  A precise understanding of mechanisms governing the dynamics of electrons in
atoms and molecules subjected to intense laser fields has a key importance for
the description of attosecond processes such as the high-harmonic generation
and ionization. From the theoretical point of view, this is still a challenging
task, as new approaches to solve the time-dependent Schr\"odinger equation with
both good accuracy and efficiency are still emerging. Until recently, the
purely numerical methods of real-time propagation of the wavefunction using
finite grids have been frequently and successfully used to capture the electron
dynamics in small one- or two-electron systems. However, as the main focus of
attoscience shifts toward many-electron systems, such techniques are no longer
effective and need to be replaced by more approximate but computationally
efficient ones. In this paper, we explore the increasingly popular method of
expanding the wavefunction of the examined system into a linear combination of
atomic orbitals and present a novel systematic scheme for constructing an
optimal Gaussian basis set suitable for the description of excited and
continuum atomic or molecular states. We analyze the performance of the
proposed basis sets by carrying out a series of time-dependent configuration
interaction calculations for the hydrogen atom in fields of intensity varying
from $5 \times 10^{13}\:\rm W/cm^2$ to $5 \times 10^{14}\:\rm W/cm^2$. We also
compare the results with the data obtained using Gaussian basis sets proposed
previously by other authors.
