5210-chap5 - Chapter 5: Slide 1 Chapter 5 Molecular...

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Unformatted text preview: Chapter 5: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Chapter 5: Slide 2 Outline The Classical Harmonic Oscillator Math Preliminary: Taylor Series Solution of Differential Eqns. The Vibrations of Diatomic Molecules The Quantum Mechanical Harmonic Oscillator Vibrational Spectroscopy Harmonic Oscillator Wavefunctions and Energies Properties of the Quantum Mechanical Harmonic Oscillator Vibrational Anharmonicity Continued on Second Page Chapter 5: Slide 3 Outline (Contd.) The Two Dimensional Harmonic Oscillator Vibrations of Polyatomic Molecules Symmetry and Vibrational Selection Rules Time Independent Perturbation Theory QM Calculations of Vibrational Frequencies and Molecular Dissociation Energies. Chapter 5: Slide 4 The Classical Harmonic Oscillator R e V(R) A B R V(R) k(R-R e ) 2 Harmonic Oscillator Approximation or V(x) kx 2 x = R-R e The Potential Energy of a Diatomic Molecule k is the force constant Chapter 5: Slide 5 m 1 m 2 x = R - R e Hookes Law and Newtons Equation Force: kx kx dx d dx dV f- = - =- = 2 2 1 Newtons Equation: 2 2 dt x d a f = = 2 1 2 1 m m m m + = where Reduced Mass Therefore: kx dt x d- = 2 2 Chapter 5: Slide 6 Solution kx dt x d- = 2 2 x k dt x d - = 2 2 Assume: ) cos( ) sin( ) ( t B t A t x + = ) cos( ) sin( t B t A dt x d - - = 2 2 2 2 x 2 - = x k x - =- 2 k = 2 = or = k 2 1 or = k c 2 1 ~ Chapter 5: Slide 7 Initial Conditions (like BCs) Lets assume that the HO starts out at rest stretched out to x = x . ) cos( ) sin( ) ( t B t A t x + = x(0) = x = = t dt dx ) sin( ) cos( t B t A dt dx - = = = = A dt dx t ) cos( ) ( t B t x = B x x = = ) ( and ) cos( ) ( t x t x = = = k 2 1 2 Chapter 5: Slide 8 Conservation of Energy ) sin( t x dt dx - = ) cos( ) ( t x t x = = k Potential Energy (V) Kinetic Energy (T) Total Energy (E) 2 2 1 kx V = ) ( cos 2 1 2 2 t kx V = 2 2 1 = dt dx T ) ( sin 2 1 2 2 2 t x T = ) ( sin 2 1 2 2 t kx T = V T E + = 2 2 1 kx E = 2 2 1 kx V = = T 2 2 1 kx E = = V 2 2 1 kx E = 2 2 1 kx T = t = 0, , 2 , ... x = x t = /2, 3 /2, ... x = 0 Chapter 5: Slide 9 Classical HO Properties Energy: E = T + V = kx 2 = Any Value i.e. no energy quantization If x = 0, E = 0 i.e. no Zero Point Energy Probability: E = kx 2 x P(x) +x-x Classical Turning Points Chapter 5: Slide 10 Outline The Classical Harmonic Oscillator Math Preliminary: Taylor Series Solution of Differential Eqns....
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5210-chap5 - Chapter 5: Slide 1 Chapter 5 Molecular...

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