Topological Versus Physical and Chemical Properties of Negatively Curved Carbon Surfaces

Some relevant physical and chemical properties of negatively curved carbon surfaces like sp 2-bonded schwarzites can be predicted or accounted for on the basis of purely topological arguments. The general features of the vibrational spectrum of complex sp 2-carbon structures depend primarily on the topology of the bond network and can be estimated, in a first approximation and for systems with only nearest-neighbor interactions, from the diagonalization of the adjacency matrix. Examples are discussed for three- and two-periodic carbon schwarzites, where a direct comparison with ab initio calculations is possible. The spectral modifications produced by the insertion of defects can also analyzed on pure topological grounds. Two-periodic (planar) schwarzites can be viewed as regular arrays of Y-shaped nanojunctions, which are basic ingredients of carbon-based nano-circuits. A special class of planar schwarzites is obtained from a modification of a graphene bilayer where the two sheets are linked by a periodic array of hyperboloid necks with a negative Gaussian curvature. Ab initio density functional calculations for some structures among the simplest planar schwarzites – (C18)2, (C26)2, and (C38)2 – are presented and discussed in light of the structural stability predictions derived from a topological graph-theory analysis based on the Wiener index. A quantum-mechanical justification is provided for the effectiveness of the Wiener index in ranking the structural stability of different sp 2-conjugated structures.