Proof-of-principle functionals for the shape function

Recently, it has been shown that the density per particle, or shape function, determines all the properties of molecules and, more generally, finite Coulomb systems (e.g., clusters). Here, we provide numerical evidence for this "existence" theorem, showing that atomic properties (specifically the kinetic energy and exchange energy) have simple approximate expressions in terms of the shape function. While the accuracy of our results is limited, this provides "proof-of-principle" evidence that shape-functional theory represents an alternative to density-functional theory for finite Coulomb systems.