Test Results

Due to symmetry of the aluminium bulk charge sloshing will not take place. Any updates in density will follow the same symmetry as one cannot distinguish one lattice direction from another, resulting an output density which obeys the same symmetry as input density, which in turn preserves the symmetry of the updated Hartree potential $\hat{V}_{\text{H}}$. Indeed for bulk calculations with either 32 atoms or 108 atoms unit cell even with no Kerker preconditioning or wave-dependent metric the calculation reached self-consistency within 12 Pulay mixing steps (with mixing parameter set at $\lambda = 0.5$).

Figure 1: Left: Total ground-state energy vs. number of self-consistency iterations. Right: Residual in electron density vs. number of self-consistency iterations.

To break the symmetry a vacancy is introduced to the aluminium bulk by removing one atom at $(0,0,0)$. The standard linear mixing method failed to reach self-consistency with mixing parameter $\lambda$ ranging from 0.1 to 0.8, however adding Kerker Preconditioning significantly improved the self-consistency convergence properties. Figure 1 shows charge sloshing behaviour in aluminium bulk system with 32 atoms unit cell and a defect (vacancy) at $(0,0,0)$ while using simple linear mixing ($q_0 = 0$ case). The effect of charge sloshing is demonstrated by the oscillations in the calculated ground-state energy. We can also see that the effect of charge sloshing is damped out with increasing $q_0$ for Kerker preconditioning.

Figure 2: Left: Number of iterations required to reach self-consistency vs. $q_0$ for Kerker preconditioning. Right: Convergence properties with different $q_0$ parameters.

Figure 2 shows the convergence properties of calculations with different Kerker pre-conditioning parameters. All calculations were done using simple linear mixing with mixing parameter of 0.5. and smearing temperature of 300K ( $\approx 0.001\text{Ha}$). Calculations with $q_0 < 0.4$ did not converge (no self-consistent solutions were found due to charge-sloshing). We found that the choice of $q_0 = 1.0 * 2\pi\text{bohr}^{-1} = 1.188\text{\AA}^{-1}$ is optimum for aluminium bulk with the chosen vacancy. This is comparable with the results of [11].

Convergence performances on the 32 atoms aluminium bulk with vacancy at (0, 0, 0) using a wave-dependent metric method were also tested. However from the test it appears that Pulay mixing alone (with 5 Pulay history steps) is enough for reaching self-consistency. And switching on/off either wave-dependent metric or Kerker-preconditioning had no noticeable effect on self-consistency. This may be because the 32 Al atom with defect system is still not difficult enough for Pulay mixing to fail. More tests may be required in future with larger systems with more complex structures may be needed to fully test the potential of Kerker preconditioning and wave-dependent metric on improving the self-consistency process.