Local pseudopotential

Like the Hartree potential, the local pseudopotential is also calculated in reciprocal-space as

$${\tilde V}_{\mathrm {ps,loc}}({\bf G}) = \sum_s {\tilde v}_{\mathrm {ps,loc}}^s ({\bf G}) {\mbox{\Bbb S}}^s({\bf G})$$ (7.14)

where the summation is over ionic species $s$, ${\tilde v}_{\mathrm {ps,loc}}^s ({\bf G})$ is the local pseudopotential for an isolated ion of species $s$ in reciprocal-space and ${\mbox{\Bbb S}}^s({\bf G})$ is the structure factor for species $s$ defined by

$${\mbox{\Bbb S}}^s({\bf G}) = \sum_{\alpha} \exp[-{\mathrm i}{\bf G} \cdot {\bf r}_{\alpha}^s]$$ (7.15)

where the sum is over all ions $\alpha$ of species $s$ with positions ${\bf r}_{\alpha}^s$. We note that in general the calculation of the structure factor is an ${\cal O}(N^2)$ operation, but since it only has to be calculated once for each atomic configuration, it is not a limiting factor of the overall calculation at this stage. Within the quantum chemistry community, work on generalised multipole expansions and new algorithms [156,157,158,159,160,161,162] has led to the development of methods to calculate Coulomb interaction matrix elements which scale linearly with system-size. The local pseudopotential energy can be calculated in reciprocal-space as

$$E_{\mathrm{ps,loc}} = \int {\mathrm d}{\bf r}~ V_{\mathrm{ps,loc}}({\bf r}) n({\bf r})$$

$$= \sum_{{\bf G} \not= 0} \sum_s {\tilde v}_{\mathrm{ps,loc}}^s({\bf... ...b S}}^s({\bf G}) {\tilde n}^{\ast} ({\bf G})+\sum_s N_s E_{\mathrm{ps, core}}^s$$

$$= 2 K^{\alpha \beta} V_{\mathrm{loc},\beta \alpha}$$ (7.16)

where $E_{\mathrm{ps, core}}^s$ is the pseudopotential core energy, and $N_s = {\mbox{\Bbb S}}^s({\bf G}=0)$ the number of ions of species $s$. The matrix elements $V_{\mathrm{loc},\alpha \beta}$ are defined by

$$V_{\mathrm{loc},\alpha \beta} = \int {\mathrm d}{\bf r}~ \ph... ...({\bf r}) V_{\mathrm{ps,loc}}({\bf r}) \phi_{\beta}({\bf r}) .$$ (7.17)

The Hartree potential and local pseudopotential can be summed and then transformed back together into real-space and added to the exchange-correlation potential to obtain the local part of the Kohn-Sham potential in real-space.

We note that the FFT is not strictly an ${\cal O}(N)$ operation but an ${\cal O}(N \log_m N)$ operation (where $m$ is some small number which depends upon the prime factors of the number of grid points), but in practice (section 9.2) this scaling is not observed.