1. Radiative Transitions between Electronic States
November 14, 2002.
Michelle

2. Quantization of energy E = h
4.1 “Paradigm” Shifts
Maxwell  light as electromagnetic waves
Blackbody Radiation
Wavelength dependence of energy distribution of light emitted by a hot object
Photoelectric Effect
Wavelength dependence of energy of electrons emitted when light strikes a metal
Planck
Quantization of energy E = h
Einstein
Photons - quantized light consisting of particles possessing “bundles” of energy
DeBroglie  light as a particle with wave properties
E = h = h(c/) = pc
From the perspective of absorption and emission it is more convenient to think of light in terms of an oscillating electromagnetic wave

3. 4.2-4.3 Absorption and Emission
Transitions between electronic energy levels accompanied by absorption or emission of light
Photochemical region of the spectrum: nm 143 kcal mol-1-41 kcal mol-1
valence orbital (, , n)  antibonding orbital (*, *)
Chromophore: atom or group acting as a light absorber
Lumophore: atom or group acting as a light emitter
C=O C=C C=C-C=C C=C-C=O aromatics
Molar absorptivity  measures “absorption strength”
units cm-1M-1 NOT cm2mol-1

4. 4.4 The Nature of Light
The classical theory of light is a convenient starting point providing a pictorial and understandable physical representation of the interaction of light and molecules
classical theory can be improved by applying quantum interpretations of basic concepts (orbital, quantized energy etc.)
Dipoles as a model for interactions between electrons and light
oscillating electric dipole field of the electromagnetic wave
oscillating dipoles due to electrons moving in orbitals
Dipole-dipole interactions are through space and don’t require orbital overlap
(Interactions requiring overlap are “exchange interactions”)
Examples of dipolar interactions: London dispersion forces, EPR, NMR, large splittings for crystals (exciton interactions)
Exciton migration: electron-hole pair hopping from molecule to molecule in a crystal

5. 4.4 Dispersion Forces
Correlation of fluctuations in electronic charge distributions in molecules
Dipole on A drives formation of dipole on B and vice versa
Fluctuating dipoles are in “resonance”
E ~ ABR-3AB
Energy of the dipole-dipole interaction falls off as A and B move apart, given by R-6AB
True for all dipole-dipole interactions, magnetic or electric

6. 4.4 Light as an Oscillating Electric Field
Frequency () of oscillating field must “match” a possible electronic oscillation frequency (conservation of energy)
There must be an interaction or coupling between the field (oscillating dipoles) and the electron
Interaction strength depends on field dipole and induced dipole strength as well as distance between the two.
Laws of conservation of angular momentum must be obeyed (electrons, nuclei, spins)
Spin change is highly resisted in absorption due to time constraints
*the most important interaction between the electromagnetic field and the electrons of a molecule can be modeled as the interaction of 2 oscillating dipole systems that behave as reciprocal energy-donor, energy-acceptors

7. 4.4 Light as an Oscillating Electric Field
H magnetic field
Direction of propogation
If the field can couple to the electrons it can exchange energy by driving the system into resonance at a frequency common to both
A light wave generates a time-dependent force field F
absorption: photons being removed from the electromagnetic field
emission: photons being added (?) to the electromagnetic field
ABSORPTION
(reverse for emission)

8. 4.4 Light as an Oscillating Electric Field
Radiative transitions between states are induced by perturbations which make the two states”look alike” by inducing a resonance
Resonance requires that the two states have the same energy and momentum characteristics and a common frequency for resonance
E = h
Energy difference between two states
Frequency of light wave oscillation
The energy of the photon must exactly match the the energy level difference in the molecule

9. 4.4 Light as an Oscillating Electric Field
Force on an electron in a molecue by a light wave
F = e + e[H v] c
Electrical force
Magnetic force
c >> velectron
Force on an electron
F ~ e
Major force on electrons is due to the oscillating electric field of the light wave
Net effect of the interaction is generation of a transitory dipole moment in the molecule

10. 4.4 Light as an Oscillating Electric Field
Electric dipole induced by an electric field generated between two plates
Direction of induced dipole is always parallel to the direction of the external electric field

11. 4.4 Light as an Oscillating Electric Field
Light and the hydrogen atom
Electric field interaction reshapes the electron distribution of the 1s orbital
s orbital has 0 units of angular momentum ( h )
p orbital has 1
 photon must have 1 unit
No node, no vibration  1 node, vibratory motion
Increase in # of nodes essential for absorption and vice versa for emission (related to nodal nature of light “wave”)

12. 4.4 Light as an Oscillating Electric Field
Light and the hydrogen molecule
Interaction involves  and  orbitals instead of s and p
Absorption is    or * and is analogous to the s  p transition in the hydrogen atom

13. 4.4 Light as a Stream of Particles: Photons
The photon as a reagent that may collide and react with molecules
Long  photons have little energy and momentum, short  photons have a lot of both
Largest cross-section of an individual chromophore is ~10 Å
Nuclei are effectively frozen in space as a photon passes
Spectroscopic Properties and Theoretical Properties
In order to use the laws of quantum mechanics to describe fundamental properties we have to consider these terms:
f oscillator strength
i transition dipole moment
P transition probabilities

14. 4.4 Oscillator Strength
Probability of light absorption is related to the oscillator strength f
Theoretical oscillator strength
Experimental absorption
f ~ 4.3x10-9 ∫ d
Area under  vs. wavenumber plot
Strong absorption => f~1
Rate constant for emission k0e is related to  by:
k0e ~ 4.3x10-9 -20 ∫ d ~ -20 f
Oscillator strength can be related to transition dipole moment by:
Transition dipole moment
f = 8me 2i ~ 10-5|eri|2
3he2
f = 8me <H>2
3he2
Relationship between experimental and quantum quantities

15. 4.5 The Shape of Absorption and Emission Specta
Electronic transitions in molecules are not as “pure” as they are in atoms, in molecules relative nuclei motions must be considered
An ensemble of nuclear configurations are observed
“most prominent vibrational progression is associated with the vibration whose eqm position is most changed by the transition” ?

16. 4.5 Franck-Condon & Absorption

17. 4.5 Franck-Condon & Emission
The most probable transitions produce an elongated ground state, while absorption initiallly produces a compressed excited state
In both cases of absorption and emission, transition occurs from the =0 level of the initial state to some vibrational level of the final state  which level is dependent on the displacement between  and *
Band spacing in the resulting spectrum is determined by the vibrational structure in the final state

18. 4.6 State Mixing < n|H| > = 0 when n and  are orthogonal
State mixing is the first- or higher-order correction to an original zero-order approximation of single orbital configurations or single spin multiplicities
Example: an n, * S1 state actually contains a finite amount of , * character mixed in so the first order wavefunction is given by:
Mixing coefficient
first order
n, *
(S1) = (n, *) + (, *)
zero order
n, *
zero order
, *
Spatial overlap of mixing states
Symmetry properties
Nature and symmetry of H
Features important to state mixing:
Energy gap between zero order configurations
Magnitude of the matrix element that mixes the states
 = <a|H|b>
Ea - Eb
< n|H| >
= 0 when n and  are orthogonal

19. 4.6 Mechanisms for Mixing Singlet and Triplet
Singlet-triplet transitions are strictly forbidden in first order but spin-orbit coupling mixes singlet and triplet states so that transitions become allowed
1. Direct coupling of T1 and Sn <T1|Hso|Sn> ≠ 0
2. Indirect electronic coupling via and intermediate triplet state (mixing of T1 with upper vibrational triplets)
3. Turning on 1. and 2. via vibrational motions of the molecule

20. 4.6 Mechanisms for Mixing Singlet and Triplet
Measurement of “forbidden” absorptions and emissions provide evidence of the identity of the mixing state
n, *
, *
Vibrational structure provides clues as to which motions are most effective in mixing states

21. 4.7 Molecular Electronic Spectroscopy
Kasha’s Rule: photochemical reactions occur from the lowest excited singlet or triplet states
absorption
emission
excitation
 ~ [log(I0/It)]/lc
Ie = 2.3 I0 AlAe[A]
for a weakly absorbing solution of A
Optical density

22. 4.7 Spin-Allowed Transitions
“allowedness” is measured by the oscillator strength f which can be dissected into:
f = (fe x fv x fs) fmax
fe - electronic factors, fv - Franck-Condon factors, fs - spin-orbit factors
A perfectly allowed transition has f = 1
A spin allowed transition has fs = 1 and for a spin-forbidden transition fs depends on spin-orbit coupling fe
Overlap forbiddeness: poor spatial overlap of orbitals involved in electronic transition
Orbital forbiddeness: wavefunctions which overlap in space but cancel because of symmetry

23. 4.7 Quantum Yields of Allowed Fluorescence
Quantum yield of emission is given by:
e = *k0e(k0e + ki)-1 = *k0e
Experimental lifetime
Formation efficiency of the emitting state
All rate constants that deactivate the excited state
ki is very sensitive to experimental conditions:
Diffusional quenching and thermal chemical reactions may compete with radiative decay
Certain molecular motions may also provide competitive decay pathways
Measurements at low temperature (77K) cause ki terms to become small relative to k0e
F = kF(kF + kST)-1 = kF 

24. 4.7 Quantum Yields of Allowed Fluorescence
Generalizations from experimental observations:
Rigid aromatic hydrocarbons are measurably fluorescent
Low value of F for these molecules is usually the result of competing ISC
Substitution of H for X generally results in a decrease in F
Substitution of C=O for H generally results in a substantial decrease in F
Molecular rigidity enhances F
F is an efficiency that compares relative transition probabilities, doesn’t relate directly to rates

25. 4.7 Quantum Yields of Allowed Fluorescence
The highest energy vibrational band in an emission spectrum usually corresponds to the 0,0 transition
ET and ES can be obtained
If there is no fine structure the onset of emission is used to guess the upper limit of E

26. 4.8 Spin-Orbit Coupling
The value of (S0T) and k0P(T  S0) are directly related to the degree of spin-orbit coupling between S0 and T
S-O coupling depends on:
Nuclear charge
Availability of transitions between “orthogonal” orbitals
Availability of a one atom center transition
Degree of S-O coupling is related to , a S-O coupling constant
 depends on the orbital configurations involved

27. 4.8 Multiplicity Change in Radiative Transitions
Greater oscillator strength than n2  , *
In general fvfe(, *) > fvfe(n, *) because (, *) > (n, *) for S-S transitions where spin is not a factor
Implies that fs (n, *) >> fs (, *) for spin-forbidden radiative transitions
a radiative transition n2 * where the electron jumps from px to py on the same atom is very favourable because of the momentum change (compensating for the spin momentum change)
In planar molecules, out of plane vibrations can cause orbital mixing and minor S-O coupling

28. 4.9 Perturbation of S0T Absorption
Compound possessing lowest energy , * or heavy atoms are usually insensitive to spin-orbit perturbations
S0T(, *) of aromatics is generally enhanced by S-O perturbation
S0T(n, *) of ketones is insensitive to S-O perturbation
S0T enhancers
Molecular oxygen
Organic halides, organometallics
Heavy atom rare gases

29. 4.9 Perturbation of S0T Absorption
Internal versus external heavy atom effect
Note position dependence
Useful for determining the nature of the excited state

30. 4.9 Triplet Sublevels
A triplet state at room temperature is actually a rapidly equilibrating mixture of 3 states (sublevels Tx Ty Tz)
Absorption initially produces only one of the three levels
Normally absorption to the sublevels is not resolved
Molecules in different sublevels have their electrons in different planes
If T’s are not rapidly equilibrated different phosphorescence parameters will be observed (above 10K this is usually not an issue)

31. 4.9 Phosphorescence
P is not a reliable parameter for characterizing T
P ~ STk0P(k0P + kTS)-1 gives the quantum yield for phosphorescence when measured at 77 K and only ISC is competing
No reports of phosphorescence from nonaromatic hydrocarbons
ISC is inefficient for flexible molecules
T1 S0 is spin forbidden AND Franck-Condon forbidden (twisted triplet)
Large kd due to surface “touching” between T1 and S0
Theoretical relationship between P or T and molecular structure is not direct
Small P may be due to low ST or to kd>>k0P
Phosphorescence may be measured at room temperature if:
Triplet quenching impurities are rigorously excluded
Unimolecular triplet deactivation must be <104 k0P at RT

32. 4.11 Excited State Structures
S1 and T1 states are electronic isomers of S0

33. 4.12 Complexes and Exciplexes
2 or more molecules may participate in a cooperative absorption or emission
Spectroscopic characteristics are:
Observation of a new absorption band not characteristic of starting components
Observation of a new emission band not characteristic of starting components
Concentration dependence of the new absorption/emission intensity
Exciplex or excimer: an excited molecular complex that is dissociated or only weakly associated in the ground state

34. 4.12 Complexes and Exciplexes
Mixtures of molecules with low IP or high EA often exhibit charge-transfer absorption bands (EDA bands)
This type of absorption is very sensitive to changes in solvent polarity
Transition from * can be thought of as D,AD+ A-

35. 4.12 Complexes and Exciplexes
Energetic considerations
Collision between M* and polarizable ground state N will usually result in a complex stablized by some charge-transfer interactions
if M*N properties are distinct this is an exciplex or excimer
Exciplex/mer emission will occur to a very weakly bound or dissociative ground state
Favourable formation is a balance with entropic considerations

36. 4.12 Complexes and Exciplexes
Exciplex/excimer emission is featureless
Excimer emission is not as solvent dependent as exciplex (less charge-transfer stabilization)
Intramolecular exciplexes/eximers are also possible when the linkage is of the appropriate length
Formation of excited state complexes can also be monitored by time-resolved spectroscopy

37. 4.13 Delayed Fluorescence
The observed F may be longer than expected based on “prompt” emission...
Thermal repopulation of S1
When the S-T gap is small and the ISC is fast the S1 state can be repopulated from Tn
Effect disappears at low temperatures
Triplet-triplet anihilation
Combination of 2 triplets to form a sinlget state from which emission is observed
Es  ET + ET

38. Emission from upper excited states
4.14 The Azulene Anomaly
Emission from upper excited states
Large S2-S1 gap slows down the interconversion which would normally cause all emission to be from S1