PhD Thesis - Universität Augsburg

22 1. Models
where ρ denotes the amount of charge transfer from the donors to the acceptors. The
ionic Hubbard model serves as an appropriate effective model for the D − A chains, with
the sites with even i representing the LUMO (Lowest Unoccupied Molecular Orbital) of an
acceptor molecule, while sites with odd i represent the HOMO (Highest Occupied Molecular
Orbital) of a donor. In the neutral state, these orbitals would be occupied by either zero
or two electrons, respectively. The model parameters U and ∆ are used for an effective
description of the microscopic parameters, like, e.g., the electron affinity of the acceptor
molecules, the ionization potential of the donors, the Madelung energy gained by the pairs
ionized due to the charge transfer ρ, and the Coulomb interaction in the D +ρ A −ρ pairs [83].
Within this picture, ∆ is interpreted as the energy necessary to move an electron from the
donor to the acceptor. The ionic Hubbard model will be studied in detail in Section 4.1 of
Chapter 4.
In quasi 1D materials like halogen-bridged transition metal chain complexes, conjugate
polymers, or inorganic blue bronzes a similar on-site energy modulation can be effectively
generated by the strong competition among the itinerancy of the electrons, electronelectron
interactions and interactions of electrons with the motion of ions. In the adiabatic
limit, the latter interactions can be taken into account within a purely electronic model by
an effective potential
H ion = ∆ ∑
(−1) i n i + KL ( ) 2 ∆
(1.44)
2
2 2
i
which in addition includes the elastic energy of a harmonic lattice with a “stiffness constant”
K. This so-called adiabatic Holstein-Hubbard model will be studied in detail in Section 4.2
of Chapter 4.
The latter two extensions, the Peierls distortion and the ionic potential, obviously
change the translation symmetry of the Hubbard model (1.21), since the extended models
are only mapped onto themselves by a lattice translation by two sites. In momentum
space, this corresponds to a reduction of the Brillouin zone by one half (from [−π/a, π/a]
to [−π/2a; π/2a], with a being the lattice constant). For U = 0, a band gap opens at the
new Brillouin zone boundaries, so these models represent effective two-band models.