Analysis of the Heyd-Scuseria-Ernzerhof density functional parameter space

TL;DR

This paper systematically maps the two-parameter space of HSE density functionals, defined by the exchange fraction $a$ and the short-range screening length $\omega^{-1}$, to assess performance across diverse properties. By comparing to sX-LDA and exploring variants such as sX-PBE, HSE12, and HSE12s, the authors identify near-1D manifolds of similarly accurate functionals and establish practical parameter choices that balance accuracy with computational cost. The work connects HSE performance to COHSEX-like screening concepts, discusses limitations in IP/EA eigenvalues due to environment-independent screening, and proposes future directions with more sophisticated screening models (e.g., HISS) and environment-dependent $W$ to extend applicability to molecules and interfaces. Overall, the study provides actionable guidance for using and improving screened-exchange hybrids, with significant implications for accurate band gaps in semiconductors and reliable total energies across diverse chemical spaces.

Abstract

The Heyd-Scuseria-Ernzerhof (HSE) density functionals are popular for their ability to improve the accuracy of standard semilocal functionals such as Perdew-Burke-Ernzerhof (PBE), particularly for semiconductor band gaps. They also have a reduced computational cost compared to hybrid functionals, which results from the restriction of Fock exchange calculations to small inter-electron separations. These functionals are defined by an overall fraction of Fock exchange and a length scale for exchange screening. We systematically examine this two-parameter space to assess the performance of hybrid screened exchange (sX) functionals and to determine a balance between improving accuracy and reducing the screening length, which can further reduce computational costs. Three parameter choices emerge as useful: "sX-PBE" is an approximation to the sX-LDA screened exchange density functionals based on the local density approximation (LDA); "HSE12" minimizes the overall error over all tests performed; and "HSE12s" is a range-minimized functional that matches the overall accuracy of the existing HSE06 parameterization but reduces the Fock exchange length scale by half. Analysis of the error trends over parameter space produces useful guidance for future improvement of density functionals.

Figures (7)

• Figure 1: Approximation of the exponential exchange screening function with a scaled complementary error function. Both functions are displayed in the upper panel and their difference is displayed in the lower panel.
• Figure 2: Semiconductor band gap errors over HSE space. A subset of functionals with approximately equivalent accuracy extend from (0,0.19) to (0.83,1).
• Figure 3: Molecular formation energy errors over HSE space. A subset of functionals with approximately equivalent accuracy extend from (0,0.3) to (0.93,1).
• Figure 4: IP and EA errors in HSE space, comparing $\Delta$SCF to eigenvalue-based estimates and including polarization corrections.
• Figure 5: IP of a dilute He$_n$ cluster, comparing $-\epsilon_\mathrm{HOMO}$(He$_n$) to $E$(He$_n^+$)$-E$(He$_n$) using the HSE06 functional and full Fock exchange with PBE correlation (HF+c), which correspond to the points $(0.5,0.25)$ and $(0.0,1.0)$ in HSE space. In HSE06, the $\Delta$SCF and eigenvalue-based results converge for $n \rightarrow \infty$. The electron-hole of He$_n^+$ is localized on one atom with HF+c and uniformly divided among all $n$ atoms with HSE06. Any $n$-dependence is a size-consistency error.