Chapter 17
Symmetry Groups
We have many opportunities to observe symmetry both macroscopic and microscopic in natural world. First, we need to formulate the symmetry appropriately. For
this purpose, we must regard various symmetry operations as mathematical elements
and classify these operations under several categories. In Part III we examined
various properties of vectors and their transformations. We also showed that the
vector transformation can be viewed as the coordinate transformation. On these
topics, we focused upon abstract concepts in various ways. On another front,
however, we have not paid attention to specific geometric objects, especially molecules. In this chapter, we study the symmetry of these concrete objects. For this, it
will be indispensable to correctly understand a variety of symmetry operations. At
the same time, we deal with the vector and coordinate transformations as group
elements. Among such transformations, rotations occupy a central place in the group
theory and related field of mathematics. Regarding the three-dimensional Euclidean
space, SO(3) is particularly important. This is characterized by an infinite group in
contrast to various symmetry groups (or point groups) we investigate in the former
parts of this chapter.
17.1 A Variety of Symmetry Operations
To understand various aspects of symmetry operations, it is convenient and essential
to consider a general point that is fixed in a three-dimensional Euclidean space and to
examine how this point is transformed in the space. In parallel to the description in
Part III, we express the coordinate of the general point P as
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_17
635
Symmetry Groups
We have many opportunities to observe symmetry both macroscopic and microscopic in natural world. First, we need to formulate the symmetry appropriately. For
this purpose, we must regard various symmetry operations as mathematical elements
and classify these operations under several categories. In Part III we examined
various properties of vectors and their transformations. We also showed that the
vector transformation can be viewed as the coordinate transformation. On these
topics, we focused upon abstract concepts in various ways. On another front,
however, we have not paid attention to specific geometric objects, especially molecules. In this chapter, we study the symmetry of these concrete objects. For this, it
will be indispensable to correctly understand a variety of symmetry operations. At
the same time, we deal with the vector and coordinate transformations as group
elements. Among such transformations, rotations occupy a central place in the group
theory and related field of mathematics. Regarding the three-dimensional Euclidean
space, SO(3) is particularly important. This is characterized by an infinite group in
contrast to various symmetry groups (or point groups) we investigate in the former
parts of this chapter.
17.1 A Variety of Symmetry Operations
To understand various aspects of symmetry operations, it is convenient and essential
to consider a general point that is fixed in a three-dimensional Euclidean space and to
examine how this point is transformed in the space. In parallel to the description in
Part III, we express the coordinate of the general point P as
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_17
635