The Geometry of $C_{60}$: A Rigorous Approach via Molecular Mechanics

Abstract

Abstract. Molecular Mechanics describes molecules as particle configurations interacting via classical potentials. These configurational energies usually consist of the sum of different phenomenological terms which are tailored to the description of specific bonding geometries. This approach is followed here to model the fullerene C 60 , an allotrope of carbon corresponding to a specific hollow spherical structure of sixty atoms. We rigorously address different modeling options and advance a set of minimal requirements on the configurational energy able to deliver an accurate prediction of the fine three-dimensional geometry of C 60 as well as of its remarkable stability. In particular, the experimentally observed truncated-icosahedron structure with two different bond lengths is shown to be a strict local minimizer.

Atomistic potentials and the Cauchy-Born rule for carbon nanotubes: a review

“…For each µ ∈ M and γ 1 = γ 2 = γ , letting λ µ 1 = µ/2 + λ µ cos α µ 1 and λ µ 2 = λ µ with λ µ and α µ 1 from Property 1. above, the configuration F λ µ 1 ,λ µ 2 ,µ is the unique minimizer of the problem (12). This can be seen by using (19) which yields along with the definition of E red…”

“…The sp 2 -type covalent bonding implies that each atom has exactly three first neighbors and that bond angles of 2π/3 are energetically preferred [9]. Similar potentials have been previously employed to describe also other carbon allotropes, e.g., graphene in [12,20,30] and fullerene C 60 in [19].…”

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

“…By Lemma 6.1 and the fact that v 3 is smooth with minimum at 2π/3 we note that one has |v 3 (β(α us , γ ))| ≤ C −2 for a constant only depending on v 3 . Consequently, multiplying the estimates in (45)-(46) by and letting → ∞ we get using (31) and (32)…”

“…The sp 2 -type covalent bonding implies that each atom has exactly three first neighbors and that bond angles of 2π/3 are energetically preferred [12]. The reader is referred to [16,20,32,57,68] for a collection of results on local and global minimizers in this setting and to [27,51] for additional results on carbon structures.…”

Carbon-Nanotube Geometries as Optimal Configurations

“…As a consequence, the stability criterion here addressed turns out to be the most restrictive. The reader is referred to [18,39] for related stability analysis for other carbon nanostructures, including carbyne, graphene, the fullerene C 60 , and diamond. Moving from the present analysis, a refined analytic discussion of the stability of carbon nanotobes under moderate stretching has been developed in [17].…”