46
3 Groups
Fig. 3.14 Dimethyl
substituted allene occurs in
two chiral forms, which are
the mirror image of each
other, and cannot be
superimposed; the point
group of each form is C 2
generates a normal C s subgroup. The full group D 3h can then be written as a direct
product of this normal subgroup and the rotational subgroup:
D 3h = C s × D 3
(3.37)
When the symmetry of a molecule is a purely rotational group, then the molecule
does not coincide with its mirror image, and there will be two copies of it, which relate to each other as do right and left hands. This is illustrated in Fig. 3.14. Molecules
with only rotational symmetry are chiral, meaning that the molecule and its reflection form optical antipodes. A synonym is enantiomeric, which literally means: on
both sides of the mirror. By contrast, when a molecular point group contains any
improper symmetry element, the molecule will be congruent to its mirror image
and is achiral. Why does the absence of an improper symmetry element prevent the
molecule’s coinciding with its mirror image? Congruence operations are whole rotations and/or translations that are performed in order to superimpose the image on
the object. As we have shown, when reflecting a molecule through a mirror, spatial
inversion is implied, and this cannot be undone by rotations or translations, but only
by another improper symmetry element that restores the inversion. Such symmetry
elements are absent in molecules with only rotational point group symmetries, and,
hence, they cannot be made to coincide with their mirror image. Molecular chirality thus derives from an obvious and basic symmetry characteristic; nonetheless, it
has far reaching physical and chemical consequences. Chiral molecules are optically active, which means that the plane of polarization of linearly polarized light
is rotated when passing through a medium with chiral molecules. The absorption
coefficients of chiral molecules for left- and right-circularly polarized light are also
different, giving rise to natural circular dichroism (CD) spectra. This will be illustrated in Sect. 6.8. The chemical consequences of chirality are of vital importance:
living organisms are based on biochemical molecules with strict chirality.
3.9 Applications: Magnetic and Electric Fields
In many spectroscopic applications external magnetic or electric fields are applied
to a molecular sample. For magnetic fields, the perturbing influence of the field
is known as the Zeeman effect. Electric fields give rise to the Stark effect. As we
have seen, uniform magnetic and electric fields have symmetries C ∞h and C ∞v ,
respectively. In this case, the symmetry group of the experiment will be constrained
to these operations, which leave the {molecule + field} combination invariant. It
3 Groups
Fig. 3.14 Dimethyl
substituted allene occurs in
two chiral forms, which are
the mirror image of each
other, and cannot be
superimposed; the point
group of each form is C 2
generates a normal C s subgroup. The full group D 3h can then be written as a direct
product of this normal subgroup and the rotational subgroup:
D 3h = C s × D 3
(3.37)
When the symmetry of a molecule is a purely rotational group, then the molecule
does not coincide with its mirror image, and there will be two copies of it, which relate to each other as do right and left hands. This is illustrated in Fig. 3.14. Molecules
with only rotational symmetry are chiral, meaning that the molecule and its reflection form optical antipodes. A synonym is enantiomeric, which literally means: on
both sides of the mirror. By contrast, when a molecular point group contains any
improper symmetry element, the molecule will be congruent to its mirror image
and is achiral. Why does the absence of an improper symmetry element prevent the
molecule’s coinciding with its mirror image? Congruence operations are whole rotations and/or translations that are performed in order to superimpose the image on
the object. As we have shown, when reflecting a molecule through a mirror, spatial
inversion is implied, and this cannot be undone by rotations or translations, but only
by another improper symmetry element that restores the inversion. Such symmetry
elements are absent in molecules with only rotational point group symmetries, and,
hence, they cannot be made to coincide with their mirror image. Molecular chirality thus derives from an obvious and basic symmetry characteristic; nonetheless, it
has far reaching physical and chemical consequences. Chiral molecules are optically active, which means that the plane of polarization of linearly polarized light
is rotated when passing through a medium with chiral molecules. The absorption
coefficients of chiral molecules for left- and right-circularly polarized light are also
different, giving rise to natural circular dichroism (CD) spectra. This will be illustrated in Sect. 6.8. The chemical consequences of chirality are of vital importance:
living organisms are based on biochemical molecules with strict chirality.
3.9 Applications: Magnetic and Electric Fields
In many spectroscopic applications external magnetic or electric fields are applied
to a molecular sample. For magnetic fields, the perturbing influence of the field
is known as the Zeeman effect. Electric fields give rise to the Stark effect. As we
have seen, uniform magnetic and electric fields have symmetries C ∞h and C ∞v ,
respectively. In this case, the symmetry group of the experiment will be constrained
to these operations, which leave the {molecule + field} combination invariant. It