18.7 Symmetry operations 525
(ii) A plane of symmetry (Oyz)that bisects the bond angle. Reflection in the plane
again results in the interchange of the hydrogens.
(iii) A plane of symmetry (Oxz)that contains all three nuclei (the molecular plane).
Reflection in this plane leaves the nuclei unmoved.
These are distinct symmetry operations and have different effects on the wave
functions that describe the states of the molecule.
The mathematical theory of symmetry is group theory.
4We give here only a brief
introduction to the concept of symmetry groups and of the matrix representations of
groups.
Symmetry groups
We consider the symmetrical plane figure formed by three points at the corners of an
equilateral triangle (Figure 18.7), with the figure in the xy-plane of a fixed coordinate
system whose origin O lies at the centroid of the triangle.
The symmetry of the figure can be described in terms of the following set of six
symmetry operations:
E the identity operation that leaves every point unmoved
Aanticlockwise rotation through 120° about the Ozaxis
B anticlockwise rotation through 240° (or clockwise through 120°) about the Ozaxis
C rotation through 180° about the Ocaxis
Drotation through 180° about the Odaxis
F rotation through 180° about the Ofaxis
4Group theory has its origins in studies of algebraic equations by Lagrange, Gauss, Abel, and Cauchy. The
relation between algebraic equations and the group of permutations was described by Évariste Galois (1811–1832),
whose short life included two failures to enter the École Polytechnique (at the second attempt he threw the blackboard
eraser at one of the examiners), expulsion from the École Normale, and two arrests, one for threatening the life of
the king and one for wearing the uniform of the dissolved National Guard. He was killed in a duel in an ‘affair of
honour’. His mathematical manuscripts were unread until published by Liouville in 1846. In a paper on quadratic
forms in 1882, Heinrich Weber (1842–1913) gave a complete axiomatic description of a finite abstract group,
defined Abelian groups, and in 1893 extended the work to infinite groups.
..........................................................................................................................................................................................................................
...
...
...
...
...
...
...
...
..
...
...
...
...
...
...
..
....
..
...
...
...
...
...
...
..
....
..
...
...
...
...
...
...
..
....
..
...
...
...
...
...
...
..
....
..
...
...
...
...
...
...
..
....
..
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
...
...
...
...
..
- •
o
y
x
c
d
f
1
2 3
...........
........
......................................
...
...
..
...
.....
...
...
..
...
.................................................................................................................................................. ..
......
......
....
.......
....
......
......
......
.....
......
.....
......
.....
......
.....
......
.....
......
.....
......
.....
......
.....
......
.....
...Figure 18.7