A clear understanding of the mechanisms that control the
electron dynamics in a strong laser field is still a challenge that requires
interpretation by advanced theory. Development of accurate theoretical and
computational methods, able to provide a precise treatment of the
fundamental processes generated in the strong field regime, is therefore
crucial. A central aspect is the choice of the basis for the wave function
expansion. Accuracy in describing multiphoton processes is strictly related
to the intrinsic properties of the basis, such as numerical convergence,
computational cost, and representation of the continuum. By explicitly
solving the 1D and 3D time-dependent Schrödinger equation for H2
+ in the
presence of an intense electric field, we explore the numerical performance
of using a real-space grid, a B-spline basis, and a Gaussian basis (improved
by optimal Gaussian functions for the continuum). We analyze the
performance of the three bases for high-harmonic generation and above-threshold ionization for H2
+. In particular, for highharmonic
generation, the capability of the basis to reproduce the two-center interference and the hyper-Raman phenomena is investigated.