Fully numerical calculations on atoms with fractional occupations and range-separated exchange functionals

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, the 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 micro-Hartree accuracy with a small number of numerical basis functions, and to reproduce the literature results for the ground states of atoms and their cations for 1Peer reviewe