Ab initio triangle maps for new insights into the crystal wave functions of carbon allotropes

Recent years have brought forth an ever-increasing number of predicted carbon allotropes. In the spirit of Heimann's original categorization scheme for carbon-only materials, we here report an ab initio method to create triangular (ternary) maps based on their valence-orbital mixing. These maps group together allotropes of similar electronic structure––and, hence, physical properties––and can thus aid in finding allotropes with specific features. Moreover, these maps can be used to classify all carbon allotropes according to their bonding nature. We suggest to extract insights about the composition of the crystal wave function as emerging from individual atomic orbitals. To do so, we develop a way to visualize the entire linear-coefficient space of an extended LCAO wave function, also based on ternary diagrams. This scheme yields that lower-level mixed states always get realized and filled first before higher-level mixed states can be created, a consequence of symmetry breaking of canonical orbitals within the solid state. For the specific case of graphite, there is more sp mixing than in sp2 mixing present, while diamond exhibits more sp and sp2 mixing than sp3 mixing. Once higher-level mixing is realized, however, these levels are more significant for the allotropes' properties than the lower-level ones. [Display omitted]