The vibrational spectrum of a two-dimensional (2D) sheet of graphite is examined using a tight-binding total-energy formalism. Motivation for this work is provided by the poor transferability of classical valence-force models for ${\mathit{sp}}^2$ carbon. A major problem with such models is the neglect of π-electron polarizability. The full tight-binding formalism considered here includes both this effect and covalent σ bonding on the same footing. Atomic force constants of arbitrary range are calculated quantum mechanically using a Green’s-function approach. Long-range interactions, resulting from delocalized π bonding, are shown to be important for in-plane vibrations. The restoring forces for out-of-plane vibrations are dominated by σ-π mixing. The resulting phonon spectrum for 2D graphite is accurate only to within 30%. This is considerably worse than previous tight-binding results for ${\mathit{sp}}^3$ solids. Some possible reasons for this are discussed. The difficulties encountered here may well impede our ability to understand the vibrational properties of more complicated π-bonded solids, particularly amorphous carbons and fullerenes.

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