesting than

2.Methodology

2.1. AB INITIO CALCULATIONS

It is feasible to carry out Hartree-Fock calculations on our available computer re- sources (an SGI Crimson Elan Workstation) using an STO-3G basis set with full ge-

ometry optimization of

but only partial geometry optimisations of the

isomers. Fig. 1 shows planar graphs of

with the carbon atoms suitably

labelled for future reference.

Whereas there are only two different bond lengths in

short between atoms 1 and

2 and long between atoms 2 and 3, there are seven

different bond lengths in

The

bond lengths have been calculated here and previously [12] by the restricted

Hartree-Fock method using an STO-3G basis set and are discussed in some detail

AB-INITIO CALCULATIONS ON MUONIUM ADDUCTS OF FULLERENES

443

later. For the present it is sufficient to split these into three groups as indicated, using an obvious notation, in Fig. 2a.

Previous calculations on have indicated that the distortion to the cage in the neighbourhood of the point of attachment is so localised that it can be well represented by allowing the positions of only six carbon atoms to relax. It is assumed here that the same will apply to muonium adducts of and addition will take place according to process (1) at sites of unsaturation (Fig. 2a).

A typical group are atoms 1 through 6 in Fig. 1, forming a type a structure using a terminolgy introduced elsewhere [4]. A type a is identified as involving atoms from two pentagonal arrangements in the fullerene structure, the connecting bond (1-2) being short and presumably unsaturated. Three distinguishable type a structures

are illustrated for

in Fig. 2b, the atoms in closed loops having the points of

muonium attachment indicated by

and

 

The groups of atoms within each closed loop is called a defect, the

defect being

different from

simply by

the point

of muonium attachment. Defect

is at right

angles to defects

and

From previous calculations [11] on

the central

two atoms of the defect are expected to

change their hybridisation

from

with

an inevitable distortion of the underlying fullerene cage. For example if the

site

is the point of muonation, the unpaired electron tends to localise on the

site,

each site changing its hybridisation from

to approximately

the associated

distortion also affecting the four nearest neighbours. Note that all defects of type a contain at their centres one of the double bonds of Fig. 2a enabling the adduct to be formed according to process (1). Although can theoretically exist in one of five isomeric forms we have already identified three isomers of type a with an alkene type of unsaturation. The rest must be limited to regions of arene type unsaturation which form a band around the equator of Even so this gives rise to a further

444 T. A. CLAXTON

four defects of which two, at least, must be discarded. All are illustrated in Fig. 2b.

Type b structures involve atoms from only one pentagonal arrangement. If type c is considered [4] (involving no pentagons) it should be noted that the point of muonium

attachment is the same as for defect

The same applies to defect which shares

the same point of attachment as

Where defects share common muonium points of

attachment it is possible that a more extensive relaxation than that considered here will be necessary to describe these isomers accurately. All these defect types have been considered in ref. [4] except type where the central bond is exceptionally unsymmetric in that it is common to pentagonal and hexagonal arrangements.

Typical structures are specified in Table 1 which uses the labelling of carbon atoms

in

defined in Fig. 1. The restricted open-shell Hartree-Fock (ROHF) method

was used in all geometry optimizations using a minimal basis set of orbitals (STO3G) [13]. These calculations are therefore exploratory in nature. Here we have chosen to use the standard ab initio ROHF method since it is well-known that the UHF

method (as used in the PRDDO

approximation [9]) does not give wave functions

which are eigenstates of the total

spin operator

The effect of spin contamination

on molecular properties is uncertain, particularly if the contamination is high (the

value obtained by Estreicher was not reported [9] but values larger than 1 have been reported [11]). The ROHF method is normally unsuitable for investigating spin properties because spin polarisation of the closed shell electrons is not allowed and so some UHF calculations after quartet spin state annihilation (UHFAA)[14] are also

reported on the geometries optimised using the ROHF method.

AB-INITIO CALCULATIONS ON MUONIUM ADDUCTS OF FULLERENES

445

2.2. RESONANCE THEORY

Resonance theory [15] contains essentially three assumptions beyond those of the valence bond method. Perhaps the most serious assumption is the contention that only unexcited canonical forms, non-polar valence bond structures or classical structures need be considered. Less serious, but no more than intuitive, is the proposition that the molecular geometry will take on that expected for the average of the classical structures. This is extended to the measurement of stability being greater the greater the number of classical structures. These concepts are still widely used in chemistry in very qualitative ways.

Molecules of the size of the fullerenes require such approaches to help rationalise the results. Even though the number of classical structures can be very large the same qualitative reasoning can be used as with the smaller molecules, typically benzene.

The enumeration of classical structures is easily accomplished using computers. It is simply a problem of determining how many ways a set of points (in our case carbon atoms) can be connected given rules governing their connectivity.

3.Results

All of the ab initio results are collected in Tables 2 and 3. The tables differ only in the theoretical method used. Table 2 used the ROHF method and Table 3 used the UHF method using optimised geometries from Table 2.

The numbering system of the carbon atoms used are given in Fig. 1 for

Further results from the ab initio calculations are collected in various figures where they are compared to the results of resonance theory. The results are displayed in Figs. 3, 4 and 5. The number of classical structures is compared with the UHFAA spin densities at each centre in Fig. 3. Correlations between the number of classical structures and spin density (Fig. 4) and bond length (Fig. 5) are plotted. The

calculated bond lengths in

are rationalised using resonance theory in Table 4 from

which a valence bond type structure of

is suggested in Fig. 2a. The structures
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