Convergence with density-matrix cut-off

In figure 9.1 we plot the total energy per atom against the support region radius $r_{\mathrm{reg}}$ for a density-kernel cut-off $r_K$ of 4.0 Å. In figure 9.2 we plot the total energy per atom against the density-kernel cut-off $r_K$ for a support region radius $r_{\mathrm{reg}}$ of 3.1 Å. In both cases, the energy converges to its limiting value from above, as expected, since the total energy is variational with respect to the density-matrix cut-off.

These results agree roughly with the calculations of Hernández et al. [127]. In their case, they used atom-centred support regions and included as many unoccupied bands as occupied bands, so that the convergence with respect to density-matrix cut-off should be more rapid in their case. They also used a local pseudopotential, which reduces the range of the Hamiltonian. Our calculations suggest a combined density-matrix cut-off of the order of 7.0 Å to obtain the same accuracy as they obtained with a cut-off of about 6.0 Å. We note that the band gap of silicon is relatively small (particularly within the LDA) so that the density-matrix decay is therefore slow. This makes silicon a difficult test case, and we can be confident that if we obtain reasonable results in this system, we shall be successful in others.

Figure 9.3: Electronic density of silicon in the (110) plane (units Å$^{-3}$).

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Figure 9.4: Diamond structure of silicon, highlighting a {110} plane.

Figure 9.5: Electronic density of silicon in the (110) plane calculated by the CASTEP code (units Å$^{-3}$).

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Figure 9.6: Difference between electronic densities of silicon calculated by the linear-scaling method and CASTEP (units Å$^{-3}$).

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