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--- physics_chemistry:point_groups [2018/04/04 23:22] – Maurits W. Haverkort
+++ physics_chemistry:point_groups [2022/11/02 15:11] (current) – Maurits W. Haverkort
@@ Line 2: / Line 2: @@
 ###
-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.
+^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.
 ###
@@ Line 52: / Line 66: @@
 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 =====

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