Reactivity of fullerenes. Quantum-chemical descriptors versus curvature

THEO CHEM Journal of Molecular Structure (Theochem) 338 (1995) 293-301

ELSEVIER

Reactivity of fullerenes. Quantum-chemical descriptors versus curvature Eenheid

Algemene

K. Choho, W. Langenaeker,

G. Van De Woude, P. Geerlings”

Chcmie

Vrtlr

IALGCt

Fukultrit

Wetmschappm,

Uttiversiteit

Brussel

( VUB),

Pleinlaan

2, 1050 Brussels,

Belgium

Received 21 November 1994: accepted 24 November 1994

Abstract The relation between the reactivity of a fullerene towards a nucleophile and the curvature of the fullerene surface, reflecting the hybridisation of the carbon atoms, is investigated. Model 3-21G calculations on distorted ethenes show that the electronic part of the molecular electrostatic potential, V,,, might be a good measure for the reactivity of a double bond towards a nucleophile when only the influence of geometrical changes is considered (increasing reactivity upon increasing curvature). STO-3G calculations on the fullerenes CbO. CT0 and CT6 again indicate the quality of V,t as a reactivity descriptor. Both the regioselectivity in C 60. CT0 and CTb (in line with local curvature), and the relative reactivity of Cc0 and CT0 calculated by Ve, are in agreement with experimental data. The local hardnessfor thesesystemsshowsa fair correlation with the global hardness calculated in a finite difference approximation.

1. Introduction

The identification of the first fullerene by Kroto [l] in 1985 initiated a large number of studies involving the synthesis and isolation of an extended series of fullerene-type systems using different techniques [2-51. Due to their particular structure, both from a geometrical as well as an electronic point of view, these systems in turn were the object of numerous experimental and theoretical studies. All this has resulted in the publication of an avalanche of papers (about 2700) and even the creation of a particular database in relation to these systems[6]. The fullerenes have their place. as all carbon clusters, in the group of non-planar conjugated systems. These systemsdiffer from planar graphite * Corresponding author. 0166-1280/95/$09.50 SSDI

in the presenceof other than &membered rings and thus pyramidalized carbon atoms [7]. The presence of 7-membered rings gives rise to a negatively curved surface [8,9] whereas the presence of pentagonal rings in the non-planar conjugated system gives rise to a positive curvature allowing, if 12 of these rings are present, a closure of the system as is the case for the fullerenes (Ref. [7], and for a beautiful illustration see Ref. [IO]). These pentagonal rings, together with 1/2(n - 20) 6-membered rings, with n ranging from 20 to a few hundred, give rise to the full range of the C, group. It was also found that the fullerenes containing only isolated Smembered rings are more stable than those containing adjacent rings of this type (isolated pentagon rule) [l l-151. As C6a is the smallest fullerene for which this is the caseit is possible to make a distinction between two series of fullerenes: those with less

:c 1995 Elsevier Science B.V. All rtghts reserved

0166-1280(94)04068-O

I-ig. 1. Ethene-like

model

s-character can reach a value of 0.25 (sp3 hybridisation for the two carbon atoms). Although strictly speaking one does not have tetrahedral carbon atoms carrying four substituents of the type studied by van? Hoff in his landmark paper [21] the hybridisation of the carbons in the systems with higher curvature approaches sp3-type, typical for the tetrahedral carbon. A systematic trend was found: the bigger the curvature the higher the reactivity. This can easily be explained as the deviation from sp’ geometry is accompanied by an increasing strain, which is eliminated upon an addition reaction as the resulting geometry is closer to that of an sp’-hybridised carbon atom. The choice of reactivity descriptors and molecular regions studied in this work is based upon considerations related to the reactivity data used. The three fullerenes (C&. CT0 and CT6) were osmylated under analogous conditions, by Hawkins and co-workers [22-261. Traditionally, obtaining a cis-diol in the reaction of a catalytic amount of free 0~0~ with a double bond is related to the electron-donating behaviour of an alkene [27]. The osmylation carried out by Hawkins with stoichiometric equivalent amounts of an 0~0~ complex [28], leading to a cyclic osmate ester. has much more in common with a cycloaddition, where the electron-withdrawing behaviour of the alkene is the driving force. It is further noteworthy that CbO.as a good electrophile. undergoes six [2 + 4]and [2 + 2]-cycloadditions respectively with cyclopentadiene [29] and with benzyne [30], while CT0 is expected to behave likewise with the latter reagent [31]. In the widest meaning of the term,

system.

than 60 atoms, of which we will briefly consider CZ4. Cj2 and C50 in the present study. and those with 60 atoms or more. of which we only consider Cbo, CT0 and CT6 m detail. A large number of theoretical studies concerning the structure of these systems (see for example Refs. [4,16.17]) have been undertaken in recent years. The reactivity of the above mentioned systems has been less frequently studied. certainly at an ab initio level. Quite often the problem has been addressed in the past in terms of a relation between reactivity and the curvature of the fullerene surface [7]. This curvature can be quantified in terms of the pyramidalization of the carbon atoms and can be expressed in terms of the deviation a0 of the angle 8. giving the angle between the p-orbital and the rr-orbitals (seeFig. l), from 90 Based on this angle it is possible to calculate the s-character of the p-orbital [7,18-201. In the caseof graphite the s-character is zero (two sp’ carbon atoms) whereas in the case of a curved system the

a

b Ftg. 2. Three

types of bonds

C present

in a fullerene.

the electron-accepting characters of C(,o and CT0 are more pronounced than their electron-donating characters. The chemistry of ChO and C7() is indeed dominated by their electron-deficient character. In both structures six distinct pyracylene units are present and, with ChO, as much as six nucleophilic additions can occur at the interpentagonal bond of each pyracylene unit (Fig. 2a). the bond being of higher order. and shorter and more reactive. than that connecting a 5-membered ring and a 6-membered-ring (see Fig. 2b). and that connecting two 6-membered rings (see Fig. Ic) [32]. A perfect octahedral array of substituents is thus achieved. Until now, the scarcity of the substance C 7h did not allow much experimental work. Although kinetic studies are not abundant in literature, the impression that Chl, is more reactive than C,, seems to be predominant [29.33,?4]. Using these experimental data. we will examine two characteristics of fullerenes directly related to their reactivity and establish the relationship with local curvature. In the first place we will investigate the reactivity sequenceof the diflerent sitesin these systems when they undergo a nucleophilic attack (intramolecular reactivity sequence). This is done using the Molecular Electrostatic Potential (MEP) [3.5], or a part of it (vide infra). as H reactivity descriptor, a quantity which RX calculated only twice before for fullerene-type systemsnot permitting yet the study of intra- and intcrmoleculai reactivity sequences,once for a series of systems on the inside of the cage (endohedral) [36] and once, while the present work was already in progress, on the outside of the C,,, cage [37]. We will. however. look at the hardness of these systems, a density functional related quantity which finds its origin in the Hard and Soft Acids and Bases(HSAB) principle by Pearson [38]. This quantity will be considered in its global form as well as in an approximated local form 1399311. thus completing a set of studies situated in the samedomain [42246].

2. Methodology In a first part of this study. the MEP was calculated starting from the well known expression given

in Ref. [35] from which the electronic part,

l,,(R)= - ..I ,f(rjp, with /j(r) being the electron density, was isolated (vide infra). In a second part, the global hardness is considered. This quantity was defined as [47]

(4 with E being the energy and N being the number of electrons in the system considered. In a finite difference approximation using integer values for N. this expression can be rewritten as: I] =

IE ~ EA 7

(3)

where IE is the Ionization Energy and EA is the Electron Affinity. Due to multiplicity problems in calculations involving fullerene cations and anions, it was chosen to approximate IE and EA by the energy of the Highest Occupied Molecular Orbital (HOMO), eHoMo, and the energy of the Lowest Unoccupied Molecular Orbital (LUMO), cLUMo, according to Koopmans’ theorem [48]. This approximation leads to the current working equation for the global hardness: I] =

f 1 I MO

- FHOMO 7

Finally we consider an approximate expression [39.41] for the local hardnessv(r)

and investigate its relation to its global counterpart. The structures considered were taken from literature data [16.49951]. They all show the advantage of having a singlet ground state. In the case of the ethene-like model systems the 3-21G basis set [52] was used in the SCF calculations of the MEP using the GAUSSIAN 97 program [.53]. For the fullerenes the STO-3G [54] basis set was used for the calculation of the above mentioned quantities, both with the GAMES [55]

program and the G-ZUSSIAN 92 program. All calculations were performed on the CRAY Y-MP I 16 computer of the Free Universities of Brussels.

3. Results and discussion As mentioned in the Introduction the influence of the curvature on the reactivity is a well know-n fact. However, one of the most easily accessible reactivity indices is the MEP. Although it was originally invoked to describe reactivity towards an electrophile [35], its use as a descriptor of reactivity towards a nucleophile has been well documented [42.56--591. We have calculated this quantity for two series of modified ethene molecules selected as model systems. in order to quantify the influences of changes in curvature and bond length. The starting structure for these ethenes is built using standard values for all angles and bond lengths. In a first series the variation in curvature is modelled by, changing the angle 0 (see Fig. 1) from 90’. the case of two pure sp’ carbons. to 110”. two carbon atoms with a tetrahedral arrangement of the “substituents” (the two hydrogens, the other carbon and the p-orbital). The value of the MEP was calculated at a distance of 1.4 A from the double bond on an axis through the middle of the bond and the minimum in the MEP.

-t i

Fig. 3. Calculated values of the mmimum for the ethene model system as a function

in the MEP (III a u ) of the curvature.

Fig 3 Calculated values of 1;, (in a.u.) swtem as a function of the curvature.

for the ethene

model

In the case of these model systems this axis clearly leads to identical contributions from the four hydrogen atoms to the MEP. As is seen in Fig. 3 the value of the MEP decreases with increasing curvature. This means that the reactivity towards a nucleophile decreases with increasing curvature, as a higher value means higher reactivity. This is not in accordance with earlier findings. An artefact due to the choice of the model system could not be ruled out at that moment. It was indeed clear that the termination of the double bond by means of hydrogen atoms could have its influence on the resulting values of the MEP. With respect to the point where the value of the MEP is taken, the magnitude of the nuclear part is only dependent on the positions of the hydrogen atoms, as the contributions of the carbon nuclei are constant. Therefore it was decided to consider only the electronic part. I&. of the MEP. As is seen in Fig. 4 the value of the I$ increases with increasing curvature. This means that the reactivity towards a nucleophile increases with increasing curvature, in accordance with earlier findings and confirming the idea of an artefact in the earlier calculations. In a second series the influence of changing bond length on the value of
-a -b -c -----)d

-5.161 1.3-I

I.36

1.38

I 40

I .42 d

1.34

146

I#

I 50

I

(a)

Fig. 5. Calculated values of L& (in a.u.) for the ethene model system as a function of the bond lengths varied between 1.34 and 1.51 A.

bond has a typical bond distance of I .34 A and the single bond has a bond distance of 1.54 A [60]. As intuitively expected the reactivity towards a nucleophile increases as the bond order decreases. It can be concluded that the electronic part of the MEP might be a good measure of the reactivity of a double bond towards a nucleophile, when only the influence of geometrical changes is considered. As mentioned before, Ci, has been recognised in the past as a very important, if not the exclusive, part of the local hardness. Therefore we will use it as such hereafter. Table 1 Calculated minimum of the MEP (in a.u.) and curvature bonds considered in C,,, CT,, and CTh System

Bond

MEP

Curvature

C 60

B

-0.00185

138.19

C 70

B D G H

-0.00173 -0.00199 -0.00048 a

136.95 139.09 146.67 149.22

A B C D E

-0.00074 -0.00200 -0.00167 -0.00302 -0.00130

136.73 13x.73 137.16 138.46 136.08

c76

a See text.

for the

(b) Fig. 6. The structure show-n is characterised handed helix if viewed two A-bonds.

of (a) CTO, (h) CT,. The Dz-enantiomer by hexacene units giving rise to a rightalong the Cz-axis joining the midpoints of

If we look at the three fullerene systems CbO, CT0 and CT6 for which experimental data are available, we see that the reactivity generally is reflected in the curvature of the surface of these systems. This curvature is quantified in Table 1 by the mean of two dihedral angles over a double bond (cl-Cl -C2-y and b~-Cl -C2-x). In the case of CbO the curvature is the same all

over the system, as there is only one type of carbon atom. Mono-osmylation with Os0,(4-tert-butylpyridine): [22] thus occurs only at 66 ring junctions, a bond, called A, for which k$ was calculated in this work. In the case of CT0 we are able to distinguish five sets of equivalent carbons (see Fig. 6a). Mono-osmylation yields only two ?) z isomers [24] which were found to c7o(oso4pY result from addition either on bond B or on bond D (see Fig. 2a). & was calculated for these two bonds. and in addition for two other bonds, G, connecting a 5-membered ring and a &membered ring (see Fig. 2b), and H connecting two 6-membered rings (see Fig. 2~). In the case of CT6, the smallest fullerene showing chirality. the racemic mixture was resolved by asymmetric mono-osmylation. Again the osmylation turns out to be regiospecific, the mono-adducts being ascribed by Hawkins et al. to osmylation at the bonds A and E (see Fig. 6b), all of the double bond type. We therefore selected these bonds for our study. together with bonds B, C and D being the other bonds with a predominant double bond character. According to the minima in the MEP, the deepest minimum in this case indicating the lowest reactivity. the reactivity sequence for the bonds in CT0 (for the “numbering” of the bonds, see Fig. 5) is G > B > D (Table 1). This sequence is very poor as B and D are found to be the least

Fig. 7. Cl, (in a.~.). on an axis through B. D. G and H (see teut) for CTO. as a function ofr (in au.): (0) B. (Cl) D, (W) G. (0) H.

ml

30

32

4-l

4h

4x

511

S?

5.4

Sh

5.x

h.l

r Fig. 8. L;, (in a.u.). on an axis through A. 8, C, D and E (see text) for Cih. as a function of r (in a.~.): (0) A, (Cl) E, (m) B, (0) C, (+) D.

reactive bonds, in contrast with the experimental data and the curvature. Looking only at the electronic part (Fig. 7). here also taken along an axis through the middle of the bond considered and the minimum in the MEP (in the case of H perpendicular to the bond as there is no minimum present), it is clear that the results of the MEP are improved significantly. The general sequence now is: B > D > G, H, in perfect agreement with experimental data (vide supra). In the case of C76 the MEP already generates a correct sequence for the selected bonds: A, E > C. B, D. This is in general agreement with the sequence according to the curvature and the assumptions on the basis of the experimental data (vide supra). qt (Fig. 8) also generates the same global sequence, but there are some minor differences. One can clearly distinguish two groups of bonds {A, E} and {B, C, D}, in contrast with the case of the MEP. Furthermore it is clear that the difference between the values for A and E is much smaller than is the case for the MEP. If we assume, based on the quality of this quantity as a reactivity descriptor for fullerenes, that the correct sequence is given by the curvature, all of these results favour the use of Q as a reactivity descriptor. We now look at the global hardness, calculated according to Eq. (4) for all of the fullerenes

4 KC1

2

44

4.6

4.8

5.0

52

5.4

5.6

5.8

r Fig. 9. Calculated values of the global hardness rl (In a.u.) as a function of the number of carbon atoms II for C,,. Cl:, CSO. C,,,. CT,,, and CTh.

Fig. 1 I. q(r) (in a.u.) at the most reactive bonds of (260, C,,, and Cih as a function of r (in a.u.): (0) C,,, (0) CTO, (0) CT6.

considered. The hardness is found to decrease with increasing molecular size (see Fig. 9) for both series, the first one consisting of fullerenes containing non-isolated pentagonal rings, the second one containing fullerenes obeying the pentagon isolation principle. More importantly, one finds a discontinuity in the “hardness versus molecular dimension” curve at CbO. This means that, in contrast with what might be expected, the reactivity of the larger fullerenes (&,. CT0 and CT6) can still be governed by the harddhard interactions

as described in the HSAB principle. This is confirmed by the first part of this study, where &, which is closely related to the local hardness, is found to be a good reactivity descriptor in this case. If we look at the values of L$ for that site which was found to be the most reactive (Fig. lo), we see that the intermolecular reactivity sequence is given as: C6o more reactive than CT0 more reactive than C&, This is in agreement with the experimental data available, and permits, based on the quality of V,t. the prediction of a lower reactivity for C&. As mentioned above. the quantity L$r is directly related to the local hardness (see Eq. (5)). We therefore compared the sequence according to the local hardness at the most reactive bonds (see Fig. 11) with the sequence of the global hardness. These were found to be identical (CeO > CT0 > C&. This result can be seen as further confirmation of the relation between I$, and the hardness.

-2x ,

1

4. Conclusions

Fig. 10. Q (in a.u.) at the most reactive bonds of C6,,, CT,, and CT6 as a function of r (in a.u.): (0) Cno. (0) C7,1, (0) Ci6,

The model calculations on distorted ethenes show that the electronic part of the MEP &, might be a good measure for the reactivity of a double bond towards a nucleophile when only the influence of geometrical changes is considered (increasing reactivity upon increasing curvature).

In C6”, C70 and CT6 bll reproduces the experimental data on both the regioselectivity in CeO, CT0 and CT6 (in line with local curvature), and the relative reactivity ( ChO versus C70). The local hardness for these systems shows a fair correlation with the global hardness calculated in a finite difference approximation.

Acknowledgements W.L. is indebted to the Research Council of the Vrije Universiteit Brussei for a position as Research Assistant. P.G. is indebted to the D.P.W.B. (National Service for Programmation of Scientific Policy ~ Impulse Program on Information Technology ~ Contract IT:‘SC/36) for a generous computer grant in support of this work. Dr. J.M.L. Martin (Limburgs Universitair Centrum, Department SBG. Diepenbeek. Belgium) is gratefully acknowledged for his kind help with the setting up of the calculations.

[I-l] [l5] 116) [I71 [IX] [19] [20] [?l] [22] [23] [?I] [25] (261 [27] [28]

[29] [?O]

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