# Dipole moment of symmetrical molecules

While studying dipole moment in organic chemistry, I got stuck at a particular statement written in the book :

Symmetrical molecules without lone pairs of electrons will have $$\mu = 0.$$

The main problem was, why would they have $$\mu =0$$ ?

What type of Symmetrical molecules are possible?

The book gave the example of $$\ce{BH3}$$. How is this a symmetrical molecule?

• I'd rather start any answer with a reminder that BH3 does not quite exist as such; B2H6 does. – Ivan Neretin Aug 27 '19 at 6:47

Yes, the statement in your book is very vague. Molecules are often classified by the symmetry elements they contain. For example, $$\ce{BH3}$$ contains a $$C_3$$ axis perpendicular to the plane containing the $$\ce{BH3}$$ molecule. "$$C_3$$" means that if you rotate the molecule around this axis by $$360^\circ/3 = 120^\circ$$, you'll get a molecule that is indistinguishable from the molecule you started with. $$\ce{BH3}$$ also contains 3 $$\sigma_\mathrm{v}$$ planes of symmetry. These planes contain the $$C_3$$ axis and one of the $$\ce{B-H}$$ bonds. $$\ce{BH3}$$ also contains one $$\sigma_\mathrm{h}$$ symmetry plane that is perpendicular to the $$C_3$$ axis and contains the $$\ce{BH3}$$ molecule. In addition to these symmetry elements, a number of other symmetry elements exist as well.
• $$C_n$$ (the molecule only contains a $$C_n$$ axis
• $$C_{n\mathrm v}$$ (the molecule contains a $$C_n$$ axis and $$n$$ total $$\sigma_\mathrm{v}$$ planes)
• $$C_{\mathrm s}$$ (the molecule only has a plane of symmetry that contains the entire molecule)
can have a permanent dipole moment. If the molecule belongs to any other point group it cannot have a dipole moment. Since $$\ce{BH3}$$ contains a $$\sigma_\mathrm{h}$$ plane of symmetry ($$\ce{BH3}$$ belongs to point group $$D_{3\mathrm h}$$) - it therefore does not belong to groups $$C_n$$, $$C_{n\mathrm v}$$ or $$C_{\mathrm s}$$ - and cannot have a dipole moment.
$$\ce{BH3}$$ is a flat triangle molecule, so it has mirror symmetry. Since the dipoles cancel out, the molecule does not have a dipole moment.