What do we learn from the classical turning surface of the Kohn–Sham potential as electron number is varied continuously?

The classical Kohn-Sham turning radius R_t of an atom can be defined as the radius where the Kohn-Sham potential is equal to the negative ionization potential of the atom, i.e., where v_s(R_t) = ϵ_h. It was recently shown [E. Ospadov et al., Proc. Natl. Acad. Sci. U. S. A. 115, E11578-E11585 (2018)] to yield chemically relevant bonding distances, in line with known empirical values. In this work, we show that extension of the concept to non-integer electron number yields additional information about atomic systems and can be used to detect the difficulty of adding or subtracting electrons. Notably, it reflects the ease of bonding in open p-shells and its greater difficulty in open s-shells. The latter manifests in significant discontinuities in the turning radius as the electron number changes the principal quantum number of the outermost electronic shell (e.g., going from Na to Na²⁺). We then show that a non-integer picture is required to correctly interpret bonding and dissociation in H₂ ⁺. Results are consistent when properties are calculated exactly or via an appropriate approximation. They can be interpreted in the context of conceptual density functional theory.