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

Radiative Transitions between Electronic States

November 14, 2002.

Michelle

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4.1 “Paradigm” Shifts•Maxwell light as electromagnetic waves

Blackbody Radiation

Wavelength dependence of energy distribution of light

emitted by a hot object

Photoelectric EffectWavelength dependence of energy of electrons emitted when light strikes a metal

PlanckQuantization of energy E =

h

EinsteinPhotons - 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

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4.2-4.3 Absorption and Emission

Transitions between electronic energy levels accompanied by absorption or emission of light

Photochemical region of the spectrum: 200-700 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

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4.4 The Nature of LightThe 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

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4.4 Dispersion Forces

Correlation of fluctuations in electronic charge distributions in molecules

E ~ ABR-3AB

•Dipole on A drives formation of dipole on

B and vice versa

•Fluctuating dipoles are in “resonance”

Energy of the dipole-dipole interaction falls off as A and B

move apart, given by R-6AB

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

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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

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4.4 Light as an Oscillating Electric Field 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

ABSORPTION(reverse for emission)

absorption: photons being removed from the electromagnetic field

emission: photons being added (?) to the electromagnetic field

A light wave generates a time-dependent force field F

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• 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 = hEnergy difference

between two statesFrequency of light wave oscillation

The energy of the photon must exactly match the the energy level difference in the

molecule

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F = e + e[H v]c

Electrical forceMagnetic force

Force on an electron in a molecue by a light wave

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

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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

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Light and the hydrogen atomElectric field interaction

reshapes the electron distribution of the 1s orbital

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”)

s orbital has 0 units of angular

momentum ( h )

p orbital has 1

photon must have 1

unit

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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

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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 passesSpectroscopic 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

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4.4 Oscillator Strength

Probability of light absorption is related to the oscillator strength f

f ~ 4.3x10-9 ∫ dTheoretical oscillator strength

Experimental absorption

Area under vs. wavenumber plot

Rate constant for emission k0e is related to by:

k0e ~ 4.3x10-9 -2

0 ∫ d ~

-20 fOscillator strength can be related to transition dipole moment by:

f = 8me 2i ~ 10-5|eri|2

3he2

Transition dipole moment

f = 8me <H>2

3he2

Relationship between experimental and quantum quantities

Strong absorption => f~1

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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” ?

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4.5 Franck-Condon & Absorption

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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

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4.6 State Mixing

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:

(S1) = (n, *) + (, *)first ordern, *

zero ordern, *

zero order, *

Mixing coefficient

= <a|H|b> Ea - Eb

Features important to state mixing:

• Energy gap between zero order configurations

• Magnitude of the matrix element that mixes the states

• Spatial overlap of mixing states

• Symmetry properties

• Nature and symmetry of H

< n|H| > = 0 when n and are orthogonal

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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

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4.6 Mechanisms for Mixing Singlet and Triplet

Measurement of “forbidden” absorptions and emissions provide evidence of the identity

of the mixing state

Vibrational structure provides clues as to

which motions are most effective in mixing states

n, * , *

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4.7 Molecular Electronic Spectroscopy

Kasha’s Rule: photochemical reactions occur from the lowest excited singlet or triplet states

~ [log(I0/It)]/lc

Optical density

Ie = 2.3 I0 AlAe[A]

for a weakly absorbing solution of A

• absorption

• emission

• excitation

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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

couplingfe

Overlap forbiddeness: poor spatial overlap of orbitals involved in electronic transition

Orbital forbiddeness: wavefunctions which overlap in space but cancel because of symmetry

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4.7 Quantum Yields of Allowed Fluorescence

Quantum yield of emission is given by:

e = *k0e(k0

e + ki)-1 = *k0e

All rate constants that deactivate the excited state

Experimental lifetime

Formation efficiency of the emitting 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 k0

e

F = kF(kF + kST)-1 = kF

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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

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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

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4.8 Spin-Orbit Coupling

• The value of (S0T) 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

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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

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4.9 Perturbation of S0T Absorption

Compound possessing lowest energy , * or heavy atoms are usually insensitive to spin-orbit perturbations

•S0T(, *) of aromatics is generally enhanced by S-O perturbation

•S0T(n, *) of ketones is insensitive to S-O perturbation

•S0T enhancers

•Molecular oxygen

•Organic halides, organometallics

•Heavy atom rare gases

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4.9 Perturbation of S0T Absorption

Internal versus external heavy atom effect

Note position dependence Useful for determining

the nature of the excited state

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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)

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4.9 Phosphorescence

P is not a reliable parameter for characterizing T

P ~ STk0P(k0

P + 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>>k0

P

• Phosphorescence may be measured at room temperature if:

•Triplet quenching impurities are rigorously excluded•Unimolecular triplet deactivation must be <104 k0

P at RT

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4.11 Excited State Structures

S1 and T1 states are electronic isomers

of S0

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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

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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,AD+ A-

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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

Energetic considerations

Favourable formation is a balance with entropic

considerations

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•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

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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

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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

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

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Transcript of Radiative Transitions between Electronic States November 14, 2002. Michelle.