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 addition to this real-space localization there is also a localization 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 that exhibits additional sparseness properties compared to the scenario where one has a basis set with real-space localization only. Taking advantage of this hypersparsity, 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. We expect that hypersparsity will pave the way for highly accurate O(N) calculations of large systems requiring many basis functions per atom. [S0163-1829(99)02904-5].