Frequency localization properties of the density matrix and its resulting hypersparsity in a wavelet representation

O(N) methods are based on the localization properties of the density matrix in real space, an effect refered to as nearsightedness. We show that, in a ddition to this real-space localization there is also a localization in Fou rier space. Using a basis set with good localization properties in both rea l and Fourier space such as wavelets, one can exploit both localization pro perties to obtain a density matrix that exhibits additional sparseness prop erties compared to the scenario where one has a basis set with real-space l ocalization only. Taking advantage of this hypersparsity, it is possible to represent very large quantum-mechanical systems in a highly compact way. T his can be done both for insulating and metallic systems. We expect that hy persparsity will pave the way for highly accurate O(N) calculations of larg e systems requiring many basis functions per atom. [S0163-1829(99)02904-5].