Conclusions

We have presented a preconditioning scheme to improve the convergence of iterative steepest descents or conjugate gradients total energy minimizations. We have derived a general expression for this preconditioning scheme for nonorthogonal basis sets. For the special case of orthogonal basis sets, we have showed that a unitary transformation may be made to a representation in which the preconditioning function is diagonal. In our linear-scaling density functional theory method, which uses an orthogonal basis set of periodic sinc (psinc) functions, this representation is accessed via discrete Fourier transformation: in other words, the preconditioning function is diagonal in reciprocal space. We have also developed an efficient and physically motivated preconditioning scheme which uses a localized convolution directly in real space, with no need for fast Fourier transforms. Both of these approaches (reciprocal space and real space) significantly improve the rate of convergence, and this improvement is found to be almost independent of the size of the basis set.