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

Abstract

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] 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 ≤ Z ≤ 86. Our results indicate that the literature values deviate by up to 10 µE h 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 ≤ Z ≤ 118.

Efficient implementation of the superposition of atomic potentials initial guess for electronic structure calculations in Gaussian basis sets

“…An improved method for determining the atomic potentials necessary for the procedure has been recently presented in ref. 8. A straightforward extension of the work in refs.…”

“…The DFA potentials used in the present work have the important property that far away the effective charge goes to zero, 1 Z A (∞) = Z A − Z el A (∞) = 0. Imposing this long-range limit translates into the condition p c p = −Z (8) where Z is the nuclear charge. (Note that atomic potentials that do not satisfy this requirement lead to molecular potentials that become worse and worse in increasing system size, see the discussion in ref.…”

“…10.) Furthermore, we can rewrite equation (7) in terms of an effective charge as Z(r) = Z − Z el (r) by using equation (8). This redefinition of Z(r) coincides with the screened nuclear potential in the case of a point nucleus, but this mathematical trick works equally well with a finite nucleus: the important thing to notice here is that the nuclear model does not enter into the fits of the potential generated by the electrons.…”

“…Although equation (9) already allows fits to the redefined Z(r), these fits may still violate the charge neutrality condition, equation (8). The condition can be enforced by treating the coefficient of the steepest function as a dependent variable…”

“…Non-relativistic calculations were performed with HelFEM 8,20,21 using 20 radial elements and a value of the practical infinity r ∞ = 40a 0 ; the resulting HelFEM energies are converged beyond nanohartree accuracy for all atoms. 8,21 The HelFEM calculations employed fractional occupations resulting in spherically symmetric densities, and the ground state for each element was found automatically by a brute force search, 8 leading e.g. to the unrestricted Hartree-Fock (HF) configurations shown in table I.…”