Geometries, stabilities, and reactions of carbon clusters: Towards a microscopic theory of fullerene formation

To clarify the microscopic formation process of C₆₀ and other fullerenes, we study the geometries and energetics of small carbon clusters and the reaction between carbon clusters using the long-range transferable tight-binding model parametrized by Omata et al. (Omata TB), the local-density-approximation (LDA) in the framework of the density-functional theory, and the constant-temperature molecular dynamics combined with Omata-TB (Omata TBMD). From the LDA geometry-optimization study, we find that the binding energy per atom of the C₁₀ ring is 0.40.3em0exeV/atom larger than that of the C₁₀ chain. This energetic preference of a ring to a chain in C₁₀ is most prominent among all Cₙ clusters studied (5n17). Moreover, the study of $sp$-hybridized small carbon clusters with Omata TBMD reveals that C₁₀ is the smallest stable ring at the temperature of 20000.3em0exK, which can explain the high abundance of C₁₀ in the experimental Cₙ^- mass spectra. From the remarkable stability of the C₁₀ ring as well as from its high abundance, it is considered that the $sp$-hybridized C₁₀ ring should play a role of major constituent units of larger clusters and fullerenes. Therefore we perform various sets of simulations of reactions between carbon clusters possessing the C atoms in multiples of 10, C₁₀ₘ+C₁₀ₙ ($m+n=2,3,4,5,6$, mn1), at several temperatures with Omata TBMD. As a result, it is found that, in most cases studied, C₂₀ and C₃₀ clusters possess the sp²-hybridized planar geometries, while C₄₀, C₅₀, and C₆₀ take the sp²-hybridized ``fullerenelike'' closed-cage geometries. These C₄₀, C₅₀, and C₆₀ cages can be formed at as low as 15000.3em0exK, which is in good accord with the experimental temperature of fullerene formation. In a few cases, even the C₃₀ cluster is found to take the cage structure. These results are in good accord with the experimental results of the gas ion chromatography. The straightforward growth process from the $sp$-hybridized ring to the sp²-hybridized plane and that from the sp²-hybridized plane to the sp²-hybridized fullerenelike cage revealed in the present study are considered to be the main road of the formation of fullerenes. Finally, the study of structural stabilities of cage geometries obtained through the reaction between a fullerenelike cage (C₄₀) or a symmetrical fullerene (D₅ₕ C₅₀ or Iₕ C₆₀) and a carbon cluster (C₁₀ or C₁₂) at various temperatures with Omata TBMD indicates that C₂ₙ fullerenelike cages larger than C₆₀ tend to decay into C_2n-2 through the C₂ loss process at the higher rate than C₆₀ or smaller fullerenelike cages. Therefore this C₂ loss process is considered to be one of the most important processes leading to the extreme abundance of C₆₀.