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30: Appendix B: Point Groups

Non axial groups

\[\begin{array}{l|c} C_1 & E \\ \hline A_1 & 1 \end{array} \label{30.1}\]

\[\begin{array}{l|cc|l|l} C_s & E & \sigma_h & & \\ \hline A & 1 & 1 & x, y , R_z & x^2, y^2, z^2, xy \\ A' & 1 & -1 & z, R_x, R_y & yz, xz \end{array} \label{30.2}\]

\[\begin{array}{l|cc|l|l} C_1 & E & i & & \\ \hline A_g & 1 & 1 & R_x, R_y, R_z & x^2, y^2, z^2, xy, xz, yz \\ A_u & 1 & -1 & x, y, z & \end{array} \label{30.3}\]

\(C_n\) groups

\[\begin{array}{l|cc|l|l} C_2 & E & C_2 & & \\ \hline A & 1 & 1 & z, R_z & x^2, y^2, z^2, xy \\ B & 1 & -1 & x, y , R_x, R_y & yz, xz \end{array} \label{30.4}\]

\[\begin{array}{l|c|l|l} C_3 & E \: \: \: \: \: C_3 \: \: \: \: \: C_3^2 & & c=e^{2\pi/3} \\ \hline A & 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 & x, R_z & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & c & c^* \\ 1 & c^* & c \end{Bmatrix} & x, y, R_x, R_y, & x^2-y^2, xy, xz, yz \end{array} \label{30.5}\]

\[\begin{array}{l|c|l|l} C_4 & E \: \: \: \: \: C_4 \: \: \: \: \: C_2 \: \: \: \: \: C_4^3 & & \\ \hline A & 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: \: 1 & z, R_z & x^2 + y^2, z^2 \\ B & 1 \: \: \: \: -1 \: \: \: \: \: \: \: \: 1 \: \: \: \: -1 & & x^2 - y^2, xy \\ E & \begin{Bmatrix} 1 & i & -1 & -i \\ 1 & -i & -1 & i \end{Bmatrix} & x, y, R_x, R_y & yz, xz \end{array} \label{30.6}\]

\(C_{nv}\) groups

\[\begin{array}{l|cccc|l|l} C_{2v} & E & C_2 & \sigma_v(xz) & \sigma_v'(yz) & & \\ \hline A_1 & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ A_2 & 1 & 1 & -1 & -1 & R_z & xy \\ B_1 & 1 & -1 & 1 & -1 & x, R_y & xz \\ B_2 & 1 & -1 & -1 & 1 & y, R_x & yz \end{array} \label{30.7}\]

\[\begin{array}{l|ccc|l|l} C_{3v} & E & 2C_3 & 3\sigma_v & & \\ \hline A_1 & 1 & 1 & 1 & z & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & -1 & R_z & \\ E & 2 & -1 & 0 & x, y, R_x, R_y & x^2 - y^2, xy, xz, yz \end{array} \label{30.8}\]

\(C_{nh}\) groups

\[\begin{array}{l|cccc|l|l} C_{2h} & E & C_2 & i & \sigma_h & & \\ \hline A_g & 1 & 1 & 1 & 1 & R_z & x^2, y^2, z^2, xy \\ B_g & 1 & -1 & 1 & -1 & R_x, R_y & xz, yz \\ A_u & 1 & 1 & -1 & -1 & z & \\ B_u & 1 & -1 & -1 & 1 & x, y & \end{array} \label{30.9}\]

\[\begin{array}{l|c|l|l}C_{3h} & E \: \: \: \: \: C_3 \: \: \: \: \: C_3^2 \: \: \: \: \: \sigma_h \: \: \: \: \: S_3 \: \: \: \: \: S_3^5 & & c = e^{2\pi/3} \\ \hline A & 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: \: 1 & R_z & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & \: \: c & \: \: c^* & \: \: 1 & \: \: c & \: \: c^* \\ 1 & \: \: c^* & \: \: c & \: \: 1 & \: \: c^* & \: \: c \end{Bmatrix} & x, y & x^2 - y^2, xy \\ A' & 1 \: \: \: \: \: \: 1 \: \: \: \: \: \: 1 \: \: \: \: -1 \: \: \: \: -1 \: \: \: \: \: -1 & z & \\ E' & \begin{Bmatrix} 1 & c & c^* & -1 & -c & -c^* \\ 1 & c^* & c & -1 & -c^* & -c \end{Bmatrix} & R_x, R_y & xz, yz \end{array} \label{30.10}\]

\(D_n\) groups

\[\begin{array}{l|cccc|l|l} D_2 & E & C_2(z) & C_2(y) & C_2(x) & & \\ \hline A & 1 & 1 & 1 & 1 & & x^2, y^2, z^2 \\ B_1 & 1 & 1 & -1 & -1 & z, R_z & xy \\ B_2 & 1 & -1 & 1 & -1 & y, R_y & xz \\ B_3 & 1 & -1 & -1 & 1 & x, R_x & yz \end{array} \label{30.11}\]

\[\begin{array}{l|ccc|l|l} D_3 & E & 2C_3 & 3C_2 & & \\ \hline A_1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & -1 & z, R_z & \\ E & 2 & -1 & 0 & x, y, R_x, R_y & x^2 - y^2, xy, xz, yz \end{array} \label{30.12}\]

\(D_{nh}\) groups

\[\begin{array}{l|cccccccc|l|l} D_{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma_xy) & \sigma(xz) & \sigma(yz) & & \\ \hline A_g & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2, y^2, z^2 \\ B_{1g} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ B_{2g} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ B_{3g} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ A_u & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ B_{1u} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ B_{2u} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ B_{3u} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \end{array} \label{30.13}\]

\(D_{nd}\) groups

\[\begin{array}{l|ccccc|l|l} D_{2d} & E & 2S_4 & C_2 & 2C_2' & 2\sigma_d & & \\ \hline A_1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & 1 & -1 & -1 & R_z & \\ B_1 & 1 & -1 & 1 & 1 & -1 & & x^2 - y^2 \\ B_2 & 1 & -1 & 1 & -1 & 1 & z & xy \\ E & 2 & 0 & -2 & 0 & 0 & x, y, R_x, R_y & xy, yz \end{array} \label{30.14}\]

\[\begin{array}{l|cccccc|l|l} D_{3d} & E & 2C_3 & 3C_2 & i & 2S_6 & 3\sigma_d & & \\ \hline A_{1g} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_{2g} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ E_g & 2 & -1 & 0 & 2 & -1 & 0 & R_x, R_y & x^2 - y^2, xy, xz, yz \\ A_{1u} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ A_{2u} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ E_u & 2 & -1 & 0 & -2 & 1 & 0 & x, y & \end{array} \label{30.15}\]

 

\(C_{\infty v}\) 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

\[\begin{array}{l|c|l|l} T & E \: \: \: 4C_3 \: \: \: 4C_3^2 \: \: \: 3C_2 & & c=e^{2\pi/3} \\ \hline A & 1 \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: 1 \: \: \: \: \: \: \: 1 & & x^2 + y^2, z^2 \\ E & \begin{Bmatrix} 1 & c & c^* & 1 \\ 1 & c* & c & 1 \end{Bmatrix} & & 2z^2 - x^2 - y^2, x^2 - y^2 \\ T & 3 \: \: \: \: \: 0 \: \: \: \: \: \: \: 0 \: \: \: -1 & R_x, R_y, R_z, x, y, z & xy, xz, yz \end{array} \label{30.19}\]

\[\begin{array}{l|ccccc|l|l} T_d & E & 8C_3 & 3C_2 & 6S_4 & 6\sigma_d & & \\ \hline A_1 & 1 & 1 & 1 & 1 & 1 & & x^2 + y^2, z^2 \\ A_2 & 1 & 1 & 1 & -1 & -1 & & \\ E & 2 & -1 & 2 & 0 & 0 & & 2z^2 - x^-2 - y^2, x^2 - y^2 \\ T_1 & 3 & 0 & -1 & 1 & -1 & R_x, R_y, R_z & \\ T_2 & 3 & 0 & -1 & -1 & 1 & x, y, z & xy, xz, yz \end{array} \label{30.20}\]

Direct product tables

 

For the point groups O and T\(_d\) (and O\(_h\))

\[\begin{array}{llllll} & \boldsymbol{A_1} & \boldsymbol{A_2} & \boldsymbol{E} & \boldsymbol{T_1} & \boldsymbol{T_2} \\ \boldsymbol{A_1} & A_1 & A_2 & E & T_1 & T_2 \\ \boldsymbol{A_2} & & A_1 & E & T_2 & T_1 \\ \boldsymbol{E} & & & A_1 + A_2 + E & T_1 + T_2 & T_1 + T_2 \\ \boldsymbol{T_1} & & & & A_1 + E + T_1 + T_2 & A_2 + E + T_1 +T_2 \\ \boldsymbol{T_2} & & & & & A_1 + E + T_1 + T_2 \end{array} \label{30.21}\]

For the point groups D\(_4\), C\(_{4v}\), D\(_{2d}\) (and \(D_{4h} = D_4 \otimes C_i\))

\[\begin{array}{llllll} & \boldsymbol{A_1} & \boldsymbol{A_2} & \boldsymbol{B_1} & \boldsymbol{B_2} & \boldsymbol{E} \\ \boldsymbol{A_1} & A_1 & A_2 & B_1 & B_2 & E \\ \boldsymbol{A_2} & & A_1 & B_2 & B_1 & E \\ \boldsymbol{B_1} & & & A_1 & A_2 & E \\ \boldsymbol{B_2} & & & & A_1 & E \\ \boldsymbol{E} & & & & & A_1 + A_2 + B_1 + B_2 \end{array} \label{30.22}\]

For the point groups D\(_3\) and C\(_{3v}\)

\[\begin{array}{llll} & \boldsymbol{A_1} & \boldsymbol{A_2} & \boldsymbol{E} \\ \boldsymbol{A_1} & A_1 & A_2 & E \\ \boldsymbol{A_2} & & A_1 & E \\ \boldsymbol{E} & & & A_1 + A_2 + E \end{array} \label{30.23}\]

For the point groups D\(_6\), C\(_{6v}\) and D\(_{3h}^*\)

\[\begin{array}{lllllll} & \boldsymbol{A_1} & \boldsymbol{A_2} & \boldsymbol{B_1} & \boldsymbol{B_2} & \boldsymbol{E_1} & \boldsymbol{E_2} \\ \boldsymbol{A_1} & A_1 & A_2 & B_1 & B_2 & E_1 & E_2 \\ \boldsymbol{A_2} & & A_1 & B_2 & B_1 & E_1 & E_2 \\ \boldsymbol{B_1} & & & A_1 & A_2 & E_2 & E_1 \\ \boldsymbol{B_2} & & & & A_1 & E_2 & E_1 \\ \boldsymbol{E_1} & & & & & A_1 + A_2 + E_2 & B_1 + B_2 + E_1 \\ \boldsymbol{E_2} & & & & & & A_1 + A_2 + E_2 \end{array} \label{30.24}\]

\(^*\)in D\(_{3h}\) make the following changes in the above table

 

\[\begin{array}{ll} \text{In table} & \text{In D}_{3h} \\ A_1 & A_1' \\ A_2 & A_2' \\ B_1 & A_1'' \\ B_2 & A_2'' \\ E_1 & E'' \\ E_2 & E' \end{array} \label{30.25}\]

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