# Why are there no dihedral mirror planes in the D3h point group?

Consider the molecule $\ce{BH3}$, which belongs to the $D_\mathrm{3h}$ point group. Why are the three mirror planes in this point group labelled as $\sigma_\mathrm v$ instead of $\sigma_\mathrm d$? Don't they bisect the angles between a pair of rotational axis $C_{2}$ axes as shown in the diagram below?

• Yes, the sigmas bisect the angles between the pairs of rotational axis C2 axes like in the picture. What is your question, really? – Ivan Neretin Feb 11 '18 at 7:45
• symmetry.jacobs-university.de/cgi-bin/… In this website and others they list $\sigma_{v}$ instead of $\sigma_{d}$. My question is why? – ChemDude Feb 11 '18 at 8:28
• Got it. They call these planes $\sigma_v$ because they are vertical mirror planes. Why would you want to call them $\sigma_d$? Just being bisectors of certain directions of twofold axes is not enough justification for that. – Ivan Neretin Feb 11 '18 at 8:37

The σ$_v$ planes in a molecule are a direct result of the C$_n$ main axis i.e. they are interchangeable if you perform the C$_n$ operation on them. On the other hand σ$_d$, which are called dihedral planes bisect the dihedral angles between members of the σ$_v$'s set. Therefore molecules with odd C$_n$ such as BH$_3$ wont have any σ$_d$ planes; only n (3 for BH$_3$) σ$_v$'s. In contrast a molecule such as benzene which has a C$_6$ main axis will have 3 σ$_v$'s (the ones including the carbon atoms) and 3 σ$_d$'s which are the ones passing between the carbon atoms. Of course it doesnt really matter which set is called dihedral and which is called vertical, it could be the other way round as well. Ref: Chemical applications of group theory by Cotton.