Electronic density

Since the minimising density-matrix is only approximately idempotent, the electronic density derived from it is not the exact ground-state density. However, in figure 9.3 we plot the electronic density in the (110) plane (containing the atoms highlighted in silver in figure 9.4) obtained from the minimising density-matrix with $r_{\mathrm{reg}} = 3.1$ Å and $r_K = 6.0$ Å, and observe that it is still qualitatively correct. Since electronic densities are used primarily for visualisation purposes, rather than for quantitative analysis, the information most commonly required can still be obtained from the minimising density-matrix.

In figure 9.5 we plot the electronic density obtained using the CASTEP plane-wave code using the same pseudopotential and energy cut-off, and equivalent Brillouin zone sampling (a $3 \times 3 \times 3$ Monkhorst-Pack mesh for an 8-atom unit cell). We observe that there is less density concentrated in the bonds for the CASTEP density compared with the linear-scaling density, and this is to be expected since in the linear-scaling calculation, the lowest energy (i.e. most bonding) orbital is over-occupied and the highest energy (i.e. least bonding) orbital is under-occupied. In figure 9.6 we plot the difference obtained by subtracting the CASTEP density from the linear-scaling density and confirm this observation.