Symmetry Elements and Operations - Molecular Symmetry - page 23

axis, a plane for a reflection and a single point for inversion. The symmetry element is
the set of points that are not moved when the corresponding symmetry operation takes
place.
Symmetry operations are the actions, such as rotations or reflections, that can be used to
transform an object in such a way that, after the operation, it is indistinguishable from the
starting point.
The symmetry elements and operations of a molecule are given standard symbols,
including:

•

Proper rotation axes , labelled C n , where n is the axis order. The highest order axis
present for a given molecule is called the principal axis and defines the vertical direction
used to orientate the molecule in space.

•

Reflection planes , labelled σ , with three possible subscripts:

- σ v , a vertical mirror plane, contains the principal axis - if there are horizontal C 2 axes,

then σ v will also contain those of highest priority;

- σ d , a dihedral mirror plane, also contains the principal axis - if there are horizontal C 2

axes, then σ d will bisect the angle between those of highest priority;

-

σ h , a horizontal mirror plane, is perpendicular to the principal axis.

•

Inversion centre , labelled i , has a single point as the symmetry element. Inversion
requires each point in the molecule to have an equivalent point on the opposite side
of the centre of symmetry and equidistant from it. A molecule may have at most one
point of inversion.

Atoms in a molecule that are linked by symmetry operations have identical chemical
environments, and so identical NMR chemical shifts.

1.5 Self-Test Questions

These questions are designed to give you practice at applying the concepts learned in this
chapter. Most of the questions for this chapter require you to visualize the geometry of a
molecule, and you may find it useful to construct models. In the illustrations here, as in the
rest of the text, we continue to use the convention that C atoms are not explicitly labelled
in organic molecules and hydrogen atoms are omitted unless the geometry is ambiguous
without them. The 'flying wedge' convention is also used to indicate bonds above and
below the plane of the paper.

1. For each of the molecules in Figure 1.27 identify all of the rotation axes present, giv-
ing the order of each axis and describing any that form a set of identical axes. You
should make sketches of the molecules viewed from different directions to illustrate
your answer.
2. Give the orders of the principal axes for the molecules illustrated in Figure 1.27.
3. Identify all the mirror planes present in the molecules of Figure 1.27 and use the rotation
axes you have found to label them as σ v , σ d or σ h .