What Do We Learn from the Classical Turning Surface of the Kohn-Sham Potential as Electron Number Is Varied Continuously?

The classical turning radius Rt of an atom can be defined as the radius where the KS potential is equal to the negative ionisation potential of the atom, i.e. where v_s(R_t)=\epsilon_h. It was recently shown [P.N.A.S. 115, E11578 (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 Na^{2+}). We then show that a non-integer picture is required to correctly interpret bonding and dissociation in H_2^+. 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.