Efficient van der Waals energy calculations via a continuum mechanics approach

Recent developments in continuum mechanics (CM) [Tao \emph{et al}, PRL{\bf 103},086401] enable the calculation of density response functions from groundstate properties only. Using the direct Random Phase Approximation (dRPA) we develop this CM approach into a third-rung van der Waals energy functional, which we dub the CM-dRPA. The functional requires as input the groundstate Kohn-Sham potential $V^{\rm{KS}}(\vec{r})$, density $n^0(\vec{r})$ and a kinetic stress tensor ${\rm{T}}^0(\vec{r})$ defined via $T^0_{\mu\nu}=Re\sum_{i\rm{ occ}} \psi_{i,\mu}^*\psi_{i,\nu} - n^0_{,\mu,\nu}/4$ where $\psi_i$ is an orbital. We present efficient algorithmic schemes for its evaluation in bulk and molecular systems using the full eigen-solutions of the bare CM equation and a second, simpler evaluation to find the interacting eigenvalues. These eigen-solutions are then used to calculate the correlation energy via a simple summation. The CM-dRPA is \emph{significantly} faster than a full dRPA calculation in systems with many electrons. We then apply the CM-dRPA functional to metallic, slab-like 2D-homogeneous jellium systems and periodic solids, with good results for vdW dispersion. In the metallic case most efficient vdW functionals would fail qualitatively.