# Point group from mono to bulk MoS2

Many articles say

even numbers of layers or bulk crystal MoS2 have D6h point group and odd numbers of layers (including monolayer) MoS2 have D3h point group

But whenever looking at the figure of MoS2, I can't recognize there is a symmetry representing the number '6' which means 6-fold(?). With the point group flow chart,

The final answer is C3h for the bulk MoS2.

If I included the inverse symmetry with C6 symmetry, then it would give D6h? Is the inversion symmetry necessary, then why? Especially, if you can point out the six ford axis in bulk case or two layers MoS2, please let us know.

I got the hind from the reference document that Mo has D3h and S has C3v. And these two are factor group of D6h. But I don't much know about group theory... So definitely it is not D6h?

Here the unit cell of bulk 2H-MoS2

Here the top and side view of 1H-MoS2

• It would help a great deal if you would add a picture or two of MoS2. Apr 18, 2019 at 13:57
• @porphyrin I added. Hope to be helpful. Apr 19, 2019 at 2:53

This is not a proper answer but can give you a clue. I arrived here looking for the same answer.

I didn't arrive to find the C6 rotation axis in bulk 2H-MoS2. On the other hand, if you look for the symmetry operations of the space group of 2H-MoS2 (P63/mmc, No. 194, D46h):

http://img.chem.ucl.ac.uk/sgp/large/194az1.htm

You can see that there are not any C6 rotation axis for this space group. Nevertheless, there are 6-fold screw axes (http://img.chem.ucl.ac.uk/sgp/misc/sixscrew.htm) and 6-fold inversion axes (http://img.chem.ucl.ac.uk/sgp/misc/barsix.htm).

You are right that the point group corresponding to this space group is D6h (https://en.wikipedia.org/wiki/Space_group#Table_of_space_groups_in_3_dimensions). And according to character tables a D6h point group should have C6 (http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=606&option=4).

Both things are contradictory. My background is Chemistry, and I am used to group theory applied to molecules rather than crystals. My guess is that the 6-fold screw axis, which is a symmetry element that is only possible in crystals (as it requires translational symmetry) might be related with labelling as D6h. But this is just my speculation.

In this article, they also talk about the symmetry operations of single-layer, multi-layer (odd and even layers) and bulk MoS2, and could be usefull for you too:

http://www1.spms.ntu.edu.sg/~qihuagroup/data/Xiong/Papers/nano%202013%20yanyuan.pdf

• Thanks a lot. I never thought the 6-fold screw and inversion axis. It's helpful but quite strange yet. There is 6-fold subgroup as you referred the 5th site. Well... It is trivial if we assume it is 6Dh. But if we apply the operator (C6), then mismatch with original positions. And I'm trying to read the last paper, thanks again. Apr 23, 2019 at 14:08
• I read a part of the paper, and I thought if the C6 involves translation symmetry along the z-axis for the bulk (not one, two, or finite layers), the atoms lie on their original positions. Because Dh6 symbol doesn't consider translation, so we are not supposed to ignore that. And I also found another example on wiki: en.wikipedia.org/wiki/Space_group See the first figure of bulk H2O. May 7, 2019 at 6:19