6.10: Tetrahedrane

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.