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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.
The rotation group SO(3) is a subgroup of O(3), the full point rotation group of the 3D Euclidean space. Correspondingly, O(3) is the direct product of SO(3) and the inversion group C i (where inversion is denoted by its matrix − I ):
Hans Bethe used characters of point group operations in his study of ligand field theory in 1929, and Eugene Wigner used group theory to explain the selection rules of atomic spectroscopy. [13] The first character tables were compiled by László Tisza (1933), in connection to vibrational spectra.
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. These are the Bravais lattices in three dimensions:
In crystallography, a crystallographic point group is a three dimensional point group whose symmetry operations are compatible with a three dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of ...
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).
Elementary abelian group; Exceptional character; F. Fibrifold; ... List of character tables for chemically important 3D point groups; List of small groups; Local ...
The (finite) list of all symmetry operations which leave the given point invariant taken together make up another group, which is known as the site symmetry group of that point. [4] By definition, all points with the same site symmetry group, or a conjugate site symmetry group, are assigned the same Wyckoff position.