We apply the method of Monte Carlo configuration interaction (MCCI) to calculate ground-state potential energy curves for a range of small molecules and compare the results with full configuration interaction. We show that the MCCI potential energy curve can be calculated to relatively good accuracy, as quantified using the non-parallelity error, using only a very small fraction of the full configuration interaction space. In most cases the potential curve is of better accuracy than its constituent single-point energies. We finally test the MCCI program on systems with basis sets beyond full configuration interaction: a lattice of 50 hydrogen atoms and ethylene. The results for ethylene agree fairly well with other computational work while for the lattice of 50 hydrogens we find that the fraction of the full configuration interaction space we were able to consider appears to be too small as, although some qualitative features are reproduced, the potential curve is less accurate.