Accuracy of a Recent Regularized Nuclear Potential

F. Gygi recently suggested an analytic, norm-conserving, regularized nuclear potential to enable all-electron plane-wave calculations [J. Chem. Theory Comput. 2023, 19, 1300--1309]. This potential $V(r)$ is determined by inverting the Schr\"odinger equation for the wave function ansatz $\phi(\boldsymbol{r})=\exp[-h(\boldsymbol{r})]/\sqrt{\pi}$ with $h(\boldsymbol{r})=r\text{erf}(ar)+b\exp(-a^{2}r^{2})$, where $a$ and $b$ are parameters. Gygi fixes $b$ by demanding $\phi$ to be normalized, the value $b(a)$ depending on the strength of the regularization controlled by $a$. We begin this work by re-examining the determination of $b(a)$ 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 of $b$, but also leads to machine-precision level satisfaction of the normalization condition. Moreover, we extend Gygi's plane-wave study by examining the accuracy of $V(r)$ 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 parameter $a$, orbital energies and shapes are indeed reproduced accurately by the regularized potential even with relatively small values of $a$, as compared to results obtained with a point nucleus. The accuracy of the potential is furthermore studied with $s$-$d$ excitation energies of Sc--Cu as well as ionization potentials of He--Kr, which are found to converge to sub-meV precision with $a=4$. 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.