I read a line today and don't get it: "Molecules with mirror symmetry like oxygen, nitrogen, carbon dioxide, and carbon tetrachloride have no permanent dipole moments."
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diph2o.html
So, why molecules with mirror symmetry have no permanent dipole moment?
While DumpsterDoofus is right, perhaps this explanation might be helpful.
A dipole is an asymmetric separation of charge, like this: $+ -$. A dipole can have many charges. The total charge must be 0. The center of charge for the $+$ charges and the center of charge for the $-$ charges must be at different places.
A dipole can exert electrical forces on nearby charges, because the nearby charge can be closer to the $-$ than the $+$. $F \propto 1/r^3$
Symmetrical arrangements like this are not dipoles. $+--+$. They are similar, in that they can exert forces. But in this example, $F \propto 1/r^4$. The forces from $+$ and $-$ cancel better than for a dipole, but not perfectly.
A CO2 molecule is a line. O=C=O. The O's are a little $-$, and the C is a little $+$. It can have no dipole moment.
H2O is similar, except that it has a V shape. H2O does have a dipole moment.
The O in the middle is slightly $-$. The H's at the ends are two slightly $+$. The center of $+$ charge is half way between the H's.
Note that a V shape does have a left-right mirror symmetry. This does not prevent a vertical dipole moment.
Inversion symmetry means that if there is something (perhaps a charge) on one side, then there is another identical something on the opposite side. Example shapes include an X, a snowflake, a circle, and a rectangle.
For a distribution of charges with inversion symmetry, the center of $+$ charge and the center of $-$ charge are always at the center. There can be no dipole.
A starfish and a tetrahedron do not have inversion symmetry. Even so, a tetrahedral molecule (carbon tetrachloride) cannot have a dipole.
In summary, there are many symmetries. Some guarantee no dipole moment. Others do not. It is possible for an asymmetrical charge distribution to have no dipole moment.
So, why molecules with mirror symmetry have no permanent dipole moment?
That's actually not true. For example, hydrogen cyanide has an infinite number of mirror planes of symmetry, but has a nonzero permanent dipole moment. Also, formaldehyde has two mirror planes of symmetry, and a permanent electrical dipole moment.
However, for any molecule with inversion symmetry, there can't really be a permanent electric dipole moment; this follows since $\boldsymbol{\mu}\rightarrow-\boldsymbol{\mu}$ under inversion, which implies $\boldsymbol{\mu}=\mathbf{0}$ if the molecule is invariant under inversion.
