Quantifying energetics of topological frustration in carbon nanostructures

We develop a graph theoretical formalism to account for the fact that sp² carbon can become spin ordered or generate free radicals for purely topological reasons. While this phenomenon has been previously considered a binary operator, we here show a quantification in discrete units of frustrations. The graph theory method is combined with open density functional theory calculations to establish the existence of an energy of frustration that is shown to greatly improve the description of carbon nanostructure energetics using classical force fields. The methodology is illustrated for a number of systems and, owing to the small computational overhead associated with its evaluation, is expected to be easily integrable into any modeling approach based on a structure's adjacency matrix.