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Density-functional study of the evolution of the electronic structure of oligomers of thiophene:
Towards a model Hamiltonian
R. Telesca,
1
H. Bolink,
2
S. Yunoki,
3
G. Hadziioannou,
2
P. Th. Van Duijnen,
1
J. G. Snijders,
1
H. T. Jonkman,
3
and
G. A. Sawatzky
3
1
Department of Theoretical Chemistry, Material Science Center, University of Groningen, Nijenborgh 4,
9747 AG Groningen, The Netherlands
2
Department of Polymer Chemistry, Material Science Center, University of Groningen, Nijenborgh 4,
9747 AG Groningen, The Netherlands
3
Department of Solid State Physics, Material Science Center, University of Groningen, Nijenborgh 4,
9747 AG Groningen, The Netherlands
共Received 19 July 2000; published 30 March 2001兲
We present density-functional and time-dependent density-functional studies of the ground, ionic, and ex-
cited states of a series of oligomers of thiophene. We show that, for the physical properties, the most relevant
highest occupied and lowest unoccupied molecular orbitals develop gradually from monomer molecular orbit-
als into occupied and unoccupied broad bands in the large length limit. We show that band gap and ionization
potentials decrease with size, as found experimentally and from empirical calculations. This gives credence to
a simple tight-binding model Hamiltonian approach to these systems. We demonstrate that the length depen-
dence of the experimental excitation spectra for both singlet and triplet excitations can be very well explained
with an extended Hubbard-like Hamiltonian, with a monomer on-site Coulomb and exchange interaction and
a nearest-neighbor Coulomb interaction. We also study the ground and excited-state electronic structures as
functions of the torsion angle between the units in a dimer, and find almost equal stabilities for the transoid and
cisoid isomers, with a transition energy barrier for isomerization of only 4.3 kcal/mol. Fluctuations in the
torsion angle turn out to be very low in energy, and therefore of great importance in describing even the
room-temperature properties. At a torsion angle of 90° the hopping integral is switched off for the highest
occupied molecular orbital levels because of symmetry, allowing a first-principles estimate of the on-site
interaction minus the next-neighbor Coulomb interaction as it enters in a Hubbard-like model Hamiltonian.
DOI: 10.1103/PhysRevB.63.155112 PACS number共s兲: 71.15.Ap, 71.15.Mb, 78.30.Jw
I. INTRODUCTION
Oligomers consist of monomer units coupled together to
form small chain length polymers. They are highly interest-
ing because of many potential applications in electronic and
optoelectronic devices. Recently bulk heterojunction, photo-
voltaic cells,
1
light-emitting diodes
2
and field-effect
transistors
3
have been demonstrated with oligomers of
thiophene as one of the active components. Because of their
interesting optoelectronic charge generating and charge
transporting properties, there is considerable interest in the
basic electronic structure of the oligomers and the fundamen-
tal interactions, which determine the charge- and spin-
density distributions in the ground, ionized, and excited
states of these systems.
4,5
There are still strong debates con-
cerning the importance of the electron-electron interactions,
electron-vibronic coupling, the size of the bandwidths in
large systems, the influence of torsion motions, and the cou-
pling with electronic degrees of freedom which determine
the existence and spatial extent of excitonic states, the local-
ization lengths of excitations, the mobility of charge carriers
in a single strand polymer, the optical oscillator strengths,
and the ionization potentials. Many of the physical properties
of
-conjugated systems, of which polyacetylene is the most
used example, can be very well explained within the Su-
Schrieffer-Heeger 共SSH兲 model,
6
in which the electronic
structure is modeled according to that of one-dimensional
strongly dimerized chain of units with one orbital of impor-
tance per unit. This models the alternation of double and
single bonds in polyacetylene and, with two electrons per
dimer, this results in a semiconducting material. In the SSH
model the electron-lattice coupling is considered to be very
strong, resulting in a strong polarization of the lattice around
the free charge carriers in the valence or conduction bands.
Moreover, because of the conjugated nature of these systems,
the actual charges and spins are bound to antiphase domain
boundaries in the lattice alternation, and move as free soli-
tons along the chain. In this model the electron-electron in-
teractions are neglected, so that the first excited states are
expected to be charged polarons or solitons rather than lo-
cally charge neutral excitons, as expected for strong local
Coulomb interactions. This model is also often applied to the
thiophenes, which have an electronic structure as depicted in
Fig. 1, where we highlight the backbone, which looks like
polyacetylene. However the sulfur atom in each monomer
breaks the symmetry present in polyacetylene, thereby re-
moving the twofold degeneracy of the ground state and cre-
ating, as a consequence, quite a different system than poly-
acetylene. In the past two decades a large number of
experimental and theoretical studies of the ground and ex-
cited states of thiophene and its oligomers has been pub-
lished, and these were reviewed in a recent book edited by
Denis Fichou.
5
An important contribution in gaining more
fundamental insight into the electronic structure of oligomers
and polymers has been in the formulation of effective low-
PHYSICAL REVIEW B, VOLUME 63, 155112
0163-1829/2001/63共15兲/155112共11兲/$20.00 ©2001 The American Physical Society63 155112-1
energy models. For example Mintmire and White
7
came to a
first-principles estimate of the optical spectra of poly-
acetylene within an Ehrenreich-Cohen approach; in other
studies
8,9
a Pariser-Parr-Pople 共PPP兲-type Hamiltonian with
an exciton basis set was used, which allowed them to char-
acterize the excited states, and which gives a better under-
standing of the photo physics of these materials. Soos et al.
10
used exact PPP results to describe the excited state structure
of oligomers. Van der Horst et al.
11
studied the electronic
and optical excitations of polythiophene using the GW 共G
stands for one-electron Green function, W for the screened
Coulomb interaction兲 approximation for the electronic self-
energy, and included excitonic effects by solving the
electron-hole Bethe-Salpeter equations. Recently a detailed
study of the polymer poly phenylene vinylene 共PPV兲 was
published using density-functional methods and the GW ap-
proximation for the self-energy.
12
The calculated optical
spectra suggest very strong electron-hole Coulomb interac-
tions, resulting in strongly bound excitonic states and a large
splitting of the singlet and triplet excitons. In our study pre-
sented here on the oligomers of thiophene, we will come to
similar conclusions concerning the importance of the Cou-
lomb interactions.
Various forms of optical spectroscopy together with the-
oretical interpretations have provided us with the location of
the lowest-energy singlet excitations,
13–26
as well as in some
cases the triplets.
27
As we will discuss below, the lowest-
energy singlet excitations show a 1/N-like dependence 共with
N the number of rings兲 on the oligomer length, whereas the
lowest triplets are considerably lower in energy. This is al-
ready an indication of a relatively strong electron-electron
interaction, which, at least for triplet excitations, forms an
excitonic state with the electron and hole in proximity to the
same monomer, reminiscent of a Frenkel-like limit for the
exciton. This in itself requires an effective electron-hole in-
teraction of the order of the one electron bandwidth in the
limit of large N.
The lowest singlet excitation seems to be much closer to
the band edge corresponding to the dissociated electron-hole
continuum, which is also the conductivity gap. Actually we
are not aware of data really pinning down the conductivity
gap in the large-N limit. Photoemission studies in the gas
phase provide us with accurate values for the ionization po-
tentials which, as the lowest singlet excitations, decrease like
1/N with the oligomer size and are in fact well described by
a simple tight-binding or Hu
¨
ckel-like model for the
molecular-orbital splitting with system size.
21
It is also inter-
esting to note that the two-photon lowest-energy singlet ex-
citation was expected by the authors of Ref. 22 to cross
below the lowest singlet optical excitation for N⬎ 6, possibly
another strong indication of the importance of electron-
electron interactions.
In this paper we present the results of a density-functional
calculation of the electronic structure up to lengths of N
⫽ 8. The advantage is that density-function theory 共DFT兲 is
at least in principle exact for determining the ground-state
properties of the molecules and ions, and also can be applied
to very large systems. In order to study the excited states we
used the time-dependent density-functional theory 共TDDFT兲
method, that can then be compared to the excitation energies
derived from the occupied and unoccupied orbital energies
obtained from the DFT calculation. A strong difference be-
tween ionization energies and excitation energies derived
from the DFT and the ⌬SCF self-consistent-field calculations
for the ionization potentials as well as the TDDFT calcula-
tions of the excitation energies is again indicative of the im-
portance of electron correlation effects in describing the sys-
tem. These results are in fact consistent with ab initio
calculations of small systems, and also with the semiempir-
ical calculations mentioned above, as well as with the experi-
mentally determined energies where available. This exhibits
the power of these methods, which can be applied to large
systems and are not dependent on empirical parameters. In
addition to this, we also study the theoretical optical oscilla-
tor strengths using TDDFT theory and the development of
these with system size. In each case we interpret the results
in terms of simplified models, which can serve as a basis for
model Hamiltonians used to describe these systems.
Another aspect, which turns out to be very interesting, is
the energy of rotational disorder and the influence of this on
the electronic structure and optical properties. For this we
studied the ground and excited states of bithiophene (
␣
-2T)
as functions of the dihedral angle between the monomer
units. Fluctuations in the dihedral angle turn out to be very
low in energy, and therefore of great importance in describ-
ing even room-temperature properties. Not only is the energy
cost low for such excitations, but the electronic structure and
optical properties are also strongly dependent on such fluc-
tuations, indicating a strong electron librational coupling.
It is also interesting that for a dihedral angle of 90° the
monomer molecular orbitals, which determine the low-
energy properties, are completely decoupled, i.e. the hopping
integrals are zero for the highest occupied molecular-orbital
共HOMO兲-based orbitals of the dimer. This gives us a direct
handle to calculate the energy of a monomer-monomer
charge-transfer excitation, which, when compared to the ex-
citation energy on a single monomer, gives us an ab initio
measure of the difference between the on-site and nearest-
FIG. 1. Schematic structure of the oligothiophenes. The ener-
getically nongenerate isomeric aromatic and quinoid structures are
displayed.
R. TELESCA et al. PHYSICAL REVIEW B 63 155112
155112-2
neighbor Coulomb interactions in an extended Hubbard-like
model Hamiltonian. In this way we arrive at what we think is
a reliable ab initio estimate of all the electronic parameters
needed for such a model Hamiltonian, including the coupling
to dihedral rotational modes. The coupling with the molecu-
lar vibration modes can in principle also be calculated in this
way, but remains a subject for future studies. However rea-
sonable estimates of these can also be obtained from the
optical
15
and photoelectron spectra presented here.
Before presenting the results, we look at some of the basic
information concerning the monomers, which will turn out to
be important in describing what happens as we increase N in
the oligomers. In Fig. 2 we represent the symmetries and
molecular-orbital buildup for the
orbitals of the thiophene
molecule. We immediately note that the problem is more
complicated than the molecular-orbital structure of the
monomer considered in polyacetylene. The most important
thing here is that the molecular orbital of the HOMO (1a
2
)
and the lowest unoccupied molecular orbital 共LUMO兲 (3b
1
)
have quite different compositions, and originate from differ-
ent symmetries. So the HOMO, which can be characterized
as an aromatic molecular orbital, and will turn out to form a
valence band in the oligomer, has no density on the sulfur
atom because of symmetry. However, the LUMO, which can
be classified as a quinoid is in this regard quite strongly
coupled to sulfur p-
orbitals. Since the LUMO of the
monomer develops into the conduction-band states in the
large oligomer limit, we see that the valence and conduction
bands are derived from strongly different monomer molecu-
lar orbitals. This means that any model Hamiltonian ap-
proach should contain at least two different bands, quite dif-
ferent from the two bands one would obtain in a Peierls-
distorted one-band system as proposed for polyacetylene.
Another point of great importance is that the HOMO has
a large component on the
␣
carbon atom, which is involved
in the bonding between the monomers in the oligomer, so
that the effective hopping integrals will be large. Surpris-
ingly enough, despite the strong difference in character be-
tween the HOMO and LUMO, the effective hopping inte-
grals are of the same magnitude, resulting in similar valence-
and conduction-band-widths in the large size limit. We also
note that the sign of the hopping integral will be opposite for
the valence and conduction bands, leading to a band struc-
ture with the minimum direct gap at k⫽ 0. Also, the inter-
monomer mixing of the HOMO and LUMO to the left and
right monomers of a central monomer will have opposite
signs, and will therefore cancel at k⫽ 0.
Another complicating factor is the close proximity of the
HOMO⫺ 1(2b
1
) and the LUMO⫹ 1(2a
2
) to the HOMO
and LUMO, respectively. This is because it will turn out that
the intermonomer molecular-orbital splitting of the HOMO
is much larger than that of the HOMO⫺ 1, which has negli-
gible electron density on the
␣
carbon atom positions, so that
the 2b
1
-derived states remain close to the center of the va-
lence band, which could have very important consequences
for the optical properties of the oligomers at somewhat
higher energies. For the LUMO this is a minor problem,
because here the splitting between the 3b
1
and 2a
2
orbitals
is larger than the expected conduction-band-width, so that
the 2a
2
-derived states remain outside of the conduction-band
states or nearly so. We should note, however, that the 2a
2
molecular orbital has considerable density on the
␣
carbon
and therefore will develop a considerable bandwidth, al-
though this turns out to be smaller than the bands derived
from the HOMO or LUMO. Of course the really low-energy
scale properties at an energy scale of the band gap in the
larger length oligomers will not be influenced much by these
other molecular orbitals, but they will be important for
higher-energy excitations. We now have a good basis for
discussing DFT and TDDFT calculations on the oligomers.
II. THEORY
In order to gain insight into the nature of the ground,
ionized and excited states of thiophene oligomers as function
of their chain length, we analyzed their electronic structure
applying 共time-dependent兲 density-functional theory to these
systems, implemented by the Amsterdam Density Functional
Program Package 共ADF兲,
28–30
which is able to provide accu-
rate solutions of the Kohn-Sham 共KS兲 equations even for
fairly long oligomers. In the ground-state calculations we
used the local-density approximation, based on the param-
etrization of the electron gas data given by Vosko, Wilk and
Nusair.
31
The basis sets used were of triple zeta plus polar-
ization Slater-type orbital function quality 共basis IV in the
ADF兲. Excited states were calculated using time-dependent
density-functional theory
32
as implemented in the RESPONSE
part of the ADF.
32–34
Since TDDFT describes 共in principle
exactly兲 how the electron density changes in time under the
influence of a time-dependent perturbation, and since this
time-dependent density will resonate at the exact excitation
energies of the system, linear-response theory based on TD-
DFT is able to provide both these excitation energies as well
as the corresponding oscillator strengths. In practice,
32
to cal-
culate excitation energies and oscillator strengths, the fol-
lowing eigenvalue equation has to be solved:
⍀F
i
⫽
i
2
F
i
, 共1兲
where the four-index matrix ⍀ has components given by
FIG. 2. Energy scheme of the low-energy occupied and unoc-
cupied
orbitals of the thiophene molecule. In this figure the phase
and amplitude of the wave functions are indicated.
DENSITY-FUNCTIONAL STUDY OF THE EVOLUTION . . . PHYSICAL REVIEW B 63 155112
155112-3
⍀
ia
,jb
⫽
␦
␦
ij
␦
ab
共
a
⫺
i
兲
2
⫹ 2
冑
共
a
⫺
i
兲
K
ia
,jb
冑
共
b
⫺
j
兲
. 共2兲
Here squared differences between occupied and virtual KS
orbital energies 共a and b refer to unoccupied energies and i
and j to occupied energies, while
and
are spin indices兲
are included and as well as a coupling K matrix, containing
Coulomb and exchange-correlation 共XC兲 parts. The square
of the desired excitation energies are the eigenvalues
i
2
,
while the oscillator strengths are simply related to the eigen-
vectors F
i
. Note that the elements of the eigenvectors F
i
are
roughly comparable to the configuration interaction coeffi-
cients in a singly excited configuration-interaction calcula-
tion, and are a measure to what extent the corresponding
excitation can be interpreted as a pure single-particle excita-
tion or if several such excitations play a crucial role in the
transition. The Coulomb part of the coupling matrix is given
by
K
lj
,kl
Coul
⫽
冕
dr
冕
dr
⬘
i
共
r
兲
j
共
r
兲
⫻
1
兩
r⫺ r
⬘
兩
k
共
r
⬘
兲
l
共
r
⬘
兲
, 共3兲
while the exchange-correlation part
K
lj
,kl
xc
共
兲
⫽
冕
dr
冕
dr
⬘
l
共
r
兲
j
共
r
兲
⫻ f
xc
共
r,r
⬘
,
兲
k
共
r
⬘
兲
l
共
r
⬘
兲
共4兲
is related to the so called exchange correlation kernel
f
xc
共
r,r
⬘
,t⫺ t
⬘
兲
⫽
␦
v
xc
共
r,t
兲
␦
共
r
⬘
,t
⬘
兲
. 共5兲
In the so called adiabatic local-density approximation
共ALDA兲 used here, the exchange-correlation kernel is simply
given by
f
xc
ALDA,
共
r,r
⬘
,
兲
⫽
␦
共
r⫺ r
⬘
兲
␦
v
xc
LDA,
共
r,t
兲
␦
冏
⫽
o,
共
r
兲
.
共6兲
Although the matrix ⍀ can become quite large, one is usu-
ally interested in the lowest few excitations, and then effi-
cient algorithms 共such as the Davidson algorithm
35,36
兲 can be
used that avoid ever having to construct the matrix
explicitly.
37
To analyze the nature of the important excitations, we
performed a so-called fragment analysis, where the molecule
is thought to be built from chemically relevant fragments,
and all the molecular orbitals are expressed as linear combi-
nations of the molecular orbitals of the constituting frag-
ments 共note that this does not change the outcome of the
calculation, but is only an analytical tool兲. In the case of
thiophene oligomers, we built the total molecule from iden-
tical thiophene monomers 共from which the
␣
hydrogen at-
oms are removed so they are biradicals兲 and hydrogen atoms
at the end of the chain. We have verified that using identical
monomer units 共rather than slightly different ones such as in
the experimental and optimized geometries兲 does not influ-
ence the results in any significant way. Moreover we will be
mainly concerned with
electrons, and it turns out that an
exact treatment of the biradical nature of the
system does
not significantly influence the form of the
electrons. For
practical reasons we therefore used fragments in which the
two spin components of the singly occupied
orbitals were
occupied with half an electron in a spin-restricted calcula-
tion. From this fragment analysis we can derive a fragment-
orbital-based Mulliken-type population analysis, which is of-
ten much more physically illuminating than a population
analysis based on the basis function, and is much less basis
set dependent. Apart from the orbitals and electron densities,
one can also analyze the transition dipole moments for the
various excitations in terms of these fragment orbitals, which
can shed light on the factors that determine the strength of a
particular transition.
III. RESULTS AND DISCUSSION
This discussion will be divided into several parts accord-
ing to the discussion in Sec. I. We start with a description of
the ground-state electronic structure and the corresponding
Kohn-Sham orbital energies as a function of the oligomer
length. We will use these orbital energies to see if a tight-
binding-like model Hamiltonian description with monomer
molecular orbitals as basis sets is an acceptable description
of the development of the electronic structure, and in this
process we will determine the tight-binding parameters re-
quired to closely simulate the DFT results. We then look at
the TDDFT calculations of the excitation energies and oscil-
lator strengths, and compare these with the experimental data
as well as with the orbital energies obtained from the Kohn-
Sham orbital energies. We note here that the DFT orbital
energies do not include additional relaxation due to the
electron-hole interaction in such an excited state, whereas the
TDDFT should at least partially include this. Also it is well
known that DFT in solid semiconductors yields band gaps
considerably lower than the experimental values, so we ex-
pect the DFT values of the excitation energies to lie consid-
erably below the experimental and TDDFT values.
We then study the total energy as a function of the dihe-
dral angle for the dimer, and obtain the energy difference of
the transoid and cisoid geometries as well as the energy bar-
rier to go from one to the other. At a dihedral angle of 90°
the intermonomer hopping integral of the HOMO and
LUMO monomer orbitals goes to zero by symmetry, allow-
ing us to compare the pure intermolecular electron-hole ex-
citations to the on-site monomer excitations in a dimer. From
this we can, in principle, extract the difference of the on-site
and nearest-neighbor Coulomb interactions which, together
with the exchange interaction determined from the singlet-
triplet splitting, gives us the full electronic part of an ex-
tended two-band Hubbard model description of the oligoth-
iophenes. We will finally use this model Hamiltonian for a
systematic study of the excited-state properties as a function
of size.
R. TELESCA et al. PHYSICAL REVIEW B 63 155112
155112-4
The Kohn-Sham orbital energies obtained from the DFT
calculation are plotted as a function of the oligomer length in
Fig. 3. We see from Fig. 3 what we anticipated in Sec. I,
namely, that the occupied 2b
1
orbitals hardly ‘‘feel’’ the
presence of neighboring monomers and remain sharp mo-
lecular levels, while the 1a
2
orbitals spread out into quite
large bandwidth valence bands. Also, the 3b
1
unoccupied
orbitals spread out into bands with widths similar to that of
the occupied 1a
2
-derived band. The 2a
2
unoccupied orbital
also spreads out into a band, but with a width considerably
less than the bands derived from 1a
2
and 3b
1
molecular
orbitals. We note here that the lowest excitation energy ob-
tained from the Kohn-Sham orbital energies is considerably
smaller than the experimental lowest singlet excitation en-
ergy. At first glance this sounds strange, because the actual
excitation energy should be lower in energy because of the
possible exciton binding energy, which is not included in the
DFT calculation. However, this is a very common and also
well-understood problem in semiconductors, and therefore
should not really be alarming.
Although we expect that the gap for an electron-hole ex-
citation obtained from the Kohn-Sham orbital energies will
be considerably smaller than the experimental gap, we might
expect that the relative energies of occupied orbitals as well
as unoccupied orbitals would be close to the experimental
values. In order to check this, in Fig. 4 we display the gas-
phase photoelectron spectrum of the oligomers of thiophene
with sizes of 2–6 thiophene units. These spectra were ob-
tained using a home-built photoelectron spectrometer with a
specially designed strongly focusing electron lens with a
high throughput, in order to collect spectra from a collimated
molecular jet beam of the oligomers. These spectra were
obtained with He
I radiation of 21.2 eV, and the binding
energies were calibrated using the Xe
2
P
3/2
line. These spec-
tra can be directly compared to the molecular-orbital ener-
gies of the occupied states as obtained in the DFT calcula-
tion.
The spectra exhibit a narrow band at a binding energy
between 9.0 and 9.5 eV, which shows only a small change as
a function of the size of the oligomer, and is derived from the
2b
1
HOMO-1 in thiophene, as discussed above. At lower
binding energies we see strong changes as function of size,
which obviously can be interpreted in terms of the develop-
ment of the 1a
2
orbital in thiophene into a rather broad band
of states corresponding very closely to the development
found in the theoretical calculation shown in Fig. 3. In fact
what we see is that the 2b
1
orbital ends up close to the center
of the band of states originating from the 1a
2
orbitals. In Fig.
5共a兲 we show the dependence of the 1a
2
-derived bandwidth
as a function of oligomer size from both DFT and experi-
ment. The extrapolation to an infinite system yields a band-
width of W⫽ 3.9 eV. The solid line drawn in Fig. 5共a兲 is the
theoretical bandwidth based on a two-band tight-binding
model, as discussed below. The good agreement with DFT
gives us confidence in using the DFT calculations for the
unoccupied orbitals, which behave in a manner similar to the
occupied 1a
2
orbitals, ending up with a conduction-band-
width of about 3.2 eV in the large size limit. This is also
shown if Fig. 5共b兲, together with the tight-binding model
calculation discussed below.
In Fig. 6 we display the calculated and experimental ion-
ization potentials as functions of the reciprocal chain length.
The experimental ionization potential decreases linearly with
the reciprocal chain length. We now compare these values
with the ground-state nonrelaxed orbital energies, as ob-
tained in the DFT calculations, although the absolute values
differ from the experimental values, as expected they show
the same dependence on the reciprocal chain length as the
experimental values. DFT calculations on the electronically
fully relaxed ionic states 共⌬SCF兲 give results which are nu-
merically closer to the experimental values for the small size
oligomers, but underestimate the ionization potentials for the
larger size oligomers 共Fig. 6兲. This discrepancy was ex-
FIG. 3. Developing band structure of the oligomers of thiophene
from sizes 1 to 8 as a result of the DFT calculations.
FIG. 4. Gas-phase photoelectron spectra of
␣
⫺ 2T,
␣
⫺ 3T,
␣
⫺ 4T, and
␣
⫺ 6T. The 2b
1
- and 1a
2
-derived band structures are
indicated.
DENSITY-FUNCTIONAL STUDY OF THE EVOLUTION . . . PHYSICAL REVIEW B 63 155112
155112-5
plained in Ref. 38, and can be attributed to an incorrect treat-
ment of the self-energy correction in the density-functional
method for charged systems; its value increases with the size
of the system.
We note in the experimental photoelectron spectra that the
widths of the photoelectron features corresponding to the
HOMO in each case are very large indeed. The experimental
resolution is 0.1 eV in these scans, which is much smaller
than the observed widths. This could be due to the electron
vibronic coupling, and if this is the case then this coupling
strength can be very large indeed. The total spread in energy
is about 0.5 eV for the dimer, and increases to 1 eV for
␣
-4T. This increase in width with the system size is not
really expected for local electron vibronic coupling, perhaps
indicating other possible origins for this broadening. From
optical photoluminescence studies it has been found that
singlet-singlet transitions are strongly coupled to vibronic
modes of various kinds with the 1470-cm
⫺1
共C-C stretch兲
mode dominating. The width of the optical transitions result-
ing from this is about 0.3 eV.
39
Of course this cannot be
directly compared with the photoemission width, since the
nature of the excited states is different. However, since the
widths and shapes are more or less independent of the size of
the oligomer in the optical data, this indicates that an addi-
tional broadening mechanism is present in the photoelectron
spectra. In any case we can conclude from the optical studies
that the electron vibronic coupling to the C-C double-bond
mode is very strong, and corresponds to an average of two
vibrational quanta, which are involved in the electronic re-
laxation energy due to bond-length changes. As we will dis-
cuss below, there are additional broadening mechanisms in
the photoelectron spectra, which may originate from the
strong change in the intermonomer hopping integrals due to
low-energy torsion modes, and the small energy difference
between the cisoid and transoid configurations of the
thiophene-thiophene bonds. In fact we suggest that this small
energy difference, and the strong influence it has on the elec-
tronic structure, is probably the main source of the so-called
defect states that seem to dominate the transport properties of
the oligomers of thiophene in the solid state.
As seen above, the progression of electronic states corre-
sponding to 1a
2
and 3b
1
molecular orbitals with size is
reminiscent of a simple tight-binding-like model prediction.
As noted by others,
21
this indicates that the molecular-orbital
structure of the monomers stays intact, and all that happens
is that the monomer levels develop into broad bands in the
long length limit in a way described by introducing nearest-
neighbor intermonomer hopping integrals but otherwise
leaving everything the same. We already suggested above
that the actual model one should use is more complicated
than a simple single-band model, since there will be hopping
integrals of comparable size, coupling the 1a
2
-1a
2
(t
hh
),
1a
2
-3b
1
(t
hl
), and 3b
1
-3b
1
(t
ll
) molecular orbitals on neigh-
bor monomers. This can be easily concluded from the
molecular-orbital structure shown in Fig. 2, and also from
the DFT calculations of Fig. 3. In fact it is easy to estimate
the relative sizes and also signs of such hopping integrals
from the phase and amplitude of the C 2p-
wave func-
tions on the two
␣
carbon atoms of the monomer. The fact
that the singlet-triplet splitting is large indicates that at least
the triplet is an excitoniclike states, which points in the di-
rection of a large monomer on-site Coulomb interaction and
also a large exchange interaction. These considerations lead
us to propose a two-band extended Hubbard model Hamil-
tonian describing the electronic properties of these systems,
H⫽ H
⫹ H
t
⫹ H
u
⫹ H
v
⫹ H
g
, 共7a兲
FIG. 5. Valence-band-width 共a兲 and conduction-band-width 共b兲
as functions of the reciprocal chainlength (1/N); the open circles
represent the results of the DFT calculations, and the solid lines the
results of a tight-binding fit with the parameters t
hh
⫽⫺0.97 eV,
t
ll
⫽ 0.76 eV, and t
hl
⫽ 0.30 eV. For the valence band we plot the
values obtained from our gas-phase UPS experiments.
FIG. 6. Comparison between ionization potentials obtained from
the gas-phase UPS experiments and the Koopmans and ⌬SCF val-
ues as obtained from the DFT calculations as functions of the re-
ciprocal chain length (1/N).
R. TELESCA et al. PHYSICAL REVIEW B 63 155112
155112-6
where
H
⫽
兺
i⫽ 1,
⫽ ↑↓
N
兺
m⫽ h,l
m
c
i,
m
⫹
c
i,
m
, 共7b兲
H
t
⫽
兺
i⫽ 1,
⫽ ↑↓
N
兺
m⫽ h,l
t
mm
c
i,
m
⫹
c
i⫹ 1,
m
⫹ t
hl
⫻
兺
i⫽ 1,
⫽ ↑↓
N
关
c
i,
h
⫹
c
i⫹ 1,
l
⫹ c
i,
l
⫹
c
i⫹ 1,
h
兴
, 共7c兲
H
u
⫽
兺
i⫽ 1
N
再
关
U⫹ ⌬U
共
␦
i,1
⫹
␦
i,N
兲
兴
⫻
冋
兺
m⫽ h,l
n
i,m↑
n
i,m↓
⫹ n
i,h
n
i,l
册
冎
, 共7d兲
H
v
⫽ V
冋
兺
i⫽ 1
N
兺
m⫽ h,l
n
i,m
n
i⫹ 1,m
⫹
兺
i⫽ 1
N
共
n
i,h
n
i⫹ 1,l
⫹ n
i,l
n
i⫹ 1,h
兲
册
,
共7e兲
H
s
⫽⫺2K
兺
i⫽ 1
N
冋
S
i
h
S
i
l
⫹
1
4
n
i
h
n
i
l
册
, 共7f兲
and
n
i,m
⫽ n
i,m↑
⫹ n
i,m↓
,
n
i,m↑
⫽ c
i,↑
m
⫹
c
i,↓
m
. 共7g兲
Here i is the site, and h and l are the HOMO and LUMO, and
h
and
l
are the one-electron thiophene HOMO and LUMO
energies. U is the on-site Coulomb interaction which we as-
sume to be the same for two electrons in the HOMO, two in
the LUMO, and one in the HOMO and one in the LUMO, as
long as they are on the same monomer. The Coulomb inter-
action at the chain ends is taken to be 0.5 eV(⌬U) larger
than at other positions, because of the reduced coordination
number there; therefore, there is a reduced screening of U. V
is the nearest-neighbor Coulomb interaction, and K is the
exchange integral. A fit of the one-electron part of this
Hamiltonian to the DFT calculations leads to the following
values for the hopping integrals: t
hh
⫽⫺0.97eV, t
ll
⫽ 0.76 eV, and t
hl
⫽ 0.30 eV 关Figs. 5共a兲 and 5共b兲兴. We should
note that the Kohn-Sham orbital energies give too small an
energy splitting (⌬⫽
h
⫺
l
⫽ 4.52 eV for the monomer兲 be-
tween the 1a
2
- and 3b
1
-based bands 共Fig. 7兲; thus to simu-
late the DFT calculations with the one-electron part of our
model Hamiltonian, we must start with a ⌬ which is smaller
than the experimental value 共5.52 eV兲 in order to obtain a
good fit and in order to extract the hopping integrals. Here
the mixing of the 1a
2
- and 3b
1
-based bands will be smaller
than in the DFT calculation because of their larger splittings.
The best way to confirm this would be with inverse photo-
emission, which unfortunately is hard to do in these systems
due to reasons of intensity. A detailed electron-energy-loss
study, in which one can separate the excitonic and interband
electron-hole transitions by varying the incident energy, may
be another way to obtain experimental values for the true
gaps.
Before we discuss the determination of these parameter
values, we first must have a good understanding of the opti-
cal spectra. We now show how the optical spectrum of the
oligomers evolves and can be understood from the basic
electronic structure of the thiophene monomer by applying
simple tight-binding theoretical concepts. Thiophene is iso-
electronic with the cyclopentadienyl anion, which has a D
5h
point group symmetry. The HOMO-LUMO transition (e
1
⬙
→e
2
⬙
) in cyclopentadienyl leads to two exited states; the
lowest one, with E
2
⬘
symmetry, is strictly forbidden by dipole
selection rules, and the second, of E
1
⬘
symmetry, is higher in
energy but is allowed. If we lower the symmetry to the sub-
group C
2
v
without changing the geometry, the degenerate
representation E
1
⬘
splits into A
1
⫹ B
2
, and E
2
⬘
reduces to A
1
⫹ B
2
. So, imposing C
2
v
symmetry on cyclopentadienyl, we
find that the lowest excitations of A
1
and B
2
symmetry are
forbidden, while the next two lowest excitations in both sym-
metries are allowed. In thiophene the D
5h
symmetry is
slightly broken, so the lowest A
1
and B
2
transitions are now
formally allowed 共and no longer at exactly the same energy兲
but in fact still weak. The thiophene HOMO-LUMO transi-
tion 1a
2
→3b
1
is the lowest of B
2
symmetry, and is there-
fore weak; the next transition 2b
1
→2a
2
is stronger. The
same is true for the A
1
transitions, which correspond to
2b
1
→3b
1
共weak兲 and 1a
2
→2a
2
共strong兲. Therefore, the
weakness of the HOMO-LUMO transition in thiophene can
be understood as a relic of the 共broken兲 D
5h
symmetry in
Fig. 7 we have plotted the calculated TDDFT optical gap as
a function of the reciprocal chain length. As for the ioniza-
tion potential, the calculated optical gap extrapolates for the
oligomers to a polymer limit, which is too low in energy
with respect to the experimental gap. The finite localization
length of the electron-hole pair created in the excitation pro-
cess can account for this discrepancy. This localization could
FIG. 7. Singlet optical gap as obtained from DFT and TDDFT
calculations as a function of the reciprocal chain length (1/N). The
results for the triplet optical gap obtained from the TDDFT calcu-
lations are also plotted, as well as the experimental singlet results
共band maximum兲共Ref. 23兲.
DENSITY-FUNCTIONAL STUDY OF THE EVOLUTION . . . PHYSICAL REVIEW B 63 155112
155112-7
be intrinsic or is perhaps a consequence of structural defects
or rotational disorder. Broken-symmetry solutions may cor-
rect a part of this problem and will be explored. But here too
the deviations may be partly inherent to the TDDFT method
used here, which does not take the self-energy correction
properly into account.
In Fig. 7 we also display the TDDFT results for the lowest
excited triplet state as a function of the reciprocal chain
length. Comparing these results with the TDDFT for the sin-
glet state one finds for the monomer a singlet-triplet splitting
of 1.8 eV and this number is a direct measure for the ex-
change integral (K⫽ 0.9 eV). The moderate dependence of
the value of the exchange splitting on the chain length for at
least the smaller oligomers suggests that at least the triplet
excitations are excitonic in nature with exciton sizes in the
order of at most a few monomer units.
If we consider the molecular orbitals of the oligomers as
being constructed from a linear combination of monomer
thiophene orbitals, we can analyze the expectation value of
the dipole operator in terms of on-site contributions and
next-neighbor contributions. The on-site contribution, which
in fact constitutes the HOMO-LUMO transition in the
thiophene molecule, is weak as explained before and the
next-neighbor contribution will be responsible for the in-
crease of the oscillator strength with the chain length. Within
this approximation and with a proper normalization of the
wave functions, we can derive a simple expression for the
value the oscillator strength f as a function of the transition
dipole moment
, chain length (N), and energy of the tran-
sition (⌬E):
f⫽
2
3
冋
ii
⫹
ij
冉
2⫺
2
N
冊
册
2
⌬E. 共8兲
For the larger oligomers we expect the oscillator strength of
the HOMO-LUMO transition to be behave like
f⫽ f
polymer
⫺
const
N
. 共9兲
Here f
polymer
is the oscillator strength of the infinite polymer.
Figure 8 very clearly shows the dependence on 1/N of the
oscillator strength from our TDDFT calculations.
In order to model the effects of rotational disorder and
structural defects on the electronic properties of the oligoth-
iophenes, we studied the ground- and excited-state properties
of
␣
-2T as a function of the dihedral angle between both
monomer units. Chadwick and Kohler
15
found experimental
evidence of the coexistence of cisoid and transoid
bithiophenes in a supersonic expansion. The ratio is depen-
dent on the temperature of the expansion, and the enthalpy
difference between the two structures was found to be 1.16
⫾ 0.13 kcal/mol. One should, of course, realize that the dihe-
dral angles in the gas phase are 72° for the cisoid form and
64° for the transoid form,
15
while those molecular structures
are flat in the solid phase.
39
Because interaction between the
molecules in the crystal are only weak, we expect a rather
shallow potential well for torsion. In Fig. 9 we give the po-
tential well for the cis-trans isomerization with the lowest
energy for a flat transoid bithiophene structure. For the most
stable cisoid structure the dihedral angle is around 70°,
which is very close to experiment, the transoid conformation
is 0.72 kcal/mol more stable than the cisoid conformation,
and the transition state barrier is 4.13 kcal/mol.
In Fig. 10 we give the molecular orbital structure of
bithiophene for torsion angles from 180° 共transoid兲 to 0°
共cisoid兲共although we do in fact a calculation of the molecule
in vacuum, we take the structural parameters for the mol-
ecule in the solid兲. In a tight-binding model the HOMO is
built from an antibonding combination of the original
thiophene 1a
2
orbitals and the HOMO-3 from the bonding
combination, and their splitting is dependent on the transfer
or hopping integral. At 90° the two-ring systems are perpen-
dicular, and the two
systems do not interact and the trans-
fer integral t⫽ 0, the two orbitals cross, and the splitting is
zero. The transfer integral is a function of the torsion angle
⌽, t(⌽)⫽ T⫻ cos(⌽).
We can describe the molecular orbitals derived from the
3b
1
thiophene LUMO in a similar way. In a solid-state de-
FIG. 8. Dependence of the oscillator strength obtained from
TDDFT calculations for oligomers of thiophene on the reciprocal
chain length (1/N). This figure clearly shows the linear dependency
on 1/N for the larger oligomers.
FIG. 9. Potential well for the cisoid-transoid isomerization as a
function of the dihedral angle. This picture clearly shows a barrier
of 4.3 kcal/mol for cisoid-transoid isomerization, and a small dif-
ference in stability of 1.16 kcal/mol between both isomers. For the
cisoid isomer we find a minimum in energy for a nonplanar confor-
mation. This should also be the case for the transoid isomer, but
this would require a further geometry optimization.
R. TELESCA et al. PHYSICAL REVIEW B 63 155112
155112-8
scription this means that, upon torsion, the transfer integral
decreases, the valence and conduction bands narrow, and the
optical gap increases. For the thiophene 2b
1
- and
2a
2
-derived molecular orbitals, almost no dispersion is ob-
served, and this can be attributed to the small transfer inte-
gral due to the small electron densities at the
␣
carbon posi-
tions.
At 90°, where binding and antibinding orbitals become
degenerate, the HOMO and LUMO are now both twofold-
degenerate orbitals, each localized to one of the molecular
entities. We have now a system of essentially noninteracting
monomers, and we can analyze the optical spectrum in terms
of pure intramolecular and intermolecular 共charge-transfer兲
excitations.
The intramolecular excited states are expected to be al-
most degenerate, and the gerade combination will have small
oscillator strength because of the corresponding weak mono-
mer transition. The two charge-transfer-excited states are
also expected to be close in energy, but will have no oscil-
lator strength because of the two mutual perpendicular
systems. The energy splitting between the intramolecular and
intermolecular excitations can in a tight-binding approach be
interpreted as U⫹ ⌬U⫺ V⫺ 2K, in which U is the on-site
interaction and V the next-neighbor Coulomb interaction,
⌬U the reduced screening of U due to end effects, and K the
exchange integral.
Our TDDFT calculations are in excellent agreement with
this simple model. At a dihedral angle of 90° we can identify
two sets of two nearly degenerate transitions. At the low-
energy end of the spectrum there are two almost degenerate
intramolecular transitions at 4.65 eV, in which the gerade
component has an oscillator strength of 0.295. About 1.0 eV
higher in energy, we calculate two almost degenerate charge-
transfer transitions at 5.63-eV energy, which as predicted do
not carry any oscillator strength. From the splitting of the
average intramolecular and intermolecular excitation ener-
gies we can estimate a value of 1.0 eV for U⫹ ⌬U⫺ V
⫺ 2K.
We have now extracted values for all the relevant param-
eters from the TDDTF calculations, which we will use as
input parameters for a two-band model Hamiltonian calcula-
tion on the oligomers of size 1–6. We also can mimic the
polymer limit by applying periodic boundary conditions to
an oligomer of size 6. In these calculations the ground state
and single excited states were included. The doubly excited
states are not important for the low-energy features because
the relative large HOMO-LUMO splitting. In these calcula-
tions the input parameters mentioned above were further op-
timized until a good fit for the singlet and triplet optical gaps
was obtained 共Fig. 11兲. In order to obtain this result one has
to take into account the reduced screening of the on-site
Coulomb interaction on the terminal thiophene rings. With a
static polarizability of about 10 Å
3
for thiophene we esti-
mated ⌬ at about 0.5 eV. The influence of the polarizability
on the on-site coulomb interaction was already described in
detail.
40
If this effect is not taken into account we find very
TABLE I. Values for the parameters used in a two-band Hub-
bard model Hamiltonian for a description of the electronic structure
of the oligomers of thiophene. The values of the parameters given
here are results of a fit to the experimental photoelectron spectro-
scopic and optical data 共Refs. 23, 24, and 27兲. Starting values for
the fit were obtained from the TDDFT calculations described in this
paper.
Parameter Value 共eV兲
U 2.4
⌬U 0.5
V 0.7
K 0.8
t
hh
⫺0.97
t
ll
0.76
t
hl
0.30
⌬ 4.72
FIG. 10. Molecular-orbital structure of
␣
⫺ 2T as a function of
the dihedral angle.
FIG. 11. The lowest singlet and triplet excited states obtained
from a two-band Hubbard model Hamiltonian calculation with the
parameter values given in Table I as a function of the reciprocal
chain length (1/N). In the calculation we included only the ground
state and all single excited states. Estimates for the polymer limit
were obtained by imposing periodic boundary conditions on
sexithiophene. For comparison available experimental data 共Refs.
23, 24, and 27兲共band maximum兲 are included.
DENSITY-FUNCTIONAL STUDY OF THE EVOLUTION . . . PHYSICAL REVIEW B 63 155112
155112-9
low-energy excitonic states, with the electron on the terminal
position and the hole next to it, this is due to the influence of
the nearest-neighbor Coulomb interaction V, which lowers
the energy of the electron hole pair if they are at the end of
the chain.
The final set of parameters is given in Table I. From this
we note the large value found for U of 2.4 eV, which is about
half the bandwidth and sufficient to strongly bind even the
singlet states into Frenkel-like excitons. In Fig. 12 we plot
the singlet optical spectrum of sexithiophene: here for sim-
plicity we assume equal transition dipole moments for the
on-site and next-neighbor transitions. From this figure it is
clear that the magnitude of the Coulomb interaction is suffi-
cient to form a bound excitonic state in which the electron
and hole are mainly on the same monomer as in a Frenkel
exciton. Of course, for such a finite chain there is no real
distinction between an exciton and an electron-hole pair,
since they are always highly confined. In a subsequent paper
we will show that these parameters indeed lead to Frenkel-
like excitons for both the singlet and triplets, with the triplets
much more tightly bound than the singlets.
41
IV. CONCLUSIONS
In this paper it has been shown that the progression of the
electronic properties with size for oligomers of thiophene
can be understood in terms of a simple tight-binding model
describing a linear system of weakly coupled monomer units,
in which the building blocks mainly retain their molecular
identity. Because the HOMO and LUMO are very different
in character, the first is aromatic with no sulfur character,
and the latter is quinoid with significant density on the sulfur,
one needs a two-band tight-binding approach in which the
next-neighbor interaction is modeled with HOMO-HOMO,
LUMO-LUMO, and intermolecular HOMO-LUMO transfer
integrals. We have been able to extract a consistent set of
tight-binding parameters from the results of the DFT calcu-
lations, which describe the experimental available data very
well. It is surprising that a simple tight-binding Hamiltonian
with only a monomer HOMO and a monomer LUMO gives
such a good description of the details of the electronic struc-
ture of the oligomers of thiophene. Of course this may not be
representative of other systems, especially polyacetylene.
Further studies will explore the generality of this approach.
From TDDFT calculations on the lowest singlet and trip-
let excited states we could estimate an effective exchange
integral of about 0.9 eV. The nature of the lowest singlet
state can be analyzed in valence bond terms in intramolecu-
lar and intermolecular contributions. We showed that almost
all the oscillator strength originates from intermolecular con-
tributions, while the intramolecular contribution is weak.
This finds its foundation in the fact that the electronic struc-
ture of the thiophene molecule is very similar to that of the
isoelectronic cyclopentadienyl anion. This also explains the
1/N dependence of the magnitude of the oscillator strength
for the larger oligomers.
We also found that rotational disorder is important in
these systems, and that there is a very shallow potential well
for torsion. These fluctuations will introduce an effective
conjugation length, and most probably will be important in
the localization of polarons and excitons, and may be an
important source of traps in these materials.
For the dimer we showed that, if we take a torsion angle
of 90°, the
systems will be perpendicular and the hopping
integral will vanish. We are now left with two sets of excited
states: one set of almost degenerate on-site excitations and
one set of almost degenerate pure charge-transfer excitations.
The splitting between both sets of excitations amounts to an
effective on-site Coulomb interaction minus the next-
neighbor Coulomb interaction U⫹ ⌬U⫺ V⫺ 2K, and that
splitting is about 1 eV.
From TDDFT calculations, numerical values could be
extracted for the relevant physical quantities, which are the
input parameters for the model Hamiltonian defined in this
paper. After a full optimization we were able to realize an
almost perfect fit for the singlet optical gap, and we predict
the positions of triplet states for longer oligomers, for which
they have not yet been observed to our knowledge. We
show that electron correlation plays an important role in
these systems, and that most of the optical spectral weight
is carried by a singlet excitoniclike state, with an electron
and hole concentrated on the same monomer or near-
neighbor monomers.
FIG. 12. The singlet optical spectrum of sexithiophene as ob-
tained from a two-band Hubbard model Hamiltonian calculation;
only the ground state and all single excited states are included. The
values for the parameters given in Table I were used. Clearly visible
is a bound excitonic state just below the conduction-band edge,
which carries most of the spectral weight. In this calculation we
have assumed equal transition dipole moments for the on-site and
next-neighbor transitions.
R. TELESCA et al. PHYSICAL REVIEW B 63 155112
155112-10
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