Vibrational Spectroscopy Flashcards
How many degrees of freedom does an atom have?
3
How many degrees of freedom are there in a diatomic molecule?
6
How many types of motion are there?
Translations - in phase displacements of atoms along cartesian coordinates - no change in internuclear separation or molecular orientation
Rotation: centre of mass is unchanged, no chang ein internuclear separation but orientation change wrt external frame of reference - ot defined
Vibration - a motion of the nuclei in which the centre of mass and orientation remain unchanged but where a change has occurred in internal coordinates - bond length/angle
How many degrees of freedom are there for N atoms?
3N
How many vibrations do linear and non-linear molecules have?
Linear: 3 translations, 2 rotation = 3N - 5 vibrations
Non-linear: 3 translation, 3 rotations = 3N - 6 vibrations
What are normal modes of vibration?
The natural vibrations of a molecule
For each normal mode all of the nuclei:
undergo simple harmonic motion
have the same frequency of oscillation
move in phase but generally with different amplitudes
What is Hook’s law and the potential energy if a classical harmonic oscillator?
F = -kx when k = force constat and x is the displacement from equilibria
restoring force - the negative gradient potential energy
How many vibrational modes does water have and what are they?
Non-linear
3N - 6 = (3x3) - 6 = 3
Stretches have higher frequencies than bends
v1 - symmetric stretch is the highest frequency in water
How do we treat each normal mode of vibration?
Each normal mode acts as a harmonic oscillator with vibrational energies
How do we find the vibrational energy levels?
Solve the Schrodinger equation for a harmonic oscialltor
What are the vibrational energies for an anharmonic oscillator?
The anharmonic oscillator levels converge to the dissociation limit De
Experimentally the dissociation energy is measured relatuve to the zero-point
ωe is the spacing that the energy levels would have if the potenial were harmonic with the curvature the actuak curve has at its minimum
How is the actual spacing between a ;pair of energy levels in an anharmonic oscillator calculated?
What are the wavefunctions and wavefunctions squared of a harmionic oscillator?
What are the wavefunctions and wavefunction squared of an anharmonic oscillator?
What are some importamt characteristics of normal modes?
In general made up of simultaneous stretching and bending of bonds
Bending vibrations generally lower vrequency than strethches
Heavy atoms move less than lioghter ones
Even at the zero point all normal modes are excited simultaneously
How do we measure IR spectra?
Measure I/I0 - Transmittence where I is the output and I0 is the incident intensity
100% transmittence = no absorption
What happens to the molecule in an IR experiment?
Absorption of an infrared photon promotes a transition from v=0 to v=1
Only photons whose energy exactly match the difference in energy between the two levels will be absorbed
What is the difference between dispersion IR and Fourier transform IR
Dispersion meaures the absorbance of IR radiation as a function of wavelength by dspersing the transmitted light through a monochrometer
A fourier transform instrument measures all IR wavelengths simultaneously using an interferometer with a key advantage being greatly increased speed of meaurement
How do we know if a transition is forbidden or allowed?
The rate of transitions between the initial and final states is proportional to the square of μv’v’’
μv’v’’ = ∫ψ’*vμ^ψ’‘vdx
where x = r-re, μ^ is the electric dipole operator, ψ’*v is the upper vibrational wavefunction and ψ’‘v is the lower vibrational wavefunction
If μv’v’’ = 0 the transition is forbidden
If μv’v’’ ≠ 0 the transition is allowed
What are the selection rules for a transition to be IR active (what allowes μv’v’’ to be non 0)?
It can be shown that
μv’v’’ = (dμ/dx)e ∫ψ’*vμ^ψ’‘vdx + ….
Part 1 (dμ/dx) implies that a transition is allowed only if the dipole moment changes on vibration
Part 2 (the integral) implies that the transition is allowed if it is between adjacent energy levels
Δv = ±1
What do allowed transitions look like for a harmonic oscillator?
All transitions the same energy
How does the selection rule get modified for an anharmonic oscillator?
Δv = ±1, ±2, ±3,…
Δv = + 1 absorption is known as the fundamental absorption
Δv = + 2 = first overtone (second harmonic)
Δv = + 3 = second overtone (third harmonic)
fundamental - v=0 —-> v=1
first overtone - v=0 —> v=2
How would a transition between v=1 and v=2 look in a spectrum?
v=1 —–> v=2 would be close in wavenumber to v=0 —-> v=1 and low intensity - the transition is at a lower energy because most molecules exist in the v=0 state - Boltzmann distribution
This transition is affected by temperature because molecules will only be at higher energy levels at higher tmepratures
These peaks are sometimes called hot bands
How many peaks show up in a specrum if two normal modes are degenerate?
Only 1
How did Raman discover elastic scattering?
First he put a violet filter on sun light to make sure only the violet light got through
The violet light then hit a liquid sample and the light was scattered
He showed that green light was given off using a green filter which would only allow green light through and not violet
These days we do Raman spectroscopy with lasers
What types of scattering can occur?
Photon can scatter elastically where the intensity is proportional to 1/λ4
Photon can lose energy (inelastic) = Stokes
Photon can gain energy (inelastic) = Anti Stokes
What does the difference between the Stokes and Rayleigh line show?
Energy difference between Stokes and Rayleigh indicates the energy difference between vibrational quantum states
What causes light to scatter?
The degree of scattering is related to the polarisability, α which is a measure of the extent to which the electrons in an atom or molecule can be displaced relative to the nuclei by an electric field
Displacement of charge produces a dipole: μind = αE where α is the polarisability and E is the electric field strength
How does light produce a change in polarisability?
The electric componenet of light induces an oscillating dipole which radiates at the frequency of oscillation
If the molecule is vibrating the polarisability changes during the vibration and the induced dipole radiates at three frequencies:
ν = Rayleigh
ν - νvib = Stokes
ν + νvib = Anti-Stokes
What transitions are allowed in Raman spectroscopy?
μv’v’’ ≠ 0 when polarisability varies with the nuclear displacement
Selection rule for harmonic oscillator - Δv ± 1
For anharmonic oscillator: Δv = ±1, ±2, ±3 ….
What do Raman transitions look like for an anharmonic oscillator?
What are the contributions to character for the symmetry operations:
E, σ, i, C2, C3, C4, C6, S3, S4, S6
E = 3 σ = +1
i = -3 C2 = -1
C3 = 0 C4 = 1
C6 = 2 S3 = -2
S4 = -1 S6 = 0
Steps to find the irreducible representation of of a molecule?
Determine point group
Determine number of unshofted atoms under every symmtery operation in the point group
Determine the contribution per unshifted atom
Multiply to find reducible
Use reduction formula to find irreducible
How do we distiguish between translations, rotations and vibrations?
Look at the character table
The symmetry operations that contain x, y, z are the translations and should be discarded
The symmetry operations that contain Rx, Ry or Rz are the rotations and should be discarded
How do we determine which symmetries belong to stretches and bends?
Γvib = Γstretch + Γbend
Work out symmetries of stretches and then take away from overall irreducible
How do we determine the symmetries of the stretches?
Determine which internal coordinates are unshifted under each symmetry operation and then reduce it
What are the symmetry reqirements for a transition to be allowed in IR?
Both vibrational states must be non-degenerate:
Γ(ψ’v) x Γ(μv) x Γ(ψ’‘v) = A where μv is a vector with 3 components one of which is non zero
Either or both vibrational states are degenerate:
Γ(ψ’v) x Γ(μv) x Γ(ψ’‘v) ⊃ A
When the transition is from v=0 for non degenerate states Γ(ψ’v) = Γ(x), (y) or (z), for degenerate states Γ(ψ’v) ⊃ Γ(x), (y) or (z)
What are the symmetry requirements for a transition to be allowed in Raman spectroscopy?
The requirement for an allowed Raman transition between non-degenerate states is
Γ(ψ’v) x Γ(αij) x Γ(ψ’‘v) = A where αij represent any components of the polarisability tensor
If v’‘=0 the requirement for a Raman transition between non-degenerate states is Γ(ψ’v) = Γ(αij)
This modifies to Γ(ψ’v) ⊃ Γ(αij) for degenerate states
How do you determine which symmetry operations in the irreducible are IR active and Raman active?
Discard the translations and rotations
Find the symmetry of the vibrations
If any of the vibrations are the same symmetry as the translations (contain the axes) the modes are IR active
IF any of the vibrational modes are the quadratic symmetry species then these are Raman active
How do we determine and show the symmetry of ivertones and combinations?
The symmetry of an overtone is determined by multiplying the symmetry of the mode being overtoned
e.g if water is vibrating with 2 quanta of v3 then Γ(ψ’v) = B2 X B2 = A1 - 1st overtone of a v3
The symmetry of a combination is determined by multiplying the symmetry operations together
e.g. if water is vibrating with 1 quanta if v2 and 1 quanta of v3 then Γ(ψ’v) = A1 X B2
Which overtones of IR and Raman active modes are IR or Raman active?
All odd overtones of IR active modes have Ag symmetry and are therefore are Raman active and IR inactive
Even and odd overtones of Raman active modes are all gerade so are Raman active only
How do we find the direction each bond is moving in in each stretch?
Use the reducible of the stretches used to find which symmetries corresponded to the stretches
Choose one bond as a reference
Apply each symmetry operation to the bond and determine which bond it becomes
Combine and factorise
Normalise
If the bond has a + it is extending and if it has a - it is compressing
How many modes of vibration does a doubly or triply degenerate symmetry operation represent?
E = two modes but peaks show up in the same place - 1 peak
T = 3 modes but only 1 peak
What are group vibrations?
While many vibrations include all the atoms in the molecule some vibrations only involve a certain group
The functional groupm region lies between 110 and 3700 cm-1
e.g. The stretching or bending vibration of a terminal -X-Y groupo where X is heavy compared to Y such as terminal OH
How do we determine what the symmetries of the group stretches?
Can be determined in the same way as stretching vibrations
Use groups as the internal coordinates
Ir and Raman activity is determined in the same way
Which bends and stretches have specific names?
