Diborane - D 2h Symmetry Diborane has 18 vibrational degrees of freedom. Nine modes are Raman active and eight are IR active. The experimental results are provided in the table below. Do a symmetry analysis to confirm the assignments given below, and identify stretches and bends. \[ \begin{pmatrix} D_{2h} & A_g & A_g & A_g & A_g & B_{1g} & B_{1g} & B_{2g} & B_{2g} & B_{3g} \\ \frac{ \text{Raman}}{ \text{cm}} & 2524 & 2104 & 1180 & 794 & 1768 & 1035 & 2591 & 920 & 1012 \\ D_{2h} & A_u & B_{1u} & B_{1u} & B_{1u} & B_{2u} & B_{2u} & B_{3u} & B_{3u} & B_{3u} \\ \frac{ \text{IR}}{ \text{cm}} & 0 & 2612 & 950 & 368 & 1915 & 973 & 2525 & 1606 & 1177 \end{pmatrix} \nonumber \] \[ \begin{matrix} \begin{array} E & & & E & C_2^z & C_2^y & C_2^x & i& \sigma_{xy} & \sigma_{xz} & \sigma_{yz} & \end{array} & ~ \\ \text{C}_{D2h} = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \end{pmatrix} & \begin{array} \text{A}_g:~ x^2,~ y^2,~ z^2 \\ \text{B}_{1g}:~ R_x,~xy \\ \text{B}_{2g}:~ R_y,~xz \\ \text{B}_{3g}:~ R_x,yx \\ \text{A}_u \\ \text{B}_{1u}:~z \\ \text{B}_{2u}:~y \\ \text{B}_{3u}:~x \end{array} & \text{D2h} = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} & \Gamma_{uma} = \begin{pmatrix} 8 \\ 0 \\ 2 \\ 2 \\ 0 \\ 4 \\ 6 \\ 2 \end{pmatrix} & \Gamma_{bonds} = \begin{pmatrix} 8 \\ 0 \\ 0 \\ 0 \\ 0 \\ 4 \\ 4 \\ 0 \end{pmatrix} \end{matrix} \nonumber \] \[ \begin{matrix} \text{A}_{g} = ( \text{C}_{D4h}^T )^{<1>} & \text{B}_{2g} = ( \text{C}_{D4h}^T )^{<2>} & \text{B}_{2g} = ( \text{C}_{D4h}^T )^{<3>} & \text{B}_{3g} = ( \text{C}_{D4h}^T )^{<4>} & ~ \\ \text{A}_{u} = ( \text{C}_{D4h}^T )^{<5>} & \text{B}_{1u} = ( \text{C}_{D4h}^T )^{<6>} & \text{B}_{2u} = ( \text{C}_{D4h}^T )^{<7>} & \text{B}_{3u} = ( \text{C}_{D4h}^T )^{<8>} & h = \sum \text{D2h} \end{matrix} \nonumber \] \[ \begin{matrix} \Gamma_{trans} = B_{1u} + B_{2u} + B_{3u} & \Gamma_{tot} = B_{1g} + B_{2g} + B_{3g} & \Gamma_{tot} = \overrightarrow{( \Gamma_{uma} \Gamma_{trans})} \\ \Gamma_{vib} = \Gamma_{tot} - \Gamma_{trans} - \Gamma_{rot} & \Gamma_{vib}^T = \begin{pmatrix} 18 & 2 & 0 & 0 & 0 & 4 & 6 & 2 \end{pmatrix} & i = 1 .. 8 \\ \Gamma_{stretch} = \Gamma_{bonds} & \Gamma_{bend} = \Gamma_{vib} - \Gamma_{stretch} \end{matrix} \nonumber \] \[ \begin{matrix} \text{Vib}_i = \frac{ \sum \overrightarrow{[ D2h ( C_{D2h}^T )^{<i>} \Gamma_{vib}]}}{h} & \text{Stretch}_i = \frac{ \sum \overrightarrow{[ D2h ( C_{D2h}^T )^{<i>} \Gamma_{stretch}]}}{h} & \text{Bend}_i = \frac{ \sum \overrightarrow{[ D2h ( C_{D2h}^T )^{<i>} \Gamma_{bend}]}}{h} \end{matrix} \nonumber \] \[ \begin{matrix} \text{Vib} = \begin{pmatrix} 4 \\ 2 \\ 2 \\ 1 \\ 1 \\ 3 \\ 2 \\ 3 \end{pmatrix} \begin{array} \text{A}_g:~ x^2,~ y^2,~ z^2 \\ \text{B}_{1g}:~ R_x,~xy \\ \text{B}_{2g}:~ R_y,~xz \\ \text{B}_{3g}:~ R_x,yx \\ \text{A}_u \\ \text{B}_{1u}:~z \\ \text{B}_{2u}:~y \\ \text{B}_{3u}:~x \end{array} & \text{Stretch} = \begin{pmatrix} 2 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 2 \end{pmatrix} \begin{array} \text{A}_g:~ x^2,~ y^2,~ z^2 \\ \text{B}_{1g}:~ R_x,~xy \\ \text{B}_{2g}:~ R_y,~xz \\ \text{B}_{3g}:~ R_x,yx \\ \text{A}_u \\ \text{B}_{1u}:~z \\ \text{B}_{2u}:~y \\ \text{B}_{3u}:~x \end{array} & \text{Bend} = \begin{pmatrix} 2 \\ 1 \\ 1 \\ 1 \\ 1 \\ 2 \\ 1 \\ 1 \end{pmatrix} \begin{array} \text{A}_g:~ x^2,~ y^2,~ z^2 \\ \text{B}_{1g}:~ R_x,~xy \\ \text{B}_{2g}:~ R_y,~xz \\ \text{B}_{3g}:~ R_x,yx \\ \text{A}_u \\ \text{B}_{1u}:~z \\ \text{B}_{2u}:~y \\ \text{B}_{3u}:~x \end{array}\end{matrix} \nonumber \] This analysis is in agreement with the experimental data. There are 9 Raman active modes and 8 IR active modes. Furthermore there are 4 Raman stretches at 2524 (Ag), 2104 (Ag), 1768 (B1g), and 2591 (B2g). The five Raman bends occur at 1180 (Ag), 794 (Ag), 1035 (B1g), 920 (B2g), and 1012 (B3g). The 4 IR stretches occur at 2612 (B1u), 1915 (B2u), 2525 (B3u), and 1606 (B3u). The bends appear at 950 (B1u), 368 (B1u), 973 (B2u), 1177 (B3u).