  Graphene sheets with regular perforations, dubbed as antidot lattices, have
theoretically been predicted to have a number of interesting properties. Their
recent experimental realization with lattice constants below 100 nanometers
stresses the urgency of a thorough understanding of their electronic
properties. In this work we perform calculations of the band structure for
various hydrogen-passivated hole geometries using both spin-polarized density
functional theory (DFT) and DFT based tight-binding (DFTB) and address the
importance of relaxation of the structures using either method or a combination
thereof. We find from DFT that all structures investigated have band gaps
ranging from 0.2 eV to 1.5 eV. Band gap sizes and general trends are well
captured by DFTB with band gaps agreeing within about 0.2 eV even for very
small structures. A combination of the two methods is found to offer a good
trade-off between computational cost and accuracy. Both methods predict
non-degenerate midgap states for certain antidot hole symmetries. The inclusion
of spin results in a spin-splitting of these states as well as magnetic moments
obeying the Lieb theorem. The local spin texture of both magnetic and
non-magnetic symmetries is addressed.
