Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.1 DIPOLE MOMENTS OF MOLECULES

(a) HCl (b) CO 2 (c) O 3

A

B

A′ B′

Figure 29.1 Three molecules, (a) hydrogen chloride, (b) carbon dioxide and

(c) ozone, for which symmetry considerations impose varying degrees of

constraint on their possible electric dipole moments.

29.1 Dipole moments of molecules

Some simple consequences ofsymmetrycan be demonstrated by considering

whether a permanent electric dipole moment can exist in any particular molecule;

three simple molecules, hydrogen chloride, carbon dioxide and ozone, are illus-

trated in figure 29.1. Even if a molecule is electrically neutral, an electric dipole

moment will exist in it if the centres of gravity of the positive charges (due to

protons in the atomic nuclei) and of the negative charges (due to the electrons)

do not coincide.

For hydrogen chloride there is no reason why they should coincide; indeed, the

normal picture of the binding mechanism in this molecule is that the electron from

the hydrogen atom moves its average position from that of its proton nucleus to

somewhere between the hydrogen and chlorine nuclei. There is no compensating

movement of positive charge, and a net dipole moment is to be expected – and

is found experimentally.

For the linear molecule carbon dioxide it seems obvious that it cannot have

a dipole moment, because of its symmetry. Putting this rather more rigorously,

we note that any rotation about the long axis of the molecule leaves it totally

unchanged; consequently, any component of a permanent electric dipole perpen-

dicular to that axis must be zero (a non-zero component would rotate although

no physical change had taken place in the molecule). That only leaves the pos-

sibility of a component parallel to the axis. However, a rotation ofπradians

about the axisAA′shown in figure 29.1(b) carries the molecule into itself, as

does a reflection in a plane through the carbon atom and perpendicular to the

molecular axis (i.e. one with its normal parallel to the axis). In both cases the two

oxygen atoms change places but, as they are identical, the molecule is indistin-

guishable from the original. Either ‘symmetry operation’ would reverse the sign

of any dipole component directed parallel to the molecular axis; this can only be

compatible with the indistinguishability of the original and final systems if the

parallel component is zero. Thus on symmetry grounds carbon dioxide cannot

have a permanent electric dipole moment.