Simple analytical wave functions satisfying appropriate boundary conditions are constructed for the ground states of one-and two-electron homonuclear molecules. Both the asymptotic condition when one electron is far away and the cusp condition when the electron coalesces with a nucleus are satisfied by the proposed wave function. For H-2(+), the resulting wave function is almost identical to the Guillemin-Zener wave function which is known to give very good energies. For the two electron systems H-2 and He-2(++), the additional electron-electron cusp condition is rigorously accounted for by a simple analytic correlation function which has the correct behavior not only for r(12)--> 0 and r(12)-->infinity but also for R --> 0 and R -->infinity, where r(12) is the interelectronic distance and R, the internuclear distance. Energies obtained from these simple wave functions agree within 2 x 10(-3) a.u. with the results of the most sophisticated variational calculations for all R and for all systems studied. This demonstrates that rather simple physical considerations can be used to derive very accurate wave functions for simple molecules thereby avoiding laborious numerical variational calculations.