When Paquette's group synthesized dodecahedrane, C 20 H 20 , they measured its infrared and Raman spectra (JACS 1983, 105, 5446-5450). Dodecahedrane: (Public Domain; Yikrazuul ). They found three IR active bands at 2945, 1298, and 728 cm -1 and eight Raman frequencies at 2924, 2938, 1324, 1164, 1092, 840, 676, and 480 cm -1 . Use group theory to show that these data are consistent with the fact that dodecahedrane has icosahedral symmetry. \[ \begin{matrix} \begin{array} E & E & C_5 & C_5^2 & & C_3& C_2& & i& & S_{10} & & S_{10}^3 & & S_6 & & \sigma \end{array} & ~ \\ \text{CIh} = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 3 & \frac{1 + \sqrt{5}}{2} & \frac{1 - \sqrt{5}}{2} & 0 & -1 & 3 & \frac{1 - \sqrt{5}}{2} & \frac{1 + \sqrt{5}}{2} & 0 & -1 \\ 3 & \frac{1 - \sqrt{5}}{2} & \frac{1 + \sqrt{5}}{2} & 0 & -1 & 3 & \frac{1 + \sqrt{5}}{2} & \frac{1 - \sqrt{5}}{2} & 0 & -1 \\ 4 & -1 & -1 & 1 & 0 & 4 & -1 & -1 & 1 & 0 \\ 5 & 0 & 0 & -1 & 1 & 5 & 0 & 0 & -1 & 1 \\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 \\ 3 & \frac{1 + \sqrt{5}}{2} & \frac{1 - \sqrt{5}}{2} & 0 & -1 & -3 & \frac{1 - \sqrt{5}}{2} & \frac{1 + \sqrt{5}}{2} & 0 & 1 \\ 3 & \frac{1 - \sqrt{5}}{2} & \frac{1 + \sqrt{5}}{2} & 0 & -1 & -3 & - \frac{1 - \sqrt{5}}{2} & - \frac{1 + \sqrt{5}}{2} & 0 & 1 \\ 4 &-1 & -1 & 1 & - & -4 & 1 & 1 & -1 & 0 \\ 5 & 0 & 0 & -1 & 1 & -5 & 0 & 0 & 1 & -1 \end{bmatrix} & \begin{array} \text{Ag: }x^2 + y^2 + z^2 \\ \text{T1g: Rx, Ry, Rz} \\ \text{T2g} \\ \text{Gg} \\ \text{Hg: }2z^2 - x^2- y^2,~x^2 - y^2, xy, yz, xz \\ \text{Au} \\ \text{T1u: x, y, z} \\ \text{T2u} \\ \text{Gu} \\ \text{Hu} \end{array} \end{matrix} \nonumber \] \[ \begin{matrix} \text{Ih} = \begin{pmatrix} 1 & 12 & 12 & 20 & 15 & 1 & 12 & 12 & 20 & 15 \end{pmatrix} & \text{Ih = Ih}^T & \Gamma \text{uma} = \begin{pmatrix} 40 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 8 \end{pmatrix} & \Gamma \text{uma} = \Gamma \text{uma}^T \end{matrix} \nonumber \] \[ \begin{matrix} \text{Ag} = ( \text{CIh}^T )^{<1>} & \text{T1g} = ( \text{CIh}^T )^{<2>} & \text{T2g} = ( \text{CIh}^T )^{<3>} & \text{Gg} = ( \text{CIh}^T )^{<4>} & \text{Hg} = ( \text{CIh}^T )^{<5>} \\ \text{Au} = ( \text{CIh}^T )^{<6>} & \text{T1u} = ( \text{CIh}^T )^{<7>} & \text{T2u} = ( \text{CIh}^T )^{<8>} & \text{Gu} = ( \text{CIh}^T )^{<9>} & \text{Hu} = ( \text{CIh}^T )^{<10>} \end{matrix} \nonumber \] \[ \begin{matrix} h = \sum Ih & h = 120 & \Gamma_{tot} = \overrightarrow{( \Gamma T1u)} & \Gamma_{vib} = \Gamma_{tot} - T1g - T1u & \Gamma_{bend} = \Gamma_{vib} - \Gamma_{stretch} & i = 1 .. 10 \end{matrix} \nonumber \] \[ \begin{matrix} \text{Vib}_i = \frac{[ \overrightarrow{ \text{Ih} ( \text{CIh}^T)^{ <i>} \Gamma \text{vib}}]}{h} & \text{Stretch}_i = \frac{[ \overrightarrow{ \text{Ih} ( \text{CIh}^T)^{ <i>} \Gamma \text{stretch}}]}{h} & \text{Bend}_i = \frac{[ \overrightarrow{ \text{Ih} ( \text{CIh}^T)^{ <i>} \Gamma \text{bend}}]}{h} \end{matrix} \nonumber \] \[ \begin{matrix} \text{Vib} = \begin{bmatrix} 2 \\ 1 \\ 2 \\ 4 \\ 6 \\ 0 \\ 3 \\ 4 \\ 4 \\ 4 \end{bmatrix} & \begin{array} \text{Ag: }x^2 + y^2 + z^2 \\ \text{T1g: Rx, Ry, Rz} \\ \text{T2g} \\ \text{Gg} \\ \text{Hg: }2z^2 - x^2- y^2,~x^2 - y^2, xy, yz, xz \\ \text{Au} \\ \text{T1u: x, y, z} \\ \text{T2u} \\ \text{Gu} \\ \text{Hu} \end{array} & \text{Stretch} = \begin{bmatrix} 2 \\ 0 \\ 0 \\ 2 \\ 3 \\ 0 \\ 2 \\ 2 \\ 2 \\ 1 \end{bmatrix} & \begin{array} \text{Ag: }x^2 + y^2 + z^2 \\ \text{T1g: Rx, Ry, Rz} \\ \text{T2g} \\ \text{Gg} \\ \text{Hg: }2z^2 - x^2- y^2,~x^2 - y^2, xy, yz, xz \\ \text{Au} \\ \text{T1u: x, y, z} \\ \text{T2u} \\ \text{Gu} \\ \text{Hu} \end{array} \end{matrix} \nonumber \] \[ \begin{matrix} \text{Bend} = \begin{bmatrix} 0 \\ 1 \\ 2 \\ 2 \\ 3 \\ 0 \\ 1 \\ 2 \\ 2 \\ 3 \end{bmatrix} & \begin{array} \text{Ag: }x^2 + y^2 + z^2 \\ \text{T1g: Rx, Ry, Rz} \\ \text{T2g} \\ \text{Gg} \\ \text{Hg: }2z^2 - x^2- y^2,~x^2 - y^2, xy, yz, xz \\ \text{Au} \\ \text{T1u: x, y, z} \\ \text{T2u} \\ \text{Gu} \\ \text{Hu} \end{array} \end{matrix} \nonumber \] According to the usual selection rules only the three T 1u vibrations are IR active. The A g and H g vibrations are Raman active giving a total of eigth frequencies in the Raman spectra. This analysis is in agreement with the experimental spectroscopic results quoted by Paquette. Note also that, as is usual for molecules with a center of inversion, there are no coincidences between the IR and Raman active modes.