Whenever you try calculating a dipole matrix element for states of the same shell (d) you will obtain zero, because the basic dipole selection rule says that Delta-l = plus orminus one. (It is actually related to parity.) 

When considering the optical transitions at K (K'), it is not correct to assuke that the valence band states are of dz^2 character. Rather, this is true for the valence band maximum, which however is located at the BZ center (i.e. Gamma).

Instead, the valence band states at K (K') possess even stronger in-plane-d-character (Y22,Y2-2) than the concuction band states in the CBM [which, however is not located at K (K')]. Spin orbit coupling is also important for these states, it produces the splitting of the top two conduction bands at K (and along KH, actually). We have discussed some of these attributions in the related compound WSe2 in our 1997 paper linked below. [Note that more recent, better resolved work seems to put in doubt our conclusion regarding the location of the VBM].

The matrix elements for the optical transitions (and the A,B excitons at K (K')) cannot derive from the Mo-4d character in both states. Rather, as we point out for the case of WSe2, it is important to understand that the bonding in the trilayers does possess a large degree of covalency, implying a substantial mixture of states. As far as I recall, this admixture of non-Mo-4d character should be stronger in the conduction band than in the valence band (at K).

Hope this is useful to you.

Article Occupied and unoccupied electronic band structure of WSe_ {2}

Hi Kai, Thanks for your answer.

I am sorry I haven't made it clear that I was talking about single layer MoS2, which is a direct band gap material. CBM and VBM are both at K point.

I agree with you that calculating the transition matrix element for the same shell will give zero. Like what you said for trilayer WSe2 considering the bonding between layers is important to make transition matrix nonzero. However, for single layer MoS2, there's no bonding between layers. So I am tentative to think that the states we use to calculate matrix element might be the mix of  d orbitals and p orbitals(from S element) . 

If this is the case(p orbitals are important to make nonzero transition matrix element),  I am wondering why most people don't mention this in their papers.

Dear Xinlin, let me add a few remarks,

when I referred to covalency, I actually primarily was talking about the TM-chalcogen bond within a/each (single) layer (which is an atomic trilayer). The elder seminal work by Wilson and Yoffe had a much more ionic understanding of the electronic structure.

Nevertheless, you are right in so far as some of the bands along Gamma-A acquire considerable dispersion along the z-direction in 3d-crystals and thus suggests some covalency in what is usually considered to be a van der Waals gap.

If it is true that single layer MoS2 is a direct gap material with the gap at K (I haven't checked) then the conduction band dispersion must considerably change upon going to the single (tri-) layer material, since in crystals the CBM is located approx. half way between Gamma and K. I have no idea whether that goes along with a change of the band character  (i.e. which atomic orbitals it is [mainly] derived from) across the Brillouin zone.

Finally, some care has to be taken in choosing the words when talking about optical matrix elements and angular momentum character of states. The Delta-l equal +/-1 dipole selection rule applies to angular momentum states located at the same point (the typical situation in atomic physics). It takes a more elaborate look when dealing with a "multi-centric situation" as in a crystal.

I cannot go into details here and now but would like to draw your attention to a more simple case. In cuprate materials, CuO2 layers are essential. Cu d states hybridize with neighboring O 2p states. the geometry is square like, within the plane each Cu atom has four oxygen neighbors. The in plane electronic motion is governed by the Cu d(x2-y2), O px and/or O py orbitals, which are participating in the hybridized band(s). Make a drawing of a Cu atom its d-orbital, and the hybridizing p-orbital of each of the oxygeb neighbors. Together, the four oxygen orbitals have the same d-symmetry as the local Cu d-orbital, when looked at with respect to the Cu location (that requires associating alternating "+" and "-" signs to the oxygen p's as you go around the Cu.). On each oxygen site, they're still p orbitals, though.

Concerning the TMDCs where also spin orbit coupling plays a stronger role, it may be adviseable to consult colleagues who are competent in the calculation of band structure and optical matrix elements as well as their interpretation. Many states have a large admixture of so-called "interstitial" character (meaning that this part of the wave function cannot be attributed to a specific atomic orbital) which doesn't easily fit with the quasi -atomic Delta-l reasoning above. If it's important to you, then it is also worth the effort of consulting someone who can help you to sort it out.

Hi, Kai. Your remarks are pretty helpful to me. Like what you said, the delta l =+- 1 dipole transition may not be suitable here. Also it is good to know there is such thing as "interstitial" character.  Thanks a lot!

Xinlin, I was not implying that the delta-l-rule was inadequate or not being useful. But one has to do the translation from atomic states to bands in a crystal. When states have mixed atomic (and, in addition, "interstitial") character, then "single site symmetry" doesn't give the whole picture. In addition, band states as elements of reciprocal space do also contain the symmetry of their k-vector, which is how translational symmetry chimes in in addition to the local atomic and bond symmetries.