4. Tests of the algorithm

A chlorine molecule of bond length 2.0 Å is placed in a cubic box of side 10 Å. With a cutoff energy of 640 eV and the basis-function radius of 4.0 Å, a total of basis functions are used. In Fig. 1 we display the convergence of the sum of Kohn-Sham eigenvalues toward the ``exact'' value obtained from direct matrix diagonalization, as a function of the iteration number. The convergence of solution is seen to be linear when the number of iterations is smaller than the number of basis functions. To investigate the effect of preconditioning on the convergence of the solution, we have used a number of fixed values. It is seen that the performance of the method improves with moderate preconditioning. Fig. 1 shows that should be about 10 eV for good convergence. We have performed another calculation with updated according to the highest kinetic energy of all approximate eigenvectors, which converges to 24 eV. This is the natural choice for used in other preconditioning schemes, and the performance of this calculation (the curve labelled by open diamonds ) is seen to be rather similar to that of the `optimal' case with eV. This method therefore allows to be chosen automatically, and optimized during the calculation, rather than being another parameter which the user must specify.

To investigate the importance of preserving the tensor nature of the search direction, we have performed calculations with the same cutoff energy of 640 eV on the molecular chlorine system, but this time with set to the identity matrix (this corresponds to the case where tensor nature of the search direction is not preserved) and the off-diagonal elements of set to zero (this corresponds to the diagonal approximation used in Ref. [14]) when we solve Eq. 22. The results of the calculations are presented in Fig. 2 where we have included the tensor-nature-preserving (TNP) curves for comparison. It is found that that the non-tensor-property-preserving (NTNP) cases fail to converge to the right solution. We conclude that it is essential to take tensor properties into account when one is dealing with a non-orthogonal basis set.

With a cutoff energy as high as 4800 eV (a total of basis functions are used in these calculations), Fig. 3 clearly indicates that it is crucial to use the preconditioning scheme. A comparison between Figs. 1 and 3 reveals that when the optimal value of is used, the number of iterations to achieve the same accuracy remains roughly the same, even though the number of basis functions has more than doubled, which shows that the preconditioning scheme is indeed working.

Similar tests are performed on the bulk crystalline silicon system. The calculations on a 64-atom silicon unit cell are performed at the equilibrium lattice parameter of 5.43 Å with an energy cutoff of 200 eV. We have chosen to be 3.1 Å which is sufficient for this purpose. These settings result in a total of basis functions for the calculations. In Fig. 4 we we note that our `best' eV is comparable with the value of 3.8 eV used by Bowler and Gillan [14]. We have performed another calculation with updated according to the highest kinetic energy of all approximate eigenvectors, which converges to 12 eV. The performance of this calculation is seen to be rather similar to that of the optimal cases with or 10 eV.