Lecture1

Principles of Inorganic Chemistry
CHM62-221

Walailak University
Department of Chemistry

Lecture 1: Introduction to Symmetry

By
Assoc. Prof. Dr. David J. Harding

Basic concepts

Molecules, orbitals, vibrations etc possess symmetry such that the atoms, orbitals or
vibrations within that molecule are identical. For example, the fluorine atoms in BF3
are all equivalent. Group theory is the mathematical treatment of symmetry.
A symmetry operation is an operation performed on an object which leaves it in a
configuration that is indistinguishable from, and superimposable on, the original
configuration.
A symmetry operation is carried with respect to points, lines or planes, and these
are known as symmetry elements.

Rotation

The symmetry operation of rotation about an n-fold axis (the symmetry element) is
denoted by the symbol Cn, in which the angle of rotation is 360/n or 2/n; n must
be integer, 2, 3, 4 etc.

By convention the rotation is always in a clockwise direction. In the BF3 molecule
the C3 rotation axis is perpendicular (⊥) to the molecular plane.

Angle of rotation = 120 = 360/n
Using traditional Cartesian coordinates (x,y,z), changes can be expressed in terms
of x, y and z (see below).

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In addition, BF3 also contains three 2-fold (C2) rotation axes, each coincident
with a B-F bond.

If a molecule possesses more than one type of n-axis, the axis with the highest
value of n is called the principal axis and is positioned coincident with the z-axis.
In the case of BF3 the C3 axis is the principal axis.

Reflection

A second type of symmetry operation is reflection. The reflection occurs through a
mirror plane given the symbol . In BF3, the plane containing the atoms is the mirror
plane. In this case the plane lies perpendicular to the vertical principal axis and is
denoted by the symbol h.

If the plane contains the principal axis, it is labeled v. Consider the H2O molecule.
This possesses a C2 axis, but it also contains two mirror planes, one containing the
H2O molecule, and one perpendicular to it. The two planes are labeled v and v.

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A special type of plane which contains the principal rotation axis, but which bisects
the angle between two adjacent 2-fold axis is labeled d. Mirror planes such as this
are present in C6H6

In the notation for planes of symmetry, , the subscripts h, v and d stand for
horizontal, vertical and dihedral respectively.
As before, we can also show rotation using Cartesian coordinates.

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Inversion

If reflection of all parts of a molecule through the centre of the molecule produces
an indistinguishable configuration, the centre is a centre of symmetry, also called a
centre of inversion designated by the symbol i. As an example SF6 and benzene
possess a centre of inversion while H2S and SiH4 do not.
i(x,y,z) = (-x, -y, -z)

Improper rotation

A further type of symmetry operation is an improper rotation sometimes called
rotation-reflection. An improper rotation involves rotation through 360/n about an
axis followed by reflection through a plane perpendicular to that axis. It is a two-
step operation and is denoted by the symbol Sn.

Tetrahedral species of the type XY4 possess three S4 axes. The S4 rotation-reflection
in the CH4 molecule is shown above.
S4(z)(x,y,z) = C4(z)(xy)(x,y,z) = (xy)(x,-y,z) = (x,-y,-z)

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Identity

A final type of symmetry operation is known as the identity operator E. This is the
simplest operator and involves either rotation through 360 or doing nothing. This
operator is required to satisfy the mathematics underpinning group theory.
E(x, y, z) = E(x, y, z)

Combined Operations

Successive operations (usually rotations or improper rotations) in molecules are
distinguished using the notation Cnm or Snm where n and m are integers.

In the case of a 3-fold rotation on the third successive rotation the molecule returns
to its initial configuration and is equivalent to the identity operation. It follows that
there are only two unique symmetry operations, C3 and C32.

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Improper rotations are the product of rotation and reflection (this should be
obvious!).

Cn x h = Sn for n even Snn = Cnn x hn = E x E = E

for n odd Snn = Cnn x hn = E x h = h

Sn2n = Cn2n x h2n = E x E = E

for m even Snm = Cnm x hm = Cnm

for m odd Snm = Cnm x hm = Snm

Finally, let’s look at the S2 operation for XeF4.

It is clear that this is the same as inversion and thus
S2 = C2  h = i

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