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Restoring the Point-and-Charge Gradient Expansion for the Strong Interaction Density Functionals

L. A. Constantin, F. Naeem, 3 E. Fabiano, F. Sarcinella, F. Della Sala

Abstract

The strong-interaction functionals $W_\infty[n]$ and ${W'}_\infty[n]$ play an important role in the adiabatic-connection method of Density Functional Theory. The strictly-correlated electron approach can be used to exactly compute these functionals, yet calculations are computationally very expensive even for small electronic systems, and thus semilocal approximations have been proposed. In this work we develop a meta-generalized gradient approximation (meta-GGA) model for the strong-interaction functionals, enhanced point-and-charge (ePC), constructed from exact constraints. In particular, the ePC restores the second-order gradient expansion of the PC model, that is relevant for the equilibrium properties of Wigner crystals, and ensures the non-negativity of ${W'}_\infty[n]$. We assess the ePC model for atoms and various model systems: Hooke's atoms, two-electron exponential densities, s- and p-hydrogenic shells, quasi-two-dimensional infinite barrier model, perturbed uniform electron gas and H$_2$ dissociation. We prove a good overall accuracy of the ePC model, that achieves a broader applicability than any previous semilocal models.

Restoring the Point-and-Charge Gradient Expansion for the Strong Interaction Density Functionals

Abstract

The strong-interaction functionals and play an important role in the adiabatic-connection method of Density Functional Theory. The strictly-correlated electron approach can be used to exactly compute these functionals, yet calculations are computationally very expensive even for small electronic systems, and thus semilocal approximations have been proposed. In this work we develop a meta-generalized gradient approximation (meta-GGA) model for the strong-interaction functionals, enhanced point-and-charge (ePC), constructed from exact constraints. In particular, the ePC restores the second-order gradient expansion of the PC model, that is relevant for the equilibrium properties of Wigner crystals, and ensures the non-negativity of . We assess the ePC model for atoms and various model systems: Hooke's atoms, two-electron exponential densities, s- and p-hydrogenic shells, quasi-two-dimensional infinite barrier model, perturbed uniform electron gas and H dissociation. We prove a good overall accuracy of the ePC model, that achieves a broader applicability than any previous semilocal models.
Paper Structure (16 sections, 25 equations, 14 figures, 4 tables)

This paper contains 16 sections, 25 equations, 14 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. The ePC model for $W_\infty$:
  4. The ePC model for $W'_\infty$:
  5. Properties of the ePC model:
  6. Applications to atoms, model systems and H$_2$ dissociation:
  7. Atoms
  8. Two-electron model density:
  9. The $s$ and $p$ hydrogenic shells:
  10. Quasi-two-dimensional infinite barrier model (quasi-2D IBM):
  11. Perturbed uniform electron gas (UEG):
  12. $H_2$ dissociation
  13. Application to atomization energies and ionization potentials:
  14. Optimized hybrid orbitals:
  15. Atomization energies and ionization potentials
  16. ...and 1 more sections

Figures (14)

  • Figure 1: The mean absolute error per electron $MAE_N$ for $W_\infty^{{ePC\,}}(p)$ (red curve) and $W_\infty^{'{ePC\,}}(p)$ (black curve), as a function of the parameter $p$. The minimum error is at $p=6.65$ for $W_\infty^{{ePC\,}}$ and at $p=11$ for $W_\infty^{'{ePC\,}}$, respectively.
  • Figure 2: The enhancement factor $F(s,z,\zeta)$ of the $W_\infty$, for $z=1$ (upper panel) and $z=0$ (lower panel) versus the reduced gradient $s$, in case of several models (ePC , PC, hPC and meta-GGA).
  • Figure 3: The enhancement factor $F'_\infty(s,z,\zeta)$ of the $W'_\infty$, for $z=1$ (upper panel), $z=0.5$ (middle panel) and $z=0$ (lower panel) versus the reduced gradient $s$, in case of several models (ePC , PC, hPC and meta-GGA).
  • Figure 4: The enhancement factors $F_\infty$ (upper panel) and $F'_\infty$ (lower panel) versus the radial distance from the nucleus $r$, for the Ne atom.
  • Figure 5: The radial density $4\pi r^2 n_\beta(r)$ of the two-electron model density in Eq. (\ref{['eqne15']}), versus the radial distance $r$, for several values of the $\beta$ parameter ($\beta=0$, 1 and 3).
  • ...and 9 more figures