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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 ...
Robert Mulliken was the first to publish character tables in English (1933), and E. Bright Wilson used them in 1934 to predict the symmetry of vibrational normal modes. [13]. For this reason, the notation used to label irreps in the above table is called Mulliken notation and for asymmetric groups it consists of letters A and B with subscripts ...
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.
From the right side of the character table, the non-vibrational degrees of freedom, rotational (R x and R y) and translational (x, y, and z), are subtracted: Γ vib = Γ 3N - Γ rot - Γ trans. This yields the Γ vib, which is used to find the correct normal modes from the original symmetry, which is either C ∞v or D ∞h, using the ...
Note that the right-hand side is a sum of characters for symmetric groups that have smaller order than that of the symmetric group we started with on the left-hand side. In other words, this version of the Murnaghan-Nakayama rule expresses a character of the symmetric group S n in terms of the characters of smaller symmetric groups S k with k<n.
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:
In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following: C n (for cyclic) indicates that the group has an n-fold rotation axis.