  We calculate from first principles the electronic structure and optical
properties of a number of transition metal dichalcogenide (TMD) bilayer
heterostructures consisting of MoS2 layers sandwiched with WS2, MoSe2, MoTe2,
BN, or graphene sheets. Contrary to previous works, the systems are constructed
in such a way that the unstrained lattice constants of the constituent
incommensurate monolayers are retained. We find strong interaction between the
\Gamma-point states in all TMD/TMD heterostructures, which can lead to an
indirect gap. On the other hand, states near the K-point remain as in the
monolayers. When TMDs are paired with BN or graphene layers, the interaction
around \Gamma-point is negligible, and the electronic structure resembles that
of two independent monolayers. Calculations of optical properties of the
MoS2/WS2 system show that even when the valence and conduction band edges are
located in different layers, the mixing of optical transitions is minimal, and
the optical characteristics of the monolayers are largely retained in these
heterostructures. The intensity of interlayer transitions is found to be
negligibly small, a discouraging result for engineering the optical gap of TMDs
by heterostructuring.
