6.4: Diborane
- Page ID
- 149274
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Diborane - D2h 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).