  We present an implementation of time-dependent density-functional theory
(TDDFT) in the linear response formalism enabling the calculation of low energy
optical absorption spectra for large molecules and nanostructures. The method
avoids any explicit reference to canonical representations of either occupied
or virtual Kohn-Sham states and thus achieves linear-scaling computational
effort with system size. In contrast to conventional localised orbital
formulations, where a single set of localised functions is used to span the
occupied and unoccupied state manifold, we make use of two sets of in situ
optimised localised orbitals, one for the occupied and one for the unoccupied
space. This double representation approach avoids known problems of spanning
the space of unoccupied Kohn-Sham states with a minimal set of localised
orbitals optimised for the occupied space, while the in situ optimisation
procedure allows for efficient calculations with a minimal number of functions.
The method is applied to a number of medium sized organic molecules and a good
agreement with traditional TDDFT methods is observed. Furthermore, linear
scaling of computational cost with system size is demonstrated on a system of
carbon nanotubes.
