Highly accurate numerical solution of Hartree–Fock equation with pseudospectral method for closed-shell atoms

Abstract

The Hartree-Fock (HF) equation for atoms with closed (sub)shells is transformed with the pseudospectral (PS) method into a discrete eigenvalue equation for scaled orbitals on a finite radial grid. The Fock exchange operator and the Hartree potential are obtained from the respective Poisson equations also discretized using the PS representation. The numerical solution of the discrete HF equation for closed-(sub)shell atoms from He to No is robust, fast and gives extremely accurate results, with the accuracy superior to that of the previous HF calculations. A very moderate number of 33 to 71 radial grid points is sufficient to obtain total energies with 14 significant digits and occupied orbital energies with 12 to 14 digits in numerical calculations using the double precision (64-bit) of the floating-point format.The electron density at the nucleus is then determined with 13 significant digits and the Kato condition for the density and s orbitals is satisfied with the accuracy of 11 to 13 digits. The node structure of the exact HF orbitals is obtained and their asymptotic dependence, including the common exponential decay, is reproduced very accurately. The accuracy of the investigated quantities is further improved by performing the PS calculations in the quadruple precision (128-bit) floating-point arithmetic which provides the total energies with 25 significant digits while using only 80 to 130 grid points.

Spatial distribution of atomic electronic density for elements 1–54 as coming from a Hartree–Fock treatment within the minimum atomic parameters paradigm

“…Taking numerical Hartree–Fock orbitals or Bunge's basis sets we see that this condition is fulfilled already at the Hartree–Fock level [7], and even individually for each atomic s orbital [26]—the only ones with a non‐vanishing electron density at the origin.…”

“…Kato's condition “measures” the orbitals in the vicinity of the nucleus. For looking at the opposite side, we may integrate energy contributions [7] from outside toward the nucleus, like kinetic energy or the electron‐nucleus attraction. In this way we see, in which region the difference in total energy between the MAP and the Bunge orbital sets are located.…”

“…As we have shown in paper I, such a setting misses about 3% in total energy with respect to numerical Hartree–Fock orbitals [7, 8], or expansions in the large Slater‐type basis sets of Bunge et al [9], which in their turn give the results at μ ‐Hartree precision with respect to the previous. Despite this loss in total energy with respect to purely numerical, basis‐set‐free Hartree–Fock calculations, Moscow–Aachen–Paris (MAP) orbitals have an overlap with their numerical analogues of more than 90%.…”

Radial Kohn-Sham problem via integral-equation approach

“…These three exchange models introduce a non-local potential vnl that makes the usual approach to solving the radial problem not applicable. Two recent studies addressed this problem with an aim to implement an efficient and precise solver [15,16]. Cinal introduced a compact basis and used it with the Hartree-Fock (HF) equations [15].…”

“…Two recent studies addressed this problem with an aim to implement an efficient and precise solver [15,16]. Cinal introduced a compact basis and used it with the Hartree-Fock (HF) equations [15]. The obtained total energies of atoms were computed in double and quadruple precision ensuring an extreme precision level.…”