vdW-DF3-mc: Optimized Molecular Crystal Functional

• The topic vdW-DF3-mc is a density functional variant that enhances molecular crystal accuracy by optimizing the nonlocal kernel and introducing a tunable GGA exchange.
• It is designed to deliver highly accurate geometries and cohesive energies for 3D molecular crystals, layered systems, and dimers through a smoothness-based kernel optimization.
• Benchmark results demonstrate minimal errors in volumes and energies, outperforming previous vdW-DF3 versions and rivaling established methods like PBE-D3.

vdW-DF3-mc is a specialized member of the vdW-DF3 family designed to give highly accurate structures and energetics for 3D molecular crystals, while preserving good performance for layered systems and dimers. It retains the constraint-based, plasmon-derived nonlocal-correlation architecture of the van der Waals density functional framework, but replaces the relatively rigid exchange used in earlier vdW-DF3 parameterizations with a new explicitly tunable GGA enhancement factor and re-optimizes the nonlocal kernel through a smoothness-based constraint on the switching function (Jenkins et al., 2 Sep 2025).

1. Formal placement within the vdW-DF family

Like other vdW-DF variants, vdW-DF3-mc decomposes the exchange-correlation energy as

$$E_{\rm xc}[n] = E_{\rm x}^{\rm GGA}[n] + E_{\rm c}^{\rm LDA}[n] + E_{\rm c}^{\rm nl}[n].$$

The nonlocal term is written as

$$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$

with a kernel derived by expanding the adiabatic connection formula to second order in an effective plasmon propagator. In the vdW-DF framework, the local plasmon dispersion is expressed through

$$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$

where $q_0(\mathbf r)$ is a local inverse length scale determined by an internal semilocal exchange-correlation model (Chakraborty et al., 2020).

This architecture preserves the defining physical interpretation of vdW-DF: dispersion is described by tracking the effects of an electrodynamic coupling among pairs of electrons and their associated exchange-correlation holes through $E_c^{\rm nl}$. Within this general framework, spatially resolved analyses show that total nonlocal-correlation binding is concentrated to pockets in the sparse electron distribution located between material fragments, rather than at atomic centers, and that these pockets provide the characteristic signature of van der Waals binding in both covalently and non-covalently bonded systems (Jiao et al., 2017).

The specific motivation for vdW-DF3-mc is the limitation identified in the original vdW-DF3 parameterizations. vdW-DF3 introduced a newly constructed form of the nonlocal correlation that more accurately modeled molecular dimers, layered structures, and surface adsorption, but also revealed an intrinsic tradeoff in vdW-DF3's parametrization and inflexibility of exchange in the generalized gradient approximation, limiting its accuracy for molecular crystals (Chakraborty et al., 2020). vdW-DF3-mc is the response to that tradeoff.

2. Exchange construction

The central modification in vdW-DF3-mc is its GGA exchange enhancement factor. The exchange energy keeps the standard GGA form

$$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$

but $F_{\rm x}(s)$ is defined piecewise as

$F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$

Here $s$ is the reduced gradient. The low-$s$ branch is a polynomial, while the high-$$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$0 branch has a fractional-power asymptotic form with leading term $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$1 (Jenkins et al., 2 Sep 2025).

The external parameters are chosen to map directly onto physically relevant features of $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$2. The parameter $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$3 is the crossover reduced gradient where the curvature changes sign,

$$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$4

and locates the shoulder in $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$5. The parameter

$$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$6

fixes the slope at the crossover, and $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$7 controls the asymptotic large-$$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$8 strength. The parameter $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$9 is fixed to the PBEsol value,

$$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$0

The actual coefficients $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$1 are not free parameters; they are determined internally by enforcing continuity of $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$2, $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$3, and $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$4 at $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$5, together with the constraints defining $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$6 and $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$7 (Jenkins et al., 2 Sep 2025).

The final exchange parameters are:

• $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$8,
• $$\omega_q(\mathbf r) = \frac{q^2}{2}\,\frac{1}{h(q/q_0(\mathbf r))},$$9,
• $q_0(\mathbf r)$0,
• $q_0(\mathbf r)$1.

The resulting dependent coefficients are:

• $q_0(\mathbf r)$2,
• $q_0(\mathbf r)$3,
• $q_0(\mathbf r)$4,
• $q_0(\mathbf r)$5,
• $q_0(\mathbf r)$6 (Jenkins et al., 2 Sep 2025).

Physically, the peak of $q_0(\mathbf r)$7 is placed at a somewhat higher $q_0(\mathbf r)$8 than in vdW-DF1 or vdW-DF2, which produces more slowly-varying corrections and a more controlled exchange repulsion in the mid-$q_0(\mathbf r)$9 region. Compared to vdW-DF3-opt1 and vdW-DF3-opt2, vdW-DF3-mc yields a larger $E_c^{\rm nl}$0 in the region $E_c^{\rm nl}$1, precisely the interval identified as the dominant contributor to exchange repulsion in X23 molecular crystals. This increase counteracts the overbinding tendency of the earlier vdW-DF3 parameterizations (Jenkins et al., 2 Sep 2025).

3. Nonlocal correlation and kernel redesign

vdW-DF3-mc keeps the vdW-DF3 switching-function form

$E_c^{\rm nl}$2

with $E_c^{\rm nl}$3. In the original vdW-DF3 construction, the parameter $E_c^{\rm nl}$4 was chosen so that the first-order adiabatic-connection expansion exactly reproduced the internal functional normalization. In vdW-DF3-mc, $E_c^{\rm nl}$5 is instead chosen to minimize the mean curvature of $E_c^{\rm nl}$6, so that the switching function is as smooth as possible (Jenkins et al., 2 Sep 2025).

This smoothness-based choice changes the role of the kernel parameters. In the original vdW-DF3-opt1/opt2 parameterizations, the constraint on $E_c^{\rm nl}$7 tended to counteract the effect of varying $E_c^{\rm nl}$8, making the nonlocal correlation less sensitive than expected to nominal kernel tuning. vdW-DF3-mc relaxes that constraint by an overall scalar factor. The result is that varying $E_c^{\rm nl}$9 again meaningfully changes the strength and shape of $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$0, while preserving the overall vdW-DF plasmon framework, the use of $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$1, and the internal-functional parameterization through $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$2 (Jenkins et al., 2 Sep 2025).

The final nonlocal parameters are:

• $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$3,
• $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$4,
• $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$5,
• $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$6 (Jenkins et al., 2 Sep 2025).

The choice $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$7 is continuous with the original vdW-DF3 development, where optimization had already found $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$8 close to zero and simplified the kernel accordingly (Chakraborty et al., 2020). What distinguishes vdW-DF3-mc is not a new analytic form for $$E_{\rm x}^{\rm GGA}[n] = \int d^3r\, n(\mathbf r)\,\varepsilon_{\rm x}^{\rm hom}(n(\mathbf r))\,F_{\rm x}(s(\mathbf r)),$$9, but a new constraint strategy that restores effective tunability to the existing vdW-DF3 ansatz. The paper describes this as improving continuity of derivatives of the exchange-correlation energy with respect to the density, improving numerical stability for forces and stresses, and improving transferability across density regimes (Jenkins et al., 2 Sep 2025).

Within the broader vdW-DF tradition, this redesign also connects to an earlier line of work showing that the plasmon-dispersion model contains an exploitable degree of freedom. Modified-kernel work on corrected $F_{\rm x}(s)$0 coefficients demonstrated that accurate long-range coefficients can be obtained without abandoning the original vdW-DF constraints, by altering the switching function and the long-range-to-short-range crossover in the plasmon dispersion (Berland et al., 2019). vdW-DF3-mc uses that same broader insight—kernel flexibility inside the constraint-based framework—but directs it toward molecular-crystal accuracy rather than toward corrected $F_{\rm x}(s)$1 coefficients as a primary target.

4. Optimization strategy

The optimization of vdW-DF3-mc uses three training components: the X23 molecular-crystal set, the layered systems graphite and hexagonal BN, and 24 molecular dimers drawn from S22×5 and S66×8. The X23 set is split into 10 dispersion-dominated and 13 H-bond-dominated systems. The dimer selection emphasizes multiple hydrogen bonds, hydrocarbons, $F_{\rm x}(s)$2-$F_{\rm x}(s)$3 stacking, and off-equilibrium separations (Jenkins et al., 2 Sep 2025).

The procedure is two-stage. In the first stage, exchange is optimized against stress tensor components of all 3D systems at fixed experimental cell shape and volume. The objective is the mean deviation

$F_{\rm x}(s)$4

with non-variable-cell relaxations of internal coordinates only. This choice emphasizes not just pressure but also shear components, which is important for anisotropic packing and cell shape (Jenkins et al., 2 Sep 2025).

In the second stage, the nonlocal correlation is optimized using the exchange obtained from the stress analysis. For solids, the cohesive energy per molecular fragment is

$F_{\rm x}(s)$5

and for dimers the binding energy is

$F_{\rm x}(s)$6

The optimization minimizes the mean absolute relative deviation of cohesive energies for solids and the weighted mean absolute relative deviation for dimers. The training weights are 50% for X23 molecular crystals, 5% for layered systems, and 45% for dimers, with H-bonded dimers weighted twice compared to dispersion and mixed complexes (Jenkins et al., 2 Sep 2025).

All reported calculations are fully self-consistent with the nonlocal correlation included. The implementation used Quantum ESPRESSO 6.x, SG15 optimized norm-conserving Vanderbilt pseudopotentials, 80 Ry for wavefunctions, 320 Ry for charge density, total-energy convergence to $F_{\rm x}(s)$7 Ry, force convergence to $F_{\rm x}(s)$8 Ry/bohr, and a variable-cell stress threshold of 0.5 kbar (Jenkins et al., 2 Sep 2025).

5. Benchmark performance

vdW-DF3-mc was proposed as a functional tailored to 3D molecular crystals, and the benchmark data are correspondingly centered on crystal geometries, cohesive energies, and polymorph energetics. The reported results show that it achieves very small average errors in both geometry and energetics on X23, generalizes to the broader G60 crystal set, remains competitive for polymorph ranking, performs particularly well for ice polymorphs, and preserves good geometry accuracy for layered systems (Jenkins et al., 2 Sep 2025).

Benchmark	Geometry	Energetics
X23	$F_{\rm x}(s)$9: MAD $F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$0, MARD $F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$1, MD $F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$2	$F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$3: MAD $F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$4, MD $F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$5, MARD $F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$6
G60	$F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$7: MARD $F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$8, MAD $F_{\rm x}(s) = \begin{cases} 1 + \mu s^2 + A s^4 + B s^6, & s < s_0,\[4pt] C + \kappa s^{2/5} + D s^{-8/5} + E s^{-18/5}, & s \ge s_0. \end{cases}$9, MD $s$0	$s$1: MARD $s$2, MAD $s$3, MD $s$4
POLY59	—	Correct lowest-energy polymorph in 3 out of 5 targets
ICE10 / DMC-ICE13	Volume errors on the order of only $s$5	Cohesive-energy errors on the order of only $s$6
9 layered solids	Interlayer separation MARD $s$7	Cohesive-energy MARD $s$8

For X23, the subset breakdown is also favorable. In the 10 vdW-bonded crystals, the MARD is $s$9 in $s$0 and $s$1 in $s$2. In the 13 H-bonded crystals, the corresponding MARDs are $s$3 and $s$4. The paper states that these results clearly outperform previous vdW-DF variants and rival PBE-D3. It also gives direct contrasts with earlier vdW-DF3 parameterizations: vdW-DF3-opt1 and vdW-DF3-opt2 overbind cohesive energies by $s$5 eV with large relative errors of $s$6, and compress crystal volumes with MARD $s$7 and $s$8, respectively (Jenkins et al., 2 Sep 2025).

For G60, vdW-DF3-mc is among the best functionals compared for cohesive energies and clearly better than most vdW-DFs on volume. For POLY59, its 3/5 success rate is below the best DFT+D performers listed in the comparison, but the failed cases remain close to experimental rankings: for target 25, the incorrect minimum is only $s$9 kJ/mol below the experimental structure, and for target 26 the competing polymorphs lie at most $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$00 kJ/mol below the experimental one. These differences are within or close to the sub-chemical-accuracy scale of approximately $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$01 kcal/mol $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$02 kJ/mol discussed in the paper (Jenkins et al., 2 Sep 2025).

The ice benchmarks are especially prominent in the presentation of vdW-DF3-mc. The abstract states that, for polymorphs of ice, errors in the volume and cohesive energy are on the order of only $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$03, and the detailed discussion emphasizes that the functional reproduces correct trends across polymorphs while reducing systematic bias relative to PBE, PBE-D3, and PBE-D3$$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$04 (Jenkins et al., 2 Sep 2025).

6. Nomenclature, relation to adjacent functionals, and scope of use

The name vdW-DF3-mc is explicitly attached to the 2025 molecular-crystal functional. This matters because earlier literature can produce a naming ambiguity. Work on corrected $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$05 coefficients from 2019 is described as a third-generation or “vdW-DF3-mc-type” modified-kernel variant in later discussions, but that paper itself explicitly refused to give its formulation a new official vdW-DF name or number (Berland et al., 2019). In strict usage, vdW-DF3-mc denotes the molecular-crystal optimization of vdW-DF3 introduced in 2025 (Jenkins et al., 2 Sep 2025).

Within the broader vdW-DF landscape, vdW-DF3-mc is one branch of a larger effort to improve sparse materials. A parallel 2025 line of work optimized a B86R-based “vdW-DF1.5” family for molecular crystals by tuning the internal parameter $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$06 between the DF1 and DF2 limits, identifying vdW-DF-B86R-1.8791 as the best overall performer on X23 in that study (Fedorov et al., 25 Nov 2025). The conceptual difference is that vdW-DF3-mc modifies both exchange flexibility and the effective tunability of the vdW-DF3 kernel, whereas the DF1.5 scheme retains the older kernel family and optimizes the internal plasmon scale.

The intended application domain of vdW-DF3-mc is pragmatic and specific. The best-suited systems listed are organic and molecular crystals, hydrogen-bonded crystals and frameworks, ice polymorphs, pharmaceutical-like polymorphic systems, and possibly hydrogen-bonded and covalent organic frameworks and porous molecular materials where H-bonding and dispersion coexist (Jenkins et al., 2 Sep 2025). The paper also states that it is not specially tuned for metals, strongly correlated systems, or pure bulk covalent and ionic solids. The training sets do not include adsorption on extended surfaces or general heterogeneous interfaces, and the authors note that vdW-DF3-opt1 and vdW-DF3-opt2 may still be preferable when surface adsorption and layered materials are the primary focus (Jenkins et al., 2 Sep 2025).

That caveat is consistent with family-level benchmarking outside the molecular-crystal setting. For example, vdW-DF3-opt2 was shown to reproduce the experimental monolayer TPD peak for $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$07 on anatase-TiO$$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$08(101) with $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$09 eV, very close to the TPD-derived $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$10 eV, while also overbinding chemisorbed $$E_{\rm c}^{\rm nl}[n] = \frac{1}{2}\int d^3r\,d^3r' \, n(\mathbf r)\,\Phi(\mathbf r,\mathbf r')\,n(\mathbf r'),$$11 on Pt(111) (Muhammady et al., 23 Jun 2025). This does not establish the performance of vdW-DF3-mc itself on catalytic adsorption, but it does underscore that the vdW-DF3 family remains application-dependent and that molecular-crystal optimization is not automatically transferable to metals or catalytic interfaces.

Implementation remains straightforward for codes that already support custom vdW-DF kernels and custom GGA exchange. The 2025 calculations were performed in a development version of Quantum ESPRESSO via the built-in vdW-DF infrastructure and user-defined kernels. At the time of that work, vdW-DF3-mc was not yet a standard named functional in mainstream releases of VASP or GPAW, but any code that allows custom vdW-DF kernels and custom exchange can implement it directly from the published parameter set (Jenkins et al., 2 Sep 2025).