  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.
