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physics_chemistry:point_groups [2018/09/19 19:14] – Maurits W. Haverkortphysics_chemistry:point_groups [2022/11/02 15:04] – Maurits W. Haverkort
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 ====== Point groups ====== ====== Point groups ======
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 ### ###
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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=$D_{\infty h}$|Civ=$C_{\infty v}$|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=$T_{d}$|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=$I_h$|Ohs=$O_h$|HHH=$\sigma_h$?|Css=$C_s$|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=$D_{nh}$|III=$n\, \sigma_v$?|Cnh=$C_{nh}$|CCC=$n \, \sigma_v$?|Cis=$C_{i}$|C1s=$C_1$
 +| | | | | | | | | | | | | YES | | NON | | YES | | NON |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
 +| | | | | | | | | | | | | |!| | | |!| | | |!| | | |!| |
 +| | | | | | | | | | | | | Dnd | | Dns | | Cnv |,| JJJ |.|Dnd=$D_{nd}$|Dns=$D_n$|Cnv=$C_{nv}$|JJJ=$S_{2n}$?
 +| | | | | | | | | | | | | | | | | | | | | | | YES | | NON |NON{border-color:#fff}=No|YES{border-color:#fff}=Yes
 +| | | | | | | | | | | | | | | | | | | | | | | |!| | | |!| |
 +| | | | | | | | | | | | | | | | | | | | | | | S2n | | Cns |S2n=$S_{2n}$|Cns=$C_{n}$
 +</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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