Conclusions

Through detailed calculations on molecules and bulk crystalline silicon, we find that the total energy and physical properties can be accurately deduced from density-functional calculations using localized spherical-wave basis sets. We find that for most purposes, the choice of $\ell_{\mathrm{max}}= 2$ and atom-centered basis spheres is sufficient to obtain an accuracy which is excellent compared to those obtained using the extended plane-wave basis set.

The dependence of the total energy on the cutoff energy of the localized basis set is found to be rather similar to that of the extended plane-wave basis set. We also find that the results converge exponentially with respect to the basis sphere radius $R$. The angular incompleteness can be improved either by increasing $\ell_{\mathrm{max}}$, or by introducing additional basis spheres at strategic locations (such as at the middle of a bond).

We find that the radii of the basis spheres depend on the bond lengths. A large bond length usually means large basis spheres are to be used. We find that it is possible to use different radii for the basis spheres depending on the relative electronegativities of the atomic species. Usually we need to use basis spheres with larger radii for atoms that are more electronegative than other atoms in a calculation.

For the bulk silicon case, the accuracy of the results obtained using $\ell_{\mathrm{max}}=1$ is marginally acceptable, which is a consequence of the $sp^3$ hybridization.

Finally, we note that one of the main advantages of this basis set is that the accuracy of the results can be systematically improved. It would be interesting to explore the possibility of using two or more basis spheres of different radii centered on an atom so that a lower cutoff energy for the larger basis spheres might be used.