Table of Contents
Point Group D5d
Character Table
| $ $ | $ \text{E} \,{\text{(1)}} $ | $ C_5 \,{\text{(2)}} $ | $ C_5^2{} \,{\text{(2)}} $ | $ C_2 \,{\text{(5)}} $ | $ \text{i} \,{\text{(1)}} $ | $ S_{10} \,{\text{(2)}} $ | $ S_{10}^3{} \,{\text{(2)}} $ | $ \sigma_d \,{\text{(5)}} $ |
|---|---|---|---|---|---|---|---|---|
| $ A_{1g} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ |
| $ A_{2g} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ |
| $ E_{1g} $ | $ 2 $ | $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ | $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ | $ 0 $ | $ 2 $ | $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ | $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ | $ 0 $ |
| $ E_{2g} $ | $ 2 $ | $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ | $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ | $ 0 $ | $ 2 $ | $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ | $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ | $ 0 $ |
| $ A_{1u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ |
| $ A_{2u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ 1 $ |
| $ E_{1u} $ | $ 2 $ | $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ | $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ | $ 0 $ | $ -2 $ | $ \frac{1}{2} \left(1+\sqrt{5}\right) $ | $ \frac{1}{2} \left(1-\sqrt{5}\right) $ | $ 0 $ |
| $ E_{2u} $ | $ 2 $ | $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ | $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ | $ 0 $ | $ -2 $ | $ \frac{1}{2} \left(1-\sqrt{5}\right) $ | $ \frac{1}{2} \left(1+\sqrt{5}\right) $ | $ 0 $ |
Product Table
| $ $ | $ A_{1g} $ | $ A_{2g} $ | $ E_{1g} $ | $ E_{2g} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_{1u} $ | $ E_{2u} $ |
|---|---|---|---|---|---|---|---|---|
| $ A_{1g} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_{1g} $ | $ E_{2g} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_{1u} $ | $ E_{2u} $ |
| $ A_{2g} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_{1g} $ | $ E_{2g} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_{1u} $ | $ E_{2u} $ |
| $ E_{1g} $ | $ E_{1g} $ | $ E_{1g} $ | $ A_{1g}+A_{2g}+E_{2g} $ | $ E_{1g}+E_{2g} $ | $ E_{1u} $ | $ E_{1u} $ | $ A_{1u}+A_{2u}+E_{2u} $ | $ E_{1u}+E_{2u} $ |
| $ E_{2g} $ | $ E_{2g} $ | $ E_{2g} $ | $ E_{1g}+E_{2g} $ | $ A_{1g}+A_{2g}+E_{1g} $ | $ E_{2u} $ | $ E_{2u} $ | $ E_{1u}+E_{2u} $ | $ A_{1u}+A_{2u}+E_{1u} $ |
| $ A_{1u} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_{1u} $ | $ E_{2u} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_{1g} $ | $ E_{2g} $ |
| $ A_{2u} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_{1u} $ | $ E_{2u} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_{1g} $ | $ E_{2g} $ |
| $ E_{1u} $ | $ E_{1u} $ | $ E_{1u} $ | $ A_{1u}+A_{2u}+E_{2u} $ | $ E_{1u}+E_{2u} $ | $ E_{1g} $ | $ E_{1g} $ | $ A_{1g}+A_{2g}+E_{2g} $ | $ E_{1g}+E_{2g} $ |
| $ E_{2u} $ | $ E_{2u} $ | $ E_{2u} $ | $ E_{1u}+E_{2u} $ | $ A_{1u}+A_{2u}+E_{1u} $ | $ E_{2g} $ | $ E_{2g} $ | $ E_{1g}+E_{2g} $ | $ A_{1g}+A_{2g}+E_{1g} $ |
Implemented Settings
Setting Zx
Details of the D5d group in with setting Zx
| Operator | Orientation |
|---|---|
| $\text{E}$ | $\{0,0,0\}$ , |
| $C_5$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
| $C_5^2$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
| $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{5+2 \sqrt{5}},0\right\}$ , $\left\{-\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{1,-\sqrt{5+2 \sqrt{5}},0\right\}$ , |
| $\text{i}$ | $\{0,0,0\}$ , |
| $S_{10}$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
| $S_{10}^3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
| $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{5+2 \sqrt{5}},0\right\}$ , $\left\{-\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{1,-\sqrt{5+2 \sqrt{5}},0\right\}$ , |
Setting Zy
Details of the D5d group in with setting Zy
| Operator | Orientation |
|---|---|
| $\text{E}$ | $\{0,0,0\}$ , |
| $C_5$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
| $C_5^2$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
| $C_2$ | $\{0,1,0\}$ , $\left\{-\sqrt{5+2 \sqrt{5}},1,0\right\}$ , $\left\{1,\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{1,-\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{\sqrt{5+2 \sqrt{5}},1,0\right\}$ , |
| $\text{i}$ | $\{0,0,0\}$ , |
| $S_{10}$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
| $S_{10}^3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
| $\sigma _d$ | $\{0,1,0\}$ , $\left\{-\sqrt{5+2 \sqrt{5}},1,0\right\}$ , $\left\{1,\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{1,-\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{\sqrt{5+2 \sqrt{5}},1,0\right\}$ , |
Table of several point groups
Return to Main page on Point Groups
| Nonaxial groups | C1 | Cs | Ci | ||||
|---|---|---|---|---|---|---|---|
| Cn groups | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
| Dn groups | D2 | D3 | D4 | D5 | D6 | D7 | D8 |
| Cnv groups | C2v | C3v | C4v | C5v | C6v | C7v | C8v |
| Cnh groups | C2h | C3h | C4h | C5h | C6h | ||
| Dnh groups | D2h | D3h | D4h | D5h | D6h | D7h | D8h |
| Dnd groups | D2d | D3d | D4d | D5d | D6d | D7d | D8d |
| Sn groups | S2 | S4 | S6 | S8 | S10 | S12 | |
| Cubic groups | T | Th | Td | O | Oh | I | Ih |
| Linear groups | C$\infty$v | D$\infty$h |
