Molecular spectroscopy

ε (cm-1) for morse oscillator

(v+½)ωe∼ - (v+½)² we∼xe

v max =

1/(2xe) - ½

De (cm-1) =

we∼/(4xe)

ω (rad s-1) = (cm-1)

2πc∼ω∼

E (tot) =

E(elec) + E(vib) + E(rot)

energy of a photon

E = hc/λ

requirement for pure rotational spectra

permanent dipole

requirement for vibrational spectra

dipole that changes with vibration (i.e. bond expands and contracts)

Rayleigh/elastic scattering

a photon is emitted at the same energy it was absorbed

stokes/inelastic scattering

photon emitted with less energy than it was absorbed

anti-stokes/inelastic scattering

photon emitted with more energy than it was absorbed

scattering that allows for molecules to be absorbed using raman spectroscopy

inelastic

requirement of raman activity

polarizability that changes upon vibration/rotation

molecules rotatioal raman doesnt work on

spherically symmetrical molecules (i.e. CH4)

moment of inertia, I

µr² (kg m²)

rotational angular momentum, L

Iω

reduced mass, mu

m1.m2/(m1+m2)

Energy (J) or rigid rotor

hBJ(J+1)

B(J) =

(h-)²/2I

B(Hz)=

h/8π²I (divide Joule eqn by h)

B (cm-1) =

h/8π²c∼I (divide Joule eqn by hc∼)

energy leves for a rigid rotor

B∼J(J+1)

Number of degenerate states for rotational rotor

2J+1

Population ratio (rigid rotor)

(2J+1)exp(-E/kT)

Jmax for population

√((kT)/2hc∼B∼) - 1/2

Allowed transitions between pure rotational levels

B∼(2J+2)

difference between pure rotational energy levels (rotational const.)

2B∼

peak strength is dependent on

how populated the energy level is

selection fule for J

∆J = ±1

effect of photon on molecule

induces a dipole

selection rules for linear molecule in raman spec

∆J = 0, ±2

∆E from v(incident) for raman spec

B∼(4J+6)

where stokes lines appear

v(incident) - B∼(4J+6) (lower frequency)

where antistokes lines appear

v(incident) + B∼(4J+6) (higher frequency)

branches seen in Raman spec

S branches

ε for harmonic oscillator

(v+1/2) ω∼

vibrational selection rule for harmonic oscillator

∆v = ±1

Vm(x) (morse potential) =

De[1-exp(-βx)]²

Vm(x) for small displacements =

Deβ²x²

km (force constant) =

2Deβ² (Vm = 1/2kx²)

D0 (dissociation energy) =

De - ε0

Fundamental transition state

v=0 to v=1

first overtone

v=0 to v=2

second overtone

v=0 to v=3

hot band

v=1 to v=2

reason for overtones being weaker

break ∆v = ±1 for HO, formally forbidden

reason for hot band being weak

low population

general expression for I for molecule

∑mr²

r for molecule I

perpendicular distance from centre of the molecule (could be the middle of a bond)

general 10^n for I

-46

each vibrational level has a series of…

rotational levels

∆J for vibrating rotor

±1

why cant ∆v=0 for vibrating rotor

due to anharmonic nature of the potential well

why does ∆J = ±1 for vibrating rotor

conservation of angular momentum (photon + molecule)

lower vibrational level

J’’

upper vibrational level

J’

relative frequency of P branches

lower

relative frequency of R branches

higher frequency

Branch for ∆J = -1

P

Branch for ∆J = 0

Q

Branch for ∆J = +1

R

branch for ∆J = +2

S (only in Raman)

branches only seen in vibrational spectra of diatomics

P and R

approximate separation of P and R branches

2B∼

Etot for vibrational rotor (in E)

Evib + Erot

Etot for vibrational rotor (v)

[(v+1/2)w∼ - (v+1/2)²w∼xe] + [BJ(J+1)]

reason B is not exactly the same between v

the more a molecule vibrates, the longer the average bond length

relationship between B and r in vib. rotor

as r increases, B decreases (inversely proportional to I)

pattern of line spacing in R branches

spacing gets smaller

pattern of line spacing in P branch

spacing gets larger

R(J) - P(J) =

2B₁(2J+1) [gradient = 2B₁]

R(J-1) - P(J+1) =

2B₀(2J+1)

Bv =

Be - α(v+1/2)

normal vibration mode

a vibration mode in which all the atoms in a molecule vibrate with the same frequency and pass through their equilibrium points simultaneously

number of coordinates for N atoms

3N (x,y,z)

what position do we use to specify how a molecule moves trhough space

its centre of mass, (x,y,z)

how many degrees of freedom do N atoms have

3N

Once the COM is tracked, how many degrees of freedom are there

3N-3 (take away the 3 coords of the COM)

Fundamental vibrations for linear molecule

3N - 5 (-2 rotation axis)

fundamental (normal mode) vibrations in non-linear molecules

3N-6 (-3 rotation axis)

labelling for vibrations (simple)

v1 = symmetric stretch
v2 = symmetric bend
v3 = anitsymmetric vibration

stretch vibrational modes for acyclic molecules with N atoms

N-1

requirement for IR activity

a change in the dipole moment during vibration

what determines whether a polyatomic molecule is IR active

whether there is a change in the dipole moment during a vibration and the symmetry of that vibration in the molecule

is CO2 symmetric stretch IR active?

no, the dipoles cancel out

how does bond length affect polarizability

the shorter the bond, the hard it is to polarise

Whats the rule of mutual exclusion

if a molecule has a COI, a raman-active vibration is IR inactive, and vice versa, but some could be neither

How can the rule of mutual exclusion be useful?

can tell the difference between some cis and tr

why do multiple, unexpected peaks occur in IR spectra

combination bands

what vibrations do the vib-rot transitions apply to

parallel vibrations

How does a Q branch arise

vibrations perpendicular to the principle axis

what is ∆J for Q branches

0

How does a Q branch occur

• bending modes of linear molecules are doubly degenerate

• rotates

• AM is conserved without having to change J (rotational angular momentum)

What is the shape of a Q branch

1 broad peak

what does having no Q branch prove

a molecule is linear

what is Te

electronic excitation energy

where is Te measured from

Bottom of PE curve

how is the strength of a transition determined

Einstein ‘B’ coefficient (proportional to the transition dipole moment)

how can we separate psi (wavefunctions of different states)

psi(E) . psi(V)

how can we separate the ransition dipole moment, µ

µ(E) + µ(V)