Nonlocal vdW-DF2 Exchange Functional

Updated 2 January 2026

Nonlocal vdW-DF2 is a Kohn-Sham DFT approach that defines exchange-correlation energy with a rigorously derived nonlocal correlation term based on the adiabatic-connection fluctuation-dissipation formalism.
It integrates PW86R exchange and LDA correlation to achieve improved predictions of equilibrium structures and energetics in systems like physisorption on metals and layered materials.
The functional offers practical insights into modeling dispersion forces in diverse systems while addressing exchange sensitivity and limitations without relying on empirical parameters.

The nonlocal vdW-DF2 exchange-correlation functional is a Kohn-Sham density functional theory (DFT) approach designed to accurately capture nonlocal van der Waals (vdW) dispersion interactions without relying on empirical fitting. It builds upon the original vdW-DF framework, integrating a rigorously nonlocal correlation term derived from the adiabatic-connection fluctuation-dissipation formalism, coupled with a revised semilocal exchange to address the shortcomings of its predecessor. The nonlocal vdW-DF2 functional is widely used for physisorption, molecular crystals, layered materials, and surface science, and demonstrates marked improvement over earlier models in reproducing both equilibrium structures and energetics for vdW-bound systems (Lee et al., 2012, Lee et al., 2011, Berland et al., 2013, Lee et al., 2010, Shukla et al., 2022).

1. Formal Structure of the vdW-DF2 Functional

The total exchange-correlation energy in the vdW-DF2 framework is decomposed as follows: $E_{xc}^{\rm vdW\text{-}DF2}[n] = E_x^{\rm GGA}[n] + E_c^{\rm LDA}[n] + E_c^{\rm nl}[n]$

$E_x^{\rm GGA}[n]$ : Generalized-gradient-approximation (GGA) exchange, specifically the PW86R form.
$E_c^{\rm LDA}[n]$ : Standard local-density-approximation (LDA) correlation.
$E_c^{\rm nl}[n]$ : Nonlocal correlation, accounting for vdW dispersion forces.

The nonlocal correlation term is written as

$E_c^{\rm nl}[n] = \frac12 \iint n(\mathbf r)\, \phi(\mathbf r, \mathbf r')\, n(\mathbf r')\, d^3r\, d^3r'$

where the universal kernel $\phi(\mathbf r, \mathbf r')$ depends on the dimensionless separation $|\mathbf r - \mathbf r'|$ and local parameters $q_0(\mathbf r), q_0(\mathbf r')$ that encode electron density $n$ and its gradient (Lee et al., 2010, Lee et al., 2011).

In vdW-DF2, the kernel parameterization is controlled by the gradient coefficient $\beta_{\rm B88}\approx0.0042$ , derived from the Becke 1988 (B88) exchange in the large- $N$ limit, replacing the original value used in vdW-DF.

2. Exchange and Correlation Components

Exchange Functional:

The PW86R exchange (a modification of the original PW86 by Murray et al.) is implemented. Its enhancement factor is

$F_x^{\rm PW86R}(s) = [1 + 0.19645\,s^2 + 0.27430\,s^4 + 0.15020\,s^6]^{1/15}$

where $s = |\nabla n| / [2k_F n]$ , and $k_F = (3\pi^2 n)^{1/3}$ (Lee et al., 2011, Lee et al., 2010). The choice of PW86R ensures correct reproduction of Hartree–Fock exchange across low and high density regimes and avoids over-repulsive behavior of earlier choices (e.g. revPBE).

Local Correlation:

The standard Perdew–Wang LDA correlation is used, via

$E_c^{\rm LDA}[n] = \int d^3r\, n(\mathbf r)\, \epsilon_c^{\rm LDA}(n(\mathbf r))$

with parameterization aligned with quantum Monte Carlo results for the homogeneous electron gas.

Nonlocal Correlation:

The kernel $\phi(\mathbf r, \mathbf r')$ incorporates a plasmon-pole approximation. In vdW-DF2, the kernel’s gradient coefficient is set by the B88 large- $N$ asymptote ( $Z_{ab}=-1.887$ ) instead of the Lund–Vosko (LV) value ( $Z_{ab}=-0.8491$ ), reducing the magnitude of nonlocal correlation and improving intermediate-range accuracy (Lee et al., 2010).
Tabulated $\phi$ values are universal, and there are no empirically adjusted parameters.

3. Implementation and Numerical Considerations

vdW-DF2 is typically implemented in plane-wave DFT software such as ABINIT and Quantum ESPRESSO:

Basis: Plane-wave with norm-conserving pseudopotentials.
Cutoffs: Energy cutoff of 70 Ry demonstrated for metallic surfaces (Lee et al., 2012).
k-point sampling: $4\times4\times1$ Monkhorst–Pack mesh for surface supercells.
Self-consistency: Nonlocal correlation is evaluated fully self-consistently, using fast algorithms for double integrals, such as the FFT convolution method of Román-Pérez and Soler.
No empirical damping functions or fitted coefficients are introduced; all parameters are dictated by theoretical constraints.

4. Performance and Experimental Benchmarking

In physisorption studies, such as H $_2$ on Cu(111), (100), and (110):

vdW-DF2 yields potential well depths within $\approx 15\%$ mean error of experimental values and reproduces equilibrium distances within $0.5$ Å.
Corrugation energies and quantum-level spacings (backscattering energies) predicted by vdW-DF2 track experimental measurements within a few meV (Lee et al., 2012, Lee et al., 2011).
Comparison with pairwise DFT-D3 and TS-vdW approaches reveals that vdW-DF2 avoids systematic overbinding and excessive short-range attraction present in those alternatives (e.g. DFT-D3(PBE) gives $D=98$ meV for H $_2$ /Cu(111); experiment: $29$ meV).
In the S22 molecular benchmark set, vdW-DF2 achieves mean absolute binding energy errors of 22 meV (vs. 41 meV for vdW-DF1) and a mean absolute deviation in equilibrium separation of 0.13 Å (Lee et al., 2010).

System	$D_{\rm exp}$ (meV)	$D_{\rm th}$ , vdW-DF2 (meV)	Error (%)
Cu(111)	29.0	39	+34
Cu(100)	31.3	≈37	+18
Cu(110)	32.1	≈36	+12

5. Role of Exchange and Sensitivity to Functional Choice

The short-range repulsion and corrugation energies are strongly exchange-sensitive:

More repulsive GGAs (e.g. revPBE) underbind and underestimate barriers.
Overly attractive variants overbind and exaggerate corrugations and adsorption energies.
The PW86R exchange employed in vdW-DF2 achieves a balance, reproducing observed facet trends and weak site dependence (e.g. benzene/Cu(111): revPBE ≈ 5 meV, PW86R ≈ 30 meV in corrugation) (Lee et al., 2012, Berland et al., 2013).
The optimal separation (equilibrium distance) and energy barriers for lateral diffusion or backscattering are exponentially sensitive to the exchange enhancement, leading to substantial differences in predicted adsorption properties across vdW-DF variants.

Subsequent functionals, including the vdW-DF2-based range-separated hybrid AHBR, combine vdW-DF2 correlation with non-empirical GGA exchange (B86R) and a screened Fock exchange component: $E_{xc}^{\rm AHBR}[n] = E_x^{\rm B86R}[n] + E_c^{\rm LDA}[n] + E_c^{\rm nl}[n] + \alpha\left(E_{\rm FX}^{\rm SR}(\gamma) - E_x^{\rm B86R,\,SR}[\gamma]\right)$ Here, $\alpha=0.25$ and $\gamma=0.106$ bohr $^{-1}$ are fixed by theoretical constraints (not fit), and the nonlocal correlation $E_c^{\rm nl}[n]$ is computed as in vdW-DF2. AHBR achieves improved agreement with high-level benchmarks for both molecular and extended systems (Shukla et al., 2022). A plausible implication is that further refinement of the exchange component and range separation may yield systematically better accuracy for both binding energies and structural properties across a spectrum of material classes.

7. Applications, Limitations, and Outlook

vdW-DF2 is particularly well-suited for:

Physisorption studies on metals, semiconductors, and insulators.
Adsorption phenomena in porous and layered materials, including MOFs.
Weakly bound molecular crystals and biological assemblies.
Surface science where both nonlocal correlation and accurate exchange description of repulsion are essential.

Limitations include:

Potential overestimation of short-range Pauli repulsion due to the choice of PW86R.
Underestimation of $C_6$ coefficients at asymptotic separations relative to some empirical approaches.
Sensitivity of predicted corrugation energies and diffusion barriers to fine details of the exchange enhancement factor.

Ongoing developments are focused on further improving the balance between exchange and nonlocal correlation, exploring hybridization with Fock exchange, and systematic benchmarking against surface and molecular experimental databases for comprehensive validation (Shukla et al., 2022).

Key references: (Lee et al., 2012, Berland et al., 2013, Lee et al., 2010, Lee et al., 2011, Shukla et al., 2022)