| Both sides previous revisionPrevious revision | |
| physics_chemistry:point_groups:ih [2018/03/23 00:36] – Maurits W. Haverkort | physics_chemistry:point_groups:ih [2018/03/23 10:52] (current) – Stefano Agrestini |
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| ====== Ih ====== | ====== Point Group Ih ====== |
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| ### | ===== Character Table ===== |
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| ===== Example ===== | |
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| | ===== Product Table ===== |
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| ^ $ G_u $ | $ G_u $ | $ G_u+H_u+T_{2u} $ | $ G_u+H_u+T_{1u} $ | $ A_u+G_u+H_u+T_{1u}+T_{2u} $ | $ G_u+2 H_u+T_{1u}+T_{2u} $ | $ G_g $ | $ G_g+H_g+T_{2g} $ | $ G_g+H_g+T_{1g} $ | $ A_g+G_g+H_g+T_{1g}+T_{2g} $ | $ G_g+2 H_g+T_{1g}+T_{2g} $ | | ^ $ G_u $ | $ G_u $ | $ G_u+H_u+T_{2u} $ | $ G_u+H_u+T_{1u} $ | $ A_u+G_u+H_u+T_{1u}+T_{2u} $ | $ G_u+2 H_u+T_{1u}+T_{2u} $ | $ G_g $ | $ G_g+H_g+T_{2g} $ | $ G_g+H_g+T_{1g} $ | $ A_g+G_g+H_g+T_{1g}+T_{2g} $ | $ G_g+2 H_g+T_{1g}+T_{2g} $ | |
| ^ $ H_u $ | $ H_u $ | $ G_u+H_u+T_{1u}+T_{2u} $ | $ G_u+H_u+T_{1u}+T_{2u} $ | $ G_u+2 H_u+T_{1u}+T_{2u} $ | $ A_u+2 G_u+2 H_u+T_{1u}+T_{2u} $ | $ H_g $ | $ G_g+H_g+T_{1g}+T_{2g} $ | $ G_g+H_g+T_{1g}+T_{2g} $ | $ G_g+2 H_g+T_{1g}+T_{2g} $ | $ A_g+2 G_g+2 H_g+T_{1g}+T_{2g} $ | | ^ $ H_u $ | $ H_u $ | $ G_u+H_u+T_{1u}+T_{2u} $ | $ G_u+H_u+T_{1u}+T_{2u} $ | $ G_u+2 H_u+T_{1u}+T_{2u} $ | $ A_u+2 G_u+2 H_u+T_{1u}+T_{2u} $ | $ H_g $ | $ G_g+H_g+T_{1g}+T_{2g} $ | $ G_g+H_g+T_{1g}+T_{2g} $ | $ G_g+2 H_g+T_{1g}+T_{2g} $ | $ A_g+2 G_g+2 H_g+T_{1g}+T_{2g} $ | |
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| | ===== Implemented Settings ===== |
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| | [[physics_chemistry:point_groups:ih:orientation_xyz| Ih_xyz ]] |
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| | ==== Setting xyz ==== |
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| | [[physics_chemistry:point_groups:ih:orientation_xyz|Details of the Ih group in with setting xyz]] |
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| | {{:physics_chemistry:pointgroup:ih_xyz.png }} |
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| ^ Operator ^ Orientation ^ | ^ Operator ^ Orientation ^ |
| ^ $\text{E}$ | $\{0,0,0\}$ , | | ^ $\text{E}$ | $\{0,0,0\}$ , | |
| ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3}\right\}$ , $\left\{\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(\sqrt{3}-1\right),\frac{1}{3} \left(-1-\sqrt{3}\right),-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(-1-\sqrt{3}\right),\frac{1}{3} \left(\sqrt{3}-1\right),-\frac{1}{3}\right\}$ , $\left\{-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3}\right\}$ , | | ^ $C_5$ | $\left\{1+\sqrt{5},2,0\right\}$ , $\left\{-1-\sqrt{5},-2,0\right\}$ , $\left\{1+\sqrt{5},-2,0\right\}$ , $\left\{-1-\sqrt{5},2,0\right\}$ , $\left\{0,1+\sqrt{5},2\right\}$ , $\left\{0,-1-\sqrt{5},-2\right\}$ , $\left\{0,1+\sqrt{5},-2\right\}$ , $\left\{0,-1-\sqrt{5},2\right\}$ , $\left\{2,0,1+\sqrt{5}\right\}$ , $\left\{-2,0,-1-\sqrt{5}\right\}$ , $\left\{-2,0,1+\sqrt{5}\right\}$ , $\left\{2,0,-1-\sqrt{5}\right\}$ , | |
| ^ $C_2$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{1+\sqrt{3},1-\sqrt{3},4\right\}$ , $\left\{\sqrt{3}-1,-1-\sqrt{3},-4\right\}$ , | | ^ $C_5^2$ | $\left\{1+\sqrt{5},2,0\right\}$ , $\left\{-1-\sqrt{5},-2,0\right\}$ , $\left\{1+\sqrt{5},-2,0\right\}$ , $\left\{-1-\sqrt{5},2,0\right\}$ , $\left\{0,1+\sqrt{5},2\right\}$ , $\left\{0,-1-\sqrt{5},-2\right\}$ , $\left\{0,1+\sqrt{5},-2\right\}$ , $\left\{0,-1-\sqrt{5},2\right\}$ , $\left\{2,0,1+\sqrt{5}\right\}$ , $\left\{-2,0,-1-\sqrt{5}\right\}$ , $\left\{-2,0,1+\sqrt{5}\right\}$ , $\left\{2,0,-1-\sqrt{5}\right\}$ , | |
| ^ $C_4$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{-\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}-\frac{1}{2},1\right\}$ , $\left\{\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{1}{2}+\frac{\sqrt{3}}{2},-1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ , | | ^ $C_3$ | $\{-1,-1,-1\}$ , $\left\{0,\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{0,\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\{1,1,1\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),0,\frac{1}{2} \left(1-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),0,\frac{1}{2} \left(\sqrt{5}-1\right)\right\}$ , $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{-\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$ , | |
| ^ $C_2$ | $\{1,1,1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ , | | ^ $C_2$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , | |
| ^ $\text{i}$ | $\{0,0,0\}$ , | | ^ $\text{i}$ | $\{0,0,0\}$ , | |
| ^ $S_4$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{-\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}-\frac{1}{2},1\right\}$ , $\left\{\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{1}{2}+\frac{\sqrt{3}}{2},-1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ , | | ^ $S_{10}$ | $\left\{1+\sqrt{5},2,0\right\}$ , $\left\{-1-\sqrt{5},-2,0\right\}$ , $\left\{1+\sqrt{5},-2,0\right\}$ , $\left\{-1-\sqrt{5},2,0\right\}$ , $\left\{0,1+\sqrt{5},2\right\}$ , $\left\{0,-1-\sqrt{5},-2\right\}$ , $\left\{0,1+\sqrt{5},-2\right\}$ , $\left\{0,-1-\sqrt{5},2\right\}$ , $\left\{2,0,1+\sqrt{5}\right\}$ , $\left\{-2,0,-1-\sqrt{5}\right\}$ , $\left\{-2,0,1+\sqrt{5}\right\}$ , $\left\{2,0,-1-\sqrt{5}\right\}$ , | |
| ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3}\right\}$ , $\left\{\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(\sqrt{3}-1\right),\frac{1}{3} \left(-1-\sqrt{3}\right),-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(-1-\sqrt{3}\right),\frac{1}{3} \left(\sqrt{3}-1\right),-\frac{1}{3}\right\}$ , $\left\{-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3}\right\}$ , | | ^ $S_{10}^3$ | $\left\{1+\sqrt{5},2,0\right\}$ , $\left\{-1-\sqrt{5},-2,0\right\}$ , $\left\{1+\sqrt{5},-2,0\right\}$ , $\left\{-1-\sqrt{5},2,0\right\}$ , $\left\{0,1+\sqrt{5},2\right\}$ , $\left\{0,-1-\sqrt{5},-2\right\}$ , $\left\{0,1+\sqrt{5},-2\right\}$ , $\left\{0,-1-\sqrt{5},2\right\}$ , $\left\{2,0,1+\sqrt{5}\right\}$ , $\left\{-2,0,-1-\sqrt{5}\right\}$ , $\left\{-2,0,1+\sqrt{5}\right\}$ , $\left\{2,0,-1-\sqrt{5}\right\}$ , | |
| ^ $\sigma _h$ | $\{1,1,1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ , | | ^ $S_6$ | $\{-1,-1,-1\}$ , $\left\{0,\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{0,\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\{1,1,1\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),0,\frac{1}{2} \left(1-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),0,\frac{1}{2} \left(\sqrt{5}-1\right)\right\}$ , $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{-\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$ , | |
| ^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{1+\sqrt{3},1-\sqrt{3},4\right\}$ , $\left\{\sqrt{3}-1,-1-\sqrt{3},-4\right\}$ , | | ^ $\sigma _h$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , | |
| |
| | ### |
| | |
| | |
| | ===== Table of several point groups ===== |
| |
| ### | ### |
| |
| ==== Input ==== | [[physics_chemistry:point_groups|Return to Main page on Point Groups]] |
| <code Quanty Example.Quanty> | |
| -- some example code | |
| </code> | |
| |
| ==== Result ==== | ### |
| <WRAP center box 100%> | |
| text produced as output | |
| </WRAP> | |
| |
| ===== Table of contents ===== | ### |
| {{indexmenu>.#2}} | |
| | ^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> | | | | | | |
| | |
| | ### |
