We have extended a form of double perturbation theory for molecules by including a screening constant in the calculation. We have computed the ground state energies for the diatomic systems H2+, HHe2+, H2 and HHe+ with inclusion of screening. A comparison of the results with those in the unscreened approximation is interesting. Thus for the one-electron systems the energies up to second order are better if we do not include screening. However, for the two-electron systems the energies up to second order are improved by screening except at small internuclear separation.