Fullerene Stability by Geometrical Thermodynamics

Schlegel projections of selected fullerenes (the non-chiral C60, C384; and the weakly-chiral C28, C76 and C380) are used to show that these fullerenes can all be represented by pairs of counter-propagating spirals featuring anti-parallel (C2) symmetry, even though C380 and C384 are nonface-spiral fullerenes. In the case of C60, the high symmetry is used to construct an analytical approximation for these spirals, demonstrating that they form a holomorphic function satisfying the Euler-Lagrange equations, and thus confirming that the entropic equivalent of the Principle of Least Action (that is, the Principle of Least Exertion) is obeyed. Hence the C60 structure has Maximum Entropy (MaxEnt), is therefore maximum likelihood, and consequently its stability is established on entropic grounds. The present MaxEnt stability criterion is general, depending only on the geometry and not the physics of the system. A Shannon entropy-based fragmentation metric is used to quantify both the intrinsic sense and the degree of chirality for C76 and C380. We have shown that the stability of C60 is a general property of the thermodynamics of the system. This is a significant methodological advance since it shows that a detailed treatment of the energetics is not always necessary: this may prove fruitful, not only for fullerenes but also for general problems of molecular stability and in other applications of conformational chemistry.