  Using density functional theory, the electronic structures of single walled
molybdenum disulfide nanotubes (MoS$_2$ NTs) were investigated as a function of
diameter. Our calculations show that the electronic structure near the band gap
is sensitive to the NT diameter: armchair MoS$_2$ NTs act as indirect gap
semiconductors for diameters up to approximately 5.0 nm, while armchair MoS$_2$
NTs with larger diameters act as direct gap semiconductors with band edges
located in the vicinity of $k = 2\pi/3$. This finding implies that MoS$_2$ NTs
with large diameters should exhibit similar photoluminescence to 2D monolayer
MoS$_2$ sheets. This indirect-to-direct band gap crossover is ascribed to the
upward shift of the valence band peak at the $\Gamma$ point in small diameter
NTs, which is caused by the tensile strain resulting from their tubular
structures.
