Calculating Intermolecular Charge Transport Parameters in ...

A thesis entitled
Calculating Intermolecular Charge Transport
Parameters in Conjugated Materials
James Kirkpatrick
Submitted for the degree of Doctor of Philosophy
of the University of London
Imperial College of Science, Technology and Medicine
February 2007

Contents
1 Introduction 4
1.1 Why is a Theoretical Study of Charge Transport Necessary to the Design
of Better Devices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Conjugated Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Charge Transport in Conjugated Materials . . . . . . . . . . . . . . . . . 8
2 Background 17
2.1 Marcus Theory in the Non-Adiabatic High Temperature Limit . . . . . . 18
2.2 Reorganisation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Transfer Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Note on Systems with Degenerate Orbitals . . . . . . . . . . . . 28
2.3.2 Splitting of the Molecular Orbitals . . . . . . . . . . . . . . . . . 29
2.4 Site Energy Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 Electrostatic Interactions in Molecules . . . . . . . . . . . . . . . 31
2.4.2 Inductive Interactions in Molecules . . . . . . . . . . . . . . . . 35
2.5 The Gaussian Disorder Model and Related Methods . . . . . . . . . . . . 36
2.5.1 Computational Methods . . . . . . . . . . . . . . . . . . . . . . 37
2.5.2 Distribution of Parameters of the Transfer Rates . . . . . . . . . . 40
2.5.3 Predictions and Results . . . . . . . . . . . . . . . . . . . . . . . 43
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Development and Verification of Methods for Calculating Charge Transport
Parameters 53
3.1 Projective Method for Calculation of Transfer Integrals . . . . . . . . . . 54
i

3.2 Molecular Orbital Overlap Method for Calculation of Transfer Integrals . 57
3.3 Calculation of Site Energy Difference from Electrostatic Interactions . . . 59
3.3.1 Ethylene: Neutral Systems . . . . . . . . . . . . . . . . . . . . . 61
3.3.2 Ethylene: Charged Systems . . . . . . . . . . . . . . . . . . . . 66
3.3.3 Pentacene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Comparison of Different Methods of Calculating Transfer Integrals . . . . 73
3.4.1 Pentacene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Hexabenzocoronene . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Reorganisation Energy in PolyPhenylenevinylene: Polaron Localisation 85
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Defect-free PPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3 Torsional Defects in PPV . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Ambipolar Transport in PCBM 98
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Transfer Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Reorganisation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 Charge Transport Parameters for Randomly Oriented Pairs of Dialkoxy Poly-
paraphenylenevinylene and Triarylamine Derivatives 111
6.1 Dialkoxy PPV Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.1.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.1.3 Results and Comment . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Spiro Linked Triarylamine Derivatives . . . . . . . . . . . . . . . . . . . 123
6.2.1 Method and Results . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
ii

7 Simulation of Charge Transport in Assemblies of Hexabenzocoronenes and
Fluorene Oligomers 130
7.1 PFO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2 HBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8 Conclusion and Future Work 151
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List of Figures
1 Probability density distribution for receiving an email from Jenny. 41%
of emails is received after five o’clock! . . . . . . . . . . . . . . . . . . . xiv
1.1 Resonant Lewis structures for benzene with single and double bonds (left)
and representation of the aromatic structure (right). . . . . . . . . . . . . 7
1.2 Lewis structure of a segment of PPV. . . . . . . . . . . . . . . . . . . . . 8
1.3 Lewis structure of a charged segment of PPV before and after charge mi-
gration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Eigenvalues of equation 1.1 as a function of the reaction coordinate Q. . . 10
1.5 Time evolution for three systems with the same reorganisation energy,
the same angular frequency ω = 5 10 12 s −1 and three different transfer
integrals J: 1 meV (top), 50 meV (middle) and 300 meV(bottom). . . . . 12
1.6 Time evolution for three systems with the same reorganisation energy, the
same transfer integral J = 10 meV and three different angular frequencies
ω: 10 12 s −1 (top), 5 10 12 s −1 (middle) and 10 14 s −1 (bottom). . . . . . . . . 14
2.1 Schematic view of a potential energy surface for asymmetric charge trans-
port. λ denotes the reorganisation energy, ∆E the difference in energy
between reactants and products in their relaxed geometries and ∆Q is the
equilibrium reaction coordinate for the products. . . . . . . . . . . . . . . 20
2.2 Pictorial representation of the vectors discussed in deriving equation 2.26 31
2.3 Pictorial representation of the vectors discussed in deriving equation 2.32 33
iv

2.4 A schematic of one dimensional transport through an energetically and
positionally disordered medium. This figure represents disorder in en-
ergy (vertical placement of the transport levels) and in transfer integrals
(thickness of arrows) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 σ versus the value of the dipole moment for five different small conju-
gated molecules (right panel)[44], σd obtained from equation 2.41 (left
panel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Pictorial representation of correlation in a 100x100x100 cubic lattice of
randomly oriented dipoles. The diameter of the spheres is proportional to
the magnitude of the dipole interaction at that point, whereas the colour
of the spheres represents the sign of the potential [45] . . . . . . . . . . . 43
2.7 Energy distribution of charges as a function of time after generation [46]. 44
2.8 Schematic of different routes from A to B in the direction of the field [46] 45
3.1 A pair of ethylene molecules at a separation of 5 Å . Also shown is the
axis around which one of the two molecules will be rotated to generate
the geometries for which ∆E is calculated. . . . . . . . . . . . . . . . . 62
3.2 Site energy difference for a pair of neutral ethylene molecules as a func-
tion of rotation of one of the molecule about the the C=C bond calcu-
lated by projection of PW91 MOs (circles) and calculated from the differ-
ence in quadrupole-monopole interactions calculated classically from the
electrical moments calculated at the PW91 level on an isolated ethylene
molecule (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Site energy difference for a pair of neutral ethylene molecules as a func-
tion of rotation of one of the molecules about the the C=C bond calculated
by projection of PW91 MOs (circles) and calculated from the difference
in quadrupole-monopole interactions corrected by the quadrupole induced
dipoles (solid lines). Polarisabilities and electrical moments were calcu-
lated at the PW91 level. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
v

3.4 Distance dependence of the contribution to ∆E from the induced dipole
in a pair of ethylene molecules as a function of separation. The solid
line corresponds to the contribution from the dipole moment induced by a
quadrupole (i.e. in a neutral system), whereas the dotted line corresponds
to that from the dipole moment induced by a monopole (i.e. in a charged
system). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Diagram showing the dipoles induced in two ethylene molecules oriented
at right angles to each other. (a) Case of point quadrupoles (neutral cal-
culation). (b) Case of point charges (charge calculation). For discussion,
please see the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 Dipoles induced on each molecule in a pair of neutral ethylene molecules
(upper pannel) and in a positively charged pair (lower panel). Notice that,
as shown in figure 3.5, in a charged system the induced dipoles are larger
than in the neutral case, but are in opposite directions. . . . . . . . . . . 70
3.7 Two cofacial pentacene molecules and the axis around which one will be
rotated and translated. The inset shows the chemical formula of pentacene. 71
3.8 ∆E for pairs of pentacene molecules, calculated using projection of the
PW91 results (red points) and electrostatic results (diamonds). The elec-
trostatic results are calculated using monopole monopole interactions only
(squares) and using all interactions up to quadrupole-quadrupole (circles). 72
3.9 Absolute value of the hole transfer integral as a function of displacement
along the x axis of one of the two pentacene molecules calculated by
projection of ZINDO orbitals (circles) and by the orbital splitting model
(triangles). The length of the molecule is 14.1 Å . . . . . . . . . . . . . . 74
3.10 HOMO for pentacene calculated at the ZINDO level. . . . . . . . . . . . 74
3.11 Absolute value of the transfer integral for a pair of pentacene molecules as
a function of rotation around the x axis. The transfer integral is calculated
by projection of the ZINDO orbitals (circles) and by the orbital splitting
method (triangles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
vi

3.12 Absolute value of the transfer integral for a pair of ethylene molecules
as a function of rotation angle. All curves were obtained by projective
method applied to PW91 calculations; the different curves are obtained
by using different basis sets. . . . . . . . . . . . . . . . . . . . . . . . . 76
3.13 Absolute value of the transfer integral as a function of displacement along
the x axis of one of the two pentacene molecules. The two curves show
the results from the projective method using the PW91 functional and
the ZINDO Hamiltonian. The PW91 calculation were performed with a
TVZP basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.14 Two cofacial HBC molecules and the cartesian axis used in the text. The
inset shows the chemical formula of HBC. . . . . . . . . . . . . . . . . . 78
3.15 Contour plots of the effective transfer integral as a function of rotation
about the x and z axis. The molecules are first rotated about the z axis,
then rotated about the x axis in such a way as to keep the minimum dis-
tance of approach between the molecules equal to 3.5 Å . . . . . . . . . . 79
3.16 Effective transfer integral and its components as a function of rotation
about the z axis calculated using the projective (PRO) and MOO (MOO)
methods. The molecules are at a distance of 3.5 Å . . . . . . . . . . . . . 80
3.17 Countour plots of the dependence on x-y displacement of two molecules
of HBC. The two molecules are at a distance of 3.5 Å. The maximum
radius of the molecule is 6.8 Å. . . . . . . . . . . . . . . . . . . . . . . . 82
4.1 Two monomers of PPV and the scheme of the labels used in the rest of the
chapter: ph1...phn will label the phenylene units and vi1...vi2 will label
the vinylene units. The first monomer also has the carbons with bonds to
hydrogen labeled 1 to 6. . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Experimental and modelled ENDOR spectra of PPV with an applied mag-
netic field parallel and perpendicular to the polymer chain axis. The figure
is from reference [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
vii

4.3 Spin density of a polaron on a PPV chain calculated with a PPP Hamil-
tonian. Sites B and C’ correspond to sites 2 and 4 from figure 4.1, sites
B’ and C correspond to sites 1 and 3 from figure 4.1 and sites E and F
correspond to sites 5 and 6 from 4.1. The figure is from reference [2]. . . 88
4.4 Mulliken spin density for different length oligomers of PPV. The x axis
labels position along the chain: the labels ph1, ph2, etc. label the first,
second, etc. phenylene unit of the PPV. Between the phenylene units phn
and phn + 1 lies the vinylene unit vin. . . . . . . . . . . . . . . . . . . . 90
4.5 Polaron localisation energy for different length oligomers of PPV calcu-
lated with BHandHLYP. . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6 Atomic Mulliken spin density for the a chain of 12 units calculated at the
BHandHLYP (top panel) and AM1 (bottom panel) level. . . . . . . . . . 93
4.7 Mulliken total spin density integrated on each phenylene and vinylene
unit as a function of chain torsion for an oligomer of length eight. . . . . 94
5.1 The seven unique pairs with greatest hole transfer integral extracted from
a cluster of twenty molecules of PCBM from its crystal structure grown
in chlorobenzene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2 Distance dependence of the electron (black circles) and hole (red squares)
transfer integrals as a function of displacement from the crystal structure
separation. The lines show exponential fits with natural lengths of respec-
tively 0.78 Å and 0.56 Å. . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Scatter plot of the average spin density for each bonded pair of atoms in
a cation (circles) or in an anion (crosses) versus the percentage change in
bond length upon charging. . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Spin density for cation and anion radicals of PCBM. The left panel shows
a front view of the molecules, the right panel a side view. Within each
panel the molecule to the left is the anion radical and the molecule to the
right is the cation radical. . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Zero field mobility for electrons (squares) and holes (circles) in PCBM
doped polystyrene as a function of PCBM concentration. . . . . . . . . . 105
viii

5.6 Distance dependance for the transfer integral for neighbour and nearest
neighbour hoppings, extracted from the cluster from the crystal structure
of PCBM discussed in the text. The transfer integral is plotted both as a
function of the distance between centres of mass of the PCBM molecules
(above) and as a function of the distance between the centres of the C60
cages (below). This is done because the C60 cages are the electronically
active parts of the molecules. . . . . . . . . . . . . . . . . . . . . . . . 107
5.7 Experimental (full symbols) and modelled (empty symbols) temperature
dependence for pristine MDMO-PPV (triangles) 1:1 MDMO-PPV:PCBM
(circles) and 1:2 MDMO-PPV:PCBM (squares). . . . . . . . . . . . . . . 107
6.1 Chemical formula of the three dialkoxy PPV derivatives studied in this
chapter. a) is di-MethoxyPPV(dMeOPPV), b) is di-hexyloxyPPV (dHeOPPV)
and finally c) is di-decyloxyPPV (dDeOPPV). . . . . . . . . . . . . . . . 113
6.2 Electric field dependence of mobility for dMeOPPV (full squares, pre-
annealed C1 ) for dHeOPPV (empty diamonds, C6) and for dDeOPPV
(empty circles, C10). The figure also shows the electric field dependence
of mobility for dMeOPPV annealed at low temperature (full triangles,
non-annealed C1) and for MDMO-PPV (dotted line). . . . . . . . . . . . 115
6.3 Representation of the three Euler angles defining relative orientation: Φ
and Θ represent the first two Euler rotations about the temporary z and x
axis, Ψ represents the last rotation about the z axis. . . . . . . . . . . . . 117
6.4 Probability distributions of the difference in site energies (left panel) and
the logarithm of the transfer integral (right panel) of two hexamers of
dMeOPPV with a minimum separation of 3.5 Å . Probability distributions
(vertical bars) are compared to a normal distribution (solid line). . . . . . 120
6.5 Distance dependence of the standard deviation in values of ∆E. . . . . . 121
6.6 Scatter plot of the absolute value of ∆E against the logarithm of the trans-
fer integral log(J) (a). Also shown is the scatter plot of log(J) versus the
centre to centre separation d (b) and the scatter plot of ∆E versus d (c). . . 122
ix

6.7 The three spiro-linked tryarilamine derivatives considered in this study.
From left to right they are 2,2’,7,7’-tetrakis-(N,N-diphenylhenylamino)-
9,9’-spirobifluorene (spiro-unsub), 2,2’,7,7’-tetrakis-(N,N-di-m-methylphenylamino)-
9,9’-spirobifluorene (spiro-Me) and finally 2,2’,7,7’-tetrakis-(N,N-di-4-
methoxyphenylamino)-9,9’-spirobifluorene (spiro-MeO) . . . . . . . . . . 123
6.8 Histogram of ∆E for spiro-unsub (dashed line), spiro-Me (dotted line)
and spiro-MeO (full line). The standard deviation for the three materials
is 8 meV for spiro-unsub, 10 meV for spiro-Me and 15 meV spiro-MeO. . 125
7.1 Schematic representation of the morphology model used by Dr. S. Athanosopou-
los to describe PFO and definitions of the parameters governing the rel-
ative position and orientation of molecules. (a) The polymer chains are
packed hexagonally, with the chain axis perpendicular to the electric field.
(b) Each trimer is allowed to rotate by a torsion φ and can be displaced
by a distace ∆r in the direction perpendicular to the chain axis and can
slip by a distance dz parallell to the chain axis. (c) Top view of the central
planes of two trimers, showing the torsion angles φ1 and φ2 (∆φ = φ2−φ1)
and the polar angle α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2 HOMO (left) and HOMO-1 (right), of the restricted open shell orbitals of
a fluorene trimer with one of the hydrogen end groups missing. . . . . . . 132
7.3 Contour plot of log(|J| 2 ) as a function of relative torsion angle dφ and the
polar angle α which describes position. Notice how the values of these
interchain transfer integrals are always much smaller than the intrachain
transfer integrals from figure 7.5. . . . . . . . . . . . . . . . . . . . . . 135
7.4 Distance dependence of the transfer integral squared on distance for φ =
150 ◦ and α = 0 ◦ (squares) and for φ = 0 ◦ and α = 0 ◦ (circles). . . . . . . 135
7.5 Intrachain transfer integrals squared as a function of relative torsion φ for
two trimers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
x

7.6 Difference between forward and backward rates for an ordered lattice
as a function the torsion angle φ at which the trimers are set. √ F =
200(V/cm) 1/2 (solid line), 400(V/cm) 1/2 (dotted line), 600(V/cm) 1/2 (dashed
line), 800(V/cm) 1/2 (chained line, one dot), 1000(V/cm) 1/2 (chained line,
two dots). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.7 Dependence of mobility on the square root of the applied electric field
for: the ordered morphology with φ = 0 ◦ (open triangles), the ordered
morphology with φ = 20 ◦ (diamonds), the morphology with disordered
torsions (full squares), the optimally disordered morphology (full trian-
gles) and the experimental data on alligned PFO from reference [3] (full
circles).The optimally ordered morphology corresponds to when neigh-
bours in the direction of the field have a relative torsion φ of 150 ◦ . The
lattice constant for the hexagonal lattice is 6.5 Å . . . . . . . . . . . . . . 138
7.8 Chemical structure of the various derivatives of HBC considered in this
study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.9 ToF results for several selected snapshots, together with the averaged over
the MD run displacement current (left). Block averages over 10, 20, ...
190 snapshots. (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.10 Displacement current for different numbers of molecules in a column.
Average over 200 snapshots at an electric field of 10 5 Vcm −1 . . . . . . . . 145
7.11 Radial distribution function along the z direction for all six derivatives
considered in this study. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.12 Distribution in logarithm of the effective transfer integrals for holes squared
for 20 columns of 100 molecules each. . . . . . . . . . . . . . . . . . . . 148
xi

List of Tables
5.1 Table of the transfer integral for holes Jh and for electrons Je. n is the
number of such pairs in a cluster of 20 molecules. dcoc is the separation
between the centres of the fullerene cages for the pair of molecules. The
labels a-g label the pairs according to figure 5.1 . . . . . . . . . . . . . . 100
5.2 Reorganisation energies for anion. . . . . . . . . . . . . . . . . . . . . . 102
5.3 Reorganisation energies for cation. . . . . . . . . . . . . . . . . . . . . . 102
6.1 GDM parameters for the various dialkoxy PPV derivatives. . . . . . . . . 115
6.2 GDM energetic disorder σGDM and standard deviation in distribution of
site energy difference calculated using distributed Multipole analysis (
σDMA ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.1 Electron (e) and hole (h) mobilities (cm 2 V −1 s −1 ) of different compounds
calculated using time-of-flight (ToF) and master equation (ME) methods,
in comparison with experimentally measured PR-TRMC mobilities. Also
shown are the nematic order parameter Q and the average vertical separa-
tion between cores of HBC molecules h (Å). . . . . . . . . . . . . . . . . 147
xii

Acknowledgments
I would like to acknowledge all the help and support from my colleagues at Imperial
College and my collaborators. None of my work would have been possible without my
supervisor, Jenny Nelson, who provided guidance and advice whilst allowing me absolute
freedom. On no scientific topic was she unable of either providing expertise or pointing
me in the correct direction. Her stakhanovist attitude to work is inspirational and is best
exemplified by the probability distribution for receiving emails from her (shown in figure
1).
I received a lot of editorial help in the writing of the thesis: from my father, Rysia,
most of the people I shared an office with in room H724 (especially Graeme) and ob-
viously from Jenny! Rysia’s help in proof-reading at the 11th hour was indispensable.
Kristian Sylvester-Hvid also provided useful insight in the introduction of the thesis.
Thanks to Bhavna and Carolyn for their efficient running of the group.
I would like to thank, or rather apologise to, those who I have pestered with program-
ming issues, especially my housemates Pete and Joe.
I would also like to thank those who helped straighten out a few of the bugs in my
code: Joe Kwiatkowski, Stavros Athanasopoulos from Bath University and David Reha
from Leeds University.
A word of thanks should go to Ian Gould and Emilio Palomares for helping me nav-
igate my first steps in Quantum Chemistry. The staff at the EPSRC centre for computa-
tional chemistry are also thanked for all the assistance in learning to use computational
packages and computer time.
Many of the investigations in this thesis were collaborative and I want to especially
acknowledge the experimental efforts of Sachetan Tuladhar, whose results are discussed
in chapter five and six . A special word of thank should go to Sachetan for being such a
xiii

Probability Density
0.1
0.05
0
0 4 8 12 16 20
Arrival Time (hr)
Figure 1: Probability density distribution for receiving an email from Jenny. 41% of
emails is received after five o’clock!
pleasant room mate in the undercroft. The time of flight code I used has been developed by
many: Jenny Nelson, Amanda Chatten, Rosemary Chandler and Joe Kwiatkowski. The
work on hexabenzorononene is the fruit of a fruitful collaboration with Denis Andrienko,
who was also a most courteous host in Mainz. For the work on poly(dioctylfluorene) I
would like to thank the group at Bath University, especially Pete Watkins who did much
of the work on developing their Monte Carlo code and morphology model.
A word of thanks goes also to the research groups which I have been priviledged to
visit and the staff who took care of me: Jérôme Cornil and Davide Beljonne from the Uni-
versity of Mons-Hainaut, Jean-Luc Brédas and Demetrio da Silvha Filho from Georgia
Institute of technology and finally Kristian Sylvester-Hvid for hosting me in Copenhagen.
Finally I would like to thank all the rest of the people in the physics deparment who
have provided lunch brakes and coffee break discussions: Graeme, the two Matts, Rob,
Peter, Thilini, Jessica, Joe, Jarvist, Dan, Rahul, Dave Johnson, Markus, Marianne, Ben,
Justin, Tom, Mariano, Jony, Ben. Paul Stavrinou also deserved mention for interesting
discussion.
xiv

Mike Bearpark and Martial Boggio-Pasqua have guided me through my PhD on CASSCF
calculations. The results are not in this thesis, but I am greatly indebted to them for the
interesting tutoring they gave me.
The last word of thanks is to the reader: I hope you do not find this too boring!
xv

Abstract
In this thesis we discuss the calculation of charge transport parameters in conjugated
solids. We aim to elucidate the role of chemical structure and molecular packing in de-
termining charge transport characteristics. In chapter two we discuss the rate equation
from Marcus theory used throughout the thesis and its parameters: electronic coupling,
reorganisation energy and site energy difference. We also discuss the Gaussian Disor-
der Model as an example of a lattice based simulation of charge transport. We show that
lattice simulations are limited because of the difficulty in relating model parameters to mi-
croscopic properties of a material. In chapter three we present fast methods of calculating
transfer integrals and site energies for pairs of molecules. We will use these methods in
chapters six and seven, in which calculations on large numbers of pairs of molecules are
carried out. These fast but approximate methods are shown to be in agreement with more
ab-initio methods. In chapter four we discuss localisation of charges along the backbone
of conjugated polymers and argue that thermal fluctuations are of greater importance than
polaron binding energies in affecting localisation. In chapter five we calculate transfer
integrals and reorganisation energies for electrons and holes in a fullerene derivative in
order to justify the choice of parameters in a lattice based simulation. In chapter six we
use the fast methods for calculating transfer integrals and site energies to rationalise the
Gaussian Disorder Model parameters of two families of materials: dialkoxy polypara-
phenylenevinylenes and triarylamine derivatives. The differences in energetic disoder
between the materials is attributed to electrostatic effects. In chapter seven we calcu-
late charge transport parameters for simulated morphologies of the crystalline phase of
poly(dioctylfluorene) and of the discotic liquid crystal mesophase of different hexabenzo-
coronene derivatives. In the case of poly(dioctylfluorene) calculations of mobility help to
shed light on molecular packing. In the case of hexabenzocoronene the availability of a
good morphology model leads, with no free parameters, to values of mobility in excellent
agreement with experimental data.

List of Symbols and Abbreviations
J: Electronic overlap
MOO: Molecular Orbital Overlap: a method to calculate J based on a simplification of
ZINDO
INDO: Intermediate Neglect of Differential Overlap method
ZINDO: Zerner’s Intermediate Neglect of Differential Overlap method [1]
∆E: Site energy difference
λ: Reorganisation energy
S CF: Self Consistent Field: a method to determine the electronic energy of a system
where the Fock matrix is dependent on the wavefunction; the procedure is therefore re-
peated until the wavefunction obtained by diagonalising the Fock matrix and the Fock
matrix obtained from the wavefunction are consistent with each other
HOMO: Highest Occupied Molecular Orbital
LUMO: Lowest Unoccupied Molecular Orbital
µ ν: Labels for molecular (or atomic) orbitals; µ is also the mobility.
A B: Labels for molecules
i j: Labels for electrons
a b: Labels for atoms
x y z α β: Labels for x,y and z coordinates or for generic coordinate labels (α β)
Ψ: Electron wavefunction
χ: Nuclear wavefunction
Ψ: Slater determinant: a combination of molecular orbitals which ensures the anti-symmetric
nature of the wavefunction
ψ: Molecular orbital
φ: Atomic orbital
S : Overlap matrix
P: The density matrix P = � µ occupied φ t µφµ
Q: Quadrupole moment
P: Polarisability
a0: Bohr radius
1

e: Magnitude of the charge of an electron
k: Boltzmann’s constant
T: Temperature
DMA: Distributed Multipole Analysis
U: An electrostatic interaction
σ: Energetic disorder or a bond formed by sp hybridised orbitals or σ overlap (i.e. the
overlap between p atomic orbitals symmetric about the vector joinining the two atomic
centres)
π: Bonds formed by p orbitals or π overlap (i.e. the overlap between p atomic orbitals
antisymmetric about the vector joinining the two atomic centres)
C: Matrix representing all the molecular orbitals in a basis set; each row corresponds to a
molecular orbital
Q: Nuclear coordinates
q: Electronic coordinates
ω: Angular frequency
F: Electric field
2

Bibliography
[1] J. Ridley and M. Zerner, An Intermediate Neglect of Differential Overlap Technique
for Spectroscopy: Pyrrole and the Azines, Theoretica Chimica Acta, 32, 111 (1973)
3

Chapter 1
Introduction
1.1 Why is a Theoretical Study of Charge Transport Nec-
essary to the Design of Better Devices?
Ever since the discovery of conductivity in trans-polyacetylene [1], study of the electronic
properties of organic molecules has been very intense. The field was further stimulated
by the discovery of light emitting polymers [2] and of photoinduced charge separation
in polymer fullerene blends [3], which stimulated the development of, respectively, light
emitting diodes and solar cells. The reason for these high levels of interest is that if it were
possible to design materials with the opto-electronic properties of inorganic semiconduc-
tors and the mechanical and processing properties of plastics, the applications would be
revolutionary. Solution processability would allow fabrication of devices in continuous
industrial processes [4, 5], thus reducing costs. Plastics are also flexible and light-weight
compared to rigid, brittle inorganic semiconductors: this could open the door to integra-
tion of electronic devices on different subtrates. It seems to us that the area which will
show the greatest long term benefits is organic photovoltaics (PV): inorganic PV is lim-
ited in applicability by its high cost, both due to material costs (solar cells are large area
devices) and processing costs (fabricating a PV module is a complex process). For both
these reasons, the potential low cost and ease of processing of organic materials would
have a great impact.
Many different designs of solar cells based on organic materials have been proposed,
4

for example: dye sensitised solar cells [6], bulk heterojunction cells [7, 8], composite
cadmium selenide/polymer solar cells [9], cells based on self organised liquid crystals
[11]. In all these systems three processes must occur in order for a photocurrent to be
collected at the electrodes: photons must be absorbed, charges must be separated and fi-
nally charges must be able to migrate to the electrodes. Charge mobility is important both
because it sets the maximum space charge limited current obtainable by the device [12]
and because, if transport is slower than charge absorption or charge separation, it causes
bottlenecks and increased recombination [13]. In this thesis, we study the parameters con-
trolling charge dynamics in organic materials. In particular we will argue for the need of
electronic structure calculations to elucidate the relationship between chemical structure
and charge transport characteristics. We want to suggest that our understanding of the
fundamental mechanism behind charge transport and the speed of computers is now suf-
ficient to allow an understanding of charge mobility based on the fundamental electronic
properties of molecules and that this is the only path which will allow disentanglement
of the various chemical and physical causes for device performance, therefore allowing
better material and device design.
The thesis is organised as follows: in chapter two we describe the non-adiabatic, Mar-
cus high temperature charge transfer equation and methods from the literature to calculate
its parameters; we also discuss some lattice based models of charge mobility. In chapter
three we discuss fast methods of calculating the parameters of the Marcus charge transfer
equation. In chapter four we discuss the nature of charged radicals on polymer chains.
The rest of the thesis is dedicated to applying the methods developed to explaining ex-
perimental charge transport characteristics of conjugated materials. In chapter five we
examine the dependence on fullerene concentration of electron and hole mobilities in
blends of fullerene and conjugated polymer. In chapter six we calculate charge transport
parameters in pairs of molecules in random orientations and compare these results to pa-
rameters from lattice models of mobility. In chapter seven we run full-blown simulations
of molecular morphology and electronic properties for polymers and discotic liquid crys-
tals in an effort to reproduce charge transport characteristics with a minimal number of
adjustable parameters. Finally we will give a brief of conclusion of the thesis. The rest of
5

this chapter introduces some concepts about charge transfer in conjugated materials.
1.2 Conjugated Materials
The properties of conjugated materials are based on the nature of covalent bonds between
carbon, nitrogen, sulfur and oxygen. Specifically, they are based on the properties of sp 2
bonds in carbon atoms. In order to better understand the different types of bonding that
carbon is capable of let us consider three compounds: ethane (C2H6), ethene (C2H4) and
ethyne (C2H2). In these three compounds, each carbon forms bonds with either four, three
or two other atoms. These three different behaviours depend on how the 2s and 2p atomic
orbitals on carbon hybridise to form bonds: in ethane four σ bonds are formed by mixing
a single 2s orbital with three 2p orbitals. These σ orbitals are therefore known as sp 3
orbitals. In ethene only three σ bonds are formed by mixing a 2s orbital with two 2p
orbitals. This is sp 2 hybridazation. In ethyne, a single σ bond is formed by mixing a 2s
and a 2p orbital. This is sp 1 hybridazation. These three types of orbitals are responsible
for the formation of the bonds which determine the fundamental geometrical features of
the three molecules. Using the principle that orbitals will tend to repel each other we can
predict the bond angles in the three materials: in ethane the bonds around each carbon
are in a tetrahedral arrangement, in ethene they are triangular and in ethyne the bonds are
linear. In the case of sp 2 and sp 1 hybridasation, one and two 2p orbitals respectively are
left un-hybridised: these orbitals participate in bonding by forming a double and a triple
bond respectively between the carbons. A conjugated material is one in which all bonds
are either sp 2 or sp 1 hybridised, resulting in an alternation of single and double bonds. As
shown in figure 1.1, in compounds such as benzene different ways of alternating single
and double bonds leads to different structures with the same energy: these structures
can mix together forming lower energy structures where instead of alternating single and
double bonds, there are only aromatic bonds. These resonances are responsible for the
high stability of aromatic compounds.
σ and π bonds are so called because they are either symmetric or antisymmetric with
respect to the axis connecting the two atoms participating in bonding. This explains
why σ bonds are strong and participate more in determining the geometrical structure
6

Figure 1.1: Resonant Lewis structures for benzene with single and double bonds (left) and
representation of the aromatic structure (right).
of molecules: σ orbitals are symmetric across the binding direction and therefore have
large overlap integrals, π orbitals, conversely, rely on overlap of the lobes of parallel 2p
orbitals and are much weaker. This observation, that 2p orbitals must be parallel to form
bonds, leads us to understand why ethene is planar: if the two triangles formed by each
carbon and the two hydrogens bonded to them are not parallel, the 2p bonds on each atom
are not capable of forming a bond. So sp 2 hybridisation implies the formation of planar
structures, with double bonds formed by the remaing 2p orbitals.
The interesting electronic properties of conjugated materials is delocalisation. Often
the concept of delocalisation is explained by arguing that when the unhybridised p atomic
orbitals on each atom interact to form molecular π orbitals, these π orbitals are delocalised
over the whole molecule. This is certainly true, however it is not a property unique to π
orbitals, also the hybridised sp 2 atomic orbitals interact to form delocalised σ orbitals. In
my opinion what is special about p conjugated systems is that charged or neutral excita-
tions are delocalised, in the sense that they can migrate along the molecule with relatively
small amounts of energy. As an example of this take polyparaphenylenevinylene (PPV).
This p conjugated polymer is shown in figure 1.2. If it becomes charged, its Lewis struc-
ture is similar to figure 1.3; in this figure the dot represents an unpaired electron and the
7

Figure 1.2: Lewis structure of a segment of PPV.
Figure 1.3: Lewis structure of a charged segment of PPV before and after charge migration.
”+” represents a charge. What we mean to represent in this figure is that, upon charging,
the bond ordering changes making the structure less aromatic and more quinoid. This
localised quinoid deformation can easily travel along the chain as it only involves the
breaking and reforming of loosely bound bonds. This therefore seems to us to be the
fundamental property of conjugated systems: the nature of π bonds is such that they can
be easily broken and reformed without the significant rearrangement of bonds that would
occur if σ bonds were broken.
1.3 Charge Transport in Conjugated Materials
In chapter two we will provide a formal description of non-adiabatic charge transfer, but in
this section we briefly describe three possible regimes for charge transfer in a conjugated
8

material: delocalisation, adiabatic transfer and non-adiabatic transfer. Since most of our
efforts are computational, rather than providing a review of analytical charge transfer
models such as can be found in references [14, 15], we will implement a simple quantum
dynamical simulation from a simple semi-classical Hamiltonian. In this section we mean
to simply exemplify different possible regimes of charge transfer.
We write a Hamiltonian for charge transfer with two non-interacting (diabatic) states
whose energies depend quadratically on a single reaction coordinate Q. Electron-phonon
coupling is introduced by displacing the potential energy parabula by a distance Q0. At
the nuclear coordinate where the energy of state |0 > is minimised the potential energy for
state |1 > is ω2 Q0 2
2 . This quantity is called the reorganisation energy and is an expression of
electron-phonon coupling. In the case of the polymers depicted in figure 1.3, the reaction
coordinate Q could represent going from an aromatic to a quinoid structure. Electronic
coupling between the two diabatic states is described by a transfer integral J. In state
representation, using mass weighted coordinates, the Hamiltonian for the system is:
ˆHe = ω2 Q 2
2 |0 >< 0| + ω2 (Q − Q0) 2
2
|1 >< 1| + (J|0 >< 1| + J ∗ |1 >< 0|) (1.1)
Figure 1.4 shows the eigenvalues of this Hamiltonian represented as a function of the
reaction coordinate Q for typical values of ω and Q0.
If the electronic coupling is of the same order of magnitude as ω2 Q0 2
2 , the system is in
the delocalised regime. In order to distinguish the other two regimes, we have to consider
the relative time scales for nuclear and electronic transitions. Nuclear oscillations occur
on a timescale of ω −1 ; electronic transitions between the diabatic states, conversely, occur
on a time scale of �/J. If the timescale for electronic transitions is much faster than
that for nuclear motion, i.e. if J is large compared to �ω, charge transfer will occur
gradually as the system oscillates along the potential for the adiabatic states (that is for
the eigenfunction of the Hamiltonian shown in equation 1.1 ). If J is small compared to
�ω, the electronic states do not have the time to rearrange themselves as the nuclei vibrate
and the system oscillates along one of the diabatic potential energy surfaces. Each time
the potential energies for each diabatic state are resonant, i.e. when Q = Q0/2 is reached,
the transfer integral J will set a finite probability for crossing from one diabatic state to
9

Energy / eV
1
0.5
0
-1 0 1 2
Nuclear coordinates Q / Q
0
Figure 1.4: Eigenvalues of equation 1.1 as a function of the reaction coordinate Q.
another.
In order to perform a simulation of the dynamics of the evolution of the system, we
will use the Ehrenfest approximation. In other words, we assume that the nuclear motion
can be treated classically and that the force on the nuclei is simply the expectation value
of the force operator ˆF = − ˆ He . The set of coupled differential equations we solve is:
Q
˙Pe = i
� [ ˆHe, Pe] (1.2a)
�
F = Tr − d ˆHe
dQ Pe
�
(1.2b)
¨Q = F (1.2c)
where Pe is the density matrix for the electronic system, the square brackets denote tak-
ing the commutator and Tr{.} denotes taking the trace. We numerically integrate these
equations of motion using the GNU octave program. In all simulations we scale the mass
weighted length scale so that ω2 Q0 2
2
= 0.3eV. In all cases we start the simulation with a
large potential energy corresponding to a displacement of −Q0 to ensure that the system
10

has sufficient energy to reach the crossing point of the two diabatic surfaces. We also pre-
pare the electronic system so that at time zero it is in the diabatic state |0 >, i.e. take the
initial condition Pe(t = 0) = |0 >< 0|. We then make two comparisons. First we keep the
angular frequency fixed at ω = 5 10 12 s −1 and increase the transfer integral from 1 meV to
500 meV to illustrate the transition of charge transfer characteristics from non-adiabatic to
adiabatic and finally to delocalised charge transfer. In the second comparison we keep the
transfer integral fixed at 10 meV and vary ω from 10 12 s −1 to 10 14 s −1 to demonstrate that
adiabaticity does not result so much from the relative values of J and the reorganisation
energy, but more from the relative values of ω and J/�. In order to exemplify the time
evolution of the system we look at two quantities: the nuclear coordinate Q and Ple f t, the
expectation value of finding the system in diabatic state |0 >. Ple f t is calculated by taking
the following trace over the density matrix: Tr {|0 >< 0|Pe}. We would like to strongly
stress that this simulation has only the purpose of illustrating graphically different transfer
regimes and that it ignores issues crucial to determining charge transfer rates, such as the
role of nuclear wavefunction overlap, the appropriateness of applying Ehrnfest dynamics
to non-adiabatic problems and the role of decoherence.
Figure 1.5 shows the time evolution for systems with the same angular frequency and
different transfer integrals. The top panel shows an example of non-adiabatic transfer,
with the timescale for transition between diabatic states begin �/J = 6.6 10 −13 s and with
the time scale for nuclear motion being 1/ω = 2.0 10 −13 s. The nuclear system is oscillat-
ing along the diabatic surface and in fact the midpoint for oscillation is 0, the position of
the bottom of the left parabula. Each time the nuclear coordinate passes through Q0
2 , Ple f t
jumps abruptly and there is a finite probability of finding the system in the diabatic state
|1 >.
The middle panel of figure 1.5 shows time evolution for the same ω but for a dif-
ferent value of J of 50 meV. In this case the timescale for diabatic transitions is �/J =
1.3 10 −14 s: electronic transitions occur faster than the nuclear motion. In fact each time
that the system goes through Q0
2 , Ple f t switches from left to right, showing that electronic
transitions take place fully and not partially as in the non-adiabatic regime. Looking at
the nuclear displacement, we notice that the midpoint of the oscillation occurs at Q0 , not 2
11

Displacement / Q 0
Displacement / Q 0
Displacement / Q 0
1.5
1
0.5
0
-0.5
-1
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
2
1.5
1
0.5
0
-0.5
-1
0 1e-12 2e-12 3e-12 4e-12 5e-12 6e-12 7e-12 8e-12 9e-12
0.88
1e-11
Time / s
Displacement
P left
0 1e-12 2e-12 3e-12 4e-12 5e-12 6e-12 7e-12 8e-12 9e-12
0
1e-11
Time / s
Displacement
P left
0 1e-12 2e-12 3e-12 4e-12 5e-12 6e-12 7e-12 8e-12 9e-12 1e-11
0
Time / s
Displacement
P left
Figure 1.5: Time evolution for three systems with the same reorganisation energy, the
same angular frequency ω = 5 10 12 s −1 and three different transfer integrals J:
1 meV (top), 50 meV (middle) and 300 meV(bottom).
12
1
0.98
0.96
0.94
0.92
0.9
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
P left
P left
P left

at 0, showing that the system is oscillating along the adiabatic potential energy surface.
Note however how the oscillation is far from harmonic and has a sharp kink at Q0 . Since
2
in this regime the system never abandons the adiabatic surface we call it adiabatic.
The bottom graph panel of figure 1.5 shows the time evolution for J = 300 meV. In
this case the coupling between the diabatic states is stronger than the electron phonon
coupling and the system is free to oscillate harmonically along the diabatic surface. Re-
gardless of what value the nuclear coordinate takes, Ple f t oscillates rapidly, in other words
the system is in the delocalised regime.
In order to underline the origin of adiabatic or non-adiabatic transfer, we will look at
three different time evolutions, all obtained using the same transfer integral J = 10 meV,
but with three different angular frequencies: ω = 10 12 s −1 , 5 10 12 s −1 and 10 14 s −1 . These
three time evolutions are show in figure 1.6. In the middle panel we show an intermediate
regime between the adiabatic and non-adiabatic transfer. The top panel shows that by de-
creasing the frequency of nuclear oscillations, transfer becomes adiabatic. In the bottom
panel we show that by making nuclear oscillations very fast (notice the change in time
scale), the electron transfer becomes non-adiabatic.
In this thesis we will describe charge transport as a non-adiabatic processes. This
can be justified by arguing that neutral to charged reactions can have contributions from
high frequency normal modes and that therefore the frequency of electronic relaxations
is in general slower than the nuclear vibrations. In chapter two we give a more detailed
explanation of the semiclassical formula that we use to describe transport and its essential
characteristics.
13

Displacement / Q 0
Displacement / Q 0
Displacement / Q 0
2
1.5
1
0.5
0
-0.5
-1
2
1.5
1
0.5
0
-0.5
-1
1.5
1
0.5
0
-0.5
-1
0 1e-12 2e-12 3e-12 4e-12 5e-12 6e-12 7e-12 8e-12 9e-12
0
1e-11
Time / s
Displacement
P left
0 1e-12 2e-12 3e-12 4e-12 5e-12 6e-12 7e-12 8e-12 9e-12
0.1
1e-11
Time / s
Displacement
P left
0 1e-13 2e-13 3e-13 4e-13 5e-13 6e-13 7e-13 8e-13 9e-13 1e-12
0.4
Time / s
Displacement
P left
Figure 1.6: Time evolution for three systems with the same reorganisation energy, the
same transfer integral J = 10 meV and three different angular frequencies ω:
10 12 s −1 (top), 5 10 12 s −1 (middle) and 10 14 s −1 (bottom).
14
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1
0.9
0.8
0.7
0.6
0.5
P left
P left
P left

Bibliography
[1] C.K. Chiang, C.R. Fincher, Y.W. Park, A.J. Heeger, H. Shirakawa, E.J. Louis, S.C.
Gau and A.G. MacDiarmid, Electrical Conductivity in Doped Polyacetylene, Physical
Review Letters, 39, 1098 (1977)
[2] J.H. Burroughes, D.D.C. Bradley, A.R. Brown, R.N. Marks, K. Mackay, R.H. Friend,
P.L. Burns and A.B. Holmes, Light Emitting Diodes based on conjugated polymers.,
Nature, 347, 539 (1990)
[3] N.S. Sariciftci , L. Smilowitz, A.J. Heeger and F. Wudl, Photoinduced charge transfer
from a conductive polymer to Buckminsterfullerene, Science, 258, 1474 (1992)
[4] J. Zamuseil, Z. Bao, Y.L. Loo, R. Cirelly and J.A. Rogers, Nanoscale organic tran-
sistors that use source drain electrodes supported by high resolution rubber stamps,
Applied Physics Letters, 82, 793 (2003)
[5] J.Z. Wang, Z.H. Zheng, H.W. Li, W.T.S. Huck and H. Sirringhaus , Dewetting of
conducting polymer inkjet droplets on patterned surfaces, Nature Materials , 3, 171
(2004)
[6] A. Hagfeldt and M. Grätzel, Molecular Photovoltaics, Account of chemical research,
33, 269 (2000)
[7] G. Yu and A.J. Heeger, Charge separation and photovoltaic conversion in polymer
composites with internal donor/acceptor heterojunctions, Journal of Applied Physics,
78, 4510 (1995)
15

[8] S.E. Shaheen, C.J. Brabec, N.S. Sariciftci, F. Padinger, T. Fromherz and J.C. Hum-
melen, 2.5 % efficient organic plastic solar cells, Applied Physics Letters, 78, 841
(2001)
[9] D.S. Ginger and N.C. Greenham, Charge Separation in Conjugated-Polymer
Nanocrystal Blends, Synthetic Metals, 101, 425 (1999)
[10] C.J. Brabec, F. Padinger, J.C. Hummelen, R.A.J. Janssen and N.S. Sariciftci, Reali-
sation of Large Area Flexible Fullerene - Conjugated Polymer Photocells: A Route to
Plastic Solar Cells, Synthetic Metals, 102,861 (1999)
[11] L. Schmidt-Mende, A. Fechtenkotter, K. Mullen, R.H. Friend and J.D. MacKen-
zie, Efficient organic photovoltaics from soluble discotic liquid crystalline materials,
Physica E, 14, 263 (2002)
[12] A. Moliton and J.M. Nunzi, How to model the behaviour of organic photovoltaic
cells, Polymer International, 55, 583 (2006)
[13] J. Nelson, J. Kirkpatrick and P. Ravirajan, Factors limiting the efficiency of molecu-
lar photovoltaic devices, Physical Review B, 69, 035337 (2004)
[14] M.D. Newton, Quantum Mechanical Probes of Electron-Transfer Kinetics: the Na-
ture of Donor-Acceptor Interactions, Chemical Reviews, 91, 767 (1991)
[15] P.F. Barbara, T.J. Meyer and M.A. Ratner, Contemporary Issues in Electron Transfer
Research, Journal of Chemical Physics, 100, 13148 (1996)
16

Chapter 2
Background
Charge transport in disordered molecular solids proceeds by discrete charge transfer events
between localised states. Because of orientational, positional and configurational disor-
der the parameters governing charge transfer are also disordered and charge dynamics in
three dimensional networks are best solved with stochastic approaches, such as Monte
Carlo methods, or by numerical solution of the underlying Master equation.
In this chapter we describe Marcus theory in the non-adiabatic high temperature limit.
This leads to an expression for the charge transfer rate in terms of three parameters: the
reorganisation energy λ, the transfer integral J and the site energy difference ∆E. We
will then discuss commonly used methods to calculate J and λ and introduce the theo-
retical framework for calculating electrostatic and inductive interactions between charge
distributions: these theoretical tools will be used in chapter three to calculate ∆E. The
last part of the chapter is devoted to describing present empirical numerical models of
charge transport in disordered materials, chiefly the Gaussian Disorder Model (GDM).
The aim of this description is both to introduce the numerical tools that will be used later
in the thesis to describe charge dynamics, and to discuss the limitations of the GDM in
identifying the physical and chemical basis for charge transport properties.
17

2.1 Marcus Theory in the Non-Adiabatic High Tempera-
ture Limit
In this section we will provide a description for charge transfer based on time dependent
perturbation theory and on the Marcus theory of oxidative-reductive reactions [1]. This
description will lead to the expression [2] for the charge transfer rate used in the remain-
der of the text which we shall refer to as the Marcus charge transfer rate. We follow the
derivation of this equation proposed by Jortner and co-workers [3, 4]. We will consider
the high temperature (classical treatment of nuclear vibrations), non-adiabatic (small elec-
tronic coupling) limit of the rate equation. The same equation can be obtained by treating
the motion of the nuclei in a fully classical way, and invoking the Landau-Zener formula
for the probability of non-adiabatic transitions [5]; however such a proof is less clear and
satisfying than the following which rests firmly on Fermi’s golden rule.
The chemical reaction which we wish to describe is a charge transfer event from
molecule DA to molecule DB:
D + A + D 0
B → D0A
+ D+ B
(2.1)
where the superscripts represent the ionic state of the molecule, + for positive and 0
for neutral. Note that although we will refer to the charged molecule with a + symbol
thoughout this section, all arguments can be applied to transport of negative charge by
changing the positive sign to a negative one.
Firstly let us list the main assumptions used in this derivation:
(a) The initial and final localised electronic states are derived from a zeroeth order
Hamiltonian which does not account for the off-diagonal elements of the Hamilto-
nian describing the interaction between the two molecules.
(b) Initially, the molecule is assumed to be in an incoherent super-position of vibrionic
states, due to its temperature.
(c) The off-diagonal elements of the Hamiltonian describing the interaction between
molecules, J, is introduced as a time dependent perturbation causing the initial and
18

final states to mix.
(d) The matrix element J is assumed to be independent of the nuclear coordinates Q.
(e) The potential energy surfaces for the initial and final states are assumed to be
parabolae with identical curvature.
Let us label the initial and final zeroeth order electronic states Ψ+0 and Ψ0+ respec-
tively. The vibrionic states will be labelled {χ+0,i} and {χ0+, f } respectively, where {} repre-
sents the fact that these states form a set and the subscripts identify the quantum numbers
for electronic and vibrionic states. Because of assumption (c) , the rate of transition from
a particular initial state Ψ+0χ+0,i to the set of final states {Ψ0+χ0+, f } can be described by a
sum of rates of the form defined by Fermi’s Golden rule:
�
W+0,i;0+ =
f
2π
� |J+0,i;0+, f | 2 δ(E+0,i − E0+, f ) (2.2)
where W+0,i;0+ represents the rate of transition from an initial state Ψ+0χ+0,i to a set of
final states {Ψ0+χ0+, f } , J+0,i;0+, f represents the matrix element for the zeroeth order initial
and final states and E+0,i and E0+, f represent the total energy of the initial and final states.
Fermi’s golden rule is derived in Landau and Lifshitz [6].
The system is assumed to be in a thermal average of vibrionic states initially. There-
fore to obtain the total rate of conversion of reactants into products W+0,0+ one must sum
over the initial vibrionic states, weighing each term by the Boltzmann probability of being
in that state:
� �
W+0,0+ =
i
f
2π
� |J+0,i;0+, f | 2 δ(E+0,i − E0+, f )exp(−E+0,i/kT)Z −1
where Z = � i exp(−E+0,i/kT) is the partition function.
(2.3)
Let us take a moment to interpret eq. 2.3. Figure 2.1 shows a schematic represen-
tation of E+0,i and E0+, f as functions of Q. These two energy surfaces are referred to in
the literature [7] as diabatic surfaces, meaning that the interactions between the zeroeth
order initial and final states have been neglected. The δ function in equation 2.3 is stating
that electronic tunnelling has to occur at resonance to conserve energy. In other words
19

λ
0
∆E
∆Q
Reactants
Products
Figure 2.1: Schematic view of a potential energy surface for asymmetric charge transport.
λ denotes the reorganisation energy, ∆E the difference in energy between reactants
and products in their relaxed geometries and ∆Q is the equilibrium
reaction coordinate for the products.
- if molecular vibrations are treated classically - the reaction occurs at the intersection
between the two curves in figure 2.1. If molecular vibrations are treated quantum me-
chanically, each quantum of vibrational energy (phonon) has energy �ω, where ω is the
frequency of vibration, and the reaction can occur at a lower or higher energy than the
intersection by absorption or emission of phonons. The matrix element |J+0,i;0+, f | 2 repre-
sents the probability of electron and nuclear tunnelling, whereas the Boltzmann factor
represents the probability of reaching the transition point. The summation is over both
the initial and the final vibrational modes.
At this point let us look at the matrix element J+0,i;0+, f . Using assumption (d), that is
that the electronic part of the matrix element is independent of nuclear coordinates, one
can rewrite equation for J:
to:
� ∞
J+0,i;0+, f =
−∞
�� ∞
J+0,i;0+, f =
−∞
dQ
� ∞
−∞
dq Ψ ∗
0+ (q; Q)χ∗0+,
f (Q) ˆH(q)Ψ+0(q; Q)χ+0,i(Q) (2.4a)
dQ χ ∗
0+, f (Q)χ+0,i(Q)
� �� ∞
dq Ψ
−∞
∗
0+ (q; Q) �
ˆH(q)Ψ+0(q; Q)
20
(2.4b)

where q represents electronic coordinates. In Dirac notation:
J+0,i;0+, f = 〈χ+0,i|χ0+, f 〉〈Ψ+0| ˆH|Ψ0+〉 = 〈χ+0,i|χ0+, f 〉J+0;0+
(2.4c)
If the results of equation 2.4 is substituted into 2.3, the rate equation can be expressed
in terms which separate the electronic and nuclear degrees of freedom. The resulting
equation will then have the form:
W+0;0+ = 2π
�
|J+0;0+| 2 �
i f
〈χ+0,i|χ0+, f 〉 2 δ(E+0,i − E0+, f )exp(−E+0,i/kT)Z −1
(2.5)
Equation 2.5 is still not at all trivial to simplify. Following [4], a sketch of the method
employed is as follows: the δ function is treated by using Fourier Transforms and an
analytic form of the vibrational coupling is obtained by assuming that all the vibrational
modes i and f are quadratic and have the same angular frequency ω. Finally, the equation
is reduced into a form similar to a standard rate equation by assuming that we are in the
semi-classical (high temperature) limit, i.e. by assuming that the energy of a quantum of
vibrational energy �ω is smaller than kT and that the electronic coupling J+0;0+ is small
when compared to the reorganisation energy λ. The final form of the rate equation is
therefore:
W+0;0+ =
|J+0;0+| 2
�
� π
λkT
+ λ)2
exp(−(∆E ) (2.6)
4λkT
This form of the equation is pleasing because of its ease of interpretation. There are
three main terms appearing in it: the quantity |J+0;0+| represents the strength of the elec-
tronic coupling, the exponential term exp(− (∆E+λ)2
) represents the Boltzmann factor for a
4λkT
state with energy equal to the intersection point between the diabatic energy surfaces de-
picted in fig 2.1 and finally the normalisation term � π
λkT
normalises the Boltzmann factor
to a density of states and weighs it by the vibrionic overlap. There are three parameters
in Equation 2.6 which depend on material properties: the reorganisation energy λ, the
electronic coupling J+0;0+ and finally the difference in site energies ∆E. In the rest of this
21

thesis the electronic coupling will sometimes be referred to as the transfer integral and
will be denoted by the symbol J, without suffixes.
There are two important conditions which must be satisfied in order for equation 2.6
to be valid: the reaction must be non-adiabatic and kT must be significantly greater than
the vibrational modes. The first condition is truly fundamental and can be cast in many,
equivalent ways: the reaction must occur between diabatic states or the reaction must be
sudden. If this is not the case, a form for the reaction rate can be obtained by assuming
that the probability of crossing diabatic surfaces is unity [2]. If the electronic coupling
is actually larger than the barrier height, it seems to me that the concept of localised
states poorly describes the physical situation and charge transfer will be a coherent, not
temperature activated process. As we will discuss in chapter five, this may be the regime
under which intrachain transport in polymers occurs. The second assumption can be
overcome by dividing the nuclear modes into low frequency and high frequency modes,
one can then obtain a formula for the electron transfer rate, the Marcus-Levich-Jortner
equation[8]:
W =
|J+0;0+| 2
�
� π
λlkT exp(−S h)
∞�
n=0
S n
h
n! exp(−(∆E + λl + n�ωh) 2
4λlkT
) (2.7)
where λl represents the contribution of low frequency modes to the reorganisation energy,
ωh represents the high frequency and S h represents the Huang-Rhys factor for that fre-
quency which is equal to S h = λh/(�ωh). This formula has pretty much the same form
as equation 2.6, with the energy in the exponential term somewhat lowered by allowing
for the absorption of n phonons. Equations 2.7 and 2.6 have the same form and the same
parameters: the only difference is that the high frequency modes are treated quantum
mechanically in equation 2.7, therefore it is necessary to distinguish a high frequency
mode that leads from the reactants energy minimum to the seam between the diabatic en-
ergy surfaces and its contribution to λh. This mode is often referred to as the promoting
mode. According to Brédas and co-workers [9], using the Marcus-Levich-Jornter formula
instead of the high temperature Marcus theory charge transfer rate leads only to a renor-
malisation of the rate not to a change in its temperature dependence. For this reason we
have in general used the high temperature form of the equation 2.6.
22

When simulating charge transport, two other forms for the charge transfer rate are
commonly used: the Miller-Abrahams and symmetric rate equations. The Miller-Abrahams
rate expression is written as:
⎧
⎪⎨ νe kT , ∆E > 0
W =
⎪⎩ ν, ∆E < 0
− ∆E
(2.8)
where ∆E represents the difference in energies, ν is the rate prefactor. The motivation for
such an expression is an assumption on the system being weakly coupled to the phonon
bath. In these cases it is easy for phonons to absorb downward steps in energy but very
hard for the opposite to occur. What this means is that the charge carriers are limited
to energy matching conditions only for upwards jumps, whereas in downward jumps the
phonon bath will absorb the excess energy without accelerating the rate. These equations
were obtained originally to describe hopping from shallow impurity traps at helium tem-
peratures and are based on a perturbational treatment in inorganic solids. More precise
non-perturbative treatment of the electron-phonon coupling [10] shows that in amorphous
solids this expression is good at low temperatures, but fails at higher temperature. The
motivation for presenting this equation is that it is used in the Gaussian Disorder Model,
an empirical model for describing charge transport that we discuss in section 2.5.
The symmetric rate has form:
∆E (−
W = νe 2kT )
(2.9)
this equation, actually, has the same form as equation 2.6. In fact, for small enough site
energy difference ∆E, equation 2.6 can be written as:
W =
|J+0;0+| 2
�
�
π
λkT exp
�
− λ
� �
exp −
4λkT
∆E
�
2kT
(2.10)
In this form, equation 2.6 has the same energy dependence as 2.9, the only difference be-
ing that the reorganisation energy leads to a temperature dependence in the rate prefactor
ν of equation 2.9. The symmetric form of the equation is used by many workers in the
field [11, 12, 13] to simulate the field dependence. In the limit of large reorganisation en-
ergy and small field, the equation gives the same field dependence as the Marcus charge
23

transfer equation. In order to recover the observed field dependence, workers using the
symmetric rate explicitally write the temperature dependence of the rate in the Marcus
theory form [14].
2.2 Reorganisation Energy
As we have shown in figure 2.1, the reorganisation energy in a reaction from an initial
electronic state to a final one corresponds to the difference in energy of the final elec-
tronic state in the final nuclear coordinates and in the initial electronic state’s minimum
energy nuclear coordinates. In principle, these energies and nuclear coordinates do not
only belong to the molecules involved in charge hopping (internal sphere reorganisation),
but also the polarisation of surrounding molecules (outer sphere reorganisation). In the
case of the earliest charge transfer systems studied, such as metal ions in acqueous solu-
tion, the outer sphere reorganisation is vastly greater than the inner sphere reorganisation,
because of the high polarity of the solvents [15, 16]. In contrast, most organic materials
have low polarity and therefore it can be argued that the main contribution to the reorgan-
isation energy is from inner sphere reorganisation. It seems that part of the motivation for
ignoring the outer reorganisation in the solid state is that the problem in the solid state is
very complicated indeed: as Barbara [17] notes in a glassy film it is hard to distinguish
the intermolecular modes which are free to move and are able to modify their geometry
during charge transport from those which are ”frozen”. The ”free” modes would be able
to participate in the polarisation of the surrounding, whereas the ”frozen” modes would
only participate in the site energy difference. Because of the low polarisabilities of the
materials considered in this thesis, outer sphere reorganisation will in general be ignored.
In order to determine the inner sphere reorganisation energy for charge transport be-
tween identical molecules we follow the method of Sakanoue [18]. In this method, the
interaction between the two molecules involved in charge transport is assumed to remain
unaltered upon charge transfer. In this approximation, therefore, the reorganisation energy
is unaffected by orientational effects: it depends uniquely on the properties of the isolated
molecule. In order to estimate it one has to consider two contributions, the relaxation of
the neutral molecule and that of the charged molecule. We can therefore write:
24

λi = λi1 + λi2
(2.11a)
where i labels the internal reorganisation energy. The relaxation energy of the neutral
molecule is then defined as:
λi1 = E0(Q+) − E0(Q0) (2.11b)
where Q+ and Q0 represent respectively the equilibrium coordinates of the charged and
neutral molecule. Similarly the relaxation energy of the charged molecule can be written
as:
λi2 = E+(Q0) − E+(Q+) (2.11c)
From a computational point of view, it is necessary to conduct two geometry opti-
misations and four energy calculations. Many different quantum chemical methods are
available, but Sakanoue [18] finds that the hybrid Density Functional B3LYP and the
semiempirical method AM1 yield consistent results. The reorganisation energy can also
be deduced experimentally from the vibrionic structure of the photoelectron spectrum. It
has been shown that results from B3LYP for pentacene are in good agreement with these
experimental findings [19]. There are also other techniques used to estimate the internal
sphere reorganisation energy directly from calculation of the electron-phonon coupling
[20]. This approach allows a better understanding of the contribution of each vibrational
mode to the reorganisation energy - although the same result can be obtained by frequency
analysis [9].
Regarding the outer sphere reorganisation, most existing theories are based on di-
electric continuum models because the first systems to be studied were donor-acceptor
charge transfer systems in solution, where the solvent can conveniently be substituted by
a continuum characterised only by static (zero frequency) and optical (infinite frequency)
dielectric constants. These models have also been expanded to treat ellipsoidal cavities,
but remain of similar form [16]. It is also possible to use Molecular Dynamics, including
25

some of the solvent in a shell around the donor acceptor molecules explicitly and substi-
tuting the rest by a continuum dielectric [21]. This kind of approach may be applicable to
the solid state, allowing one to distinguish the free modes from the frozen ones.
2.3 Transfer Integral
The overall rate of charge transfer is dictated by the transfer integral J0+;+0 which we have
defined in equation 2.4. In order to evaluate this matrix element, we must have some
knowledge of the electronic Hamiltonian and of the wavefunctions Ψ+0 and Ψ0+. The
electronic Hamiltonian can be written as:
H e = T e + U = Σi(T e i
ZA
− ΣA
riA
1
) + ΣiΣ j>i
ri j
(2.12)
where T e i represents the kinetic energy of the ith electron, ZA the nuclear charge of the
A th nucleus, riA the distance between the i th electron and the A th nucleus and ri j represents
the distance between the i th and j th electrons. The first term of this Hamiltonian is a one
electron property and will be represented from now on by O1 = Σi(T e i
ZA − ΣA ); the second
riA
term is a two electron property and will be represented by the symbol O2 = ΣiΣ j>i 1
ri j .
It is also necessary to have forms for the wavefunctions representing the molecular
system with charge localised on either the components. In order to correctly represent
the anti-symmetric nature of the wavefunction, Slater determinants of the molecular spin-
orbitals are used [22]. Slater determinants are anti-symmetrised sums of all permutations
of a particular set of one electron wavefunctions (molecular orbitals). Imagine that these
orbitals are all localised on one molecule or on the other and label the ones localised on
molecule A ψ Ai , and the ones localised on molecule B ψ Bi , where i is a label enumerating
the spatial part of the wavefunction. A bar will be used to label spin state. A possible
Slater determinant representing a state with the charge on molecule A would therefore be:
Ψ+0 = |ψ A0 ¯ψ A0 ψ B0 ¯ψ B0 . . . ψ An−1 ¯ψ An−1 ψ Bn−1 ¯ψ Bn−1 ¯ψ An ψ Bn ¯ψ Bn > (2.13)
where the bold Ψ represents a Slater determinant and ψ represents a molecular orbital. If
the charge is localised on molecule B, the state might be represented as:
26

Ψ0+ = |ψ A0 ¯ψ A0 ψ B0 ¯ψ B0 . . . ψ An−1 ¯ψ An−1 ψ Bn−1 ¯ψ Bn−1 ψ An ¯ψ An ¯ψ Bn > (2.14)
It is easy to notice that these two Slater determinants differ in only one spin-orbital:
ψ An and ψ Bn , assuming that all spin orbitals are orthonormal. We are therefore easily able
to use Slater matrix rules [22] and write the form of the matrix element:
+
�
µ=A0,B0...An,Bn
J0+;+0 =< Ψ0+|O1 + O2|Ψ+0 >= (2.15)
< ψ An |T e − ΣA
ZA
rA
|ψ Bn >
(< ψ An ψ Bn | 1
r |ψµ ψ µ > − < ψ An ψ µ | 1
r |ψBnψ µ >)
The right hand side of the equation is now a matrix element of the Fock operator [22]. In
other words we have shown that in going from a multi-electron Hamiltonian to a single
electron picture of molecular orbitals, the transfer integral of the Hamiltonian becomes
equal to the matrix element of the Fock operator:
J0+;+0 =< ψ An | ˆF|ψ Bn > (2.16)
So long as we can represent the charge localised states as Slater determinants with only
one singly occupied orbital, the problem of finding the transfer integral corresponds to
finding the singly occupied orbitals on each molecule in the pair then finding the Fock
matrix for the whole dimer and taking the matrix element. When we perform such analysis
with density functional methods, we use the Kohn-Sham matrix instead of the Fock matrix
and the Kohn-Sham orbitals instead of the molecular orbitals. Three methods of achieving
this will be discussed: by performing SCF calculations on a pair of molecules and taking
the splitting between the frontier orbitals, by projecting the orbitals of the pair onto the
molecular orbitals of the single molecules and finally by direct calculation of the Fock
matrix. The latter two methods were designed and implemented as part of the work for
this thesis and will be discussed in the next chapter.
27

2.3.1 Note on Systems with Degenerate Orbitals
It is interesting to discuss the validity of equation 2.16 for the case of molecules with
degenerate frontier orbitals. This is relevant to chapter 7, in which we discuss charge
transport in highly symmetric molecules such as hexabenzocoronene (HBC). Because of
its S6 symmetry HBC possesses doubly degenerate highest occupied molecular orbitals
(HOMOs). In this case we argue that representing the charged state with a single Slater
determinant of the type represented in equation 2.13 is not correct, as the choice of which
degenerate orbital to leave singly occupied is arbitrary. We should note that, upon charg-
ing, Jahn-Teller instabilities would cause the molecule to deform along one of its three
symmetry axes, lose its S6 symmetry and lift the degeneracy of the HOMOs. Presumably
a pair of such molecules would then both be able to deform and charge hopping would
occur along one of nine possible seams on the diabatic energy surfaces describing charge
transport. This description is overly complicated and not fully consistent with the assump-
tions made in deriving the Marcus charge transfer rate, as J will be different along each
of these nine seams; we wish to describe charge transfer with a single effective transfer
integral Je f f . Following Newton [7] we wish to show that it is appropriate to simply pick
the root mean square value of the transfer integrals for each pair of degenerate orbitals.
Instead of accounting for the lifting of the degeneracy, we assume that the charged states
can be expressed as a superposition of two Slater determinants:
ΨA = cos(θ)Ψ HOMOA
ΨB = cos(ω)Ψ HOMOB
+ sin(θ)Ψ HOMO−1A
+ sin(ω)Ψ HOMO−1B
(2.17a)
(2.17b)
where ΨA represents the wavefunction with charge localised on A, ΨB represents the wave-
function with charge localised on B, the bold Ψ represents Slater determinants, the labels
HOMO A
represent the orbital which is singly occupied in the Slater determinant and θ and
ω represent two mixing angles. Using the same arguments we used to write equation 2.16
28

we can calculate a transfer integral for a particular pair of superpositions ΦA and ΦB:
J(θ, ω) = cos(θ)cos(ω)Jhomo A ;homo B + cos(θ)sin(ω)Jhomo A ;homo−1 B
+ sin(θ) cos(ω)Jhomo−1 A ;homo B + sin(θ)sin(ω)Jhomo−1 A ;homo−1 B
where the expressions Jµ;ν represent the matrix element of the Fock operator for molecular
orbitals µ and ν. In order to obtain such an expression for an effective integral this equation
is squared and averaged over all possible angles θ and ω. One then obtains the expression:
J 2
e f f
= J2
homo A ;homo B + J 2
homo A ;homo−1 B + J 2
homo−1 A ;homo B + J 2
homo−1 A ;homo−1 B
4
this is the expression that we use when treating highly symmetric systems.
2.3.2 Splitting of the Molecular Orbitals
(2.18)
A method for calculating transfer integrals often used in the literature [23, 24, 9] consists
in equating half the splitting in the energies of the frontier orbitals of a pair of identical
molecules to the coupling between the frontier orbitals of the constituent molecules. In
the case of hole transport, for example, one assumes that the molecular orbitals which are
singly occupied in equations 2.13 and 2.14 are the HOMOs of either molecule: this is
equivalent to invoking the frozen orbitals approximation. It is then assumed that the only
non-zero elements of the Fock matrix involving the HOMOs of the isolated molecules are
the diagonal elements and one transfer integral, which connects the two HOMOs on either
molecule. This latter assumption is equivalent to assuming that the HOMO and HOMO-1
of the pair of molecules are simply formed by a unitary transformation of the HOMOs of
the isolated molecules. In light of these assumptions the problem of determining the two
highest molecular orbitals of the pair can be expressed as the following matrix problem,
in the basis set of the HOMOs of the isolated molecules:
H = EA J
J EB
29
(2.19)

where J is the shorthand for the transfer integral defined in equation 2.16 and EA and
EB represent the expectation values of Fock operator of the pair of molecules for the
HOMOs on molecules A and B respectively. Solving this eigenvalue problem, we find
the expression for the two HOMOs of the pair:
E± = EA + EB
2
± 1
2 ( � (EA − EB) 2 + (2J) 2 ) (2.20)
If EA and EB are the same, by taking the difference between the HOMOs of the pair
one gets 2J. This method was widely applied to a variety of organic materials[23, 24,
25, 9], both electron conductors and hole conductors; electron conductors are treated
by looking at the lowest unoccupied molecular orbital. The model chemistry used by
Brédas and co-workers [25, 9] was independent neglect of differential overlap (INDO),
with Zerner’s parametrisation for spectroscopy (ZINDO) [27]. The ZINDO parametri-
sation was used both because it allows one to treat very large systems and because the
spectroscopic parametrisation guarantees good treatment of both the occupied and unoc-
cupied frontier orbitals.
The main limitation of considering the transfer integral as half the difference in orbital
energies lies in the fact that it cannot deal with cases in which the diagonal elements EA
and EB are not the same. In this case other methods must be used such as the fragment
orbitals methods [28]: in the next chapter we will discuss how to obtain similar results
from projection of the orbitals onto localised basis set.
2.4 Site Energy Difference
One of the aims of this thesis is to find accurate and efficient methods to calculate the
difference in site energy ∆E and to elucidate its origin. There are many possible origins
for such driving force: if the molecules have different ionisation potential or electron
affinity or if an external electric field was present. We have shown how, using the frozen
orbital approximation, we can define transfer integrals of a multi-electron wave-function
in terms of matrix elements of the Fock matrix. Similarly the site energies can be obtained
by looking at the diagonal elements of the Fock matrix in a localised basis set, and we
30

Figure 2.2: Pictorial representation of the vectors discussed in deriving equation 2.26
can argue that the difference between these two energies is the site energy difference
that appears in equation 2.6. In this thesis we are mainly concerned with charge transfer
between identical molecules and in section 3.3 we show that, for these cases, it is sufficient
to consider electrostatic and inductive effects to explain the observed variations in site
energy difference. In this section we explain how to calculate the electrostatic interaction
between two charge distributions. We also explain how to calculate inductive interactions,
that is how to calculate the dipole induced by a particular electric field and its influence
on the interaction energy between two molecules.
2.4.1 Electrostatic Interactions in Molecules
SCF quantum chemical calculations will produce a form of the electron density in terms
of atomic orbitals. Rather than calculating the electron density of two molecules at close
points in space and calculating a huge number of charge charge interactions, we want to
integrate the various moments of the charge density and deduce a simpler form for the
interaction energy. The resource I have found most useful in studying this topic has been
a set of notes by A.J. Stone, available from the Cambridge website [29], still I have found
this topic so confusing that I decided to repeat the derivation and obtained a somewhat
different approach.
Consider the potential V(r) of a charge distribution ρ(r) at a point r as illustrated in
figure 2.2. The potential will be described as a sum of potential from all points at distances
r + δr:
�
V(r) =
1
4πɛ0
ρ(δr)
|r + δr| d3 (δr) (2.21)
where the vertical bars || represent taking the modulus, ρ represents charge density and ɛ0
31

is the permittivity of free space. Let us introduce a function v such that:
v(r + δr) = 1
4πɛ0
1
|r + δr|
(2.22)
So long as |δr| is smaller than |r| we can write the Taylor expansion of this function, and
know that it will converge:
�
v(r + δr) = v(r) + δrα(∇αv)r + 1 �
δrαδrβ(∇α∇βv)r + ... (2.23)
2
α
where α and β represents the summation of the coordinates x, y, z. Substituting this ex-
pression back into 2.21 we arrive at sums of integrals of the charge density multiplied by
derivatives of the distance vector r:
αβ
�
V(r) = v(r) d 3 (δr)ρ(δr) (2.24)
+ � �
α (∇αv)r δrαd 3 (δr)ρ(δr)
+ 1
�
�
αβ (∇α∇βv)r δrαδrβd
2
3 (δr)ρ(δr) + ...
The integrals in equation 2.24 are readily manipulated to yield the first three electrical
moments of the charge distribution depicted on the right of figure 2.2:
c = �
dα = �
Qαβ = �
d 3 (δr)ρ(δr) (2.25)
δrαd 3 (δr)ρ(δr)
( 3
2 δrαδrβ − 1
2 |r|2 δαβ)d 3 (δr)ρ(δr)
The first two quantities are readily recognizable: they are the total charge c and the
dipole moment d. The third quantity is the so-called traceless quadrupole moment. The
reader will notice that, compared to the quantity shown in equation 2.24, we have multi-
plied it by three and subtracted a quantity from the diagonal elements to ensure that the
trace of the matrix is zero. We are justified in doing this because a quadrupole moment
which is proportional to the unit matrix does not contribute to the potential of a charge
32

Figure 2.3: Pictorial representation of the vectors discussed in deriving equation 2.32
distribution. To see why this is justified, we rewrite equation 2.24 in terms of the quanti-
ties defined in equation 2.25, adding a term proportional to the unit matrix to make up for
the fact that we have defined a traceless form of the quadrupole moment:
�
V(r) = v(r)c + (∇αv)rdα + 1 �
�
(∇α∇βv)rQαβ + Γ (∇α∇αv)r + ... (2.26)
3
α
αβ
where Γ collects the proportionality factors and the trace of the quadrupole moment which
we are going to ignore. The fourth term of this equation is equal to zero. To see why,
evaluate the expression (∇α∇αv)r for α corresponding to x. This term will have the value:
(∇x∇xv)r = 1
� 2 3x 1
−
4πɛ0 |r| 5 |r| 3
�
α
(2.27)
If we add the xx, yy, and zz contributions they will sum up to zero. It is therefore pre-
ferrable to write the quadrupole moment in a traceless basis as shown in equation 2.25.
Equation 2.26 therefore shows the first few terms of the potential due to a charge
distribution at a particular point in space. However we want to be able to calculate the
interaction energy between two charge distributions. This is better described by figure 2.3
than by figure 2.2. In order to distinguish two charge distributions we use labels A and B
to distiguish the electrical moments of one molecule from those of the other.
Again let us write the interaction energy UAB as an integral:
�
UAB =
ρ(δrA)V B (r − δrA)d 3 (δrA) (2.28)
Once again assume that |δrA| is smaller than r to ensure the convergence of a Taylor series:
33

�
UAB =
d 3 ⎛
(δrA)ρ(δrA) ⎜⎝ V B �
(r) −
α
(∇αV B )rδrAα + 1
2
�
αβ
(∇α∇βV B )rδrAαδrAβ + ...
⎞
⎟⎠
(2.29)
Once again we can take the terms in equation 2.29 that depend on δrA and collect them
as electrical moments of molecule A:
UAB = c A V B �
(r) − d A α(∇αV B � 1
)r +
3 QA αβ(∇α∇βV B )r + ... (2.30)
α
Now we substitute the expression for V B from equation 2.26 into equation 2.30. Before
showing this equation, however, let us introduce another handy piece of notation. In equa-
tion 2.26 the only part which depends on δr are the derivatives of v(r), so for convenience
let us define the operator Tαβγ... as:
αβ
Tαβγ... = (∇α∇β∇γ∇...v)(r)
Using this convenient notation the interaction energy UAB can be written as:
UAB = c A Tc B + c
−
+
A �
Tαd B α + c
A �
1
Tαβ
3 QB αβ
α
αβ
�
d
α
A αTαc B �
− d
αβ
A αTαβd B �
β − d
αβγ
A 1
αTαβγ
3 QB βγ
� 1
3
αβ
QA αβTαβc B � 1
+
3
αβγ
QA αβTαβγd B � 1
γ +
3
αβγδ
QA αβTαβγδ
1
3 QB γδ + ...
(2.31)
(2.32)
So finally we have an expression for the interaction between two charge densities in
terms of interactions between electrical moments.
Note that if each electrical moment is to correspond to a whole molecule, this ex-
pression is valid only for those cases in which the two molecules are further apart than the
largest distance in each molecule. In this case the electrical moments can be obtained from
a quantum chemical calculation on one isolated molecule; most quantum chemistry pack-
ages will calculate these quantites. In the case in which two molecules are closer than the
largest distance inside the molecule, we have to partition the space in each molecule and
34

substitute each partitioned area with a collection of monopoles, dipoles, quadrupoles etc.:
this scheme is known as distributed multipole analysis. Some computational chemistry
packages, such as GAMESS-UK [30], have modules to do this. In the case of Gaussian 03
[31] a helper program gdma by A. Stone [32] is available. Distributed multipole analysis
is the technique we will use to determine the electrostatic interaction of large molecules at
short distances. The distribute multipoles are obtained from the gdma program and C++
libraries were written to calculate the interactions from equation 2.33.
2.4.2 Inductive Interactions in Molecules
In the treatment described so far, we have assumed that it is possible to take the elec-
trical moments from a calculation on an isolated molecule and apply them to a pair of
molecules: we have assumed that the electrical field due to one molecule does not influ-
ence the charge distribution of the other. In order to improve on this model it is possible
to calculate the dipole-polarisability of a molecule: the tensor that determines the dipole
induced on one molecule by the electric field due to the other. This polarisability can
be obtained from a quantum chemical calculation, provided that the first derivatives of
the energy are calculated. The polarisability returned by Gaussian corresponds to a static
polarisability, but excludes rotational and vibrational effects. The polarisability will in
general be a 3x3 matrix Pαβ. The induced dipole at molecule A will therefore have form:
d A
ind α = PαβF B β
(2.33)
where FB is the electrical field from molecule B and dA ind is the induced dipole moment.
This induced dipole will in turn change the value of the electric field due to molecule A
at molecule B and induce a different dipole on B, which will in turn change F B β
and dA
ind .
In the implementation of the protocol that we use in section 3.3, a self consistent solution
to this iterative procedure is implemented. In other words corrections are calculated until
they are smaller than a token value. It is intuitively obvious that this series will tend to
converge rather rapidly: induced dipoles tend to lower the electric field and therefore each
term will be decreasing in value.
In the previous paragraph, we have described how to treat polarisabity by using a
35

single polarisability for each molecule. A caveat is that we have implicitly assumed the
electric field to be uniform in the molecule; if the molecules are close relative to their
overall size, this will not be a fair assumption. Schemes to treat this situation exist: for
example G.A. Kaminski et al [33] developed models of molecules where partitioned zones
each have their own polarisability which is deduced by performing DFT calculations in
the presence of point charges, measuring the change in the electrostatic potential of a
molecule and fitting this potential to particular values of the polarisability. E.V. Tsiper
and Z.G. Soos, in their description of the binding energy of pentacene crystals [34], de-
duce atom polarisabilities by performing semi-empirical calculation in the presence of an
electric field, then separate the induced dipole into two contributions: one from charge
reorganisation in the molecule, the other from the creation of atomic dipoles. From this
latter contribution an atomic polarisability is deduced. Yet another technique is that of
A. Stone [35] where distributed multipole analysis is performed for a variety of applied
fields, leading to the deduction of a polarisability for each zone considered in the mul-
tipole analysis. We have not implemented any of these approaches to calculate dipole-
polarizabilities in non-uniform fields and correct our calculations by an inductive term
only in the case when molecules are far enough apart to consider the electric field across
them uniform.
2.5 The Gaussian Disorder Model and Related Methods
All the methods discussed in this chapter treat charge transport events as hops between
different localised states. In this section we give an overview of the methods to obtain
macroscopic charge mobilities from the microscopic rates that we discuss in the previous
sections. Charge hops between molecules can be represented as hops between different
sites on a lattice. Figure 2.4 shows how charge transport can be envisioned to occur in
a conjugated material. In this figure we are trying to represent two possible forms of
disorder: the position of the sites where the charges are localised and and the coupling
between the sites (i.e. the rate of transfer of charge from site to site). Disorder in the
energies of the charge transporting site is represented in the figure by the distribution
in heights of energy levels and disorder in the electronic coupling by a distribution in
36

Figure 2.4: A schematic of one dimensional transport through an energetically and positionally
disordered medium. This figure represents disorder in energy (vertical
placement of the transport levels) and in transfer integrals (thickness of
arrows)
orientations and separations which leads to different transfer integrals (represented by the
thickness of the arrows). A typical model of charge transport in a material will therefore
comprise three elements:
(a) an appropriate computational or analytical method to solve charge dynamics
(b) an appropriate microscopic equation for hopping rates
(c) a distribution function for the energies of the sites and, in some cases, the electronic
coupling between sites according to a particular distribution.
The second of these three aspects is discussed in section 2.1, the other two are discussed
separately in the next sections. Finally some results from the literature are reviewed
briefly.
2.5.1 Computational Methods
Two numerical methods are commonly used to simulate microscopic charge transport: the
Master equation and Monte Carlo approach. In order to model mobility, the experiment
modelled is usually a time of flight (ToF) experiment. In a time of flight experiment charge
pairs are generated near a semi-transparent, non-injective electrode. Once separated by
an electric fied F either electrons or holes (depending on the sign of the applied field)
will drift towards the other electrode. This drift causes a displacement current which falls
to zero as the carriers reach the counter electrode. Thus a time for carriers to reach the
counter electrode electrode can be extracted and knowing the thickness of the film allows
the measurement of the charge’s velocity and hence the charge mobility for a particular
field and temperature. More information can be found in references [36] [37] .
37

The Master equation is a differential equation that describes the time behaviour of
stochastic systems [38], i.e. one whose state can only be described in terms of probabil-
ities. If the states of a system are labelled 0, 1, 2, ..., m , for example, the state of the
system at time t will be described by a set of probabilities P0(t), P1(t), Pm(t). The Master
equation simply states that:
dP(n, t)
dt
�
= (AmnPm(t) − AnmPn(t)) (2.34)
m�n
where Amn is the transition rate from state m to state n. The problem can conveniently be
cast in matrix form if one writes the diagonal elements of the matrix A as:
�
Ann = −
m�n
Anm
(2.35)
If the rates A are independent of the probabilities, the problem is linear and can be readily
solved.
The power of this method is that it is applicable to a vast number of systems and
that it is possible to determine the steady state behavior by simply setting the right side
of equation 2.34 to zero. For the particular case of charge transport, in the most formal
approach Pm(t) would describe the probability of being in the m th many-electron state at
time t. For example, for a system containing two states, each of which is allowed to carry
either one electron or none, there would be four many-electron states: two where either
of the states is occupied, one where neither is occupied and one where both are occupied.
This approach becomes unwieldy very quickly, as the number of many-electron states
increases as 2 N where N is the number of sites. A common approach [39] [13] is to use
Pm(t) as the probability of occupation of the m th site on the lattice at time t, the rates Amn
are then assumed to be independent of P. For a one-dimensional model the steady state
Master equation can be solved analytically, even in the presence of random disorder in
the hopping rates[39]. For higher dimensionalities, and with more physical values for
disorder, the problem can be treated as a linear problem so long as carrier concentrations
38

are low. Then, if the probabilities are written as a vector and the rates as a matrix:
AP = 0 (2.36a)
�
Pi = 1 (2.36b)
i
The second equation states normalisation. In such an approach, equation 2.36a can be
solved for the Pi and the velocity can be measured from the steady state values of P and
A:
v = d¯r
dt =
�
¯rmnAmnPm
m�n
where rmn is the distance between sites m and n. Mobility is then obtained as µ = v
F .
(2.37)
Monte Carlo methods are another way of calculating the properties of stochastic sys-
tems. The term Monte Carlo refers to any method that uses random numbers. In this
case the system is modelled as a large regular lattice or as a disordered assembly of lat-
tice points. Once energies are distributed according to the particular distribution function
used, carriers are generated and are assigned waiting times and a destination according to
the particular formula for the rates which is used [41] [42]. In the adaptive time step ver-
sion of the method, time is advanced by the shortest waiting time and the carrier with the
shortest waiting time is moved. If necessary, boundary conditions in the directions per-
pendicular to the field are applied. When a carrier reaches the counter electrode its time
of arrival is recorded. The simulation continues until all or most carriers reach the counter
electrode. Simulations are repeated over different initial carrier distributions and realisa-
tions of the lattice until a representative photocurrent transient is obtained. Then the mean
arrival time is calculated and mobility is deduced. The great advantage of the adaptive
time step Monte Carlo method is that it can deal with extremely non-linear, asymmetric
hopping rates. For example, we have discussed how, in the Master equation approach, the
rates Amn are routinely treated as independent of the occupation probabilities in order to
make the problem linear. This makes it impossible to include effects where multiple occu-
pation of a site is not allowed. In a Monte Carlo program, adding this feature is extremely
easy; one simply does not assign as a possible destination a site which is occupied! Monte
39

Carlo simulations are then run at a number of different fields and temperatures to model
the results of ToF experiments.
2.5.2 Distribution of Parameters of the Transfer Rates
In this section, we will discuss some of the different models that have been used to assign
distributions of energies. Two types of disorder can be considered: disorder in the energy
of the hopping sites and disorder in the transfer integrals between molecules. These two
types of disorder are sometimes called energetic and positional, they are also referred to
as diagonal and off-diagonal disorder. In Equation 2.6, these two types of disorder would
represent distributions in the values of ∆E and J respectively. The simplest assumption
in the treatment of energetic disorder is that a material has no long range order and the
only source of disorder is the random local energetic landscape. In this case the energetic
disorder is simply treated as a Gaussian distribution of width σ. The off-diagonal disorder
is modelled in the following way by H. Bässler and co-workers [41] [42]. The transfer
integral is assumed to vary exponentially with distance, with a wave function overlap
parameter γi j for sites i and j. So:
|J| 2 = |J0| 2 e −2γi jRi j (2.38)
where Ri j is the distance between the sites. The wave function overlap γi j parameter
represents the inverse of the spatial extent of the wave function. In the model γi j is allowed
to vary in the following way:
2γi j = Γi + Γ j
a
= Γi j
a
(2.39)
where Γi is a parameter specific to site i drawn from a gaussian distribution of width Σ
and a is the lattice constant for the simulation. We can already see how the interpretation
of off-diagonal disorder in real materials will be difficult: whereas in the model it is
assumed to simply represent the variation in intersite separations, in a real material it will
also depend on orientational effects. These forms for the distributions of site energy and
transfer integral, together with the usage of a Miller-Abraham rate equation and a Monte
Carlo routine, form the basis for the well known Gaussian Disorder Model (GDM) due to
40

H. Bässler and coworkers [41]. The GDM was used to determine an empirical equation
[42] that describes the temperature and field dependences of ToF mobility data in terms
of the parameters σ and Σ. It was especially succesful in describing the Poole-Frenkel
like field dependence of mobility under high fields:
⎧
⎪⎨ µ0e
µ =
⎪⎩ µ0e
2 −( 3kT σ)2 σ C[( e kT )2−Σ2 ] √ F , Σ > 1.5
2 −( 3kT σ)2 σ C[( e kT )2−2.25] √ F , Σ < 1.5
(2.40)
where µ0 represents the zero field mobility, high temperature mobility and C is a lattice
related constant. Once data for a particular material are fitted with equation 2.40, that
material is assigned particular values of the empirical parameters µ0 , Σ, σ and C.
The value of σ has been interpreted in terms of the dipole moment of the constituents
of the system by several groups[44, 47, 11]. The assumption made by [44] is that the
diagonal disorder has two contributions, from van der Waals and dipole interactions.
σ 2 = σ 2
d + σ2vdw
(2.41)
The effect of randomly oriented dipoles on the energetic landscape was modelled and it
was found that σd is proportional to the dipole moment [44]. When the dipole moment
of small conjugated molecules was plotted against the values of the diagonal disorder
extracted by GDM analysis of measured mobility, the plot figure 2.5 was obtained [44].
If the value of σvdW is extrapolated from the intercept at p=0 and σd is extrapolated with
equation 2.41 the graph on the left of figure 2.5 is obtained, indicating that the dipole
induced disorder is proportional to the magnitude of the dipole moment, as expected. The
observation that a possible source of disorder could be dipole-dipole interactions led S.V.
Novikov and coworkers to challenge the view that long range order is not present in these
materials: dipole interactions in fact are long range and would lead to a spatial correlation
in site energies. Novikov calculated the value of the correlation [45] for a cubic lattice as:
< E(r)E(0) >= 0.74σ 2 a
d
r
(2.42)
where the triangular brackets denote averaging, r is the vector separating the sites and
41

Figure 2.5: σ versus the value of the dipole moment for five different small conjugated
molecules (right panel)[44], σd obtained from equation 2.41 (left panel)
a is the lattice constant. Figure 2.6 [45] shows a pictorial representation of the dipole
potential in a lattice of randomly oriented dipoles. It can be seen clearly from figure 2.6
that there are rather large domains of similar magnitudes of energy, i.e. that the energies
are correlated.
Other models have been proposed to explain how correlation in energy could arise
in non-polar materials: including the quadrupolar glass model [48] and the Los Alamos
model [13]. The quadrupolar glass model is similar to the correlated dipole disorder
model, but substitutes dipole interactions with higher electric moment interactions (quadrupole)
interaction. Both these approaches are formally correct only in the case of small, well
separated molecules. In the case of large densily packed molecules the electrostatic in-
teractions should be treated using distributed multipole analysis. The Los Alamos [13]
model assigns correlations in polymers such as polyphenylenevinylene to fluctuations in
the torsional angle between neighbouring benzene rings. The necessity for introducing
correlations in the models will be explained in the next section, where we discuss the
successful predictions and short-comings of these models.
42

Figure 2.6: Pictorial representation of correlation in a 100x100x100 cubic lattice of randomly
oriented dipoles. The diameter of the spheres is proportional to the
magnitude of the dipole interaction at that point, whereas the colour of the
spheres represents the sign of the potential [45]
2.5.3 Predictions and Results
In this section we discuss the correct predictions and short-comings of the models de-
scribed so far. The first model to be truly successful at explaining the results from ToF
experiments was the GDM. It successfully explains the process of energetic relaxation
in non-dispersive materials, through which the energy of a packet of carriers relaxes at
long times to a constant value. This process is shown schematically in figure 2.7. It
also explains anomalous broadening, that is the fact that in materials with large values
of σ, eD
µkT
>> 1 , contrary to what would be predicted by the Einstein relation between
diffusivity D and drift mobility µ.
The GDM also explained that dispersive transport, where mobility depends on the
thickness of the material and energetic relaxation is not achieved, results from high values
of σ. The model invokes a critical temperature Tc, below which transport is dispersive.
Importantly, the GDM leads to equations 2.40, expressing a non-Arrhenius temperature
dependence and strong electric field dependence of mobility. When off-diagonal disorder
is present, the GDM also explains the odd phenomenon of negative field dependence of
mobility, that is the phenomenon by which mobility can decrease with increasing fields.
43

Figure 2.7: Energy distribution of charges as a function of time after generation [46].
This was explained by arguing that if the transfer integrals along the route parallel to the
field are smaller than those along a route that goes against the field, increasing the field
will have the effect of slowing down the charge transfer process by making the route with
steps against the field slower.
The correlated disorder models were introduced mainly to explain a shortcoming of
the GDM: the fact that it can only explain Poole-Frenkel like behavior for electric fields
greater than 10 6 V/cm, whereas such behaviour is observed experimentally for fields ten
times smaller [47]. Y.N. Garstein and E.M. Conwell’s simulations of the correlated dipole
disorder model, where energetic disorder contains a contribution from dipolar interactions
and no off-diagonal disorder is used, do in fact show Poole-Frenkel behaviour for far
lower fields. They also show that field dependence is not as dependent on the particular
type of hopping equation used. The quadrupolar glass model and the Los Alamos model,
are simply explanations of how correlations can occur for non polar materials. The Los
Alamos model predicts a different dependence on temperature: ln(µ) ∝ 1
kT
as opposed to
the behaviour predicted by the GDM: ln(µ) ∝ 1 2
however it is very difficult to recog-
kT
nize which of the two expressions is most in agreement with experiment. An important
achievement of these models is in explaining how Poole-Frenkel behavior is observed in
such a large class of materials, and in achieving this by using a relatively small set of
tunable parameters.
44

Figure 2.8: Schematic of different routes from A to B in the direction of the field [46]
The main limitation of the GDM is the difficulty to relate the parameters σ and Σ to
microscopic physical and chemical properties of a material. This difficulty is due to two
factors: the use of a Miller-Abrahams rate and of a cubic lattice. Using a Miller-Abrahams
rate makes it difficult to calculate the paramters for charge transport because the correct
models for non-adiabatic charge transport in conjugated materials must be consisted with
Marcus theory, as described in section 2.1. This problem has been corrected by P.E. Parris
and co-workers [14] by using a rate equation equivalent to equation 2.6. P.E. Parris’s
approach, which we shall call polaronic GDM, was compared to the GDM by T. Kreouzis
and co-workers [49]. They found that charge transport in polyfluorene is better described
by the polaronic version than by the GDM, confirming that Marcus theory is indeed the
correct form for charge transfer rate in conjugated materials. In chapter six of this thesis
we will compare the GDM parameters of different materials to microscopic calculations
of site energy differences and transfer integrals in order to show that the trends in GDM
parameters are nevertheless consistent with the microscopic parameters.
The use of a cubic lattice in the GDM is also a shortcoming, as it makes it impossible
to distinguish the effects of electronic structure from those deriving from morphology.
We try and approach this problem in chapter seven where we describe charge transport
in polyfluorene and in hexabenzocoronene by using models of the morphology of the
material to determine the irregular lattice that charges will hop in, to calculate the mi-
croscopic parameters of equation 2.6 and finally to run Monte Carlo and Master equation
45

simulations of the charge dynamics. This approach allows us to pinpoint exactly which
microscopic parameters determine charge transport properties.
The GDM and the other models we have briefly discussed have the great power of
being able to describe charge transport in many different materials in a unified theoret-
ical framework. We believe that this advantage is also a shortcoming as a microscopic
understanding of the chemical and physical properties which lead to charge transport
characteristics in a particular material must be based on the properties of that particular
material.
2.6 Conclusions
In this chapter we have introduced the theoretical framework for treating microscopic in-
termolecular charge hopping events: the non-adiabatic high temperature Marcus theory
charge transfer rate equation. We have described the three parameters of this equation:
the transfer integral J, the reorganisation energy λ and the site energy differene ∆E. We
have also presented some background information of how to calculate interactions be-
tween charge distributions, which we will use to calculate site energy differences in the
next chapter. In the second part of the chapter we have discussed empirical models of
charge transport, especially the Gaussian Disorder Model. We have argued that its main
shortcoming is the difficulty of relating its parameters with chemical and physical char-
acterstics of molecules. In the next chapters we will further discuss methods to calculate
the Marcus charge transfer parameters, with an emphasis on finding efficient methods for
calculating J and ∆E and on defining the charge transporting unit in a polymer. We will
also consider three different types of uses of these methods: to justify choice of param-
eters in Monte Carlo simulation of ambi-polar mobility in fullerene doped polystyrene,
to compare the GDM parameters of materials to the distribution in Marcus charge trans-
fer parameters of randomly oriented pairs of molecules and finally to parametrise Monte
Carlo simulations on non-cubic lattices derived from morphology simulations of polyflu-
orene and hexabenzocoronene.
46

Bibliography
[1] R.A. Marcus, On the Theory of Oxidation Reduction Reaction Involving Electron
Transfer, Journal of Chemical Physics, 24, 966 (1955)
[2] R.A. Marcus and N. Sutin, Electron transfer in chemistry and biology, Biochimica et
Biophysica acta 811, 265 (1985)
[3] N.R. Kestner, J. Logan and J. Jortner, Thermal Electron Transfer Reactions in Polar
Solvents, Journal of Chemical Physics, 21 , 2148 (1974)
[4] K.F. Freed and J. Jortner, Multiphonon Processes in the Nonradiative Decay of Large
Molecules, Journal of Chemical Physics, 52 , 6272 (1970)
[5] B.S. Brunshwig, J. Logan, M.D. Newton and N. Sutin, A Semiclassical Treatment of
Electron Exhange Reactions. Application to the Hexaaquoiron(II)-Hexaaquoiron(III)
System, Journal of the American Chemical Society, 102, 5798 (1980)
[6] L.D Landau, E.M Lifshitz, Quantum Mechanics non-relativistic theory, Pergamon
Press (1965), 42
[7] M.D. Newton, Quantum Mechanical Probes of Electron-Transfer Kinetics: the Na-
ture of Donor-Acceptor Interactions, Chemical Reviews, 91, 767 (1991)
[8] J. Jortner, M. Bixon, T. Lagenbacher and M.E. Michele-Beyerle, Charge Transfer and
transport in DNA, Proceedings of the National Academy of Science 95, 12759 (1997)
[9] J.L. Brédas, D. Beljonne, V. Coropceanu and J. Cornil, Charge Transfer and Energy
Transfer processes in π-conjugated Oligomers and Polymers: A Molecular picture,
Chemical Reviews, (104), 4971, (2004)
47

[10] D. Emin, Phonon-Assisted Jump Rate in Non-Crystalline Solids, Physics Review
Letters 32, 303 (1973)
[11] S.V. Novikov, D.H. Dunlap, V.M. Kenkre, P.E. Parris and A.V. Vannikov, Essential
Role of Correlations in Governing Charge Transport in Disordered Organic Materials,
Physical Review Letters 81, 4472 (1998)
[12] Z.G. Yu, D.L. Smith, A. Saxena, R.L. Martin and A.R. Bishop, Molecular Geometry
Fluctuation Model for the Mobility of Conjugated Polymers, Physical Review Letters
84, 721 (2000)
[13] Z.G. Yu, D.L. Smith, A. Saxena, R.L. Martin, and A.R. Bishop, Molecular Geometry
and field-dependent mobility in conjugated polymers, Physical Review B 63, 085202
(2001)
[14] P.E. Parris, V.M. Kenkre and D.H. Dunlap, Nature of Charge Carriers in Disordered
Molecular Solids: Are Polarons Compatible with Observations?, Physical Review Let-
ters 87, 126601 (2001)
[15] R.A. Marcus and N. Sutin, Electron Transfer in chemistry and biology, Biochimica
et biophysica Acta, (811), 265 (1985)
[16] B.S. Bruceschwig, S. Ehrenson and N. Sutin, Solvent reorganisation in Optical and
Thermal Electron-Transfer Processes, Journal of Physical Chemistry, 90, 3657 (1986)
[17] P.F. Barbara, T.J. Mayer and M.A. Ratner, Contemporary Issues in Electron Trans-
fer, Journal of Physical Chemistry, 100, 13148 (1996)
[18] K. Sakanoue, M. Motoda, M. Sugimoto and S. Sakaki, A Molecular Orbital Study
on the Hole Transport Property of Organic Amine Compounds, Journal of Chemical
Physics A, 103, 551 (1999)
[19] N.E. Gruhn, D.A. da Silvha Filho, T.G. Bill, M. Malagoli, V. Coropceanu, A. Kahn
and J.L Brédas, The Vibrational Reorganisation Energy in Pentacene: Molecular In-
fluence on Charge Transport, Journal of the American Chemical Society, 124, 7918
(2001)
48

[20] T. Kato and T. Yamabe, Electron-phonon Coupling interactions in charged cubic
fluorocarbon cluster (CF)8, Journal of Chemical Physics, 120, 1006 (2004)
[21] I.V. Leontyev, M.V. Basilevski and M.D. Newton, Theory and computation of elec-
tron transfer reorganisation energies with continuum and molecular solvent models.,
Theretical Chemistry Accounts, 111, 110 (2004)
[22] A. Szabo and N.S. Ostlund, Modern Quantum Chemistry, Dover Publications, 1989
[23] M.N. Paddon-Row and S.W. Wong, A Correlation between the Rates of Pho-
toinduced Long-Range Intramolecular Electron Transfer in Rigidly Linked Donor-
Acceptor Systems and Computed π and π∗ Splitting Energies in Structurally Related
Dienes, Chemical Physics Letters 167, 432 (1990)
[24] C. Lian and M.D. Newton, Ab-initio Studies of Electron Transfer: Pathway Analysis
of Effective Transfer Integral, Journal of Physical Chemistry 96, 2855 (1992)
[25] J.L. Brédas, J.P. Calbert, D.A. da Silva Filho and J. Cornil, Organic Semiconductors:
A Theoretical Characterisation of the Basic Parameters Governing Charge Transport,
Proceedins of the National Accademy of Science 99, 5804 (2002)
[26] P.G. da Costa, R.G. Dandrea and E.M. Conwell, First-principles calculation of the
three-dimensional band structure of poly(phenylene vinylene), Physycal Review B, 47,
1800, (1993)
[27] J. Ridley and M. Zerner, An Intermediate Neglect of Differential Overlap Technique
for Spectroscopy: Pyrrole and the Azines, Theoretica Chimica Acta, 32, 111 (1973)
[28] K. Senthilkumar, F.C. Groezema, F.M. Bickelhaupt and L.D.A. Siebbeles, Charge
transport in columnar stacked triphenylenes: Effects of conformational fluctuations
on charge transfer integrals and site energies, Journal of Chemical Physics, 32, 9809
(2003)
[29] A.J. Stone, http://www-teach.ch.cam.ac.uk/teach/M11/handout.pdf, Intermolecular
Forces, University of Cambridge (2004)
49

[30] M.F. Guest, I.J. Bush, H.J.J. van Dam, P. Sherwood, J.M.H. Thomas, J.H. van
Lenthe, R.W.A Havenith and J. Kendrick, The GAMESS-UK electronic structure pack-
age: algorithms, developments and applications, Molecular Physics, 103,719 (2005)
[31] Gaussian 03, Revision C.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuse-
ria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C.
Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G.
Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R.
Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene,
X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R.
Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W.
Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V.
G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick,
A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S.
Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi,
R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M.
Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J.
A. Pople, Gaussian, Inc., Wallingford CT, 2004.
[32] A.J. Stone, Distributed multipole analysis: Stability for large basis sets, Journal of
chemical theory and computation, 1, 1128 (2005)
[33] G.A. Kaminski, H.A. Stern, B.J. Berne, R.A. Friesner, Y.X. Cao, R.B. Murphy, R.
Zhou and T.A. Halgren, Development of a polarisable Force Field For Proteins via Ab
Initio Quantum Chemistry: First Generation Model and Gas Phase Tests, Journal of
Computational Chemistry, 23, 1515 (2002)
[34] E.V. Tsiper and Z.G. Soos, Charge redistribution and polarisation energy of organic
molecular crystalst, Physics Review B, 64, 195124 (2001)
[35] A.J. Stone, Distributed polarisabilities, Molecular Physics, 56, 1065 (1985)
[36] I.H. Campbell and D.L. Smith, Physics of organic electronic devices, Chapter 4
(2001)
50

[37] H. Scher and E. Montroll, Anomalous transit-time dispersion in amorphous solids ,
Physical Review B, 12, 2455 (1975)
[38] I. Oppenheim, K.E. Shuler and G.H. Weiss, Stochastic processes in chemical reac-
tions, MIT press (1977)
[39] B. Derrida, Velocity and diffusion constant of a periodic one-dimensional hopping
model , Journal of Statistical Physics, 31, 433 (1982);
[40] J. Nelson, J. Kirkpatrick and P. Ravirajan, Factors limiting the efficiency of molecu-
lar photovoltaic devices, Physical Review B, 69, 035337 (2004)
[41] L. Pautmeier , R. Richert and H. Bässler , Poole-Frenkel behaviour of Charge Trans-
port in organic solids with off-diagonal disorder studied by Monte Carlo simulation,
Synthetic Metals, 37, 271 (1990)
[42] P.M. Borsenberger, L. Pautmeier and H. Bässler, Charge Transport in disordered
solids, Journal of Chemical Physics, 94, 5447 (1991)
[43] R.A. Marcus , Electron Transfer Reactions in chemistry. Theory and experiment,
Reviews of Modern Physics 65, 599 (1993)
[44] A. Dieckamnn, H. Bässler and P.M. Borsenberger, An Assessment of the role of
dipoles on the density of states of function of disordered molecular solics, Journal of
Chemical Physics, 99, 8136 (1993)
[45] S.V. Novikov and A.V. Vannikov, Cluster Structure and Distribution of the Elec-
trostatic in a Lattice of Randomly oriented dipoles, Journal of Physical Chemistry 99,
14573 (1995)
[46] P.M. Borsenberger, D. Weiss, Organic Photoreceptors for Xerography, Marcel
Dekker inc., 1998
[47] Y.N. Garstein, E.M. Conwell, High Field Hopping mobility in molecular systems
with spatially correlated energetic disorder, Chemical Physics Letters 245, 351 (1995)
[48] S.V. Novikov, Charge-Carrier Transport in Disordered Polymers, Journal of Poly-
mer Science: Part B: Polymer Physics 41, 2584 (2003)
51

[49] T. Kreouzis, D. Poplavskyy, S.M. Tuladhar, M. Campoy-Quiles, J. Nelson, A.J.
Campbell and D.D.C. Bradley, Temperature and field dependence of hole mobility in
poly(9,9-dioctylfluorene), Physical Review B 73, 235201 (2006)
52

Chapter 3
Development and Verification of
Methods for Calculating Charge
Transport Parameters
In chapter 2 we have shown how hopping transport in disordered molecular assemblies
can be described in terms of a number of fundamental electronic parameters of molecules.
We introduced the parameters that determine the charge transfer rate between two molecules:
the transfer integral J, the site energy difference ∆E and the reorganisation energy λ. Of
these three parameters, J and ∆E will in general have different values for each pair of
molecules in a slab of material, depending on the relative position and conformation of
molecules. In order to calculate charge transfer rates for all such pairs of molecules, J
and ∆E have to be calculated many times, and therefore fast and efficient methods are
desirable. This chapter presents the methods that we developed in this thesis for this
purpose.
In section 2.3 we described a method for calculating the transfer integral J from the
splitting in orbital energies that has been widely used in the literature. We also intro-
duced the theoretical framework that can be used to describe the electrostatic and in-
ductive interactions between two charge distributions. In this chapter we explain how
to express the Fock matrix for a pair of molecules in a basis set of orbitals localised on
each molecule. This allows the transfer integrals and energies of the localised orbitals to
be obtained directly. We then present two methodologies for determining J and ∆E for
53

pairs of molecules in which SCF calculations have to be carried out only on the isolated
molecules. The approximate method to calculate J is based on a simplified version of
the ZINDO [1] Hamiltonian and will involve calculation of the Molecular Orbital Over-
lap, this method will be referred to as MOO. The method to calculate ∆E is based on the
classical tools for calculating interaction energies between charge distributions that were
introduced in section 2.4. The results of these classical calculations will be compared to
the difference in site energies obtained by the projective method.
The last section of the chapter is devoted to comparing the results of the three methods
available for calculating J: the projective and orbital splitting methods are shown to be
in agreement in the cases where site energies are the same. The projective method is
used to compare results from the DFT Kohn-Sham and the ZINDO Fock matrix, to show
the ZINDO method gives very similar values for the transfer integral to the pure DFT
calculations. The projective method with the ZINDO Fock matrix is then used as the
reference to evaluate the accuracy of the approximate MOO method. The systems used for
this comparison are ethylene, pentacene and hexabenzocoronene. Ethylene and pentacene
are chosen as typical conjugated molecules; hexabenzocoronene is selected because of
its degenerate orbitals and because it is used as a model system for the study of charge
transport in chapter seven.
The notation and description of methods used in this chapter applies to charge transfer
between identical molecules, as they will be used on identical molecules in the rest of the
thesis. It would be straightforward to generalise these methods to be applicable to different
molecules.
3.1 Projective Method for Calculation of Transfer Inte-
grals
The orbital splitting method, where the transfer integral is approximated by half the split-
ting of the HOMO energies of a pair of molecules, is only valid if EA and EB, the site
energies of equation 2.19, are the same. In order to find the general solution of equa-
tion 2.16 K. Senthilkumar and co-workers [2] have exploited a feature of the Amsterdam
54

Density Functional [3] package that allows the orbitals of molecular fragments of a larger
system to be used as the basis set in which to solve the Kohn Sham equations. In order to
be able to perform fragment type calculations using the ZINDO method as well as DFT
functionals, we developed a procedure whereby the orbitals of a pair of molecules are pro-
jected onto a basis set defined by the orbitals of each individual molecule. The molecular
orbital energies of the pair of molecules are then used to rewrite the Fock matrix in the new
localised basis set. We call this method the projective method. This method is based on
application of the spectral theorem [4]. In the case of model chemistries such as Hartree
Fock or Density Functional Theory, where the atomic orbitals do not form an orthogonal
set, symmetric Lowdin renormalisation [5] is used to re-orthogonalise the orbitals. Com-
pared to the orbital splitting method, the projective method has several advantages. First
it is not necessary to assume that the two energies of the HOMOs are equal and therefore
the projective method allows us to look at pairs of different molecules, or pairs of identical
molecules in such arrangments as to lead to differences in EA and EB due to polarisation.
Another advantage is that we are no longer limited in the type of orbitals we use to define
the isolated molecule’s Slater determinant: we could, for example, use the orbitals from
a restricted open shell calculation, or the natural orbitals from an unrestricted open shell
calculation, or from a multi-reference method. The third advantage is that it is possible,
by looking at the diagonal elements of the Fock matrix in the localised basis set, to deter-
mine the expectation values of orbitals of the isolated molecules; if the orbitals chosen are
the HOMO (or the LUMO) of each molecule this allows us to determine the site energies
for the positive (or negative) charge localised states.
Let us show how this projection is carried out. The orbitals of the pair of molecules
will be written as a matrix Cpair, where each column corresponds to a molecular orbital
and each row to an atomic orbital. The vector space that we wish to project these orbitals
onto will be written in terms of the molecular orbitals of the isolated molecules. We write
these orbitals as Cloc where the first N/2 columns correspond to the molecular orbitals
of the first molecule and the last N/2 columns correspond to the molecular orbitals on
the second molecule, similarly the first N/2 rows correspond to the atomic orbitals on the
first molecule and the second N/2 to those on the second molecule. The localised basis
55

set Cloc can be written as:
Cloc =
φ 1
1
φ 1
2
φ 2
1 ... φ N 2
1 0 0 0 0
φ 2
2 ... φ N 2
2 0 0 0 0
... ... ... ... 0 0 0 0
φ 1 N 2
φ2 N
2
... φ N 2
N
2
0 0 0 0
0 0 0 0 φ N 2 +1
N
2 +1 φ N 2 +2
N
2 +1
... φN N
2 +1
0 0 0 0 φ N 2 +1
N 2 +2 φ N 2 +2
N 2 +2
... φ N N 2 +2
0 0 0 0 ... ... ... ...
0 0 0 0 φ N 2 +1
N
φ N 2 +2
N ... φN N
where φ µ ν labels the contribution of molecular orbitals µ from atomic orbital (AO) ν.
(3.1)
If the AOs are orthogonal, as they are in the ZINDO method, the orbitals of the pair of
molecules are written in the new basis set simply by calculating the dot products between
the localised orbitals and the orbitals of the pair of molecules. In matrix form this is
written as:
where C loc
pair
C loc
pair = CTlocCpair
(3.2)
represents the orbitals of the pair of molecules in the localised basis set of
the isolated molecule orbitals, and the superscript T represents transposition. Once an
orthogonal set of orbitals for the pair of molecules is found, we can rewrite the Fock
matrix for the pair of molecules, Floc pair , as:
F loc
pair
T loc
= Cloc pair ɛpairCpair (3.3)
where the molecular orbital energies of the pair of molecules have been written as a diag-
onal matrix ɛpair.
The off-diagonal element µ, ν of this Fock matrix will represent the transfer integral
of the µ th and ν th isolated molecule molecular orbitals. The µ, µ diagonal elements will be
equal to the expectation value of the µth molecular orbital of the isolated molecule with
the Fock operator of the pair of molecules: if µ labels the HOMO of molecule A, this will
correspond to the quantity EA from equation 2.19.
56

If the molecular orbitals are expressed in a non-orthogonal basis set they can be or-
thogonalised by using Lowdin orthogonalisation and defining the matrix D :
S = D T D (3.4)
where S is the overlap matrix, the µ th , ν th element S µ,ν of the overlap matrix corresponds
to the integral over all space of the product of the µ th and ν th atomic orbitals. The non-
orthogonal orbitals C ′ loc are then orthogonalised by writing:
Cloc = DC ′ loc
(3.5)
To see why this is the case, consider the definition of the orthonormality condition for
the canonical molecular orbitals C ′ loc in a non-orthogonal basis set with overlap matrix S:
C ′ T
loc SC′ loc = 1 (3.6a)
C ′ T
loc DT DC ′ loc = 1 (3.6b)
C T
loc Cloc = 1 (3.6c)
showing that our new orbitals are orthonormal. In the case of DFT calculation the or-
thogonalisation procedure is carried out on all isolated molecule and system orbitals. We
can then apply equations 3.2 and 3.3 to the orthogonalised orbitals and obtain transfer
integrals and site energies.
3.2 Molecular Orbital Overlap Method for Calculation
of Transfer Integrals
The fast method we use to calculate transfer integrals is based on calculating the ZINDO
Fock matrix of a pair of molecules directly. We use an SCF ZINDO calculation to ob-
tain the isolated molecule’s molecular orbitals and wrote custom code to calculate atomic
overlaps and derive an approximate value for the transfer integral. Because this method is
based on calculating Fock matrix elements from atomic overlap, we dub this method and
57

the program we devised to carry it out molecular orbital overlap (MOO). The approxima-
tions made in MOO are discussed in this section.
We use the same naming convention for atomic orbitals which we have used in writing
equation 3.1. In this convention molecular orbitals localised on molecule A will have
zeroes as the first N/2 elements, whilst orbitals localised on molecule B will have zeroes
as their last N/2 elements. We are going to calculate transfer integrals between orbitals
localised on either molecule, therefore the only elements of the Fock matrix we need to
calculate are the off-diagonal blocks with one index running from 0 to N/2 and the other
running from N/2 to N. This is because the diagonal blocks of the Fock matrix will be
cancelled out by the rows of zeros present on either localised molecular orbitals. The
elements of the Fock matrix we need to calculate are those for atomic orbitals on different
atoms, therefore the form of these elements in the INDO approximation is [1]:
(βa
Fµν = ¯S
+ βb) γab
µν + Pµν
2 2
(3.7)
where ¯S represents the overlap matrix of atomic orbitals (with σ and π overlap between
p orbitals weighted differently, see below), a and b label the two atomic centres that the
µ and ν atomic orbitals are centred on, βa is the ionisation potential of atom a, Pµν labels
the density matrix and γ is the Mataga-Nashimoto potential. The approximation we make
in this calculation is to assume that Pµν is block diagonal, in the sense that all elements
entering equation 3.7 are equal to zero; the density matrix and Mataga-Nashimoto poten-
tial therefore do not contribute to the elements of the Fock matrix we are interested in
calculating. This assumption for the orbitals will hold either if the orbitals of the pair are
all localised or if each pair of orbitals can be written as a constructive/destructive combi-
nation of a corresponding pair of orbitals in the isolated molecule. To see why the latter is
the case, consider two particular doubly occupied orbitals ψ µ and ψ µ+1 which are formed
from bonding and anti-bonding combinations of two of the isolated molecule’s occupied
orbitals ψ Aν and ψ Bν . The contribution δP of these two orbitals to the density matrix will
be of the form:
δP = 2(ψ µ T ψ µ + ψ µ+1 T ψ µ+1 ) (3.8a)
58

= 2((ψ Aν + ψ Bν ) T (ψ Aν + ψ Bν ) + (ψ Aν − ψ Bν ) T (ψ Aν − ψ Bν )) (3.8b)
= 2(2ψ Aν T ψ Aν + 2ψ Bν T ψ Bν ) (3.8c)
Because all isolated molecule orbitals ψ Aν and ψ Bν are localised on one molecule only, this
contribution will be block diagonal. Furthermore, since all contributions to the density
matrix will be of this form, the density matrix will be overall block diagonal. Assuming
that the density matrix is block diagonal can be thought of as assuming that both the
bonding and antibonding combination of molecular orbitals are doubly occupied, that is
assuming that the two molecules are not covalently bonded to each other.
The task of determining values for the Fock matrix has therefore been reduced to
the comparatively simple task of determining ¯S , the weighted atomic orbital overlap.
Atomic orbital overlaps can be determined analytically using the expressions derived by
Mulliken [6] and the π and σ components of the p-orbital overlaps must be weighed by
the appropriate proportionality factors, in accordance with the scheme devised by Zerner
and coworkers [1].
The great advantage of this approach is that it requires only one SCF calculation to
be carried out to obtain the orbitals of the isolated molecule. All other quantities can be
derived simply from the relative geometry of the two molecules. Additionally, since the
heaviest computational task is computing the atomic overlaps, the s, σ and π overlaps can
be tabulated as a function of distance for the most common atom types encountered in
organic molecules, carbon and hydrogen, allowing a significant increase in computational
speed for a modest increase in memory usage.
3.3 Calculation of Site Energy Difference from Electro-
static Interactions
In this section we show how to determine the contribution to ∆E from classical electric
interactions only and compare the results with those from ab-initio calculation. The elec-
trostatic contribution will be determined from the electrical moments of two molecules
in their charged and neutral states. Assuming that the two molecules are identical, the
59

procedure to determine the electrostatic contribution to the site energy difference will be
as follows:
(a) distributed multipole analysis (DMA) is carried out for a charged and a neutral
molecule
(b) the two sets of distributed multipoles are rotated to match the orientations of molecule
A and molecule B
(c) equation 2.32 is used to calculate the interaction energies UA0B+ between the mul-
tipoles calculated for the neutral molecule in the orientation of molecule A and the
charged radical in the orientation of molecule B, and UA+B0 between the multipoles
calculated for the charged radical in the orientation A and the neutral molecule in
the orientation of B
(d) these two interaction energies are subtracted to obtain ∆E
If the calculation of the charged radical is performed using the orbitals of the neutral
molecule as a guess without optimising the orbitals, this corresponds to the frozen orbital
approximation and can be compared to the results from the projective method.
The only aspect we have not already discussed in this scheme is the rotation of elec-
trical multipoles. Charges are clearly independent of coordinate transformations, dipoles
transform as vectors and the elements of the quadrupole moment Qαβ will transform as
rαrβ, that is Q behaves as a rank two tensor:
Q ′ αβ =
� �
γ
δ
RγαRδβQγδ
(3.9)
The multipole expansion is truncated so that the highest moments included are quadrupoles
because in conjugated materials the π system creates two layers of negative charge sand-
wiching the positive nuclei. This leads to a large quadrupole moment which we expect
to be important in interactions between π systems; for example, it is known that simple
models of π stacking can be made which treat molecules as a collection of quadrupoles
[7].
60

In the following sections we carry out this procedure for two molecules: ethylene
and pentacene. Ethylene is chosen because it is small enough to permit substituting each
molecule with a single set of multipoles even at relatively small intermolecular distances.
The projective method will be used on systems with small intermolecular separations,
because under these conditions differences in site energy are larger than the error in the
SCF calculations. In the first subsection, we perform DFT calculations on neutral pairs
of ethylene molecules and show that electrostatics can be used to calculate the difference
in site energies between the localised HOMOs on the two molecules. We also show
that the small discrepancy between the density functional and electrostatic results can
be accounted for by including the induction energy. In the second part we address the
question of how the presence of charge would modify the induced dipoles and ∆E in
pairs of ethylene molecules. The nature of the site energy difference and the role of
asymmetry in the properties that determine it will also be discussed. In the third part
we will present a comparison of ∆E calculated from projection of MOs calculated from
pure DFT results with ∆E calculated from the electrostatic interaction between pairs of
pentacene molecules. In this case the size of the molecules is large compared to their
separation and therefore we must use distributed multipole analysis (DMA) to calculate
the electrostatic interaction, as discussed in section 2.4.1.
3.3.1 Ethylene: Neutral Systems
In this section we calculate site energy difference for the HOMO of neutral pairs of ethy-
lene molecules. The aim is to show that both projection of DFT MOs and classical calcu-
lations of electric interactions lead to similar values of the site energy difference. Ethylene
is chosen as an example of a molecule small enough that it can be substituted entirely by
a single set of electrical moments. This has the advantage that the multipole moments can
be obtained exactly from a SCF calculation on an isolated molecule, without the need for
the somewhat arbitrary partitioning schemes necessary to perform DMA. A represention
of the pair of ethylene molecules which we consider is shown in figure 3.1; this figure
also shows the x axis, around which one of the molecules will be rotated to generate the
geometries in which we will calculate ∆E. The two molecules are at centre of mass sep-
61

Figure 3.1: A pair of ethylene molecules at a separation of 5 Å . Also shown is the axis
around which one of the two molecules will be rotated to generate the geometries
for which ∆E is calculated.
aration of 5 Å . This large separation justifies the use of a single electrical moment to
describe each molecule: the neutral molecule will be described by a point quadrupole, the
charged one by a point charge. In the remainder of this chapter we refer to point charges
as monopoles to keep the terminology consistent.
The density functional used to determine the value of the quadrupole moment of ethy-
lene of was PW91, the basis set employed was a triple zeta basis set with extra polari-
sation basis sets (TVZP) and all quantum chemical calculations were performed in the
GAMESS-UK [8] suite. In atomic units (units of ea 2
0 , where a0 is the Bohr radius) the
non-zero values of the quadrupole moment are:
Qxx = 1.51 (3.10)
Qyy = 1.34
Qzz = −2.85
�
|Q| = Q2 xx + Q2 yy + Q2 zz = 3.49
where the x axis is defined along the C=C double bond and the z axis is perpendicu-
lar to the plane of the molecule. Consideration of these three numbers shows that the
quadrupole of the molecule is essentially a linear quadrupole aligned in the z direction.
62

site energy difference / eV
0.2
0.15
0.1
0.05
PW91 ∆E
Quadrupole-Monopole ∆E
0
0 50 100
rotation angle / degrees
150 200
Figure 3.2: Site energy difference for a pair of neutral ethylene molecules as a function of
rotation of one of the molecule about the the C=C bond calculated by projection
of PW91 MOs (circles) and calculated from the difference in quadrupolemonopole
interactions calculated classically from the electrical moments calculated
at the PW91 level on an isolated ethylene molecule (solid line).
A linear quadrupole corresponds to two dipole moments emanating from the same point,
but pointing in opposite directions. We can therefore think of ethylene as having a partial
positive charge in the plane of the atoms, sandwiched between two negative charge clouds
caused by the π electrons.
In figure 3.2 we compare the site energy difference computed using the projective
method applied to MOs from a PW91/TVZP DFT calculation with that obtained from a
classical calculation based on treating the neutral molecule as point quadrupole and the
charged molecule as a point charge. It is immediately clear that the two distributions
follow the same form and are indeed rather close in value: the greatest disagreement is
roughly 10%. Also we notice that, having excluded the inductive term, we have overes-
timated the value of the site energy difference. Let us compare this last point to a recent
study by E.M. Valeev and coworkers [9]. In that study, calculations of ∆E are carried out
for pairs of ethylene molecules and for an isolated ethylene molecule in the presence of
63

point charges on each atom chosen to fit the electrostatic potential of the molecule as best
as possible. The aim was to establish whether ∆E is electrostatic in origin and can there-
fore be reproduced by using point charges. The fact that their calculation underestimates
∆E, can be explained by the neglect of the most important electrical moment of the charge
distribution of ethylene: the zz element of the quadrupole moment. In fact, it is impos-
sible for 6 point charges in the xy plane to replicate the zz component of the quadrupole
moment, which is due to sheets of negative charge being present above and below of the
xy plane.
In order to establish whether inductive interactions are responsible for the discrep-
ancy in the two curves in figure 3.2, we now compute the polarisability of an ethylene
molecule and use it to compute the induced dipole on each molecule. We recalculate ∆E
taking these induced dipoles into account. PW91 calculations are run on neutral systems,
therefore we must calculate the dipoles induced by the electrostatic moments of a neutral
molecule. For this reason, we calculate the quadrupole induced dipoles, then repeat the
procedure iteratively until the induced dipoles converge. The convergence criterion we
used was when the root mean square difference of the n th and (n + 1) th iteration of the
calculated induced dipole should be smaller than 10 − 10ea0. As mentioned in subsection
2.4.2, convergence should be very rapid. In fact, less than 10 cycles were always suffi-
cient. In atomic units (units of a3 0 ) the non-zero elements of the polarisability calculated
at the PW91 level are:
Pxx = 34.80 (3.11)
Pyy = 21.82
Pzz = 13.93
Using these values for the polarisability, we were able to find the correction to the
quadrupole monopole interaction energy. Comparing the results thus obtained with those
obtained by the projective method yields figure 3.3: the two curves are now much closer.
The main purpose of this exercise was to show that using simple classical methods,
we can obtain results which are very close to those obtained from a SCF calculation on
64

site energy difference / eV
0.15
0.1
0.05
Quadrupole+Induced Dipole
PW91 ∆E
0
0 50 100
rotation angle / degrees
150
Figure 3.3: Site energy difference for a pair of neutral ethylene molecules as a function of
rotation of one of the molecules about the the C=C bond calculated by projection
of PW91 MOs (circles) and calculated from the difference in quadrupolemonopole
interactions corrected by the quadrupole induced dipoles (solid
lines). Polarisabilities and electrical moments were calculated at the PW91
level.
65

a pair of molecules, whilst performing a single SCF calculation on an isolated molecule
only. The next section will be devoted to making the calculations somewhat more relevant
to the case of charge transport by considering a charged pair of molecules.
3.3.2 Ethylene: Charged Systems
In this section we intend to calculate ∆E for charged pairs of molecules. It would be
difficult and beyond the scope of our investigation to run DFT calculations on charged
systems while ensuring the localisation of a charge on one molecule only, therefore we
will only calculate the site energy difference from electrostatic and inductive interactions.
In particular, we ignore the change in quadrupole moment upon charging and only look at
the change in induced dipole in going from a neutral pair of molecules (where the dipole
is induced by a quadrupole) to a charged pair of molecule (where the dipole is induced by
a quadrupole and a monopole).
We imagine that two effects might come into play upon charging a molecule in a pair:
• charging a molecule will modify its electrical moments
• a point charge will induce a much larger dipole than a point quadrupole
We ignore the first point, as this is expected to be a small effect. For ethylene the lead-
ing term in the electrostatic interaction is unchanged as we do not expect any intramolec-
ular charge transfer to occur upon charging; the second largest electrical moment after
the monopole will still be the quadrupole. We also ignore the small change in quadrupole
moment.
Regarding the second point, putting a monopole on one of the two molecules is ex-
pected to have a strong influence on the value of the induced dipole, and thereby on the
contribution of the induced dipole to ∆E.
In order to test this hypothesis, we consider two ethylene molecules with their π sys-
tems perpendicular and consider the separation dependance of the various induced-dipole
monopole interactions. We focus on the point in figure 3.3 where ∆E is the greatest in
neutral molecules (θ = 90 ◦ ). We compare the induced dipole monopole interaction in
neutral and charged systems, calculated using classical arguments. In the neutral system,
66

Energy / eV
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
positive pair
neutral pair
4 6 8 10 12
Separation / A
Figure 3.4: Distance dependence of the contribution to ∆E from the induced dipole in a
pair of ethylene molecules as a function of separation. The solid line corresponds
to the contribution from the dipole moment induced by a quadrupole
(i.e. in a neutral system), whereas the dotted line corresponds to that from the
dipole moment induced by a monopole (i.e. in a charged system).
67

Figure 3.5: Diagram showing the dipoles induced in two ethylene molecules oriented at
right angles to each other. (a) Case of point quadrupoles (neutral calculation).
(b) Case of point charges (charge calculation). For discussion, please see the
text.
the induced dipole is induced by a quadrupole. In the charged system the induced dipole
is induced both by a monopole and a quadrupole. Figure 3.4 shows that, contrary to
our expectations, the contribution to ∆E of dipoles induced in a neutral and in a charged
system are not very different in magnitude. This is because it is not the magnitude of
the molecules’ electric fields and their polarisabilities which determines the magnitude of
∆E, but their asymmetry. The reason for this is rather subtle and is best explained with
the aid of a diagram, which is shown in figure 3.5. The sign and magnitude of the induced
dipole for perpendicular ethylene molecules is represented by the direction and width of
arrows. Two cases are considered: a) shows the dipoles induced by point quadrupoles
(corresponding to a neutral system) and b) shows the dipole induced by point charges
(corresponding to a charged system).
Let us consider case a). The quadrupole of ethylene is depicted as a linear quadrupole,
which is represented by a circle with a ”+” surrounded by two circles with ”-”s. The shape
to the left represents an ethylene molecule with the molecular plane lying on the xy plane
and therefore its π electrons are in two sheets above and below the plane of the page; the
68

ethylene molecule to the right is rotated by 90 ◦ and has its π electrons lying to the left and
to the right of the figure. The molecule to the left has the larger dipole, represented by a
wider arrow, for two reasons:
• the field caused by the molecule to the right is larger
• the polarisability Pyy is larger than Pzz
The most important aspect about this part of the diagram is that both dipoles are lying
in the same direction: therefore their contributions to the components UA0B+ and UA+B0
have opposite signs and, when subtracted to obtain ∆E, would increase the site energy
difference by the sum of their magnitudes.
Let us compare this case to that represented in b). In this case the two dipoles act in
opposite directions. Therefore ∆E would be increased only by the difference in magni-
tudes of the contributions from the dipole interactions to UA0B+ and UA+B0. There will still
be a net contribution to ∆E, because the two components of the polarisability are different
for the two different orientations and hence the molecule to the left will have the larger
induced dipole.
In order to demonstrate this effect, in figure 3.6 we plot the dipoles induced on each
molecule in a neutral and in a positive ethylene system as function of separation. This
figure shows that the dipoles induced by monopoles are indeed larger but act in opposite
direction, whereas the dipoles induced by quadrupoles are smaller but act in the same
direction. In our calculation for the charged system, we include the inductive effects of
both the point charge and the point quadrupole and therefore we observe a sum of these
two effects. Our explanation why the induced dipole has a slightly smaller effect on ∆E
in the charged system than in the neutral system is as follows:
• the monopole induced dipoles contribute to ∆E only because of the asymmetry in
the polarisability
• the quadrupole induced dipole is asymmetric because of the asymmetry in the field
of the differently oriented molecules
69

Figure 3.6: Dipoles induced on each molecule in a pair of neutral ethylene molecules
(upper pannel) and in a positively charged pair (lower panel). Notice that, as
shown in figure 3.5, in a charged system the induced dipoles are larger than in
the neutral case, but are in opposite directions.
• for positively charged systems, the quadrupole induced dipole actually diminishes
the asymmetry in the monopole induced dipoles and reduces the overall contribu-
tion to ∆E, making it similar in magnitude to the quadrupole induced dipole.
The lesson from this investigation is that one has to take great care in predicting the
inductive contributions to ∆E, because the site energy difference is determined not only by
the magnitude of the interactions involved but also, most importantly, by their asymmetry.
3.3.3 Pentacene
In this section we will compare the site energy difference calculated using distributed mul-
tipole analysis for pairs of pentacene molecules with that by the projective method applied
to PW91/TVZP calculations. Pentacene is chosen as a typical conjugated molecule that
has been studied in the literature [11]. The projection of the DFT MOs is carried out
exactly as in the case of ethylene: namely calculations of the overlap matrix, the orbitals
of the pair of molecules and of each molecule separately are carried out in GAMESS-
UK; the orbitals are then orthogonalized and projected; finally the Kohn-Sham matrix is
recreated using the mathematical package Octave with some custom written scripts. The
70

Figure 3.7: Two cofacial pentacene molecules and the axis around which one will be rotated
and translated. The inset shows the chemical formula of pentacene.
calculation of the isolated molecule’s electrical moments is carried out using Gaussian ’03
with the same functional and basis set specified in GAMESS-UK; the DMA is then car-
ried out by the helper program gdma [10]. The calculations on a cation were carried out
using the guess wavefunction from the calculation on the neutral molecule and without
carrying out the SCF procedure, in an effort to replicate the frozen orbital approximation
assumed in the projective method. It was necessary to carry out the DMA in gdma, rather
than using the values given by GAMESS-UK because the latter gave multipoles which
did not reflect the symmetry of the molecule.
The geometry of the pair of pentacenes considered is shown in figure 3.7: the two
pentacene molecules are kept at a distance of minimum approach of 3.5 Å, and one of
them is rotated along its long axis. The results for ∆E as a function of rotation angle are
shown in figure 3.8. This figure shows how the classical DMA results are in relatively
good agreement with the DFT results, and how including only the monopole interactions
accounts for only half of the electrostatic interaction energy calculated using DMA. It
also shows that the DMA results somewhat underestimate the PW91 results; this is a
rather surprising result as we would expect that including inductive interactions would
further lower the electrostatically calculated ∆E. Three possible reasons could explain this
discrepancy: using Gaussian instead of GAMESS-UK might mean we are using slightly
71

Figure 3.8: ∆E for pairs of pentacene molecules, calculated using projection of the PW91
results (red points) and electrostatic results (diamonds). The electrostatic results
are calculated using monopole monopole interactions only (squares) and
using all interactions up to quadrupole-quadrupole (circles).
different implementations of the functional; in fact, Gaussian calculates a total energy
for a molecule of pentacene which is 57 meV smaller than GAMESS-UK, demonstrating
the two programs are not exactly identical. It is also possible that the multipole expansion
does not actually reproduce the field of the molecule predicted by PW91, either because of
the method gdma uses to partition the molecule into atoms that the moments are integrated
over or because the expansion is truncated at quadrupoles. It should be noted, however,
that the agreement between the DFT results and the classical results is still rather good:
the relative difference between the two method is approximately 10%, for an angle of 90 ◦ .
The conclusion of the past three sections is that most of the site energy difference can
be explained in terms of the electrostatic interaction between two molecules, which is
best calculated using distributed multipole analysis. The great advantage of electrostatic
interaction is that they are pairwise additive. Therefore it is easy to go from describing
site energies in pairs of molecules to describing them in large assemblies of molecules by
adding the contributions to the energy from all the pairs in the assembly.
The role of inductive interactions is still difficult to identify exactly. So far we have
been able to perform these calculations only for molecules at large separation, as we have
not implemented any method to calculate polarisabilities in non-uniform electric fields.
72

For ethylene pairs we found that the contributions to ∆E from inductive interactions is
much smaller than from the electrostatic interactions. Also, it is not strongly dependent
on whether we consider the dipole induced in neutral pairs or in charged ones, although
this observation might be specific to the system considered. Another complication is that
inductive interactions are not at all additive, therefore the self-consistent procedure used
to determine interaction for pairs of molecules would have to be carried out for the whole
assembly of molecules.
3.4 Comparison of Different Methods of Calculating Trans-
fer Integrals
In this section we will compare the three methods described to calculate the transfer in-
tegral J: splitting of frontier orbital energies, projective method and MOO. The aim is to
show that the use of MOO is well justified when compared to both SCF and more ab-initio
methods. We will use three different molecules to carry out these comparisons: pentacene,
hexabenzocoronene and ethylene. Ethylene is included because the previous calculations
of ∆E furnished us with the information necessary to study the basis set dependence of
transfer integrals.
3.4.1 Pentacene
In this section, pentacene will be used to compare the orbital splitting method with the
projective method. We will also show that the projective method can be used with either
a ZINDO Hamiltonian or a PW91 functional to yield very similar results. The geome-
tries we will investigate are represented in figure 3.7. We consider the cases where one
molecule is translated along the long axis of the pentacene molecule, which we label the
x-axis, and where one molecule is rotated around that axis.
Because of the D2h symmetry of pentacene, a pair of two parallel molecules with
one molecule translated along the x axis possesses a point of inversion: that is, there is
a unitary transformation which brings one molecule onto the other. This means that the
site energies EA and EB in equation 2.19 must be the same: we should therefore be in
73

Figure 3.9: Absolute value of the hole transfer integral as a function of displacement along
the x axis of one of the two pentacene molecules calculated by projection of
ZINDO orbitals (circles) and by the orbital splitting model (triangles). The
length of the molecule is 14.1 Å .
Figure 3.10: HOMO for pentacene calculated at the ZINDO level.
74

Figure 3.11: Absolute value of the transfer integral for a pair of pentacene molecules as a
function of rotation around the x axis. The transfer integral is calculated by
projection of the ZINDO orbitals (circles) and by the orbital splitting method
(triangles).
the regime when the projective and orbital splitting methods give the same results. The
hole transfer integral as calculated by each method is shown in figure 3.9. The origin
for the oscillations in |J| with displacement in the x direction can be understood in terms
of orbital overlap. As the two molecules are translated over each other, the nodes and
antinodes of the HOMOs on each molecule pass over each other; if the nodes of the
HOMOs of one molecule are over the nodes, the contributions add and one obtains a
large transfer integral. If the nodes are over the antinodes the contributions cancel out and
the transfer integral vanishes [12]. This explanation is consistent with the period of the
oscillation, which is approximately the same as the width of a benzene ring in pentacene,
and with the shape of the HOMO of pentacene - shown in figure 3.10. The reason why
the peak amplitude is decreasing with displacement is because of the diminished overlap
between the two molecules. It is evident from figure 3.9 that, for this symmetric geometry,
the projective and splitting methods are in very close agreement.
Now let us look at the two pentacene molecules separated by 4.5 Å where one molecule
is rotated relative to the other, as shown in figure 3.7. For any angle different from zero
there will be no unitary transformation that moves the atoms of one molecule onto the
other; therefore symmetry will not ensure that the site energies will be the same. Further-
75

Transfer integrals / eV
0.06
0.05
0.04
0.03
0.02
0.01
sto-3g
DVZP
TVZP
CC-PVQZ
0
0 50 100
Rotation angle / degrees
150 200
Figure 3.12: Absolute value of the transfer integral for a pair of ethylene molecules as a
function of rotation angle. All curves were obtained by projective method
applied to PW91 calculations; the different curves are obtained by using
different basis sets.
more the electric field of pentacene is asymmetric and as we discussed in the previous
section this will lead to a difference in site energies. In this regime we do not expect the
two methods to agree and in fact they do not, as shown in figure 3.11. The results from
the projective method are sensible: the transfer integral falls to zero as the π systems on
the two molecules become orthogonal. The orbital splitting method, conversely, suggests
incorrectly that the transfer integral would increase at large angles. The discrepancy is
explained by the fact that at larger angles the site energy difference increases.
Having confirmed the accuracy of calculating |J| using the projective method, we now
address the accuracy of results obtained by the projective method applied to ZINDO re-
sults and by the projective method applied to PW91 calculations. We made a preliminary
study on the pairs of ethylene molecules as a function of rotation angle using the pure DFT
functional PW91 and studying the dependence of the transfer integral on the basis set. A
strong dependence is expected because the transfer integral is intimately connected to or-
bital overlap, and therefore the ability of a basis set to describe charge density far from
76

Figure 3.13: Absolute value of the transfer integral as a function of displacement along
the x axis of one of the two pentacene molecules. The two curves show
the results from the projective method using the PW91 functional and the
ZINDO Hamiltonian. The PW91 calculation were performed with a TVZP
basis set.
a molecule is important. We compare four different basis sets: minimal Slater type or-
bital (sto-3g), a double and a triple zeta basis set with extra polarisation functions (DVZP
and TVZP) and finally a huge correlation-corrected quintuple zeta set with extra polari-
sation basis sets (cc-QVZP). The results of this study show that a TVZP set is sufficient
to describe transfer integrals appropriately: as shown in figure 3.12, both the TVZP and
the cc-QVZP sets give similar results. Having established this, we compare the results
for the case of pentacene molecules with relative displacement using the ZINDO and
PW91/TVZP model chemistries. The results from this comparison are shown in figure
3.13. We can see that the two curves are very similar, justifying the use of ZINDO. It is
pretty astonishing that the ZINDO Hamiltonian, with its minimal basis set, produces such
good agreement with the results from a much more time consuming PW91 calculation! It
should also been noted that these results are at odds with the investigation of Huang and
coworkers [13] who find significant discrepancies in the predicted value of J using PW91
and ZINDO methods for pairs of ethylene molecules. It is possible that these differences
are less pronounced for larger molecules than for smaller oleculas such as ethylene.
In this section we have validated the use of the projective method by comparison to
77

Figure 3.14: Two cofacial HBC molecules and the cartesian axis used in the text. The
inset shows the chemical formula of HBC.
the orbital splitting method when the site energies of two molecules are the same. When
site energies are different, the projective method gives sensible results. Finally, we show
that the projective method can be applied both to ZINDO and PW91 results and, in both
cases, the transfer integral obtained is similar: for the rest of the thesis we will use the
ZINDO method because of its greater computational efficiency.
3.4.2 Hexabenzocoronene
In this section we compare transfer integrals calculated using the projective and MOO
methods, using hexabenzocoronene (HBC) as an example molecule. The structure of two
cofacial molecules and the definition of the axes are shown in figure 3.14.
Before showing the comparisons of the projective and MOO methods, we briefly re-
view some of the literature on HBC. HBC has been studied in the literature by Lemaur
and co-workers [14]: in that work, HBC and other discotic liquid crystals were compared
by calculating at the relative values of the reorganisation energy and the dependence of
the transfer integral on rotation about the z axis. The rotational potential energy for ro-
tation about the z axis was obtained from molecular mechanics calculations on pairs of
molecules and an expression for the relative mobility in the various compounds was ob-
tained using a mean field treatment. We would like to expand on this work by considering
78

Figure 3.15: Contour plots of the effective transfer integral as a function of rotation about
the x and z axis. The molecules are first rotated about the z axis, then rotated
about the x axis in such a way as to keep the minimum distance of approach
between the molecules equal to 3.5 Å .
the dependence of the transfer integral on the degrees of freedom neglected in this study,
rotation about the x axis and slip in the x-y plane. In section 7.2 we combine calculation of
transfer integrals with accurate atomistic molecular dynamics modelling of entire stacks
of HBC. It is in order to carry out this modelling that we aim to show the projective and
MOO methods to be equivalent. Since the HOMO of HBC is doubly degenerate, in this
section and in 7.2 we use an effective transfer integral Je f f , as defined in equation 2.18,
to determine the charge transfer rates. In reference [14], instead of the effective transfer
integral the splitting between the degenerate HOMOs of a pair of HBC molecules is taken
as the electronic coupling. In this section we explain why, for certain relative orientations,
this approach is equivalent to ours except for a difference in a constant scaling factor.
We wish to carry out two comparisons of the effective transfer integral: as a function
of x-y displacement and as a function of rotation about the z axis and about the x axis.
First we show the results for rotation about the x and z axes. The relative geometry of the
molecules is obtained by starting with two identical molecules A and B: first molecule
B is rotated about the z axis, then it is rotated about the x axis, finally molecule B is
translated along the z axis by as much as is necessary to make the minimum distance of
approach equal to 3.5 Å. The results for the effective transfer integral Je f f are shown in
figure 3.15. It can immediately be seen that both methods predict very similar results. We
have chosen to limit the z rotation to between 0 ◦ and 180 ◦ in order to show that the transfer
integral varies with a period of 120 ◦ , as suggested by the S6 symmetry of the molecule. If
we had constrained the HBC molecule to a planar geometry, the overall symmetry would
have been D6h and the period of Je f f would have been 60 ◦ . Rotations about the x axis are
79

Transfer integral / eV
0.4
0.2
0
-0.2
MOO J (HOMO HOMO)
MOO J (HOMO HOMO-1)
MOO J (HOMO-1 HOMO)
MOO J (HOMO-1 HOMO-1)
PRO Jeff MOO J
eff
-0.4
0 50 100
Rotation angle / degrees
150 200
Figure 3.16: Effective transfer integral and its components as a function of rotation about
the z axis calculated using the projective (PRO) and MOO (MOO) methods.
The molecules are at a distance of 3.5 Å .
limited to ±30 ◦ because atomistic modelling of various HBC derivatives has suggested
that this is the average tilt in the HBC derivatives with most disorder[15]. Figure 3.16
also shows that, for a particular rotational angle about the x axis, rotation about the z axis
can cause a variation in the transfer integral of roughly a factor of 6. In contrast figure
3.15 shows that a rotation of 30 ◦ about the x axis produces a variation in transfer integral
of a factor of 20, because of the greater variation in π system overlap. In a discotic liquid
crystal such as HBC we expect that rotations about the x axis should be small, therefore
we expect that the reduction in order caused by increasing the temperature or by having
shorter side chain lengths will have drastic effects on the charge transport characteristics.
It is interesting at this point to compare these results with the results calculated by
Lemaur and co-workers [14]. In that work, the transfer integral was calculated from the
splitting between the top two orbitals of the pair of molecules and the next two. Sur-
prisingly, this treatment produced similar results for the dependence of J on z rotation as
those from equation 2.18 shown in figure 3.16. This can be explained by considering the
values of the transfer integral as a function of rotation about the z axis. These results are
shown in figure 3.16 for the MOO method, with the results for Je f f calculated from the
projective method shown for comparison. This figure clearly shows how - for this geome-
try - JHOMO,HOMO is equal to JHOMO−1,HOMO−1 and JHOMO,HOMO−1 is equal to JHOMO−1,HOMO,
as expected for a system with C3 symmetry such as these systems [2]. Therefore, using
80

similar arguments to those used for systems with non-degenerate orbitals, we can write
the matrix governing the orbitals of the pair of molecules as:
F =
E 0 JHOMO,HOMO JHOMO,HOMO−1
0 E −JHOMO,HOMO−1 JHOMO,HOMO
JHOMO,HOMO −JHOMO,HOMO−1 E 0
JHOMO,HOMO−1 JHOMO,HOMO 0 E
(3.12)
Diagonalising the Fock matrix F, we obtain two sets of doubly degenerate eigenvalues
for the pair of molecules, with energies corresponding to:
E± = E ±
�
J2 homo,homo + J2
homo,homo−1
(3.13)
This explains why the difference between the splitting from a pair of molecules re-
ported by Lemaur in [14] and the values shown in figure 3.16 is a factor of 2 √ (2). In the
case when JHOMO,HOMO is different from JHOMO−1,HOMO−1 and JHOMO−1,HOMO is different
from −JHOMO,HOMO−1, for example if x rotations are included, equation 3.13 does not hold
anymore and it is not straightforward to deduce the transfer integral from the splitting of
the frontier orbitals, especially if polarisation makes the diagonal terms of the 4x4 matrix
from equation 3.12 different. Therefore considering the splitting of the orbitals seems
valid only in the specific cases where the pair of molecules has a particular symmetry.
It is also interesting to look at figure 3.16 and note how neither the JHOMO,HOMO nor
the JHOMO−1,HOMO possess the S6 symmetry of the molecule, this gives a graphic explana-
tion of why - if Jahn-Teller distortions are ignored - it is necessary to look at the effective
transfer integral in order to respect the symmetry of the system analysed. The degener-
acy of the frontier orbitals is not accidental, but reflects the symmetry properties of the
molecules involved.
Finally let us look briefly at displacements in the x-y direction. A comparison of the
results for the projective and MOO methods is shown in figure 3.17. Again it can be seen
that the two contour plots are very similar. We do not expect HBC molecules in a discotic
liquid phase to be able to slip freely in the x-y direction, as later movements should
81

Figure 3.17: Countour plots of the dependence on x-y displacement of two molecules of
HBC. The two molecules are at a distance of 3.5 Å. The maximum radius of
the molecule is 6.8 Å.
be restricted by the interactions with the surrounding columns. However a variation in
transfer integral of a factor of two is obtained by displacements of less than 2 Å.
To conclude this section we have shown how the MOO method adequately reproduces
results from the projective method applied to the ZINDO Hamiltonian. This is at a consid-
erable computational advantage, since all calculations on pairs of HBC molecules require
only a single ZINDO calculation to be performed on an isolated HBC molecules, whereas
the projective method requires a SCF calculation for each pair considered. We have also
shown that tilting of the disks will has a tremendous effect on transfer integral and have
given a graphical demonstration of the significance of molecular orbital degeneracy in the
calculation of transfer integrals.
3.5 Conclusion
In this chapter we have presented fast methods which allow the calculation of J and ∆E
for very large numebers of relative orientations and positions of molecules. The method
to calculate ∆E is based on the classical treatment of electrostatic interaction discussed
in chapter two, whereas the method to calculate J is based on an approximation of the
ZINDO Hamiltonian. The site energy differences thus calculated were found to be in
agreement with site energy differences deduced from projection of DFT calculations,
whereas the MOO method for calculating J was found to be in agreement with the pro-
jective method applied to ZINDO claculations.
82

Bibliography
[1] J. Ridley and M. Zerner, An Intermediate Neglect of Differential Overlap Technique
for Spectroscopy: Pyrrole and the Azines, Theoretica Chimica Acta, 32, 111 (1973)
[2] K. Senthilkumar, F.C. Groezema, F.M. Bickelhaupt and L.D.A. Siebbeles, Charge
transport in columnar stacked triphenylenes: Effects of conformational fluctuations
on charge transfer integrals and site energies, Journal of Chemical Physics, 32, 9809
(2003)
[3] G. te Velde, F. M. Bickelhaupt, E. J. Baerends, C. Fonseca Guerra, S.J.A. Van Gis-
bergen, J.G. Snijders, and T. Ziegler, Chemistry with ADF, Journal Computational
Chemistry, 22, 931 (2001).
[4] L.D. Landau and E.M. Lifshitz, Quantum Mechanics non-relativistic theory, Perga-
mon Press (1965)
[5] P.O. Lowdin, On the Non-Orthogonality Problem Connected with the Use of Atomic
Wave Functions in the Theory of Molecules and Crystals, Journal of Chemical Physics,
18, 365 (1950)
[6] R.S. Mulliken, C.A. Reike, D. Orloff and H. Orloff, Formulas and numerical table
for overlap integrals, Journal of Chemical Physics, 17, 1248 (1949)
[7] C.A. Hunter and J.K.M. Sanders, The nature of π − π interactions, Journal of the
American Chemical Society, 112, 5525 (1990)
[8] M.F. Guest, I.J. Bush, H.J.J. van Dam, P. Sherwood, J.M.H. Thomas, J.H. van Lenthe,
R.W.A. Havenith and J. Kendrick, The GAMESS-UK electronic structure package:
algorithms, developments and applications, Molecular Physics, 103, 719 (2005)
83

[9] E.F. Valeev, V. Coropceanu, D.A. da Silva Filho, S. Salman, and J.L. Brédas, Ef-
fect of Electronic Polarisation on Charge-Transport Parameters in Molecular Organic
Semiconductors, Journal of the American Chemical Society, 128, 9882 (2006)
[10] A.J. Stone, Journal of chemical theory and computation, Distributed multipole anal-
ysis: Stability for large basis sets, 1, 1128 (2005)
[11] O. Kwon, V. Coropceanu, N.E. Gruhn, J.C. Durivage, J.G. Laquindanum, H.E. Katz,
J. Cornil and J.L. Brédas, Characterisation of the molecular parameters determining
charge transport in anthradithiophene, Journal of Chemical Physics 120, 8186 (2004)
[12] J.L. Brédas, J.P. Calbert, D. A. da Silva Filho and J. Cornil, Organic semiconductors:
A theoretical characterisation of the basic parameters governing charge transport,
Proceedings of the National Academy of Science 59, 5804(2002)
[13] J. Huang, M. Kertesz, Validation of intermolecular transfer integral and band-
width calculations for organic molecular materials, Journal of Chemical Physics, 122,
234707 (2005)
[14] V. Lemaur, D.A. da Silva Filho, V. Coropcenau, M. Lehmann, Y. Geerts, J. Piris,
M.G. Debije, A.M. van der Craats, K. Senthilkumar, L.D.A. Siebbeles, J.M. Warman,
J.L. Brédas and J. Cornil, Journal of the American Chemical Society, Charge Transport
Properties in Discotic Liquid Crystals: A Quantum-Chemical Insight into Structure-
Property Relationships, 126126, 3271 (2004)
[15] D. Andrienko, V. Marcon, K. Kremer, Atomistic simulation of structure and dy-
namics of columnar phases of hexabenzocoronene derivatives, Journal of Chemical
Physics, 125, 124902 (2006)
84

Chapter 4
Reorganisation Energy in
PolyPhenylenevinylene: Polaron
Localisation
In the previous chapters, we have described ways of calculating the parameters of the
Marcus electron transfer equation in terms of single molecule properties. We discussed
how the reorganisation energy can be calculated from the relaxation energy of radical
cations (or anions) for single molecules and how the site energy difference and transfer
integral can be computed from the diagonal and off-diagonal elements of the Fock matrix
written in the localised basis set of the molecular orbitals of each molecule. A problem
arises when we want to extend this treatment to polymers: what segment length should
be considered as the charge transporting unit? The oligomer segment length considered
is important as one would expect that increasing its length would decrease the ionisa-
tion potential and increase the electron affinity. In analogy to a potential well, if the
well is made wider the energy levels become shallower. In this chapter we will consider
electron-phonon coupling as a process which can limit the localisation of charge in a poly-
mer, thereby creating an effective chain length beyond which charged radical excitations
do not spread. The polymer we take as an example is polyparaphenylenevinylene (PPV),
shown in figure 4.1. The figure also shows the labelling convention of the phenylene units
and vinylene units and of atoms we use in the rest of the chapter. In particular, we will
look at two possible models for polarons: those predicted by semi-empirical calculations
85

Figure 4.1: Two monomers of PPV and the scheme of the labels used in the rest of the
chapter: ph1...phn will label the phenylene units and vi1...vi2 will label the
vinylene units. The first monomer also has the carbons with bonds to hydrogen
labeled 1 to 6.
and those predicted by hybrid density functional methods. We will show that even though
the spin density distributions obtained from DFT calculations are more delocalised than
those obtained from semi-empirical methods, neither method is in disagreement with the
spin distribution observed by electron nuclear double resonance (ENDOR) experimental
spectra on PPV. We will then discuss the influence of torsional fluctuations on polaron lo-
calisation to show how the charge transporting units are liable to be affected by fluctuation
in torsional angles small enough to be easily accessible at room temperature.
4.1 Introduction
A process that can limit the spatial extent of a charge on a polymer chain is polaron
self-localisation, that is an interaction of a charge with the polymer backbone such that
the charged excitation and the chain deformation localise. The term polaron was origi-
nally used in the context of describing quasi-particles created by the interaction between
lattice vibrations (phonons) and electrons in inorganic polar crystals. In the case of or-
ganic solids, there is evidence for the formation of polarons in stretched films of poly
para phenylenevinylene (PPV) from electron nuclear double resonance spectroscopy (EN-
DOR) measurements on PPV crystals [1, 2, 3]. This technique measures the hyperfine
interaction between carbon and hydrogen atoms and is therefore very sensitive to the ab-
solute value of the spin density on the carbon atoms bonded to hydrogen atoms. The
ENDOR spectrum consists of a series of resonance peaks measured at different magnetic
86

Figure 4.2: Experimental and modelled ENDOR spectra of PPV with an applied magnetic
field parallel and perpendicular to the polymer chain axis. The figure is from
reference [2].
fields, as shown in the lowest panel of figure 4.2. The position of each peak can be deter-
mined by the equation:
ν± =
� �
i=x,y,z
(ν0 ± Ai) 2 p 2 i
(4.1)
where ν0 is the free proton frequency, ± labels the shift on either branch of the ENDOR
spectra, pi is the i th component of the magnetic field and finally Ai is the i th component
of the hyperfine coupling tensor, which is proportional to the absolute value of the spin
density on the carbon atom multiplied by a geometry dependent proportionality factor.
This expression is derived and discussed in reference [4]. Figure 4.2 shows three dis-
tinct ENDOR peaks observed experimentally, therefore because of equation 4.1 the spin
densities of all carbon atoms bonded to hydrogens must take three distinct values. In ref-
erence [2], Shimoi and co-workers find that such spin density values can be obtained by
using a Pariser-Parr-Pople (PPP) Hamiltonian, which takes electron-phonon coupling as
a parameter. The authors find that the parameters which allow the experimental data to
be fitted also lead to localisation of the charged excitation on a segment of roughly four
units of PPV, as shown in figure 4.3. An important point to make is that these ENDOR
measurements are made in the dark, in the absence of doping ions and at low temperature:
in other words, the polaron seems to be an intrinsic property of the polymer.
The PPP Hamiltonian is an ad-hoc Hamiltonian, which has to be parametrised to
match the experimental ENDOR data. Attempts have been made by Geskin and co-
87

Figure 4.3: Spin density of a polaron on a PPV chain calculated with a PPP Hamiltonian.
Sites B and C’ correspond to sites 2 and 4 from figure 4.1, sites B’ and C
correspond to sites 1 and 3 from figure 4.1 and sites E and F correspond to
sites 5 and 6 from 4.1. The figure is from reference [2].
workers [5, 6] to repeat such calculations using methods which are more ab-initio and
do not require such parametrisation. Three classes of methods were used by Geskin
and co-workers for the geometry optimisation of radical cations on different lengths of
polymer: pure DFT methods, hybrid DFT methods and semi-empirical methods (Austin
Model 1). It was found that pure DFT methods show no localisation, semi-empirical
methods show strong localisation and hybrid DFT methods an intermediate level of lo-
calisation. In the first section of this chapter we repeat the calculation from reference [6]
using hybrid DFT, with particular emphasis on the variation in polaron localisation energy
with increasing oligomer length. We also argue that hybrid DFT, even though it does not
show significant polaron localisation, still predicts variations in site spin denisities which
are not inconsistent with the ENDOR spectra of PPV. In the second section of the chapter
we introduce torsional defects in the middle of an oligomer of PPV and show how they
would cause polaron localisation by effectively breaking conjugation. We also show that
the thermal energy required for such breaks of conjugation is relatively low. The conclu-
sion we draw from this chaper is that polaron localisation is likely to be weak factor in
defect-free chains of PPV. A suitable model for charge transport in three dimensional PPV
networks is one where torsional defects create conjugated segments within which charge
88

is delocalised and between which charge transport occurs in the temperature activated,
incoherent Marcus regime. The boundaries of these conjugated segments would have to
be defined dynamically, as the breaks in conjugation would be thermally activated.
4.2 Defect-free PPV
In this section we study polaron localisation in different length oligomers of PPV by
calculating the spin density distribution and polaron binding energy. The calculations
were performed with the BHandHLYP functional and the 6-31g* basis set, in accordance
with the work of [5, 6]. Using a hybrid density functional with a large proportion of
Hartree Fock exchange leads to qualitative localisation of the spin density for chains of
length smaller than ten, as shown in figure 4.4. In figure 4.4 we show the total Mulliken
spin density per phenylene and vinylene unit on the chain. To obtain these numbers, we
perform Mulliken analysis [7] on the density matrix for each spin, then subtract the up and
down spin populations and finally sum the spin density over all the atoms belonging to the
same phenylene or vinylene unit. Mulliken analysis tends to overlocalise charge densities
on each atom, so that the difference in spin density between neighbouring phenylene and
vinylene units might be somewhat exaggerated. Nevertheless the qualitative observation
is clear: spin density is peaked around the middle of the oligomer chain. The spatial
extent of the spin density is approximately four phenylene units, in accordance with [2].
On the other hand it is also evident that for the longest oligomers considered the
spin density is more evenly distributed along the chain. In order to assess whether this
functional is predicting polaron localisation we calculate the relaxation energy of a radical
cation ( λi2 from equation 2.11). Figure 4.5 shows the polaron binding energy for different
length oligomers: as the oligomer becomes longer this quantity decreases and seems to
tend to values smaller than 75 meV for infinite chains. This suggests that, although figure
4.4 suggests some degree of localisation of the spin density in short chains, the polaron is
actually only very weakly localised in a longer, defect-free chain.
In order to show that the hybrid DFT results are not inconsistent with ENDOR data,
we compare the total spin density on each carbon atom bonded to a hydrogen for a PPV
oligomer of length 12 obtained with the BHandHLYP functional and the AM1 (Austin
89

total spin density
0.4
0.3
0.2
0.1
0
12PV
10PV
8PV
6PV
4PV
2PV
ph1 ph2 ph3 ph4 ph5 ph6 ph7 ph8 ph9 ph10 ph11 ph12
ring number
Figure 4.4: Mulliken spin density for different length oligomers of PPV. The x axis labels
position along the chain: the labels ph1, ph2, etc. label the first, second, etc.
phenylene unit of the PPV. Between the phenylene units phn and phn + 1 lies
the vinylene unit vin.
90

polaron localization energy / eV
0.25
0.2
0.15
0.1
polaron localization energy
0 2 4 6 8 10 12
length of oligomer
Figure 4.5: Polaron localisation energy for different length oligomers of PPV calculated
with BHandHLYP.
Model 1) Hamiltonian. In accordance with reference [8] we use a restricted open shell
wavefunction when using AM1. It should also be noted that on a polymer of such a
great length, the convergence of the geometry optimisation using AM1 is very difficult,
as the potential energy surface of such a large polymer is very flat. Therefore we were
obliged to relax the convergence criteria of Gaussian 1 . The comparison of the AM1 and
BHandHLYP results is shown in figure 4.6. AM1 predicts a very similar spin density
distribution to that from figure 4.3, obtained by Shimoi and co-workers [2] using the PPP
Hamiltonian. Even though the extent of the polaron obtained by PPP and AM1 are similar,
the values of the spin density are much greater with AM1 than with PPP. It should be noted
that because we are using a restricted open shell wavefunction for AM1, we are unable to
reproduce the small negative spin densities seen with the PPP Hamiltonian. BHandHLYP,
conversely, shows a very large alternation of positive and negative Mulliken spin densities,
but with absolute values which are smaller than those necessary to reproduce the ENDOR
1 The geometry we shows had a maximum force of 4.15 10 −4 a.u., a root-mean-square force of
4.2 10 −5 a.u., a maximum displacement of 2.345 10 −3 a.u. and a root-mean-square displacement of
2.55 10 −4 a.u.
91

peaks seen by Shimoi [2]. Additionally, the BHandHLYP functional shows negative spin
densities which are not picked up in ENDOR triple resonance (TRIPLE) experiments [3]:
ENDOR is not sensitive to the sign of the spin density on the carbon atoms, whereas
TRIPLE can determine whether different shifts are caused by spin densities of different
signs. Still, one could argue that since a lot of positive and negative spin densities have
similar values, TRIPLE might simply not be able to pick out the fact that the different
ENDOR shifts are caused by spin densities of both signs of different magnitudes.
In conclusion, we find that AM1 reproduces the PPP spin density distribution but not
its magnitude, while BHandHLYP reproduces the magnitude of the spin densities slightly
better than AM1 but has a very different spin distribution. Neither method is entirely
consistent with the ENDOR data, but even so neither set of results can be excluded on
the basis of the ENDOR data. Evidence of a strongly localised polaron was found in
reference [2] using PPP simulations, which are not ab-initio and are designed to create
localised excitation. In my opinion, the ENDOR data does not unambigously demonstrate
the existence of strongly localised charged excitations: all ENDOR suggests is that dif-
ferent spin densities exist in the charged radical and all that TRIPLE suggests is that the
two largest spin densities are either both of the same sign or they are both positive and
negative.
4.3 Torsional Defects in PPV
In order to model the effect of defects in PPV on polaron localisation we consider an
oligomer of PPV of length eight and rotate the chain around the bond connecting the
central vinylene unit to a phenylene unit by a certain torsion angle θ. We then perform
constrained geometry optimisations for different values of θ and study the variation of
the spin density distribution. This evolution is shown in figure 4.7: it can be clearly seen
that increasing the torsion in the chain above an angle of 55 ◦ critically affects the spin
distrribution. An angle of ≈ 45 ◦ is sufficient to heavily affect this localisation, this is
consistent with studies [9] that show such an angle is sufficient to effectively break the
conjugation in a π system. If a charge was travelling down such a polymer, it would have
to hop over such a defect.
92

Mulliken Spin Density
Mulliken Spin Density
0.1
0.05
0
0.2
0.15
0.1
0.05
BHandHLYP
0 10 20 30 40 50 60 70
AM1
0
0 10 20 30 40 50 60 70
Carbon Atom Number
Figure 4.6: Atomic Mulliken spin density for the a chain of 12 units calculated at the
BHandHLYP (top panel) and AM1 (bottom panel) level.
In order to estimate how much thermal energy would be necessary to achieve such
angles we consider second order perturbation theory MP2 calculation of the oligomer of
length two of PPV (stillbene) [10]. The barrier of rotation calculated in reference [10] is
just 0.17eV and angles of 45 ◦ can be achieved with just 40 meV; similar values were ob-
tained by using using DFT. Therefore thermal fluctuations should be able to partition the
polymer chain into conjugated segments within which charge is effectively delocalised.
This underlines the need to be able to dynamically describe the charge transporting unit in
conjugated polymers. Quasi-coplanar segments would have charge delocalised on them
and transport inside these segments would be treated as a coherent quantum mechanical
process [10], whereas transport between such segments would be treated in the temper-
ature activated, non-adiabatic, incoherent limit described by the Marcus charge transfer
equation.
Another observation is that AM1, which predicts a very sharp localisation in space,
also predicts a twisted structure for neutral state of PPV, because it predicts shorter bond
lengths between vinylene and phenylene units which lead to steric interaction between
93

total spin density
0.25
0.2
0.15
0.1
0.05
0
θ=0
θ=18
θ=36
θ=47
θ=55
θ=90
ph1 ph2 ph3 ph4 ph5 ph6 ph7 ph8
ring number
Figure 4.7: Mulliken total spin density integrated on each phenylene and vinylene unit as
a function of chain torsion for an oligomer of length eight.
the hydrogen groups on these units and torsion of ≈ 22 ◦ between the units. BHandHLYP
predicts angles of ≈ 8 ◦ degrees between phenylene and vinylene units, whereas B3LYP
predicts planar structures. MP2 predicts planar structures for neutral stillbene. Maybe the
fact that DFT results are in better agreement with MP2 results suggests that they represent
the electronic structure of PPV better than AM1 does.
4.4 Conclusion
We have shown how even the hybrid density funtional which was previously shown to lead
to most polaron localisation, BHandHLYP, shows polaron localisation energies which
tend to diminishing values for long chains. We have argued that the spin density distribu-
tions predicted by this functional are consistent with ENDOR data. We have also shown
how introducing torsions in an oligomer chain tends to localise the charged excitation to
one side of the defect. Whether or not charged radicals are localised and the magnitude
of the polaron localisation energy remains an open question which depends on the model
94

chemistry chosen. The fact the DFT methods reproduces the geometry of stillbene calcu-
lated at the MP2 level more closely than AM1 leads us to tentatively suggest that polarons
may not be intrinsically localised on perfect chains of PPV, or at least that their localisa-
tion length is very long. This view is supported by calculations of the electronic structure
of crystalline PPV, which show that polarons should not exist in perfect crystals of PPV
[11] and therefore the actual existence of polarons in crystalline PPV must be explained
by the presence of chemical or physical defects in the conjugated backbone. Similarly,
the difference between the high intrachain charge mobilities measured for single chains
of PPV in solution and the lower values obtained for solid films is attributed to defects
due to chemical or structural irregularities [12], a conclusion supported by tight-binding
calculations of the effects of geometric defects on intrachain mobilities in single polymer
chains [10]. Recent measurements of metallic conductivity in highly ordered polyaniline
[13] reinforces the hypothesis that in perfect polymers, polaron localisation is not a strong
effect.
The relevance of these observations to charge transport in polymers is gleaned from
the realisation that the kinds of torsional defects sufficient to break conjugation are easily
accessible at room temperature. Even if polarons were not intrinsic properties of perfect
PPV chains, the picture of a polymer as a series of segments that charge is delocalised
would still apply. Transport events between such segments should be treated in the tem-
perature activated regime, whereas within the coplanar segments values of the transfer
integral would be much larger than the reorganisation energy and coherent conduction
could occur.
95

Bibliography
[1] S. Kuroda, T. Noguchi and T. Ohnishi, Electron Nuclear Double Resonance Obser-
vation of π -Electron defect States in Undoped poly(paraphenylenevinylene), Physics
Review Letters 72, 286 (1994)
[2] Y. Shimoi, S. Abe, S. Kuroda and K. Murata, Polarons and their ENDOR spectra in
poly p-phenylene vinylene, Solid State Communications 95, 137 (1995)
[3] S. Kuroda, Y. Shimoi, S. Abe, T. Noguchi and T. Ohnishi, Electron Nuclear Double
Resonance Spectra of Polarons in poly (Phenylene Vinylene) , Journal of the Physical
Society of Japan 67, 3936 (1998)
[4] H. Muto and M. Iwasaki, ENDOR studies of the superhyperfine couplings of
hydrogen-bonded protons. I. Carboxyl radical anions in irradiated L-analine single
crystals, Journal of Chemical Physics, 59, 4821 (1973)
[5] V.M. Geskin, A. Dkhissi and J.L. Brédas, Oligothioohene radical cations: polaron
structure in hybrid DFT and MP2calculations , Internation Journal of Quantum Chem-
istry 91, 350 (2003)
[6] V.M. Geskin, F.C Grozema, L.D.A. Siebbeles, D. Beljonne, J.L. Brédas and J.
Cornil, Impact of computational method on the geometric and electronic properties
of oligo(phenylene) vinylene radical cations, Journal of Physical Chemistry B 109,
20237 (2005)
[7] R.S. Mulliken, Electronic Population Analysis on LCAO-MO Molecular Wave Func-
tions, Journal of Chemical Physics 23, 1833 (1955)
96

[8] V.M. Geskin, J. Cornil and J.L. Brédas, Comment on ’Polaron formation and symme-
try breaking’ by L. Zuppiroli at al[Chem Phys Lett. 374 (2003) 7], Chemical Physics
Letters 403, 228 (2005)
[9] J.L. Brédas, G.B. Street, B. Themans and J.M. Andre, Organic Polymers Based
on aromatic rings (polyparaphenylen, polypyrrole, polythiophene) - evolution of the
elctronic-properties as a function of the torsion angle between adjacent rings., Journal
of Chemical Physics 83, 1323, 1985
[10] F.C. Grozema, P.T. van Duijnen, Y.A. Berlin, M.A. Ratner and L.D.A. Siebbeles,
Intramolecular charge transport along isolated chains of conjugated polymers: effect
of torsional disorder and polymerisation defects, Journal of Physical Chemistry B 106,
7791 (2002)
[11] P.G. da Costa, R.G. Dandrea and E.M. Conwell, First-principles calculation of the
three-dimensional band structure of poly(phenylene vinylene), Physical Review B 47,
1800, (1993)
[12] R.J.O. Hoofman, M.P. de Haas, L.D.A. Siebbeles and J.M. Warman, Highly mobile
electrons and holes on isolated chains of the semiconducting polymer poly(phenylene
vinylene), Nature 392, 54, (1998)
[13] K. Lee, S. Cho, S.H. Park, A.J. Heeger, C.W. Lee and S.H. Lee, Metallic transport
in polyaniline, Nature 441, 65 (2006).
97

Chapter 5
Ambipolar Transport in PCBM
In this chapter we present calculations of transfer integrals and reorganisation energies for
electrons and holes in a methanofullerene. The aim is to support experimental observation
of ambipolar transport in this material and to provide order of magnitude estimates for the
parameters which should be used in lattice simulations of charge transport in polymer
blends doped with methanofullerene.
5.1 Introduction
[6,6]-phenyle-C61 butyric acid methyl ester (PCBM) is a popular choice as electron ac-
ceptor for organic photo-voltaic devices, when used in combination with a donor polymer
such as the PPV derivative poly[2-methoxy-5-(3,7-dimethyloctyloxy)-1,4-phenylenevinylene]
(MDMO-PPV). A curious property of such cells is that they achieve maximum efficiency
at a relatively high concentration of PCBM of 80% by weight. This is especially confusing
as one would think that a higher concentration of MDMO-PPV would be more desirable
as MDMO-PPV is the light absorbing material. The high concentration of PCBM needed
for optimal devices has been explained by the observation that as the PCBM concentra-
tion is increased, both electron and hole transport become faster[1]. This explains the in-
creased efficiency of photovoltaic cells but leaves a major question open: how can PCBM,
traditionally viewed as an electron conductor, enhance hole mobility in MDMO-PPV?
Recent time of flight measurements of PCBM doped polystyrene have shown that
PCBM has similar electron and hole mobilities [2] and that PCBM acts as an ambipolar
98

transporter. Our colleagues Prof. Nelson and Dr. Chatten wanted to run simulations of
hole transport on a cubic lattice partially occupied by PCBM and MDMO-PPV, with the
aim of explaining the experimentally observed PCBM concentration dependence of mo-
bilty by allowing PCBM occupied sites to participate in hole transport. In order to help
parametrise such simulations, in this chaper we calculate the Marcus charge trasfer equa-
tion parameters for electrons and holes. We calculate the reorganisation energy for radical
cations and anions and compare them. Pairs of PCBM molecules are extracted from the
crystal structure of PCBM [3] and their transfer integral for both electrons and holes are
compared. The crystal structure used is that of crystals grown from chlorobenzene solu-
tions, this structure is more tightly packed than the one grown from ortho-dichlorobenzene
solutions and is therefore thought to be responsible for the higher efficiency of devices
fabricated from that solvent [3]. In the Monte Carlo simulations of charge transport, in-
cluding hopping to next-nearest neighbours proved necessary. We therefore calculate the
distance dependence of the transfer integral for PCBM, to provide an order of magnitude
estimate for the natural length of decay of the transfer integral with distance.
PCBM has a similar electronic structure to fullerene (C60). C60 belongs to the icosa-
hedral point group and because of this its HOMO is fivefold degenerate and its LUMO
threefold degenerate [4]. It has also been demonstrated via cyclic voltammetry that C60
has multiple reduction potentials [5]. PCBM, because of its sidechain, does not pos-
sess icosahedral symmetry and therefore its frontier orbitals are quasi degenerate. At the
ZINDO level the separation between the top two HOMOs is 64 meV and the separation
between the bottom two LUMOs is 132 meV. Even though these energy differences are
not very large we feel that, since we are only interested in order of magnitude estimates
of the magnitudes of hole and electron transfer rates, it is sufficient to look at the reor-
ganisation energies for single ionisation and to the transfer integrals for the HOMO and
LUMO only of the isolated molecule only.
5.2 Transfer Integrals
A cluster of 20 PCBM molecules is generated from the crystal structure obtained in ref-
erence [3]. We generate all possible pairs within this cluster and calculate their transfer
99

Figure 5.1: The seven unique pairs with greatest hole transfer integral extracted from
a cluster of twenty molecules of PCBM from its crystal structure grown in
chlorobenzene.
pair n Jh / meV Je /meV dcoc / Å
a 4 26 5 6.78
b 4 23 32 6.76
c 4 20 5 6.93
d 4 16 9 6.86
e 2 14 26 6.87
f 8 5 11 6.71
g 1 4 3 6.71
Table 5.1: Table of the transfer integral for holes Jh and for electrons Je. n is the number
of such pairs in a cluster of 20 molecules. dcoc is the separation between the
centres of the fullerene cages for the pair of molecules. The labels a-g label the
pairs according to figure 5.1
integrals using the projective method described in section 3.1 applied to ZINDO Hamil-
tonian. All calculations are performed with Gaussian 03. We consider the seven unique
pairs with the greatest magnitude hole transfer integrals; these seven unique pairs also
have the highest electron transfer integrals. These pairs are shown in figure 5.1 and their
transfer integral values are shown in table 5.1. The average value of the transfer integrals
for these pairs of molecules was found to be 15meV for holes and 14meV for electrons.
These two values are very similar therefore electron and hole transfer rates should be
similar if the reorganisation energies for electrons and holes are similar.
In order to allow Monte Carlo simulations to include both nearest and next nearest
100

neighbour hops, it is necessary to know the distance dependence of the transfer integral.
In order to calculate this we consider pairs of PCBM in the relative orientation b) from
figure 5.1, displaced them by a certain distance from their crystal centre of mass dis-
tance (keeping the angles describing their position and orientation constant) and calculate
transfer integrals using the projective method applied to the ZINDO Hamiltonian. This
particular pair was chosen because it has similar values of the hole and electron transfer
integral. The results of these calculation are shown in figure 5.2. For displacements from
the crystal separation greater than 2 Å, both transfer integrals fall exponentially with a
natural length scale of 0.5 Å and 0.53 Å for electrons and hole respectively. At shorter
displacements the electron transfer integral falls far more slowly, with a decay length of
0.70 Å . The difference in decay lengths between electron and hole transfer integrals for
short displacement can be explained by noticing that PCBM is not a planar molecule. To
illustrate how this might affect the value of the natural decay length of transfer integrals
let us assume that two particular faces of PCBM are closest together. If the two molecules
are close, as the separation between molecules is increased the distance and overlap be-
tween atomic orbitals on those faces becomes smaller; but maybe some of the other faces
will find themselves in such positions as to have better overlap. The net result of this effect
is that the decay would be slightly less steep at short distances, depending on which faces
of PCBM are closest and whether the molecular orbitals are involved. We have therefore
established a lower bound for the natural length of decay of the transfer integrals of ap-
proximataly 0.5 Å; an exact value for short distances will be more difficult to ascertain
and would depend heavily on the relative orientation of the molecules involved.
5.3 Reorganisation Energy
We have already stated that in order to calculate the reorganisation energy it is necessary
to compute the geometries of both the charged and neutral molecules. We will follow
Sakanoue [6] and use both semiempirical and DFT methods to compute these geometries.
The semiempirical method we pick is AM1 and the DFT functional is the hybrid function
B3LYP. The values of λ1, λ2 and λ in the methods are tabulated in tables 5.2 and 5.3.
It can be clearly seen that both methods predict similar reorganisation energies for
101

|J| / eV
0.01
0.001
0.0001
exp(-r/0.7)
exp(-r/0.53)
exp(-r/0.5)
|J| electrons
|J| holes
1e-05
0 1 2 3
Displacement from crystal position r / A
Figure 5.2: Distance dependence of the electron (black circles) and hole (red squares)
transfer integrals as a function of displacement from the crystal structure separation.
The lines show exponential fits with natural lengths of respectively
0.78 Å and 0.56 Å.
method λ1 λ2 λ
B3LYP 0.062 0.059 0.121
AM1 0.113 0.93 0.206
Table 5.2: Reorganisation energies for anion.
method λ1 λ2 λ
B3LYP 0.064 0.060 0.124
AM1 0.086 0.082 0.168
Table 5.3: Reorganisation energies for cation.
102

positive or negative charge transport, but that AM1 predicts larger values of the reorgan-
isation energy compared to B3LYP. A good illustration of the relation between charge
localisation and lattice distortion can be obtained by comparing the change in geome-
try of the molecule that occurs upon charging to the spin density distribution. In both the
cases of positive and negative charging, some bond lengths change up to 1-2%. In order to
show the correlation between spin density and bond length deformation, in figure 5.3 we
show a scatter plot of the percentage change in absolute value of a bond length versus the
average Mulliken spin density on the two atoms involved in the bond. This figure clearly
shows that there is a very strong correlation indeed between spin density and bond length
deformation; for an anion the Pearson’s product moment correlation 95 % interval is 0.76
to 0.87 and for cation it is 0.63 to 0.80. In order to show where the bond deformation
occurs and the spin density is localised, in figure 5.4 we plot the total spin isosurface for
an anion radical (left) and a cation (right) radical. It can be seen that in the anion radical
the spin density is mainly concentrated around the equator of the molecule, whereas in
the cation radical the spin density is mainly located at the top pole, near the sp 3 carbon
which links the fullerene cage to the side chain. When the bonds with the highest spin
density rearrange in a radical anion, the bonds which cross the equator of the molecule
are elongated and the molecule becomes more ”rugby ball” shaped. In a radical cation,
the bonds which rearrange the most are near the sp 3 carbon, resulting in an opening of
the fullerene cage.
The conclusions from these calculations are that in PCBM the reorganisation energies
for both cations and anions are very similar and that therefore the magnitude of charge
transfer rates should be similar for both positive and negative charges. This explains the
similarity of zero field electron and hole mobilities for PCBM doped polystyrene, shown
in figure 5.5.
5.4 Monte Carlo Simulations
In this section we describe the results of the Monte Carlo simulations run by our col-
leagues Dr. Chatten and Prof. Nelson on charge transport in blends of MDMO-PPV:PCBM.
The aim of this exercise is to fit the experimental temperature dependence of hole trans-
103

Absolute value of the bond length distorsion / %
2
1.5
1
0.5
anion (Pearson’s correlation=0.76-0.87)
cation (Pearson’s correlation=0.63-0.80)
0
-0.02 0 0.02 0.04 0.06 0.08
Average Mulliken spin density
Figure 5.3: Scatter plot of the average spin density for each bonded pair of atoms in a
cation (circles) or in an anion (crosses) versus the percentage change in bond
length upon charging.
Figure 5.4: Spin density for cation and anion radicals of PCBM. The left panel shows a
front view of the molecules, the right panel a side view. Within each panel
the molecule to the left is the anion radical and the molecule to the right is the
cation radical.
104

zero field mobility / cm 2 V -1 s -1
0.0001
1e-05
1e-06
hole mobility
electron mobility
1e-07
20 25 30 35
PCBM concentration / %
40 45 50
Figure 5.5: Zero field mobility for electrons (squares) and holes (circles) in PCBM doped
polystyrene as a function of PCBM concentration.
port in pristine MDMO-PPV and in blends of MDMO-PPV:PCBM using the same pa-
rameters. The model used is very similar to that used to describe hole transport at low
PCBM concentration [7]. Its main features are: use of the Marcus form for setting charge
hopping rates, use of a distribution of energies based on a Gaussian with an exponential
distribution of traps and use of a cubic lattice. Holes are allowed to be transported in the
polymer donor material, in the fullerene and between the fullerene and the polymer; hole
hops from the donor to the acceptor are decelerated by inclusion of an additional energy
difference ∆ between the two materials. The lattice is allowed to be partially occupied
by MDMO-PPV sites and PCBM sites. The density of states for hole transport sites are
picked from a Gaussian of width 75 meV for PCBM [8] and 25 meV for MDMO-PPV;
for MDMO-PPV 2% of site energies are drawn from an exponential trap distribution with
characteristic energy 65 meV [7]. The reorganisation energy for PCBM was 0.5 eV, or
roughly twice the internal sphere reorganisation energy we have calculated. This value
can be justified by arguing the importance of the outer sphere reorganisation energy. The
value of the reorganisation energy used to model hole transport in MDMO-PPV was also
105

0.5 eV, following reference [9], although a value of 0.25 eV could also reproduce the ex-
perimental data. The transfer integral for nearest neighbours in PCBM which fitted the
experimental mobility is 28 meV, which correlates rather well with the values we have
calculated.
The lattice used is cubic with a lattice constant of 1nm, whilst hops within a 3nm
radius are allowed. In order to calculate the transfer integrals to next nearest neighbours,
we assume that the transfer integral falls exponentially with a natural length of 2 Å; this
length is somewhat longer than the one we calculated. The localisation length is very
important in simulations on partially occupied lattices, as it will have a very large ef-
fect on the percolation threshold and on the formation of efficient pathways for charge
transport. The fact that such a large natural length is required to reproduce experimental
data might suggest that cubic lattices do not represent the packing of PCBM very well:
next nearest neighbours are actually closer in a real solid than a cubic lattice is capable
of representing. Also, at the relatively short distance variations between nearest and next
nearest neighbours the exponential dependence of transfer integrals on distance might not
hold, as orientational effects become stronger. To show this we plot the transfer integral
for pairs of molecules from the cluster we ran the calculations in table 5.1 from, and plot
them against centre of mass distance between each pair. We also plot them against the
distance between the centre of the C60 cages, as these are the parts of the molecules that
contain most of the HOMOs and LUMOs and therefore it is the proximity of these areas
which is more relevant to transfer integrals. This plot is shown in figure 5.6. An expo-
nential fit is not terribly good for either the dependence on centre of mass distance or for
distance between the centres of the C60 cages, but it is somewhat better in the latter case.
This is easily understood by looking at inset c) in figure 5.1. In orientations of that kind,
with the side chains opposite the closest point of the two cages, looking at the distance be-
tween the centres of mass overestimates separation because of the effect of the side chain
in determining the centre of mass. However, the rate of decrease with distance of transfer
integral is certainly not as fast as with a natural length of 0.5 Å, the transfer integral falls
two orders of magnitude over approximately 3 Å , which would correspond to a natural
length of approximately 1.5 Å.
106

J /eV
J / eV
0.01
0.001
0.0001
0.01
0.001
0.0001
distance between centres of mass
8 9 10 11 12 13 14
6 7 8 9 10
distance between c60 cage centres
Figure 5.6: Distance dependance for the transfer integral for neighbour and nearest neighbour
hoppings, extracted from the cluster from the crystal structure of PCBM
discussed in the text. The transfer integral is plotted both as a function of
the distance between centres of mass of the PCBM molecules (above) and
as a function of the distance between the centres of the C60 cages (below).
This is done because the C60 cages are the electronically active parts of the
molecules.
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Figure 5.7: Experimental (full symbols) and modelled (empty symbols) temperature dependence
for pristine MDMO-PPV (triangles) 1:1 MDMO-PPV:PCBM (circles)
and 1:2 MDMO-PPV:PCBM (squares).
107
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The most crucial parameter in determining whether the simulation is capable of re-
producing experimental observation is the offset in site energies between PCBM and
MDMO-PPV ∆. Nominally, the difference between the HOMO of PCBM and the HOMO
of MDMO-PPV is 0.9 eV. If simulations are run with such a high offset, charges cannot
hop from the polymer to the fullerene and hole mobilities are decreased as the PCBM
concentration is increased. If, however, a far smaller value for ∆ is picked of between 0.1
and 0.2 eV, the experimental temperature dependence of the zero field mobility can be
reproduced, as figure 5.7 shows. Values of ∆ as large as 0.25 eV could also be made con-
sistent with the experimental data. The temperature dependence is shown because hops
between MDMO-PPV and PCBM are temperature activated: if our model of transport
was correct and if the barrier to hopping between the materials ∆ was large, we would
predict that there would be a very strong temperature dependence of the mobility, but this
is not observed experimentally. The conclusion from this investigation is that in order for
ambipolar transport in PCBM to explain the increased hole mobility in PCBM:MDMO-
PPV blends, the offset between the hole transport levels must be much smaller than one
would imagine from looking at the PCBM HOMO. Such small offset could be justified
by invoking the formation of an interface state, with a much lower ionisation potential
compared to PCBM.
5.5 Conclusion
In this chapter we have shown results of calculations of transfer integrals and reorganisa-
tion energies for electron and hole transport. These calculations imply that hole transport
is plausible in PCBM. Simulation of hole transport in MDMO-PPV:PCBM blends show
that experimental mobility data can be consistent with ambipolar transport in PCBM only
if the energy offset between the hole transporting states on MDMO-PPV and on PCBM
is between 0.1 and 0.25 eV.
108

Bibliography
[1] V.D. Mihailetchi, B. de Boer, C. Melzer, L.J.A. Koster and P.W.M. Blom, Electron
and hole transport in poly(para-phenylenevinylene):methanofullerene bulk heterojunc-
tion solar cells, proceeding of the SPIE conference (2004)
[2] S.M. Tuladhar, D. Poplavskyy, S.A. Choulis, J.R. Durrant, D.D.C. Bradley
and J. Nelson, Ambipolar charge transport in films of methanofullerene and
poly(phenylenevinylene) blends, Advanced functional Materials, 15, 1171 (2005)
[3] M.T. Rispens, A. Meetsma, R. Rittberger, C.J. Brabec, N.S. Sarificiftci and J.C. Hum-
melen, Influence of the solvent on the crystal structure of PCBM and the efficiency of
MDMO-PPV:PCBM ’plastic’ solar cells Chemical Communications, 2006 (2003)
[4] P.J. Benning, J.L. Martins, J.H. Weaver, L.P.F. Chibante and R.E. Smalley, Electronic
structure of C60: Insulating Metallic and Superconducting character, Science, 252,
1417 (1991)
[5] P.M. Allemand, A. Koch, F. Wudl, Y. Rubin, F. Diederich, A.M. Alvarez, S.J. Anz
and R.L. Whetten, Two different fullerenes have the same cyclic voltametry, Journal of
the American Chemical Society, 113, 1050 (1991)
[6] K. Sakanoue, M. Motoda, M. Sugimoto and S. Sakaki, A Molecular Orbital Study
on the Hole Transport Property of Organic Amine Compounds, Journal of Chemical
Physics A, 103, 551 (1999)
[7] A.J. Chatten, S.M. Tuladhar, S.A. Choulis, D.D.C. Bradley and J. Nelson, Monte
Carlo modelling of hole transport in MDMO-PPV:PCBM blends, Journal of Material
Science, 40, 1393 (2005)
109

[8] V.D. Mihailetchi, J.K.J. van Duren, P.W.M. Blom, J.C. Hummelen, R.A.J. Janssen,
J.M. Kroon, M.T. Rispens, W.J.H. Verhees and M.M. Wienk, Electron Transport in a
Methanofullerene, Advanced Functional Materials, 13, 43
[9] A.J. Chatten, S.M. Tuladhar, D.D.C. Bradley and J. Nelson, Modelling charge trans-
port in gated polymers for solar cell applications, proceedings of the Photovoltaic
Solar Energy Conference and Exhibition (2004)
110

Chapter 6
Charge Transport Parameters for
Randomly Oriented Pairs of Dialkoxy
Poly-paraphenylenevinylene and
Triarylamine Derivatives
This chapter reports calculations of J and ∆E on huge samples of pairs of molecules in
random relative orientations and positions. We use the fast methods of calculating these
parameters described in chapter three, namely molecular orbital overlap based calcula-
tions for J and distributed multipole electrostatic calculations for ∆E. We will compare
the distributions of J and ∆E from these calculations to the trends in GDM parameters for
two different series of materials: dialkoxy PPV with different length side chains and spiro
linked triarylamine derivatives with different side groups. Hole transport in these two
groups of materials has been studied experimentally at Imperial College with the explicit
intention of studying the relationship between charge mobility and chemical structure.
Our aim is to provide an explanation in terms of microscopic properties of the physical
origin of the GDM parameters. As mentioned in chapter two, the GDM is a model which
provides an expression for the temperature and field dependence of mobility to equation
2.40, namely:
111

⎧
⎪⎨ µ0e
µ =
⎪⎩ µ0e
2 −( 3kT σ)2 σ C[( e kT )2−Σ2 ] √ F , Σ > 1.5
2 −( 3kT σ)2 σ C[( e kT )2−2.25] √ F , Σ < 1.5
(6.1)
this equation takes four parameters: µ0, σ, Σ and C. The GDM is developed from an
idealised model of charge transport on a cubic lattice, where the form for the transfer rates
is the Miller-Abrahams rate equation (equation 2.8). Within this model the four parame-
ters σ, Σ, µ0 and C have very specific meanings: the energetic disorder σ represents the
width of the distribution of site energies, the positional disorder Σ represents the width of
the distribution of the logarithm of the transfer integral, µ0 scales the absolute value of the
mobility and C is related to the lattice constant. When applied to experimental data, how-
ever, it is not always possible to relate these parameters to the microscopic properties they
represent in the model. Strictly speaking, the GDM parameters are really fitting parame-
ters for the Poole-Frenkel like dependence of mobility on electric field and temperature:
a more positive field dependence and temperature dependence of mobility corresponds to
greater energetic disorder and a more negative field dependence but unchanged tempera-
ture dependence of mobility corresponds to a greater positional disorder. In this chapter
we calculate distributions of site energy differences and of transfer integrals and compare
the results to GDM parameters in order to make a comparison of the GDM with chemical
structure grounded on physical computation.
6.1 Dialkoxy PPV Series
6.1.1 Introduction
This part of the study is aimed at explaining the trends in GDM hole transport parameters
for a series of dialkoxy polyparaphenylenevinylene (PPV) derivatives with different length
side chains. Figure 6.1 shows the three derivatives considered in this study: they all have
the same conjugated backbone and all are symmetrically substituted with alkoxy side
chains which vary only by their lengths.
In polymers with a conjugated backbone such as PPV, the influence of side chains on
electronic properties is often thought of as arising from their role in determining chain
112

Figure 6.1: Chemical formula of the three dialkoxy PPV derivatives studied in this chapter.
a) is di-MethoxyPPV(dMeOPPV), b) is di-hexyloxyPPV (dHeOPPV) and
finally c) is di-decyloxyPPV (dDeOPPV).
packing, which in turn affects the degree of coplanarity of the backbone. This effect is
certainly observed in regio-regular polythiophenes with alkyl side groups [1], where pro-
cessing methods which lead to high charge mobility also lead to the formation of ordered
micro-crystallites. X-ray diffraction studies [2] proved that these crystallites are formed
thanks to interdigitation of the alkyl side chains, which aid the formation of long, coplanar
segments of the backbone and to supramolecular arrangements of the backbones in lamel-
lae. Martens et al. [3] measure hole mobilities for symmetrically and asymmetrically sub-
stituted PPV polymers and extract the corresponding parameters for the correlated GDM:
symmetrically substituted PPV are shown to have lower energetic disorder and greater
mobility, which were both attributed to the hypothesis that symmetrical side chains aid
ordering of the backbones. Kemerink and co-workers [4] also observe that asymmetric
substitution leads to stronger intrachain interactions and in some cases intrachain aggre-
gation in PPV polymers, whereas symmetric subsitution leads to interchain aggregation
and greater mobilities.
In this study we are not comparing the effect of symmetric versus asymmetric substi-
tution on backbone conformation, but rather the influence of side chain length on charge
transport parameters. We use the simplest possible interpretation of the role of side chains:
longer side chains keep the backbones of different polymers at larger separations, but oth-
erwise they do not affect the relative orientation of molecules or their electronic structure.
Our aim in this investigation is to evaluate whether or not this naïve assumption can be
consistent with the observed trends in charge transport characteristics of the polymers
with different length side chains. The hypothesis we want to test in this study is that the
113

qualititative dependence of charge mobility on side chain length for symmetrically sub-
stituted PPV derivatives can be explained uniquely in terms of changes in the minimum
distance of separation between the π conjugated backbones, without invoking changes in
backbone conformation.
Of the three materials considered and defined in figure 6.1, dHeOPPV and dDeOPPV
can be processed from solution, whereas dMeOPPV cannot. In order to make films of
this materials for time-of-flight charge mobility measurements the technique described
in [5] was used, namely in-situ polymerisation of dMeOPPV by thermal conversion of a
soluble precursor. A study by Dr Marc Sims [6] and coworkers at Imperial College also
demonstrates by Raman spectroscopy that converting dMeOPPV at a high temperature
(185 ◦ C) produces films with micron-sized domains showing high Raman signal contrast,
compared to lower contrast for films converted at lower temperatures. The authors relate
these high contrast Raman regions to the presence of an ordered, tightly packed mor-
phology caused by the interdigitation of methoxy group on neighbouring polymer chains.
Such interdigitation has also been postulated to explain the high degree of translational
ordering observed by electron diffraction in crystals of dMeOPPV [7]. Ellipsometry mea-
surements also reveal that the films converted at high temperatures have a density greater
by 6 ± 2% than the films converted at a lower temperature.
The time of flight mobility of these three materials was measured by our colleague Mr
Sachetan Tuladhar. He found that as the side chains get longer:
• the overall mobility becomes smaller
• the field dependence becomes less positive (see figure 6.2 )
These observations are reflected in the GDM parameters of the three materials. These
parameters are shown in table 6.1. Note that the stronger field dependence of mobility
of dMeOPPV is reflected both by its higher energetic disorder and its lower positional
disorder, and its higher mobility is reflected by higher mobility prefactor µ0.
The trend in the mobility prefactor µ0 can be intuitively explained by arguing that
longer side chains lead to greater interchain distances and hence smaller interchain trans-
fer rates. The trend in positional disorder Σ seems also to be in agreement with the emer-
gence of more crystalline regions in dMeOPPV: in a crystal the positions and orientations
114

Figure 6.2: Electric field dependence of mobility for dMeOPPV (full squares, preannealed
C1 ) for dHeOPPV (empty diamonds, C6) and for dDeOPPV (empty
circles, C10). The figure also shows the electric field dependence of mobility
for dMeOPPV annealed at low temperature (full triangles, non-annealed C1)
and for MDMO-PPV (dotted line).
material µ0/cm 2 V −1 s −1 σ /eV Σ C/cmV −1
dMeOPPV 2.5 10 −1 0.110 2.0 5.10 10 −4
dHeOPPV 4.25 10 −3 0.091 3.05 3.20 10 −4
dDeOPPV 2.73 10 −4 0.085 3.92 3.26 10 −4
Table 6.1: GDM parameters for the various dialkoxy PPV derivatives.
115

of molecules are more ordered and therefore the logarithms of the transfer integral will
also be more narrowly distributed. The trend in energetic disorder is somewhat harder
to explain. When comparing the energetic disorder of symmetrically and asymmetrically
subsituted PPV derivatives earlier in the introduction of this chapter, we quoted Martens’
and coworkers’ observation that this could be due to differences in backbone conforma-
tion. Applying the same reasoning to the comparison of the more crystalline dMeOPPV
with the longer side chain derivatives, we would conclude that stronger interchain inter-
actions in dMeOPPV and the postulated interdigitation of side chains would lead to a
stiffer polymer backbone with less energetic disorder than the more amorphous polymers
with longer side chains. However, the GDM parameters indicate that energetic disorder in
dMeOPPV is greater than in both dHeOPPV and dDeOPPV. In this study we argue that
electrostatic interactions can explain the observed trend in energetic disorder.
In this investigation we assume that all three polymers can be modelled as dMeOPPV
oligomers and that longer side chains only lead to greater interchain separations without
affecting any other aspect of the relative geometry of pairs of oligomers. Calculations
of the transfer integral and site energy difference are performed on pairs of oligomers
at fixed minimum distances of approach and the results are compared to the mobility
prefactor and the energetic disorder respectively. The conclusion of the study is that the
variations in energetic disorder are consistent with electrostatic calculations of ∆E, which
predict that more tightly packed molecules will have stronger electrostatic interactions
and hence greater site energy differences. We also show an unexpected result concerning
the correlation between ∆E and J.
6.1.2 Method
We consider pairs of hexamers of dMeOPPV. Calculations on shorter oligomers were car-
ried out and yielded similar results to those described here. The geometry of the isolated
oligomer is obtained at the B3LYP/6-31g* level using Gaussian 03 [8]. The site energy
difference ∆E is calculated as the difference in electrostatic interaction energy estimated
using Distributed Multipole Analysis, as described in section 3.3. Briefly, two interaction
energies are calculated corresponding respectively to the charge being localised on either
116

Figure 6.3: Representation of the three Euler angles defining relative orientation: Φ and
Θ represent the first two Euler rotations about the temporary z and x axis, Ψ
represents the last rotation about the z axis.
oligomer. The difference in these interaction energies is then the site energy difference
∆E. Calculations of the charge density for the radical cation and neutral state are per-
formed at the B3LYP/6-31g* level using Gaussian 03[8]. The distributed multipoles for
either molecule are obtained using Stone’s gdma program [9]. Inductive interactions are
ignored in the calculation of ∆E. The transfer integral J is calculated using the MOO
method, as described in section 3.2. ZINDO/S orbitals for the individual oligomers were
calculated using the ZINDO/S Hamiltonian, as implemented in Gaussian 03[8]. In order
to model the increase in side chain length we increase the separation dmin between the pair
of oligomers, defined as the minimum distance of approach between any two atoms on
either oligomer.
In order to introduce disorder we orient one of the two chains at random with respect
to the other. Five quantities define the relative orientation of the two chains: an azimuthal
and a polar angle define the unit dispalcement vector of the centre of the second molecule
from the first molecule; three angles define the rotation matrix that, if the two molecules
had the same centre, would bring the coordinates of one onto the other. The rotation
matrix is defined in terms of Euler angles, that is three successive rotations about the z,x
and z axes respectively. In terms of two sets of internal Cartesian coordinates on either
molecule, the first two Euler angles define the position of the z axis of the internal coordi-
117

nate system of one molecule with respect to the other, while the third Euler angle places
the x axis of the internal coordinate system of a molecule in a particular position; the three
Euler angles Φ, Θ and Ψ are represented pictorially in figure 6.3. In order to achieve a
uniform distribution of the position of the centre of the molecule, the azimuthal angle and
the cosine of the polar angle are varied uniformly. To achieve a uniform distribution of
orientations, we guarantee that the internal z axis of the second molecule is distributed
uniformly on the surface of a sphere by varying the Euler angle Φ and the sine of the
Euler angle Θ uniformly; the Euler angle Ψ is also varied uniformly. In order to obtain a
particular minimum separation between the two molecules, we vary the distance between
the centres of mass of the two molecules until the minimum separation has the required
value. A binomial search algorithm is used to search the separation d between the centres
of the two molecules in order to achieve greater computational efficiency. A separation d
is picked and the distance of closest approach dmin is calculated; if this distance is smaller
than the minimum separation desired then the separation d is doubled, if it is bigger then
d is halved. The procedure is iterated until the desired value of dmin is achieved.
Another aspect which should be mentioned is the computational efficiency of the
methods used: for each separation dmin the 10 5 calculations took roughly 90 minutes
on a single processor of an Opteron 24x workstation. For comparison, on the same com-
puter a single SCF calculation with the ZINDO/S Hamiltonian on a pair of oligomers of
PPV takes roughly 2 minutes. Performing 10 5 calculations using the SCF method would
require roughly one month: therefore repeating such calculations for many separations
would require a huge amount of computational resources.
6.1.3 Results and Comment
For each minimum distance of separation, we generate 10 5 configurations of two oligomers
with random orientations; for each of these configurations we calculate the site energy dif-
ference ∆E and the transfer integral J. We then calculate the root-mean-square average of
the tranfer integral Jrms and the standard deviation of the distribution of energy differences
σDMA. The distributions of log(J) and ∆E obtained for a minimum separation dmin of 3.5
Å are shown in figure 6.4. A separation of 3.5 Å was chosen to match the sum of two van
118

der Waals radii of carbon (the van der Waals radius of carbon is approximately 1.7 Å ).
The distribution of site energy differences and the distribution of logarithms of the trans-
fer integral can both be approximated by normal distributions. One should bear in mind
that we are considering only the electrostatic interaction of pairs of molecules, not the
interactions in the whole solid, therefore we are unable to observe the spatial correlations
in site energies discussed in section 2.5.2 which would be expected to arise because of the
long range nature of electrostatic interactions. When comparing to values of the energetic
disorder σ one must bear in mind two factors: in the first place the GDM predicts values
for the distribution in site energy, not in site energy difference. Our values of σDMA should
therefore be divided by √ 2 1 . In the second place, the site energy difference we calculate
is only for a pair of molecules and therefore does not take into account interactions with
the rest of the molecules in the solid. In a solid several pairs of molecules are present
and thus several contributions to ∆E would be summed and we expect that the site energy
would be distributed more widely than for a single pair.
Next, we plot the dependence of the standard deviation of ∆E and of the root mean
square value of J on dmin, shown in figure 6.5. Both Jrms and σDMA decrease with in-
creasing separation, J falls exponentially with a natural length scale of 0.42 Å , while ∆E
falls far more slowly and can be fitted to a polynomial. The dependence on separation of
Jrms can be explained by the decrease in orbital overlap, whereas the separation depen-
dence of ∆E can be explained by arguing that the difference in electrostatic interactions
is correlated to the absolute values of the two electrostatic interactions and therefore falls
with distance. With regard to site energy difference this study indicates that, in the pres-
ence of orientational disorder, tighter packing leads to greater energetic disorder because
of stronger electrostatic interactions. This observation would be unchanged by including
inductive interactions.
Once again, we want to stress that the site energy difference we calculated reflects the
distribution of site energies of pairs of molecules, not of a solid. In a solid, in fact, several
1To see why this is the case, consider that f (E) is the site energy distribution and is distributed normally:
f (E) = 1
√ exp(−
2πσ E2
2∗σ2 ) . To estimate the distribution of the difference of two site energies E1 and E2 we
must therefore estimate the integral: f (u) = � � f (E1) f (E2)δ((E1 − E2) − u)dE1dE2. If this expression is
evaluated, the resulting distribution will be: f (u) = 1
2 √ u2 exp(−
πσ 4σ2 ), or in other words a normal distribution
with a standard deviation σ ′ = √ (2)σ
119

f(∆E)
15
10
5
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
∆E / eV
f(log 10 J)
0.5
0.4
0.3
0.2
0.1
0
-8 -6 -4
log J / eV 10
-2 0
Figure 6.4: Probability distributions of the difference in site energies (left panel) and the
logarithm of the transfer integral (right panel) of two hexamers of dMeOPPV
with a minimum separation of 3.5 Å . Probability distributions (vertical bars)
are compared to a normal distribution (solid line).
pairs of molecules would be present and the site energies would depend on the sum of
all pairwise interaction energies. We have already calculated the separation dependent
distributions of electrostatic interactions between pairs of molecules (which we used to
generate figure 6.4). If the radial distribution function of hexamers in the solid film was
known, we could have used that information to reconstruct the distribution of site energies
for the whole solid. This approach would be questionable for two reasons: first there are
many different ways of subdividing a polymer into such conjugated segments and hence a
radial distribution function of hexamers in the solid is somewhat arbitrary. Second, simply
picking interaction energies at random from a separation dependent distribution ignores
the long range nature of electrostatic interactions and the correlations in interaction ener-
gies to which this would lead. For both these reasons it seems to us that the best approach
would be to generate realistic morphologies for slabs of polymer, use the relative orienta-
tion and position of polymer segments to calculate all pairwise electrostatic interactions
and thereby obtain the distribution of site energies for the slab. This approach depends on
the ability to generate accurate morphologies for polymers, a formidable computational
task which we have not attempted. Nevertheless, the pairwise interactions we have cal-
culated would be the fundamental building block for such an approach. It can be safely
120

σ DMA / eV
0.05
0.04
0.03
0.02
0.01
0
e -d/λ λ=0.42A
10 -10
10 -12
10 -14
10 -16
5 10
distance of closest approach d / A min
15
10
20
-18
Figure 6.5: Distance dependence of the standard deviation in values of ∆E.
assumed that if the distribution in these pairwise interactions is wider, so will be the dis-
tribution in site energies for the whole solid. We expect that, all else being equal, tighter
packing leads to wider distributions of site energies.
Another aspect we investigate is whether, for a particular separation, a correlation ex-
ists between J and ∆E. We know from section 2.5 that spatial correlations in site energies
have a significant effect on charge transport. A correlation between J and ∆E would have
an effect on transport because in the Marcus equation the transfer integral sets the overall
rate, whereas the site energy difference sets the ratio of forward and backward rates for
hops. We consider the absolute value of the site energy difference and the logarithm of
J. The decision to pick the absolute value of ∆E is that its sign is arbitrary: if we swap
the labels of each molecule, ∆E changes sign. The decision to consider the logarithm of
J is that its distribution seemed similar ot that of ∆E. If we calculate the Pearson coef-
ficient of these two quantities for a separation between dMeOPPV molecules of 3.5 Å ,
we find a value of 0.11, which suggests a weak, but not zero, correlation. Also, this does
not exclude a correlation between the two quantities which is not linear. A scatter plot
of the absolute value of ∆E versus log(J) is shown in figure 6.6 (a). This figure shows
how non-linear the correlation between the two quantities is. Let us also make a scatter
plot of log(J) and ∆E as a function of the centre to centre distance, shown in figures 6.6
(b) and (c). The assumption behind these plots is that, for a given minimum distance of
121
10 -2
10 -4
10 -6
10 -8
J rms / eV

Figure 6.6: Scatter plot of the absolute value of ∆E against the logarithm of the transfer
integral log(J) (a). Also shown is the scatter plot of log(J) versus the centre
to centre separation d (b) and the scatter plot of ∆E versus d (c).
separation, shorter centre to centre distances correspond to closer packed, more coplanar
morphologies. The scatter plot of transfer integrals suggests an isotropic distribution for
distances over 10 Å and a positive correlation for shorter distances. The scatter plot for
∆E shows that the site energy difference increases with decreasing centre to centre dis-
tances, but that for centre to centre distances inferior to 8 Å or so ∆E becomes smaller
with shorter centre of mass distances. This suggests that as packing becomes denser the
electrostatic interactions become stronger. Below certain centre to centre distances, how-
ever, the site energy differences decrease because more coplanar packing increases the
symmetry in electrostatic interactions. It seems therefore that the fact that both ∆E and
J have a dependence - albeit a complex one - on the centre of mass separation, this leads
to some degree of correlation between these two values. This relationship is also highly
non-linear: large transfer integrals would correspond to either very small or very large
∆E values, but not to intermediate ones; for this reason the correlation between the two
quantities is probably badly represented by Pearson’s coeffiecient.
Symmetrically substituted derivatives of PPV were studied because of their high mo-
bilities compared to asymmetrically substituted derivatives. It was expected that shorter
side chains would lead to greater mobilities. However, an unexpected result was that the
122

Figure 6.7: The three spiro-linked tryarilamine derivatives considered in this study.
From left to right they are 2,2’,7,7’-tetrakis-(N,N-diphenylhenylamino)-
9,9’-spirobifluorene (spiro-unsub), 2,2’,7,7’-tetrakis-(N,N-di-mmethylphenylamino)-9,9’-spirobifluorene
(spiro-Me) and finally 2,2’,7,7’tetrakis-(N,N-di-4-methoxyphenylamino)-9,9’-spirobifluorene
(spiro-MeO)
.
field dependence of mobility also became more positive as side chains became shorter.
Analysis using the GDM attributes such a change in field dependence to greater energetic
disorder and lower positional disorder. However, the physical origin of these changes in
disorder is unclear. Without performing electronic structure calculations one would have
to speculate about the origin of this difference in energetic disorder. For example it could
have been tempting to attribute the differences in charge transport characteristics to those
differences between the films that were observed with the Raman technique, namely the
fact that films made from the dMeOPPV chains showed domains of high density and were
therefore less homogeneous than films made from the longer sidechain derivatives. Elec-
tronic structure calculations revealed that it is entirely reasonable to attribute the changes
in energetic disorder to the denser packing in the shorter side chain derivatives and to
the fact that electrostatic interactions are stronger. They also confirm that the increase in
overall mobility with denser packing can be attributed to greater root-mean-square trans-
fer integrals.
6.2 Spiro Linked Triarylamine Derivatives
In this section, we analyse the dependence on side group substitution of energetic dis-
order for the three differently substituted spiro-linked tryarilamine derivatives shown in
figure 6.7. The three compounds considered all share the same core: two spiro-linked
bi-phenyl units, where all four phenyl units are bonded to nitrogen atoms, each of which
in turn is bonded to two more phenyl rings. The eight outer phenyl rings are either unsub-
123

stitued (spiro-unsub), or substituted with methyl groups in the para position (spiro-Me),
or substituted with methoxy groups in the meta position (spiro-MeO). These compounds
benefit both from high hole mobilities and from high glass transition temperatures and
have found application in LEDs [13] and dye sensitised solar cells [14]. Other than the
technological interest in these materials, the motivation for comparing their charge trans-
port characteristics is similar to the motivation for studying different dialkoxy PPVs: to
understand the relationship between chemical structure and charge transport characteris-
tics. The time-of-flight hole mobility of spiro-MeO and spiro-unsub was measured by U.
Bach et al [11], whereas the time of flight hole mobility of spiro-MeO was measured by
D. Poplavskyy and J. Nelson [12]. D. Poplavskyy observes that the energetic disorder
calculated via the GDM for the three materials is larger for spiro-MeO (0.101eV) than for
either spiro-Me (0.080eV) or spiro-unsub (0.080eV). They attribute this observation to
two possible interpretations: either the methoxy group increases the permanent dipole of
the spiro-MeO, or the different techniques used to produce the samples for time of flight
measurements (evaporation for spiro-Me and spiro-unsub and solution casting for spiro-
MeO) lead to different amounts of disorder. Because all molecules belong to the S 4 point
group, the hypothesis of a permanent dipole moment can be excluded. Even though the
molecules in their ground state have no permanent dipole, they can still have electrostatic
interactions. In the case of well separated small polar molecules, the leading component
of the electrostatic interaction is the permanent dipole-monopole interaction; however in
the case of densely packed large non-polar molecules, electrostatic interactions should
be calculated from distributed multipole analysis. The aim of this study was to assess
whether the differences in energetic disorder extracted via the GDM from mobility data
for the different spiro linked compounds can be attributed to electrostatic interactions.
6.2.1 Method and Results
Geometries for the spiro-MeO, spiro-Me, and spiro-unsub were obtained at the AM1 level
using Gaussian 03 and assuming S 4 symmetry. Calculation of the charge density was
done using the B3LYP functional with the 6-31g* basis set and the distributed multipole
analysis was performed using Prof. Stone’s gdma program. Geometries were generated
124

number of calculations
7000
6000
5000
4000
3000
2000
1000
spiro-unsub
spiro-Me
spiro-MeO
0
-0.06 -0.04 -0.02 0
∆E / eV
0.02 0.04
Figure 6.8: Histogram of ∆E for spiro-unsub (dashed line), spiro-Me (dotted line) and
spiro-MeO (full line). The standard deviation for the three materials is 8 meV
for spiro-unsub, 10 meV for spiro-Me and 15 meV spiro-MeO.
material σGDM/eV σDMA/eV
spiro-unsub 0.080 0.08
spiro-Me 0.080 0.010
sprio-MeO 0.101 0.015
Table 6.2: GDM energetic disorder σGDM and standard deviation in distribution of site
energy difference calculated using distributed Multipole analysis ( σDMA )
in the same way as in the previous section, using a minimum distance of approach of
3.5 Å and ensuring that pairs of molecules are in relative orientations which are evenly
distributed in orientational phase space. A minimum separation of 3.5 Å was chosen
taking into account the van der Waals radius of carbon. The distributions of values of ∆E
are shown in figure 6.8, whereas the values of standard deviations are shown in table 6.2.
The standard deviation for spiro-MeO is considerably larger than for spiro-Me or spiro-
unsub, while those for spiro-Me and spiro-unsub are similar, reflecting the trend in the
value of the energetic disorder parameters with chemical structure. We have also carried
out these calculations for a constant centre to centre separation of 18.5 Å while varying
125

the relative orientation at random to ensure that the minimum distance of approach is 3.4
Å to 3.6 Å. These calculations show the same trend in variations of the standard deviation
of ∆E for the three materials.
The values of σDMA are rather different in magnitude from the energetic disorder pa-
rameter σ. Several possible reasons could be responsible for this: first, considering inter-
actions with more neighbours might increase ∆E. Second, local minimum energy ground
state structures are likely to exist in such a large molecule, and these would contribute to
the distribution of ∆E. Third, without a good morphology model it is impossible to deter-
mine which particular pair geometries exist in the solid and would therefore contribute to
the site energies. What is clear, however, is that adding polar side groups to a molecule
will increase its short range electric field and therefore increase the magnitude of ∆E even
if it does not modify the molecule’s permanent dipole.
6.3 Conclusion
In this chapter we have shown how the fast methods developed for calculating ∆E and
J can be used to improve our understanding of charge transport in disordered solids by
interpreting the origin of GDM parameters in terms of molecular properties. In particu-
lar, the use of DMA for calculating ∆E allows us to make qualitative statements on the
strength and asymmetry of electrostatic interaction and allows us to understand the factors
contributing to energetic disorder.
126

Bibliography
[1] H. Sirringhaus, P.J. Brown, R.H. Friend, M.M. Nielsen, K. Bechgaard, B.M.W.
Langeveld-Voss, A.J.H. Spiering, R.A.J. Janssen, E.W. Meijer, P. Herwig and D.M.
de Leeuw, Two-dimensional charge transport in self-organized, high-mobility conju-
gated polymers, Nature, 401, 605 (1999)
[2] M.J. Winokur and W. Chunwachirasiri, Nanoscale StructureProperty Relationships
in Conjugated Polymers: Implications for Present and Future Device Applications,
Journal of Polymer Science: Part B: Polymer Physics, 41, 2630, (2003)
[3] H.C.F. Martens, P.W.M. Blom and H.F.M. Schoo, Comparative study of hole trans-
port in poly(p-para-phenylne vinylene) derivatives, Physical Review B, 61, 7489
(2000)
[4] M. Kemerink, J.K.J. van Duren, A.J.J.M. van Breemen, J. Wildeman, M.M. Wienk,
P.W.M. Blom, H.F.M. Schoo and R.A.J. Janssen, Substitution and Preparation Effects
on the Molecular-Scale Morphology of PPV Films, Macromolecules 38, 7784 (2005)
[5] P.L. Burn, D.D.C. Bradley, R.H. Friend, D.A. Halliday, A.B. Holmes, R.W. Jack-
son and A. Kraft, Precursor route chemistry and electronic properties of poly(p-
phenylnevinylene), poly[(2,5-dimethyl-p-phenylne)vinylene] and poly[(2,5-dimethoxy-
p-phenylne)vinylene], Journal of the Chemical Society, Perkin Transaction 1 23, 3231
(1992)
[6] M. Sims, S.M. Tuladhar, R.C. Maher, M. Campoy-Quiles, S.A. Choulis, J. Nelson,
D.D.C. Bradley, P.G. Etchegoin, C. Jones, A. Connor, K. Suhling, D.R. Richards,
P. Massiot, C. Nielsen and J.H.G Steinke, Structure-charge transport correlations in
poly(2,5-dimethoxy-1,4-phenylnevinylene) (PDMeOPV). , manuscript in preparation
127

[7] J.H.F. Martens, E.A. Marseglia, D.D.C. Bradley, R.H. Friend, P.L. Burn and A.B.
Holmes The Effect of Side Groups on the Structure and Ordering of poly(p-Phenylene
Vinylene) Derivatives, Synthetic Metals, 55, 449 (1993)
[8] Gaussian 03, Revision C.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuse-
ria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C.
Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G.
Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R.
Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene,
X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R.
Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W.
Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V.
G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick,
A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S.
Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi,
R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M.
Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J.
A. Pople, Gaussian, Inc., Wallingford CT, 2004.
[9] A.J. Stone, Distributed multipole analysis: Stability for large basis sets, Journal of
chemical theory and computation, 1, 1128 (2005)
[10] J. Caillet and P. Claverie, Theoretical Evaluation of the Intermolecular Interaction
Energy of a Crystal: Application to the Analysis of Crystal Geometry, Acta Crystallo-
graphica, A31, 448 (1975)
[11] U. Bach, K. de Cloedt, H. Spreitzer and M. Grätzel Characterisation of Hole Trans-
port in a New Class of Spiro-Linked Oligotriphenylamine Compounds, Advanced Ma-
terials, 12, 1060 (2000)
[12] D. Poplavskyy and J. Nelson, Nondispersive hole transport in amorphous films of
methoxy-spirofluorene-arylamine organic compounds, Journal of Applied Physics, 93,
341 (2003)
128

[13] F. Steuber, J. Staudigel, M. Stossel, J. Simmerer, A. Winnacker, H. Spreitzer, F.
Weissortel and J. Salbeck, White Light Emission from Organic LEDs Utilizing Spiro
Compounds with High-Temperature Stability, Advanced Materials 12, 130, (2000)
[14] J. Kruger, R. Plass, L. Cevey, M. Piccirelli, M. Grätzel and U. Bach, High efficiency
solid-state photovoltaic device due to inhibition of interface charge recombination,
Applied Physics Letters 79, 2085, (2001).
129

Chapter 7
Simulation of Charge Transport in
Assemblies of Hexabenzocoronenes and
Fluorene Oligomers
In the previous chapters, calculation of charge transfer parameters are carried out to give
qualitative understanding, either by helping parametrize lattice Monte Carlo simulations
(chapter five) or to explain GDM parameters (chapter six). In this chapter we present two
studies in which calculations of transfer integrals are used to model charge mobility in as-
semblies of molecules, as models of solid films of two particular materials. The transfer
integrals are used to calculate charge transfer rates using the Marcus charge transfer equa-
tion and these rates are in turn used to perform Monte Carlo simulations of charge dynam-
ics. Since the relative position and orientation of neighbouring molecules must be known
to calculate J, it is necessary to have a model of the molecular packing of a material.
The two material systems we consider were the crystalline phase of poly(dioctylfluorene)
(PFO) and the hexagonal mesophase of the discotic liquid crystal hexabenzocoronene
(HBC). In both cases the morphology model is provided by collaborators: in the case
of PFO Dr. S. Athanosopoulos and Dr. A. Walker from the University of Bath designed
morphologies using an empirical morphology model. In the case of HBC classical molec-
ular dynamics methods were used by Dr. V. Marcon and Dr. D. Andrienko from the Max
Planck Institute for Polymer Research in Mainz to obtain an atomistic simulation of the
morphology of columns of HBC. For the PFO calculations, Monte Carlo simulations
130

(a)
z
Electric field
x
(b)
φ 1
∆∆∆∆r
Figure 7.1: Schematic representation of the morphology model used by Dr. S.
Athanosopoulos to describe PFO and definitions of the parameters governing
the relative position and orientation of molecules. (a) The polymer chains
are packed hexagonally, with the chain axis perpendicular to the electric field.
(b) Each trimer is allowed to rotate by a torsion φ and can be displaced by a
distace ∆r in the direction perpendicular to the chain axis and can slip by a
distance dz parallell to the chain axis. (c) Top view of the central planes of
two trimers, showing the torsion angles φ1 and φ2 (∆φ = φ2 −φ1) and the polar
angle α.
were performed by our colleagues in Bath, while for HBC we used some in-house code,
adapted from Dr. A. Chatten’s and Prof. J. Nelson’s [1] code to be able to accomodate
lattices where the transport sites are arbitrarily distributed. By applying calculations of
transfer integrals to simulations of hopping transport the work described in this chapter
goes beyond previous studies in relating charge transport parameters to charge mobilities.
7.1 PFO
Charge transport in polyfluorene derivatives is heavily dependent on morphology and
orientation. For example, in poly(9,9 dioctylfluorene) (PFO) studies have shown [2] [3]
that changes in thermal treatment and alignment can lead to changes in mobility of over
two orders of magnitude. The aim of this part of the study is to examine the effect of
variations in the morphology of a film of crystalline polyfluorene on simulated charge
transport behaviour. The morphology simulation was developed by our colleagues in
Bath, Dr. A. Walker, Dr. P. Watson and Dr. S. Anthopoulos. This morphology simulation
is inspired by studies of PFO crystal structure by S. Kawana and co-workers [4]. In
that work, the structure of crystalline PFO that best fitted the observed X-ray spectra of
aligned films of PFO was a hexagonal arrangement of parallel chains. In our morphology
131
φ 2
y
(c)
φ 1
α
φ 2
x

Figure 7.2: HOMO (left) and HOMO-1 (right), of the restricted open shell orbitals of a
fluorene trimer with one of the hydrogen end groups missing.
model, polymer chains are packed in a hexagonal lattice with a particular lattice constant
a. The chains are segmented into charge transporting units, which we choose to represent
as trimers. This length oligomer is to 23 Å and is consistent with the proposed length of
polarons in PPV (25 Å [5]) and in polythiophene (20 Å [6]). The excluded volume of octyl
side chains was calculated and corresponds to a minimum distance of approach between
neighbouring polymer chains of 6.3 Å. It must be noted that the morphology model we
use is not truly ab initio. In particular, the hexagonal lattice constant was allowed to vary.
In fact the fundamental difference between this approach and models of charge transport
on lattices such as the GDM is that whereas in the lattice models the parameters of the
transfer rate equation are picked from arbitrary distributions, in our model the morphology
is chosen in a parametrized fashion but the transfer rates are then derived rigorously.
Figure 7.1 shows the model of morphology we use. The position and orientation of
a trimer is defined by its torsion angle φ (defined as the angle between the plane of the
central monomer to with respect to the central axis of the hexagonal axis), and its x,y,z
coordinates. In a torsionally disordered system φ is chosen at random for each monomer
and in a laterally disordered system the displacement in x and y are chosen at random,
132

ensuring that neighbouring trimers never come closer than 6.3 Å. Then, the relative po-
sition of two trimers is therefore uniquely defined by a displacement ∆r perpendicular to
the chain axis, a displacement dz parallel to the chain axis, the difference dφ between the
torsion angles and the polar angle α defining the position of the second trimer with respect
to the plane of the central monomer of the first trimer.
For each pair of nearest neighbouring trimers we use the Marcus charge transfer equa-
tion (equation 2.6) to find the transfer rates, having calculated the transfer integrals using
MOO and the reorganisation energy using a scheme such as that described in section 2.2
and using B3LYP/6-31g* for the geometry optimisations. The site energy difference ∆E
is calculated from the external electric field F and the relative position of the centres dr
as: ∆E = F.dr. We exclude electrostatic effects from our calculation of ∆E. Note that
since we treat all transport sites as trimers, no site energy disorder due to a distribution in
conjugation lengths is allowed. By eliminating such sources of site energy disorder this
approach should help to clarify the influence of morphology on charge transport charac-
teristics.
Four different morphologies are compared in this study:
• an ordered morphology where the centre of each trimer lies on the lattice points and
each trimer has the same torsion angle φ
• a torsionally disordered morphology where the torsion angles are varied at random
• a regular morphology with a unit cell composed of several trimers with the relative
torsion angles chosen to maximise transfer rates. We refer to this morphology as
optimally ordered
• a laterally disordered morphology where the trimers are displaced by random amounts
x, y.
Once the morphology has been generated, it is necessary to compute two different
types of transfer integral to describe charge transport in such assemblies of molecules:
intramolecular transfer integrals between neighbouring trimers in the same chain and in-
termolecular integrals between neighbouring trimers on different chains. Both were cal-
culated using the MOO method described in section 3.2. In this paragraph we discuss
133

how the end groups of the oligomers were treated. When calculating the transfer inte-
grals for intermolecular transfer, we added two hydrogen end groups to each trimer to
obtain the molecular orbitals. For the intramolecular transfer integrals we could only add
a single hydrogen end group to each trimer and had to therefore calculate the trimer’s
molecular orbitals in a doublet spin state. We performed this calculation of the orbitals
using a restricted open shell wavefunction: this way we obtain a singly occupied HOMO
which corresponds to the unpaired electron which, in the presence of another hydrogen
end group, would form a σ bond. We also obtain a doubly occupied HOMO-1 which is
virtually indistinguishable from the HOMO of the closed shell singlet with two hydrogen
end group. These orbitals are shown in figure 7.2. Since the end group hydrogens hardly
contribute to the HOMO of the closed shell trimer and the HOMO-1 of the open shell
trimer, the choice of whether or not to include these end groups is somewhat arbitrary, re-
membering that in the MOO scheme described in section 3.2 the density matrix does not
contribute to the calculation of the Fock matrix and hence the transfer integral depends
only on the orbitals of interest.
In this paragraph we show the results for the calculations of transfer integrals per-
formed on pairs of trimers on the same chain and on neighbouring chains. Figure 7.3
shows the dependence of the transfer integral on the polar and torsional angles for a dis-
tance of 6.3 Å, while figure 7.4 shows its dependence on separation for particular values
of α and ∆φ. When calculating the intramolecular integrals, the trimers are assumed to be
colinear and hence the transfer integral is assumed to depend only on the relative torsion
angle. Figure 7.5 shows the dependence of the modulus square of the transfer integral
on ∆φ. It should be noted that our treatment of intramolecular transport is still somewhat
unsatisfactory, because the largest intramolecular transfer integrals obtained are compa-
rable in size to the reorganisation energy λ. Therefore using the non-adiabatic Marcus
equation is not appropriate in this case. However the hopping rate predicted by the Mar-
cus transport equation along the polymer chain are still significantly faster than that for
interchain hopping and since we are measuring transport in the direction perpendicular to
the chain axis, interchain hops limit the mobility. Also, no model exists, to the best of our
knowledge that allows the treatment of both coherent and incoherent transport.
134

Polar angle α
350
300
250
200
150
100
50
0
0 100 200 300
Torsion angle dφ
Figure 7.3: Contour plot of log(|J| 2 ) as a function of relative torsion angle dφ and the
polar angle α which describes position. Notice how the values of these interchain
transfer integrals are always much smaller than the intrachain transfer
integrals from figure 7.5.
|J| 2 / eV 2
0.0001
1e-06
1e-08
1e-10
1e-12
6 7 8
centre to centre separation / A
9 10
Figure 7.4: Distance dependence of the transfer integral squared on distance for φ = 150 ◦
and α = 0 ◦ (squares) and for φ = 0 ◦ and α = 0 ◦ (circles).
135
−10
−9
−8
−7
−6
−5
−4
−3
log(|J| 2 ) /eV 2

transfer integral |J| 2 / eV 2
0.04
0.03
0.02
0.01
0
0 100 200
torsion angle ∆φ / degrees
300 400
Figure 7.5: Intrachain transfer integrals squared as a function of relative torsion φ for two
trimers.
The reorganisation energy for the trimer was calculated using the protocol described
in section 2.2 with the hybrid density functional B3LYP and Pople’s double split basis set
with extra polarisation basis functions 6-31g*. λ was found to be equal to 0.21eV.
Once the parameters for the Marcus charge transfer equation have been calculated
for all interchain and intrachain neighbours, charge transport can be simulated using the
Monte Carlo method. These transport simulations were carried out by Dr. S. Athanosopou-
los using a continuous time random walk algorithm with continuous boundary conditions
as described in section 2.5.1. First we simulate hole transport in an ordered lattice where
all the trimers have the same torsion φ. To choose the optimal torsion, we study the net
transfer in the field direction as a function of φ for an ordered lattice. The overall mobility
will be higher when the rates along the field (forward rates) are faster compared to rates
against the field (backward rates). The difference between forward and backward rates as
a function of φ is shown in figure 7.6 for several different fields. It can be seen that the
fastest net forward rate is obtained for φ = 20 ◦ . Thus, for the ordered morphology we
pick a torsion of 20 ◦ . The results for the hole mobility µh as a function of electric field
136

net rate (s -1 )
2.0×10 10
1.5×10 10
1.0×10 10
5.0×10 9
0.0
0 20 40 60 80
torsion angle φ (degrees)
Figure 7.6: Difference between forward and backward rates for an ordered lattice as
√
a function the torsion angle φ at which the trimers are set. F =
200(V/cm) 1/2 (solid line), 400(V/cm) 1/2 (dotted line), 600(V/cm) 1/2 (dashed
line), 800(V/cm) 1/2 (chained line, one dot), 1000(V/cm) 1/2 (chained line, two
dots).
are shown in figure 7.7 (diamonds) for a lattice constant of 6.5 Å (the choice of lattice
constant is justified below). Also shown are the results for φ = 0 ◦ (open triangles) and
a torsionally disordered system (squares). Results for lateral disorder are not shown, as
it has little effect on muh at this lattice constant because the trimers have little freedom
to move outside the excluded volume of PFO, which has a diamater of 6.3 Å. The fourth
plot (closed triangles) is for optimally ordered systems. The optimally ordered system
was designed ensuring that each trimer has a neighbour in the direction of the field with
∆φ = 150 ◦ (this is the difference in torsion angles that ensures greatest possible J 2 for
α = 0 ◦ as can be seen in figure 7.3). Also shown in figure 7.7 is the experimental mobility
for aligned films of PFO from reference [3]. As can be seen, the disordered morphology
with a lattice constant of 6.5 Å produces the best fit with experimental data.
The experimental hole mobility is weakly dependent on electric field. Equation 2.10
shows that for small fields the Marcus transfer rates are proportional to exp(−∆E/2kT);
137

Hole mobility µ h (cm 2 /Vs)
10 -1
10 -2
10 -3
0 200 400 600 800 1000
F 1/2 (V/cm) 1/2
10 -4
Figure 7.7: Dependence of mobility on the square root of the applied electric field for:
the ordered morphology with φ = 0 ◦ (open triangles), the ordered morphology
with φ = 20 ◦ (diamonds), the morphology with disordered torsions (full
squares), the optimally disordered morphology (full triangles) and the experimental
data on alligned PFO from reference [3] (full circles).The optimally
ordered morphology corresponds to when neighbours in the direction of the
field have a relative torsion φ of 150 ◦ . The lattice constant for the hexagonal
lattice is 6.5 Å .
138

therefore if the only contribution to the site energy is the electric field via ∆E = qF.r and if
the fields is small, we expect the velocity of charges to be proportional to the electric field
and the mobility to be field independent. Since our simulations were made in the absence
of disorder, the weak electric field dependence can be explained this way. To fit the
magnitude of the hole mobility we treat the lattice constant as a fitting parameter. There
are two interesting observations emerging from this study: first we notice that disorder
in the transfer integrals can increase mobility and second that the lattice constant (6.5 Å)
which needed to fit the experimental data is much shorter than the lattice constant (9.5 Å)
expected from a density of 1 gcm −3 .
The fact that disorder in the intermolecular transfer integral distribution can increase
mobility can be readily understood if one bears in mind that intramolecular charge transfer
is almost always faster than intermolecular charge transfer. Since charges are therefore
free to travel quickly along the chains, they will always be able to find locations with
highest transfer integrals for hopping to a neighbouring chain in the field direction. It ap-
pears therefore that in this system mobility is controlled by hot spots where the interchain
transfer integral takes the largest values. The large variability of transfer integrals on α
and ∆φ means that introducing disorder leads to wider distributions in |J| 2 . Since mobility
is set by the greatest transfer integrals in the distribution and not by the average value of
|J| 2 , a wide distribution in J 2 can lead to higher mobilities than a narrower one with a
greater average.
In order to fit the experimental data a lattice constant used of 6.5 Å was used. Such a
lattice constant is much higher than that which is necessary for PFO to have a density of
1 gcm −3 (approximately 9 Å). There are different possible interepretations of this obser-
vation. The mobility could have been underestimated because of an underestimation of
the transfer integrals or an overestimation of the reorganisation energy. Alternatively, one
could accept that in the polymer pathways with such high densities and short interchain
distances do indeed form. Another possible interpretation would be that the morphology
model we employed was fundamentally incorrect. We will discuss these three possibilities
in order.
We argue that the discrepancy cannot be explained by underestimation of the transfer
139

ates. The transfer integral would have to be greater by a factor of 1000 (based on the nat-
ural length of the decay of the transfer integral) to obtain similar mobilities to those we ob-
tained with a lattice constant of 6.5 Å with a lattice constant of 9.0 Å. Even if the reorgani-
sation energy had a value of 0.1 eV instead of the 0.21 eV we calculated, this would lead to
an increase in transfer rates for mobilities of a factor of √ (0.21/0.1)exp((0.21−0.1)/4kT)
which would be equal to roughly a factor of five, not the factor of 10 6 needed. Smaller
values of the reorganisation energy would easily lead to the electric field causing energy
differences greater than the reorganisation energy, under which the Marcus equation pre-
dicts the so-called inverted regime and rates become smaller with greater driving forces.
This would lead to negative field dependences which are not experimentally observed.
Also, we have ignored outer sphere reorganisation and as such it is highly unlikely that
we underestimated the reorganisation energy.
The hypothesis that mobility is dominated by high density regions is more compelling.
Studies show that the same thermal treatment which leads to higher µh also leads to an in-
creased average density [2]. Unfortunately this experimental technique cannot distinguish
between an even or an uneven increase in packing density. In order to test this hypothesis,
we design morphologies with lattice constants in the range of 6.5 Å to 10 Å, with maximal
lateral disorder. The increase in average interchain distance from 6.5 Å to 10 Å should
lead to a decrease in mobility of five orders of magnitude in an ordered system. However,
in the presence of lateral disorder, we find that the mobility is reduced by less than three
orders of magnitude; the reduction would be even smaller if longer polymer chains were
used to run the Monte Carlo simulations.
The third hypothesis, that the morphology of PFO is not hexagonal, rests on transmis-
sion electron microscopy by S.H. Chen and coworkers [7], which came to our attention
after our model had been designed. They find that the unit cell of PFO is rather large (or-
thorhombic a=2.56 nm, b=2.34 nm, c=3.32 nm) compared to the unit cell we assumed.
When they perform molecular mechanics simulation of 8 chains of PFO in such a unit
cell, they obtain a morphology with interchain distances as short as approximately 6 Å.
Our mobility calculation would be entirely consistent with such interchain distances. The
last two hypotheses are not in disagreement with each other: in both cases we are argu-
140

ing that there is experimental evidence refuting a perfect hexagonal lattice structure in
PFO. At the moment it is still unclear whether the crystal structure from S.H. Chen and
coworkers would fit experimental data, or whether lateral disorder on larger assemblies
of molecules would be sufficient. In fact an interesting possibility from this study is that
calculations of mobility can be used to test the validity of a morphology model.
In conclusion, our investigation strongly suggests that charge transport in PFO is de-
termined by interchain hops where the conjugated backbones are separated by a distance
of approximately 6.5 Å. In order for charge transport to be determined by the fastest in-
terchain, only a minority of interchain distances between charge transporting units need
to be so short.
7.2 HBC
Discotic thermotropic liquid crystals are formed by flat molecules with a central aromatic
core and several aliphatic chains attached at its edges [8, 9, 10, 11, 12]. Discotics can
form columnar phases, where the molecules stack on top of each other and the resulting
columns are arranged in a regular lattice[9]. The self-organisation into stacks with aro-
matic cores surrounded by saturated hydrocarbons results in the onedimensional transport
of charge within the core, along columns [13, 14]. By modifying the periphery or shape
of the core it is possible to obtain materials with different electronic characteristics and
phase behaviour. Charge transport occurs preferentially along the columns and the mo-
bility in directions perpendicular to the column axis is much smaller, since in this case
the charges have to tunnel through the insulating side chains. Therefore the discotics in
a columnar phase can be seen as nano-wires and an alternative to conjugated polymers,
where intrachain transport is fast but where charge mobility in devices is often limited by
inefficient interchain hops.
In this part of the study we simulate charge transport in different derivatives (shown
in figure 7.8) of the discotic liquid crystal hexabenzocoronene (HBC) in a column and
compare the results to pulse radiolysis time resolved microwave conductivity (PR-TRMC)
studies of the mobility in these materials. The structure of the simulation we employ is
very similar to that of the previous simulation on PFO: a morphology for stacks of HBC
141

R
R
R
R
R
R
C12
PhC12
C10-6
Figure 7.8: Chemical structure of the various derivatives of HBC considered in this study
is generated, the transfer integrals are calculated for the nearest neighbours in the stack,
the transfer integrals are used to calculate charge transfer rates using the Marcus charge
transfer rate equation and finally the charge transfer rates are used to simulate charge
transport using Monte Carlo and Master Equation methods. The main difference between
this study and the study on polyfluorene is that the morphology model we use to determine
the relative position and orientation of HBC molecules is not an empirical model, but is the
result of an atomistic molecular dynamics simulation of HBC molecules. The resulting
simulation is able to replicate the trends and magnitudes of mobilities measured in the
different derivatives considered in this study.
The use of the Marcus equation is justified by experiments by T. Kreouzis and co-
workers [15] who found that Holstein’s small polaron model of charge transport is best
capable of describing charge transport in the discotic liquid crystal triphenylene. The
parameters required for the Marcus electron transfer equation, i.e. transfer integrals for
nearest neighbours and reorganisation energies, are calculated in a manner very similar to
the previous study on polyfluorene. The reorganisation energies for electrons and holes
are calculated using the scheme described in section 2.2 using the functional B3LYP with
the basis set 6-31g**. We find that the reorganisation energies are of 0.13 eV and 0.11
eV for negative and positive charges respectively. A technical difference from the case
of polyfluorene is that the HOMO and LUMO of HBC are doubly degenerate, thus we
use an effective transfer integral as defined in section 2.3.1; that is we consider all four
transfer integral between the HOMOs (LUMOs) and HOMO-1s (LUMO+1s) on either
molecule then take the root mean square of the set.
The molecular dynamics simulation of the morphology of HBC derivatives was car-
142

ied out by Dr. V. Marcon and Dr. D. Andrienko at the Max Planck Institute for Polymer
Research in Mainz. Molecular Dynamics (MD) simulation is based on using parametrised
potentials (force fields) to describe the bonded interactions (harmonic bond, angle and tor-
sions) and non-bonded interaction (van der Waals and electrostatic interactions) between
the atoms of several different molecules of HBC. The potentials are then used to integrate
Newton’s equations. In our case some atoms were substituted by pseudoatoms, namely
the hydrogens in the aliphatic side chains were incorporated into the carbons they were
bonded to. The exact force field used is described in reference [16]. The molecules were
arranged in 16 columns with 10, 20, 60 or 100 molecules in each column. The time step
for integration of Newton’s equations of motion was 2 f s and each run lasted 100 ns. The
temperature and pressure of the simulation were kept fixed at 300 K and 0.1MPa using
the Berendsen method. A snapshot of the system was taken every 20 ps; such large time
intervals are necessary in order to ensure that the configurations are not correlated with
one another. For each of these snapshots we recorded the relative position and orienta-
tion between nearest neighbours in a column. The relative position and orientation are
then used to calculate the effective transfer integral between two conjugated cores with
a fixed internal geometry: in other words, before the MOO method is used to calculate
transfer integrals the cores from the MD simulation are substituted with rigid copies of
ones obtained from geometry relaxation of the neutral geometry. We consider only near-
est neighbours on the same column because the distance between conjugated cores on
separate columns is much larger than the distance between cores on the same column
and therefore the transfer integral between columns will be negligibly small compared to
transfer along the columns.
Having calculated the transfer integrals for a particular snapshot, we simulate time
of flight (ToF) experiments using a continuous time random walk algorithm. The details
of the simulation are as follows. A column of HBC molecules of a height of 1µm (cor-
responding to approxiately 2800 molecules) is obtained by stacking copies of the same
stack from the MD simulation on top of each other. At the beginning of the simulation
a charge is positioned at random near the beginning of the column. The motion of the
charge and the resulting displacement current is calculated using the continuous time ran-
143

Displacement current
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
TOF, single configurations
C12, 300K, 10V
1
10
20
30
40
50
60
70
80
90
100
average (190)
0
0 5e-10 1e-09 1.5e-09
time (sec)
2e-09 2.5e-09 3e-09
Displacement current
0.8
0.6
0.4
0.2
TOF block averages
C12, 300K, 1000 Nsims
0
0 1e-09 2e-09
time (sec)
Figure 7.9: ToF results for several selected snapshots, together with the averaged over
the MD run displacement current (left). Block averages over 10, 20, ... 190
snapshots. (right)
dom walk Monte Carlo (MC) algorithm described in section 2.5.1. Two thousand such
displacement currents are then averaged to ensure that the displacement current reflects
the properties of that column.
Several displacement currents are shown in the left panel of figure 7.9. They were
obtained for different columns from different snapshots for the C12 derivative with stacks
10 molecules high. It can be noticed that the transit time is rather different between
different columns, hence it will be necessary to average the displacement currents for
different columns. The averaged current transients fall off less rapidly than the transients
for individual columns; this dispersion is due to the distribution in arrival times for the
different columns. The right panel of figure 7.9 shows that 200 snapshots are sufficient
to obtain a converged displacement current. The observation of variation in mobility for
different columns is the result of the one dimensional nature of transport along columns
of HBC: mobility is heavily determined by the smallest transfer integral, whose value
fluctuates between different columns.
Figure 7.10 shows the variation of displacement currents for columns composed of
different length stacks of the C12 derivative. Note that the size of column that the MC
simulation is run over is unchanged, what changes is the number of molecules in each
column over which MD is carried out. As the number of molecules in a stack becomes
larger, the transit time increases slightly. This again can be understood by remembering
that the smallest transfer integral sets the mobility: as stack length increases the chances
144
10
30
50
70
90
110
130
150
170
190

Displacement current
0.8
0.6
0.4
0.2
0
TOF, column length dependence
C12, 400K, 200 configurations average, 10V
10 molecules
20 molecules
100 molecules
0 1e-09 2e-09 3e-09
time (sec)
Figure 7.10: Displacement current for different numbers of molecules in a column. Average
over 200 snapshots at an electric field of 10 5 Vcm −1 .
of smaller transfer integrals occuring increases. A stack length of 100 molecules seems
sufficient to converge a value of the transit time and will be used throughout in the rest of
the study. In the next part of the study we calculate mobilities for holes and electrons for
all the derivatives shown in figure 7.8.
Figure 7.11: Radial distribution function along the z direction for all six derivatives considered
in this study.
Having established the size of the system to use for the Monte Carlo simulation, let us
compare the morphology of the different materials we look at in this study. In particular
we consider two measures, the centre of mass separation between neighbouring molecules
in a column, h, and the nematic order parameter, Q. Figure 7.11 shows the radial distri-
bution function for all six derivatives, that is the probability of finding a molecule at a
particular distance along the z axis, defined as the symmetry axis of the stack. The four
145

derivatives with unbranched aliphatic chains C10, C12, C14 and C16 all show a high de-
gree of order and have a similar mode and average intermolecular separation of 3.6 Å.
The derivative with the branched side chain C10-6 is considerably more disordered. Al-
though the mode distance to the nearest neighbour (the position of the first peak in figure
7.11) is identical to that for unbranched side chains, the probability does not fall to zero
for positions away from the crystal positions. As we shall see in the next paragraph this
is a consequence of misalignment. The derivative with phenyl rings Ph-C12 shows larger
nearest neighbour distances than the unbranched side chains, but is more crystalline than
the branched side chain.
The nematic order parameter Q is obtained by calculating the vector normal to the
molecular plane of the i th HBC molecule u (i) . A matrix Qαβ is then calculated by taking:
Qαβ =
� 1
N
N�
i=1
�
3
2 u(i) α u (i) 1
β −
2 δαβ
��
(7.1)
This matrix is diagonalized and the greatest eigenvalue is called the nematic order
paramater Q. The maximum value that the order parameter can take is 1, denoting perfect
ordering. The minimimum is 0, denoting an isotropic arrangements. Table 7.1 shows the
order parameter Q for the various HBC derivatives considered. Q takes very large val-
ues for all the unbranched aliphatic derivatives, and slightly smaller ones for the C10-6
and Ph-C12 derivatives respectively. This confirms the information from the radial distri-
bution function, that C10-6 and Ph-C12 form less well ordered stacks than the aliphatic
unbranched side chain derivatives. The fact that columns of C10-6 have a slightly higher
order parameter than Ph-C12, suggests that C10-6 stacks are composed of short segments
of well ordered molecules, which do not stack well with each other. The small mode
distance in C10-6 stacks is explained by the fact that most molecules are in well ordered
segments, but the badly defined peaks at greater distances correspond to the fact that these
segments are misaligned.
Figure 7.12 shows histograms of the values of the calculated hole transfer integrals
for the six derivatives. As can be seen, the distribution of transfer integrals for each of the
unbranched side chain derivatives is very similar. Ph-C12 has a wider peak and a smaller
mean than the aliphatic side chains reflecting the greater degree of disorder in columns
146

compound µPR−TRMC µ e
ToF µ h
ToF µ e
ME µ h
C10 0.5 [17] 0.22 0.75 0.14
ME
0.49
Q
0.98
h
3.6
C12 0.9 [17] 0.23 0.76 0.14 0.49 0.98 3.6
C14 1 [17] 0.27 0.80 0.17 0.59 0.98 3.6
C16 - 0.29 0.91 0.17 0.58 0.98 3.6
C10−6 0.08 [18] - 0.01 6e-4 0.003 0.96 3.7
PhC12 0.2 [17] 0.036 0.13 0.012 0.045 0.95 4.1
Table 7.1: Electron (e) and hole (h) mobilities (cm 2 V −1 s −1 ) of different compounds calculated
using time-of-flight (ToF) and master equation (ME) methods, in comparison
with experimentally measured PR-TRMC mobilities. Also shown are the
nematic order parameter Q and the average vertical separation between cores
of HBC molecules h (Å).
and the greater separation between molecules. C10-6 has a peak in values of log(|J 2 |)
with values which are actually slightly larger than those of the aliphatic unbranched side
chains, reflecting the similar value of the mode of the separation between nearest neigh-
bours. However, C10-6 has also a tail of low values of log(|J 2 |), reflecting the high degree
of disorder in the columns. These histograms explain the trend in calculated mobilities
shown in table 7.1. Because of the one dimensional nature of transport in HBC columns,
mobility is determined by the lowest transfer integrals in the column. For this reason
C10-6 has smaller mobilities than any other derivative, even though it has the highest
mode transfer integrals. Table 7.1 shows mobility as calculated by both a Monte Carlo
and a Master Equation formalism and compares the mobilities to results from PR-TRMC,
which probes the sum of electron and hole mobility. Our calculations match both the
trends and magnitudes of the charge mobility, with no adjustable parameters.
7.3 Conclusion
In this chapter we have shown calculations of electric field dependence of mobility for two
materials. We argue that the fact that we are able to reproduce experimental mobility for
HBC with no free parameters proves the validity of the methods used to calculate transfer
integrals and reorganisation energies. Agreement with experiment could also be obtained
in the case of PFO, however the morphology model required empirical parameters.
147

counts
200
150
100
50
200
150
100
C10
0
-8 -6 -4 -2 0
50
200
150
100
0
-8 -6 -4 -2 0
50
C12
C14
0
-8 -6 -4 -2 0
200
150
100
50
C10_6
0
-8 -6 -4 -2 0
70
60
50
40
30
20
10
0
-8 -6
C12_Ph
-4 -2 0
200
150
100
50
2
log (J ) 10 eff
C16
0
-8 -6 -4 -2 0
Figure 7.12: Distribution in logarithm of the effective transfer integrals for holes squared
for 20 columns of 100 molecules each.
148

Bibliography
[1] A.J. Chatten, S.M. Tuladhar, S.A. Choulis, D.D.C. Bradley, and J. Nelson, Monte
Carlo modelling of hole transport in MDMO-PPV:PCBM blends, Journal of Materials
Science, 40, 1393 (2005)
[2] T. Kreouzis, D. Poplavskyy, S.M. Tuladhar, M. Campoy-Quiles, J. Nelson, A.J.
Campbell, and D.D.C. Bradley, Temperature and field dependence of hole mobility
in poly(9,9-dioctylfluorene), Physical Review B 73, 235201 (2006)
[3] M. Redecker, D.D.C. Bradley, M. Inbasekaran and E.P. Woo, Mobility enhancement
through homogeneous nematic alignment of a liquid-crystalline polyfluorene, Applied
Physics Letters 74, 1400 (1999)
[4] S. Kawana, M. Durrell, J. Lu, J.E. MacDonald, M. Grell, D.D.C. Bradley, P.C. Jukes,
R.A.L. Jones and S.L. Bennett, X-Ray diffraction study of the structure of thin polyflu-
orene films, Polymer, 43, 1907 (2002)
[5] Y. Shimoi, S. Abe, S. Kuroda and K. Murata, Polarons and their ENDOR spectra in
poly p-phenylene vinylene, Solid State Communications 95, 137 (1995)
[6] M. Knupfer, J. Fink and D. Fichou, Strongly confined polaron excitations in charged
organic semiconductors, Physical Review B, 63, 165203 (2001)
[7] S. H. Chen, H.L. Chou, A.C. Su and S.A. Chen, Molecular Packing in Crystalline
Poly(9,9-di-n-octyl-2,7-fluorene), Macromolecules, 37, 6833 (2004)
[8] M. Müller, C. Kubel and K. Müllen, Giant polycyclic aromatic hydrocarbons, Chem-
istry -A European Journal 4, 2099 (1998)
149

[9] S. Chandrasekhar and G.S. Ranganath, Discotic liquid-crystals, Reports on Progress
in Physics, 53, 57 (1990)
[10] J.D. Brand, C. Kubel, S. Ito and K. Müllen, Functionalized hexa-peri-
hexabenzocoronenes: Stable supramolecular order by polymerisation in the discotic
mesophase, Chemistry Of Materials , 12, 1638 (2000)
[11] R.J. Bushby and O.R. Lozman, Discotic liquid crystals 25 years on, Current Opinion
In Colloid and Interface Science, 7, 343 (2002)
[12] S. Kumar, Recent developments in the chemistry of triphenylene-based discotic liq-
uid crystals, Liquid Crystals, 31, 1037 (2004)
[13] A.M. van de Craats, J.M. Warman, A. Fechtenkotter, J.D. Brand, M.A. Harbison
and K. Müllen, Record charge carrier mobility in a room-temperature discotic liquid-
crystalline derivative of hexabenzocoronene, Advanced Materials, 11, 1469 (1999)
[14] L. Schmidt-Mende, A. Fechtenkotter, K. Müllen, E. Moons, R. H. Friend and J. D.
MacKenzie, Self-organized discotic liquid crystals for high-efficiency organic photo-
voltaics, Science, 293, 1119 (2001)
[15] T. Kreouzis, K.J. Donovan, N. Boden, R.J. Bushby and O.R. Lozman, Temperature-
independent hole mobility in discotic liquid crystals, Journal of Chemical Physics, 114,
1797 (2001)
[16] D. Andrienko, V. Marcon and K. Kremer, Atomistic simulation of structure and
dynamics of columnar phases of hexabenzocoronene, Journal of Physical Chemistry,
125, 124902 (2006)
[17] A.M. van de Craats, L.D.A. Siebbeles, I. Bleyl, D. Haarer, Y.A. Berlin, A.A.
Zharikov and J.M. Warman, Mechanism of Transport along Columnar Stacks of a
Triphenylene Dimer, Journal of Physical Chemistry B, 102, 9625 (1998)
[18] W. Pisula, M. Kastler, D. Wasserfallen, M. Mondeshki, J. Piris, I. Schnell and
K. Müllen, Relation between Supramolecular Order and Charge Carrier Mobility of
Branched Alkyl Hexa-peri-hexabenzocoronenes, Chemical Materials, 18, 3634 (2006)
150

Chapter 8
Conclusion and Future Work
We have discussed the role of electronic structure theory calculations in helping to under-
stand charge transport characteristics of conjugated materials. The starting point of this
approach is the Marcus charge transfer theory rate equation, which takes as parameters
three quantities: the transfer integral, the reorganisation energy and the site energy dif-
ference. Ideally we would calculate the charge transfer rates for all nearest neighbours
in a material and would then simulate charge dynamics in a realistic simulation. Since
charge dynamics simulation volumes can be very large, it is of paramount importance
to be able to calculate the parameters of the Marcus charge transfer equation using fast
methods. For this reason we developed methods to calculate transfer integrals and site
energy differences for pairs of molecules without performing SCF calculations.
In chapter five we applied these methods to justify the choice of parameters to be used
in a lattice simulation of charge transport in blends of MDMO-PPV:PCBM. In chapter
six, calculations were performed on pairs of molecules in random relative orientation,
with the aim of comparing the distributions in transfer integrals and site energy difference
to the distributions obtained using the Gaussian Disorder Model. In both these applica-
tions, the aim of our calculations was qualitative. Chapter seven contains two full-blown
simulations of charge transport in a conjugated polymer (polyfluorene) and in a liquid
crystal (hexabenzocoronene). The study on hexabenzocoronene demonstrates that if a
good morphology model is available, it is possible to quantitatively predict the mobility
of a material.
Future work will be directed towards two main areas: extending the work on hex-
151

abenzocoronene to polymers and simulating temperature as well as field dependence of
mobility. Two challenges will have to be resolved for a treatment of charge transport in
polymers: first, calculating morphologies for polymers will be substantially harder than
for hexabenzocoronene and second the issue of charge delocalisation along the polymer
chain discussed in chapter four will have to be resolved. In order to simulate the tem-
perature dependence of mobility it will be necessary to calculate site energies accurately,
calculating electrostatic and inductive interactions. This will require extending the meth-
ods from chapter two to treat slabs of material rather than pairs of molecules only.
152