List of character tables for chemically important 3D point groups

List of character tables for chemically important 3D point groups

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 as more extensive lists of them, can be found in the references. [cite book | last = Drago | first = Russell S. | title = Physical Methods in Chemistry | publisher = W.B. Saunders Company | date = 1977 | isbn = 0-7216-3184-3] [ cite book | last=Cotton | first = F. Albert | title = Chemical Applications of Group Theory | publisher = John Wiley & Sons: New York | date = 1990 | isbn = 0-4715-1094-7] [cite web | last = Gelessus | first = Achim | title = Character tables for chemically important point groups | publisher = Jacobs University, Bremin; Computational Laboratory for Analysis, Modeling, and Visualization | date=2007-07-12 | url=http://symmetry.jacobs-university.de/ | accessdate=2007-07-12 ] cite journal | last=Shirts | first=Randall B. | title=Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables | journal=Journal of Chemical Education | volume=84 | issue=1882 | publisher=American Chemical Society | date=2007 | url=http://jchemed.chem.wisc.edu/Journal/Issues/2007/Nov/abs1882.html | accessdate= 2007-10-16]

Notation

For each group, the order of the group (its number of invariant symmetry operations) is given (except the linear groups), followed by its character table.

The rows of the character tables correspond to the irreducible representations of the group, with their conventional names in the left margin. The naming conventions are as follows:

* "A" and "B" are singly degenerate representations, with the former transforming symmetrically around the principal axis of the group, and the latter asymmetrically. "E", "T", "G", "H", ... are doubly, triply, quadruply, quintuply, ... degenerate representations.
* "g" and "u" subscripts denote symmetry and antisymmetry, respectively, with respect to a center of inversion. Subscripts "1" and "2" denote symmetry and antisymmetry, respectively, with respect to a nonprincipal rotation axis. Higher numbers denote additional representations with such asymmetry.
* Single primes (') and double primes (") superscripts denote symmetry and antisymmetry, respectively, with respect to a horizontal (normal to the principal rotation axis) mirror plane σh.

All but the two rightmost columns correspond to the symmetry operations which are invariant in the group. In the case of sets of similar operations with the same characters for all representations, they are presented as one column, with the number of such similar operations noted in the heading.

The body of the tables contain the characters in the respective irreducible representations for each respective symmetry operation, or set of symmetry operations.

The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates ("x", "y" and "z"), rotations about those three coordinates ("Rx", "Ry" and "Rz"), and functions of the quadratic terms ("x"2, "y"2, "z"2, "xy", "xz", and "yz").

The symbol "i" used in the body of the table (as opposed to a column heading where it denotes inversion) denotes the imaginary unit ("i" 2 = −1). A superscripted uppercase "C" denotes complex conjugation.

Character tables

Nonaxial symmetries

These groups are characterized by a lack of a proper rotation axis, noting that a "C"1 rotation is considered the identity operation. These groups have involutional symmetry: the only nonidentity operation, if any, is its own inverse.

In the group "C"1, all functions of the Cartesian coordinates and rotations about them transform as the "A" irreducible representation.

- style="background:#fafafa;"
"C"s || 2
align="left"

-
"C"4 || 4
align="left"

-
"C"8 || 8
align="left"

-
"C"4"h" || 8
align="left"

-

Pyramidal groups ("C"nv)

The pyramidal groups are denoted by "C"nv. These groups are characterized by i) an "n"-fold proper rotation axis "C"n; ii) "n" mirror planes "σv" which contain "C"n. The "C"1"v" group is the same as the "C"s group in the nonaxial groups section.

-
"C"4"v" || 8
align="left"

-

Improper rotation groups ("S"n)

The improper rotation groups are denoted by "S"n. These groups are characterized by an "n"-fold improper rotation axis "S"n, where "n" is necessarily even. The "S"2 group is the same as the "C"s group in the nonaxial groups section.

The S8 table reflects the 2007 discovery of errors in older references. Specifically, ("Rx", "Ry") transform not as E1 but rather as E3.

-
"S"8 || 8
align="left"

-
"D"4 || 8
align="left"

-

Prismatic groups ("D"nh)

The prismatic groups are denoted by "D"nh. These groups are characterized by i) an "n"-fold proper rotation axis "C"n; ii) "n" 2-fold proper rotation axes "C"2 normal to "C"n; iii) a mirror plane "σh" normal to "C"n and containing the "C"2s. The "D"1"h" group is the same as the "C"2"v" group in the pyramidal groups section.

The D8"h" table reflects the 2007 discovery of errors in older references. Specifically, symmetry operation column headers 2S8 and 2S83 were reversed in the older references.

-
"D"4"h" || 16
align="left"

-
"D"8"h" || 32
align="left"

-
"D"4"d" || 16
align="left"

-


= Polyhedral symmetries =

These symmetries are characterized by having more than one proper rotation axis of order greater than 2.

Cubic groups

These polyhedral groups are characterized by not having a "C"5 proper rotation axis.

-
"Th" || 24
align="left"

-

Icosahedral groups

These polyhedral groups are characterized by having a "C"5 proper rotation axis.

-

Linear (cylindrical) groups

These groups are characterized by having a proper rotation axis "C"∞ around which the symmetry is invariant to "any" rotation.

-

References

See also


*Character tables in character theory
*Linear combination of atomic orbitals molecular orbital method
*Raman spectroscopy
*Vibrational spectroscopy (molecular vibration)


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