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#

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.

Arash Mostofi
2003-10-28