  The Bethe-Salpeter equation (BSE) based on GW quasiparticle levels is a
successful approach for calculating the optical gaps and spectra of solids and
also for predicting the neutral excitations of small molecules. We here present
an all-electron implementation of the GW+BSE formalism for molecules, using
numeric atom-centered orbital (NAO) basis sets. We present benchmarks for
low-lying excitation energies for a set of small organic molecules, denoted in
the literature as "Thiel's set". Literature reference data based on
Gaussian-type orbitals are reproduced to about one meV precision for the
molecular benchmark set, when using the same GW quasiparticle energies and
basis sets as the input to the BSE calculations. For valence correlation
consistent NAO basis sets, as well as for standard NAO basis sets for ground
state density-functional theory with extended augmentation functions, we
demonstrate excellent convergence of the predicted low-lying excitations to the
complete basis set limit. A simple and affordable augmented NAO basis set
denoted "tier2+aug2" is recommended as a particularly efficient formulation for
production calculations. We finally demonstrate that the same convergence
properties also apply to linear-response time-dependent density functional
theory within the NAO formalism.
