  We have carried out a density functional study of unoccupied, resonance
states in a single Au atom, dimers, a trimer and infinite Au chains on the
NiAl(110) surface. Two inequivalent orientations of the ad-chains with
substantially different interatomic distances were considered. From the study
of the evolution of the electron states in an Au chain from being isolated to
adsorbed, we find that the resonance states derive from the 6$s$ states of the
Au atoms, which hybridize strongly with the substrate states and develop a
$p$-like polarization. The calculated resonance states and LDOS images were
analyzed in a simple tight-binding, resonance model. This model clarifies (1)
the physics of direct and substrate-mediated adatom-adatom interactions and (2)
the physics behind the enhancements of the LDOS at the ends of the adatom
chains, and (3) the physical meaning of the "particle-in-box" model used in the
analysis of observed resonance states. The calculated effective mass and band
bottom energy are in good agreement with experimental data obtained from
scanning tunnelling spectroscopy.
