Symmetrical tops can be divided into two categories based on the relationship between the inertia of the unique axis and the inertia of the two axes with equivalent inertia. typically reflected in an $$3 \times 3$$ inertia tensor. 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 $$M$$. Three principal moments of inertia IA , IB , and IC designated. Techniques and instrumentation. U. V. Spectra of Diphenylselenides and Benzyl-Phenyl-Selenides The Influence of a Solvent upon the Electronic Spectra of Polyatomic Molecules Spettri nel Vicino U. V. del p-Fenossi-Difenilsolfuro e Corrispondenti Metil-, Cloro-, Nitro-, Amino- ed Acetilamino-Derivati The rotational energy in Equation $$\ref{Ediatomic}$$ can be expressed in terms of the moment of inertia $$I$$, $I =\sum_i m_i R_i^2 \label{Idiatomic}$. Chapter 5 Structure and spectra of polyatomic molecules 5.1 Structure of polyatomic molecules Thesameapproximationscanbeusedforthestationarystatesofapolyatomicmoleculeas Pure rotational Raman spectra. Molecular Spectra and Molecular Structure III - Electronic Spectra and Electronic Structure of Polyatomic Molecules | Gerhard Herzberg | download | Z-Library. 12Jan2018 Chemistry21b – Spectroscopy Lecture# 5 – Rotation of Polyatomic Molecules The rotational spectra of molecules can be classiﬁed according to their “principal moments of inertia”. 4- Raman spectroscopy. As discussed previously, the Schrödinger equation for the angular motion of a rigid (i.e., having fixed bond length $$R$$) diatomic molecule is, $\dfrac{\hbar^2}{2 μ} \left[ \dfrac{1}{R^2 \sin θ} \dfrac{∂}{∂θ} \left(\sin θ \dfrac{∂}{∂θ} \right) + \dfrac{1}{R^2 \sin^2 θ} \dfrac{∂^2}{∂φ^2} \right] |ψ \rangle = E | ψ \rangle$, $\dfrac{L^2}{2 μ R^2 } | ψ \rangle = E | ψ\rangle$, The Hamiltonian in this problem contains only the kinetic energy of rotation; no potential energy is present because the molecule is undergoing unhindered "free rotation". Symmetrical tops are molecules with two rotational axes that have the same inertia and one unique rotational axis with a different inertia. 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 the energy now depends on K, these levels are only $$2J + 1$$ degenerate due to the $$2J + 1$$ different $$M$$ values that arise for each $$J$$ value. 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. Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry. 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. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 2;:::;R~. $I =\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix} \label{inertiamatrix}$, 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 $$c$$. The eigenfunctions of $$J^2$$, $$J_Z$$ and $$J_a$$, $$|J,M,K>$$ are given in terms of the set of rotation matrices $$D_{J,M,K}$$ : $|J,M,K \rangle = \sqrt{ \dfrac{2J + 1}{8 π^2}} D^* _{J,M,K} ( θ , φ , χ )$, $J^2 |J,M,K \rangle = \hbar^2 J(J+1) | J,M,K \rangle$, $J_a |J,M,K \rangle = \hbar K | J,M,K \rangle$, $J_Z |J,M,K \rangle = \hbar M | J,M,K \rangle$. For this reason accurate determinations of vibration-rotation interactions in polyatomic molecules are more difficult to make experimentally. Download books for free. The influence of rotation on spectra of polyatomic molecules. This matrix will not be diagonal because the $$|J, M, K \rangle$$ functions are not eigenfunctions of the asymmetric top $$H_{rot}$$. 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. Generally, polyatomic molecules have complex rotational spectra. Vibrational Raman spectra. The vector coefficients express the asymmetric top eigenstates as, $\psi_n ( θ , φ , χ ) = \sum_{J, M, K} C_{n, J,M,K} |J, M, K \rangle$. 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: $\psi _{n,J} ( θ , φ , χ ) = \sum_{M, K} C_{n, J,M,K} |J, M, K \rangle$. typically reflected in an $$3 \times 3$$ inertia tensor. 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, $I =\left(\begin{array}{ccc}I_{a}&0&0\\0&I_{b}&0\\0&0&I_{c}\end{array}\right)$, $H_{rot} = \dfrac{J_a^2}{2I_a} + \dfrac{J_b^2}{2I_b} + \dfrac{J_c^2}{2I_c} \label{genKE}$. CHAPTER I: ROTATION AND ROTATION SPECTRA 13 1. $$B$$ is the rotational constant. 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