**Plane-wave Psuedopotential Electronic Structure Calculations on
Parallel Supercomputers**

J. S. Nelson and S. J. Plimpton, chapter in Topics in Computational Materials Science, edited by C. Y. Fong, published by World Scientific, 61-95 (1998).

We describe recent algorithmic advances in the numerical solution of the electronic structure for large systems. Numerous proof of principle calculations that demonstrate linear-scaling already exist. Most have focused on thigh-binding calculations not DFT. Nevertheless, much of the developments do carry over to DFT. Furthermore, the few DFT proof-of-principle calculations are equally encouraging not withstanding the significant remaining technical challenges. It is noteworthy that there is not one, now two, but numerous approaches to eliminating the N3 scaling with an N-scaling algorithm. In this chapter, we attempt to highlight the underlying theme of N-scaling approaches while noting the many differences. Evidently, the verdict is still out with regards to which methods will endure and survive the test of time. It is also less than obvious that there will ever exist one clear superior method. In all likelihood, the preferred method will depend on the nature of the application. Our goal is to present basic principles so that whatever the continued development in this active field, what we say herein will remain relevant long into the future. We operate from the biased expectation that a "good" algorithm will be robust, fairly general, mathematically sound; preferable variational; accurate and that it will strike a balance between efficiency and algorithmic simplicity.

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