Hartree energy and potential

The Hartree and exchange-correlation terms are calculated by determining the electronic density on a real-space grid $n({\bf r})$, and Fast Fourier Transforms (FFTs) are used to transform between real- and reciprocal-space^7.1to obtain ${\tilde n}({\bf G})$. The Hartree energy is then given by

$$E_{\mathrm H} = \frac{1}{2} \int {\mathrm d}{\bf r}~{\mathrm... ...t= 0} \frac{\left\vert {\tilde n}({\bf G}) \right\vert^2}{G^2}$$ (7.9)

where $\Omega_{\mathrm{cell}}$ is the volume of the supercell and the (infinite) ${\bf G} = 0$ term is omitted because the system is charge neutral overall. This term is therefore cancelled by similar terms in the ion-ion and electron-ion interaction energies. The Hartree potential in real-space is given by

$$V_{\mathrm H}({\bf r}) = \int {\mathrm d}{\bf r'}~ \frac{n({\bf r'})}{\left\vert {\bf r} - {\bf r'} \right\vert}$$ (7.10)

but is calculated in reciprocal-space as

$${\tilde V}_{\mathrm H}({\bf G}) = \frac{4 \pi {\tilde n}({\bf G})}{\Omega_{\mathrm {cell}} G^2}$$ (7.11)

and then transformed back into real-space by a FFT.