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This lists the character tables for the more common molecular point groups used in the study of molecular symmetry. These tables are based on the group-theoretical treatment of the symmetry operations present in common molecules, and are useful in molecular spectroscopy and quantum chemistry. Information regarding the use of the tables, as well ...
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types C 2, C i, C s. This total number is one of the characteristics helping to distinguish the various abstract group types, while their isometry type helps to distinguish the various isometry groups of the same abstract group.
Character table properties. In summary, for any group, its character table gives a tabulation (for the classes of the group) of the characters (the sum of the diagonal elements) of the matrices of all the irreducible representations of the group. As the number of irreducible representations equals the number of classes, the character table is ...
In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type.Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.
The space of complex-valued class functions of a finite group G has a natural inner product: , := | | () ¯ where () ¯ denotes the complex conjugate of the value of on g.With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table:
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In other cases there is no point around which the point group applies. The notation is somewhat ambiguous, without a table giving more information. For example, space groups I23 and I2 1 3 (nos. 197 and 199) both contain two-fold rotational axes as well as two-fold screw axes. In the first, the two-fold axes intersect the three-fold axes ...