Vibrational Spectroscopy

Question 1

How many degrees of freedom does an atom have?

Answer 1

3

Question 2

How many degrees of freedom are there in a diatomic molecule?

Answer 2

6

Question 3

How many types of motion are there?

Answer 3

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

Question 4

How many degrees of freedom are there for N atoms?

Answer 4

3N

Question 5

How many vibrations do linear and non-linear molecules have?

Answer 5

Linear: 3 translations, 2 rotation = 3N - 5 vibrations

Non-linear: 3 translation, 3 rotations = 3N - 6 vibrations

Question 6

What are normal modes of vibration?

Answer 6

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

Question 7

What is Hook’s law and the potential energy if a classical harmonic oscillator?

Answer 7

F = -kx when k = force constat and x is the displacement from equilibria

restoring force - the negative gradient potential energy

Question 8

How many vibrational modes does water have and what are they?

Answer 8

Non-linear

3N - 6 = (3x3) - 6 = 3

Stretches have higher frequencies than bends

v₁ - symmetric stretch is the highest frequency in water

Question 9

How do we treat each normal mode of vibration?

Answer 9

Each normal mode acts as a harmonic oscillator with vibrational energies

Question 10

How do we find the vibrational energy levels?

Answer 10

Solve the Schrodinger equation for a harmonic oscialltor

Question 11

What are the vibrational energies for an anharmonic oscillator?

Answer 11

The anharmonic oscillator levels converge to the dissociation limit D_e

Experimentally the dissociation energy is measured relatuve to the zero-point

ω_e is the spacing that the energy levels would have if the potenial were harmonic with the curvature the actuak curve has at its minimum

Question 12

How is the actual spacing between a ;pair of energy levels in an anharmonic oscillator calculated?

Answer 12

Question 13

What are the wavefunctions and wavefunctions squared of a harmionic oscillator?

Answer 13

Question 14

What are the wavefunctions and wavefunction squared of an anharmonic oscillator?

Answer 14

Question 15

What are some importamt characteristics of normal modes?

Answer 15

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

Question 16

How do we measure IR spectra?

Answer 16

Measure I/I₀ - Transmittence where I is the output and I₀ is the incident intensity

100% transmittence = no absorption

Question 17

What happens to the molecule in an IR experiment?

Answer 17

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

Question 18

What is the difference between dispersion IR and Fourier transform IR

Answer 18

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

Question 19

How do we know if a transition is forbidden or allowed?

Answer 19

The rate of transitions between the initial and final states is proportional to the square of μ_v’v’’

μ_v’v’’ = ∫ψ’*_vμ^ψ’‘_vdx

where x = r-r_e, μ^ is the electric dipole operator, ψ’*_v is the upper vibrational wavefunction and ψ’‘_v is the lower vibrational wavefunction

If μ_v’v’’ = 0 the transition is forbidden

If μ_v’v’’ ≠ 0 the transition is allowed

Question 20

What are the selection rules for a transition to be IR active (what allowes μ_v’v’’ to be non 0)?

Answer 20

It can be shown that

μ_v’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

Question 21

What do allowed transitions look like for a harmonic oscillator?

Answer 21

All transitions the same energy

Question 22

How does the selection rule get modified for an anharmonic oscillator?

Answer 22

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

Question 23

How would a transition between v=1 and v=2 look in a spectrum?

Answer 23

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

Question 24

How many peaks show up in a specrum if two normal modes are degenerate?

Answer 24

Only 1

Question 25

How did Raman discover elastic scattering?

Answer 25

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

Question 26

What types of scattering can occur?

Answer 26

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

Question 27

What does the difference between the Stokes and Rayleigh line show?

Answer 27

Energy difference between Stokes and Rayleigh indicates the energy difference between vibrational quantum states

Question 28

What causes light to scatter?

Answer 28

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: μ_ind = αE where α is the polarisability and E is the electric field strength

Question 29

How does light produce a change in polarisability?

Answer 29

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 - ν_vib = Stokes ν + ν_vib = Anti-Stokes

Question 30

What transitions are allowed in Raman spectroscopy?

Answer 30

μ_v'v'' ≠ 0 when polarisability varies with the nuclear displacement Selection rule for harmonic oscillator - Δv ± 1 For anharmonic oscillator: Δv = ±1, ±2, ±3 ....

Question 31

What do Raman transitions look like for an anharmonic oscillator?

Answer 31

Question 32

What are the contributions to character for the symmetry operations: E, σ, i, C2, C3, C4, C6, S3, S4, S6

Answer 32

E = 3 σ = +1 i = -3 C2 = -1 C3 = 0 C4 = 1 C6 = 2 S3 = -2 S4 = -1 S6 = 0

Question 33

Steps to find the irreducible representation of of a molecule?

Answer 33

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

Question 34

How do we distiguish between translations, rotations and vibrations?

Answer 34

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

Question 35

How do we determine which symmetries belong to stretches and bends?

Answer 35

Γ_vib = Γ_stretch + Γ_bend Work out symmetries of stretches and then take away from overall irreducible

Question 36

How do we determine the symmetries of the stretches?

Answer 36

Determine which internal coordinates are unshifted under each symmetry operation and then reduce it

Question 37

What are the symmetry reqirements for a transition to be allowed in IR?

Answer 37

Both vibrational states must be non-degenerate: Γ(ψ'_v) x Γ(μ_v) x Γ(ψ''_v) = A where μ_v is a vector with 3 components one of which is non zero Either or both vibrational states are degenerate: Γ(ψ'_v) x Γ(μ_v) 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)

Question 38

What are the symmetry requirements for a transition to be allowed in Raman spectroscopy?

Answer 38

The requirement for an allowed Raman transition between non-degenerate states is Γ(ψ'_v) x Γ(α_ij) x Γ(ψ''_v) = A where α_ij 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(α_ij) This modifies to Γ(ψ'_v) ⊃ Γ(α_ij) for degenerate states

Question 39

How do you determine which symmetry operations in the irreducible are IR active and Raman active?

Answer 39

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

Question 40

How do we determine and show the symmetry of ivertones and combinations?

Answer 40

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

Question 41

Which overtones of IR and Raman active modes are IR or Raman active?

Answer 41

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

Question 42

How do we find the direction each bond is moving in in each stretch?

Answer 42

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

Question 43

How many modes of vibration does a doubly or triply degenerate symmetry operation represent?

Answer 43

E = two modes but peaks show up in the same place - 1 peak T = 3 modes but only 1 peak

Question 44

What are group vibrations?

Answer 44

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

Question 45

How do we determine what the symmetries of the group stretches?

Answer 45

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

Question 46

Which bends and stretches have specific names?

Answer 46