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Harmonic Oscillator

Harmonic Oscillator. Harmonic Oscillator. . +q. -q. R e. Selections rules. Permanent Dipole moment.

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Harmonic Oscillator

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  1. Harmonic Oscillator

  2. Harmonic Oscillator

  3. +q -q Re Selections rules Permanent Dipole moment An electric dipole consists of two electric charges q and -q separated by a distance R. This arrangement of charges is represented by a vector, the electric dipole moment  with a magnitude: = Re q Unit: Debye, 1D = 3.33×10-30Cm When the molecule is at its equilibrium position, the dipole moment is called “permanent dipole moment” 0.

  4. Selections rules Electric dipole moment operator  The probability for a vibrational transition to occur, i.e. the intensity of the different lines in the IR spectrum, is given by the transition dipole momentfibetween an initial vibrational state i and a vibrational final state f : The electric dipole moment operator depends on the location of all electrons and nuclei, so its varies with the modification in the intermolecular distance “x”. 0 is the permanent dipole moment for the molecule in the equilibrium position Re

  5. 0 The higher terms can be neglected for small displacements of the nuclei The two states i and f are orthogonal. Because they are solutions of the operator H which is Hermitian

  6. Second condition: First condition: fi= 0, if ∂/ ∂x = 0 In order to have a vibrational transition visible in IR spectroscopy: the electric dipole moment of the molecule must change when the atoms are displacedrelative to one another. Such vibrations are “ infrared active”. It is valid for polyatomic molecules. By introducing the wavefunctions of the initial state i and final state f , which are the solutions of the SE for an harmonic oscillator, the following selection rules is obtained:  = ±1

  7. Note 1: Vibrations in homonuclear diatomic molecules do not create a variation of  not possible to study them with IR spectroscopy. Note 2: A molecule without a permanent dipole moment can be studied, because what is required is a variation of  with the displacement. This variation can start from 0.

  8. IR Stretching Frequencies of two bonded atoms: What Does the Frequency, , Depend On? • = frequency k = spring strength (bond stiffness)  = reduced mass (~ mass of largest atom) • is directly proportional to the strength of the bonding between the two atoms ( k) • is inversely proportional to the reduced mass of the two atoms (v  1/) 51

  9. Stretching Frequencies Frequency decreases with increasing atomic weight. Frequency increases with increasing bond energy. 52

  10. IR spectroscopy is an important tool in structural determination of unknown compound

  11. IR Spectra: Functional Grps Alkane C-C -C-H Alkene Alkyne 11

  12. IR: Aromatic Compounds (Subsituted benzene “teeth”) C≡C 12

  13. IR: Alcohols and Amines O-H broadens with Hydrogen bonding CH3CH2OH C-O N-H broadens with Hydrogen bonding Amines similar to OH 13

  14. CO2, A greenhouse gas ?

  15. Electromagnetic Spectrum • Over 99% of solar radiation is in the UV, visible, and near infrared bands • Over 99% of radiation emitted by Earth and the atmosphere is in the thermal IR band (4 -50 µm) Near Infrared Thermal Infrared

  16. What are the Major Greenhouse Gases? N2 = 78.1% O2 = 20.9% H20 = 0-2% Ar + other inert gases = 0.936% CO2 = 370ppm CH4 = 1.7 ppm N20 = 0.35 ppm O3 = 10^-8 + other trace gases

  17. Molecular vibrations • The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations. • Transition occur for v = ±1 • This potential does not apply to energies close to dissociation energy. • In fact, parabolic potential does not allow molecular dissociation. • Therefore more consider anharmonic oscillator. PY3P05

  18. Vibrational modes of CO2

  19. Anharmonic oscillator • A molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations. • At high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit. • Must therefore use a asymmetric potential. E.g., The Morse potential: where Deis the depth of the potential minimum and PY3P05

  20. Anharmonic oscillator • The Schrödinger equation can be solved for the Morse potential, giving permitted energy levels: where xeis the anharmonicity constant: • The second term in the expression for E increases • with v => levels converge at high quantum numbers. • The number of vibrational levels for a Morse oscillator is finite: v = 0, 1, 2, …, vmax PY3P05

  21. Energy Levels: Basic Ideas Basic Global Warming: The C02 dance … About 15 micron radiation

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