First, exploratory calculations were performed to investigate the presence of non-nuclear maxima (NNMs) in ground-state electron densities of homonuclear diatomic molecules from hydrogen up to calcium at their equilibrium geometries. In a second stage, only for the cases in which these features were previously detected, a rigorous analysis was carried out by several combinations of theoretical methods and basis sets in order to ensure that they are not only calculation artifacts. Our best results support that Li-2, B-2, C-2, and P-2 are molecules that possess true NNMs. A NNM was found in values obtained from the largest basis sets for Na-2, but it disappeared at the experimental geometry because optimized bond lengths are significantly inaccurate for this case (deviations of 0.10 angstrom). Two of these maxima are also observed in Si-2 with CCSD and large basis sets, but they are no longer detected as core-valence correlation or multiconfigurational wave functions are taken into account. Therefore, the NNMs in Si-2 can be considered unphysical features due to an incomplete treatment of electron correlation. Finally, we show that a NNM is encountered in LiNa, representing the first discovery of such electron density maxima in a heteronuclear diatomic system at its equilibrium geometry, to our knowledge. Some results for LiNa, found in variations in internuclear distances, suggest that molecular electric moments, such as dipole and quadrupole, are sensitive to the presence of NNMs.