For simplification think of these two categories as either frisbees for oblate tops or footballs for prolate tops. Introduction. Have questions or comments? If the unique rotational axis has a lower inertia than the degenerate axes the molecule is called a prolate symmetrical top. Generally, polyatomic molecules have complex rotational spectra. In this case, the total rotational energy Equation \(\ref{genKE}\) can be expressed in terms of the total angular momentum operator \(J^2\), As a result, the eigenfunctions of \(H_{rot}\) are those of \(J^2\) (and \(J_a\) as well as \(J_Z\) both of which commute with \(J_2\) and with one another; \(J_Z\) is the component of \(J\) along the lab-fixed Z-axis and commutes with \(J_a\) because, act on different angles. Legal. Measured in the body frame the inertia matrix (Equation \(\ref{inertiamatrix}\)) is a constant real symmetric matrix, which can be decomposed into a diagonal matrix, given by, \[ I =\left(\begin{array}{ccc}I_{a}&0&0\\0&I_{b}&0\\0&0&I_{c}\end{array}\right)\], \[H_{rot} = \dfrac{J_a^2}{2I_a} + \dfrac{J_b^2}{2I_b} + \dfrac{J_c^2}{2I_c} \label{genKE}\]. Only the molecules that have permenant electric dipole moment can absorb or emit the electromagnetic radiation in such transitions. The components of the quantum mechanical angular momentum operators along the three principal axes are: \[ \begin{align} J_a &= -i\hbar \cos χ \left[\cot θ \dfrac{∂}{∂χ} - (\sin θ )^{-1} \dfrac{∂}{∂φ} \right] - -i\hbar \sin χ \dfrac{∂}{∂θ} \\[4pt] J_b &= i\hbar \sin χ \left[\cot θ \dfrac{∂}{∂χ} - (\sin θ )^{-1} \dfrac{∂}{∂φ} \right] - -i\hbar \cos χ \dfrac{∂}{∂θ} \\[4pt] J_c &= - \dfrac{ih ∂}{∂χ} \end{align}\], The angles \(θ\), \(φ\), and \(χ\) are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. Since most of the larger polyatomic molecules possess internal rotors with low-lying torsional energy levels, their vapour phase spectra should exhibit influence of torsion on the vibrationalrotational levels. The eigenfunctions \(|J, M,K>\) are the same rotation matrix functions as arise for the spherical-top case. Consequently, organic compounds will absorb infrared radiation that corresponds in energy to these vibrations. Two simple parallel bands were observed at 8870A and 11590A. Each energy level is therefore \((2J + 1)^2\) degenarate because there are \(2J + 1\) possible K values and \(2J + 1\) possible M values for each J. In addition to rotation of groups about single bonds, molecules experience a wide variety of vibrational motions, characteristic of their component atoms. Vibrational-Rotational Spectroscopy Vibrational-Rotational Spectrum of Heteronuclear Diatomic Absorption of mid-infrared light (~300-4000 cm-1): • Molecules can change vibrational and rotational states • Typically at room temperature, only ground vibrational state populated but several rotational levels may be populated. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. To form the only non-zero matrix elements of \(H_{rot}\) within the \(|J, M, K\rangle\) basis, one can use the following properties of the rotation-matrix functions: \[\langle j, \rangle = \langle j, \rangle = 1/2

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