| Nonaxial groups | C1 - $1$ | Cs - $m$ | Ci - $\bar{1}$ | ||||
|---|---|---|---|---|---|---|---|
| Cn groups | C2 - $2$ | C3 - $3$ | C4 - $4$ | C5 - $5$ | C6 - $6$ | C7 - $7$ | C8 - $8$ |
| Dn groups | D2 - $222$ | D3 - $32$ | D4 - $422$ | D5 - $52$ | D6 - $622$ | D7 - $722$ | D8 - $822$ |
| Cnv groups | C2v - $mm2$ | C3v - $3m$ | C4v - $4mm$ | C5v - $5m$ | C6v - $6mm$ | C7v - $7m$ | C8v - $8mm$ |
| Cnh groups | C2h - $2/m$ | C3h - $\bar{6}$ | C4h - $4/m$ | C5h - $\bar{10}$ | C6h - $6/m$ | ||
| Dnh groups | D2h - $mmm$ | D3h - $\bar{6}m2$ | D4h - $4/mmm$ | D5h - $\bar{10}m2$ | D6h - $6/mmm$ | D7h - $\bar{14}m2$ | D8h - $8/mmm$ |
| Dnd groups | D2d - $\bar{4}2m$ | D3d - $\bar{3}m$ | D4d - $\bar{8}2m$ | D5d - $\bar{5}m$ | D6d - $\bar{12}2m$ | D7d - $\bar{7}m$ | D8d - $\bar{16}2m$ |
| Sn groups | S2 - $\bar{2}$ | S4 - $\bar{4}$ | S6 - $\bar{6}$ | S8 - $\bar{8}$ | S10 - $\bar{10}$ | S12 - $\bar{12}$ | |
| Cubic groups | T - $23$ | Th - $m\bar{3}$ | Td - $\bar{4}3m$ | O - $432$ | Oh - $m\bar{3}m$ | I - $532$ | Ih - $\bar{5}\bar{3}m$ |
| Linear groups | C$\infty$v | D$\infty$h |
The following set of pages list properties of the different point groups and their irreducible representations. The table above links to the main page of each of the different point group. For each group we list the character and product table. Often one needs to answer the question how a potential in a given point group looks like and what the eigen-states of that potential are. The eigen states can be grouped according to the irreducible representations of the group and for each of these representations one can give representing functions. The form of these functions and the potential however do depend on the orientation of the point group. We therefor list for each point group different orientations.
As we are interested in explicit representations we do need to specify the orientation of the symmetry operators. This results in several tables for the same point group but with different choices for the symmetry operations. For example the cubic $O_h$ point group can be represented with the $C_4$ axes in the $x$, $y$ and $z$ direction, or with a $C_3$ axis in the $z$ direction. We list several orientations of the different point-groups available.
We use the following notation for symmetry operations.
E = identity
Cn = n-fold rotation
Sn = n-fold rotation plus reflection through a plane perpendicular to the axis of rotation
i = inversion through a centre of symmetry
$\sigma$v = reflection through a mirror plane (called “vertical”) parallel to the principal axis
$\sigma$h = reflection through a mirror plane (called “horizontal”) perpendicular to the principal axis
$\sigma$d = reflection through a vertical mirror plane bisecting the angle between two C2 axes
We use the following notation for the irreducible representations.
A = one-dimensional irreducible representation with character +1 under the principal rotation
B = one-dimensional irreducible representation with character -1 under the principal rotation
E = two-dimensional irreducible representation
T = three-dimensional irreducible representation
Point groups with inversion symmetry are separated into even (g) and odd (u) irreducible representations
| A point in space | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Linear ($C_{\infty}$)? | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Yes | No | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inversion ($i$)? | Two or more $C_n$ with $n>2$? |
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Yes | No | Yes | No | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $D_{\infty h}$ | $C_{\infty v}$ | Inversion ($i$)? | $C_n$? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Yes | No | Yes | No | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $C_5$? | $T_{d}$ | $n \, C_2 \, \perp \, C_n$? | $\sigma_h$? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Yes | No | Yes | No | Yes | No | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $I_h$ | $O_h$ | $\sigma_h$? | $\sigma_h$? | $C_s$ | Inversion ($i$)? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Yes | No | Yes | No | Yes | No | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $D_{nh}$ | $n\, \sigma_v$? | $C_{nh}$ | $n \, \sigma_v$? | $C_{i}$ | $C_1$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Yes | No | Yes | No | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $D_{nd}$ | $D_n$ | $C_{nv}$ | $S_{2n}$? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Yes | No | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $S_{2n}$ | $C_{n}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
These pages and tables on point groups are generated from a small code written in Quanty and Mathematica developed and tested by Maurits W. Haverkort, Vincent Vercamer and Stefano Agrestini.
| Nonaxial groups | C1 | Cs | Ci | ||||
|---|---|---|---|---|---|---|---|
| Cn groups | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
| Dn groups | D2 | D3 | D4 | D5 | D6 | D7 | D8 |
| Cnv groups | C2v | C3v | C4v | C5v | C6v | C7v | C8v |
| Cnh groups | C2h | C3h | C4h | C5h | C6h | ||
| Dnh groups | D2h | D3h | D4h | D5h | D6h | D7h | D8h |
| Dnd groups | D2d | D3d | D4d | D5d | D6d | D7d | D8d |
| Sn groups | S2 | S4 | S6 | S8 | S10 | S12 | |
| Cubic groups | T | Th | Td | O | Oh | I | Ih |
| Linear groups | C$\infty$v | D$\infty$h |