6.2: Dodecahedrane
- Page ID
- 149259
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When Paquette's group synthesized dodecahedrane, C20H20, 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 T1u vibrations are IR active. The Ag and Hg 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.