6.3: Xenon Tetrafluoride (2024)

Tetrahedral Symmetry - T_d

\[ \begin{matrix} \begin{array} E & E & C_3 & C_2 & & S_4 & \sigma \end{array} & ~ \\ \text{CIh} = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & -1 & -1 \\ 2 & -1 & 2 & 0 & 0 \\ 3 & 0 & -1 & 1 & -1 \\ 3 & 0 & -1 & -1 & 1 \end{bmatrix} & \begin{array} \text{A}_1: ~ x^2 + y^2 + z^2 \\ \text{A}_2 \\ \text{E}:~2z^2-x^2y^2, x^2-y^2 \\ \text{T}_1:~(R_x,~R_y,~R_z) \\ \text{T}_2: \text{ (x, y, z)(xy, xz, yz)} \end{array} & \text{Td} = \begin{bmatrix} 1 \\ 8 \\ 3 \\ 6 \\ 6 \end{bmatrix} & \Gamma_{uma} = \begin{bmatrix} 5 \\ 2 \\ 1 \\ 1 \\ 3 \end{bmatrix} \end{matrix} \nonumber \]

\[ \begin{matrix} A_1 = (C_{Td}^T)^{<1>} & A_2 = ( C_{Td}^T )^{<2>} & E = ( C_{Td}^T )^{<3>} & T_1 = (C_{Td}^T )^{<4>} \\ T_2 = (C_{Td}^T )^{<5.} & \Gamma_{tot} = \overrightarrow{ \Gamma_{uma} T_2} & h = \sum Td & \Gamma_{tot}^T = \begin{pmatrix} 15 & 0 & -1 & -1 & 3 \end{pmatrix} \end{matrix} \nonumber \]

\[ \begin{matrix} \Gamma_{vib} = \Gamma_{tot} - T_1 - T_2 & \text{Vib}_i = \frac{ \sum \overrightarrow{ [Td (C Td^T)^{<i>} \Gamma_{vib}]}}{h} & \text{Vib} = \begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \\ 2 \end{bmatrix} \begin{array} \text{A}_1:~ x^2 + y^2 +z^2 \\ \text{A}_2 \\ \text{E: } 2z^2 -x^2 - y^2, x^2 - y^2 \\ \text{T}_1:~ (R_x,~R_y,~R_z) \\ \text{T}_2:~ (x,~y,~z)(xy,~xz,~yz) \end{array} \end{matrix} \nonumber \]

This analysis predicts two IR active modes (2T₂) and four Raman active modes (A₁, E, 2T₂). It also suggests that the IR and Raman should have the two T₂ modes in common. Thus, tetrahedral geometry is not consistent with the spectroscopic data. Further detail can be obtained by noting that the stretching vibrations have the same symmetry properties as the chemical bonds. This allows the vibrational modes to be decomposed further into the symmetry of the stretches and bends.

Square Planar Geometry - D_4H

\[ \begin{matrix} \begin{array} E & & E & C_4 & C_2 & C_2' & C_2"& i& S_4& & \sigma_h & \sigma_v & \sigma_v & \end{array} & ~ \\ \text{CIh} = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 \\ 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 \\ 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 \\ 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1\\ 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 \end{bmatrix} & \begin{array} \text{A1g: }x^2 + y^2 + z^2 \\ \text{A2g: Rz} \\ \text{B1g: } x^2-y^2\\ \text{B2g: xy} \\ \text{Eg: (Rx, Ry), xz, yz)} \\ \text{A1u} \\ \text{A2u: z} \\ \text{B1u} \\ \text{B2u} \\ \text{Eu: (x, y)} \end{array} & \text{D4h} = \begin{bmatrix} 1 \\ 2 \\ 1 \\ 2 \\ 2 \\ 1 \\ 2 \\ 1 \\ 2 \\ 2 \end{bmatrix} & \Gamma_{uma} = \begin{bmatrix} 5 \\ 1 \\ 1 \\ 3 \\ 1 \\ 1 \\ 1 \\ 5 \\ 3 \\ 1 \end{bmatrix} & \Gamma_{bonds} = \begin{bmatrix} 4 \\ 0 \\ 0 \\ 2 \\ 0 \\ 0 \\ 0 \\ 4 \\ 2 \\ 0 \end{bmatrix} \end{matrix} \nonumber \]

\[ \begin{matrix} \text{A}_{1g} = ( \text{C}_{D4h}^T )^{<1>} & \text{A}_{2g} = ( \text{C}_{D4h}^T )^{<2>} & \text{B}_{1g} = ( \text{C}_{D4h}^T )^{<3>} & \text{B}_{2g} = ( \text{C}_{D4h}^T )^{<4>} & \text{E}_g = ( \text{C}_{D4h}^T )^{<5>} \\ \text{A}_{1u} = ( \text{C}_{D4h}^T )^{<6>} & \text{A}_{2u} = ( \text{C}_{D4h}^T )^{<7>} & \text{B}_{1u} = ( \text{C}_{D4h}^T )^{<8>} & \text{B}_{2u} = ( \text{C}_{D4h}^T )^{<9>} & \text{E}_u = ( \text{C}_{D4h}^T )^{<10>} \end{matrix} \nonumber \]

\[ \begin{matrix} h = \sum D4h & \Gamma_{trans} = A_{2u} + E_u & \Gamma_{rot} = A_{2g} + E_g & \Gamma_{tot} = \overrightarrow{( \Gamma_{uma} \Gamma_{trans})} \end{matrix} \nonumber \]

\[ \begin{matrix} \Gamma_{vib} = \Gamma_{tot} - \Gamma_{trans} - \Gamma_{rot} & \Gamma_{stretch} = \Gamma_{bonds} & \Gamma_{bend} = \Gamma_{vib} - \Gamma_{stretch} & i = 1 .. 10 \end{matrix} \nonumber \]

\[ \begin{matrix} \text{Vib}_i = \frac{ \sum \overrightarrow{[ D4h ( C_{D4h}^T )^{<i>} \Gamma_{vib}]}}{h} & \text{Stretch}_i = \frac{ \sum \overrightarrow{[ D4h ( C_{D4h}^T )^{<i>} \Gamma_{stretch}]}}{h} & \text{Bend}_i = \frac{ \sum \overrightarrow{[ D4h ( C_{D4h}^T )^{<i>} \Gamma_{bend}]}}{h} \end{matrix} \nonumber \]

\[ \begin{matrix} \text{Vib} = \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1 \\ 2 \end{bmatrix} \begin{array} \text{A1g: }x^2 + y^2 + z^2 \\ \text{A2g: Rz} \\ \text{B1g: } x^2-y^2\\ \text{B2g: xy} \\ \text{Eg: (Rx, Ry), xz, yz)} \\ \text{A1u} \\ \text{A2u: z} \\ \text{B1u} \\ \text{B2u} \\ \text{Eu: (x, y)} \end{array} & \Gamma_{uma} = \begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \begin{array} \text{A1g: }x^2 + y^2 + z^2 \\ \text{A2g: Rz} \\ \text{B1g: } x^2-y^2\\ \text{B2g: xy} \\ \text{Eg: (Rx, Ry), xz, yz)} \\ \text{A1u} \\ \text{A2u: z} \\ \text{B1u} \\ \text{B2u} \\ \text{Eu: (x, y)} \end{array} & \Gamma_{bonds} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1 \\ 1 \end{bmatrix} \begin{array} \text{A1g: }x^2 + y^2 + z^2 \\ \text{A2g: Rz} \\ \text{B1g: } x^2-y^2\\ \text{B2g: xy} \\ \text{Eg: (Rx, Ry), xz, yz)} \\ \text{A1u} \\ \text{A2u: z} \\ \text{B1u} \\ \text{B2u} \\ \text{Eu: (x, y)} \end{array} \end{matrix} \nonumber \]

This analysis predicts three IR active modes (A_2u, 2E_u) and three Raman active modes (A_1g, B_1g, B_2g). It also indicates that there are no coincidences between the IR and Raman spectra. Thus, square planar geometry is consistent with the spectroscopic data. Examining the symmetry of the stretches and bends adds further support to this conclusion.

This analysis also predicts two Raman (A_1g, B_1g) and one IR (E_u) active stretches, and two IR (A_2u, E_u) and one Raman (B_2g) active bends. This is also consistent with the experiemental data.

References

• J. Am. Chem. Soc. 1963, 85, 1927;
• J. Phys. Chem. 1971, 54, 5247.