Fully numerical Hartree-Fock and density functional calculations. II. Diatomic molecules

We present the implementation of a variational finite element solver in the HelFEM program for benchmark calculations on diatomic systems. A basis set of the form $\chi_{nlm}(\mu,\nu,\phi)=B_{n}(\mu)Y_{l}^{m}(\nu,\phi)$ is used, where $(\mu,\nu,\phi)$ are transformed prolate spheroidal coordinates, $B_{n}(\mu)$ are finite element shape functions, and $Y_{l}^{m}$ are spherical harmonics. The basis set allows for an arbitrary level of accuracy in calculations on diatomic molecules, which can be performed at present with either nonrelativistic Hartree--Fock (HF) or density functional (DF) theory. Hundreds of DFs at the local spin-density approximation (LDA), generalized gradient approximation (GGA) and the meta-GGA level can be used through an interface with the Libxc library; meta-GGA and hybrid DFs aren't available in other fully numerical diatomic program packages. Finite electric fields are also supported in HelFEM, enabling access to electric properties. We introduce a powerful tool for adaptively choosing the basis set by using the core Hamiltonian as a proxy for its completeness. HelFEM and the novel basis set procedure are demonstrated by reproducing the restricted open-shell HF limit energies of 68 diatomic molecules from the $1^{\text{st}}$ to the $4^{\text{th}}$ period with excellent agreement with literature values, despite requiring \emph{orders of magnitude} fewer parameters for the wave function. Then, the electric properties of the BH and N2 molecules under finite field are studied, again yielding excellent agreement with previous HF limit values for energies, dipole moments, and dipole polarizabilities, again with much more compact wave functions than what were needed in the literature references. Finally, HF, LDA, GGA, and meta-GGA calculations of the atomization energy of N2 are performed, demonstrating the superb accuracy of the present approach.