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 [Gygi J. Chem. Theory Comput. 2023, 19, 1300-1309.]. This potential V ( r ) is determined by inverting the Schrödinger equation for the wave function Ansatz ϕ( r ) = exp[- h ( r )]/√π with h ( r ) = r erf( ar ) + b exp(- a 2 r 2 ), where a and b are parameters. Gygi fixes b by demanding ϕ to be normalized, with 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.