Topographical features of the Laplacian of the momentum density of atoms and ions most indicative of shell structure

The Laplacian of the spherically averaged momentum density, ∇2Π̄(p), is computed from nonrelativistic self-consistent-field wave functions for the neutral atoms from hydrogen to uranium, and the singly positive ions, from helium to barium and lutetium to radium. In particular, the topography of ∇2Π̄(p) reveals that there are seven distinct types of behavior for the studied atomic and ionic cases, based on the number of zeros and extrema present and the value of ∇2Π̄(p) at the origin. Shell structure within ∇2Π̄(p) is shown to be exhibited by the zeros that occur as ∇2Π̄(p) crosses from negative values to positive values. These zeros have been found to be the topographical feature most intimately related to the radii predicted by the Bohr theory of an atom. At most four shells per atom or ion are revealed based on this criterion.