SCAN Meta-GGA: Constraint-Satisfying DFT Functional

• SCAN meta-GGA is a rigorously constructed density functional that satisfies 17 exact constraints for accurately modeling diverse bonding environments.
• It dynamically distinguishes between covalent, metallic, and weakly bonded regimes using iso-orbital indicators and kinetic energy density dependency.
• Regularized variants like rSCAN and r²SCAN mitigate grid-sensitivity issues, enabling efficient high-throughput and solid-state simulations.

The SCAN (Strongly Constrained and Appropriately Normed) meta-generalized gradient approximation (meta-GGA) is a semilocal exchange–correlation functional within density functional theory (DFT) that is rigorously constructed to satisfy all known exact constraints for functionals at the meta-GGA rung. SCAN achieves a balance between computational efficiency and transferability, with enhanced accuracy for diverse bonding environments and solid-state phases. Its "restored and regularized" variants, particularly r²SCAN, retain SCAN’s constraint-satisfaction while curing numerical difficulties, making high-throughput and solid-state calculations practical. This article details the mathematical construction, foundational principles, extensions, performance assessments, and implementation best practices underlying SCAN and r²SCAN, drawing on primary research literature.

1. Mathematical Formulation and Constraint Satisfaction

The SCAN functional is defined for a spin-unpolarized system by

$$E_{xc}^{\text{SCAN}}[\rho, \nabla \rho, \tau] = \int d^3r\, \rho(\mathbf{r})\, \varepsilon_x^{\text{unif}}[\rho(\mathbf{r})]\, F_x[s(\mathbf{r}),\alpha(\mathbf{r})] + \int d^3r\, \rho(\mathbf{r})\, \varepsilon_c^{\text{SCAN}}[\rho(\mathbf{r}),\nabla\rho(\mathbf{r}),\tau(\mathbf{r})].$$

Key ingredients are:

• $\rho(\mathbf{r})$: electron density
• $\nabla \rho(\mathbf{r})$: gradient of $\rho$
• $$\tau(\mathbf{r}) = \frac{1}{2} \sum_i |\nabla \psi_i(\mathbf{r})|^2$$: Kohn–Sham kinetic energy density
• $\tau_W(\mathbf{r}) = |\nabla \rho|^2/(8\rho)$: von Weizsäcker kinetic energy density
• $$\tau_{\text{unif}}(\mathbf{r}) = (3/10)(3\pi^2)^{2/3} \rho^{5/3}$$: uniform electron gas kinetic energy density

Dimensionless variables:

• $s(\mathbf{r}) = |\nabla \rho|/[2k_F \rho]$ with $k_F = (3\pi^2 \rho)^{1/3}$
• $$\alpha(\mathbf{r}) = [\tau(\mathbf{r}) - \tau_W(\mathbf{r})] / \tau_{\text{unif}}(\mathbf{r})$$

The exchange enhancement factor $F_x[s,\alpha]$ interpolates between "single-orbital" regions ($\alpha\to 0$), slowly varying metallic regions ($\alpha \approx 1$), and weakly bonded regions ($\alpha \gg 1$). The SCAN construction ensures exact recovery of all 17 known meta-GGA constraints, including uniform density limits, second-order gradient expansion, tight Lieb–Oxford bound, one-electron self-interaction freedom, exact scaling behavior, and spin-scaling relations (Yin et al., 2024, Ehlert et al., 2020).

2. Physical Principles and Design Strategy

SCAN’s foundational strategy is the nonempirical satisfaction of constraints, with no empirical fitting to bonded or molecular data. The functional leverages the iso-orbital indicator $\alpha$ to distinguish between different bonding regimes dynamically. In the context of band gap formation and chemical bonding, SCAN’s kinetic-energy dependence allows it to "recognize" and properly interpolate between covalent, metallic, ionic, hydrogen-bonded, and van der Waals (vdW) environments. Mechanistically, SCAN suppresses spurious metallic $d$–$d$ overlap (as in quantum spin-Hall TMDs), shifts orbital energies in line with experimental gaps, and improves localization in Mott or correlated systems (Yin et al., 2024, Zhang et al., 2017, Buda et al., 2017).

3. Numerically Robust Variants: rSCAN and r²SCAN

While SCAN is highly accurate, its switching functions $f(\alpha)$ generate sharp curvature near $\alpha \approx 1$, leading to severe grid-sensitivity and convergence issues. rSCAN replaces the piecewise analytic switching with a smooth 7th-degree polynomial at the price of breaking some exact constraints (Yamamoto et al., 2020). r²SCAN ("restored and regularized SCAN") further improves upon rSCAN by introducing a regularized iso-orbital indicator

$$\bar{\alpha} = \frac{\tau - \tau_W}{\tau_U + \eta\,\tau_W},\quad \eta = 10^{-3},$$

and reconstructs the interpolation functions to rigorously restore all critical meta-GGA constraints (GE2, coordinate scaling, uniform limit). The enhancement factor and correlation functional are correspondingly regularized. r²SCAN exhibits MAEs and performance nearly identical to SCAN while being stable and efficient for plane-wave and real-space implementations. In practice, r²SCAN requires 2–5 times fewer grid points than SCAN to achieve the same accuracy and is robust for large-scale simulations and pseudopotential generation (Furness et al., 2020, Ehlert et al., 2020).

Functional	Accuracy (MAE, WTMAD₂, typical)	Grid Robustness	Constraint Satisfaction
SCAN	2.8–6 kcal/mol (atomization), 8.6 kcal/mol (WTMAD₂)	Requires very fine grids	Full (17/17)
rSCAN	Similar to SCAN (slight degradation in AE)	Coarser grids sufficient	Violates a few
r²SCAN	Nearly SCAN-level, 7.5 kcal/mol (WTMAD₂)	Coarse grids, stable	Full (restored)

4. Performance Across Materials and Chemistry

SCAN and r²SCAN have been extensively benchmarked for both molecular and solid-state properties:

• Lattice constants: SCAN (MAE = 0.015 Å, LC20), r²SCAN (0.027 Å) (Furness et al., 2020)
• Thermochemistry: r²SCAN-D4 yields bond lengths within 0.4–1.9 pm, WTMAD₂ = 7.5 kcal/mol (GMTKN55), main group and transition metal complexes are accurately described (Ehlert et al., 2020)
• Band gaps: SCAN improves over GGA but does not fully correct underestimation; gKS implementation necessary for realistic gaps; in 1T′-WTe₂, SCAN and r²SCAN yield gaps of –62 meV and –44 meV (no SOC), outperforming GGA but not matching the result of MVS or HSE06+SOC (Yin et al., 2024)
• Ferroelectrics and multiferroics: SCAN corrects super-tetragonality and provides accurate structural distortion, polarization, and energetics for prototypical oxides and hydrogen-bonded ferroelectrics (Zhang et al., 2017)
• Magnetism: mSCAN (magnetic extension) corrects SCAN’s over-magnetization, matching experimental moments and band gaps in itinerant and localized magnets; r²SCAN yields Néel temperatures in antiferromagnets with Pearson R=0.98, MAPE=22% versus experiment, dramatically outperforming GGA and GGA+U (Desmarais et al., 2024, Rezaei et al., 10 Jan 2025)
• Dispersion: SCAN and r²SCAN lack explicit long-range vdW interactions, but with D4 correction, r²SCAN-D4 achieves chemically accurate lattice energies and noncovalent binding energies (e.g., MAD = 0.7 kcal/mol, DMC8 molecular crystals) (Ehlert et al., 2020)

5. Extensions and Adaptations

Several SCAN-based extensions have been developed:

• SCAN-L: Deorbitalized version using Laplacian-dependent approximations (replacing explicit $\tau$ with Laplacian-based kinetic energy density), enables ordinary KS calculations and runs up to 3× faster with near-parity in bulk structure and energetics (Mejia-Rodriguez et al., 2018).
• mSCAN and JSCAN: Magnetic and spin-current–gauge–invariant generalizations achieve correct magnetic responses, SU(2) invariance, and improved descriptions of systems with spin currents or SOC, correcting SCAN’s tendency toward spurious moment formation and enhancing accuracy for spin–orbit–split bands and related observables (Desmarais et al., 2024, Desmarais et al., 2024).
• SCAN-U: Empirical tuning of self-interaction correction on specific subspaces (e.g., transition-metal $d$ levels) can recover physical properties when SCAN overcorrects delocalization (as in Ni$_2$MnGa) (Baigutlin et al., 2020).

6. Computational Considerations and Practicality

• Cost: SCAN is ~1.2–1.3× GGA for each SCF iteration, r²SCAN is slightly higher per-step but often requires fewer iterations due to smoother SCF behavior. r²SCAN is especially well-suited for plane-wave codes and high-throughput applications, avoiding SCAN’s grid-induced instabilities (Yin et al., 2024, Furness et al., 2020).
• Grid Sensitivity: SCAN requires extremely dense quadrature (≥$10^5$ points per atom), r²SCAN requires only standard GGA-level grids. Failure to converge SCAN may result in spurious energy or band gap artifacts (Yamamoto et al., 2020, Yang et al., 2016).
• High-throughput/large-scale: r²SCAN and SCAN-L enable efficient, accurate high-throughput searches in materials screening, ferroelectric and magnetic discovery, and molecular databases. For QSH and TMD systems with small band gaps, MVS or hybrid functionals plus SOC are currently required for quantitative accuracy; for broad surveys, r²SCAN provides a practical universal alternative (Yin et al., 2024, Ehlert et al., 2020).

7. Perspective and Limitations

SCAN and its descendants embody the principle of nonempirical constraint satisfaction, positioning them as physically grounded reference functionals for both development and applications. The regularized r²SCAN achieves SCAN-level transferability at GGA-like cost and stability, now extended to effective dispersion correction (r²SCAN-D4), magnetism (mSCAN), and spin–orbit physics (JSCAN). Limitations include grid sensitivity (for SCAN), moderate underestimation of atomization energies in regularized variants (rSCAN), and incomplete correction of self-interaction in certain $d$-electron systems. For strongly correlated or small-gap systems, controlled symmetry breaking or localized “+U” corrections may sometimes be required (Perdew et al., 2022, Baigutlin et al., 2020). Overall, SCAN and r²SCAN represent highly transferable, physically-motivated meta-GGAs central to modern quantum chemistry and condensed matter computational practice (Yin et al., 2024, Ehlert et al., 2020, Rezaei et al., 10 Jan 2025).