  We derive a low-scaling $G_0W_0$ algorithm for molecules, using pair atomic
density fitting (PADF) and an imaginary time representation of the Green's
function and describe its implementation in the Slater type orbital (STO) based
Amsterdam density functional (ADF) electronic structure code. We demonstrate
the scalability of our algorithm on a series of water clusters with up to 432
atoms and 7776 basis functions and observe asymptotic quadratic scaling with
realistic threshold qualities controlling distance effects and basis sets of
triple-$\zeta$ (TZ) plus double polarization quality. Also owing to a very
small prefactor, with these settings a $G_0W_0$ calculation for the largest of
these clusters takes only 240 CPU hours. With errors of 0.24 eV for HOMO
energies in the GW100 database on the quadruple-$\zeta$ level, our
implementation is less accurate than canonical all-electron implementations
using the larger def2-QZVP GTO-tpye basis set. Apart from basis set errors,
this is related to the well-known shortcomings of the GW space-time method
using analytical continuation techniques as well as to numerical issues of the
PADF-approach of accurately representing diffuse AO-products. We speculate,
that these difficulties might be overcome by using optimized auxiliary fit sets
with more diffuse functions of higher angular momenta. Despite these
shortcomings, for subsets of medium and large molecules from the GW5000
database, the error of our approach using basis sets of TZ and augmented DZ
quality is decreasing with system size. On the augmented DZ level we reproduce
canonical, complete basis set limit extrapolated reference values with an
accuracy of 80 meV on average for a set of 20 large organic molecules. We
anticipate our algorithm, in its current form, to be very useful in the study
of single-particle properties of large organic systems such as chromophores and
acceptor molecules.
