  A recently developed finite element approach for fully numerical atomic
structure calculations [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)]
is extended to the description of atoms with spherically symmetric densities
via fractionally occupied orbitals. Specialized versions of Hartree-Fock as
well as local density and generalized gradient approximation density
functionals are developed, allowing extremely rapid calculations at the basis
set limit on the ground and low-lying excited states even for heavy atoms.
  The implementation of range-separation based on the Yukawa or complementary
error function (erfc) kernels is also described, allowing complete basis set
benchmarks of modern range-separated hybrid functionals with either integer or
fractional occupation numbers. Finally, computation of atomic effective
potentials at the local density or generalized gradient approximation levels
for the superposition of atomic potentials (SAP) approach [S. Lehtola, J. Chem.
Theory Comput. 15, 1593 (2019)] that has been shown to be a simple and
efficient way to initialize electronic structure calculations is described.
  The present numerical approach is shown to afford beyond microhartree
accuracy with a small number of numerical basis functions, and to reproduce
literature results for the ground states of atoms and their cations for $1 \leq
Z \leq 86 $. Our results indicate that the literature values deviate by up to
10 {\mu}Eh from the complete basis set limit. The numerical scheme for the erfc
kernel is shown to work by comparison to results from large Gaussian basis set
calculations from the literature. Spin-restricted ground states are reported
for Hartree-Fock and Hartree-Fock-Slater calculations with fractional
occupations for $1 \leq Z \leq 118$.
