We have recently described the implementation of atomic
electronic
structure calculations within the finite element method with numerical
radial basis functions of the form χμ(r) =r–1Bμ(r), where high-order Lagrange
interpolating polynomials (LIPs) were used as the shape functionsBμ(r). In this work, we
discuss how χμ(r) can be evaluated
in a stable manner at smallrand also revisit the
choice of the shape functionsBμ(r). Three kinds of shape functions are considered:
in addition to thecontinuous LIPs, we consider the analytical
implementation of first-order Hermite interpolating polynomials (HIPs)
that arecontinuous, as well as numerical implementations
ofn-th order (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 nonuniform
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-σ local kinetic energy τσin
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 τσ. Moreover, we discuss
the implications of HIPs on calculations with meta-GGA functionals
with a number of recent meta-GGA functionals, and we 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 we find that confining potentials are
still likely a better option.