7. Optical Processes in Molecules
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
American Dye Source, Inc.
http://www.adsdyes.com/
7.1. The intensities of the spectral lines
7.1.1 Beer-Lambert law
Transmittance: T=  /0
 how much it is transmitted
or
Absorbance: A=log (0/ )
 how much it is absorbed
[J], 
0

l
Empirical Beer-Lambert law
d = - [J]  dl

A= -log T
d / = - [J] dl

 is a proportionality coefficient

0

d

l
0
 - d
   J  dl
l
0
If the concentration [J] is uniform, the integration gives:

ln
  J  l 
0
  0 e
  J  l
dl
 =  ln10
 = molar absorption coefficient
in L.mol-1.cm-1
Exponential decay
h 0
h
 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
  10  0 e  J  l
 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
b
I    ( ) d
a
a
b
frequency 
7.1.2. Absorption and emission processes
state f
The energy density of radiation  is the energy
per unit of volume per unit of frequency range:
 =I()/c
state i
StA
StE
SpE
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.
 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
 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
molecules in the higher state f.
i
+ Bfi ), where Nf is the number of
Important points:
 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 and
there is no net absorption.
 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

 fi   *f   i d
B
 fi
2
6 0  2
 The SpE increases dramatically with the frequency
compared to the StE
Af i
 8h 3 
 B f i
 
3
 c 
 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
7.2. Linewidths
A. Doppler broadening: only for gaseous samples
B. Lifetime broadening
From 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:
Af i   3
 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
7.3. Characteristics of electronic transitions
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
-
N = 20
+
1Ag 1Bu

 fi   *f   i d
0.15
Transition density (|e|)
0.10
INDO/SCI
0.05
Atomic transition densities
 qK = 0
0.00
K
-0.05
 qK rK =  1A
-0.10
K
-0.15
0
2
4
6
8
10
12
14
16
18
g 1Bu
20
Site number
 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
Colors
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.
The chlorophyll
7.3.1 The vibrational structure: Franck-Condon principle
E
Classical picture:
*s
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* > Regs 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.
gs
Reg Re*
s
Separation
distance between
atoms in the molecule
QM picture:
Initial state: the lowest vibrational state of gs
(nuclei at Regs). 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.
*s
E
gs
Regs
However, it's not the only accessible vibrational state
because several nearby states have an appreciable
probability of the nuclei being at Regs. 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
Dipole moment operator is a sum over all nuclei “j” and electrons “k” in the molecules
  e rk  e Z j R j
k
 i (r , R)   i (r , R)i ( R)
j
Born-Oppenheimer

 fi   *f   i d


 fi    f  f  e rk  e Z j R j   ii d

k

j



Electronic states
are orthogonal
 fi  e   f rk  i d   f i d  e Z j   f R ji d   f  i d
k
j




 fi   e   f rk  i d    f i d  

f
, i =

k
f
, i
S ( f ,i )
Transition dipole moment arising from the redistribution of electrons
S ( f ,i ) = Overlap integral between the vibrational state i in the initial electronic
state and the vibrational state f in the final electronic state
S ( f ,i )
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.
 fi  
f
, i
S ( f ,i )  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.
7.3.2. * 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*
In Ethene, the energy needed to excite
electronically the molecule, from the
ground state 12 to the first excited state
11 2*1 is provided by 7 eV: Ethene
absorbs the UV light (=170 nm).
E (eV ) 
1240
 (nm)
HOMO= 1
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.
-carotene
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.
Collisions
Molecule B
Molecule A
 Dissociation and chemical reaction
 Radiative decay = the excitation energy is
discarded
as
a
photon
(fluorescence,
phosphorescence)
Quantum yields
M. Pope M.Swenberg, in ”Electronic processes in organic crystal and polymers”
7.4.1. Selection rules
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.
 fi  e   f r i d  e  f r  i dr  S f Si d
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.
S  T: not allowed
T  S: not allowed
S  S: allowed
T  T: allowed
 fi  e   f r i d  e  f r  i dr  S f Si d
Symmetry selection rule
Molecular orbitals of butadiene
LUMO +1
LUMO
HOMO
HOMO-1
The function is unpair (u) if there is an inversion center
otherwise it is pair (g)
 Homo-1=g ;Homo=u; Lumo=g; Lumo+1=u
The operator ”r” is unpair.

The integral  f r  i dr 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 : allowed
HOMO LUMO+1 : not allowed
7.4.2 Fluorescence
Energy
*s
v=0 gs
v=2
v=1
v=0
Regs Re*
a
R
max b
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
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.
1.0
2.5
0.8
2.0
0.6
1.5
0.4
1.0
0.2
0.5
Polyfluorene (F8)
-1
3.0
5
1.2
a (x 10 cm )
PL Intensity (arb. units)
Fluorescence
0.0
n
2.5
1.0
- Weak self-absorption
0.8
2.0
- Vibronic structure
0.4
1.0
Carlos Silva, University of Cambridge
-1
1.5
5
0.6
a (x 10 cm )
ntensity (arb. units)
1.2
7.4.3. Phosphorescence
*S
*T
gs
S
Conditions:
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 gsS 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
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.