Charged systems with density functional theory

On this page, we briefly describe technical issues caused by computing the energies of charged systems with periodic density functional theory calculations. We then discuss why the energies of charged systems diverge for systems with lower dimensionality, such as with surfaces (2D), nanowires (1D) and molecules (0D) while potentially providing useful information for bulk (3D) systems. Finally, we present methods implemented in VASP which allow for calculations of charged 3D, 2D and 0D systems.

Cancelled divergences in charged systems

VASP makes use of efficient fast Fourier transforms (FFT) to compute the electrostatic potential from the charge density using the Poisson equation,

$$V(\mathbf{r}) = 4\pi \int \frac{\rho(\mathbf{r}^\prime)}{\left | \mathbf{r} - \mathbf{r}^\prime \right|} d\mathbf{r}^\prime$$

where ${\textstyle \mathbf{r}}$ and ${\textstyle \mathbf{r}^\prime}$ are all points in real space. Fourier transforming the Poisson equation to reciprocal space,

$$V(\mathbf{g}) = \frac{4\pi}{\mathrm{G}^2} \varrho(\mathbf{G})$$

where ${\textstyle \mathbf{G}}$ is the reciprocal lattice vector and ${\textstyle \mathrm{G}}$ is its norm.