F. Gygi recently
suggested an analytic, norm-conserving, regularized
nuclear potential to enable all-electron plane-wave calculations[GygiJ. Chem. Theory Comput.2023,19,1300–1309.]36757291. This potentialV(r) is determined by inverting the Schrödinger equation
for the wave function Ansatz ϕ(r) = exp[−h(r)]/√π withh(r) =rerf(ar) +bexp(−a2r2), whereaandbare
parameters. Gygi fixesbby demanding ϕ to
be normalized, with the valueb(a) depending on the strength of the regularization controlled bya. We begin this work by re-examining the determination
ofb(a) and find that the original
10-decimal tabulations of Gygi are only correct to 5 decimals, leading
to normalization errors in the order of 10–10. In
contrast, we show that a simple 100-point radial quadrature scheme
not only ensures at least 10 correct decimals ofbbut also leads to machine-precision level satisfaction of the normalization
condition. Moreover, we extend Gygi’s plane-wave study by examining
the accuracy ofV(r) with high-precision
finite element calculations with Hartree–Fock and LDA, GGA,
and meta-GGA functionals on first- to fifth-period atoms. We find
that although the convergence of the total energy appears slow in
the regularization parametera, orbital energies
and shapes are indeed reproduced accurately by the regularized potential
even with relatively small values ofa, as compared
to results obtained with a point nucleus. The accuracy of the potential
is furthermore studied withs-dexcitation
energies of Sc–Cu as well as ionization potentials of He–Kr,
which are found to converge to sub-meV precision witha= 4. The findings of this work are in full support of Gygi’s
contribution, indicating that all-electron plane-wave calculations
can be accurately performed with the regularized nuclear potential.