524 Chapter 18Matrices and linear transformations
(iii) The determinant of Ais
0 Exercise 61
Orthogonal transformations
An orthogonal transformation is a linear transformation
x′ 1 = 1 Ax (18.65)
whose transformation matrix Ais orthogonal. Orthogonal transformations are
important because they preserve the scalar product of vectors; that is, the lengths of
vectors and the angles between them are unchanged by an orthogonal transformation.
All the transformations in Examples 18.18 (except for D), 18.19, and 18.20 are
orthogonal. The preservation of lengths and angles is demonstrated in Figure 18.4
of Example 18.19; the size and shape of the figure (square) is not changed by the
orthogonal transformation (rotation). All orthogonal transformations in a plane or
in a three-dimensional space are either rotations or reflections, or combinations of
these, and such transformations are important for the mathematical description of
the symmetry properties of molecules.
18.7 Symmetry operations
The symmetry of a physical system is characterized by a set of symmetry elements,
with each of which is associated one or more transformations called symmetry
operations. These are transformations that leave the description of the system
unchanged. Most important in molecular chemistry are the spatial transformations
that result in the interchange of identical nuclei. The possible symmetry elements are
then axes of symmetry, planes of symmetry, and a centre of inversion.
EXAMPLE 18.23Symmetry of the water molecule
The water molecule in its ground state has the nonlinear
equilibrium nuclear geometry illustrated in Figure 18.6, with
bond angle 105°. The system has three symmetry elements each
with one associated operation:
(i) A two-fold axis of symmetry (Ozin the figure). Rotation
through 180° about the axis results in the interchange of the
hydrogen nuclei.
detA=
−
−=−
1
3
212
221
122
1
3H
H
O
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Figure 18.6