# d-orbital splittings in WS2 monolayer

The formerly degenerate $d$ orbitals of the tungsten atoms in the $\ce{WS2}$ monolayer are split into three groups: (1) $\mathrm{d}_{z^2}$, (2) $\mathrm{d}_{x^2-y^2}, \mathrm{d}_{xy}$ and (3) $\mathrm{d}_{xz}, \mathrm{d}_{yz}$ in the order of increasing energy with the large band gap between the first and the second groups.

Now the valence band maximum consists dominantly of $\mathrm{d}_{x^2-y^2}, \mathrm{d}_{xy}$ orbitals, whereas the conduction band minimum is composed mainly of $\mathrm{d}_{z^2}$ metal orbital.

So, the order of $\mathrm{d}$ orbitals in the first two splitting groups now is reversed, since the valence band is lower in energy than the conduction band. Where is the correspondence between these two facts?

A prismatic arrangement of atoms (first figure) does not capture fully all symmetries of a hexagonal arrangement in a crystal lattice (which is the case of the second figure). However, the most overlap (in terms of the number of symmetries) between the two occurs at the $\Gamma$ point, since it is the point of the highest symmetry. Thus, the first figure resembles mostly the orbitals at the $\Gamma$ point in the first Brillouin zone of the crystal lattice. Please, if I am thinking in a right direction, refine my answer to make it more strict and rigid.