  We present a scheme to controllably improve the accuracy of tight-binding
Hamiltonian matrices derived by projecting the solutions of plane-wave ab
initio calculations on atomic orbital basis sets. By systematically increasing
the completeness of the basis set of atomic orbitals, we are able to optimize
the quality of the band structure interpolation over wide energy ranges
including unoccupied states. This methodology is applied to the case of
interlayer and image states, which appear several eV above the Fermi level in
materials with large interstitial regions or surfaces such as graphite and
graphene. Due to their spatial localization in the empty regions inside or
outside of the system, these states have been inaccessible to traditional
tight-binding models and even to ab initio calculations with atom-centered
basis functions.
