1998/06/05 by S. Goedecker, Goedecker, S., O. V. Ivanov +1
Physics and Astronomy · #Condensed Matter (cond-mat) #FOS: Physical sciences #cond-mat
paper · pdf · doi:10.48550/arxiv.cond-mat/9806069
4 color figures, 3 normal figures
arxiv created 1998/06/05 · arxiv updated 2009/11/30
O(N) methods are based on the decay properties of the density matrix in real space, an effect sometimes refered to as near-sightedness. We show, that in addition to this near-sightedness in real space there is also a near-sightedness in Fourier space. Using a basis set with good localization properties in both real and Fourier space such as wavelets, one can exploit both localization properties to obtain a density matrix which exhibits additional sparseness properties compared to the scenario where one has a basis set with real space localization only. We will call this additional sparsity hyper-sparsity. Taking advantage of this hyper-sparsity, it is possible to represent very large quantum mechanical systems in a highly compact way. This can be done both for insulating and metallic systems and for arbitrarily accurate basis sets. We expect that hyper-sparsity will pave the way for O(N) calculations of large systems requiring many basis functions per atom, such as Density Functional calculations.