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Harmonic Oscillator Harmonic Oscillator 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: +q -q Re = 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. 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 fi between an initial vibrational state i and a vibrational final state f : fi f ˆ i d f ˆ i 1 2 2 ( x) 0 x 2 x ... 2 x 0 x 0 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 1 2 fi 0 f i d f xi d 2 f x 2i d ... 2 x 0 x 0 0 The two states i and f are orthogonal. Because they are solutions of the operator H which is Hermitian The higher terms can be neglected for small displacements of the nuclei fi f xi d x 0 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 displaced relative to one another. Such vibrations are “ infrared active”. It is valid for polyatomic molecules. Second condition: f x i d 0 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 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. IR Stretching Frequencies of two bonded atoms: What Does the Frequency, , Depend On? E h clas h 2 k = 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 Stretching Frequencies • Frequency decreases with increasing atomic weight. • Frequency increases with increasing bond energy. 52 IR spectroscopy is an important tool in structural determination of unknown compound IR Spectra: Functional Grps Alkane -C-H C-C Alkene Alkyne 11 IR: Aromatic Compounds (Subsituted benzene “teeth”) C≡C 12 IR: Alcohols and Amines O-H broadens with Hydrogen bonding CH3CH2OH C-O Amines similar to OH N-H broadens with Hydrogen bonding 13 CO2, A greenhouse gas ? Electromagnetic Spectrum Near Infrared Thermal Infrared • • 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) 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 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 anharmonic oscillator. consider PY3P05 Vibrational modes of CO2 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: a(R R ) hcDdepth where De isV the potential e 1 e of the minimum and e 2 1/ 2 2 a 2hcDe PY3P05 Anharmonic oscillator • The Schrödinger equation can be solved for the Morse potential, giving permitted energy levels: 2 1 1 E hc~ hcxe~ ; 0,1,2,... max 2 2 a 2 ~ xe 2meff 4 De where xe is 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 Energy Levels: Basic Ideas Basic Global Warming: The C02 dance … About 15 micron radiation