In classical MD simulations of lattice structures, whenever we are trying to incorporate vdW corrections to the atomic force calculation, we need to set up cutoff length beyond which the vdW correction is not considered to keep the computational cost manageable. As a result, the classical simulation requires reasonably large supercell(usually with its dimensions twice as long as the cutoff length) to capture vdW interactions.
When it comes to DFT calculations, however, the cutoff lengths for nonlocal vdW corrections, such as the rVV10 functional, are not explicitly set in DFT software like Quantum ESPRESSO. It is also common practice to use primitive cells in DFT calculations along with nonlocal vdW functionals. By looking at the formalism of the rVV10 vdW correction, the integration of the functional is clearly performed within a volume, $$ E_{c}^{n l}=\frac{\hbar}{2} \iint n(\mathbf{r}) \Phi\left(n, n^{\prime},|\nabla n|,\left|\nabla n^{\prime}\right|,\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right) n\left(\mathbf{r}^{\prime}\right) d \mathbf{r} d \mathbf{r}^{\prime},\tag{1} $$ where $n(\mathbf{r})$ is the charge density at location of $\mathbf{r}$. So my question is what are the lower and upper limits for Eq. (1), and how do we derive them?
To attack this question, I tried to understand Eq.(1) in terms of its Fourier transform, $$ E_{c}^{n l}=\frac{\hbar}{2} \Omega \sum_{i j} \sum_{\mathbf{G}} \tilde{\theta}_{i}^{*}(\mathbf{G}) \tilde{\Phi}^{\mathrm{rVV} 10}\left(q_{i}, q_{j},|\mathbf{G}|\right) \tilde{\theta}_{j}(\mathbf{G}),\tag{2} $$ where $\tilde{\Phi}^{\mathrm{rVV} 10}\left(q_{i}, q_{j},|\mathbf{G}|\right)$ are the Fourier transforms of the rVV10 kernel evaluated on a bidimensional grid of $q$ values, and $\tilde{\theta}_{i}$ are the Fourier transforms of $\theta_{i}$ introduced in the rVV10 paper here. But this left me with more questions than answers. I have a feeling that I might be pursuing the answer in a wrong line of thinking, so any hint will be appreciated!
