Chapter 3
Groups
Abstract The concept of a group is introduced using the example of the symmetry
group of the ammonia molecule. In spite of its tiny size, this molecule has a structural symmetry that is the same as the symmetry of a macroscopic trigonal pyramid.
From the mathematical point of view, a group is an elementary structure that proves
to be a powerful tool for describing molecular properties. Three ways of dividing
(and conquering) groups are shown: subgroups, cosets, and classes. An overview
of molecular symmetry groups is given. The relationship between rotational groups
and chirality is explained, and symmetry lowerings due to applied magnetic and
electric fields are determined.
Contents
3 . 1
T h eS y m m e t r yo fA m m o n i a............................
2 1
3 . 2
T h eG r o u pS t r u c t u r e................................
2 4
3.3
Some Special Groups ...............................
2 7
3.4
Subgroups .....................................
2 9
3 . 5
C o s e t s .......................................
3 0
3 . 6
C l a s s e s.......................................
3 2
3.7
Overview of the Point Groups . . . ........................
3 4
S p h e r i c a lS y m m e t r ya n dt h eP l a t o n i cS o l i d s ...................
3 4
C y l i n d r i c a lS y m m e t r i e s ..............................
4 0
3.8
Rotational Groups and Chiral Molecules . . . ..................
4 4
3 . 9
A p p l i c a t i o n s :M a g n e t i ca n dE l e c t r i cF i e l d s....................
4 6
3 . 1 0
P r o b l e m s......................................
4 7
R e f e r e n c e s...........................................
4 8
3.1 The Symmetry of Ammonia
The umbrella shape of the ammonia molecule has trigonal symmetry with, in addition, three vertical reflection planes through the hydrogen atoms. Together these
symmetry elements form a point group, which, in the Schoenflies notation, is denoted as C 3v . It is good practice to start the treatment by making a simple sketch of
the molecule and putting it in a right-handed Cartesian frame, as shown in Fig. 3.1.
By convention, the z-axis is defined as the principal threefold axis. One of the hydrogens is put in the xz plane as shown in the figure. We attach labels A, B, C to distinA.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_3,
© Springer Science+Business Media Dordrecht 2013
21
Groups
Abstract The concept of a group is introduced using the example of the symmetry
group of the ammonia molecule. In spite of its tiny size, this molecule has a structural symmetry that is the same as the symmetry of a macroscopic trigonal pyramid.
From the mathematical point of view, a group is an elementary structure that proves
to be a powerful tool for describing molecular properties. Three ways of dividing
(and conquering) groups are shown: subgroups, cosets, and classes. An overview
of molecular symmetry groups is given. The relationship between rotational groups
and chirality is explained, and symmetry lowerings due to applied magnetic and
electric fields are determined.
Contents
3 . 1
T h eS y m m e t r yo fA m m o n i a............................
2 1
3 . 2
T h eG r o u pS t r u c t u r e................................
2 4
3.3
Some Special Groups ...............................
2 7
3.4
Subgroups .....................................
2 9
3 . 5
C o s e t s .......................................
3 0
3 . 6
C l a s s e s.......................................
3 2
3.7
Overview of the Point Groups . . . ........................
3 4
S p h e r i c a lS y m m e t r ya n dt h eP l a t o n i cS o l i d s ...................
3 4
C y l i n d r i c a lS y m m e t r i e s ..............................
4 0
3.8
Rotational Groups and Chiral Molecules . . . ..................
4 4
3 . 9
A p p l i c a t i o n s :M a g n e t i ca n dE l e c t r i cF i e l d s....................
4 6
3 . 1 0
P r o b l e m s......................................
4 7
R e f e r e n c e s...........................................
4 8
3.1 The Symmetry of Ammonia
The umbrella shape of the ammonia molecule has trigonal symmetry with, in addition, three vertical reflection planes through the hydrogen atoms. Together these
symmetry elements form a point group, which, in the Schoenflies notation, is denoted as C 3v . It is good practice to start the treatment by making a simple sketch of
the molecule and putting it in a right-handed Cartesian frame, as shown in Fig. 3.1.
By convention, the z-axis is defined as the principal threefold axis. One of the hydrogens is put in the xz plane as shown in the figure. We attach labels A, B, C to distinA.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_3,
© Springer Science+Business Media Dordrecht 2013
21