6.10: Tetrahedrane
- Page ID
- 149288
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Symmetry Analysis for Tetrahedrane
Tetrahedrane, C4H4, belongs to the Td point group. Use group theory to predict the number of IR and Raman acitive vibrational modes it has. To date tetrahedrane has not been synthesized.
\[ \begin{matrix} ~ & \begin{array} E E & C_3 & C_2 & S_4 & \sigma \end{array} \\ C_{Td} = & \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & -1 & -1 \\ 2 & -1 & 2 & 0 & 0 \\ 3 & 0 & -1 & 1 & -1 \\ 3 & 0 & -1 & -1 & 1 \end{pmatrix} & \begin{array} A_1:~x^2+y^2+z^2 \\ A_2 \\ E:~ 2z^2-x^2-y^2,~x^2-y^2 \\ T_1:~ (R_x,~R_y,~R_z) \\ T_2:~(x,~y,~z),~(xy,~xz,~yz) \end{array} & Td = \begin{pmatrix} 1 \\ 8 \\ 3 \\ 6 \\ 6 \end{pmatrix} & \Gamma_{uma} = \begin{pmatrix} 8 \\ 2 \\ 0 \\ 0 \\ 4 \end{pmatrix} & \Gamma_{bonds} = \begin{pmatrix} 10 \\ 1 \\ 2 \\ 0 \\ 4 \end{pmatrix} \end{matrix} \nonumber \]
\[ \begin{matrix} A_1 = (C_{Td}^T )^{<1>} & A_2 = ( C_{Td}^T )^{<2>} & E = ( C_{Td}^T )^{<3>} & T_1 = ( C_{Td}^T )^{<4>} \\ T_2 = ( C_{Td}^T )^{<5>} & \Gamma_{tot} = \overrightarrow{ ( \Gamma_{uma} T_2)} & h = \sum Td & \Gamma_{tot}^T = \begin{pmatrix} 20 & 0 & 0 & 0 & 4 \end{pmatrix} \\ i = 1 .. 5 \end{matrix} \nonumber \]
\[ \begin{matrix} \Gamma_{vib} = \Gamma_{tot} - T_1 - T_2 & \text{Vib}_i = \frac{ \sum \overrightarrow{[Td \left( C_{Td}^T \right)^{<i>} \Gamma_{vib}]}}{h} & \text{Vib} = \begin{pmatrix} 2 \\ 0 \\ 2 \\ 1 \\ 3 \end{pmatrix} & \begin{array} A_1:~x^2+y^2+z^2 \\ A_2 \\ E:~ 2z^2-x^2-y^2,~x^2-y^2 \\ T_1:~ (R_x,~R_y,~R_z) \\ T_2:~(x,~y,~z),~(xy,~xz,~yz) \end{array} \\ \Gamma_{stretch} = \Gamma_{bonds} & \text{Stretch}_i = \frac{ \sum \overrightarrow{[Td \left( C_{Td}^T \right)^{<i>} \Gamma_{stretch}]}}{h} & \text{Stretch} = \begin{pmatrix} 2 \\ 0 \\ 1 \\ 0 \\ 2 \end{pmatrix} & \begin{array} A_1:~x^2+y^2+z^2 \\ A_2 \\ E:~ 2z^2-x^2-y^2,~x^2-y^2 \\ T_1:~ (R_x,~R_y,~R_z) \\ T_2:~(x,~y,~z),~(xy,~xz,~yz) \end{array} \\ \Gamma_{bend} = \Gamma_{vib} - \Gamma_{stretch} & \text{Bend}_i = \frac{ \sum \overrightarrow{[Td \left( C_{Td}^T \right)^{<i>} \Gamma_{bend}]}}{h} & \text{Bend} = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 1 \\ 1 \end{pmatrix} & \begin{array} A_1:~x^2+y^2+z^2 \\ A_2 \\ E:~ 2z^2-x^2-y^2,~x^2-y^2 \\ T_1:~ (R_x,~R_y,~R_z) \\ T_2:~(x,~y,~z),~(xy,~xz,~yz) \end{array} \end{matrix} \nonumber \]
According to the selection rules, tetrahedrane should have three IR active modes (3T2) and seven Raman active modes (2A1 + 2E + 3T2). Two of the IR modes are stretches, while five of the Raman modes are stretches.