Upper bounds to atomic electron densities in position and momentum spaces

Modified functions r−αρ(r) and p−β (p) of the spherically averaged electron densities ρ(r) in position space and (p) in momentum space are found to be convex (i.e., the second derivatives are nonnegative everywhere) for all the 103 ground-state atoms from hydrogen (atomic number Z = 1) to lawrencium (Z = 103), if the parameters are chosen to be α 0.6 and β 1.4. The convex property of r−αρ(r) and p−β (p) is used to derive upper bounds to the density functions ρ(r) and (p) in terms of their radial moments 〈rs 〉 and 〈ps 〉 or frequency moments 〈ρt 〉 and 〈 t 〉. In most cases, the present bounds are shown to be more general and more accurate than those reported in the literature.