Effect Transition Moment Integral Can be evaluated analytically - - PowerPoint PPT Presentation

Lecture 14: Anharmonic Oscillator and Raman Effect

Transition Moment Integral

• Can be evaluated analytically
• Often simplified by symmetry
• Gives rise to selection rules if recursion formulae exist

The chemical bond as a simple harmonic oscillator SHO: a good approximation for small displacements Parabolic potential: V x

( ) = 1

2 kx2

Schrödinger equation: Boundary condition:  = 0 at x = + ∞ Ev = (v + ½)ħω

The chemical bond as a simple harmonic oscillator v = Vibrational quantum number = 0, 1, 2, 3, …

w = k m

Force constant Reduced mass Ev+1 – Ev =   Ev=0 = ½ħω Zero point energy = h /2฀ = hn

n = 1 2p k m n = 1 2pc k m

Spectrum of a harmonic oscillator v = Vibrational quantum number = 0, 1, 2, 3, …

w = k m

Force constant Reduced mass Ev+1 – Ev =   = h /2฀ = hn

n = 1 2p k m n = 1 2pc k m

Energy of transition Intensity

1000 2000 3000

IR Spectrum: Bond strength Polyatomic molecule: Different Bond strengths  Functional groups Dv=1

High resolution IR spectrum of HCl Rotational fine structure Isotope effect Dv=1, DJ=+1

Anharmonic oscillator SHO: a good approximation

• nly for small displacements
• The bond breaks at large displacements
• Bond dissociation energy

Morse potential

Anharmonic oscillator

• The bond breaks at large displacements
• Bond dissociation energy
• Energy levels come closer for higher values of v
• Fundamental and overtones in IR spectra

SHO: a good approximation

• nly for small displacements

Morse potential

Anharmonic oscillator: Energies

• The bond breaks at large displacements
• Bond dissociation energy
• Energy levels come closer for higher values of v
• Fundamental and overtones in IR spectra

SHO: a good approximation

• nly for small displacements

ev = (v + ½)n - (v + ½)2xen Morse potential xe = n/4De Dv=+1, +2, +3,….

Anharmonic oscillator: “Selection” rules

• The bond breaks at large displacements
• Bond dissociation energy
• Energy levels come closer for higher values of v
• Fundamental and overtones in IR spectra

ev = (v + ½)n - (v + ½)2xen Morse potential xe = n/4De Dv=+1, +2, +3,…. Fundamental 1st Overtone 2nd Overtone

Anharmonic oscillator: Position of spectral lines ev = (v + ½)n - (v + ½)2xen xe = n/4De Dv=+1, +2, +3,…. Fundamental 1st Overtone 2nd Overtone

e0 = 1 2 n - 1 4 xen e1 = 3 2n - 9 4 xen e2 = 5 2 n - 25 4 xen e3 = 7 2 n - 49 4 xen

Anharmonic oscillator: Position of spectral lines xe = n/4De Dv=+1, +2, +3,…. Fundamental 1st Overtone 2nd Overtone

e0 = 1 2 n - 1 4 xen e1 = 3 2n - 9 4 xen e2 = 5 2 n - 25 4 xen e3 = 7 2 n - 49 4 xen

e1 -e0 =n -2xen =n(1-2xe) e2 -e0 = 2n -6xen = 2n(1-3xe) e3 -e0 = 3n -12xen = 3n(1-4xe)

Anharmonic oscillator: Position of spectral lines xe = n/4De Dv=+1, +2, +3,…. Fundamental 1st Overtone 2nd Overtone

e0 = 1 2 n - 1 4 xen e1 = 3 2n - 9 4 xen e2 = 5 2 n - 25 4 xen e3 = 7 2 n - 49 4 xen

Intensity Wavenumber

e1 -e0 =n -2xen =n(1-2xe) e2 -e0 = 2n -6xen = 2n(1-3xe) e3 -e0 = 3n -12xen = 3n(1-4xe)

IR Spectrum of Carbon Monoxide Fundamental Peak First Overtone 2143 cm-1 4260 cm-1

IR Spectrum of Carbon Monoxide: High resolution Fundamental Peak First Overtone 2143 cm-1 4260 cm-1

Population of states and hot band Morse potential Typical energy gap: 100s and 1000s of cm-1

v

n µ exp -ev / kT

( )

Boltzmann distribution:

v=1

n

v=0

n

= 0.008

for energy gap of 1000 cm-1 High temperature: v =1 to v = 2 ….. are possible Hot band Intensity of hot band: Population of v =1 at that temperature

How to find out bond length of H2 ?

• Polarizability: Induced dipole moment.
• Molecular rotation or vibration: Oscillating induced dipole
• Scattering of (usually) visible monochromatic light by

molecules of a gas, liquid or solid

• Two kinds of scattering :

– Rayleigh (1 in every 10,000) : No change in frequency – Raman (1 in every 10,000,000): Change in frequency Raman Spectroscopy Diatomic molecule, NO permanent dipole moment

Raman scattering: different from original Raman Spectroscopy Rayleigh scattering: no change in energy of light Anti- Stokes shift Stokes shift Virtual level Dn (= n ฀ nex)  nex n

• Dn : energy gaps in molecule
• Dn : No dpendence on nex
• Stokes strong, anti-Stokes

weak for vibrational levels

Rotational Raman Spectroscopy: CO2

0, 2 J D = 

Selection Rule:

CH 107 in a nutshell: Quantum mechanics in Chemistry: Theory and its manifestations

ˆ Hy = Ey

CH 107 in a nutshell: Bold thoughts from great minds

CH 107 in a nutshell: Don’t be a frog in the well. “Seek, and ye shall find”

history.cultural-china.com vkaisthaaseem.blogspot.com

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