corrections. These methodologies require the definition of a ‘unit of transport’, i.e., a
region of the polymer where the charge is assumed to reside. For transport in small
molecules, liquid crystals, and some relatively short polymers it is straightforward to
consider the full molecular unit as the unit of transport. It is also reasonable to
consider long oligomers when studying transport in the direction perpendicular to
the π–π stacking in semicrystalline phases. For amorphous regions in fully amorphous polymers, or if one is interested in the transport along the π-conjugated
backbone, further approximations are needed. In the case of amorphous polymers
like poly( p-phenylene vinylene) (PPV), the structure determined by classical simulations [121–123] contains regions where there is a strong deviation from planarity
and it was proposed that the polymeric chain can be divided into conjugated portions
defined as the regions between these conjugation-breaking distortions [124]. This
idea has been used to interpret spectroscopic data but it is very difficult to apply it to
computational systems. First of all, it is not clear how one can rigorously define a
sharp threshold separating the complete conjugation breaking from the full conjugation between monomers. Moreover, there are many interesting polymers (including
P3HT) that form crystalline domains so extended that no conjugation breaks are
found for hundreds of nanometers [125]. The charge is certainly more localized than
that, either by the disorder present in the semicrystalline phase or by electron–phonon
coupling (evidence of charge localization can be found for example from chargemodulated spectroscopy [126, 127]). Determination of the localization characteristic
of charge carriers in polymers is another important area of computational investigation that has been pursued by several groups over the past few years.
To determine the charge localization in a large system there is no other alternative than computing the electronic wavefunction of a large model system. The
model systems are generated by classical simulation of polymers containing thousands of atoms, so it is not practical to study them in a charged state using quantummechanical methods. A calculation of the electronic structure of such a large model
with an electron less than the neutral state would only yield the ground state,
whereas one is generally interested in the energy distribution and localization of
many charged states. The excited state of the charged simulation box cannot be
computed with modern computational methods and therefore one cannot even
evaluate the nuclear relaxation (reorganization energy) for the hopping between
two states by starting with a large model system. For these reasons, all attempts to
evaluate the wavefunction of a large model of polymers have focused on neutral
systems and have interpreted the one-electron states (the orbitals) as the possible
sites where the excess charge can be localized [107, 108]. Results are normally
presented in terms of a density of states (DOS) and localization length but it should
be noted that these two quantities are computed for systems with orbitals fully filled
and empty above and below the band gap, respectively. Therefore, they cannot be
directly compared with the models in Sect. 4, which include electron and nuclear
polarization but have to make assumptions regarding the localization length.
For the one-electron states to be representative of the actual localization of the
charge carrier, a further condition needs to be satisfied. The charge needs to be
localized predominantly by the conformational disorder of the polymer and not by
170
C. Poelking et al.
region of the polymer where the charge is assumed to reside. For transport in small
molecules, liquid crystals, and some relatively short polymers it is straightforward to
consider the full molecular unit as the unit of transport. It is also reasonable to
consider long oligomers when studying transport in the direction perpendicular to
the π–π stacking in semicrystalline phases. For amorphous regions in fully amorphous polymers, or if one is interested in the transport along the π-conjugated
backbone, further approximations are needed. In the case of amorphous polymers
like poly( p-phenylene vinylene) (PPV), the structure determined by classical simulations [121–123] contains regions where there is a strong deviation from planarity
and it was proposed that the polymeric chain can be divided into conjugated portions
defined as the regions between these conjugation-breaking distortions [124]. This
idea has been used to interpret spectroscopic data but it is very difficult to apply it to
computational systems. First of all, it is not clear how one can rigorously define a
sharp threshold separating the complete conjugation breaking from the full conjugation between monomers. Moreover, there are many interesting polymers (including
P3HT) that form crystalline domains so extended that no conjugation breaks are
found for hundreds of nanometers [125]. The charge is certainly more localized than
that, either by the disorder present in the semicrystalline phase or by electron–phonon
coupling (evidence of charge localization can be found for example from chargemodulated spectroscopy [126, 127]). Determination of the localization characteristic
of charge carriers in polymers is another important area of computational investigation that has been pursued by several groups over the past few years.
To determine the charge localization in a large system there is no other alternative than computing the electronic wavefunction of a large model system. The
model systems are generated by classical simulation of polymers containing thousands of atoms, so it is not practical to study them in a charged state using quantummechanical methods. A calculation of the electronic structure of such a large model
with an electron less than the neutral state would only yield the ground state,
whereas one is generally interested in the energy distribution and localization of
many charged states. The excited state of the charged simulation box cannot be
computed with modern computational methods and therefore one cannot even
evaluate the nuclear relaxation (reorganization energy) for the hopping between
two states by starting with a large model system. For these reasons, all attempts to
evaluate the wavefunction of a large model of polymers have focused on neutral
systems and have interpreted the one-electron states (the orbitals) as the possible
sites where the excess charge can be localized [107, 108]. Results are normally
presented in terms of a density of states (DOS) and localization length but it should
be noted that these two quantities are computed for systems with orbitals fully filled
and empty above and below the band gap, respectively. Therefore, they cannot be
directly compared with the models in Sect. 4, which include electron and nuclear
polarization but have to make assumptions regarding the localization length.
For the one-electron states to be representative of the actual localization of the
charge carrier, a further condition needs to be satisfied. The charge needs to be
localized predominantly by the conformational disorder of the polymer and not by
170
C. Poelking et al.