An algorithm for atom-centered lossy compression of the atomic orbital basis in density functional theory calculations

Large atomic-orbital (AO) basis sets of at least triple and preferably quadruple-zeta (QZ) size are required to adequately converge Kohn-Sham density functional theory (DFT) calculations towards the complete basis set limit. However, incrementing the cardinal number by one nearly doubles the AO basis dimension, and the computational cost scales as the cube of the AO dimension, so this is very computationally demanding. In this work, we develop and test a natural atomic orbital (NAO) scheme in which the NAOs are obtained as eigenfunctions of atomic blocks of the density matrix in a one-center orthogonalized representation. The NAO representation enables one-center compression of the AO basis in a manner that is optimal for a given threshold, by discarding NAOs with occupation numbers below that threshold. Extensive tests using the Hartree-Fock functional suggest that a threshold of $10^{-5}$ can yield a compression factor (ratio of AO to compressed NAO dimension) between 2.5 and 4.5 for the QZ pc-3 basis. The errors in relative energies are typically less than 0.1 kcal/mol when the compressed basis is used instead of the uncompressed basis. Between 10 and 100 times smaller errors (i.e., usually less than 0.01 kcal/mol) can be obtained with a threshold $10^{-7}$, while the compression factor is typically between 2 and 2.5.