The Geometry of Nanoscale Carbon

Abstract

Carbon is an unusual element. The isolated carbon atom has filled 1s and 2s states and two electrons in the 2p state for a configuration of (1s22s22p2). Since carbon is a first-row element, the atom is very small and the Coulomb potential felt by the valence electrons is corre- spondingly high (remember the Coulomb potential en- ergy varies as 1/r). When carbon atoms are assem- bled into a larger structure, the potentials from nearby atoms perturb the 2s and 2p atomic orbitals and cre- ate bonding, nonbonding, and antibonding molecular or- bitals formed from linear combinations (i.e. sums and differences) of the 2s and 2p states. Bonding occurs when the charge density of the electronic wavefunction occupies favorable areas, wherein the attractive atomic potentials of neighboring atoms overlap. Normally, the bonding orbitals between neighboring atoms pile up electron charge in the space that lies directly between the atoms, since this is the region where the attractive atomic potentials overlap most strongly. Such bonds are called σ states.However, for carbon, an accident of the fundamental constants (i.e. the mass of the electron, Planck’s constant, the charge of an electron) implies that two neighboring atoms can also bond strongly by piling up charge in the regions above and below the line of in- tersection between the atoms, the so-called π states (Figure 1). Because carbon can bond “sideways” in this manner, using the p states that point perpendicular to the line connecting the neighboring atoms, it can form highly anisotropic and stable two-dimensional layered structures.