Nonlocal van der Waals density functional made faster

TL;DR

The authors present a fast, low-rank approximation to the VV10 nonlocal van der Waals density functional by expressing the six-dimensional dispersion integral as a sum of products of density-like distributions and distance-dependent functions, enabling fast multipole or density-fitting evaluation. They introduce a two-stage approximation that transforms the integrand into functions of interelectronic distance and local densities via $\eta(\mathbf{r})$ and $\mu(\mathbf{r})$, and then further expand the distance kernel into $\tilde{v}(s,\mu,\nu,\beta)$ using asymptotic constraints, yielding a set of four density-like $q_i$ and three potential-like $u_n$ that express $\tilde{E}_6$ through $U_{ij}^{(n)}$ integrals. The resulting LPBEVVV functional balances accuracy and speed, achieving VV10-like results with substantially reduced computational cost in molecular systems, as demonstrated on noble-gas dimers and real-world organic reactions. The approach is generalizable to other functionals of this class and highlights the practical impact of combining density-fitting with low-rank dispersive kernels for scalable nonlocal correlation corrections.

Abstract

A simplification of the VV10 van der Waals density functional [J. Chem. Phys. 133, 244103 (2010)] is made by an approximation of the integrand of the six-dimentional integral in terms of a few products of three-dimensional density-like distributions and potential-like functions of the interelectronic distance only, opening the way for its straightforward computation by fast multipole methods. An even faster computational scheme for molecular systems is implemented where the density-like distributions are fitted by linear combinations of usual atom-centered basis functions of Gaussian type and the six-dimensional integral is then computed analytically, at a fraction of the overall cost of a typical calculation. The simplicity of the new approximation is commensurate with that of the original VV10 functional, and the same level of accuracy is seen in tests on molecules.

Figures (1)

• Figure 1: Function $v(r^2,\mu,\nu)$ and its approximation $\tilde{v}(r^2,\mu,\nu,1)$ of Eq. (\ref{['eq:va']}) for $(\mu,\nu) = (\tfrac{1}{2},\tfrac{1}{2}), (\tfrac{1}{2},1), (1,1), (1,\tfrac{3}{2}), (\tfrac{3}{2},\tfrac{3}{2})$.