character tables & vibrational spectroscopy

In a character table, what does the final two columns mean?

what does A mean?

a singly degenerate state which is symmetric with respect to rotation about the principal C_n axis

what does B mean?

a singly degenerate state which is antisymmetric with respect to rotation about the principal C_n axis

In Mulliken symbols - what does E mean?

what does T(F) mean?

a doubly degenerate state

triply degenerate - G,H etc continue this sequence of degeneracy

In Mulliken symbols - what is Xg?

symmetric with respect to inversion through the centre of the molecule

In Mulliken symbols - what is Xu?

antisymmetric with respect to inversion through the centre of the molecule

In Mulliken symbols - what is X₁?

symmetric with respect to a vertical mirror plane perpendicular to the principal axis

In Mulliken symbols - what is X₂?

anti symmetric with respect to a vertical mirror plane perpendicular to the principal axis

In Mulliken symbols - what is X’?

symmetric with respect to a horizontal mirror plane

In Mulliken symbols - what is X’’?

anti symmetric with respect to a horizontal mirror plane

how many degrees of freedom does a molecular system with N nuclei and n electrons have?

how can these be treated and therefore how many nuclear degrees of freedom are there?

3(N+n) degrees of freedom

• can be treated separately due to Born Oppenheimer approx.

consider 3N nuclear degrees of freedom

• N is number of atoms

in linear molecules, what are the degrees of freedom?

3 DOF are translation in x,y,z

2 DOF are rotational (rotating around central principal axis of molecule causes no change in energy)

3N-5 = vibrations

in non linear molecules, what are the degrees of freedom?

3 DOF are translation in x,y,z

3 DOF are rotational (rotating around x,y,z - all require energy)

3N-6 = vibrations

is antisymmetric higher/lower/the same freq as symmetric? why?

intensity?

higher

takes more energy to distort the molecule

• higher intensity as distorts dipole moment a lot

show harmonic oscillator graph

show different springs and their position on the graph

what is on axes?

in a harmonic oscillator, how is force related to potential energy (E_PE)?

show potential energy vs displacement (x) graph and show how it changes with increasing k_f

what is k in this graph? (equation)

show rigid body rotor (rotational)

• energy levels (what symbol?)

• what are the gaps between ELs

• what are the E values?

show energy vs percentage of radiation transmitted

show radiation and then absorption

what is energy of photon (general)?

show emission

what is energy of photon?

what is a key parameter to consider when thinking about absorption/emission?

excited state lifetime

show transmission vs frequency (Beer-Lambert_Bouguer Law) graph

explain about width

when does this apply?2

• measurement of gas where reference is empty gas cell

• measurement of KBr pellets

what is usually calculated?

integrated intensity = area under the peak

• used rather than peak height

what is a gross selection rule? (general - how does this then apply to vibrational and rotational transitions?)

something about the molecule that decides if a transition is allowed

• vibrational = must be a change in dipole during vibration

• rotational = must be a permanent dipole

what is a specific selection rule (i.e. where does it come from)

• how does this apply to Harmonic oscillator and to Diatomic Rigid Rotor

comes from quantum mechanical understanding of ELs and tells which transitions are allowed

• harmonic oscillator - Δv = ±1

• diatomic rigid rotor - ΔJ = ±1

how does concentration change over these three plots?

why are the spectra different? 4

same amount of water so concentration is the same in each spectra

• central frequencies move, number of modes change, peak widths change, change in spectral intensity

draw S=0 and S=1 energy levels

show up to V=4 for each

where are the J levels?

show these different transitions

what is Rayleigh scattering? why does Stokes Raman scattering not come all the way back to ground state?

Rayleigh = elastic scattering, energy doesn’t change

Stokes Raman loses some E corresponding to vibration

what are the Raman gross selection rules? 2(vibrational and rotational)

vibrational = must be a change in polarisability during vibration

rotational = polarisability must be anisotropic (depends on orientation of molecule)

what are the Raman specific selection rules? 2

harmonic oscillator Δv=±1

diatomic rigid rotor ΔJ=±2

what is equation for induced dipole

show Δμ vs Q (dipole moment) and Δɑ vs Q (polarisability) for A₁ stretch

show Δμ vs Q (dipole moment) and Δɑ vs Q (polarisability) for B₂ stretch

show Δμ vs Q (dipole moment) and Δɑ vs Q (polarisability) for A₁ bend

for CO₂, which has the biggest peak and why?

(asymmetric/symmetric, bend, stretch)

asymmetric stretch as largest change in dipole

for CO₂, show Δμ vs Q (dipole moment) and Δɑ vs Q (polarisability) for asymmetric stretch

explain shape of polarisability graph

is CO₂ Raman active?

no because the gradient change of polarisability is 0

for CO₂, show Δμ vs Q (dipole moment) and Δɑ vs Q (polarisability) for symmetric stretch

for CO₂, show Δμ vs Q (dipole moment) and Δɑ vs Q (polarisability) for bend in y direction

for CO₂, show Δμ vs Q (dipole moment) and Δɑ vs Q (polarisability) for bend in z direction

for harmonic oscillator, how does equilibrium bond length change as more E is put in?

show graph of E vs internuclear separation

• where is average bond length

• what does length of each line mean

• ZPE?

equilibrium bond length stays the same

what does ZPE show?

vibration E is never zero

• still oscillates around central position at absolute zero

what is ZPE for E_v and G^~(v)?

G^~(v) is vibrational

what is the difference between the first two equations?

what is F^~(J)?

first has c in ms^-1, second has it in cms^-1

F^~(J) is rotational

for harmonic oscillator - how are energy levels space? is there dissociation? vibrational coupling? how does it change w temp?

all ELs are evenly spaced

peak position doesn’t change with temperature

bonds are not allowed to break = no dissociation

all vibrations are independent meaning no vibrational coupling

show harmonic oscillator with morse potential - where is dissociation E?

where is De and D₀?

how does anharmonic oscillator differ from harmonic?5

energy levels are lower than corresponding harmonic levels

ELs converge

allows bond dissociation

peak position changes with temp

vibrations can couple

how do gases in the atmosphere heat the earth by the greenhouse effect?

photons are absorbed after bouncing off the earth and remitted back down and heat the earth

what is integrated intensity(area under peak) of a vibrational peak show?

proportional to change in dipole moment/polarisability as a function of vibration

specific selection rules for harmonic vs anharmonic oscillator

Δv = ±1 (harmonic)

Δv = 0,±1,±2 …

why can nuclei and electrons be treated separately?

nuclei are an order of magnitude heavier than electrons

when a nucleus moves, the electrons move instantaneously