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

2kx2

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

mn =

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

mn =

1

2pc

k

m

Energy of transition

Inte

nsi

ty

0 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 only 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 only 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 only 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

2n -

1

4xen

e1 =3

2n -

9

4xen

e2 =5

2n -

25

4xen

e3 =7

2n -

49

4xen

Anharmonic oscillator: Position of spectral lines

xe = n/4De

Dv=+1, +2, +3,….

Fundamental

1st Overtone 2nd Overtone

e0 =1

2n -

1

4xen

e1 =3

2n -

9

4xen

e2 =5

2n -

25

4xen

e3 =7

2n -

49

4xen

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

2n -

1

4xen

e1 =3

2n -

9

4xen

e2 =5

2n -

25

4xen

e3 =7

2n -

49

4xen In

ten

sity

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

vn µ exp -ev / kT( )Boltzmann distribution:

v=1nv=0n

= 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

0

• Dn : energy gaps in molecule

• Dn : No dpendence on nex

• Stokes strong, anti-Stokes weak for vibrational levels

Rotational Raman Spectroscopy: CO2

0, 2JD = Selection Rule:

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

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”

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