Isomers of Small Fullerenes

	ring
	bowl
	cage

Since its proposed existence in 1987 by Kroto, the C₂₀ caged-fullerene has received considerable attention by the scientific community. The interest stems from the fact that the C₂₀ cage is the smallest and least stable of the fullerenic structures and may be a possible precursor to C₆₀, and the other larger fullerenes, or serve has fragments in the structure of narrow nanotubes. Although first observed in the gas phase in 2000, the primary tool with which C₂₀ is being studied is through the application of semi-empirical and ab initio calculations. The focus of these theoretical calculations is the characterization of the geometric and electronic structures of the fullerene (C₂₀, C₂₄, C₂₈, etc.) and to evaluate the energetic stability of the fullerene (C_n) relative to other possible isomeric structures within the n-carbon atom system.

Initial studies have focused on the C₂₀ fullerene; however, we have initiated the investigation into the structure of the PE surface for the C₂₄ fullerene and have obtained some preliminary results pertaining to the C₂₈ system. Our investigations into the smaller fullerenes can be separated into three inter-related projects.

1. The structure of the potential energy (PE) surface that exists between isomers is described by performing saddle-point calculations at the semi-empirical level. These calculations enable us to compare the relative energy of two isomers as well as any PE barriers that may separate the isomers from one another of the PE surface. A typical result is shown above, where the energy profile between the ring and bowtie isomers is depicted. This graph depicts a 2D cross-section of the PE surface that is proposed to exist between these two isomers. Note that there exist energy minima on either side of the saddle point (0 on the x axis), thus indicating that these two isomers do not lie adjacent to one another of the PE surface, but are separated from one another by an additional (halfbow) isomer. Through this process, we are able to not only compare the relative energies of the isomers, but to also determine which isomer lie adjacent to one another. Since it has been reported that the stability of the isomers is a function of temperature, we can suggest a step-wise evolution, from one isomer to another, which may occur as the temperature of the system changes.
   
   
2. Once we have identified a series of significantly stable isomers and performed multiple saddle point calculations between these structures, we can develop of 2D plot of the relative positions of these isomers on the PE surface. These plots are created through a process of triangulation and provide a map-like description of the positioning of the isomers relative to one another.
   
   
3. The most stable (granted, a qualitative conclusion) isomers identified with semi-empirical techniques are subsequently subjected to higher-order theoretical calculations, such as Hartree-Fock (HF), density functional theory (DFT), Møller-Plesset (MP), etc. Equilibrium geometries are performed at a moderately large basis set (6-31G*, for example) in addition to vibrational analysis in order to insure that the geometry corresponds to a true stationary state. Single-point calculations at a larger basis set (6-311+G**, for example) are then employed to further fine-tune the energetics of the isomer. These calculations provide a more accurate picture of the relative energies of the isomers under investigation (although isomeric stability appears to have some dependence upon the theoretical model employed). These results provide basic information in terms of identifying isomers (energetically stable) that may possibly serve as precursors or intermediate structures in the synthesis of the larger fullerenes or narrow nanotubes.

1. Optimize "known" isomers of C₂₄, C₂₈, etc.
2. Perform saddle-point calculations between these isomers
3. Identify unknown isomers
4. Determine energy relationships between all isomers
5. Determine structure of the PE surface
6. Determine relative position of isomers on the PE surface

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