Analysis of numerical methods for evaluating the Fock exchange integral in a plane-wave basis

In this Brief Report, we analyze methods for evaluating the singular integrals of the Fock exchange interaction with the help of a simple analytic model. We find that Brillouin zone sampling can be important even for systems with large unit cells.

Figure 1

Magnitude of the exchange-energy differences relative to exact result (22), calculated using the Duchemin-Gygi (“D-G”) and Spencer-Alavi (“S-A”) schemes. The horizontal axis represents the $\mathbf{k}$-point sampling in terms of the partitioning numbers $N_1{N}_2{N}_3$. Note that results for the D-G (corr) scheme do not show up on the plot.

Figure 2

Plot of the exact exchange kernel function $X_{n\mathbf{k}}\left(\mathbf{r}\right)$ for $\mathbf{k}=0$ plotted along a cube edge, compared with the numerical differences obtained using single $\mathbf{k}$-point sampling with the approximations of Spencer and Alavi ($\Delta X\equiv X_{n\mathbf{k}}^{\mathrm{S}-\mathrm{A}}-X_{n\mathbf{k}}$) and Duchemin and Gygi ($\Delta X\equiv X_{n\mathbf{k}}^{\mathrm{D}-\mathrm{G}}-X_{n\mathbf{k}}$). To help the visualization, the exchange kernel differences ($\Delta X$) have been multiplied by 100.