Convergence of atomic calculations for singular confinement with exponent n=3 at very narrow transition widths

Determine whether self-consistent atomic electronic-structure calculations can be converged for atoms confined by the generalized singular confinement potential V_c(r)=0 for r≤r_i; V_0·exp(−(r_c−r_i)/(r−r_i))/(r_c−r)^n for r_i<r<r_c; and V_c(r)=∞ for r≥r_c, when the exponent is n=3 and the transition width satisfies r_c−r_i≤0.5 Å, in the regime used to approach the hard-wall limit.

Background

The paper studies a family of singular confinement potentials used for generating numerical atomic orbitals, defined by a smooth exponential factor divided by a power (r_c−r)^n, which enforces strict localization at r=r_c while being continuous at r=r_i. The authors analyze how these potentials approach the hard-wall limit by reducing the transition width r_c−r_i.

For exponents n∈{1,2,3}, they systematically decrease r_c−r_i and observe that the confined orbitals closely match hard-wall solutions when the transition width becomes very small. However, they explicitly report a failure to converge calculations for n=3 when r_c−r_i≤0.5 Å, indicating an unresolved convergence issue in this steep-potential regime.