  Interest in layered two dimensional materials, particularly stacked
heterostructures of transition metal dichalcogenides, has led to the need for a
better understanding of the structural and electronic changes induced by
stacking. Here, we investigate the effects of idealized heterostructuring, with
periodic commensurate stacking, on the structural, electronic, and vibrational
properties, when compared to the counterpart bulk transition metal
dichalcogenide. We find that in heterostructures with dissimilar chalcogen
species there is a strong compression of the inter-layer spacing, compared to
the bulk compounds. This compression of the heterostructure is caused by an
increase in the strength of the induced polarization interaction between the
layers, but not a full charge transfer. We argue that this effect is real, not
due to the imposed commensurability, and should be observable in
heterostructures combining different chalcogens. Interestingly, we find that
incommensurate stacking of Ti-based dichalcogenides leads to the stabilization
of the charge density wave phonon mode, which is unstable in the 1T phase at
low temperature. Mixed Ti- and Zr-heterostructures are still unstable, but with
a charge density wave or a ferroelectric instability.
