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Regularized SCAN functional
Albert P. Bartók, Jonathan R. Yates
TL;DR
SCAN's numerical instabilities complicate reliable pseudopotential generation and grid-based DFT implementations. The paper regularizes SCAN's iso-orbital indicator and switching behavior, then finds that rSCAN remains close in accuracy while improving numerical smoothness and stability.
Problem
Numerical instabilities in SCAN hinder self-consistent generalized Kohn-Sham calculations and reliable generation of consistent pseudopotentials for plane-wave DFT.
Method
The paper introduces rSCAN by regularizing SCAN's kinetic-energy-density inputs, iso-orbital indicator, and switching function to smooth problematic behavior while preserving the original functional form closely.
Results
rSCAN remains transferable and accurate across broad solid-state and molecular systems, with an Ar-dimer relative-binding-energy mean absolute error of 1.1 kcal/mol versus below 1 kcal/mol for SCAN.
Takeaways & Limitations
rSCAN supports more straightforward pseudopotential and PAW dataset generation and may improve stability when exchange-correlation potentials are represented on grids.
Abstract
from arXiv · showhide
We propose modifications to the functional form of the SCAN density functional to eliminate numerical instabilities. This is necessary to allow reliable, automatic generation of pseudopotentials (including PAW potentials). The regularized SCAN is designed to match the original form very closely, and we show that its performance remains comparable.
I. INTRODUCTION
SCAN offers greater flexibility and accuracy than GGA functionals, but numerical instabilities and limited implementation hinder consistent plane-wave pseudopotential calculations. The paper proposes rSCAN to smooth these instabilities while retaining SCAN's exchange-correlation accuracy.
- Motivation: SCAN extends GGA functionals by incorporating orbital kinetic energy density alongside the electron density and its gradients.This additional local property gives meta-GGAs more flexibility in their approximate functional form.
- Implementation gap: SCAN-based pseudopotentials were limited, with the available norm-conserving library lacking kinetic energy density augmentation terms and nonlinear core corrections.Consequently, many condensed-phase calculations used PBE pseudopotentials as an uncontrolled approximation.
- Problem: Severe numerical instabilities appeared both in solving atomic generalized Kohn-Sham equations and in constructing pseudopotentials.These problems motivated developing a regularized SCAN form suitable for pseudopotential generation.
- Approach: The paper analyzes SCAN's iso-orbital indicator and modifies it to remove divergent exchange-correlation potentials while keeping it close to the original in most regions.The modification targets unphysical divergences and problematic behavior in the switching construction.
- Evaluation: rSCAN uses smoother switching while retaining SCAN's exchange-correlation energy description, and is tested for closeness to SCAN and in consistent plane-wave pseudopotential calculations.The evaluation combines functional comparison with benchmark calculations using rSCAN-generated pseudopotentials.
A. The iso-orbital indicator function
The iso-orbital indicator α distinguishes local bonding environments and switches among corresponding exchange-correlation approximations, but its derivatives can diverge in low-density regions. rSCAN regularizes the indicator and its inputs to improve numerical behavior while preserving the original interpretation over most relevant regions.
- Indicator role: The iso-orbital indicator α detects covalent single, metallic, and weak-bond environments and switches among corresponding local exchange-correlation approximations.It is defined from orbital kinetic-energy-density quantities and is central to SCAN and related meta-GGAs.
- SCAN instability: Divergent derivatives of α at rapidly decreasing densities can make exchange-correlation potentials diverge, including for hydrogen-like 1s densities.The derivatives enter the potential expressions and are insufficiently damped in these cases.
- Kinetic-density regularization: The worst divergence occurs in low-density single-orbital regions where α approaches zero, partly because τU decreases rapidly in α's denominator.The first regularization replaces τU with τ′U = τU + τr, using τr = 1 × 10−4 to affect α only at very low densities.
- Indicator regularization: The regularized indicator α′ differs from α only at small values and has vanishing derivatives with respect to n, ∇n, and τ in single-orbital regions.This reduces interference between switching and the physically motivated exchange-correlation expressions.
- Physical interpretation: For isolated noble-gas atoms, rSCAN returns the indicator toward zero in the far-density tail instead of SCAN's α ≫ 1 behavior.The regularized behavior is described as more similar to helium, where α = 0 everywhere by construction.
- Figure 2: Figure 2 compares SCAN and rSCAN iso-orbital indicators versus distance from the nucleus and relates them to the dominant single-orbital contribution in isolated Kr.The top panel shows the indicator; the bottom panel shows the highest-contributing orbital's fraction of the total density.
B. The switching function
The authors identify numerical instabilities in SCAN’s switching function near α ≈1 and replace that region with a smoother polynomial while preserving the functional’s performance closely.
- B. The switching function: SCAN’s switching function introduces rapidly oscillating regions in the exchange-correlation potential near α ≈1.The switching function’s original form was fitted for accurate energies, but its specific shape creates numerical instability.
- B. The switching function: Replacing 0 < α < 2.5 with a 7th-degree polynomial removes the oscillatory behavior while retaining SCAN-like performance.The polynomial preserves selected endpoint derivatives and imposes f(1) = 0.
- B. The switching function: Figure 3 compares the SCAN and rSCAN switching functions and their derivatives.The comparison focuses on the smoother behavior produced by the modified switching function.
- B. The switching function: Figure 4 compares SCAN and rSCAN exchange-correlation potentials for isolated He and Ge atoms using PBE self-consistent densities, with PBE as a reference.The figure provides a practical test of the improved smoothness in atomic calculations.
III. RESULTS
The authors implement rSCAN in plane-wave and quantum-chemistry codes and benchmark it across atomic, solid, magnetic, dimer, and water systems. The results show performance comparable to SCAN, with close agreement for several tested properties.
- III. RESULTS: rSCAN was implemented in CASTEP and PySCF, with self-consistent generalized Kohn-Sham calculations and on-the-fly ultrasoft pseudopotentials in CASTEP.The implementation includes pseudization of the τ-dependent potential contribution.
- III. RESULTS: 1.1 kcal/mol mean absolute error was obtained for rSCAN Ar-dimer relative binding energies at 1.6 Å, 1.8 Å, and 2.0 Å, versus below 1 kcal/mol for SCAN.The compared switching-function tests include isolated noble-gas atoms and compressed Ar dimers.
- III. RESULTS: Across the reported model-system benchmarks, rSCAN demonstrates performance comparable to the original SCAN functional.The literature comparisons are qualified because published values use different codes, basis sets, and sometimes inconsistent PAW pseudopotentials.
- III. RESULTS: rSCAN gives a bcc-iron spin moment of 2.62 µB at a 2.84 Å optimized lattice constant, close to SCAN’s 2.60 µB at 2.85 Å.This indicates similar performance for the reported ferromagnetic test despite SCAN’s known magnetic-energy overestimation.
- III. RESULTS: rSCAN recovers the CCSD(T)- and SCAN-consistent energetic ordering of four low-energy water hexamers and somewhat improves the absolute energies.The water monomer geometry and isolated-molecule dipole moments are also close to the original SCAN results.
IV. CONCLUSIONS
The implemented regularized SCAN functional improves numerical stability while retaining the accuracy of original SCAN. Benchmarking indicates broad transferability and supports easier pseudopotential and PAW dataset generation.
- SCAN was implemented in a plane-wave DFT program with ultrasoft pseudopotentials generated consistently using the same functional.
- The regularized form improves numerical stability while retaining the accuracy of the original SCAN functional.
- Table IV reports water-hexamer dissociation energies alongside water molecular geometry and dipole properties, with reference and SCAN values included.
- Benchmark calculations show rSCAN remains transferable and accurate across a broad range of solid-state and molecular systems.
- rSCAN is expected to simplify pseudopotential and PAW dataset generation and improve stability when exchange-correlation functionals are represented on grids.