Ana Sanz
Matías
*^{a},
Remco W. A.
Havenith
^{bc},
Manuel
Alcamí
^{de} and
Arnout
Ceulemans
*^{a}
^{a}Department of Chemistry, University of Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium. E-mail: ana.sanzmatias@chem.kuleuven.be; arnout.ceulemans@chem.kuleuven.be; Fax: +32-16-327992
^{b}Theoretical Chemistry, Zernike Institute for Advanced Materials and Stratingh Institute for Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands. E-mail: r.w.a.havenith@rug.nl; Tel: +31-50-3637754
^{c}Ghent Quantum Chemistry Group, Department of Inorganic and Physical Chemistry, Ghent University, Krijgslaan 281 (S3), B-9000 Gent, Belgium
^{d}Departamento de Química, Módulo-13, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain. E-mail: manuel.alcami@uam.es; Tel: +34-91-4973857
^{e}Instituto Madrileño de Estudios Avanzados en Nanociencias (IMDEA-Nanociencia), Cantoblanco 28049 Madrid, Spain. E-mail: manuel.alcami@imdea.org
First published on 28th September 2015
The fullerene-50 is a ‘magic number’ cage according to the 2(N + 1)^{2} rule. For the three lowest isomers of C_{50} with trigonal and pentagonal symmetries, we calculate the sphericity index, the spherical parentage of the occupied π-orbitals, and the current density in an applied magnetic field. The minimal energy isomer, with D_{3} symmetry, comes closest to a spherical aromat or a superaromat. In the D_{5h} bond-stretch isomers the electronic structure shows larger deviations from the ideal spherical shells, with hybridisation or even reversal of spherical parentages. It is shown that relative stabilities of fullerene cages do not correlate well with aromaticity, unlike the magnetic properties which are very sensitive indicators of spherical aromaticity. Superaromatic diamagnetism in the D_{3} cage is characterized by global diatropic currents, which encircle the whole cage. The breakdown of sphericity in the D_{5h} cages gives rise to local paratropic countercurrents.
Fullerenes are hollow carbon cages formed by hexagons and pentagons which, in spite of allowing closure, bring strain to the structure.^{5–7} Although a plethora of fullerene cages is mathematically possible, only a few have been detected experimentally.^{8} In view of their approximate globular shape fullerenes could be expected to be promising targets for spherical aromaticity. However, the two most prominent peaks in the mass spectra of evaporated graphite correspond to C_{60} and C_{70},^{9} neither of which holds a magic electron count.^{10,11} The exceptional stability of icosahedral C_{60} has been attributed to it being the first leapfrog cage with isolated pentagons.^{7} The frontier orbitals of leapfrog cages are based on entangled spherical shells,^{12,13} which is exactly the opposite of the spherical shell model. Likewise C_{70} represents the first case of a cylindrical fullerene with isolated pentagons.^{14} It too corresponds to a severe symmetry breaking of the spherical model.
Another intense peak in the mass spectrum corresponds to C_{50}. This is a magic electron count which fulfils the 2(N + 1)^{2} rule with N = 4. Despite theoretical research has already been carried out on its isomers,^{15–18} little attention has been paid to what is mentioned as a cause of their different stability: spherical aromaticity.^{19} This paper will be devoted, through a distinct approach, to a detailed examination of the electronic structure of 50-fullerenes, with special attention to the role of spherical aromaticity.
For the D_{3} isomer a modified Schlegel diagram was adopted to obtain a better visualization of the symmetry. Since the main rotational axis of the molecule, C_{3}, crosses through the top and bottom carbon atoms instead of being centred in a hexagon, a standard Schlegel projection in a hexagon will not offer a suitable representation of the trigonal symmetry. Instead we erase the bottom carbon atom (atom 5 in Fig. 1, situated in the center of three fused hexagons) and replace it by a triangle linking its three surrounding atoms. Consequently, when the Schlegel projection is performed through this triangle, the symmetry of the structure is preserved. The D_{3} isomer consists of six isolated pentagon–pentagon pairs which are arranged in a trigonal way. Top and bottom of the trigonal axis are occupied by a carbon atom lying in the centre of three adjacent hexagons, and linked to three pentagon–pentagon pairs.
The D_{5h} structure on the other hand is a regular pentagonal prism, with five isolated pentagon–pentagon pairs arranged around the equator, and a top and bottom isolated pentagon. For the D_{5h} structure, Lu et al.^{17} found two stable bond-stretch isomers,^{20} denoted hereafter as A and B, with different bond lengths within the same connectivity. Both isomers differ in their frontier orbitals: while in the most stable one (A) the HOMO corresponds to an orbital of a_{1}′ symmetry and the LUMO to an orbital of a_{2}′ symmetry, for the B isomer the HOMO and LUMO orbitals are exchanged (HOMO a_{2}′ and LUMO a_{1}′). As we will discuss later these different electronic distributions yield very different structures and values of aromaticity.
According to previous calculations^{15,18} the D_{3} structure is the most stable, despite the presence of six adjacent pentagons (AP) as compared to only five AP's in the D_{5h} isomer. This represents an exception to the Pentagon Adjacency Penalty Rule (PAPR),^{21,22} which states that the most stable fullerene isomer corresponds to the one with the lowest number of adjacent pentagons and for every extra AP a destabilization of the system between 0.8–0.9 eV is expected.
So far experimental isolation of 50-fullerenes has been limited to the preparation of the chloro-derivative, decachlorofullerene[50] C_{50}Cl_{10}.^{23} It was demonstrated by a variety of techniques that this derivative most likely has a D_{5h} structure, with the ten chlorine atoms attached to the equatorial pentagon–pentagon fusions.^{24} This compound was nicknamed ‘saturnene’ in view of the chlorine ring around its equator. In contrast the ground state of unsubstituted C_{50} has not yet been characterized experimentally.
(1) |
(2) |
(3) |
Nucleus independent chemical shifts (NICS, ppm)^{32} were computed at the centre of the cage using the GIAO method. Current density maps were calculated at the BLYP/6-31G* level using the ipsocentric choice of origin as implemented in GAMESS-UK^{33} and SYSMO.^{34} Diatropic and paratropic currents are represented, respectively, by anticlockwise and clockwise circulations.
Isomer | AP | E _{rel} | Gap | S | NICS |
---|---|---|---|---|---|
D _{3} | 6 | 0 | 2.27 | 0.024 | −40.3 |
D _{5h} (A) | 5 | 5.58 | 1.27 | 0.037 | −32.5 |
D _{5h} (B) | 5 | 2.23 | 1.37 | 0.051 | −2.7 |
The difference in geometry between D_{5h} A and B isomers is most pronounced on the equatorial belt. The calculated length of C–C bonds for the equatorial pentagon–pentagon fusions is 1.396 Å for the A isomer, versus 1.478 Å for the B isomer. As noted by Lu et al., in the A isomer the pentaphenyl belt has a quinoid-like valence-bond structure, while in the B isomer it is a sequence of aromatic benzenes,^{17} as shown in Fig. 2. Accordingly the two bonding schemes can be characterized as a Fries structure for A, and a Clar structure for B.
Fig. 2 Bond alternation in the equatorial belts for the A (Fries) and B (Clar) isomers of C_{50}, and orbital pictures of the frontier molecular orbitals (A isomer: a_{1}′ HOMO, a_{2}′ LUMO). |
While the two bond-stretch isomers are close in energy, their HOMO and LUMO are interchanged: for A, the HOMO has a_{1}′ orbital symmetry and the LUMO a_{2}′, while for B it is the opposite (HOMO a_{2}′ and LUMO a_{1}′). These MOs are mainly concentrated along the equator of the molecule. The a_{1}′ orbital is bonding on the C–C bonds at the pentagon–pentagon fusions, and thus will be occupied in the Fries structure. In contrast the a_{2}′ is antibonding and thus will favor single bonds on these sites, yielding a chain of isolated aromatic sextets which is typical for a Clar structure. Since the two structures are close both in energy and in geometry, we have investigated whether electron correlation effects could stabilize an intermediate resonant structure.
First it should be noted that both D_{5h} isomers A and B have, as shown by a CASSCF(18,12) 6-31G calculation, an undoubtedly monodeterminantal wavefunction. The obtained energy difference between them (3.06 kcal mol^{−1}, Table S9, ESI†) is in agreement with the DFT results, which yield a 3.35 kcal mol^{−1} gap (Table 1). In order to explore the possibility of the above mentioned resonant isomer lying lower in energy than A and B, further calculations are performed on intermediate structures. The ground state singlet and a state average calculation including the GS and first excited state singlet are carried out using CASSCF(6,6) with a 6-31G basis set. The intermediate coordinates are obtained through linear combinations (interpolation) of A and B leading to the interconversion of A and B. While no energy well is observed, there seems to be an avoided crossing at 0.6 B + 0.4 A (Fig. S1, ESI†), a coordinate in which both A and B electronic structures coexist in a resonant structure. The barrier between B and A amounts to 33 kcal mol^{−1}.
The Hückel calculation for the D_{5h} graph, with all nearest neighbour interactions equal, invariably yields the ground state of the B isomer. However by changing the orbital occupation numbers of the frontier orbitals, one can study the bonding scheme in the A isomer as well. The Hückel π bond order of C–C bonds in equatorial pentagon–pentagon fusions clearly reproduces the DFT results: in the A isomer this bond order is 0.528 in agreement with the proposed Fries structure, while for B it is 0.385, showing a preponderant Clar structure.
MO | g_{z4} | g_{z3x} | g_{z3y} | g_{z2(x2−y2)} | g_{z2xy} | g_{zx3} | g_{zy3} | g_{x4+y4} | g_{xy(x2−y2)} | ∑ (%) |
---|---|---|---|---|---|---|---|---|---|---|
17 | 0.00 | 0.00 | 0.00 | 0.00 | 0.08 | 89.75 | 0.00 | 0.00 | 0.05 | 89.88 |
18 | 0.00 | 0.08 | 0.90 | 66.80 | 3.08 | 0.00 | 0.11 | 25.79 | 1.22 | 97.97 |
19 | 0.00 | 2.48 | 0.07 | 3.08 | 65.66 | 0.01 | 0.02 | 1.21 | 25.84 | 98.36 |
20 | 96.85 | 0.01 | 0.34 | 0.00 | 0.00 | 0.00 | 0.06 | 0.00 | 0.00 | 97.25 |
21 | 0.00 | 0.02 | 0.01 | 4.47 | 18.48 | 0.06 | 0.00 | 12.44 | 51.16 | 86.63 |
22 | 0.00 | 0.00 | 0.04 | 18.47 | 4.46 | 0.03 | 0.00 | 51.20 | 12.44 | 86.62 |
23 | 0.07 | 18.65 | 47.60 | 0.14 | 0.31 | 0.00 | 14.51 | 0.26 | 0.13 | 81.66 |
24 | 0.06 | 35.71 | 30.63 | 0.20 | 0.49 | 0.00 | 13.99 | 0.32 | 0.26 | 81.64 |
25 | 0.03 | 23.35 | 0.43 | 0.09 | 0.42 | 0.00 | 64.12 | 0.06 | 0.21 | 88.70 |
∑ (%) | 97.01 | 80.31 | 80.00 | 93.24 | 92.97 | 89.87 | 92.81 | 91.28 | 91.30 |
l | HMO | D _{3} | D _{5h} (A) | D _{5h} (B) |
---|---|---|---|---|
0 (s) | 1 | 100 | 100 | 100 |
1 (p) | 2–4 | ∼99.7 | ∼99.8 | ∼99.8 |
2 (d) | 5–9 | >95 | >97 | >97 |
3 (f) | 10–16 | >91 | >92 | >92 |
4 (g) | 17–24 | >81 | >81 | >81 |
4 (g) | HOMO | 88 (a_{1}) | 92.2 (a_{1}′) | 0 (a_{2}′) |
For the D_{3} cage the valence orbital shell shows a close match with the particle on a sphere model: the 25 occupied orbitals correspond to a closed spherical shell up to l = 4. This is confirmed by the energy diagram in Fig. 3. Since the symmetry and shape of the Hückel and DFT MOs are similar, the spherical shell structure is also retrieved in the DFT orbital ordering.
For D_{5h}, on the other hand, the results show a large difference between the A and B isomers. The Hückel result for the D_{5h} cage yields a HOMO with a_{2}′ symmetry, which corresponds to the HOMO of the B-isomer. For this orbital the sign of the eigenvector in the equator alternates ten times when going around, indicating that it has an l = 5 parentage. Indeed the overlap of this HOMO with the l = 4 g-harmonics is exactly zero, as the g-orbitals do not subduce an a_{2}′ symmetry:
Γ_{l=4}(D_{5h}) = a_{1}′ + e_{1}′ + e_{2}′ + e_{1}′′ + e_{2}′′ | (4) |
As a result the B isomer is found to be characterized by a pseudo-spherical shell, in which one of the g-orbitals is replaced by a h-orbital. The ground state of the B isomer thus is not a spherical aromat, and consequently, the filling rule for spherical shells does not apply. In contrast, in the A isomer the HOMO has a_{1}′ symmetry, and can be identified as the LUMO of the Hückel calculation. This orbital has a close resemblance to the z^{4} harmonic function of the g-shell. By occupying this orbital instead of the a_{2}′ HOMO, the occupation of the g-shell is completed, and the sphericity of the electronic structure is nicely restored. The results of overlap calculations thus indicate that for the D_{3} and D_{5h} (A) C_{50} isomers the projection of the occupied π-orbitals on the spherical harmonics yields complete spherical shells, while for the D_{5h} (B) isomer it does not. However, as we pointed out in the introduction while the mapping on closed spherical shells is a criterion for spherical aromaticity, it is not sufficient. The degree of distortion of sphericity is equally important. Ring current density plots provide a way to investigate this further.
The case for the D_{5h} geometry is less clear cut and the two isomers behave differently. For the D_{5h} (A) isomer currents are still diatropic on the poles and tropics, but near the equator paratropic counter currents are observed. The B isomer shows analogous characteristics but the paratropic aspects are more pronounced in line with the incomplete filling of the spherical g-shell. In both cases, the centered paratropic current that appears at 4.42 a.u. (Fig. 4) corresponds to the bottom pentagon of the cage. Since the projection plane is parallel to it, the plot obtained is comparable to those of annulenes. Thus it is not a global current but a local one, not representative of the character of the molecule as a whole.
A more detailed understanding can be obtained by analyzing the separate excitations to virtual orbitals that contribute to the magnetic response. Both D_{5h} isomers are characterized by a small HOMO–LUMO gap. The corresponding excitations have the symmetry of the z-component of the magnetic dipole operator:
a_{1}′ × a_{2}′ = A_{2}′ = Γ(R_{z}) | (5) |
The matrix element in this operator appears to be strong since both levels share an l = 5 parentage.
As we have already indicated the a_{2}′ orbital has no g-character, but its Hückel projection overlaps to almost 99% with the a_{2}′ component of the l = 5 shell. The relevant h-orbital is expressed as:
sin^{5}θsin5ϕ ∼ 5x^{4}y − 10x^{2}y^{3} + y^{5} | (6) |
The remaining a_{1}′ orbital shows a strong overlap with the g-shell, 92.2% according to Table 3, nevertheless it also overlaps to a significant extent of 26% with the a_{1}′ component of the l = 5 shell. The corresponding function is given by:
sin^{5}θcos5ϕ ∼ 5xy^{4} − 10x^{3}y^{2} + x^{5} | (7) |
The large simultaneous overlaps with equisymmetric l = 4 and l = 5 parent harmonics indicate that we have arrived at a point in the spectrum where orthogonality of the projected spherical harmonics breaks down as a result of the finite size of the atomic basis set. The affiliation of the HOMO and the LUMO to two complementary components of the h-shell explains the large paratropic contribution of this excitation, since the mixing of both by the magnetic dipole operator will create a large orbital moment along the z-axis.
As for the D_{5h} A isomer, it is clear that the weak surrounding diatropic current is generated by small contributions of many different excitations from the π molecular orbitals below the HOMO (from the MOs 142–149 to the LUMO and above). However, besides the main HOMO–LUMO excitation, other transitions seem to be also highly contributing to the global paratropic picture, such as 146 → 151, 147 → 151 (mainly) and 150 → 156.
Compatibility between magic counts and symmetry numbers thus would require a different strategy, by adjusting electron counts through doping with hetero-atoms. A successful example is the endohedral U@C_{28} metallo-fullerene, where the carbon cage has tetrahedral symmetry and attains the magic electron count of 32 by doping with four electrons from uranium.^{37}
The link between fullerene stability and magic electron counts is not well established either. The most stable fullerenes C_{60} and C_{70} are not aromats. Clearly the stability of the neutral fullerenes is dictated more by the frequency of occurrence of stable motifs, i.e. the fact that pentagons are isolated as in C_{60} and C_{70} or the absence of pentagon triplets as in C_{50}, rather than by electron count. Nevertheless the higher aromatic character of some isomer can affect the expected stability by taking only into account the distribution of motifs. For the three analysed structures of C_{50} the D_{3} isomer complies best with a spherical aromat. It has a higher degree of sphericity and a neat closed shell spherical parentage. According to DFT calculations it is slightly more stable than the D_{5h} isomers, even though the PAPR rule would predict it to be around 20 kcal mol^{−1} less stable. Therefore a most favourable electronic structure (i.e. enhanced aromaticity) can stabilize some isomers of neutral fullerenes and favour structures with a worst distribution of structural motifs.
These effects should be more evident for negatively charged fullerenes, and in these systems electronic effects can determine the stability instead of strain. For instance in the case of endohedral metallo fullerenes, where stability is mainly dictated by the charge transfer to the carbon cage and the most stable isomers correspond to the most stable negatively charged carbon cages, the structure observed experimentally does not fulfil in many cases the isolated pentagon rule.^{38} The stability of these isomers has been directly related to the electronic structure of the π shell^{39} and with maximum aromaticity.^{40}
Of the two D_{5h} isomers, the A isomer fulfils the criteria for spherical aromaticity, while the B isomer is an anti-aromat as far as spherical parentage is concerned. While the relative stabilities do not typically correlate with aromaticity, the magnetic properties are very sensitive to the spherical shell nature of the cages. The aromaticity of the D_{3} isomer is confirmed by a substantial diamagnetic NICS value, and a global diatropic ring current. The NICS values of the D_{5h} isomers also reflect their spherical shell character (Table 1). The main difference between A and B isomers lies in the currents surrounding the equator, since the cap current density plots look quite similar (diatropic except for the apical pentagons, which are paratropic). The current density maps however indicate paratropic currents even for the A isomer, which is more aromatic. The reason is the strong hybridization between the spherical shells in the frontier orbital region, made possible by the prismatic symmetry of these isomers.
Footnote |
† Electronic supplementary information (ESI) available: CASSF(18,12) calculation results for C_{50}D_{5h}; detailed results of spherical parentages for C_{50}D_{3} and D_{5h}. See DOI: 10.1039/c5cp04970a |
This journal is © the Owner Societies 2016 |