evaluated to a value that scales linearly with the size of the system [135–137]. For
organic polymers, it seems natural to describe the polymer orbitals as a linear
combination of monomer orbitals. It is possible to reduce the number of orbitals
per monomer to be considered (e.g., from HOMO-1 to LUMO+1 in P3HT), thus
reducing the size of the one-electron effective Hamiltonian. Different schemes have
been proposed for evaluating the diagonal and off-diagonal elements of such a
reduced Hamiltonian, including a fragment molecular orbital approach [138] and a
charge patching method [107, 139, 140]. Their calculation of the matrix elements
shares many points with the methodology described in Sect. 4.3, except that the
calculations are always performed for the neutral state. A general formalism for
such methods has been developed by the quantum chemistry community [137, 141–
143] and has the obvious advantage of being very intuitive: All matrix elements
entering in the Hamiltonian have a clear physical meaning.
The various partitioning schemes proposed for polymer systems are very often
complemented by further approximations that take into account the chemical
structure of the investigated system. As the frontier orbitals are invariably localized on the conjugated fragment of the monomer, while a large fraction of the
molecular weight is taken up by the polymer side chains that do not contribute to
the charge transport states, it is customary to remove the side chains from the
calculation, sometimes substituting them with effective point charges that simulate the missing electrostatic effects [133]. For lamellar systems such as P3HT and
PBTTT, it is an excellent approximation to neglect completely the electronic
coupling between lamellae, therefore describing a system that is essentially
two-dimensional [108, 144].
A relatively simple approach that can be considered the simplest possible
Hamiltonian partitioning technique has been proposed recently for the calculation
of the electronic properties of MEH-PPV [133]. Here the amorphous-polymer
model is simply partitioned into subsystems containing single chains (surrounded
by point charges reproducing the local electrostatic potential and with shortened
side chains). The full wavefunction of each single chain can be computed with
routine DFT calculations and the electronic couplings between chains can be
evaluated similarly. The coupling between polymer orbitals localized on different
chains is very small because only a few atoms localized on different chains are in
contact while the orbitals are typically localized over several monomers. A further
reduction in computational cost can be achieved using a small basis set, which, at
least for this type of polymer, does not seem to affect the shape of the DOS or the
localization length.
An alternative approach to full or ad hoc linear scaling DFT methods is calculation through approximate DFT methods such as the tight binding DFT (DFTB)
[145]. This family of methods has been developed over the past few years and the
most convenient version of the methodology is the self-consistent charge-density
functional tight binding (SCC-DFTB) [146], as implemented for example in the
software DFTB+. An acceleration of one to two orders of magnitudes for the
electronic structure calculation is achieved by an approximate evaluation of the
Kohn–Sham–Fock matrix elements in the atomic-orbital basis. A very broad range
172
C. Poelking et al.
organic polymers, it seems natural to describe the polymer orbitals as a linear
combination of monomer orbitals. It is possible to reduce the number of orbitals
per monomer to be considered (e.g., from HOMO-1 to LUMO+1 in P3HT), thus
reducing the size of the one-electron effective Hamiltonian. Different schemes have
been proposed for evaluating the diagonal and off-diagonal elements of such a
reduced Hamiltonian, including a fragment molecular orbital approach [138] and a
charge patching method [107, 139, 140]. Their calculation of the matrix elements
shares many points with the methodology described in Sect. 4.3, except that the
calculations are always performed for the neutral state. A general formalism for
such methods has been developed by the quantum chemistry community [137, 141–
143] and has the obvious advantage of being very intuitive: All matrix elements
entering in the Hamiltonian have a clear physical meaning.
The various partitioning schemes proposed for polymer systems are very often
complemented by further approximations that take into account the chemical
structure of the investigated system. As the frontier orbitals are invariably localized on the conjugated fragment of the monomer, while a large fraction of the
molecular weight is taken up by the polymer side chains that do not contribute to
the charge transport states, it is customary to remove the side chains from the
calculation, sometimes substituting them with effective point charges that simulate the missing electrostatic effects [133]. For lamellar systems such as P3HT and
PBTTT, it is an excellent approximation to neglect completely the electronic
coupling between lamellae, therefore describing a system that is essentially
two-dimensional [108, 144].
A relatively simple approach that can be considered the simplest possible
Hamiltonian partitioning technique has been proposed recently for the calculation
of the electronic properties of MEH-PPV [133]. Here the amorphous-polymer
model is simply partitioned into subsystems containing single chains (surrounded
by point charges reproducing the local electrostatic potential and with shortened
side chains). The full wavefunction of each single chain can be computed with
routine DFT calculations and the electronic couplings between chains can be
evaluated similarly. The coupling between polymer orbitals localized on different
chains is very small because only a few atoms localized on different chains are in
contact while the orbitals are typically localized over several monomers. A further
reduction in computational cost can be achieved using a small basis set, which, at
least for this type of polymer, does not seem to affect the shape of the DOS or the
localization length.
An alternative approach to full or ad hoc linear scaling DFT methods is calculation through approximate DFT methods such as the tight binding DFT (DFTB)
[145]. This family of methods has been developed over the past few years and the
most convenient version of the methodology is the self-consistent charge-density
functional tight binding (SCC-DFTB) [146], as implemented for example in the
software DFTB+. An acceleration of one to two orders of magnitudes for the
electronic structure calculation is achieved by an approximate evaluation of the
Kohn–Sham–Fock matrix elements in the atomic-orbital basis. A very broad range
172
C. Poelking et al.