r²SCAN+rVV10 vdW Functional

• r²SCAN+rVV10 is a density functional method that integrates the r²SCAN meta-GGA with a nonlocal rVV10 kernel to accurately capture both covalent bonding and van der Waals interactions.
• The functional achieves numerical stability and efficient self-consistency through refined parameterization, reducing grid sensitivity and convergence issues.
• Benchmark tests show that r²SCAN+rVV10 improves equilibrium volume and binding energy predictions, particularly for layered solids and molecular assemblies.

The r$^2$SCAN+rVV10 van der Waals (vdW) functional is a density functional theory (DFT) approach designed to deliver strong numerical stability and broad accuracy for the prediction of energetic and structural properties in a wide range of materials, particularly those where noncovalent (vdW) interactions are significant. It combines the regularized strongly constrained and appropriately normed meta-generalized gradient approximation (meta-GGA) known as r$^2$SCAN with a nonlocal vdW correction using the rVV10 kernel. This combination leverages the high fidelity of meta-GGA for covalent and weak interactions while accounting for long-range electronic correlation effects essential for layered solids and molecular assemblies (Ning et al., 2022, Kothakonda et al., 2022).

1. Mathematical Structure

The r$^2$SCAN+rVV10 functional adds a nonlocal van der Waals term, $E_{\mathrm{disp}}$, to the semilocal exchange-correlation energy of r$^2$SCAN. The exchange-correlation energy takes the form

$$E_{\mathrm{xc}} = \int d^3r\, n(\mathbf r)\, \epsilon_{\mathrm{xc}}^{r^2\mathrm{SCAN}}[n, \nabla n, \tau] + \frac{1}{2} \iint d^3r\,d^3r'\,n(\mathbf r)\,\Phi(|\mathbf r-\mathbf r'|; n, \nabla n, \tau)\, n(\mathbf r')$$

where $n(\mathbf r)$ is the electron density, $\nabla n$ its gradient, and $\tau$ the positive-definite kinetic energy density.

The nonlocal rVV10 part, $E_{\mathrm{disp}}$, uses a kernel $\Phi(|\mathbf r-\mathbf r'|; \cdots)$ capturing density–density interactions modulated by parameters $C$ (universal, $C=0.0093$ Hartree) and short-range damping parameter $b$. For r$^2$SCAN+rVV10, $b$ is refitted to $11.95$ to avoid double-counting intermediate-range correlation present in the meta-GGA (Kothakonda et al., 2022, Ning et al., 2022).

The r$^2$SCAN component regularizes the dimensionless variable $\alpha = [\tau - \tau_W]/\tau_{\text{unif}}$ (with $\tau_W = |\nabla n|^2/(8n)$ and $\tau_{\text{unif}} \propto n^{5/3}$) using a smooth interpolating function $\tilde\alpha$, ensuring stability in self-consistency and the exchange–correlation potential.

2. Parameterization and Numerical Stability

Optimal parameterization for r$^2$SCAN+rVV10 was achieved by minimizing the mean absolute error (MAE) with respect to the Ar$_2$ binding-energy curve (CCSD(T) reference) and secondary validation on the S22 molecular interaction set. The best-fit value $b=11.95$ yields robust agreement without large empirical adjustment.

A central advantage is numerical stability: r$^2$SCAN eliminates the divergence in $\partial\epsilon/\partial\alpha$ for small $\alpha$, preventing slow or erratic self-consistent field (SCF) convergence and grid-dependent artifacts seen in SCAN+rVV10. This robustness holds even for moderate plane wave cutoffs and standard real-space FFT grids, with no special grid refinement required for stable geometries and energies (Ning et al., 2022).

3. Computational Implementation

Practical calculations employ self-consistent routines in VASP (versions $\geq$5.4.4), with projector augmented-wave (PAW) pseudopotentials. Typical simulation settings include:

• Plane-wave cutoff energy (ENCUT) of 600–900 eV for solids; ENAUG up to 2000 eV for dense augmented grids in molecular systems.
• PAW “hard” setups for meta-GGA accuracy.
• $\Gamma$-centered Monkhorst–Pack $k$-point meshes with spacing $\leq 0.20\,\mathrm{\AA}^{-1}$.
• SCF convergence criteria at $10^{-6}$–$10^{-7}$ eV and force convergence to $0.01$ eV/Å.
• Methfessel–Paxton smearing or tetrahedron method for Brillouin-zone sampling as appropriate.
• The rVV10 kernel is evaluated on the standard real-space grid of VASP.
• Phonon dispersion relations for solids are obtained using the PHONOPY finite-displacement approach, with atomic displacements $\sim$0.015 Å.

Routine calculations require no special grid refinements beyond meta-GGA defaults, and r$^2$SCAN+rVV10 remains free of self-consistency failures or spurious energy/force oscillations (Ning et al., 2022, Kothakonda et al., 2022).

4. Benchmark Performance

Benchmarking on diverse datasets demonstrates the capabilities and limitations of r$^2$SCAN+rVV10:

Formation Enthalpies (1015 solids)

Functional	MAE (meV/atom)
PBE	186
SCAN	107
SCAN+rVV10	114
r$^2$SCAN	92
r$^2$SCAN+rVV10	99

• r$^2$SCAN outperforms SCAN and PBE; r$^2$SCAN+rVV10 yields only a minor MAE increase (by $\sim$7 meV/atom) but offers improved equilibrium volume prediction.
• The dispersion correction in SCAN+rVV10 often worsens formation-enthalpy accuracy for non-vdW solids; r$^2$SCAN+rVV10 avoids this degradation due to reduced $b$ (Kothakonda et al., 2022).

Equilibrium Volumes

Functional	Mean Error (Å$^3$/atom)	MAE (Å$^3$/atom)	RMSE (Å$^3$/atom)
PBE	+0.77	0.98	1.80
SCAN	–0.11	0.58	0.96
SCAN+rVV10	–0.32	0.59	0.95
r$^2$SCAN	+0.24	0.59	1.04
r$^2$SCAN+rVV10	–0.11	0.50	0.88

• The addition of rVV10 to r$^2$SCAN reduces equilibrium volume MAE by $\sim$15%. Layered (vdW) solids show the largest improvement (Kothakonda et al., 2022).

Molecular and Layered System Benchmarks

• On the S22 molecular set, r$^2$SCAN+rVV10 achieves a total MAE of $0.30$ kcal/mol (fully converged), outperforming PBE+D3, vdW-DF2, and SCAN+rVV10.
• For 28 layered materials, r$^2$SCAN+rVV10 systematically overbinds ($\sim$+2.7 meV/Å$^2$) relative to RPA benchmarks but achieves c-lattice constant MAD of $0.13$ Å (a $\sim$30–40% error reduction compared to SCAN+rVV10). Phonon dispersion curves for systems such as graphite and MoS$_2$ are reproduced accurately (Ning et al., 2022).

Bandgaps

• For 130 solids, r$^2$SCAN+rVV10 bandgaps closely match those of r$^2$SCAN (typical MAE $1.15$ eV), with persistent underestimation characteristic of semilocal DFT and occasional slight overestimation for small-gap materials.

Decomposition Enthalpies

• For convex hull decomposition MAEs: Meta-GGA functionals halve the Type 1 and Type 3 errors relative to PBE. Type 2 (compound decomposition) errors are similar across functionals.

5. Applicability and Limitations

The r$^2$SCAN+rVV10 functional is recommended as a general-purpose, numerically robust meta-GGA for high-throughput materials discovery (Kothakonda et al., 2022, Ning et al., 2022). Key application domains include layered solids (e.g., transition metal dichalcogenides, graphene), molecular crystals, and systems with significant nonlocal correlation effects. The combination of r$^2$SCAN's numerical stability and reduced double-counting of intermediate-range vdW with the rVV10 kernel's efficient nonlocality underpins its broad adoption.

However, several caveats are noted:

• Systematic overestimation of interlayer binding energies ($\sim$2–3 meV/Å$^2$ above RPA/QMC) occurs, attributed to the pairwise form of the rVV10 kernel and neglect of many-body (e.g., Axilrod–Teller) contributions.
• For transition metal intermetallics, PBE or PBEsol (GGA-level DFAs) deliver lower errors in formation enthalpy than either SCAN or r$^2$SCAN variants.
• Properties of systems with complex electronic order (e.g., charge-density wave, superconductivity) require beyond-DFT methods.
• Large molecular clusters, surface/adsorption systems, and liquid-phase environments need further validation.

6. Comparative Analysis and Outlook

Collectively, r$^2$SCAN+rVV10 offers a minimal-empiricism approach with only two system-independent parameters ($C$ and refitted $b$), granting it a unique position among nonlocal vdW-corrected meta-GGAs. In most benchmarks, it exhibits superior energetic and geometric accuracy compared to PBE+D3, SCAN+rVV10, and vdW-DF2, especially for molecular and weakly bonded layered systems. Its numerical tractability—manifest in smooth forces and reliable self-consistency even on reasonable computational grids—distinguishes it as a practical choice for both molecular and solid-state calculations at scale.

A plausible implication is that, while r$^2$SCAN+rVV10 is the "workhorse" for weak and mixed-type bonding, careful consideration of the system class (e.g., metallic vs vdW bound) and property of interest (e.g., energetics vs lattice dynamics) should guide the selection of exchange–correlation functional for computational materials screening.

Future advances may address the remaining overbinding bias via improved nonlocal kernels or incorporation of many-body dispersion. Ongoing validation against larger and more diverse datasets (e.g., S66, L7, surface adsorption, liquid-phase) is anticipated to further delineate the practical range and limits of applicability of the r$^2$SCAN+rVV10 functional.