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--- physics_chemistry:point_groups [2018/03/23 18:06] – Stefano Agrestini
+++ physics_chemistry:point_groups [2022/11/02 15:11] (current) – Maurits W. Haverkort
@@ Line 2: / Line 2: @@
 ###
-alligned paragraph text
+^Nonaxial groups      | [[physics_chemistry:point_groups:c1|C]]<sub>[[physics_chemistry:point_groups:c1|1]]</sub> - [[physics_chemistry:point_groups:c1|$1$]] | [[physics_chemistry:point_groups:cs|C]]<sub>[[physics_chemistry:point_groups:cs|s]]</sub> - [[physics_chemistry:point_groups:cs|$m$]] | [[physics_chemistry:point_groups:ci|C]]<sub>[[physics_chemistry:point_groups:ci|i]]</sub> - [[physics_chemistry:point_groups:ci|$\bar{1}$]] | | | | |
+^C<sub>n</sub> groups | [[physics_chemistry:point_groups:c2|C]]<sub>[[physics_chemistry:point_groups:c2|2]]</sub> - [[physics_chemistry:point_groups:c2|$2$]] | [[physics_chemistry:point_groups:c3|C]]<sub>[[physics_chemistry:point_groups:c3|3]]</sub> - [[physics_chemistry:point_groups:c3|$3$]] | [[physics_chemistry:point_groups:c4|C]]<sub>[[physics_chemistry:point_groups:c4|4]]</sub> - [[physics_chemistry:point_groups:c4|$4$]] | [[physics_chemistry:point_groups:c5|C]]<sub>[[physics_chemistry:point_groups:c5|5]]</sub> - [[physics_chemistry:point_groups:c5|$5$]] | [[physics_chemistry:point_groups:c6|C]]<sub>[[physics_chemistry:point_groups:c6|6]]</sub> - [[physics_chemistry:point_groups:c6|$6$]] | [[physics_chemistry:point_groups:c7|C]]<sub>[[physics_chemistry:point_groups:c7|7]]</sub> - [[physics_chemistry:point_groups:c7|$7$]] | [[physics_chemistry:point_groups:c8|C]]<sub>[[physics_chemistry:point_groups:c8|8]]</sub> - [[physics_chemistry:point_groups:c8|$8$]] |
+^D<sub>n</sub> groups | [[physics_chemistry:point_groups:d2|D]]<sub>[[physics_chemistry:point_groups:d2|2]]</sub> - [[physics_chemistry:point_groups:d2|$222$]] | [[physics_chemistry:point_groups:d3|D]]<sub>[[physics_chemistry:point_groups:d3|3]]</sub>  - [[physics_chemistry:point_groups:d3|$32$]] | [[physics_chemistry:point_groups:d4|D]]<sub>[[physics_chemistry:point_groups:d4|4]]</sub>  - [[physics_chemistry:point_groups:d4|$422$]] | [[physics_chemistry:point_groups:d5|D]]<sub>[[physics_chemistry:point_groups:d5|5]]</sub>  - [[physics_chemistry:point_groups:d5|$52$]] | [[physics_chemistry:point_groups:d6|D]]<sub>[[physics_chemistry:point_groups:d6|6]]</sub>  - [[physics_chemistry:point_groups:d6|$622$]] | [[physics_chemistry:point_groups:d7|D]]<sub>[[physics_chemistry:point_groups:d7|7]]</sub>  - [[physics_chemistry:point_groups:d7|$722$]] | [[physics_chemistry:point_groups:d8|D]]<sub>[[physics_chemistry:point_groups:d8|8]]</sub>  - [[physics_chemistry:point_groups:d8|$822$]] |
+^C<sub>nv</sub> groups | [[physics_chemistry:point_groups:c2v|C]]<sub>[[physics_chemistry:point_groups:c2v|2v]]</sub> - [[physics_chemistry:point_groups:c2v|$mm2$]] | [[physics_chemistry:point_groups:c3v|C]]<sub>[[physics_chemistry:point_groups:c3v|3v]]</sub> - [[physics_chemistry:point_groups:c3v|$3m$]] | [[physics_chemistry:point_groups:c4v|C]]<sub>[[physics_chemistry:point_groups:c4v|4v]]</sub> - [[physics_chemistry:point_groups:c4v|$4mm$]] | [[physics_chemistry:point_groups:c5v|C]]<sub>[[physics_chemistry:point_groups:c5v|5v]]</sub> - [[physics_chemistry:point_groups:c5v|$5m$]] | [[physics_chemistry:point_groups:c6v|C]]<sub>[[physics_chemistry:point_groups:c6v|6v]]</sub> - [[physics_chemistry:point_groups:c6v|$6mm$]] | [[physics_chemistry:point_groups:c7v|C]]<sub>[[physics_chemistry:point_groups:c7v|7v]]</sub> - [[physics_chemistry:point_groups:c7v|$7m$]] | [[physics_chemistry:point_groups:c8v|C]]<sub>[[physics_chemistry:point_groups:c8v|8v]]</sub> - [[physics_chemistry:point_groups:c8v|$8mm$]] |
+^C<sub>nh</sub> groups | [[physics_chemistry:point_groups:c2h|C]]<sub>[[physics_chemistry:point_groups:c2h|2h]]</sub> - [[physics_chemistry:point_groups:c2h|$2/m$]] | [[physics_chemistry:point_groups:c3h|C]]<sub>[[physics_chemistry:point_groups:c3h|3h]]</sub> - [[physics_chemistry:point_groups:c3h|$\bar{6}$]] | [[physics_chemistry:point_groups:c4h|C]]<sub>[[physics_chemistry:point_groups:c4h|4h]]</sub> - [[physics_chemistry:point_groups:c4h|$4/m$]] | [[physics_chemistry:point_groups:c5h|C]]<sub>[[physics_chemistry:point_groups:c5h|5h]]</sub> - [[physics_chemistry:point_groups:c5h|$\bar{10}$]] | [[physics_chemistry:point_groups:c6h|C]]<sub>[[physics_chemistry:point_groups:c6h|6h]]</sub> - [[physics_chemistry:point_groups:c6h|$6/m$]] | | |
+^D<sub>nh</sub> groups | [[physics_chemistry:point_groups:d2h|D]]<sub>[[physics_chemistry:point_groups:d2h|2h]]</sub> - [[physics_chemistry:point_groups:d2h|$mmm$]] | [[physics_chemistry:point_groups:d3h|D]]<sub>[[physics_chemistry:point_groups:d3h|3h]]</sub> - [[physics_chemistry:point_groups:d3h|$\bar{6}m2$]] | [[physics_chemistry:point_groups:d4h|D]]<sub>[[physics_chemistry:point_groups:d4h|4h]]</sub> - [[physics_chemistry:point_groups:d4h|$4/mmm$]] | [[physics_chemistry:point_groups:d5h|D]]<sub>[[physics_chemistry:point_groups:d5h|5h]]</sub> - [[physics_chemistry:point_groups:d5h|$\bar{10}m2$]] | [[physics_chemistry:point_groups:d6h|D]]<sub>[[physics_chemistry:point_groups:d6h|6h]]</sub> - [[physics_chemistry:point_groups:d6h|$6/mmm$]] | [[physics_chemistry:point_groups:d7h|D]]<sub>[[physics_chemistry:point_groups:d7h|7h]]</sub> - [[physics_chemistry:point_groups:d7h|$\bar{14}m2$]] | [[physics_chemistry:point_groups:d8h|D]]<sub>[[physics_chemistry:point_groups:d8h|8h]]</sub> - [[physics_chemistry:point_groups:d8h|$8/mmm$]] |
+^D<sub>nd</sub> groups | [[physics_chemistry:point_groups:d2d|D]]<sub>[[physics_chemistry:point_groups:d2d|2d]]</sub> - [[physics_chemistry:point_groups:d2d|$\bar{4}2m$]] | [[physics_chemistry:point_groups:d3d|D]]<sub>[[physics_chemistry:point_groups:d3d|3d]]</sub> - [[physics_chemistry:point_groups:d3d|$\bar{3}m$]] | [[physics_chemistry:point_groups:d4d|D]]<sub>[[physics_chemistry:point_groups:d4d|4d]]</sub> - [[physics_chemistry:point_groups:d4d|$\bar{8}2m$]] | [[physics_chemistry:point_groups:d5d|D]]<sub>[[physics_chemistry:point_groups:d5d|5d]]</sub> - [[physics_chemistry:point_groups:d5d|$\bar{5}m$]] | [[physics_chemistry:point_groups:d6d|D]]<sub>[[physics_chemistry:point_groups:d6d|6d]]</sub> - [[physics_chemistry:point_groups:d6d|$\bar{12}2m$]] | [[physics_chemistry:point_groups:d7d|D]]<sub>[[physics_chemistry:point_groups:d7d|7d]]</sub> - [[physics_chemistry:point_groups:d7d|$\bar{7}m$]] | [[physics_chemistry:point_groups:d8d|D]]<sub>[[physics_chemistry:point_groups:d8d|8d]]</sub> - [[physics_chemistry:point_groups:d8d|$\bar{16}2m$]] |
+^S<sub>n</sub> groups | [[physics_chemistry:point_groups:S2|S]]<sub>[[physics_chemistry:point_groups:S2|2]]</sub> - [[physics_chemistry:point_groups:S2|$\bar{2}$]] | [[physics_chemistry:point_groups:S4|S]]<sub>[[physics_chemistry:point_groups:S4|4]]</sub> - [[physics_chemistry:point_groups:S4|$\bar{4}$]] | [[physics_chemistry:point_groups:S6|S]]<sub>[[physics_chemistry:point_groups:S6|6]]</sub> - [[physics_chemistry:point_groups:S6|$\bar{6}$]] | [[physics_chemistry:point_groups:S8|S]]<sub>[[physics_chemistry:point_groups:S8|8]]</sub> - [[physics_chemistry:point_groups:S8|$\bar{8}$]] | [[physics_chemistry:point_groups:S10|S]]<sub>[[physics_chemistry:point_groups:S10|10]]</sub> - [[physics_chemistry:point_groups:S10|$\bar{10}$]] | [[physics_chemistry:point_groups:S12|S]]<sub>[[physics_chemistry:point_groups:S12|12]]</sub> - [[physics_chemistry:point_groups:S12|$\bar{12}$]] |  |
+^Cubic groups | [[physics_chemistry:point_groups:T|T]] - [[physics_chemistry:point_groups:T|$23$]] | [[physics_chemistry:point_groups:Th|T]]<sub>[[physics_chemistry:point_groups:Th|h]]</sub> - [[physics_chemistry:point_groups:Th|$m\bar{3}$]] | [[physics_chemistry:point_groups:Td|T]]<sub>[[physics_chemistry:point_groups:Td|d]]</sub> - [[physics_chemistry:point_groups:Td|$\bar{4}3m$]] | [[physics_chemistry:point_groups:O|O]] - [[physics_chemistry:point_groups:O|$432$]] | [[physics_chemistry:point_groups:Oh|O]]<sub>[[physics_chemistry:point_groups:Oh|h]]</sub> - [[physics_chemistry:point_groups:Oh|$m\bar{3}m$]] | [[physics_chemistry:point_groups:I|I]] - [[physics_chemistry:point_groups:I|$532$]] | [[physics_chemistry:point_groups:Ih|I]]<sub>[[physics_chemistry:point_groups:Ih|h]]</sub> - [[physics_chemistry:point_groups:Ih|$\bar{5}\bar{3}m$]] |
+^Linear groups      | [[physics_chemistry:point_groups:cinfv|C]]<sub>[[physics_chemistry:point_groups:cinfv|$\infty$v]]</sub> | [[physics_chemistry:point_groups:cinfv|D]]<sub>[[physics_chemistry:point_groups:dinfh|$\infty$h]]</sub> | | | | | |
+###
+###
+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.
+###
 ###
@@ Line 27: / Line 52: @@
 ###
-alligned paragraph text
+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 =====
+<diagram>
+| | | | | | | | | AAA | | | | | | |AAA=A point in space
+| | | | | | | | | |!| | | | | | | |
+| | | |,|-|-|-|-| BBB |-|-|-|-|.| | | |BBB=Linear ($C_{\infty}$)?
+| | | YES | | | | | | | | | | NON | | |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
+| | | |!| | | | | | | | | | | |!| | | |
+| |,| CCC |.| | | |,|-|-|-|-| DDD |-|-|-|-|-|.| |CCC=Inversion ($i$)?|DDD=Two or more $C_n$ \\ with $n>2$?
+| YES | | NON | | YES | | | | | | | | | | | NON |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
+| |!| | | |!| | | |!| | | | | | | | | | | | |!| |
+| Dih | | Civ |,| CCC |.| | | | | |,|-|-|-| GGG |-|-|-|.| |Dih{border-color:800;background-color:#faa}=$D_{\infty h}$|Civ{border-color:800;background-color:#faa}=$C_{\infty v}$|CCC=Inversion ($i$)?|GGG=$C_n$?
+| | | | | | | YES | | NON | | | | YES | | | | | | | | NON |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
+| | | | | | | |!| | | |!| | | | | |!| | | | | | | | | |!| |
+| | | | | |,| EEE |.| Tds |,|-|-| FFF |-|-|.| | | |,| HHH |.| |EEE=$C_5$?|Tds{border-color:800;background-color:#faa}=$T_{d}$|FFF=$n \, C_2 \, \perp \, C_n$?|HHH=$\sigma_h$?
+| | | | | YES | | NON | | YES | | | | | | NON | | YES | | NON |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
+| | | | | |!| | | |!| | | |!| | | | | | | |!| | | |!| | | |!| |
+| | | | | Ihs | | Ohs |,| HHH |.| | | |,| HHH |.| Css |,| CCC |.| |Ihs{border-color:800;background-color:#faa}=$I_h$|Ohs{border-color:800;background-color:#faa}=$O_h$|HHH=$\sigma_h$?|Css{border-color:800;background-color:#faa}=$C_s$|CCC=Inversion ($i$)?
+| | | | | | | | | | | YES | | NON | | YES | | NON | | YES | | NON |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
+| | | | | | | | | | | |!| | | |!| | | |!| | | |!| | | |!| | | |!| |
+| | | | | | | | | | | Dnh |,| III |.| Cnh |,| CCC |.| Cis | | C1s |Dnh{border-color:800;background-color:#faa}=$D_{nh}$|III=$n\, \sigma_v$?|Cnh{border-color:800;background-color:#faa}=$C_{nh}$|CCC=$n \, \sigma_v$?|Cis{border-color:800;background-color:#faa}=$C_{i}$|C1s{border-color:800;background-color:#faa}=$C_1$
+| | | | | | | | | | | | | YES | | NON | | YES | | NON |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
+| | | | | | | | | | | | | |!| | | |!| | | |!| | | |!| |
+| | | | | | | | | | | | | Dnd | | Dns | | Cnv |,| JJJ |.|Dnd{border-color:800;background-color:#faa}=$D_{nd}$|Dns{border-color:800;background-color:#faa}=$D_n$|Cnv{border-color:800;background-color:#faa}=$C_{nv}$|JJJ=$S_{2n}$?
+| | | | | | | | | | | | | | | | | | | | | | | YES | | NON |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
+| | | | | | | | | | | | | | | | | | | | | | | |!| | | |!| |
+| | | | | | | | | | | | | | | | | | | | | | | S2n | | Cns |S2n{border-color:800;background-color:#faa}=$S_{2n}$|Cns{border-color:800;background-color:#faa}=$C_{n}$
+</diagram>
+<flow>
+graph TD;
+    A[A point in space] %%-->%% B["Linear (C<sub>&infin;</sub>) ?"];
+    B %%-->%% |yes|C[i?];
+    B %%-->%% |no|D[Two or more C<sub>n</sub>,<br> with n>2?];
+    C %%-->%% |yes|Dinfh(("D<sub>&infin; h</sub>"));
+    C %%-->%% |no|Cinfv(("C<sub>&infin; v</sub>"));
+    D %%-->%% |yes|E[i?];
+    D %%-->%% |no|F[C<sub>n</sub>?];
+    E %%-->%% |yes|G[C<sub>5</sub>?];
+    E %%-->%% |no|Td((T<sub>d</sub>));
+    G %%-->%% |yes|Ih((I<sub>h</sub>));
+    G %%-->%% |no|Oh((O<sub>h</sub>));
+    F %%-->%% |yes|H[Are there n C<sub>2</sub> <br> perpendicular to C<sub>n</sub> <br> for the largest n?];
+    F %%-->%% |no|I["&sigma;?"];
+    H %%-->%% |yes|J["&sigma;<sub>h</sub>?"];
+    H %%-->%% |no|K["&sigma;<sub>h</sub>?"];
+    J %%-->%% |yes|Dnh((D<sub>nh</sub>));
+    J %%-->%% |no|L["n &sigma;<sub>d</sub>?"];
+    L %%-->%% |yes|Dnd((D<sub>nd</sub>));
+    L %%-->%% |no|Dn((D<sub>n</sub>));
+    K %%-->%% |yes|Cnh((C<sub>nh</sub>));
+    K %%-->%% |no|M["n &sigma;<sub>v</sub>?"];
+    M %%-->%% |yes|Cnv((C<sub>nv</sub>));
+    M %%-->%% |no|N[S<sub>2n</sub>?];
+    N %%-->%% |yes|S2n((S<sub>2n</sub>));
+    N %%-->%% |no|Cn((C<sub>n</sub>));
+    I %%-->%% |yes|Cs((C<sub>s</sub>));
+    I %%-->%% |no|O[i?];
+    O %%-->%% |yes|Ci((C<sub>i</sub>));
+    O %%-->%% |no|C1((C<sub>1</sub>));
+    click Ci "https://www.quanty.org/physics_chemistry/point_groups/ci";
+    click Cs "https://www.quanty.org/physics_chemistry/point_groups/cs";
+    click C1 "https://www.quanty.org/physics_chemistry/point_groups/c1";
+    click Td "https://www.quanty.org/physics_chemistry/point_groups/td";
+    click Oh "https://www.quanty.org/physics_chemistry/point_groups/oh";
+    click Ih "https://www.quanty.org/physics_chemistry/point_groups/ih";
+    click Dinfh "https://www.quanty.org/physics_chemistry/point_groups/dinfh";
+    click Cinfv "https://www.quanty.org/physics_chemistry/point_groups/cinfv";
+</flow>
+===== Acknowledgements =====
+These pages and tables on point groups are generated from a small code written in Quanty and Mathematica developed and tested by [[http://www.thphys.uni-heidelberg.de/~haverkort/|Maurits W. Haverkort]], [[https://tel.archives-ouvertes.fr/tel-01458771|Vincent Vercamer]] and [[https://www.cpfs.mpg.de/person/29151/1675485| Stefano Agrestini]].
 ===== Table of several point groups =====
@@ Line 43: / Line 157: @@
 ^Cubic groups | [[physics_chemistry:point_groups:T|T]] | [[physics_chemistry:point_groups:Th|T]]<sub>[[physics_chemistry:point_groups:Th|h]]</sub> | [[physics_chemistry:point_groups:Td|T]]<sub>[[physics_chemistry:point_groups:Td|d]]</sub> | [[physics_chemistry:point_groups:O|O]] | [[physics_chemistry:point_groups:Oh|O]]<sub>[[physics_chemistry:point_groups:Oh|h]]</sub> | [[physics_chemistry:point_groups:I|I]] | [[physics_chemistry:point_groups:Ih|I]]<sub>[[physics_chemistry:point_groups:Ih|h]]</sub> |
 ^Linear groups      | [[physics_chemistry:point_groups:cinfv|C]]<sub>[[physics_chemistry:point_groups:cinfv|$\infty$v]]</sub> | [[physics_chemistry:point_groups:cinfv|D]]<sub>[[physics_chemistry:point_groups:dinfh|$\infty$h]]</sub> | | | | | |
-###
-===== Different orientations =====
-###
-bla
 ###
@@ Line 55: / Line 162: @@
 ===== Table of contents =====
-{{indexmenu>.#3}}
+{{indexmenu>.#1}}

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