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6.5: Cubane

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    149276
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    Cubane, C8H8, has 42 vibrational degrees of freedom, but only three IR active modes. Cubane belongs to the octahedral point group. Show that group theory predicts three IR active modes. Determine how many vibrational modes will be Raman active. Will there be any coincidences between the IR and Raman active modes? The synthesis and characterization of cubane was reported in 1964 by Philip Eaton and Thomas Cole in JACS 1964, 86, 3157-3158. They reported three IR bands at 3000, 1231, and 851 cm-1.

    \[ \begin{matrix} \begin{array} E & & E & C_3 & C_2 & C_4 & C_2" & i & S_{4} & S_{6} & \sigma_h & \sigma_d \end{array} & ~ \\ \text{C}_{Oh} = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & 1 & 1 \\ 2 & -1 & 0 & 0 & 2 & 2 & 0 & -1 & 2 & 0 \\ 3 & 0 & -1 & 1 & -1 & 3 & 1 & 0 & -1 & 0 \\ 3 & 0 & 1 & -1 & -1 & 3 & -1 & 0 & -1 & 1 \\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 \\ 2 & -1 & 0 & 0 & 2 & -2 & 0 & 1 & -2 & 0 \\ 3 & 0 & -1 & 1 & -1 & -3 & -1 & 0 & 1 & 1 \\ 3 & 0 & 1 & -1 & -1 & -3 & 1 & 0 & 1 & -1 \end{bmatrix} & \begin{array} \text{A1g: }x^2 + y^2 + z^2 \\ \text{A2g} \\ \text{Eg: } 2z^2-x^2-y^2,~x^2-y^2 \\ \text{T1g: Rx, Ry, Rz} \\ \text{T2g: }xy,~xz,~yz \\ \text{A1u:} \\ \text{A2u} \\ \text{Eu} \\ \text{T1u: x, y, z} \\ \text{T2u} \end{array} \end{matrix} \nonumber \]

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

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

    \[ \begin{matrix} \text{Vib} = \begin{bmatrix} 2 \\ 0 \\ 2 \\ 1 \\ 4 \\ 0 \\ 2 \\ 2 \\ 3 \\ 2 \end{bmatrix} & \begin{array} \text{A1g: }x^2 + y^2 + z^2 \\ \text{A2g} \\ \text{Eg: } 2z^2-x^2-y^2, x^2-y^2 \\ \text{T1g: Rx, Ry, Rz} \\ \text{A1u:} \\ \text{A2u} \\ \text{Eu} \\ \text{T1u: x, y, z} \\ \text{T2u} \end{array} & \text{Stretch} = \begin{bmatrix} 2 \\ 0 \\ 1 \\ 0 \\ 2 \\ 0 \\ 1 \\ 0 \\ 2 \\ 1 \end{bmatrix} \begin{array} \text{A1g: }x^2 + y^2 + z^2 \\ \text{A2g} \\ \text{Eg: } 2z^2-x^2-y^2, x^2 - y^2 \\ \text{T1g: Rx, Ry, Rz} \\ \text{A1u:} \\ \text{A2u} \\ \text{Eu} \\ \text{T1u: x, y, z} \\ \text{T2u} \end{array} & \text{Bend} = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \\ 2 \\ 0 \\ 1 \\ 2 \\ 1 \\ 1 \end{bmatrix} \begin{array} \text{A1g: }x^2 + y^2 + z^2 \\ \text{A2g} \\ \text{Eg: } 2z^2-x^2-y^2, x^2-y^2 \\ \text{T1g: Rx, Ry, Rz} \\ \text{A1u:} \\ \text{A2u} \\ \text{Eu} \\ \text{T1u: x, y, z} \\ \text{T2u} \end{array} \end{matrix} \nonumber \]

    Only the three T1u vibrational modes are IR active, which is consistent with the spectroscopic data. The character table indicates that the A1g, Eg, and T2g modes are Raman active. Thus there should be 8 Raman active modes. These have been observed at 2996, 2978, 1185, 1081, 1002, 912, 826, and 665 cm-1 (J. Phys. Chem. 1981, 85, 2186). There should be no coincidences between the IR and Raman modes because cubane has a center of inversion.

    The vibrational modes can be sorted into stretches and bends by determining how the chemical bonds transform under the symmetry operations of the octahedral group. The symmetry of the stretching modes is the same as the symmetry of the bonds.

    This analysis tells us that there are two IR active stretches (2T1u) and five Raman active stretches (2A1g, Eg, and 2T2g). This is consistent with the experimental spectra in that stretches generally occur at a higher frequency than bends.


    This page titled 6.5: Cubane is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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