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.
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Notes
- 1.
Named after the Norwegian mathematician Niels Henrik Abel (1802–1829).
- 2.
This group is isomorphic to Felix Klein’s four-group (Vierergruppe).
- 3.
The sum of the angles subtended at a vertex of a Platonic solid must be smaller than a full angle of 2π. Hence, no more than five triangles, three squares, or three pentagons can meet in a vertex; regular hexagons are already excluded since the junction of three such hexagons already gives rise to an angle of 2π at the shared vertex.
- 4.
Van’t Hoff published his findings in 1874 in Utrecht. In the same year, Le Bel came to the same conclusion, based on the investigation of optical rotatory power. An English translation of the original papers of both chemists can be found in: [1].
- 5.
In graph theory the graph of a simplex with n vertices is the complete n-graph, K n . In such a graph, each of the n vertices is connected to all the other (n−1) vertices. There is only one simplex for each dimension.
- 6.
Dihedral means literally “having two planes.” The dihedral angle is an important molecular descriptor. The dihedral angle of the central B−C bond in an A−B−C−D chain is the angle between the ABC and BCD faces. In the present context, the term dihedral originates from crystallography, such as when two plane faces meet in an apex of a crystal.
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Ceulemans, A.J. (2013). Groups. In: Group Theory Applied to Chemistry. Theoretical Chemistry and Computational Modelling. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6863-5_3
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DOI: https://doi.org/10.1007/978-94-007-6863-5_3
Publisher Name: Springer, Dordrecht
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