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physics_chemistry:point_groups [2018/04/05 09:34] – Maurits W. Haverkortphysics_chemistry:point_groups [2022/11/02 15:11] (current) – Maurits W. Haverkort
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 ====== Point groups ====== ====== Point groups ======
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-^Nonaxial groups      | [[physics_chemistry:point_groups:c1|C]]<sub>[[physics_chemistry:point_groups:c1|1]]</sub> | [[physics_chemistry:point_groups:cs|C]]<sub>[[physics_chemistry:point_groups:cs|s]]</sub> | [[physics_chemistry:point_groups:ci|C]]<sub>[[physics_chemistry:point_groups:ci|i]]</sub> | | | | | +^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:c3|C]]<sub>[[physics_chemistry:point_groups:c3|3]]</sub> | [[physics_chemistry:point_groups:c4|C]]<sub>[[physics_chemistry:point_groups:c4|4]]</sub> | [[physics_chemistry:point_groups:c5|C]]<sub>[[physics_chemistry:point_groups:c5|5]]</sub> | [[physics_chemistry:point_groups:c6|C]]<sub>[[physics_chemistry:point_groups:c6|6]]</sub> | [[physics_chemistry:point_groups:c7|C]]<sub>[[physics_chemistry:point_groups:c7|7]]</sub> | [[physics_chemistry:point_groups:c8|C]]<sub>[[physics_chemistry:point_groups:c8|8]]</sub> |  +^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:d3|D]]<sub>[[physics_chemistry:point_groups:d3|3]]</sub> | [[physics_chemistry:point_groups:d4|D]]<sub>[[physics_chemistry:point_groups:d4|4]]</sub> | [[physics_chemistry:point_groups:d5|D]]<sub>[[physics_chemistry:point_groups:d5|5]]</sub> | [[physics_chemistry:point_groups:d6|D]]<sub>[[physics_chemistry:point_groups:d6|6]]</sub> | [[physics_chemistry:point_groups:d7|D]]<sub>[[physics_chemistry:point_groups:d7|7]]</sub> | [[physics_chemistry:point_groups:d8|D]]<sub>[[physics_chemistry:point_groups:d8|8]]</sub> |  +^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:c3v|C]]<sub>[[physics_chemistry:point_groups:c3v|3v]]</sub> | [[physics_chemistry:point_groups:c4v|C]]<sub>[[physics_chemistry:point_groups:c4v|4v]]</sub> | [[physics_chemistry:point_groups:c5v|C]]<sub>[[physics_chemistry:point_groups:c5v|5v]]</sub> | [[physics_chemistry:point_groups:c6v|C]]<sub>[[physics_chemistry:point_groups:c6v|6v]]</sub> | [[physics_chemistry:point_groups:c7v|C]]<sub>[[physics_chemistry:point_groups:c7v|7v]]</sub> | [[physics_chemistry:point_groups:c8v|C]]<sub>[[physics_chemistry:point_groups:c8v|8v]]</sub> |  +^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:c3h|C]]<sub>[[physics_chemistry:point_groups:c3h|3h]]</sub> | [[physics_chemistry:point_groups:c4h|C]]<sub>[[physics_chemistry:point_groups:c4h|4h]]</sub> | [[physics_chemistry:point_groups:c5h|C]]<sub>[[physics_chemistry:point_groups:c5h|5h]]</sub> | [[physics_chemistry:point_groups:c6h|C]]<sub>[[physics_chemistry:point_groups:c6h|6h]]</sub> | | |  +^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:d3h|D]]<sub>[[physics_chemistry:point_groups:d3h|3h]]</sub> | [[physics_chemistry:point_groups:d4h|D]]<sub>[[physics_chemistry:point_groups:d4h|4h]]</sub> | [[physics_chemistry:point_groups:d5h|D]]<sub>[[physics_chemistry:point_groups:d5h|5h]]</sub> | [[physics_chemistry:point_groups:d6h|D]]<sub>[[physics_chemistry:point_groups:d6h|6h]]</sub> | [[physics_chemistry:point_groups:d7h|D]]<sub>[[physics_chemistry:point_groups:d7h|7h]]</sub> | [[physics_chemistry:point_groups:d8h|D]]<sub>[[physics_chemistry:point_groups:d8h|8h]]</sub> |  +^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:d3d|D]]<sub>[[physics_chemistry:point_groups:d3d|3d]]</sub> | [[physics_chemistry:point_groups:d4d|D]]<sub>[[physics_chemistry:point_groups:d4d|4d]]</sub> | [[physics_chemistry:point_groups:d5d|D]]<sub>[[physics_chemistry:point_groups:d5d|5d]]</sub> | [[physics_chemistry:point_groups:d6d|D]]<sub>[[physics_chemistry:point_groups:d6d|6d]]</sub> | [[physics_chemistry:point_groups:d7d|D]]<sub>[[physics_chemistry:point_groups:d7d|7d]]</sub> | [[physics_chemistry:point_groups:d8d|D]]<sub>[[physics_chemistry:point_groups:d8d|8d]]</sub> |  +^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:S4|S]]<sub>[[physics_chemistry:point_groups:S4|4]]</sub> | [[physics_chemistry:point_groups:S6|S]]<sub>[[physics_chemistry:point_groups:S6|6]]</sub> | [[physics_chemistry:point_groups:S8|S]]<sub>[[physics_chemistry:point_groups:S8|8]]</sub> | [[physics_chemistry:point_groups:S10|S]]<sub>[[physics_chemistry:point_groups:S10|10]]</sub> | [[physics_chemistry:point_groups:S12|S]]<sub>[[physics_chemistry:point_groups:S12|12]]</sub> |  |  +^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: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> | +^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> | | | | | | ^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> | | | | | |
  
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-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.+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.
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 +===== 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 ===== ===== Acknowledgements =====

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