  By means of first-principles density functional calculations, we study the
maximally localized Wannier functions for the 2D transition metal
dichalcogenides MX2 (M=Mo,W X=S,Se,Te). We found a M^+4-like ionic charge and a
single occupied d-band. The center of the d-like maximally localized Wannier
function associated with this band is distributed among three M sites. Part of
the energy gap is opened by the crystal field splitting induced by the
X^-2-like atoms. We extract the hopping parameters for the Wannier functions
and provide a perspective on tight-binding model. From the analysis of the
tight binding model, we have found an inversion of the band character between
the Gamma and the K points of the Brillouin zone due to the M-M hybridization.
The consequence of this inversion is the closure of the gap. The M-X
hybridization is the only one that tends to open the gap at every k-point, the
change in the M-X and M-M hybridization is the main responsible for the
difference in the gap between the different dichalcogenide materials. The
inversion of the bands gives rise to different spin-orbit splitting at Gamma
and K point in the valence band. The different character of the gap at Gamma
and K point offers the chance to manipulate the semiconductive properties of
these compounds. For a bilayer system, the hybridizations between the out of
plane orbitals and the hybridizations between the in plane orbitals split the
valence band respectively at the Gamma and K point. The splitting in the
valence band is opened also without spin-orbit coupling and comes from the M-M
and X-X hybridization between the two monolayers.
