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## $C_{\infty\mathrm{v}}$ and $D_{\infty\mathrm{h}}$

$$\begin{array}{c|cccc|cc} \hline
C_{\infty\mathrm{v}}  & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} &  & \\ \hline
\mathrm{A_1} \equiv \Sigma^+ & 1 & 1 & \cdots & 1 & z & x^2 + y^2, z^2 \\
\mathrm{A_2} \equiv \Sigma^- & 1 & 1 & \cdots & -1 & R_z & \\
\mathrm{E_1} \equiv \Pi      & 2 & 2 \cos\phi & \cdots & 0 & (x,y), (R_x,R_y) & (xz,yz) \\
\mathrm{E_2} \equiv \Delta   & 2 & 2 \cos 2\phi & \cdots & 0 & & (x^2-y^2,xy) \\
\mathrm{E_3} \equiv \Phi     & 2 & 2 \cos 3\phi & \cdots & 0 & & \\
\vdots & \vdots & \vdots & \ddots & \vdots &  & \\ \hline
\end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline
D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline
\mathrm{A_{1g}} \equiv \Sigma^+_{\mathrm{g}} & 1 & 1 & \cdots & 1 & 1 & 1 & \cdots & 1 &  & x^2 + y^2, z^2 \\
\mathrm{A_{2g}} \equiv \Sigma^-_{\mathrm{g}} & 1 & 1 & \cdots & -1 & 1 & 1 & \cdots & -1 & R_z & \\
\mathrm{E_{1g}} \equiv \Pi_{\mathrm{g}} & 2 & 2\cos\phi & \cdots & 0 & 2 & -2\cos\phi & \cdots & 0 & (R_x,R_y) & (xz,yz) \\
\mathrm{E_{2g}} \equiv \Delta_{\mathrm{g}} & 2 & 2\cos 2\phi & \cdots & 0 & 2 & 2\cos 2\phi & \cdots & 0 & & (x^2-y^2,xy) \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & & \\
\mathrm{A_{1u}} \equiv \Sigma^+_{\mathrm{u}} & 1 & 1 & \cdots & 1 & -1 & -1 & \cdots & -1 & z & \\
\mathrm{A_{2u}} \equiv \Sigma^-_{\mathrm{u}} & 1 & 1 & \cdots & -1 & -1 & -1 & \cdots & 1 & & \\
\mathrm{E_{1u}} \equiv \Pi_{\mathrm{u}} & 2 & 2\cos\phi & \cdots & 0 & -2 & 2\cos\phi & \cdots & 0 & (x,y) & \\
\mathrm{E_{2u}} \equiv \Delta_{\mathrm{u}} & 2 & 2\cos 2\phi & \cdots & 0 & -2 & -2\cos 2\phi & \cdots & 0 & & \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & & \\ \hline
\end{array}$$

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