UNDER CONSTRUCTION Low-symmetry groups (\(C_1, \; C_s, \; C_i\)) \(\begin{array}{|l|c|l|l|} \hline \bf{C_1} & \ E & h=1\\ \hline A_1 & 1 \\ \hline \end{array} \nonumber\) \(\begin{array}{|l|lc|ll|} \hline \bf{C_s} & E & \sigma_h & h=2& \\ \hline A & 1 & 1 & x, y , R_z & x^2, y^2, z^2, xy \\ A' & 1 & -1 & z, R_x, R_y & yz, xz \\ \hline \end{array}\nonumber\) \(\begin{array}{|l|cc|ll|} \bf{C_1} & E & i & h=3 & \hline \\ \hline A_g & 1 & 1 & R_x, R_y, R_z & x^2, y^2, z^2, xy, xz, yz \\ A_u & 1 & -1 & x, y, z & \\ \hline \end{array} \nonumber\) The groups \(C_n\) \(\begin{array}{|l|cc|ll|} \hline \bf{C_2} & E & C_2 & h=2 & \\ \hline A & 1 & 1 & z, R_z & x^2, y^2, z^2, xy \\ B & 1 & -1 & x, y , R_x, R_y & yz, xz \\ \hline \end{array} \nonumber\) \(\begin{array}{|l|c|ll|} \hline \bf{C_3} & E \: \: \: \: \: C_3 \: \: \: \: \: C_3^2 & h = 3 & \\ \hline A & 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 & x, R_z & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & \varepsilon & \varepsilon^* \\ 1 & \varepsilon^* & \varepsilon \end{Bmatrix} & (x, y), \; (R_x, R_y) & (x^2-y^2, xy), \; (xz, yz) \\ \hline \end{array} \\ \nonumber \\ \) \(\varepsilon = e^{(2\pi i)/3}\) \(\begin{array}{|l|c|ll|} \hline \bf{C_4} & E \: \: \: \: \: C_4 \: \: \: \: \: C_2 \: \: \: \: \: C_4^3 &h=4& \\ \hline A & 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 & z, R_z & x^2 + y^2, z^2 \\ B & 1 \: \: \: \: -1 \: \: \: \: \: \: \: \: 1 \: \: \: \: -1 & & x^2 - y^2, xy \\ E & \begin{Bmatrix} 1 & i & -1 & -i \\ 1 & -i & -1 & i \end{Bmatrix} & (x, y), \;(R_x, R_y) & (yz, xz) \\ \hline \end{array} \) The groups \(C_{nv}\) \(\begin{array}{|l|cccc|ll|} \hline \bf{C_{2v}} & E & C_2 & \sigma_v(xz) & \sigma_v'(yz) & h=4 & \\ \hline A_1 & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ A_2 & 1 & 1 & -1 & -1 & R_z & xy \\ B_1 & 1 & -1 & 1 & -1 & x, R_y & xz \\ B_2 & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}\) \(\begin{array}{|l|ccc|ll|} \hline \bf{C_{3v}} & E & 2C_3 & 3\sigma_v & & \\ \hline A_1 & 1 & 1 & 1 & z & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & -1 & R_z & \\ E & 2 & -1 & 0 & x, y, R_x, R_y & x^2 - y^2, xy, xz, yz \\ \hline \end{array}\) The groups \(C_{nh}\) \[\begin{array}{l|cccc|l|l} C_{2h} & E & C_2 & i & \sigma_h & & \\ \hline A_g & 1 & 1 & 1 & 1 & R_z & x^2, y^2, z^2, xy \\ B_g & 1 & -1 & 1 & -1 & R_x, R_y & xz, yz \\ A_u & 1 & 1 & -1 & -1 & z & \\ B_u & 1 & -1 & -1 & 1 & x, y & \end{array} \label{30.9}\] \[\begin{array}{l|c|l|l}C_{3h} & E \: \: \: \: \: C_3 \: \: \: \: \: C_3^2 \: \: \: \: \: \sigma_h \: \: \: \: \: S_3 \: \: \: \: \: S_3^5 & & c = e^{2\pi/3} \\ \hline A & 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 & R_z & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & \: \: c & \: \: c^* & \: \: 1 & \: \: c & \: \: c^* \\ 1 & \: \: c^* & \: \: c & \: \: 1 & \: \: c^* & \: \: c \end{Bmatrix} & x, y & x^2 - y^2, xy \\ A' & 1 \: \: \: \: \: \: 1 \: \: \: \: \: \: 1 \: \: \: \: -1 \: \: \: \: -1 \: \: \: \: \: -1 & z & \\ E' & \begin{Bmatrix} 1 & c & c^* & -1 & -c & -c^* \\ 1 & c^* & c & -1 & -c^* & -c \end{Bmatrix} & R_x, R_y & xz, yz \end{array} \label{30.10}\] The groups \(D_n\) \[\begin{array}{l|cccc|l|l} D_2 & E & C_2(z) & C_2(y) & C_2(x) & & \\ \hline A & 1 & 1 & 1 & 1 & & x^2, y^2, z^2 \\ B_1 & 1 & 1 & -1 & -1 & z, R_z & xy \\ B_2 & 1 & -1 & 1 & -1 & y, R_y & xz \\ B_3 & 1 & -1 & -1 & 1 & x, R_x & yz \end{array} \label{30.11}\] \[\begin{array}{l|ccc|l|l} D_3 & E & 2C_3 & 3C_2 & & \\ \hline A_1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & -1 & z, R_z & \\ E & 2 & -1 & 0 & x, y, R_x, R_y & x^2 - y^2, xy, xz, yz \end{array} \label{30.12}\] The groups \(D_{nd}\) \[\begin{array}{|l|ccccc|ll|} \hline D_{2d} & E & 2S_4 & C_2 & 2C_2' & 2\sigma_d & & \\ \hline A_1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & 1 & -1 & -1 & R_z & \\ B_1 & 1 & -1 & 1 & 1 & -1 & & x^2 - y^2 \\ B_2 & 1 & -1 & 1 & -1 & 1 & z & xy \\ E & 2 & 0 & -2 & 0 & 0 & (x, y), (R_x, R_y) & (xz, yz) \\ \hline \end{array} \label{30.14}\] \[\begin{array}{l|cccccc|l|l} D_{3d} & E & 2C_3 & 3C_2 & i & 2S_6 & 3\sigma_d & & \\ \hline A_{1g} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_{2g} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ E_g & 2 & -1 & 0 & 2 & -1 & 0 & R_x, R_y & x^2 - y^2, xy, xz, yz \\ A_{1u} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ A_{2u} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ E_u & 2 & -1 & 0 & -2 & 1 & 0 & x, y & \end{array} \label{30.15}\] The Groups \(D_{nh}\) \[\begin{array}{|c|rrrrrrrr|cc|} \hline \bf{D_{2h}} & E & C_2(z) & C_2(y) &C_2(x) & i &\sigma(xy) & \sigma(xz) & \sigma(yz) & h=8 & \\ \hline A_{g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2, \; y^2, \; z^2\\ B_{1g} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy\\ B_{2g} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & zx \\ B_{3g} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ A_{u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ B_{1u} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ B_{2u} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ B_{3u} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}\] \[\begin{array}{|c|rrrrrr|cc|} \hline \bf{D_{3h}} & E & 2C_3 & 3C_2 &\sigma_h & 2S_3 & 3\sigma_v & h=8 & \\ \hline A_{1}' & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, \; z^2\\ A_{2}' & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ E' & 2 & -1 & 0 & 2 & -1 & 0 & (x,\;y) & (x^2-y^2,\; xy) \\ A_{1}" & 1 & 1 & 1 & -1 & -1 & -1 & & \\ A_{2}" & 1 & 1 & -1 & -1 & -1 & 1 & R_z & \\ E" & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,\;R_y) & (xz,\; yz) \\ \hline \end{array}\] \begin{array}{|c|rrrrrrrrrr|cc|} \hline \bf{D_{4h}} & E & 2C_4 & C_2 & 2C_2' & 2C_2" & i & 2S_4 & \sigma_h & 2\sigma_v & 2\sigma_d & h=16 & \\ \hline A_{1g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, \; z^2\\ A_{2g} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ B_{1g} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ B_{2g} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ E_{g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,\;R_y) & (xz,\; yz) \\ A_{1u} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ A_{2u} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ B_{1u} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ B_{2u} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ E_{u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & & (x, \; y)\\ \hline \end{array} \begin{array}{|c|rrrrrrrr|cc|} \hline \bf{D_{5h}} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma _h & 2S_5 & 2S_5^2 & 5 \sigma_h & h=20 & \\ \hline A_{1}’ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, \; z^2\\A_{2}’ & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ E_{1}’ & 2 & 2cos(72 ^{\circ}) & 2cos(144 ^{\circ}) & 0 & 2 & 2cos(72 ^{\circ}) & 2cos(144 ^{\circ}) & 0 & (x, \; y) & \\ E_{2}’ & 2 & 2cos(144 ^{\circ}) & 2cos(72 ^{\circ}) & 0 & 2 & 2cos(144 ^{\circ}) & 2cos(72 ^{\circ}) & 0 & & (x^2-y^2,\; xy) \\ A_{1}” & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ A_{2}” & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ E_{1}” & 2 & 2cos(72 ^{\circ}) & 2cos(144 ^{\circ}) & 0 & -2 & -2cos(72 ^{\circ}) & -2cos(144 ^{\circ}) & 0 & (R_x,\;R_y) & (xz,\; yz) \\ E_{2}” & 2 & 2cos(144 ^{\circ}) & 2cos(72 ^{\circ}) & 0 & -2 & -2cos(144 ^{\circ}) & -2cos(72 ^{\circ}) & 0 & & \\ \hline \end{array} High-symmetry groups \[\begin{array}{|c|cccccccc|c|c|} \hline \bf{D_{\infty h}} & \mathrm{E} & 2 \mathrm{C}_{\infty}^{\phi} & ... & \infty \sigma_v & i & 2S_{\infty}^{\phi} & ... & \infty C_2 & & \\ \hline {A}_{1g} & 1 & 1 & ... & 1 & 1 & 1 & ...& 1 & & x^{2}+y^{2}, \, z^{2} \\ {A}_{2g} & 1 & 1 & ... & -1 & 1 & 1 & ...& -1 & R_z & \\ {E}_{1g} & 2 & 2\cos \phi & ... & 0 & 2 & -2\cos\phi & ...& 0 & (R_z, \, R_y) & (xz, \, yz) \\ {E}_{2g} & 2 & 2\cos 2\phi & ... & 0 & 2 & 2\cos2\phi & ...& 0 & & (x^{2}-y^{2}, \, xy) \\ ... & ... & ... & ... & ... & ... & ... & ...& ... & & \\ {A}_{1u} & 1 & 1 & ... & 1 & -1 & -1 & ...& -1 & z & \\ {A}_{2u} & 1 & 1 & ... & -1 & -1 & -1 & ...& 1 & & \\ {E}_{1u} & 2 & 2\cos \phi & ... & 0 & -2 & 2\cos\phi & ...& 0 & (x, \, y) & \\ {E}_{2u} & 2 & 2\cos 2\phi & ... & 0 & -2 & -2\cos2\phi & ...& 0 & & \\ ... & ... & ... & ... & ... & ... & ... & ...& ... & & \\ \hline \end{array}\] \(C_{\infty v}\) and \(D_{\infty h}\) \[\begin{array}{l|cccccccc|l|l} D_{\infty h} & E & 2C_\infty^\Phi & \ldots & \infty \sigma_v & i & 2S_\infty^\Phi & \ldots & \infty C_2 & & \\ \hline \Sigma_g^+ & 1 & 1 & \ldots & 1 & 1 & 1 & \ldots & 1 & & x^2 + y^2, z^2 \\ \Sigma_g^- & 1 & 1 & \ldots & -1 & 1 & 1 & \ldots & -1 & R_z & \\ \Pi_g & 2 & 2cos \Phi & \ldots & 0 & 2 & -2cos \Phi & \ldots & 0 & R_x, R_y & xz, yz \\ \Delta_g & 2 & 2cos 2\Phi & \ldots & 0 & 2 & 2cos 2\Phi & \ldots & 0 & & x^2 - y^2, xy \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & & \\ \Sigma_u^+ & 1 & 1 & \ldots & 1 & -1 & -1 & \ldots & -1 & z & \\ \Sigma_u^- & 1 & 1 & \ldots & -1 & -1 & -1 & \ldots & 1 & & \\ \Pi_u & 2 & 2cos \Phi & \ldots & 0 & -2 & 2cos \Phi & \ldots & 0 & x, y & \\ \Delta_u & 2 & 2cos 2\Phi & \ldots & 0 & -2 & -2cos 2\Phi & \ldots & 0 & & \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & & \end{array} \label{30.16}\] \(S_n\) groups \[\begin{array}{l|c|l|l} S_4 & E \: \: \: \: \: S_4 \: \: \: \: \: C_2 \: \: \: \: \: S_4^3 & & \\ \hline A & 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 & R_z & x^2 + y^2, z^2 \\ B & 1 \: \: \: \: -1 \: \: \: \: \: \: \: \: 1 \: \: \: \: -1 & z & x^2 - y^2, xy \\ E & \begin{Bmatrix} 1 & i & -1 & -i \\ 1 & -i & -1 & i \end{Bmatrix} & x, y, R_x, R_y & xz, yz \end{array} \label{30.17}\] \[\begin{array}{l|c|l|l} S_6 & E \: \: \: \: \: C_3 \: \: \: \: \: C_3^2 \: \: \: \: \: i \: \: \: \: \: S_6^5 \: \: \: \: \: S_6 & & c=e^{2\pi/3} \\ \hline A_g & 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 & R_z & x^2 + y^2, z^2 \\ E_g & \begin{Bmatrix} 1 & \: \: c & \: \: c^* & \: \: 1 & \: \: c & \: \: c^* \\ 1 \: \: & \: \: c^* & \: \: c & \: \: 1 & \: \: c^* & \: \: c \end{Bmatrix} & R_x, R_y & x^2 - y^2, xy, xz, yz \\ A_u & 1 \: \: \: \: \: \: 1 \: \: \: \: \: \: 1 \: \: \: \: -1 \: \: \: \: -1 \: \: \: \: \: -1 & z & \\ E_u & \begin{Bmatrix} 1 & c & c^* & -1 & -c & -c^* \\ 1 & c^* & c & -1 & -c^* & -c \end{Bmatrix} & x, y & \end{array} \label{30.18}\] Cubic groups \[\begin{array}{l|c|l|l} T & E \: \: \: 4C_3 \: \: \: 4C_3^2 \: \: \: 3C_2 & & c=e^{2\pi/3} \\ \hline A & 1 \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: 1 & & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & c & c^* & 1 \\ 1 & c* & c & 1 \end{Bmatrix} & & 2z^2 - x^2 - y^2, x^2 - y^2 \\ T & 3 \: \: \: \: \: 0 \: \: \: \: \: \: \: 0 \: \: \: -1 & R_x, R_y, R_z, x, y, z & xy, xz, yz \end{array} \label{30.19}\] \[\begin{array}{l|ccccc|l|l} T_d & E & 8C_3 & 3C_2 & 6S_4 & 6\sigma_d & & \\ \hline A_1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & 1 & -1 & -1 & & \\ E & 2 & -1 & 2 & 0 & 0 & & 2z^2 - x^-2 - y^2, x^2 - y^2 \\ T_1 & 3 & 0 & -1 & 1 & -1 & R_x, R_y, R_z & \\ T_2 & 3 & 0 & -1 & -1 & 1 & x, y, z & xy, xz, yz \end{array} \label{30.20}\]