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physics_chemistry:point_groups:d3d [2018/03/21 18:36] – created Stefano Agrestini | physics_chemistry:point_groups:d3d [2018/03/23 10:57] (current) – Stefano Agrestini |
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====== D3d ====== | ====== Point Group D3d ====== |
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| ===== Character Table ===== |
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### | ### |
alligned paragraph text | |
| | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_3 \,{\text{(2)}} $ ^ $ C_2 \,{\text{(3)}} $ ^ $ \text{i} \,{\text{(1)}} $ ^ $ S_6 \,{\text{(2)}} $ ^ $ \sigma_d \,{\text{(3)}} $ ^ |
| ^ $ A_{1g} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | |
| ^ $ A_{2g} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | |
| ^ $ E_g $ | $ 2 $ | $ -1 $ | $ 0 $ | $ 2 $ | $ -1 $ | $ 0 $ | |
| ^ $ A_{1u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | |
| ^ $ A_{2u} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ 1 $ | |
| ^ $ E_u $ | $ 2 $ | $ -1 $ | $ 0 $ | $ -2 $ | $ 1 $ | $ 0 $ | |
### | ### |
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===== Example ===== | ===== Product Table ===== |
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### | ### |
description text | |
| | $ $ ^ $ A_{1g} $ ^ $ A_{2g} $ ^ $ E_g $ ^ $ A_{1u} $ ^ $ A_{2u} $ ^ $ E_u $ ^ |
| ^ $ A_{1g} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | |
| ^ $ A_{2g} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | |
| ^ $ E_g $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | |
| ^ $ A_{1u} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | |
| ^ $ A_{2u} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | |
| ^ $ E_u $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | |
### | ### |
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==== Input ==== | ===== Implemented Settings ===== |
<code Quanty Example.Quanty> | |
-- some example code | |
</code> | |
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==== Result ==== | [[physics_chemistry:point_groups:d3d:orientation_111| D3d_111 ]] [[physics_chemistry:point_groups:d3d:orientation_zx| D3d_Zx ]] [[physics_chemistry:point_groups:d3d:orientation_zx_a| D3d_Zx_A ]] [[physics_chemistry:point_groups:d3d:orientation_zx_b| D3d_Zx_B ]] [[physics_chemistry:point_groups:d3d:orientation_z(x-y)| D3d_Z(x-y) ]] [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_a| D3d_Z(x-y)_A ]] [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_b| D3d_Z(x-y)_B ]] [[physics_chemistry:point_groups:d3d:orientation_zy| D3d_Zy ]] [[physics_chemistry:point_groups:d3d:orientation_zy_a| D3d_Zy_A ]] [[physics_chemistry:point_groups:d3d:orientation_zy_b| D3d_Zy_B ]] |
<WRAP center box 100%> | |
text produced as output | |
</WRAP> | |
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===== Table of contents ===== | ==== Setting 111 ==== |
{{indexmenu>.#2}} | |
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| [[physics_chemistry:point_groups:d3d:orientation_111|Details of the D3d group in with setting 111]] |
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| {{:physics_chemistry:pointgroup:d3d_111.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , | |
| ^ $C_2$ | $\{1,-1,0\}$ , $\{0,1,-1\}$ , $\{1,0,-1\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , | |
| ^ $\sigma _d$ | $\{1,-1,0\}$ , $\{0,1,-1\}$ , $\{1,0,-1\}$ , | |
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| ### |
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| ==== Setting Zx ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_zx|Details of the D3d group in with setting Zx]] |
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| {{:physics_chemistry:pointgroup:d3d_zx.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , | |
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| ### |
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| ==== Setting Zx_A ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_zx_a|Details of the D3d group in with setting Zx_A]] |
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| {{:physics_chemistry:pointgroup:d3d_zx_a.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , | |
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| ### |
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| ==== Setting Zx_B ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_zx_b|Details of the D3d group in with setting Zx_B]] |
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| {{:physics_chemistry:pointgroup:d3d_zx_b.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , | |
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| ### |
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| ==== Setting Z(x-y) ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_z(x-y)|Details of the D3d group in with setting Z(x-y)]] |
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| {{:physics_chemistry:pointgroup:d3d_z(x-y).png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , | |
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| ### |
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| ==== Setting Z(x-y)_A ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_a|Details of the D3d group in with setting Z(x-y)_A]] |
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| {{:physics_chemistry:pointgroup:d3d_z(x-y)_a.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , | |
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| ### |
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| ==== Setting Z(x-y)_B ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_b|Details of the D3d group in with setting Z(x-y)_B]] |
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| {{:physics_chemistry:pointgroup:d3d_z(x-y)_b.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , | |
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| ### |
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| ==== Setting Zy ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_zy|Details of the D3d group in with setting Zy]] |
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| {{:physics_chemistry:pointgroup:d3d_zy.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , | |
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| ### |
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| ==== Setting Zy_A ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_zy_a|Details of the D3d group in with setting Zy_A]] |
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| {{:physics_chemistry:pointgroup:d3d_zy_a.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , | |
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| ### |
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| ==== Setting Zy_B ==== |
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| [[physics_chemistry:point_groups:d3d:orientation_zy_b|Details of the D3d group in with setting Zy_B]] |
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| {{:physics_chemistry:pointgroup:d3d_zy_b.png }} |
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| ### |
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| ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $C_2$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | |
| ^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , | |
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| ### |
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| ===== Table of several point groups ===== |
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| ### |
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| [[physics_chemistry:point_groups|Return to Main page on Point Groups]] |
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| ### |
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| ### |
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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> | | | | | |
| ^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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| ### |