6.10: Tetrahedrane
- Page ID
- 149288
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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.