Lecture 14: Anharmonic Oscillator and Raman Effect
Transcript of Lecture 14: Anharmonic Oscillator and Raman Effect
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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