1. Prerequisites
1.1.The Schrödinger Equation for molecules
 All properties of a molecule (= M nuclei + n electrons) can be evaluated if we find the
wavefunction (R1,….Rm, x1, x2,..., xn) by solving the Schrödinger equation (SE): H =E.
H= Ttot + Vtot= (TN + Te )+ (VeN + Vee + VNN)
Rj are the nucleus coordinates, xi= (ri, σi) the electronic coordinates (ri=spatial
coordinates, σi=spin coordinate).
The total kinetic operator of the molecule is composed of a part for the nuclei TN and one
for the electrons Te. The total potential energy operator is the sum of the
electron/electron (Vee), electron/nucleus (VeN) and nucleus/nucleus (VNN) interactions.
2
ˆ
Ttot = 
2
Vˆtot  
1
4 0
n
M

i=1 j 1
2
M

j=1
j
2
2

2
mj
1
Z je
+
4 0
ri  R j
n

i=1 i>i
i

i=1 mi
n
2
e2
1
+
ri  ri '
4 0
M

j=1 j   j
2
Z j Z j e
R j  R j'
Mj is the mass of the nucleus j and mi is the electron mass. Zj is the number of protons in
the nucleus j and e is the charge of an electron.
 Solving with the best approximation the SE is the challenge of quantum chemistry (W.
Kohn and J. Pople: Nobel Prize in Chemistry 1998).
http://staffwww.itn.liu.se/~xavcr/TNE078/
1.2. The Born-Oppenheimer approximation
 The electrons are much lighter than the nuclei (me/mH1/1836)  their motion is much
faster than the vibrational and rotational motions of the nuclei within the molecule.
 A good approximation is to neglect the coupling terms between the motion of the
electrons and the nuclei: this is the Born-Oppenheimer approximation. The Schrödinger
equation can then be divided into two equations:
1) One describes the motion of the nuclei: the “nuclear Schrödinger equation” (see next
slide). The eigenvalues of this nuclear part of the SE gives the discrete energetic levels of
the vibration and rotation, translation of the molecule.
 The vibrational and rotational spectroscopies are used to observe transition between
these energetic levels.
2) The other one describes the motion of the electrons around the nuclei whose positions
are fixed. This electronic part of the SE is the “electronic Schrödinger equation”:
Tˆ Vˆ
e

ˆ  elect(R, r) = E (R)  elect(R, r)

V
eN
ee
elect
 The knowledge of the electronic wavefunction is necessary to understand chemical
bonding, electronic and optical properties of the matter.
2. Molecular Vibrations
2.1. The electronic Schrödinger equation
Tˆ Vˆ
e
eN

 Vˆee  elect(R, r) = Eelect  elect(R, r)
R
 The nuclear coordinate R appears as a parameter in the
expression of the electronic wave function.
 An electronic wave function elect(R,r) and an energy Eelect
are associated to each structure of the molecule (set of
nuclei coordinates R).
 For each variation of bond length in the molecule (each
new R), we can solve the electronic SE and evaluate the
energy that the molecule would have in this structure, i.e.
Etot(R)=Eelect+VNN(R): the molecular potential energy curve
Etot(R) is obtained (see Figure).
D0
 The molecule is the most stable (minimum of energy) for
one specific position of the nuclei: the equilibrium position
Re.
 The zero energy corresponds to the dissociated molecule.
 The depth of the minimum, De, gives the bond dissociation energy, D0, considering the
fact that vibrational energy is never zero, but ½ħ :
D0=De- ½ ħ
Appendix: More on The Born-Oppenheimer Approximation
After solving the electronic Schrödinger equation Eelect(R,r) for a specific position of
nuclei, one can estimate the total energy for fixed nuclei: Etot(R) = Eelect(R,r) + VNN(R)
Varying the position of nuclei and solving the electronic SE allows to calculate potential
felt by the nuclei: potential energy surface Etot(R)=f(R). Since electrons move faster than
nuclei, the nuclei feel this average potential Etot(E)=f(R) created by the surrounding
electrons.
The nuclear Hamiltonian describing the motion of nuclei in this average field of
electrons is:
Hnucl(R) = TNN (R) + Etot(R)
Thus, the nuclear Schrödinger equation is Hnucl(R)ψnucl(R)=Eψnucl(R)
The nuclear wavefunction ψnucl(R) describes vibration, rotation, translation motions of
the molecule.
The eigenvalue of that equation is the total energy in the BO approximation of the total
Schrödinger equation: Htot =Etot since Hnucl includes the nuclear energies TNN, VNN and
the electronic energies Tee and Vee, as well as VeN (indirectly in the calculated potential
Etot(R) estimated after solving the electronic SE).
In the BO approximation: tot (R,r)=ψnucl(R)* ψelect(R, r)
2.2. Diatomic molecules
A. The harmonic approximation: F  
V
 kx
x
V 
1 2
kx
2
x= R-Re = the deformation of the bond
k= the force constant of the bond
1   2V  2
 V 
V ( x)  V (0)  
 x   2  x  ...
2  x 0
 x 0
Zero of energy:
V(0)=0
Energy minimum
1   2V
V ( x )   2
2  x
 2
 x
0
  2V 
k   2 
 x 0
V 
1 2
kx
2
 The steeper the walls of the potential (the
stiffer the bond), the greater the force constant.
B. The Schrödinger equation
The relative motion of 2 atoms of masses m1 and m2 with a
parabolic potential energy is a problem mathematically equivalent
to the motion of an effective mass meff in the harmonic potential.
This is described with the Schrödinger Equation:
 2 2 1 2 
 kx   E

2
2
m
eff

x
2


meff 
m2 m1
m1  m2
meff is the effective mass (or reduced mass).
The eigenvalues or permitted vibrational energy
levels are:
1

E      ;   0,1,2,...
2

1/ 2
 k 
 

m
eff


C. The form of the wavefunctions
1/ 4
  ( x)  N H ( y )e  y
Hermite Polynomial
2
/2
;
 2 
x

y
and   

 mk
Gaussian-type function
N is the normalization constant
 The higher the quantum number  , the larger the number of nodes
in the wavefunction
D. Quantum behavior of the oscillator
The probability to find an oscillator (in its ground
state: =0) beyond the turning point xtp (the classical

limit), is:
P    20 ( x) dx  0.08
xtp
1
V  Vmax  E  kxtp2
2
Quantum
behavior
Quantum
behavior
Classical
behavior
xtp 0 -xtp
1/ 2
 2E 
 xtp   

 k 
0
xtp
Quantum
behavior
xtp
Classical
behavior
In the harmonic approximation, a diatomic
molecule in the vibration state = 0 has a
probability of 8% to be stretched (and 8% to be
compressed) beyond its classical limit. These
tunnelling probabilities are independent of the
force constant and the mass of the oscillator.
This effect will be significant in some electron
transfer reactions
2.3. Anharmonicity
 An simple analytical expression, called the Morse potential, represents the main
features of the real potential for a molecule:
- close to the minimum potential of depth De, the potential is close to be harmonic.
- for large displacement, the potential represents the bond dissociation

V  hcD e 1  e  a ( R  Re )
1/ 2
 meff  2 

a
 2hcD 
e 


2
1/ 2
 k 

 
 2hcDe 
 The Schrödinger equation can be solved with the
Morse potential and the permitted energy levels are:
2
1
1


E     hc~     hcxe~ ;   0,1,2,... max
2
2


a 2
~
xe 

2meff 4 De
When  increases, the second term becomes quickly more negative than the first
term
 the energy levels become less widely spaced at high excitation
2.4. Vibrations in polyatomic molecules
A. Number of vibrational modes
 A molecule composed of N atoms. The total number of
coordinates to specify the locations of N atoms is 3N (e.g:
x, y, z per atom).
 But, we are interested with the relative motion of these
atoms with respect to each other,
 we can remove the 3 coordinates (Xc.m, Yc.m, Zc.m) of
the center of gravity, which characterize the translation
of the molecule.
 we can remove the 3 angular coordinates (, , )
which specify the global rotation of the molecule in the
space. (2 angles are enough for linear molecules)
 The remaining coordinates are directly related to the
vibrations between atoms. There are 3N-6 displacements
of the atoms relative to one another: these are the 3N-6
independent vibrational modes (3N-5 for linear
molecules)
B. Combinations of displacements
L
 Example: CO2 3 atoms, 3N-5= 4 modes
R
2 modes can describe the variation of the CO bond lengths
(stretching modes) and 2 modes the variations of the bond
angles (bending modes).
If the stretching modes are L and R (each bond is
considered separately), then one see that if one is excited, it
gives the energy to the other mode. The modes are not
independent.
Stretching
1 symmetric
2
 It’s possible to find specific modes that are independent,
that is if one is excited, it does not excite the other: these
are the normal modes. For CO2, these are: 1, 2, 3 and 4.
 Each normal mode q behaves like an independent
harmonic oscillator (approximation), so has a series of
terms Gq() where ~q is the wavenumber of the mode q and
depends on the force constant kq and the effective mass of
the mode mq.
1

Gq ( )     ~q ;   0,1,2,...
2

1  k q 
~
q 
2c  mq 
1/ 2
Stretching
Anti-symmetric
3
4
3. Electron Transfer (ET)
3.1. Model system:
H2, H*2+
H2+, H*2
The position of the molecules
is fixed. The only structural
parameter that can change is
the bond length.
2 diatomic molecules
Transition state
or coincidence
event geometry
Total energy of the system
composed of 2 molecules:
H2, H*2+
H2+, H*2
The process by which the
reactants become products
depends on precisely how
the barrier is passed over,
tunneled
through,
or
otherwise avoided. This is
the topic of ET kinetics and
rate theory.
Figure by F. Barbara et al., J. Phys.
Chem. 100, 13148 (1996).
The internal reorganization energy (i)
i reflects the energy released upon geometric changes in the two molecules after
the electron transfer. (i ↔ electron-vibration coupling): H2 gives 2 , H2* gives 1
H2 + H*2+
H2+ + H*2
q
H*2← H*2+
Potential energy curve
of one molecule H2 in
the neutral state and
its ionized state H2+.
H2  H2+
H
H
i= 1+ 2
Interatomic distance q
3.2. Marcus theory: ET from a donor (D) to an acceptor (A)
(Nobel Prize in Chemistry in 1992)
G= Gibbs energy
D + A  D+ +A-
G
G= H+T S
Reactants
Products

Reorganization energy 
G#
G°
Activation energy
i
s
Internal + External
Geometry
Solvent
Generalized coordinates (solvent + D + A)
ket = A exp
-G# / kT
-(G°+ )2 / 4  kT
= (42/h) HRP2 (4kT)-1/2 exp
Ket is the rate of electron transfer
A. Solvent Reorganization
Example: a self-exchange reaction
Fe*2+ + Fe3+  Fe*3+ + Fe2+
 The solvent molecules rearrange to screen at best the charges.
This phenomenon is accompanied with a reorganization energy s
B. Two expected regimes for ET from Macrus theory
G
G
G

G*
G°
G*
G°=
Q
G°

G° = 
(b) G°=
(c) G°>
G° > 
Activationless case
(kET independent on T)
ed
ert
No
rm
a
v
In
l
log (kET)
ket
G° < 
(a)
G°<
G°
<
Thermodynamically
favorable,
but
not
kinetically
favorable:
the reaction is blocked
-G°
C. Marcus Theory / The Inverted Region
 Validity: The inverted region predicted by Marcus is demonstrated experimentally
 Limit: The evolution of the rate with G0 is not exactly following a parabola
Figure by J.R. Miller et al.,
JACS, 106, 3047 (1984)
ket = A exp
-(G°+ )2 / 4  kT
 Failure of Marcus’ theory: if T  0, kET 0: not observed experimentally
because it is assumed that the barrier is crossed… while nuclar tunnel effects occurs
3.3. Quantum Mechanical Corrections: Vibronic Theory
Considering that the nuclei can tunnel through the barrier, the rate of electron
transfer becomes according to the golden rule:
Energy match
criterion
Electronic coupling
Overlap of the vibrational
wavefunctions
Reactants
Figure by DeVault, D. Quantum Mechanical
Tunneling in Biological Systems; Cambridge
University Press (Cambridge, 1984).
Products
 4 2  2  1 

 H rp 
krp  
 h   4s kT 
Frank-Condon
FC term
S
i
 
1
2
Marcus-Levich-Jortner
Theory:
 (G 0  s   '   ) 2 
S '

exp(S )
exp 


 '!
4s kT
'


Huang-Rhys factor
3.4. Distance Dependence of ET Rates
A.Tunnelling
If the energy E of the electron is below a finite
barrier of potential V, the wavefunction of the
electron is non-zero inside the barrier and
outside the barrier.
 there is certain probability to find the
electron outside the barrier, even though
according to classical mechanics the electron
has insufficient energy to escape: this effect is
called “tunnelling”.
X=0
X=L
wavefunction
 Even though valence electrons in molecular orbitals are not represented by this
simple plane-wave, their wavefunction also undergoes an exponential-type decay
towards the outside of the molecule.
R0
D
A
R
k ET  H RP
2
V0 = donor/acceptor electronic coupling matrix element at van der Waals separation R0.
 is a constant that determines the rate of falloff of HRP with distance [0.8-1.2 Å-1].
According to the Marcus-Levich-Jortner expression:
if s is small, the (FC) term does not change much with R.
 KET decreases exponentially with R
4. The Semiconducting Phase:
Undoped Conjugated Materials
A. Polymers with chromophoric pendant groups
 Microscopic disorder
B. Molecular crystals
poly(N-vinylcarbazole)
C. Conjugated polymers
4.1. Introduction: classes of solids
σ = electrical conductivity
Eg = Band gap
1 S (Siemens)=1/ Ω
For Eg> ~2eV
 Insulator
For 0< Eg < 2eV  Semiconductor
For Eg0
 Metal
 σRT < 10-10 Ω-1cm-1= 10-10 S/cm
 10-10 S/cm < σ RT < 102 S/cm
 σRT > 102 S/cm
A. Definitions
 The mobility  of the charge carriers is the average speed of diffusion ||,
or net drift velocity, of the charge carrier (cm/s) as a function of applied electric
field (V/cm)
 = ||/E

in cm2/Vs
 is positive even though e- and h+ travel
in opposite direction.
||
 The electrical conductivity σ can be defined as a sum of two terms:
σ = (ne e + pe h )
in 1/Ωcm
n and p = density of charge carriers (n for electrons and p for holes) in cm-3
e = unitary charge (C)
E
B. Temperature dependence
The behavior of the electrical conductivity () vs. Temperature (T) of solids is
one criterion used to classify them as:
 metal:  decreases as T is raised
 semiconductor:  increases as T is raised
Note that an insulator appears as a semiconductor with very low conductivity.
Metals
 The excitation energy can be provided via an
chem/kT
increase of temperature. The population of the orbitals is
given by the Fermi-Dirac distribution:
P
1
e  E  chem  / kT  1
chem is the electron chemical potential, that is -EF for metals (T=0)
 When T increases, the charge carrier density increases…. However the
conductivity decreases because there are more collisions between the
transported electrons and the nuclei (phonon scattering)  less efficient
transport.
Semiconductors
In order to have a net electrical current: electrons must jump
from filled levels to empty levels across the band gap
If Eg is not too large, upon applying an external electric
field, few electrons at room T have the necessary energy to
jump from valence band to conduction band
Thermal energy: kT; at 300K, kT~0.025 eV~0.6 kcal/mol
 In crystals of intrinsic inorganic semiconductors, the
band gap can be small, thermal excitations promote e- to the
conduction band. The concentration in charge carriers
produced is proportional to exp(-Eg/2kBT), leading to an
increase of σ with T. The delocalized electrons/holes are not
strongly bound to each other because of the high dielectric
constant and participate efficiently to the electrical current.
 In undoped conjugated polymers, Eg is large. So, the
thermal excitation are negligible, i.e the concentration of
carrier does not increase with T. However, the conductivity
increases with T like in organic crystals. This is the subject
of this chapter.
C. Undoped Conjugated Polymers
 From the order of magnitude of the band gap and the conductivity, most
undoped conjugated polymers are rather like ”insulators”
However, these organic polymers do have a conjugated π-system:
 As a result, they have a low ionization potential (usually lower than ~6eV)
And/or a high electron affinity (lower that ~2eV)
 They will be easily oxidized by electron accepting molecules (I2, AsF5,
SbF5,…) and/or easily reduced by electron donors (alkali metals: Li, Na, K)
Charge transfer between the polymer chain and dopant molecules is easy
(see next chapter on doped polymers)
4.2. Mobility vs. Molecular Order
A. Molecular Crystals
 Molecular order can be controlled via deposition conditions
 Charge carrier mobility depends strongly on the molecular order
pentacene
Dimitrakopoulos & Mascaro, IBM J. Res. Dev. (2001) 45 11
slow
transport
slow
transport
Anisotropy of the charge transport
Fast
transport
tetracene
Fast
transport
Layer structure; thickness of a
monolayer = 1 to 3 nm.

Charge carrier mobility is anisotropic:
High mobility along the layers

Sexithiophene
Crystal structure
pentacene
sexithiophene
B. Conjugated Polymers
The mobility is limited by the slowest steps (bottle neck):
 At the macroscopic scale: defects and lack of crystallinity are the
limiting parameters.
 At the microscopic scale: the mobility is limited by π-π interchain
rather than intrachain transport (fast).
 The challenge: to create order on the macroscopic scale and
maximize the interchain transport
Self-Organized Polymer Thin Films
Direction of the current flow
measured in a FET
High regioregularity (96%)
Low regioregularity (81%)
Film formation
mechanisms are not
understood!
Poly-3-hexylthiophene (spin-coated on SiO2/Si substrates)
H. Sirringhaus et al., Nature 1999, 401, 685
4.3. π-π intermolecular interactions
 due to the overlap between π-orbitals of adjacent molecules
 creation of a narrow π-band in the neutral ground state of the organic crystal.
2tLUMO
W=4tLUMO  Electron mobility
2tHOMO
W=4tHOMO  Hole mobility
By J.Cornil et al., Adv. Mater. 2001, 13, 1053
 The strength of the interaction, i.e. the electronic coupling, is measured by the
transfer integral: t = <Psi/H/Psi>
 “t” is estimated from the splitting of the frontier levels in a dimer
4.4. Charge transport mechanism
Carrier residency time, τ, on a molecule:
τ ↔ 1/kET and W↔ 4t
W= full effective bandwidth
 If W > 0.1–0.2 eV, τ < time for a molecular vibration (10-14 s)
 the molecules do not have the time to geometrically relax and trap the charge:
This is a condition for a band like motion.
Temperature
Band regime
Hopping Regime
 Vibrations introduce a loss of coherence among the interacting units, leading to a
decrease of W upon temperature increases.
 Upon temperature increase, charge carriers can go from band motion to hopping
regime
Hopping Regime
μ ÷ f(T) exp(-Ea/kbT)
Ea = activation energy
Temperature
Band regime
(diffusion limited)
μ ÷ T-n , n>1.
Holstein’s theory for transport in
one dimension
Band motion regime in molecular crystals
4.5. The fundamental events in Hopping transport
A. Electron-Phonon Coupling
R
E
H2C===CH2
LUMO= 2*
-1
Ionization
GS
2||
ReqGS Req-1
HOMO= 1
Relaxation
effects
R
An electron injected in the LUMO
introduces
antibonding
character
between the 2 carbon atoms the C=C
bond length increases. ReqGS < Req-1
B. Charge carriers: radical-cation or polarons
 Molecular crystal
Example
of
positive
polaron created after
electron injection from an
electrode
In the hopping regime,
the polaron is a localized
charge associated with a
structure distortion on
one molecule or part of
one conjugated chain
Screening of the surrounding electron density
 Conjugated polymer
Example
of
positive
polaron created after hole
injection from an electrode
+
C. The localized polaron hopping events
b)
E
 An electron
(Self-exchange) = polaron hop
a) transfer between two similar molecules
Charged
E
 In the self-exchange model, the electric
kET
field is neglected.
The presence of an electric field 
would
2
stabilize one of the potential well.
A+B
However, this model gives
the main
Neutral
molecular
parameters
governing
the
hopping transport.
AB+
Q
1
 At high temperature, the motion of the carriers can be modeled by a
sequence of uncorrelated hops, which gives a mobility:
Q
||
a = average spacing between molecules or chain
segments
E
The rate for electron transfer kET is given by the semi-classical theory of electron
transfer (see previous chapter)
Two major parameters determine the self-exchange ET rate and ultimately the
charge mobility:
1)
The electronic coupling between adjacent molecules/segments represented by
the transfer integral t (or HRP), which needs to be maximized
2)
The reorganization energy, , which needs to be small for efficient transport
The transfer integral t
estimated from the splitting of the frontier levels in a dimer
2tLUMO
 kET for an negative polaron hop
electron mobility
2tHOMO
 kET for an positive polaron hop
hole mobility
Distance dependence of ”t”(HRP) is
exponential as demonstrated for
other ET (see previous chapter):
t=
By J.Cornil et al., Adv. Mater. 2001, 13, 1053
Example: pentacene
There are significant electronic splittings only along the a axis and the d1 and d2
axes. Interactions between molecules located in adjacent layers (along c) are
negligible  charge transport has a dominant two-dimensional character and
takes place within the layers in directions that are nearly perpendicular to the long
molecular axes.
J. Cornil et al (2001) J. Am. Chem. Soc. 123, 1250–1251.
The internal reorganization energy (i)
i reflects the geometric changes in the molecules when going from the neutral to
the ionized state or vice versa. (i ↔ electron-vibration coupling)
i= 1+ 2
i (defined in the ET theory)
corresponds to the polaron binding
energy (Epol= 2~ i/2) defined in
transport theory of solids
Internal reorganization energy vs. molecular size
For those three molecules, the polaron
(charge+
structure
distortion)
is
delocalized over the whole molecule.
To modify slightly many bond-lengths
cost less than modifying a lot few bondlengths (see the shape of the Morse
potential)
The larger the molecule, the lower
the reorganization energy
By V. Coropceanu et al. Theor Chem Acc (2003) 110:59–69