  It has been suggested that the metal-insulator transitions in a number of
spinel materials with partially-filled t_2g d-orbitals can be explained as
orbitally-driven Peierls instabilities. Motivated by these suggestions, we
examine theoretically the possibility of formation of such orbitally-driven
states within a simplified theoretical model, a two-dimensional checkerboard
lattice with two directional metal orbitals per atomic site. We include orbital
ordering and inter-atom electron-phonon interactions self-consistently within a
semi-classical approximation, and onsite intra- and inter-orbital
electron-electron interactions at the Hartree-Fock level. We find a stable,
orbitally-induced Peierls bond-dimerized state for carrier concentration of one
electron per atom. The Peierls bond distortion pattern continues to be period 2
bond-dimerization even when the charge density in the orbitals forming the
one-dimensional band is significantly smaller than 1. In contrast, for carrier
density of half an electron per atom the Peierls instability is absent within
one-electron theory as well as mean-field theory of electron-electron
interactions, even for nearly complete orbital ordering. We discuss the
implications of our results in relation to complex charge, bond, and
orbital-ordering found in spinels.
