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 . Indeed for bulk calculations with either 32 atoms or 108 atoms unit cell even with no Kerker preconditioning or wavedependent metric the calculation reached selfconsistency within 12 Pulay mixing steps (with mixing parameter set at ).

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

Figure 2 shows the convergence properties of calculations with different Kerker preconditioning parameters. All calculations were done using simple linear mixing with mixing parameter of 0.5. and smearing temperature of 300K ( ). Calculations with did not converge (no selfconsistent solutions were found due to chargesloshing). We found that the choice of 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 wavedependent metric method were also tested. However from the test it appears that Pulay mixing alone (with 5 Pulay history steps) is enough for reaching selfconsistency. And switching on/off either wavedependent metric or Kerkerpreconditioning had no noticeable effect on selfconsistency. 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 wavedependent metric on improving the selfconsistency process.
Lianheng Tong 20110302