symmetry and molecular polarity

A molecule is polar if its centers of positive and negative charge do not coincide — it has a net dipole moment, a tiny arrow pointing from plus to minus. Water is polar (which is why it dissolves salts and sticks to itself); carbon dioxide, despite having two polar C=O bonds, is not, because the two bond dipoles point in exactly opposite directions and cancel. Symmetry is what decides whether the cancellation happens, and it does so by a simple geometric argument.

Here is the reasoning. A permanent dipole is a fixed vector built into the molecule. Every symmetry operation must leave the molecule (and therefore that vector) looking unchanged. A dipole can only survive if it lies along a direction that every symmetry operation preserves. The consequences are sharp: any molecule with a center of inversion must be nonpolar (inversion would flip the dipole into its opposite, which is only allowed if the dipole is zero); a dipole cannot lie perpendicular to a Cn axis (the rotation would point it somewhere else), so it must lie on the principal axis if one exists; and a dipole cannot survive if it would be reflected to a different direction by a mirror plane. In practice this means only the lower-symmetry point groups — Cn, Cnv, and Cs — can be polar.

This gives a fast structural verdict. Tetrahedral CH4 (Td), octahedral SF6 (Oh), square-planar PtCl4 (2-) (D4h), trigonal-planar BF3 (D3h), and linear CO2 (D-infinity-h) are all nonpolar by symmetry, even though several contain polar bonds. Water (C2v), ammonia (C3v), and the bent SO2 molecule (C2v) are polar because their symmetry leaves a unique axis along which a dipole can point. So before measuring or computing anything, the point group already tells you whether a permanent dipole is even possible.