Chapter 5 Structure and spectra of polyatomic molecules 5.1 Structure of polyatomic molecules Thesameapproximationscanbeusedforthestationarystatesofapolyatomicmoleculeas Influence of Vibration-Rotation Interaction on Line Intensities in Vibration-Rotation Bands of Diatomic Molecules The Journal of Chemical Physics 23 , 637 (1955); 10.1063/1.1742069 Algebraic approach to molecular spectra: Two-dimensional problems For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The spectra of polar molecules can be measured in absorption or emission by microwave spectroscopy or by far infrared spectroscopy. They produce much more complex spectra, but basic relationships, analogous to those for a diatomic molecule, exist between their moments and the observed absorption lines. Thus each energy level is labeled by \(J\) and is \(2J+1\)-fold degenerate (because \(M\) ranges from \(-J\) to \(J\)). The rotational eigenfunctions and energy levels of a molecule for which all three principal moments of inertia are distinct (a asymmetric top) can not easily be expressed in terms of the angular momentum eigenstates and the \(J\), \(M\), and \(K\) quantum numbers. 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, \[ I =\left(\begin{array}{ccc}I_{a}&0&0\\0&I_{b}&0\\0&0&I_{c}\end{array}\right)\], \[H_{rot} = \dfrac{J_a^2}{2I_a} + \dfrac{J_b^2}{2I_b} + \dfrac{J_c^2}{2I_c} \label{genKE}\]. We can divide these molecules into four classes in order to interpret the spectra. 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". typically reflected in an \(3 \times 3\) inertia tensor. Thus each energy level is labeled by \(J\) and is \(2J+1\)-fold degenerate (because \(M\) ranges from \(-J\) to \(J\)). Each of the elements of \(J_c^2\), \(J_a^2\), and \(J_b^2\) must, of course, be multiplied, respectively, by \(1/2I_c\), \(1/2I_a\), and \(1/2I_b\) and summed together to form the matrix representation of \(H_{rot}\). 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. Electronic Spectroscopy of Polyatomics We shall discuss the electronic spectroscopy of the following types of polyatomic molecules: 1. general AH2 molecules, A = first-row element 2. formaldehyde 3. benzene & aromatic complexes (Hückel theory) 4. transition metal complexes In considering the electronic spectra of polyatomic molecules, Splitting in P and R branch due to a difference in (A-B) in upper and lower vib. Watch the recordings here on Youtube! The rotational structure of the two bands was analyzed yielding Vibrational bands, vibrational spectra A-axis N H This moment of inertia replaces \(μR^2\) in the denominator of Equation \(\ref{Ediatomic}\): \[E_J= \dfrac{\hbar^2J(J+1)}{2I} = B J(J+1) \label{Ediatomic2}\]. grating) in the photographic infra‐red with an absorbing path of up to 60 meters, obtained by multiple reflection according to the method of J. U. This matrix will not be diagonal because the \(|J, M, K \rangle\) functions are not eigenfunctions of the asymmetric top \(H_{rot}\). In addition, with the same path length the spectrum from 1.2 to 2.4μ was obtained under low resolution with a photoelectric infra‐red spectrometer. For simplification think of these two categories as either frisbees for oblate tops or footballs for prolate tops. where \(m_i\) is the mass of the \(i^{th}\) atom and \(R\) is its distance from the center of mass of the molecule. The eigenfunctions \(|J, M,K>\) are the same rotation matrix functions as arise for the spherical-top case. Find books Each of the elements of \(J_c^2\), \(J_a^2\), and \(J_b^2\) must, of course, be multiplied, respectively, by \(1/2I_c\), \(1/2I_a\), and \(1/2I_b\) and summed together to form the matrix representation of \(H_{rot}\). The eigenfunctions of \(J^2\), \(J_Z\) and \(J_a\), \(|J,M,K>\) are given in terms of the set of rotation matrices \(D_{J,M,K}\) : \[|J,M,K \rangle = \sqrt{ \dfrac{2J + 1}{8 π^2}} D^* _{J,M,K} ( θ , φ , χ ) \], \[J^2 |J,M,K \rangle = \hbar^2 J(J+1) | J,M,K \rangle \], \[J_a |J,M,K \rangle = \hbar K | J,M,K \rangle \], \[J_Z |J,M,K \rangle = \hbar M | J,M,K \rangle \]. When the potential energy surface V(R~. 13.8: Rotational Spectra of Polyatomic Molecules, [ "article:topic", "moment of inertia tensor", "Rotational of Polyatomic Molecules", "Spherical Tops", "Asymmetric Tops", "Symmetric Tops", "prolate top", "oblate top", "showtoc:no" ], These labels are assigned so that \(I_c\) is the, The rotational kinetic energy operator for a rigid non-linear polyatomic molecule is then expressed as, The assignment of semi-axes on a spheroid. The influence of the normal mode rotation (i.e. Structure of the Spectra of Diatomic Molecules Vibration-Rotation Spectra 129 ... Rotations and Vlbratlons of Polyatomic Molecules 203 Transformation From the Laboratory System to the Molecule-fixed where \(m_i\) is the mass of the \(i^{th}\) atom and \(R\) is its distance from the center of mass of the molecule. 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. 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 Pure rotation spectra of polyatomic molecules : 24 (S) Energy levels of a rigid rotor : 25: Polyatomic vibrations: normal mode calculations : 26: Polyatomic vibrations II: s-vectors, G-matrix, and Eckart condition : 27: Polyatomic vibrations III: s-vectors and H 2 O : 28: Polyatomic vibrations IV: symmetry : 29: A sprint through group theory : 30 • For a polyatomic, we often like to think in terms of the stretching or bending of a bond. Rotation of Polyatomic Molecules In contrast to diatomic molecules (Equation \ref{Idiatomic}), the rotational motions of polyatomic molecules in three dimensions are characterized by multiple moments of inertia. Splitting in Q branch due to difference in B in upper and lower vib. Rotational spectra of polyatomic molecules 4. 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. Bibliography. The first part concentrates on the theoretical aspects of molecular physics, such as the vibration, rotation, electronic states, potential curves, and spectra of molecules. If the unique rotational axis has a lower inertia than the degenerate axes the molecule is called a prolate symmetrical top. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. If the unique rotational axis has a lower inertia than the degenerate axes the molecule is called a prolate symmetrical top. typically reflected in an \(3 \times 3\) inertia tensor. The energies associated with such eigenfunctions are, \[E(J,K,M) = \dfrac{\hbar^2 J(J+1)}{2I^2}\]. Consequently, organic compounds will absorb infrared radiation that corresponds in energy to these vibrations. - Rotational spectroscopy is called pure rotational spectroscopy, to distinguish it from roto-vibrational spectroscopy (the molecule changes its state of vibration and rotation simultaneously) and vibronic spectroscopy (the molecule changes its electronic state and vibrational state simultaneously) If the unique rotational axis has a greater inertia than the degenerate axes the molecule is called an oblate symmetrical top (Figure \(\PageIndex{1}\)). 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. \(B\) is the rotational constant. White. for all K (i.e., J a quantum numbers) ranging from -J to J in unit steps and for all M (i.e., J Z quantum numbers) ranging from -J to J. Polyatomic, we often like to think in terms of the stretching or bending of bond... 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