The eigenfunctions of \(J^2\), \(J_Z\) and \(J_a\), \(|J,M,K>\) are given in terms of the set of rotation matrices \(D_{J,M,K}\) : \[|J,M,K \rangle = \sqrt{ \dfrac{2J + 1}{8 π^2}} D^* _{J,M,K} ( θ , φ , χ ) \], \[J^2 |J,M,K \rangle = \hbar^2 J(J+1) | J,M,K \rangle \], \[J_a |J,M,K \rangle = \hbar K | J,M,K \rangle \], \[J_Z |J,M,K \rangle = \hbar M | J,M,K \rangle \]. The absorption spectrum of CD 3 H has been investigated under high resolution in the photographic infrared with absorbing paths of up to 7 m atmos obtained by multiple reflection. levels 2. 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, \[ I =\left(\begin{array}{ccc}I_{a}&0&0\\0&I_{b}&0\\0&0&I_{c}\end{array}\right)\], \[H_{rot} = \dfrac{J_a^2}{2I_a} + \dfrac{J_b^2}{2I_b} + \dfrac{J_c^2}{2I_c} \label{genKE}\]. Which of the following molecules would show rotational spectrum: Br2 , HBr and CS2? White. again for K and M (i.e., \(J_a\) or \(J_c\) and \(J_Z\) quantum numbers, respectively) ranging from \(-J\) to \(J\) in unit steps. The electromagnetic field exerts a torque on the molecule. The resultant rotational energies are given as: \[E_J= \dfrac{\hbar^2J(J+1)}{2μR^2} = B J(J+1) \label{Ediatomic}\], and are independent of \(M\). The eigenvalues corresponding to each eigenfunction are straightforward to find because \(H_{rot}\) is proportional to the \(L^2\) operator whose eigenvalues have already been determined. Vibrational bands, vibrational spectra A-axis N H \[I =\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix} \label{inertiamatrix} \], The components of this tensor can be assembled into a matrix given by, \[ I_{xx}=\sum _{k=1}^{N}m_{k}(y_{k}^{2}+z_{k}^{2})\], \[ I_{yy}=\sum _{k=1}^{N}m_{k}(x_{k}^{2}+z_{k}^{2})\], \[ I_{zz}=\sum _{k=1}^{N}m_{k}(x_{k}^{2}+y_{k}^{2})\], \[ I_{yx}=I_{xy}=-\sum _{k=1}^{N}m_{k}x_{k}y_{k}\], \[ I_{zx}=I_{xz}=-\sum _{k=1}^{N}m_{k}x_{k}z_{k}\], \[ I_{zy}=I_{yz}=-\sum _{k=1}^{N}m_{k}y_{k}z_{k}.\], The rotational motions of polyatomic molecules are characterized by moments of inertia that are defined in a molecule based coordinates with axes that are labeled \(a\), \(b\), and \(c\). This matrix will not be diagonal because the \(|J, M, K \rangle\) functions are not eigenfunctions of the asymmetric top \(H_{rot}\). In contrast to diatomic molecules (Equation \ref{Idiatomic}), the rotational motions of polyatomic molecules in three dimensions are characterized by multiple moments of inertia. The angles \(θ\) and \(φ\) describe the orientation of the diatomic molecule's axis relative to a laboratory-fixed coordinate system, and \(μ\) is the reduced mass of the diatomic molecule. \(B\) is the rotational constant. When the potential energy surface V(R~. The rotational structure of the two bands was analyzed yielding 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. typically reflected in an \(3 \times 3\) inertia tensor. The eigenvalues corresponding to each eigenfunction are straightforward to find because \(H_{rot}\) is proportional to the \(L^2\) operator whose eigenvalues have already been determined. Vibrational Modes of Polyatomic Molecules Let N > 2 be the number of nuclei in a polyatomic molecule with 3N degrees of freedom. The influence of the normal mode rotation (i.e. Assume that the molecule rotates as a rigid body, that is, the relative nuclear positions are fixed. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Three principal moments of inertia IA , IB , and IC designated. If the unique rotational axis has a lower inertia than the degenerate axes the molecule is called a prolate symmetrical top. It is common in rigid body mechanics to express in these moments of inertia in lab-based Cartesian coordinates via a notation that explicitly identifies the \(x\), \(y\), and \(z\) axes such as \(I_{xx}\) and \(I_{xy}\), for the components of the inertia tensor. levels 3. Splitting in Q branch due to difference in B in upper and lower vib. Thus each energy level is labeled by \(J\) and is \(2J+1\)-fold degenerate (because \(M\) ranges from \(-J\) to \(J\)). Because the total angular momentum \(J^2\) still commutes with \(H_{rot}\), each such eigenstate will contain only one J-value, and hence \(Ψ_n\) can also be labeled by a \(J\) quantum number: \[\psi _{n,J} ( θ , φ , χ ) = \sum_{M, K} C_{n, J,M,K} |J, M, K \rangle \]. Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry. In addition to rotation of groups about single bonds, molecules experience a wide variety of vibrational motions, characteristic of their component atoms. The corresponding square of the total angular momentum operator \(J^2\) can be obtained as, \[ \begin{align} J^2 &= J_a^2 + J_b^ 2 + J_c^2 \\[4pt] & = - \dfrac{∂^2}{∂θ^2} - \cot θ \dfrac{∂}{∂θ} - \left(\dfrac{1}{\sin θ} \right) \left( \dfrac{∂^2}{∂φ^2} + \dfrac{∂^2}{∂χ^2} - 2 \cos θ \dfrac{∂^2}{∂φ∂χ} \right) \end{align}\], and the component along the lab-fixed \(Z\) axis is, When the three principal moment of inertia values are identical, the molecule is termed a spherical top. 2;:::;R~. N) is known we can com- pute the energy levels of the molecule. The spectra of polar molecules can be measured in absorption or emission by microwave spectroscopy or by far infrared spectroscopy. For prolate tops, Equation \(\ref{genKE}\) becomes, \[H_{rot} = \dfrac{J^2}{2I} + J_a^2 \left( \dfrac{1}{2I_a} - \dfrac{1}{2I} \right)\], For oblate tops, Equation \(\ref{genKE}\) becomes, \[H_{rot} = \dfrac{J^2}{2I} + J_c^2 \left( \dfrac{1}{2I_c} - \dfrac{1}{2I} \right)\]. It is oblate if, 13.9: Normal Modes in Polyatomic Molecules, Telluride Schools on Theoretical Chemistry, information contact us at info@libretexts.org, status page at https://status.libretexts.org. Classification of polyatomic molecules 3. Rotations and vibrations of polyatomic molecules. The diagonalization of this matrix then provides the asymmetric top energies and wavefunctions. \[I =\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix} \label{inertiamatrix} \], The components of this tensor can be assembled into a matrix given by, \[ I_{xx}=\sum _{k=1}^{N}m_{k}(y_{k}^{2}+z_{k}^{2})\], \[ I_{yy}=\sum _{k=1}^{N}m_{k}(x_{k}^{2}+z_{k}^{2})\], \[ I_{zz}=\sum _{k=1}^{N}m_{k}(x_{k}^{2}+y_{k}^{2})\], \[ I_{yx}=I_{xy}=-\sum _{k=1}^{N}m_{k}x_{k}y_{k}\], \[ I_{zx}=I_{xz}=-\sum _{k=1}^{N}m_{k}x_{k}z_{k}\], \[ I_{zy}=I_{yz}=-\sum _{k=1}^{N}m_{k}y_{k}z_{k}.\], The rotational motions of polyatomic molecules are characterized by moments of inertia that are defined in a molecule based coordinates with axes that are labeled \(a\), \(b\), and \(c\). Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. The eigenfunctions of \(J^2\), \(J_Z\) and \(J_a\), \(|J,M,K>\) are given in terms of the set of rotation matrices \(D_{J,M,K}\) : \[|J,M,K \rangle = \sqrt{ \dfrac{2J + 1}{8 π^2}} D^* _{J,M,K} ( θ , φ , χ ) \], \[J^2 |J,M,K \rangle = \hbar^2 J(J+1) | J,M,K \rangle \], \[J_a |J,M,K \rangle = \hbar K | J,M,K \rangle \], \[J_Z |J,M,K \rangle = \hbar M | J,M,K \rangle \]. The eigenfunctions \(|J, M,K>\) are the same rotation matrix functions as arise for the spherical-top case. LINEAR MOLECULES 13 Energy levels, 14-—Symmetry properties, 15—Statistical weights and influence of nuclear spin and statistics, 16—Thermal distribu­ tion of rotational levels, 18—Infrared rotation spectrum, 19—• Rotational Raman spectrum, 20 2. The corresponding square of the total angular momentum operator \(J^2\) can be obtained as, \[ \begin{align} J^2 &= J_a^2 + J_b^ 2 + J_c^2 \\[4pt] & = - \dfrac{∂^2}{∂θ^2} - \cot θ \dfrac{∂}{∂θ} - \left(\dfrac{1}{\sin θ} \right) \left( \dfrac{∂^2}{∂φ^2} + \dfrac{∂^2}{∂χ^2} - 2 \cos θ \dfrac{∂^2}{∂φ∂χ} \right) \end{align}\], and the component along the lab-fixed \(Z\) axis is, When the three principal moment of inertia values are identical, the molecule is termed a spherical top. typically reflected in an \(3 \times 3\) inertia tensor. Splitting in P and R branch due to a difference in (A-B) in upper and lower vib. Watch the recordings here on Youtube! 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 \ ) are the same inertia and one unique rotational axis with different! 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