# How to find a symmetry group of a system if all the symmetry transformations do not obey closure and don't form a group?

For instance, consider a system with $$p_x$$ and $$p_z$$ orbitals at the vertices of a square (on xy-plane). A square by itself would have $$D_4$$ symmetry. However, because of the $$p_x$$ orbital; the $$90^\circ$$ rotation ($$C_4$$) and $$270^\circ$$ rotation ($$C_4^{-1}$$) are no longer symmetry operations. Now the rest of the $$D_4$$ members do preserve the symmetry, but do not form a group since the subset is not closed.

How does one go about formulating a symmetry group for cases like these?

• Never saw a rotation denoted with $r$. Normally, 90° and 270° rotations are denoted with $C_4$ and $C_4^{-1}$ (or $3C_4),$ respectfully. This should give you a hint when you look up the point group using a character table. $r$ and $r^3$ in your context would refer to the distance and volume, respectfully. – andselisk Jun 28 '20 at 17:01
• Orbitals are not relevant to molecular symmetry. Only atoms are. Are the atoms at the four corners of the square identical? If so, then 90 degree rotation does return the original molecule. – Andrew Jun 28 '20 at 17:05
• @andselisk Sorry about the issue with notations, I've fixed it. Thanks! – Feynfan Jun 28 '20 at 17:19
• I am mildly confused: a square should be $D_\mathrm{4h}$, surely? – orthocresol Jun 28 '20 at 17:23
• @Andrew Actually this is sort of a hypothetical problem from physics. It's not really a molecule, and I'm considering a system that contains hypothetical atoms with only $p_x$ and $p_z$ like orbitals at the four vertices. I'm afraid a $C_4$ rotation will turn $p_x$ into $p_y$ breaking the symmetry. – Feynfan Jun 28 '20 at 17:24

Hmm, so I'm not entirely sure what notation you're using ($$D_4$$ sounded like Schönflies to me), but I'm sure you can figure out the equivalent.
In general I agree with Andrew's point that molecular symmetry is determined by atomic positions and not orbitals. But at least to me as of now, this seems to be a thought exercise more than anything (after all, no real atom has only two p-orbitals), so I'll play along. In Schönflies notation your system would be labelled $$D_\mathrm{2h}$$. The three $$C_2$$ axes and three mirror planes are illustrated below. I left out the identity operation $$E$$ and the centre of inversion $$i$$. 