Differences

This shows you the differences between two versions of the page.

--- physics_chemistry:point_groups [2018/09/19 17:28] – Maurits W. Haverkort
+++ physics_chemistry:point_groups [2022/11/02 15:11] (current) – Maurits W. Haverkort
@@ Line 1: / Line 1: @@
 ====== Point groups ======
 ###
 ^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: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> | | | | | |
@@ Line 17: / Line 16: @@
 ###
-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.
 ###
@@ Line 68: / Line 67: @@
 ###
+===== 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 =====

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