The density per particle can be used as the fundamental descriptor for systems with rapidly decaying external potentials.

For systems of electrons bound by potentials that decay faster than 1/r asymptotically, the density per particle determines the number of electrons and therefore the electron density. The density per particle, commonly called the shape function, can thus be used as the fundamental descriptor of systems with rapidly decaying external potentials. This result is analogous to a result that is known for Coulomb potentials. Possible extensions of the result to include broader classes of external potentials and alternative density-like descriptors are discussed.