Table of Contents

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

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.

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

A flow diagram to determine the point group
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
$S_{2n}$ $C_{n}$

graph TD; A[A point in space] %%-->%% B["Linear (C∞) ?"]; B %%-->%% |yes|C[i?]; B %%-->%% |no|D[Two or more Cn,
with n>2?]; C %%-->%% |yes|Dinfh(("D∞ h")); C %%-->%% |no|Cinfv(("C∞ v")); D %%-->%% |yes|E[i?]; D %%-->%% |no|F[Cn?]; E %%-->%% |yes|G[C5?]; E %%-->%% |no|Td((Td)); G %%-->%% |yes|Ih((Ih)); G %%-->%% |no|Oh((Oh)); F %%-->%% |yes|H[Are there n C2
perpendicular to Cn
for the largest n?]; F %%-->%% |no|I["σ?"]; H %%-->%% |yes|J["σh?"]; H %%-->%% |no|K["σh?"]; J %%-->%% |yes|Dnh((Dnh)); J %%-->%% |no|L["n σd?"]; L %%-->%% |yes|Dnd((Dnd)); L %%-->%% |no|Dn((Dn)); K %%-->%% |yes|Cnh((Cnh)); K %%-->%% |no|M["n σv?"]; M %%-->%% |yes|Cnv((Cnv)); M %%-->%% |no|N[S2n?]; N %%-->%% |yes|S2n((S2n)); N %%-->%% |no|Cn((Cn)); I %%-->%% |yes|Cs((Cs)); I %%-->%% |no|O[i?]; O %%-->%% |yes|Ci((Ci)); O %%-->%% |no|C1((C1)); click Ci "<a href='https://www.quanty.org/physics_chemistry/point_groups/ci' class='urlextern' title='https://www.quanty.org/physics_chemistry/point_groups/ci' rel='ugc nofollow'>https://www.quanty.org/physics_chemistry/point_groups/ci</a>"; click Cs "<a href='https://www.quanty.org/physics_chemistry/point_groups/cs' class='urlextern' title='https://www.quanty.org/physics_chemistry/point_groups/cs' rel='ugc nofollow'>https://www.quanty.org/physics_chemistry/point_groups/cs</a>"; click C1 "<a href='https://www.quanty.org/physics_chemistry/point_groups/c1' class='urlextern' title='https://www.quanty.org/physics_chemistry/point_groups/c1' rel='ugc nofollow'>https://www.quanty.org/physics_chemistry/point_groups/c1</a>"; click Td "<a href='https://www.quanty.org/physics_chemistry/point_groups/td' class='urlextern' title='https://www.quanty.org/physics_chemistry/point_groups/td' rel='ugc nofollow'>https://www.quanty.org/physics_chemistry/point_groups/td</a>"; click Oh "<a href='https://www.quanty.org/physics_chemistry/point_groups/oh' class='urlextern' title='https://www.quanty.org/physics_chemistry/point_groups/oh' rel='ugc nofollow'>https://www.quanty.org/physics_chemistry/point_groups/oh</a>"; click Ih "<a href='https://www.quanty.org/physics_chemistry/point_groups/ih' class='urlextern' title='https://www.quanty.org/physics_chemistry/point_groups/ih' rel='ugc nofollow'>https://www.quanty.org/physics_chemistry/point_groups/ih</a>"; click Dinfh "<a href='https://www.quanty.org/physics_chemistry/point_groups/dinfh' class='urlextern' title='https://www.quanty.org/physics_chemistry/point_groups/dinfh' rel='ugc nofollow'>https://www.quanty.org/physics_chemistry/point_groups/dinfh</a>"; click Cinfv "<a href='https://www.quanty.org/physics_chemistry/point_groups/cinfv' class='urlextern' title='https://www.quanty.org/physics_chemistry/point_groups/cinfv' rel='ugc nofollow'>https://www.quanty.org/physics_chemistry/point_groups/cinfv</a>";

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

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