Calculating large systems with plane waves: Is it a N³ or N² scaling problem?

It is commonly asserted that in performing large scale plane wave calculations with N plane waves, and indirect diagonalization, the N³ orthogonalization step quickly becomes the limiting factor. Using a pre-condition Lanczos subspace diagonalization algorithm we find that while the orthogonalization will eventually become the dominate cpu restriction, this is not the limiting factor. Typically the limitation resides not with the cpu, but with the memory limitations of the current computer systems. The memory needed to store the wave functions scales as N_atom². This scaling "limitation" is reached before the cpu time becomes the dominate factor. We illustrate this scaling with a Na vacancy calculation using up to 2000 atoms.