Fully numerical Hartree-Fock and density functional calculations. I. Atoms

Although many programs have been published for fully numerical Hartree--Fock (HF) or density functional (DF) calculations on atoms, we are not aware of any that support hybrid DFs, which are popular within the quantum chemistry community due to their better accuracy for many applications, or that can be used to calculate electric properties. Here, we present a variational atomic finite element solver in the HelFEM program suite that overcomes these limitations. A basis set of the type $\chi_{nlm}(r,\theta,\phi)=r^{-1}B_{n}(r)Y_{l}^{m}(\hat{\boldsymbol{r}})$ is used, where $B_{n}(r)$ are finite element shape functions and $Y_{l}^{m}$ are spherical harmonics, which allows for an arbitrary level of accuracy. HelFEM supports nonrelativistic HF and DF including hybrid functionals, which are not available in other commonly available program packages. Hundreds of functionals at the local spin density approximation (LDA), generalized gradient approximation (GGA), as well as the meta-GGA levels of theory are included through an interface with the Libxc library. Electric response properties are achievable via finite field calculations. We introduce an alternative grid that yields faster convergence to the complete basis set than commonly used alternatives. We also show that high-order Lagrange interpolating polynomials yield the best convergence, and that excellent agreement with literature HF limit values for electric properties, such as static dipole polarizabilities, can be achieved with the present approach. Dipole moments and dipole polarizabilities at finite field are reported with the PBE, PBEh, TPSS, and TPSSh functionals. Finally, we show that a recently published Gaussian basis set is able to reproduce absolute HF and DF energies of neutral atoms, cations, as well as anions within a few dozen microhartrees.