5210-chap5

# 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 (Cont’d.) • 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) ≈ ½k(R-R e ) 2 Harmonic Oscillator Approximation or V(x) ≈ ½kx 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 Hooke’s Law and Newton’s Equation Force: kx kx dx d dx dV f- = - =- = 2 2 1 Newton’s Equation: 2 2 dt x d a f μ μ = = 2 1 2 1 m m m m + = μ where Reduced Mass Therefore: kx dt x d- = 2 2 μ Chapter 5: Slide 6 Solution kx dt x d- = 2 2 μ x k dt x d μ- = 2 2 Assume: ) cos( ) sin( ) ( t B t A t x ϖ + ϖ = ) cos( ) sin( t B t A dt x d ϖ ϖ- ϖ ϖ- = 2 2 2 2 x 2 ϖ- = x k x μ ϖ- =- 2 μ ϖ k = πν 2 = or μ π = ν k 2 1 or μ π = ν k c 2 1 ~ Chapter 5: Slide 7 Initial Conditions (like BC’s) Let’s assume that the HO starts out at rest stretched out to x = x . ) cos( ) sin( ) ( t B t A t x ϖ + ϖ = x(0) = x = = t dt dx ) sin( ) cos( t B t A dt dx ϖ ϖ- ϖ ϖ = = ϖ = = A dt dx t ) cos( ) ( t B t x ϖ = B x x = = ) ( and ) cos( ) ( t x t x ϖ = μ π = π ϖ = ν k 2 1 2 Chapter 5: Slide 8 Conservation of Energy ) sin( t x dt dx ϖ ϖ- = ) cos( ) ( t x t x ϖ = μ = ϖ k Potential Energy (V) Kinetic Energy (T) Total Energy (E) 2 2 1 kx V = ) ( cos 2 1 2 2 t kx V ϖ = 2 2 1 = dt dx T μ ) ( sin 2 1 2 2 2 t x T ϖ μϖ = ) ( sin 2 1 2 2 t kx T ϖ = 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 = ϖ t = 0, π , 2 π , ... x = ± x ϖ t = π /2, 3 π /2, ... x = 0 Chapter 5: Slide 9 Classical HO Properties Energy: E = T + V = ½kx 2 = Any Value i.e. no energy quantization If x = 0, E = 0 i.e. no Zero Point Energy Probability: E = ½kx 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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## This note was uploaded on 04/11/2011 for the course CHEM 5210 taught by Professor Staff during the Spring '08 term at North Texas.

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

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