In a search for quasi-bound states, or resonances, high-resolution measurem
ents (DeltaE similar to 25 meV) of the total positron scattering cross sect
ions have been carried out in the energy region of the first electronically
excited states of H-2, N-2, CO and Ar. In the case of H-2, a recent calcul
ation by Varella et al predicts such a resonance in the total elastic cross
section near the excitation threshold for the B (1)Sigma (+)(u) state. We
find no experimental evidence for the existence of this resonance and also
find similar negative results for the other atomic and molecular targets.