Point Group Character Tables
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character table In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters ...
s for the more common molecular point groups used in the study of
molecular symmetry Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain m ...
. These tables are based on the group-theoretical treatment of the
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
operations present in common
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s, and are useful in molecular
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
and
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
. Information regarding the use of the tables, as well as more extensive lists of them, can be found in the references.


Notation

For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the order of the group (number of invariant symmetry operations). The finite group notation used is: Zn:
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order ''n'', Dn:
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
isomorphic to the symmetry group of an ''n''–sided regular polygon, Sn:
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
on ''n'' letters, and An:
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic prop ...
on ''n'' letters. The character tables then follow for all groups. The rows of the character tables correspond to the irreducible representations of the group, with their conventional names, known as Mulliken symbols, 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 prime ( ' ) and double prime ( '' ) superscripts denote symmetry and antisymmetry, respectively, with respect to a horizontal mirror plane σh, one perpendicular to the principal rotation axis. All but the two rightmost columns correspond to the
symmetry operation In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...
s 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 symbol ''i'' used in the body of the table denotes the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
: ''i'' 2 = −1. Used in a column heading, it denotes the operation of inversion. A superscripted uppercase "C" denotes
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. 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 of the coordinates(''x''2, ''y''2, ''z''2, ''xy'', ''xz'', and ''yz''). A further column is included in some tables, such as those of Salthouse and Ware For example, The last column relates to cubic functions which may be used in applications regarding ''f'' orbitals in atoms.


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.


Cyclic symmetries

The families of groups with these symmetries have only one rotation axis.


Cyclic groups (''C''n)

The cyclic groups are denoted by ''C''n. These groups are characterized by an ''n''-fold proper rotation axis ''C''n. The ''C''1 group is covered in the nonaxial groups section.


Reflection groups (''C''nh)

The reflection groups are denoted by ''C''nh. These groups are characterized by i) an ''n''-fold proper rotation axis ''C''n; ii) a mirror plane ''σh'' normal to ''C''n. The ''C''1''h'' group is the same as the ''C''s group in the nonaxial groups section.


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.


Improper rotation groups (''S''n)

The improper rotation groups are denoted by ''Sn''. These groups are characterized by an ''n''-fold improper rotation axis ''Sn'', where ''n'' is necessarily even. The ''S''2 group is the same as the ''C''i group in the nonaxial groups section. ''Sn'' groups with an odd value of ''n'' are identical to C''n''h groups of same ''n'' and are therefore not considered here (in particular, S1 is identical to Cs). The S8 table reflects the 2007 discovery of errors in older references. Specifically, (''Rx'', ''Ry'') transform not as E1 but rather as E3.


Dihedral symmetries

The families of groups with these symmetries are characterized by 2-fold proper rotation axes normal to a principal rotation axis.


Dihedral groups (''D''n)

The dihedral groups are denoted by ''D''n. 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. The ''D''1 group is the same as the ''C''2 group in the
cyclic groups In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
section.


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.


Antiprismatic groups (''D''nd)

The antiprismatic groups are denoted by ''D''nd. 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) ''n'' mirror planes ''σd'' which contain ''C''n. The ''D''1''d'' group is the same as the ''C''2''h'' group in the
reflection groups Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in s ...
section.


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.


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.


See also

* Linear combination of atomic orbitals (molecular orbital method) *
Raman spectroscopy Raman spectroscopy () (named after Indian physicist C. V. Raman) is a spectroscopic technique typically used to determine vibrational modes of molecules, although rotational and other low-frequency modes of systems may also be observed. Raman sp ...
* Vibrational spectroscopy (molecular vibration) *
List of small groups The following list in mathematics contains the finite groups of small order up to group isomorphism. Counts For ''n'' = 1, 2, … the number of nonisomorphic groups of order ''n'' is : 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, ...
*
Cubic harmonic In fields like computational chemistry and solid-state and condensed matter physics the so-called atomic orbitals, or spin-orbitals, as they appear in textbooks on quantum physics, are often partially replaced by cubic harmonics for a number ...
s


Notes


External links


Character tables for many more point groups
(includes symmetry transformations of Cartesian products up to sixth order)


Further reading

* {{cite book , last = Bunker , first = Philip , author2=Jensen, Per , title = Molecular Symmetry and Spectroscopy, Second edition , publisher = NRC Research Press , year = 2006 , location =
Ottawa Ottawa (, ; Canadian French: ) is the capital city of Canada. It is located at the confluence of the Ottawa River and the Rideau River in the southern portion of the province of Ontario. Ottawa borders Gatineau, Quebec, and forms the core ...
, isbn = 0-660-19628-X Theoretical chemistry Physical chemistry Group theory Finite groups Spectroscopy