Chemistry 330
Symmetry Properties of Molecules
Symmetry Elements
Some of the
symmetry elements
of a cube. The
twofold, threefold,
and fourfold axes
are labelled with the
conventional
symbols.
Proper Rotations
An NH3 molecule
has a threefold (C3)
axis
An H2O molecule
has a twofold (C2)
axis.
Mirror Planes
An H2O molecule
has two mirror
planes. They are
both vertical (that is,
contain the principal
axis) and so are
denoted v and ‘v.
Dihedral mirror planes
Dihedral mirror
planes (d) bisect
the C2 axes
perpendicular to the
principal axis.
The Inversion Centre
A regular
octahedron has a
centre of inversion
(i).
Improper Rotation Axes
A CH4 molecule has
a fourfold improper
rotation axis (S4)
The staggered form
of ethane has an S6
axis composed of a
60 rotation followed
by a reflection.
The Inversion Centre
The presence of a
twofold axis and a
horizontal mirror
plane jointly imply
the presence of a
centre of inversion
in the molecule.
The Dn Point Group
A molecule with n
twofold rotation axes
perpendicular to an
n-fold rotation axis
belongs to the group
Dn.
The Dnh Point Group
A molecule with a
mirror plane
perpendicular to a
Cn axis, and with n
twofold axes in the
plane, belongs to
the group Dnh.
Cubic Point Groups
Tetrahedral and
octahedral
molecules are
drawn in a way that
shows their relation
to a cube: they
belong to the cubic
groups Td and Oh,
respectively.
The Point Group Flow Diagram
A flow diagram for
determining the
point group of a
molecule. Start at
the top and answer
the question posed
in each diamond (Y
= yes, N = no).
Molecules and Point Groups
A summary of the
shapes
corresponding to
different point
groups.
Equivalent Elements
Some symmetry
elements are
implied by the other
symmetry elements
in a group
Presence of an
inversion means at
least an S2 element
Hence, i and S2 are
equivalent
Characters of Rotations
A rotation through 180
about the internuclear
axis leaves the sign of a
 orbital unchanged but
the sign of a  orbital is
changed.
The characters of the
C2 rotation are +1 and 1 for the  and 
orbitals, respectively.
Classes and Groups
Symmetry
operations in the
same class are
related to one
another by the
symmetry
operations of the
group.
The px Orbitals and Symmetry
A px orbital on the
central atom of a C2v
molecule and the
symmetry elements
of the group.
Reflections
The two orbitals
shown here have
different properties
under reflection
through the mirror
plane: one changes
sign (character -1),
the other does not
(character +1).
SALC’s
Two symmetryadapted linear
combinations
Each span a onedimensional
irreducible
representation, and
their symmetry
species are
different.
More SALC’s
Typical symmetryadapted linear
combinations of
orbitals in a C3v
molecule.
Symmetry Species and Orbital
Overlap
Orbitals of the same
symmetry species
may have nonvanishing overlap
The a1 and the two
components of the
doubly degenerate e
orbitals