6.9: P₄

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T_d - Tetrahedral Symmetry for P₄

The following Raman and IR frequencies have been observed for the tetrahedral P₄ molecule. Is the assignment of tetrahedral geometry to this molecule in agreement with the spectroscopic data? Explain.

\[ \begin{pmatrix} R & R & R,IR \\ \frac{614}{ \text{cm}} & \frac{372}{ \text{cm}} & \frac{466}{ \text{cm}} \end{pmatrix} \nonumber \]

Screen Shot 2019-05-02 at 4.13.31 PM.png

\[ \begin{matrix} ~ & \begin{array} E E & C_3 & C_2 & S_4 & \sigma \end{array} \\ C_{Td} = & \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & -1 & -1 \\ 2 & -1 & 2 & 0 & 0 \\ 3 & 0 & -1 & 1 & -1 \\ 3 & 0 & -1 & -1 & 1 \end{pmatrix} & \begin{array} A A_1:~x^2 + y^2+z^2 \\ A_2 \\ E:~2z^2-x^2-y^2,~x^-y^2 \\ T_1:~(R_x,~R_y,~R_z) \\ T_2:~ (x,~y,~z),~(xy,~xz,~yz) \end{array} & Td = \begin{pmatrix} 1 \\ 8 \\ 3 \\ 6 \\ 6 \end{pmatrix} & \Gamma_{uma} = \begin{pmatrix} 4 \\ 1 \\ 0 \\ 0 \\ 2 \end{pmatrix} & \Gamma_{bonds} = \begin{pmatrix} 6 \\ 0 \\ 2 \\ 0 \\ 2 \end{pmatrix} \end{matrix} \nonumber \]

\[ \begin{matrix} A_1 = (C_{Td}^T)^{<1>} & A_2 = (C_{Td}^T)^{<2>} & E = (C_{Td}^T)^{<3>} & T_1 = (C_{Td}^T)^{<4>} \\ T_2 = (C_{Td}^T)^{<5>} & \Gamma_{tot} = \overrightarrow{( \Gamma_{uma} T_2)} & h = \sum Td & \Gamma_{tot}^T = \begin{pmatrix} 12 & 0 & 0 & 0 & 2 \end{pmatrix} & i = 1 .. 5 \end{matrix} \nonumber \]

\[ \begin{matrix} \Gamma_{vib} = \Gamma_{tot} - T_1 - T_2 & \text{Vib}_i = \frac{ \sum \overrightarrow{ \left[ Td (C_{Td}^T )^{<i>} \Gamma_{vib} \right]}}{h} & \text{Vib} = \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ 1 \end{pmatrix} \begin{array} A A_1:~x^2 + y^2+z^2 \\ A_2 \\ E:~2z^2-x^2-y^2,~x^-y^2 \\ T_1:~(R_x,~R_y,~R_z) \\ T_2:~ (x,~y,~z),~(xy,~xz,~yz) \end{array} \\ \Gamma_{stretch} = \Gamma_{bonds} & \text{Stretch}_i = \frac{ \sum \overrightarrow{ \left[ Td (C_{Td}^T )^{<i>} \Gamma_{stretch} \right]}}{h} & \text{Stretch} = \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ 1 \end{pmatrix} \begin{array} A A_1:~x^2 + y^2+z^2 \\ A_2 \\ E:~2z^2-x^2-y^2,~x^-y^2 \\ T_1:~(R_x,~R_y,~R_z) \\ T_2:~ (x,~y,~z),~(xy,~xz,~yz) \end{array} \\ \Gamma_{bend} = \Gamma_{vib} - \Gamma_{stretch} & \text{Bend}_i = \frac{ \sum \overrightarrow{ \left[ Td (C_{Td}^T )^{<i>} \Gamma_{bend} \right]}}{h} & \text{Bend} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \begin{array} A A_1:~x^2 + y^2+z^2 \\ A_2 \\ E:~2z^2-x^2-y^2,~x^-y^2 \\ T_1:~(R_x,~R_y,~R_z) \\ T_2:~ (x,~y,~z),~(xy,~xz,~yz) \end{array} \end{matrix} \nonumber \]

The group theoretical analysis assuming tetrahedral geometry is in excellent agreement with the spectroscopic data. Group theory predicts one IR active mode, and that it is coincident with a Raman frequency. This is observed with the T₂ vibration. In addition theory predicts that there are two additional Raman active modes A₁ and E.