7. Optical Processes in Molecules

American Dye Source, Inc. http://www.adsdyes.com/

7.1. The intensities of the spectral lines

7.2. Linewidths

7.3. The characteristics of electronic transitions

7.3.1 The vibrational structure

7.3.2 * transition

7.4. The fates of electronically excited states

7.4.1 Fluorescence

7.4.2 phosphorescence

7.4.3 Dissociation

7.1. The intensities of the spectral lines

[J],

0

Transmittance: T= /0 how much it is transmittedorAbsorbance: A=log (0/ ) how much it is absorbed

A= -log TEmpirical Beer-Lambert law

d = - [J] dl d / = - [J] dl

is a proportionality coefficient

l

dlJd

0

0

If the concentration [J] is uniform, the integration gives:

l

dl

- d

0

lJ

0

ln lJe 0

= ln10 = molar absorption coefficientin L.mol-1.cm-1

l

7.1.1 Beer-Lambert law

0

lJe 010

Exponential decay We consider only one specific frequency of the incident photon beam and we look what happens to the intensity of this beam.

The concentration [J] and the thickness l have a strong impact on the intensity of the transmitted light

The molar absorption coefficient is specific to the molecules in the sample !!

The molar absorption coefficient is a function of the frequency of the incident photon h: = f()

is large at the frequency corresponding to an absorption, i.e an excitation of the molecules by the incident photons. is related to the transition dipole moment fi.

The intensity of a transition is

h h

b

a

dI

)(

frequency

a b

7.1.2. Absorption and emission processes

The transition rate wif for StA is proportional to the energy density of radiation , i.e. “ the intensity of the incident light I()”: wif=Bif , via a constant Bif, the Einstein coefficient for StA (related to the transition dipole moment fi).

For StE: wfi=Bfi .

For SpE, transition rate is independent of the intensity of the incident light: w*fi=Afi

state i

state f

The energy density of radiation is the energy per unit of volume per unit of frequency range: =I()/c

Stimulated emission: the molecule in an excited state can be stimulated by an incoming photon in order to come down to the lower energetic state. Only radiation of the same frequency as the transition gives rise to the stimulate emission.

StAStE SpE

The total rate of absorption Wif is the transition rate of a single molecule multiplied by the number of molecules Ni in the lower state: Wi f = Ni wi f

The total rate of emission Wfi= Nf (Af i + Bfi ), where Nf is the number of molecules in the higher state f.

The Einstein coefficients for the stimulated absorption and emission are the same

Bif =Bfi =B

If two states f and i have equal population: Nf = Ni, the StA rate = StE rate andthere is no net absorption.

Important points:

The Einstein coefficients for the stimulated absorption and emission give the intensity of lines in absorption or emission spectroscopy. They are related to the square of the transition dipole moment

20

2

6 fi

B

diffi *

ifif Bc

hA

3

38 The SpE increases dramatically with the frequency compared to the StE

The higher the energy difference between the state f and i, the higher the rate of SpE for this high photon energy h.

For rotational and vibrational transition (low frequency), SpE can be neglected. Then, the net rate of absorption is:

Wnet= Ni Bif - Nf Bfi = (Ni - Nf)B

Wnet is proportional to the population difference (Ni - Nf) between the 2 states f and i

B. Lifetime broadeningFrom the Heisenberg uncertainty principle: if a system survives in a state for a time , the lifetime of the state, then its energy levels are blurred to an extent of order E

E

The shorter the lifetime of the states involved in the transition, the broader the corresponding spectral lines.

Lifetime factors:

The rate of spontaneous emission, w*fi=Afi, determines the natural limit of the lifetime of an excited state. This results in a natural linewidth of the transition directly related to Afi, which increases strongly with the frequency:

Natural lifetime for different transitions: electronic<vibrational<rotational

Natural linewidth: Eelectronic> Evibrational> Erotational

3 ifA

7.2. Linewidths

A. Doppler broadening: only for gaseous samples

Atomic transition densities

1Ag 1BuN = 20

INDO/SCI

+-

qK rK = K

1Ag 1Bu

qK = 0 K

diffi *

For a transition to be allowed, a dipole should be formed during the transition. This is properly represented in QM with the transition dipole moment μfi

7.3. Characteristics of electronic transitions

The size of the transition dipole can be regarded as a measure of the charge redistribution that accompanies a transition: a transition will be active only if the accompanying charge redistribution is dipolar

Sunlight, a white light, is composed of all the colors of the visible spectrum. Our eyes work like spectrometer: light goes from the source (the sun) to the object (the apple), and finally to the detector (the eye and brain).The surface of a green apple absorbs all the colored light rays, except for those corresponding to green, and reflects this color to the human eye. The green apple absorbs in the blue-violet and in the red. Green contributes in the complementary colors of violet and red. The dye comes from the chlorophyll molecules on the skin of the apple, they absorb photons with a wave-length around 400-450nm and 650-750nm. The dye molecules reach an electronic excited state, they are mainly deactivated by a quenching process, a non-radiative decay.

Colors

The chlorophyll

7.3.1 The vibrational structure: Franck-Condon principle

Classical picture:Because the nuclei are heavier than electrons, an electronic transition takes place much faster than the nuclei can respond. This is represented by the vertical green arrow in the graph: during the vertical electronic transition, the molecule has the same geometry as before the excitation.

During the transition, the electron density is rapidly built up in new regions of the molecule and removed from others, and the nuclei experience suddenly a new force field, a new potential (upper curve). They respond to this new force by beginning to vibrate.

Re* > Re

gs because an excited state is characterized by an electron in an anti-bonding molecular orbital, which gives rise to an elongation of one or several bonds in the molecule.

E

Separation distance between atoms in the molecule

Regs Re

*

*s

gs

QM picture:

Initial state: the lowest vibrational state of gs (nuclei at Re

gs). The vertical transition cuts through several vibrational levels of *s. The level marked * is the vibrational excited state of *s

that has a maximum amplitude at Regs, so this

vibrational state is the most probable state for the termination of the transition. Therefore, the electronic transition occurs the most intensely to the state *, that is the origin of the maximum in the absorption spectrum max.

E

Regs

*s

gs

However, it's not the only accessible vibrational state because several nearby states have an appreciable probability of the nuclei being at Re

gs. Therefore, transitions occur to all the vibrational states in this region, but with lower probabilities, that is the origin of the structure (small peaks) in an absorption spectrum: they are feature of the vibrational levels of *s.

Franck-Condon factors

diffi *

j

jjk

k RZere

dRZere iij

jjk

kfffi

)(),(),( RRrRr iii

j

ifijfjk

ifikffi ddRZeddre

Dipole moment operator is a sum over all nuclei “j” and electrons “k” in the molecules

= 0

Electronic states are orthogonal

),(, ififk

ikffi Sddreif

= Transition dipole moment arising from the redistribution of electronsif ,

),( ifS = Overlap integral between the vibrational state i in the initial electronic state and the vibrational state f in the final electronic state

Born-Oppenheimer

),( ifS is a measure of the match between the vibrational wavefunctions in the upper and lower electronic states: S=1 for a perfect match, S=0 where there is no similarity.

),(, iffi Sif

Intensity ÷ |μfi|2 Intensity ÷ |S(f, i)|2

At max, S >>, there is a good matching between the vibrational levels f and I

At a and b, S is small, there is a poor matching between f and i

The greater the overlap of the vibrational state wavefunction in the upper electronic state with the vibrational wavefunction in the lower electronic state, the greater the absorption intensity of that particular simultaneous electronic and vibrational transition.

A C=C double bond in a molecule acts as a chromophore. One of its important transitions is the * transition, in which an electron is promoted from a orbital to the corresponding antibonding orbital.

LUMO= 2*

HOMO= 1

-carotene

When the -conjugation pathway in the molecule is extended, the HOMO-LUMO separation energy, EL-H decreases. If EL-H is on the order of the energy of visible light E=h, then the molecules, such as the long carotenoid molecules, absorb visible light at a certain frequency (in the green). The photons with another energy, i.e. the radiations with other frequencies, are reflected towards our eyes and that gives the “orange” color of carrots that contains a lot of -carotenes.

In Ethene, the energy needed to excite electronically the molecule, from the ground state 12 to the first excited state 11 2*1 is provided by 7 eV: Ethene absorbs the UV light (=170 nm).

)(

1240)(

nmeVE

7.3.2. * transition

7.4. The fates of electronically excited states

Nonradiative decay = the excitation energy is transferred into the vibration, rotation, translation of the surrounding molecules via collisions.

Radiative decay = the excitation energy is discarded as a photon (fluorescence, phosphorescence)

Dissociation and chemical reaction

Molecule A

Molecule B

Collisions

Quantum yields

M. Pope M.Swenberg, in ”Electronic processes in organic crystal and polymers”

For a close-shell system in its ground state (all the electrons are paired), a simple electronic transition transforms the electronic wavefunction of a molecule into an excited state represented with 2 electrons in two different molecular orbitals (similar to the system of 2 electrons).

The probability for a transition is given by the transition dipole moment fi between an initial state i and a final state f :

Both states of the molecule are characterized by a spatial function and a spinfunction S.

Spin selection rule

If the initial and final states have both a spinfunction of the same symmetry, the transition dipole moment is non-zero: the transition is allowed.

If the two states have different spinfunction symmetries, the transition is forbidden.

dSSdrredre ifififfi

S T: not allowed S S: allowedT S: not allowed T T: allowed

7.4.1. Selection rules

dSSdrredre ifififfi

Symmetry selection rule

Molecular orbitals of butadiene

HOMO

LUMO +1

LUMO

HOMO-1

The function is unpair (u) if there is an inversion centerotherwise it is pair (g) Homo-1=g ;Homo=u; Lumo=g; Lumo+1=u

The operator ”r” is unpair.

The integral is zero if the product of the three functions is an unpair function. For a molecule with a center of inversion, this occurs if the final and initial state do not have the same parity.

Let’s consider two transitions in the monoelectronic picture:

HOMO LUMO : allowedHOMO LUMO+1 : not allowed

drr if

a bmax

Energy

R

gs

*s

v=0v=1v=2

v=0

Regs Re

*

7.4.2 Fluorescence

In accord with the Franck-Condon principle, the most probable transition occurs from *s to the vibrational state of gs, for which the molecule has the same inter-atomic separation Re

*. This vibrational state (v=1 in the Figure) is characterized by a maximum intensity of its vibrational wavefunction at Re

*. This is the origin of the maximum in the fluorescence or emission spectrum.

The excited molecule collides with the surrounding molecules and steps down the ladder of vibrational levels to v=0 of *s. The surrounding molecules, however, might now be unable to accept the larger energy difference needed to lower the molecule to gs. It might therefore survive long enough to undergo spontaneous emission. As a consequence, the transitions in the emission process have lower energy compared to the absorption transition

1.2

1.0

0.8

0.6

0.4

0.2

0.0

PL

Int

ens

ity (

arb

. uni

ts)

3.0

2.5

2.0

1.5

1.0

0.5

(x 1

05 cm

-1)1.2

1.0

0.8

0.6

0.4

0.2

0.0

PL

Int

ens

ity (

arb

. uni

ts)

4.03.53.02.52.0Photon Energy (eV)

2.5

2.0

1.5

1.0

0.5

(x 1

05 cm

-1)n

Fluorescence

Polyfluorene (F8)

Carlos Silva, University of Cambridge

- Weak self-absorption

- Vibronic structure

7.4.3. PhosphorescenceConditions:1) The potential felt by the atoms when the molecule is in its electronic singlet excited state (↑↓) crosses the potential for the molecule in its triplet excited state (↑↑). In other words, the structure of the molecule in both states is similar for specific vibrational levels of both states.

2) If there is a mechanism for unpairing two electron spins (and achieving the conversion of ↑↓ to ↑↑), the molecule may undergo intersystem crossing and becomes in *T. This is possible if the molecule contains heavy atoms for which spin-orbit coupling is important.

When the molecule reaches the vibrational ground state of *T, it is trapped!The solvent cannot absorb the final, large quantum of electronic energy, and the molecule cannot radiate its energy because return to gs

S is spin-forbidden….. However, it is not totally spin-forbidden because the spin-orbit coupling mixed the S and T states, such that the transition becomes weakly allowed.

weak intensity and slow radiative decay (can reach hours!!).Note: Phosphorescence more efficient for the solid phase

*S

*T

gsS

7.4.4. Dissociation

A dissociation is characterized by an absorption spectrum composed of two parts:

(i) a vibrational progression (ii) a contiuum absorption

For some molecules, the potential surface of the excited state is strongly shifted to the right compared to the potential of the ground state.

As a consequence, lot of vibrational states of the electronic excited state are accessible (vibrational progression described by the Franck-Condon principle), and the dissociation limit can be reached.

Beyond this dissociation limit, the absorption is continuous because the molecule is broken into two parts. The energy of the photon is used to break a bond and the rest in transformed in the unquantized translational energy of the two parts of the molecule.