vdW-DF2 Exchange–Correlation Functional

• vdW-DF2 is a nonempirical DFT exchange–correlation functional that separates energy into semilocal and nonlocal components to capture dispersion interactions.
• It refines the PW86r exchange and employs a plasmon-based nonlocal correlation kernel, leading to improved predictions of binding energies and equilibrium geometries.
• Efficient convolution algorithms enable self-consistent implementations in modern DFT codes, making vdW-DF2 invaluable for modeling weakly bound molecular and layered systems.

The vdW-DF2 (van der Waals density functional, version 2) exchange–correlation functional is a nonempirical density functional theory (DFT) approach designed to account for long-range dispersion (van der Waals) interactions within a seamless first-principles framework. Developed as a successor to the original vdW-DF1 functional, vdW-DF2 implements methodological refinements in both the semilocal exchange and nonlocal correlation components. These modifications yield significantly improved predictions of binding energies, equilibrium structures, and potential-energy curves for systems governed by weak interactions, including molecular crystals, physisorbed molecules, and layered materials. vdW-DF2 is defined by a rigorous split of the exchange–correlation energy into semilocal (GGA) exchange, local (LDA) correlation, and a nonlocal correlation term derived from a plasmon-based response model, all free of empirical parameters and combining efficiently with modern plane-wave DFT codes (Lee et al., 2010, Berland et al., 2010).

1. Energy Decomposition and Defining Equations

The total exchange–correlation energy in vdW-DF2 is given by: $$E_{\mathrm{xc}}^{\mathrm{vdW}\textrm{-}\mathrm{DF2}}[n] = E_{x}^{\mathrm{GGA}}[n] + E_{c}^{\mathrm{LDA}}[n] + E_{c}^{\mathrm{nl}}[n]$$ where:

• $E_{x}^{\mathrm{GGA}}$ is the semilocal exchange, specifically a refitted variant of the Perdew–Wang 1986 (PW86) functional (also denoted as PW86r or rPW86).
• $E_{c}^{\mathrm{LDA}}$ is the local density approximation to correlation, with standard parametrizations (e.g., Perdew–Wang 1992).
• $E_{c}^{\mathrm{nl}}$ is the fully nonlocal van der Waals correlation energy: $$E_{c}^{\mathrm{nl}}[n] = \frac{1}{2}\int d^3r \int d^3r'\; n(\mathbf r)\,\phi(q_0(\mathbf r),q_0(\mathbf r'),|\mathbf r-\mathbf r'|)\,n(\mathbf r')$$ Here, the universal kernel $\phi$ couples density fluctuations at two spatial locations, parameterized by the local “plasmon wavevectors” $q_0(\mathbf r)$ and $q_0(\mathbf r')$.

The exchange enhancement factor $F_x(s)$ for PW86r is: $$F_x^{\textrm{PW86r}}(s) = (1 + \mu s^2 + \alpha s^4 + \beta s^6)^{1/15}$$ with recommended parameters $\mu=0.21951$, $\alpha=0.136$, $\beta=-0.287$ (Lee et al., 2010, Larsen et al., 2017).

2. Nonlocal Correlation: Kernel Construction and Innovations

The hallmark of vdW-DF2 is its nonlocal correlation term, which systematically accounts for dispersion interactions:

• The kernel $\phi$ is derived from a plasmon-pole model within the adiabatic-connection fluctuation–dissipation theorem and is tabulated as a universal function of $q_0(\mathbf r)$, $q_0(\mathbf r')$, and $|\mathbf r-\mathbf r'|$.
• The pivotal modification in vdW-DF2 compared to vdW-DF1 is the use of an internal response parameter calibrated to the large-$N$ neutral-atom asymptote, reflected in the gradient coefficient $Z_{ab} = -1.887$ (vdW-DF2) vs. $-0.8491$ (vdW-DF1).
• The local plasmon characteristic wavevector,

$$q_0(\mathbf r) = k_F(\mathbf r)\left[1+\frac{Z_{ab}}{9}s^2(\mathbf r)\right],\quad k_F(\mathbf r) = (3\pi^2 n(\mathbf r))^{1/3}$$

is modified in vdW-DF2 to reflect improved gradient corrections, weakening intermediate-range attraction and sharpening the nonlocal kernel (Lee et al., 2010, Larsen et al., 2017).

No damping, empirical switching, or additional cutoffs are introduced beyond those dictated by the analytic kernel form; all parameters are fixed by the constraint to physical response and asymptotic limits.

3. GGA Exchange: PW86r and Its Role

In vdW-DF2, the exchange term is constructed from a refitted PW86 enhancement factor:

• The PW86r exchange is designed to interpolate between the second-order gradient expansion at small reduced gradients and the correct $s^{2/5}$ asymptote at large $s$, consistent with Hartree–Fock exchange for neutral atoms.
• This choice avoids the over-repulsion at binding separations found in revPBE exchange (used in vdW-DF1) and corrects the spurious stabilization of noncovalent complexes.
• The result is a more accurate description of equilibrium distances and interaction energies for weakly bound systems (Berland et al., 2013).

The analytic fit to $F_x^{\mathrm{PW86r}}(s)$, as given above, ensures that the exchange hole is normalized and avoids unphysical binding at large gradients.

4. Implementation Algorithms and Computational Strategy

Efficient evaluation of the nonlocal correlation energy is achieved through fast convolution algorithms:

• The Román-Pérez–Soler FFT-based method recasts the double integral as $M^2$ convolutions in reciprocal space (with typically $M=20$ $q_0$ spline grid points), reducing computational complexity to $O(N \log N)$ (Lee et al., 2010, Larsen et al., 2017).
• In real-space grid implementations, the kernel is split into an inner sphere and outer shell for efficient parallel evaluation, scaling linearly with system size in the bulk limit (Berland et al., 2010).
• Libraries such as libvdwxc provide reference implementations, MPI-based parallelization, and integration with standard DFT codes (e.g., GPAW, Octopus) (Larsen et al., 2017).

Self-consistent inclusion of $E_{c}^{\mathrm{nl}}$ and its potential is standard, and norm-conserving pseudopotentials (e.g., Troullier–Martins) are tested to within $1\%$ of all-electron references (Lee et al., 2010, Lee et al., 2011).

5. Physical Validation and Benchmark Results

vdW-DF2 systematically improves equilibrium separations and interaction energies in weakly bound systems:

• For molecular crystals (hexamine, dodecahedrane, C60, graphite), vdW-DF2 reduces the $\sim$0.2–0.3 Å overestimation of lattice constants by vdW-DF1 and brings internal cohesion energies closer to experiment: | System | DF1 | DF2 | Experiment | | -------------- | ------------- | ------------- | -------------- | | Hexamine ($a,\,$Å) | 7.16 / 1.01 eV | 6.96 / 0.93 eV | 6.91 / 0.83 eV | | Dodecahedrane ($a,\,$Å) | 10.92 / 1.46 eV | 10.64 / 1.35 eV | 10.60 / – | | C₆₀ ($a,\,$Å) | 14.38 / 1.70 eV | 14.30 / 1.30 eV | 14.04 / 1.6–1.9 eV | | Graphite ($c,\,$Å, eV/atom) | 7.24 / 0.053 | 6.96 / 0.053 | 6.67 / 0.052 | (Berland et al., 2010)
• For the S22 noncovalent benchmark, mean absolute deviations in binding energies are reduced from $41$ meV (vdW-DF1) to $22$ meV (vdW-DF2), and structural deviations drop from $0.23$ Å to $0.13$ Å (Lee et al., 2010).
• Physisorption of $\mathrm{H}_2$ on Cu(111) is captured within $8$ meV of experiment, outperforming both GGA and empirical DFT-D3 functionals (Lee et al., 2011).
• In layered materials (graphene, h-BN), vdW-DF2 yields equilibrium interlayer spacings overestimated by $\sim$4–6%, yet when the experimental spacing is enforced, it delivers shear and out-of-plane response within 10% of experiment or LMP2/RPA (Lebedeva et al., 2017).

6. Comparison to Alternative Functionals and Limitations

vdW-DF2 is fully first-principles and parameter-free, contrasting with empirical dispersion-corrected approaches (DFT-D2/D3, PBE-TS, optPBE-vdW):

• Advantages include correct asymptotic power-law behavior, transferability across wide classes of weakly-bound systems, and robust performance for stacking energetics and vibrational properties (Lebedeva et al., 2017).
• Limitations manifest as a $0.05$–$0.1$ Å overestimation of noncovalent separations at self-consistent geometries, leading to underestimation of in-plane shear modulus and corrugation energies for layered crystals at predicted equilibrium spacing. Enforcing the experimental geometry addresses these discrepancies (Berland et al., 2013, Lebedeva et al., 2017).
• The balance of exchange and nonlocal correlation is sensitive to details of the kernel and gradient corrections; deviations at short range are ultimately limited by the plasmon-pole approximation and neglect of explicit many-body screening effects (Lee et al., 2011).

7. Recommendations for Practical Application

• For molecular and layered materials systems, use of vdW-DF2 at its self-consistent geometry is preferred for out-of-plane and stacking energetics; experimental spacings should be imposed for accurate in-plane properties and corrugation.
• The functional is recommended as a baseline for studies where stacking registry and correct ordering of metastable polymorphs are critical, especially in heterostructures and physisorbed molecular assemblies (Lebedeva et al., 2017).
• When interpreting corrugation, friction, and diffusion barriers, care should be taken to compare potential energy surfaces at fixed experimental separation (Berland et al., 2013, Lebedeva et al., 2017).
• For quantitative benchmarking, decomposition of adsorption and cohesion energies into exchange, LDA correlation, and nonlocal correlation at fixed geometry enables tighter constraint of the functional balance and facilitates theory-experiment cross-validation strategies (Berland et al., 2013).

In summary, the vdW-DF2 exchange–correlation functional is a mathematically robust, physically motivated, and computationally efficient framework for incorporating dispersion interactions in DFT, enabling reliable quantitative prediction of structures and energetics in a wide spectrum of sparse and weakly bound materials (Lee et al., 2010, Berland et al., 2010, Larsen et al., 2017, Berland et al., 2013, Lebedeva et al., 2017, Lee et al., 2011).