lecture34_umn

# lecture34_umn - Lecture 34 Vibrational Spectroscopy Matrix...

This preview shows pages 1–6. Sign up to view the full content.

Lecture 34 December 7, 2009 Vibrational Spectroscopy Matrix Isolation Spectroscopy

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

View Full Document
Molecular Vibrations With a defined PES, it is possible to formulate and solve Schrödinger equations for nuclear motion (as opposed to electronic motion) (33-6) N is the number of atoms , m is the atomic mass , V is the potential energy from the PES as a functions of the 3 N nuclear coordinates q , Ξ is the nuclear wave function that is expressed in those coordinates. Solution of eq. 33-6 provides entry into the realms of rotational and vibrational spectroscopy. " 1 2 m i i N # \$ i 2 + V q ( ) % & ( ) * + q ( ) = E + q ( )
Shrödinger Vibrational Equation Diatomic case: eq. 33-6 is a function of only a single variable , the interatomic distance r . Challenge: we do not know the potential energy function V ( r) . At a level of theory compute V point by point . Those points may then be fit to a polynomial, Morse , etc. One-dimensional Schrödinger equation solved using standard numerical recipes. Harmonic oscillator equation would be recovered if we fit V to a parabolic form. " 1 2 m i i N # \$ i 2 + V q ( ) % & ( ) * + q ( ) = E + q ( ) " 1 2 μ # 2 r 2 + 1 2 k ( r " r eq ) 2 \$ % & ( ) * nuc ( r ) = E * nuc ( r ) 33-6

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

View Full Document
One-Dimensional Case Simplification when the potential is harmonic . The quantum mechanical harmonic oscillator : analytical solutions to the Schrödinger equation Products of Hermite polynomials and gaussian functions with eigenvalues (34-1) e n is the vibrational quantum number beginning at 0 and (34-2) μ reduced mass; k is the bond force constant, second derivative of the energy with respect to bond stretching at the equilibrium bond length (see eq. 9-3). E = n + 1 2 " # \$ % & h ( " = 1 2 # k μ
Polyatomic Case Multi-dimensional Taylor expansion: (34-3) q : vector of atomic coordinates. q eq vector at equilibrium structure. H : hessian matrix. Difficult to solve (34-3) in the indicated coordinates.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.

### Page1 / 19

lecture34_umn - Lecture 34 Vibrational Spectroscopy Matrix...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online