Table of Contents
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Point groups
| 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 |
There are several good websites listing the point-groups and character tables http://gernot-katzers-spice-pages.com/character_tables/ or http://www.cryst.ehu.es/cryst/get_point_genpos.html for example. So why do we add another page on point groups? The question one often needs to answer is how does my Hamiltonian that is represented by a potential look like in a given symmetry. Where most pages list which angular momenta $l$ are allowed, i.e. contain an $a_1$ representation they generally do not list the specific form of the allowed function. We here present tables that explicitly list the symmetric representation and present forms that can be used in Quanty.
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.
Symmetry operations
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
Irreducible representations
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
Acknowledgements
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.
Table of several point groups
| 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 |
