  We explore the computation of high-harmonic generation spectra by means of
Gaussian basis sets in approaches propagating the time-dependent
Schr{\"o}dinger equation. We investigate the efficiency of Gaussian functions
specifically designed for the description of the continuum proposed by Kaufmann
et al. [J. Phys. B 22, 2223 (1989)]. We assess the range of applicability of
this approach by studying the hydrogen atom, i.e. the simplest atom for which
"exact" calculations on a grid can be performed. We notably study the effect of
increasing the basis set cardinal number, the number of diffuse basis
functions, and the number of Gaussian pseudo-continuum basis functions for
various laser parameters. Our results show that the latter significantly
improve the description of the low-lying continuum states, and provide a
satisfactory agreement with grid calculationsfor laser wavelengths $\lambda$0 =
800 and 1064 nm. The Kaufmann continuum functions therefore appear as a
promising way of constructing Gaussian basis sets for studying molecular
electron dynamics in strong laser fields using time-dependent quantum-chemistry
approaches.
