  The dispersion energy between extended molecular chains (or equivalently
infinite wires) with non-zero band gaps is generally assumed to be expressible
as a pair-wise sum of atom-atom terms which decay as $R^{-6}$. Using a model
system of two parallel wires with a variable band gap, we show that this is not
the case. The dispersion interaction scales as $z^{-5}$ for large interwire
separations $z$, as expected for an insulator, but as the band gap decreases
the interaction is greatly enhanced; while at shorter (but non-overlapping)
separations it approaches a power-law scaling given by $z^{-2}$, \emph{i.e.}
the dispersion interaction expected between \emph{metallic} wires. We
demonstrate that these effects can be understood from the increasing length
scale of the plasmon modes (charge fluctuations), and their increasing
contribution to the molecular dipole polarizability and the dispersion
interaction, as the band gaps are reduced. This result calls into question
methods which invoke locality assumptions in deriving dispersion interactions
between extended small-gap systems.
