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

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 126

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

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online