Interpolation Grid Convergence

As with all plane-wave calculations the usual convergence tests for the kinetic energy cutoff and number of $k$-points must be performed. Experience shows that for a given system the specific convergence values will be the same for all functionals, including the WDA, so it is easiest to use the LDA for these tests.

The first quantity to establish in a WDA calculation is the maximum weighted density interpolation value ${\tilde{n}_{max}}$ required for the system under study. There is no hard and fast rule to predict this value for a given system, although the greater the inhomogeneity in the density, and the larger the number of electrons in the unit cell, will generally mean that a higher value of ${\tilde{n}_{max}}$ will be needed. Once ${\tilde{n}_{max}}$ is established, the second stage is to perform a convergence test on the number of weighted density grid points $N_{\tilde{n}}$ - single-point energy calculations are performed, and the results are plotted against increasing $N_{\tilde{n}}$. The converged value is determined in the usual fashion, i.e. when the curve becomes flat. Fig. 4.1 shows the convergence of $N_{\tilde{n}}$ for the solids chosen here.

The third and final stage it to converge the three-dimensional spatial grid to obtain $N_x, N_y$ and $N_z$. For bulk solids this can be performed for just one of the unit cell directions and the same converged value used for the other two directions. Obviously for systems that are distinctly different in particular directions, such as a surface calculation where the bulk directions are very different from that perpendicular to the surface, then convergence tests will be required in the other directions. Fig. 4.2 shows the convergence test of $N_x$ for the primitive cells of C-diamond, Si, and GaAs, and the unit cell of Al. It can be seen from this figure that the systems with more rapidly varying densities such as GaAs require more grid points than other systems. Note that the convergence for Al was performed on the $4$ atom non-primitive unit cell, and so actually requires the least number of spatial grid points per unit volume. This is to be expected since the metallic density of Al varies most slowly.

Figure 4.1: Convergence of the total energy, $E[n({\bf r})]$, calculated using the WDA, with respect to the number of WDA weighted density interpolation points $N_{\tilde{n}}$, for bulk C-diamond, Si, GaAs and Al.

Figure 4.2: Convergence of the total energy, $E[n({\bf r})]$, calculated using the WDA, with respect to the number of WDA spatial interpolation grid points $N_i$ along one of the crystal directions, for C, Si, GaAs and Al. A non-primitive unit cell is used for Al.