104
5 What has Quantum Chemistry Got to Do with It?
components of the dipole moment are strictly forbidden. This is in agreement with
the Neumann principle. A dipole moment corresponds to a displacement of charge,
and the only displacement that does not destroy the molecular point group is along
the z-direction. Hence, this is the only direction that is compatible with the Neumann
principle.
Pierre Curie realized that this principle is not limited to crystals, but applies to
physical phenomena in general, and not only to isolated systems, but also to systems subject to external perturbations. His proposition is known as the principle of
dissymmetry.
Theorem 10 The symmetry of a phenomenon is the maximal symmetry compatible
with the existence of the phenomenon. In order for a phenomenon to exist, it is
necessary that certain elements of symmetry are absent: dissymmetry creates the
phenomenon.
Curie understood that under stress, or in the presence of external electric or magnetic fields, the symmetry of a system is changed. The Neumann principle still applies but should no longer be based on the symmetry of the isolated crystal, but on
that of the combined system of crystal and external field, as we have considered in
Sect. 3.9. In the case of ammonia, application of an electric field has the C ∞v symmetry of a polar vector. The symmetry that results from the superposition of the field
with the molecular point group C 3v depends on the orientation (see Appendix B).
In the coordinate frame of Fig. 3.1 one has:
z : C ∞v ∩ C 3v = C 3v
x : C ∞v ∩ C 3v = C s
y : C ∞v ∩ C 3v = C 1
(5.1)
If the field is oriented along the z-direction, it will keep the C 3v symmetry of
the molecule. This is also in line with the existence of a permanent dipole in the
z-direction according to the Neumann principle since a charge dipole has the same
symmetry as an electric field. If an external field is applied in the x-direction, the
symmetry is reduced to the reflection group, C s ={ ˆ
E, ˆ
σ 1 }. In the presence of such
a field, displacement of charge in the x-direction is compatible with the extended
symmetry principle, which means that ammonia can acquire an induced dipole moment in the x-direction. However, a field along x cannot induce a dipole moment in
the y-direction since the σ 1 reflection plane is incompatible with the displacement
of charge across the plane of symmetry.
The symmetry principles of Neumann and Curie can be recast in the language
of irreducible representations. The requirement that physical properties be invariant
under the symmetry elements of the point group simply means that they should
transform as the totally symmetric irrep. For the dipole moment, the components
5 What has Quantum Chemistry Got to Do with It?
components of the dipole moment are strictly forbidden. This is in agreement with
the Neumann principle. A dipole moment corresponds to a displacement of charge,
and the only displacement that does not destroy the molecular point group is along
the z-direction. Hence, this is the only direction that is compatible with the Neumann
principle.
Pierre Curie realized that this principle is not limited to crystals, but applies to
physical phenomena in general, and not only to isolated systems, but also to systems subject to external perturbations. His proposition is known as the principle of
dissymmetry.
Theorem 10 The symmetry of a phenomenon is the maximal symmetry compatible
with the existence of the phenomenon. In order for a phenomenon to exist, it is
necessary that certain elements of symmetry are absent: dissymmetry creates the
phenomenon.
Curie understood that under stress, or in the presence of external electric or magnetic fields, the symmetry of a system is changed. The Neumann principle still applies but should no longer be based on the symmetry of the isolated crystal, but on
that of the combined system of crystal and external field, as we have considered in
Sect. 3.9. In the case of ammonia, application of an electric field has the C ∞v symmetry of a polar vector. The symmetry that results from the superposition of the field
with the molecular point group C 3v depends on the orientation (see Appendix B).
In the coordinate frame of Fig. 3.1 one has:
z : C ∞v ∩ C 3v = C 3v
x : C ∞v ∩ C 3v = C s
y : C ∞v ∩ C 3v = C 1
(5.1)
If the field is oriented along the z-direction, it will keep the C 3v symmetry of
the molecule. This is also in line with the existence of a permanent dipole in the
z-direction according to the Neumann principle since a charge dipole has the same
symmetry as an electric field. If an external field is applied in the x-direction, the
symmetry is reduced to the reflection group, C s ={ ˆ
E, ˆ
σ 1 }. In the presence of such
a field, displacement of charge in the x-direction is compatible with the extended
symmetry principle, which means that ammonia can acquire an induced dipole moment in the x-direction. However, a field along x cannot induce a dipole moment in
the y-direction since the σ 1 reflection plane is incompatible with the displacement
of charge across the plane of symmetry.
The symmetry principles of Neumann and Curie can be recast in the language
of irreducible representations. The requirement that physical properties be invariant
under the symmetry elements of the point group simply means that they should
transform as the totally symmetric irrep. For the dipole moment, the components