The evolution of Rydberg states of hydrogen and alkali-metal atoms subject to short half-cycle pulses is studied. The convergence of the numerical solutions of the time-dependent Schrödinger equation based on an expansion of the electronic wave function in a finite basis set of Sturmian functions is analyzed in detail. It is shown that the accuracy of such calculations can be established by investigating the stabilization of the transition probabilities with respect to the parameters that define the basis set. The dependence of the quantum and classical ionization thresholds on the pulse shape is investigated. The calculations are compared with experimental data for various pulse profiles, which feature slow or fast rise times. The results show that the ionization thresholds for long pulses are very sensitive to the rise time of the electric field. © 1998 The American Physical Society.