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--- physics_chemistry:point_groups:oh [2018/03/21 18:47] – created Stefano Agrestini
+++ physics_chemistry:point_groups:oh [2018/03/23 01:33] (current) – Maurits W. Haverkort
@@ Line 1: / Line 1: @@
-====== Oh ======
+====== Point Group Oh ======
+===== Character Table =====
 ###
-alligned paragraph text
+|    ^  $ \text{E} \,{\text{(1)}}  $^$  C_3 \,{\text{(8)}}  $^$  C_2 \,{\text{(6)}}  $^$  C_4 \,{\text{(6)}}  $^$  C_2 \,{\text{(3)}}  $^$  \text{i} \,{\text{(1)}}  $^$  S_4 \,{\text{(6)}}  $^$  S_6 \,{\text{(8)}}  $^$  \sigma_h \,{\text{(3)}}  $^$  \sigma_d \,{\text{(6)}} $  ^
+^  $A_{1g}$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |
+^  $A_{2g}$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |   $1$  |   $-1$  |   $1$  |   $1$  |   $-1$  |
+^  $E_g$  |   $2$  |   $-1$  |   $0$  |   $0$  |   $2$  |   $2$  |   $0$  |   $-1$  |   $2$  |   $0$  |
+^  $T_{1g}$  |   $3$  |   $0$  |   $-1$  |   $1$  |   $-1$  |   $3$  |   $1$  |   $0$  |   $-1$  |   $-1$  |
+^  $T_{2g}$  |   $3$  |   $0$  |   $1$  |   $-1$  |   $-1$  |   $3$  |   $-1$  |   $0$  |   $-1$  |   $1$  |
+^  $A_{1u}$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |
+^  $A_{2u}$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |   $-1$  |   $1$  |   $-1$  |   $-1$  |   $1$  |
+^  $E_u$  |   $2$  |   $-1$  |   $0$  |   $0$  |   $2$  |   $-2$  |   $0$  |   $1$  |   $-2$  |   $0$  |
+^  $T_{1u}$  |   $3$  |   $0$  |   $-1$  |   $1$  |   $-1$  |   $-3$  |   $-1$  |   $0$  |   $1$  |   $1$  |
+^  $T_{2u}$  |   $3$  |   $0$  |   $1$  |   $-1$  |   $-1$  |   $-3$  |   $1$  |   $0$  |   $1$  |   $-1$  |
 ###
-===== Example =====
+===== Product Table =====
 ###
-description text
+|    ^   $A_{1g}$   ^   $A_{2g}$   ^   $E_g$   ^   $T_{1g}$   ^   $T_{2g}$   ^   $A_{1u}$   ^   $A_{2u}$   ^   $E_u$   ^   $T_{1u}$   ^   $T_{2u}$   ^
+^  $A_{1g}$   |  $A_{1g}$   |  $A_{2g}$   |  $E_g$   |  $T_{1g}$   |  $T_{2g}$   |  $A_{1u}$   |  $A_{2u}$   |  $E_u$   |  $T_{1u}$   |  $T_{2u}$   |
+^  $A_{2g}$   |  $A_{2g}$   |  $A_{1g}$   |  $E_g$   |  $T_{2g}$   |  $T_{1g}$   |  $A_{2u}$   |  $A_{1u}$   |  $E_u$   |  $T_{2u}$   |  $T_{1u}$   |
+^  $E_g$   |  $E_g$   |  $E_g$   |  $A_{1g}+A_{2g}+E_g$   |  $T_{1g}+T_{2g}$   |  $T_{1g}+T_{2g}$   |  $E_u$   |  $E_u$   |  $A_{1u}+A_{2u}+E_u$   |  $T_{1u}+T_{2u}$   |  $T_{1u}+T_{2u}$   |
+^  $T_{1g}$   |  $T_{1g}$   |  $T_{2g}$   |  $T_{1g}+T_{2g}$   |  $A_{1g}+E_g+T_{1g}+T_{2g}$   |  $A_{2g}+E_g+T_{1g}+T_{2g}$   |  $T_{1u}$   |  $T_{2u}$   |  $T_{1u}+T_{2u}$   |  $A_{1u}+E_u+T_{1u}+T_{2u}$   |  $A_{2u}+E_u+T_{1u}+T_{2u}$   |
+^  $T_{2g}$   |  $T_{2g}$   |  $T_{1g}$   |  $T_{1g}+T_{2g}$   |  $A_{2g}+E_g+T_{1g}+T_{2g}$   |  $A_{1g}+E_g+T_{1g}+T_{2g}$   |  $T_{2u}$   |  $T_{1u}$   |  $T_{1u}+T_{2u}$   |  $A_{2u}+E_u+T_{1u}+T_{2u}$   |  $A_{1u}+E_u+T_{1u}+T_{2u}$   |
+^  $A_{1u}$   |  $A_{1u}$   |  $A_{2u}$   |  $E_u$   |  $T_{1u}$   |  $T_{2u}$   |  $A_{1g}$   |  $A_{2g}$   |  $E_g$   |  $T_{1g}$   |  $T_{2g}$   |
+^  $A_{2u}$   |  $A_{2u}$   |  $A_{1u}$   |  $E_u$   |  $T_{2u}$   |  $T_{1u}$   |  $A_{2g}$   |  $A_{1g}$   |  $E_g$   |  $T_{2g}$   |  $T_{1g}$   |
+^  $E_u$   |  $E_u$   |  $E_u$   |  $A_{1u}+A_{2u}+E_u$   |  $T_{1u}+T_{2u}$   |  $T_{1u}+T_{2u}$   |  $E_g$   |  $E_g$   |  $A_{1g}+A_{2g}+E_g$   |  $T_{1g}+T_{2g}$   |  $T_{1g}+T_{2g}$   |
+^  $T_{1u}$   |  $T_{1u}$   |  $T_{2u}$   |  $T_{1u}+T_{2u}$   |  $A_{1u}+E_u+T_{1u}+T_{2u}$   |  $A_{2u}+E_u+T_{1u}+T_{2u}$   |  $T_{1g}$   |  $T_{2g}$   |  $T_{1g}+T_{2g}$   |  $A_{1g}+E_g+T_{1g}+T_{2g}$   |  $A_{2g}+E_g+T_{1g}+T_{2g}$   |
+^  $T_{2u}$   |  $T_{2u}$   |  $T_{1u}$   |  $T_{1u}+T_{2u}$   |  $A_{2u}+E_u+T_{1u}+T_{2u}$   |  $A_{1u}+E_u+T_{1u}+T_{2u}$   |  $T_{2g}$   |  $T_{1g}$   |  $T_{1g}+T_{2g}$   |  $A_{2g}+E_g+T_{1g}+T_{2g}$   |  $A_{1g}+E_g+T_{1g}+T_{2g}$   |
 ###
-==== Input ====
+===== Implemented Settings =====
-<code Quanty Example.Quanty>
--- some example code
-</code>
-==== Result ====
+[[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z| Oh_0sqrt2-1z ]] [[physics_chemistry:point_groups:oh:orientation_0sqrt21z| Oh_0sqrt21z ]] [[physics_chemistry:point_groups:oh:orientation_111z| Oh_111z ]] [[physics_chemistry:point_groups:oh:orientation_sqrt20-1z| Oh_sqrt20-1z ]] [[physics_chemistry:point_groups:oh:orientation_sqrt201z| Oh_sqrt201z ]] [[physics_chemistry:point_groups:oh:orientation_xyz| Oh_XYZ ]]
-<WRAP center box 100%>
-text produced as output
-</WRAP>
-===== Table of contents =====
+==== Setting 0sqrt2-1z ====
-{{indexmenu>.#2}}
+[[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z|Details of the Oh group in with setting 0sqrt2-1z]]
+{{:physics_chemistry:pointgroup:oh_0sqrt2-1z.png }}
+###
+^ Operator ^ Orientation ^
+^  $\text{E}$  |  $\{0,0,0\}$  , |
+^  $C_3$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$  ,  $\left\{0,-\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{0,\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$  ,  $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$  , |
+^  $C_2$  |  $\left\{-\sqrt{2},0,0\right\}$  ,  $\left\{-\sqrt{2},-\sqrt{6},0\right\}$  ,  $\left\{-\sqrt{2},\sqrt{6},0\right\}$  ,  $\left\{0,-\sqrt{2},-2\right\}$  ,  $\left\{-\sqrt{6},-\sqrt{2},4\right\}$  ,  $\left\{-\sqrt{6},\sqrt{2},-4\right\}$  , |
+^  $C_4$  |  $\left\{0,-\sqrt{2},1\right\}$  ,  $\left\{0,\sqrt{2},-1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$  , |
+^  $C_2$  |  $\left\{0,-\sqrt{2},1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_4$  |  $\left\{0,-\sqrt{2},1\right\}$  ,  $\left\{0,\sqrt{2},-1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$  , |
+^  $S_6$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$  ,  $\left\{0,-\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{0,\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$  ,  $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$  , |
+^  $\sigma _h$  |  $\left\{0,-\sqrt{2},1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$  , |
+^  $\sigma _d$  |  $\left\{-\sqrt{2},0,0\right\}$  ,  $\left\{-\sqrt{2},-\sqrt{6},0\right\}$  ,  $\left\{-\sqrt{2},\sqrt{6},0\right\}$  ,  $\left\{0,-\sqrt{2},-2\right\}$  ,  $\left\{-\sqrt{6},-\sqrt{2},4\right\}$  ,  $\left\{-\sqrt{6},\sqrt{2},-4\right\}$  , |
+###
+==== Setting 0sqrt21z ====
+[[physics_chemistry:point_groups:oh:orientation_0sqrt21z|Details of the Oh group in with setting 0sqrt21z]]
+{{:physics_chemistry:pointgroup:oh_0sqrt21z.png }}
+###
+^ Operator ^ Orientation ^
+^  $\text{E}$  |  $\{0,0,0\}$  , |
+^  $C_3$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$  ,  $\left\{0,\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{0,-\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$  ,  $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$  , |
+^  $C_2$  |  $\left\{\sqrt{2},0,0\right\}$  ,  $\left\{\sqrt{2},\sqrt{6},0\right\}$  ,  $\left\{\sqrt{2},-\sqrt{6},0\right\}$  ,  $\left\{0,\sqrt{2},-2\right\}$  ,  $\left\{\sqrt{6},\sqrt{2},4\right\}$  ,  $\left\{\sqrt{6},-\sqrt{2},-4\right\}$  , |
+^  $C_4$  |  $\left\{0,\sqrt{2},1\right\}$  ,  $\left\{0,-\sqrt{2},-1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$  , |
+^  $C_2$  |  $\left\{0,\sqrt{2},1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_4$  |  $\left\{0,\sqrt{2},1\right\}$  ,  $\left\{0,-\sqrt{2},-1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$  , |
+^  $S_6$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$  ,  $\left\{0,\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$  ,  $\left\{0,-\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$  ,  $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$  , |
+^  $\sigma _h$  |  $\left\{0,\sqrt{2},1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$  ,  $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$  , |
+^  $\sigma _d$  |  $\left\{\sqrt{2},0,0\right\}$  ,  $\left\{\sqrt{2},\sqrt{6},0\right\}$  ,  $\left\{\sqrt{2},-\sqrt{6},0\right\}$  ,  $\left\{0,\sqrt{2},-2\right\}$  ,  $\left\{\sqrt{6},\sqrt{2},4\right\}$  ,  $\left\{\sqrt{6},-\sqrt{2},-4\right\}$  , |
+###
+==== Setting 111z ====
+[[physics_chemistry:point_groups:oh:orientation_111z|Details of the Oh group in with setting 111z]]
+{{:physics_chemistry:pointgroup:oh_111z.png }}
+###
+^ Operator ^ Orientation ^
+^  $\text{E}$  |  $\{0,0,0\}$  , |
+^  $C_3$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3}\right\}$  ,  $\left\{\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right\}$  ,  $\left\{\frac{1}{3} \left(\sqrt{3}-1\right),\frac{1}{3} \left(-1-\sqrt{3}\right),-\frac{1}{3}\right\}$  ,  $\left\{\frac{1}{3} \left(-1-\sqrt{3}\right),\frac{1}{3} \left(\sqrt{3}-1\right),-\frac{1}{3}\right\}$  ,  $\left\{-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right\}$  ,  $\left\{\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3}\right\}$  , |
+^  $C_2$  |  $\{1,-1,0\}$  ,  $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$  ,  $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$  ,  $\{1,1,-2\}$  ,  $\left\{1+\sqrt{3},1-\sqrt{3},4\right\}$  ,  $\left\{\sqrt{3}-1,-1-\sqrt{3},-4\right\}$  , |
+^  $C_4$  |  $\{1,1,1\}$  ,  $\{-1,-1,-1\}$  ,  $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$  ,  $\left\{-\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}-\frac{1}{2},1\right\}$  ,  $\left\{\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{1}{2}+\frac{\sqrt{3}}{2},-1\right\}$  ,  $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$  , |
+^  $C_2$  |  $\{1,1,1\}$  ,  $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$  ,  $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_4$  |  $\{1,1,1\}$  ,  $\{-1,-1,-1\}$  ,  $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$  ,  $\left\{-\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}-\frac{1}{2},1\right\}$  ,  $\left\{\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{1}{2}+\frac{\sqrt{3}}{2},-1\right\}$  ,  $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$  , |
+^  $S_6$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3}\right\}$  ,  $\left\{\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right\}$  ,  $\left\{\frac{1}{3} \left(\sqrt{3}-1\right),\frac{1}{3} \left(-1-\sqrt{3}\right),-\frac{1}{3}\right\}$  ,  $\left\{\frac{1}{3} \left(-1-\sqrt{3}\right),\frac{1}{3} \left(\sqrt{3}-1\right),-\frac{1}{3}\right\}$  ,  $\left\{-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right\}$  ,  $\left\{\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3}\right\}$  , |
+^  $\sigma _h$  |  $\{1,1,1\}$  ,  $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$  ,  $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$  , |
+^  $\sigma _d$  |  $\{1,-1,0\}$  ,  $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$  ,  $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$  ,  $\{1,1,-2\}$  ,  $\left\{1+\sqrt{3},1-\sqrt{3},4\right\}$  ,  $\left\{\sqrt{3}-1,-1-\sqrt{3},-4\right\}$  , |
+###
+==== Setting sqrt20-1z ====
+[[physics_chemistry:point_groups:oh:orientation_sqrt20-1z|Details of the Oh group in with setting sqrt20-1z]]
+{{:physics_chemistry:pointgroup:oh_sqrt20-1z.png }}
+###
+^ Operator ^ Orientation ^
+^  $\text{E}$  |  $\{0,0,0\}$  , |
+^  $C_3$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$  ,  $\left\{-\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$  ,  $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$  ,  $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$  ,  $\left\{\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$  ,  $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$  , |
+^  $C_2$  |  $\left\{0,\sqrt{2},0\right\}$  ,  $\left\{-\sqrt{6},\sqrt{2},0\right\}$  ,  $\left\{\sqrt{6},\sqrt{2},0\right\}$  ,  $\left\{-\sqrt{2},0,-2\right\}$  ,  $\left\{-\sqrt{2},\sqrt{6},4\right\}$  ,  $\left\{\sqrt{2},\sqrt{6},-4\right\}$  , |
+^  $C_4$  |  $\left\{-\sqrt{2},0,1\right\}$  ,  $\left\{\sqrt{2},0,-1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$  , |
+^  $C_2$  |  $\left\{-\sqrt{2},0,1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_4$  |  $\left\{-\sqrt{2},0,1\right\}$  ,  $\left\{\sqrt{2},0,-1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$  , |
+^  $S_6$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$  ,  $\left\{-\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$  ,  $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$  ,  $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$  ,  $\left\{\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$  ,  $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$  , |
+^  $\sigma _h$  |  $\left\{-\sqrt{2},0,1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$  , |
+^  $\sigma _d$  |  $\left\{0,\sqrt{2},0\right\}$  ,  $\left\{-\sqrt{6},\sqrt{2},0\right\}$  ,  $\left\{\sqrt{6},\sqrt{2},0\right\}$  ,  $\left\{-\sqrt{2},0,-2\right\}$  ,  $\left\{-\sqrt{2},\sqrt{6},4\right\}$  ,  $\left\{\sqrt{2},\sqrt{6},-4\right\}$  , |
+###
+==== Setting sqrt201z ====
+[[physics_chemistry:point_groups:oh:orientation_sqrt201z|Details of the Oh group in with setting sqrt201z]]
+{{:physics_chemistry:pointgroup:oh_sqrt201z.png }}
+###
+^ Operator ^ Orientation ^
+^  $\text{E}$  |  $\{0,0,0\}$  , |
+^  $C_3$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$  ,  $\left\{\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$  ,  $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$  ,  $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$  ,  $\left\{-\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$  ,  $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$  , |
+^  $C_2$  |  $\left\{0,-\sqrt{2},0\right\}$  ,  $\left\{\sqrt{6},-\sqrt{2},0\right\}$  ,  $\left\{-\sqrt{6},-\sqrt{2},0\right\}$  ,  $\left\{\sqrt{2},0,-2\right\}$  ,  $\left\{\sqrt{2},-\sqrt{6},4\right\}$  ,  $\left\{-\sqrt{2},-\sqrt{6},-4\right\}$  , |
+^  $C_4$  |  $\left\{\sqrt{2},0,1\right\}$  ,  $\left\{-\sqrt{2},0,-1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$  , |
+^  $C_2$  |  $\left\{\sqrt{2},0,1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_4$  |  $\left\{\sqrt{2},0,1\right\}$  ,  $\left\{-\sqrt{2},0,-1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$  , |
+^  $S_6$  |  $\{0,0,1\}$  ,  $\{0,0,-1\}$  ,  $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$  ,  $\left\{\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$  ,  $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$  ,  $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$  ,  $\left\{-\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$  ,  $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$  , |
+^  $\sigma _h$  |  $\left\{\sqrt{2},0,1\right\}$  ,  $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$  ,  $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$  , |
+^  $\sigma _d$  |  $\left\{0,-\sqrt{2},0\right\}$  ,  $\left\{\sqrt{6},-\sqrt{2},0\right\}$  ,  $\left\{-\sqrt{6},-\sqrt{2},0\right\}$  ,  $\left\{\sqrt{2},0,-2\right\}$  ,  $\left\{\sqrt{2},-\sqrt{6},4\right\}$  ,  $\left\{-\sqrt{2},-\sqrt{6},-4\right\}$  , |
+###
+==== Setting XYZ ====
+[[physics_chemistry:point_groups:oh:orientation_xyz|Details of the Oh group in with setting XYZ]]
+{{:physics_chemistry:pointgroup:oh_xyz.png }}
+###
+^ Operator ^ Orientation ^
+^  $\text{E}$  |  $\{0,0,0\}$  , |
+^  $C_3$  |  $\{1,1,1\}$  ,  $\{1,1,-1\}$  ,  $\{1,-1,1\}$  ,  $\{-1,1,1\}$  ,  $\{-1,-1,1\}$  ,  $\{-1,1,-1\}$  ,  $\{1,-1,-1\}$  ,  $\{-1,-1,-1\}$  , |
+^  $C_2$  |  $\{1,1,0\}$  ,  $\{1,-1,0\}$  ,  $\{1,0,-1\}$  ,  $\{1,0,1\}$  ,  $\{0,1,1\}$  ,  $\{0,1,-1\}$  , |
+^  $C_4$  |  $\{0,0,1\}$  ,  $\{0,1,0\}$  ,  $\{1,0,0\}$  ,  $\{0,0,-1\}$  ,  $\{0,-1,0\}$  ,  $\{-1,0,0\}$  , |
+^  $C_2$  |  $\{0,0,1\}$  ,  $\{0,1,0\}$  ,  $\{1,0,0\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_4$  |  $\{0,0,1\}$  ,  $\{0,1,0\}$  ,  $\{1,0,0\}$  ,  $\{0,0,-1\}$  ,  $\{0,-1,0\}$  ,  $\{-1,0,0\}$  , |
+^  $S_6$  |  $\{1,1,1\}$  ,  $\{1,1,-1\}$  ,  $\{1,-1,1\}$  ,  $\{-1,1,1\}$  ,  $\{-1,-1,1\}$  ,  $\{-1,1,-1\}$  ,  $\{1,-1,-1\}$  ,  $\{-1,-1,-1\}$  , |
+^  $\sigma _h$  |  $\{1,0,0\}$  ,  $\{0,1,0\}$  ,  $\{0,0,1\}$  , |
+^  $\sigma _d$  |  $\{1,1,0\}$  ,  $\{1,-1,0\}$  ,  $\{1,0,-1\}$  ,  $\{1,0,1\}$  ,  $\{0,1,1\}$  ,  $\{0,1,-1\}$  , |
+###
+===== Table of several point groups =====
+###
+[[physics_chemistry:point_groups|Return to Main page on Point Groups]]
+###
+###
+^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> | | | | |
+^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> |
+^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> |
+^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>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> | | |
+^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>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> |
+^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> |  |
+^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> | | | | | |
+###

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