The interest in fullerenes lies in their unique energy levels and high electron affinity which makes these molecules promising candidates for biological applications such as carriers for drug and gene therapy or pharmaceutical applications such as being used as carriers for drug delivery and increasing the acceptance of artificial implants, writes Carl Pichon.

Introduction

A fullerene is an allotrope of carbon that is a closed carbon-cage containing only hexagonal and pentagonal rings. The most common, and one of the most vital, fullerenes is the much celebrated archetype, icosahedral C₆₀, also known as Buckminster fullerene after the architect who popularised geodesic domes Buckminster Fuller.

The discovery of fullerenes arose when the mass spectrum observed from the sooty residue produced by vaporising carbon within a helium atmosphere consisted of two primary groups of molecules who’s mass corresponded to the exact weight of 60 or 70 carbon atoms.

This experiment was carried out by Robert F Curl, Sir Harold W Kroto and Richard E Smalley and led to the three receiving the Nobel prize in 1996. Since their discovery in 1985, fullerenes have been the subject of numerous studies including in-depth investigations into their complex structures, formation and use which have been conducted in an attempt to grasp the full potential of this family of molecules.

The interest in fullerenes lies in their unique energy levels and high electron affinity which makes these molecules promising candidates for biological applications such as carriers for drug and gene therapy or pharmaceutical applications such as being used as carriers for drug delivery and increasing the acceptance of artificial implants.

The epidemic of new research into fullerenes infected chemists, physicists and even material researchers around the globe. Based on this fact, it would be nearly impossible to summarise the work of each individual who has crossed the path of fullerenes.

The purpose of this work was to completed a literature review of this class of materials combined with a computational study of the first 1,200 structures.

The M06-2X density functional was used in combination with a basis set known as 6-31G(d), which are computational quantum mechanical modelling methods used to describe the electronic structure of many-body systems and to describe the core and valence orbitals, respectively.

With the use of services provided by the Irish Centre for High-End Computing 800 of the original 1,200 structures were optimised due to time restrictions created by COVID-19.

This fresh combination of computational methods will create a precious data set which can be used in future work to gain an understanding of fullerenes compared to past computational methods and experimental data on physical fullerenes. 

Results 

A sample of the results produced can be seen in the table below along with their accompanying structures as seen in the book An Atlas of Fullerenes by PW Fowler and DE Manolopoulos (Fowler & Manolopoulos, 2006).

In the above table, ‘order’ refers to the number of carbon atoms present within the molecule and repeated orders represent isomers of the IPR fullerenes.

In order to investigate the stability of these molecules, I was in contact with Professor Patrick W Fowler, a professor of theoretical chemistry at the University of Sheffield to gain insight into their complex structures and stability factors. The above categories were chosen to represent the data for the following reasons:

The total energy, or the zero-point energy, represents the total energy of each of the carbon atoms in a fullerene at 0 kelvin. It is the sum of the electronic energy and the zero-point vibrational energy of each of the atoms in a system.

These values are in atomic units, which correspond to 1 Hartree (2625.5 kj/mol) and hence were consequently changed to kJ/mol and electron volts for simplicity of viewing.

When looking at these values, dividing it by the number of carbon atoms in the molecule itself will give you the amount of energy per carbon atom.

The total energy for a fullerene system provides information about the thermodynamic stability of the fullerene cage, ie, a greater gain by forming a particular isomer from the isolated nuclei and electrons.

In correspondence with my communications with Prof Fowler, it was brought to my attention that it is very difficult to compare structures with different number of atoms as they would involve the formation of different bonds, and hence, the total energy will only be used to compare isomers of fullerenes of the same order.

Hence for comparative reasons, the thermodynamic stability was only used to compare the thermodynamic stability of isomers of fullerenes of the same order. For example, you can see that the C₇₆ molecule with D₂ symmetry is more thermodynamically stable than the C₇₆ fullerene with C₁ symmetry.

The column titled HOMO refers to the Highest Occupied Molecular Orbital. This is regarded as the discriminating criterion when completing the analysis of fullerenes for which is the most stable.

More stable fullerenes have a lower HOMO. This means that the highest energy of an electron in said fullerene is more stable within a molecule. This also contributes to the reactivity of a molecule due to the fact that the lower the HOMO, the more stable the molecule.

More specifically, this values contribution to the HOMO-LUMO gap gives indication to its kinetic stability. The LUMO refers to the Lowest Unoccupied Molecular Orbital.

This is the next available orbital after the HOMO. It is higher in energy than the HOMO and is the level to which electrons would climb if the molecule were to be excited by an addition of energy.

The HOMO-LUMO gap refers to the energy difference between the two respective molecular orbitals. The HOMO is considered nucleophilic which essentially means that it is electron donating, whereas the LUMO is considered electrophilic, or electron accepting.

As aforementioned, this gap represents the most likely jump made by an electron when it becomes excited. Frontier molecular orbital theory postulates that the HOMO-LUMO gap plays a dominant role in the kinetic stability of molecules and associated chemical reactions.

Nobel prize winner Kenichi Fukui published a paper titled 'A molecular orbital of reactivity in aromatic hydrocarbons' (Fukui et al., 1952) in which he looked at the effect of frontier molecular orbitals (HOMO & LUMO) on reaction mechanisms.

In a paper published by (Aihara, 2000), it is stated that a lower HOMO-LUMO gap leads to a lower kinetic stability due to the fact that it is energetically favourable for electrons to join a high-lying, lowest-unoccupied molecular orbital.

That being said, it is conversely favourable for an electron to be extracted from a low-lying, highest-occupied molecular orbital. This leads to an understanding that the most reactive site of any IPR fullerene, corresponds to the lowest HOMO-LUMO gap in the molecule.

In terms of the information that we gathered as seen in the table above, it is clear that the C₆₀ molecule is least reactive, or has the highest kinetic stability of the molecules in the table due to the fact that it has the highest HOMO-LUMO gap.

So in summary, the smaller the HOMO-LUMO gap, the less the kinetic stability and higher the reactivity. The limiting case for this analysis is if the gap is zero, the molecule will be a radical and highly reactive.

The HOMO-LUMO gap for each of the structures optimised were represented graphically and each of the peaks below represents the most stable fullerenes.

Along with this analysis of the energy of each of the respective fullerene structures, visualisation software (Avogadro) was used to observe the symmetry of the various structures such as those seen below.

From the above structures, I was able to observe the orbital hybridisation and the corresponding symmetry for each of the fullerenes which were optimised.

It was important to view how the electrons were shared about the surface of the fullerene cages to again gain a deeper understanding of the relative stability of each of the structures. As you can see the symmetry in C₆₀ & C₇₀ is much better than that of C₈₀ suggesting increased stability. 

Conclusion 

Using the optimisations completed, we were able to generate a good understanding of the family of molecules known as fullerenes, including information such as the kinetic stability, the thermodynamic stability, the symmetry and many others not mentioned in the review completed here.

Future work needs to be completed on the basis of the stability of fullerenes, in which the thermodynamic stability could be better defined in a manner that generalises the information between different fullerene structures. 

References 

1.) Aihara, J I (2000). Correlation found between the HOMO-LUMO energy separation and the chemical reactivity at the most reactive site for isolated-pentagon isomers of fullerenes. Physical Chemistry Chemical Physics, 2(14), 3121–3125. https://doi.org/10.1039/b002601h

2.) Fowler, P, and Manolopoulos, D (2006). An atlas of fullerenes. New York: Dover Publications. Retrieved from http://scholar.google.es/scholar?start=10&q=fullerenes&hl=es&as_sdt=0,5#2

3.) Fukui, K, Yonezawa, T, and Shingu, H (1952). A molecular orbital theory of reactivity in aromatic hydrocarbons. The Journal of Chemical Physics, 20(4), 722–725. https://doi.org/10.1063/1.1700523 

Author: Carl Pichon, University of Limerick, Chemical and Biochemical Engineering. Superviser: Matthias Vandichel