Atomic electronic structure calculations with Hermite interpolating polynomials

We have recently described the implementation of atomic electronic structure calculations within the finite element method with numerical radial basis functions of the form $\chi_{\mu}(r)=r^{-1}B_{\mu}(r)$, where high-order Lagrange interpolating polynomials (LIPs) were used as the shape functions $B_{\mu}(r)$ [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)]. In this work, we discuss how $\chi_{\mu}(r)$ can be evaluated in a stable manner at small $r$ and also revisit the choice of the shape functions $B_{\mu}(r)$. Three kinds of shape functions are considered: in addition to the $\mathcal{C}^{0}$ continuous LIPs, we consider the analytical implementation of first-order Hermite interpolating polynomials (HIPs) that are $\mathcal{C}^{1}$ continuous, as well as numerical implementations of $n$-th order ($\mathcal{C}^{n}$ continuous) HIPs that are expressed in terms of an underlying high-order LIP basis. Furnished with the new implementation, we demonstrate that the first-order HIPs are reliable even with large numbers of nodes and that they also work with non-uniform element grids, affording even better results in atomic electronic structure calculations than LIPs with the same total number of basis functions. We demonstrate that discontinuities can be observed in the spin-$\sigma$ local kinetic energy $\tau_{\sigma}$ in small LIP basis sets, while HIP basis sets do not suffer from such issues; however, either set can be used to reach the complete basis set limit with smooth $\tau_{\sigma}$. Moreover, we discuss the implications of HIPs on calculations with meta-GGA functionals with a number of recent meta-GGA functionals, and find most Minnesota functionals to be ill-behaved. We also examine the potential usefulness of the explicit control over the derivative in HIPs for forming numerical atomic orbital basis sets, but find that confining potentials are still likely a better option.