Table 2 Character table of the group \(D_{3h}\). Two common notations are used for the IRs of the single and double group.
\(D_{3h}\) | E | \(\sigma _{h}\) | \(2C_{3}\) | \(2S_{3}\) | \(3C_{2}\) | \(3\sigma _{v}\) | Linear | Quadratic | |
|---|---|---|---|---|---|---|---|---|---|
\(A^{\prime }_{1}\) | \(\Gamma _1\) | 1 | 1 | 1 | 1 | 1 | 1 | Â | \(x^2+y^2\), \(z^2\) |
\(A^{\prime }_{2}\) | \(\Gamma _2\) | 1 | 1 | 1 | 1 | −1 | −1 | \(R_z\) |  |
\(A^{\prime \prime }_{1}\) | \(\Gamma _3\) | 1 | −1 | 1 | −1 | 1 | −1 |  |  |
\(A^{\prime \prime }_{2}\) | \(\Gamma _4\) | 1 | −1 | 1 | −1 | −1 | 1 | z |  |
\(E^{\prime }\) | \(\Gamma _6\) | 2 | 2 | −1 | −1 | 0 | 0 | (x, y) | \((x^2-y^2,xy)\) |
\(E^{\prime \prime }\) | \(\Gamma _5\) | 2 | −2 | −1 | 1 | 0 | 0 | \((R_x,R_y)\) | (xz, yz) |
\(E_{1/2}\) | \(\Gamma _7\) | \({\pm } 2\) | 0 | \(\pm 1\) | \(\pm \sqrt{3}\) | 0 | 0 | \(\left| \frac{1}{2},\pm \frac{1}{2}\right\rangle ,\left| \frac{3}{2},\pm \frac{1}{2}\right\rangle\) | |
\(E_{3/2}\) | \(\Gamma _9\) | \(\pm 2\) | 0 | \({\mp } 2\) | 0 | 0 | 0 | \(\left| \frac{3}{2},\pm \frac{3}{2}\right\rangle ,\left| \frac{5}{2},\pm \frac{3}{2}\right\rangle\) | |
\(E_{5/2}\) | \(\Gamma _8\) | \(\pm 2\) | 0 | \(\pm 1\) | \({\mp }\sqrt{3}\) | 0 | 0 | \(\left| \frac{5}{2},\pm \frac{5}{2}\right\rangle\) | |
