Molecular Vibration

6 Key excerpts on "Molecular Vibration"

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  Modern Vibrational Spectroscopy and Micro-Spectroscopy

  Theory, Instrumentation and Biomedical Applications

  • Max Diem(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Chapter 1 Molecular Vibrational Motion

  The atoms in matter – be it in gaseous, liquid, or condensed phases – are in constant motion. The amplitude of this motion increases with increasing temperature; however, even at absolute zero temperature, it never approaches zero or perfect stillness. Furthermore, the amplitude of the atomic motion is a measure of the thermodynamic heat content as measured by the product of the specific heat times the absolute temperature. If one could observe the motion in real time – which is not possible because the motions occur at a timescale of about 1013 Hz – one would find that it is completely random and that the atoms are most likely to be found in ellipsoidal regions in space, such as the ones depicted in X-ray crystallographic structures. Yet, the random motion can be decomposed into distinct “normal modes of vibration.” These normal modes can be derived from classical physical principles (see Section 1.2 ) and are defined as follows: in a normal mode, all atoms vibrate, or oscillate, at the same frequency and phase, but with different amplitudes, to produce motions that are referred as symmetric and antisymmetric stretching, deformation, twisting modes, and so on. In general, a molecule with N atoms will have 3N − 6 normal modes of vibrational normal modes.

  At this point, a discrepancy arises between the classical (Newtonian) description of the motion of atoms in a molecule and the quantum mechanical description. While in the classical description the amplitude of the motion, and thereby the kinetic energy of the moving atoms, can increase in arbitrarily small increments, the quantum mechanical description predicts that the increase in energy is quantized, and that infrared (IR) photons can be absorbed by a vibrating molecular system to increase the energy along one of the normal modes of vibration.

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  Structural Methods in Molecular Inorganic Chemistry

  • D. W. H. Rankin, Norbert Mitzel, Carole Morrison(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Figure 2.14 , alongside those for the harmonic potential. For the anharmonic oscillator, it is clear that the vibrational energy levels associated with a particular vibration have diminishing separation with increasing quantum number, but there is a finite number of them below the bond dissociation limit. In contrast, the harmonic model has equally spaced vibrational levels that in principle go on for ever. Around the equilibrium structure, however, the two functions are broadly similar. This is the validation for the harmonic approximation, which is routinely used in the calculation of vibrational frequencies by quantum mechanical methods (Section 3.3).

  8.3 Observing Molecular Vibrations

  As chemical bonds involve electrostatic attraction between positively charged atomic nuclei and negatively charged electrons, the displacements of atoms during a Molecular Vibration lead to distortions in the electric charge distribution of the molecule, which can be resolved into dipole, quadrupole, etc. terms in various directions. Molecular Vibrations therefore lead to oscillations of electric charge, with frequencies governed by the normal vibration frequencies of the system.

  Oscillating molecular dipoles are observed by infrared spectroscopy. Raman spectroscopy, which depends on the polarization of the electron cloud induced by an incident light beam, reveals the changes in energy due to vibrational excitations when light is scattered. These are by far the most important ways of studying vibrations, and are described in detail in the following sections, but vibrational frequencies may also be determined by analysis of high-resolution electronic spectra (Section 9.4.2; see also the on-line supplement to chapter 8), and by inelastic scattering of neutrons and electrons (Section 8.3.4).

  8.3.1 Absorption in the Infrared

  An oscillating molecular dipole can interact directly with the oscillating electric vector of electromagnetic radiation of the same frequency, leading to resonant absorption of radiation. The quantum energy of the radiation (hv ) is equal to the quantum energy of the oscillator (hω ) if ν ≡ ω, so resonant absorption introduces exactly enough energy to raise the oscillator from a level v to the level (v + 1) (Figure 8.2(a) ). Molecular Vibrations, which have frequencies that are generally between 1011 and 1013 Hz, corresponding to 30–3000 cm−1

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  Quantum Mechanics

  • T Y Wu, T D Lee;C N Yang;;(Authors)
  • 1986(Publication Date)
  • WSPC

    (Publisher)

  Chapter 10 Quantum Mechanics of Molecules 1. Introduction A stable molecule in the normal state may be viewed either as a system of electrons in the field of the atomic nuclei, or as a number of atomic nuclei imbedded in a distribution of electrons. Just as in atoms, the electrons in a molecule can exist in different energy states. These states are quantized, as in the case of atoms. This was established directly by experiments on the excita-tion by electron impact—the Franck-Hertz experiment, or by the analysis of the electronic spectra of molecules. Now, in any given electronic state of a stable molecule, the atomic nuclei are capable of executing small oscillations about their equilibrium position. By virtue of the very much larger masses of the atomic nuclei compared with that of the electron, one may picture these vibrational motions as those of the atomic ions (formed by the nuclei with their more tightly bound electrons) in the (time) average field of the electrons that are responsible for the formation of the stable molecular state. To a first approximation, the vibrational motions of a molecule are describable as the normal vibrations of a system of particles from their equilibrium position in classical dynamics. The vibrational states are quantized. Now a non-vibrating molecule certainly can rotate as a whole, the rota-tional motion being also quantized. To a zeroth-order approximation, one regards the electronic, vibrational and rotational motion as separate and independent, and their energies are additive* £ = £„ + £„ + E r . Here n, v, r denote the totality of electronic, vibrational and rotational quan-tum numbers. E a corresponds to the electronic states and is of the order of a few electron volts (eV); E v , depending on the molecule and the modes of vibrations, ranges from 0.4 eV for such light chemical bonds as CH (in C 2 H 2 , say) to 0.03-0.10 eV in CC1 4 .

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  Mathematics for Chemistry and Physics

  • George Turrell(Author)
  • 2001(Publication Date)
  • Academic Press

    (Publisher)

  9 Molecular Mechanics A molecule is composed of a certain number N of nuclei and usually a much larger number of electrons. As the masses of the electrons and the nuclei are significantly different, the much lighter electrons move rapidly to create the so-called electron cloud which “sticks” the nuclei into relatively fixed equilibrium positions. The resulting geometry of the nuclear configuration is usually referred to as the molecular structure. The vibrational and rotational spectra of a molecule, as observed in its infrared absorption or emission and the Raman effect, are determined by this molecular geometry. Within the framework of the Born–Oppenheimer approximation (see Chapter 12) the energy of a molecule can be written in the form ε = ε elec + ε nucl , ( 1 ) where ε elec is the energy associated with the electronic configuration of the molecule and ε nucl is the energy of displacements of the nuclei. In general this approximation is an excellent one, hence, interaction terms can be neglected. The electronic energy serves as the effective potential function that governs the movement of the nuclei. As a first approximation, it is possible to decompose the energy associated with the nuclear displacements by writing ε nucl = ε trans + ε rot + ε vib + interactions . ( 2 ) Furthermore, it is convenient to consider separately the kinetic-and potential-energy contributions. 9.1 KINETIC ENERGY The kinetic energy of a polyatomic molecule is a function of the atomic masses and the velocities of the atoms with respect to a space-fixed origin, O in Fig. 1. The center of mass of the molecule (cm) is located by the vector R . The instantaneous position of each atom, α , with respect to the center of mass is specified by r α and the corresponding equilibrium position by a α . Thus the vectors ρ α = r α − a α represent instantaneous displacements from equilibrium 216 MATHEMATICS FOR CHEMISTRY AND PHYSICS r a r a a a R z y x Z Y X cm O Fig.

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  Spectroscopy of Pharmaceutical Solids

  • Harry G. Brittain(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  Molecular Motion and Vibrational Spectroscopy 209 the bonds. The atomic motions for the asymmetric stretching vibration gives rise to a compression of one bond and a stretching of the other bond, whereas the atomic motions for the bending vibration yield an opening and closing of the bond angle. ROTATION AND VIBRATION IN DIATOMIC MOLECULES In order to develop a more rigorous treatment of the wave functions associated with rotational and vibrational motion, the application of wave mechanics to diatomic molecules will be considered. In the approximation of Born and Oppenheimer, it is recognized that the time frame for nuclear motion is very slow when compared with that of electronic motion, and hence the two may be separated Figure 3 Rotational motion in a non-linear triatomic molecule about its center of mass. 210 Brittain Figure 4 Vibrational motion in a linear triatomic molecule. Figure 5 Vibrational motion in a nonlinear triatomic molecule. Molecular Motion and Vibrational Spectroscopy 211 from each other. In other words, the molecule will remain in the same electronic state during the entire lifetime of nuclear motion. As a result, one can develop the wave mechanical expressions for nuclear motion independent of the expressions for electronic motion. For a diatomic molecule, the Hamiltonian operator for the nuclear motion is given by the sum of the kinetic and the potential energy operators. One can then write the Schro ¨dinger equation for nuclear motion as: ½ T N þ V ( r ) Š c ¼ E c (1) where T N is the kinetic energy operator, V ( r ) is the potential energy function, and it is understood that c is the wave function for nuclear motion. For the diatomic molecule A-B, Equation (1) becomes: 1 m A r A 2 c þ 1 m B r B 2 c þ 2 ½ E V ( r ) Š c ¼ 0 (2) where m A and m B are the masses of the two atoms comprising the molecule.

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  Molecular Properties V4

  • Douglas Henderson(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Chapter 3 The Vibrations of Molecules GERALD W . KING I. The Classical Theory of Molecular Vibrations 68 A. The Lagrangian Equations of Motion 68 B. Generalized Coordinates 68 C. Kinetic and Potential Energies 69 D. Normal Vibrations 70 E. The Diatomic Molecule as an Example 73 F. Degenerate Vibrations 75 G. Matrix Formulation 76 H. Internal Coordinates 77 II. Introductory Group Theory 79 A. Symmetry Properties of Molecules 79 B. Group Representations 86 C. Bases of Group Representations 93 III. The Quantum-Mechanical Theory of Vibrations 100 A. Factorization of the Molecular Wave Function 100 B. Harmonic Oscillator Wave Functions 101 C. Energy Levels of Polyatomic Molecules 103 D. Vibrational Symmetries of Polyatomic Molecules 103 IV. Vibrational Spectra of Polyatomic Molecules 108 A. Infrared Spectra 108 B. Transition Probabilities for Infrared Spectra 109 C. Selection Rules for Infrared Spectra Ill D. Raman Spectra 112 E. Transition Probabilities for Raman Spectra 113 F. The Rule of Mutual Exclusion 118 G. Depolarization of Raman Lines 119 References 121 67 68 Gerald W. King I. The Classical Theory of Molecular Vibrations A. T H E LAGRANGIAN EQUATIONS OF MOTION The dynamics of a system containing only one or two particles in motion can usually be determined by solving the Newtonian equations of motion. These are expressed in terms of forces and accelerations, which can be readily obtained for simple systems in terms of, for example, Cartesian or spherical polar coordinates. For more complex systems it is easier to work with more general coordinate systems, and to employ an alternative formulation of the classical dynamical laws that is based on the energy of the system, instead of the forces acting within it.

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