The variationnals approaches based on the fractional form of the schwinger variationnel principle appears to be a powerful tool that relieves the perturbatif treatment to investigate the direct excitation processes of atoms by the impact of ions at intermediate energies. Furthermore, this procedure was shown to be very successful in predicting the direct excitation cross sections for the hydrogen-like lithium (Li2+) and beryllium (Be3+) ions by protons (H+) impact in the energy range between 16 to 4000 keV/u. These characteristics are illustrated by a good agreement between our theoretical results when the wave function is well represented (5, 10, and 14 basic sets) and with available theoretical calculations obtained by close-coupling method like those of two-center atomic orbital expansion (TCAO) of Ermolaev et al, single center expansion (SCE) of Hall et al. and the Coupled-Sturmian cross sections T. G. Winter et al.

Coupled-Sturmian, atomic excitation, Variational Schwinger Principle, Cross section.