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-ζ (QZ) size are required to adequately converge Kohn–Sham
density functional theory (DFT) calculations toward 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 threshold-based natural atomic
orbital (NAO) scheme in which ϵ-NAOs are obtained as eigenfunctions
of atomic blocks of the density matrix in a one-center orthogonalized
representation. This enables compression of the AO basis that is optimal
for a given threshold, 10–ϵ, by discarding
NAOs with occupation numbers below that threshold. Extensive pilot
test calculations using the Hartree–Fock functional and taking
the converged density matrix as input suggest that a threshold of
10–5 can yield a compression factor (ratio of AO
to compressed ϵ-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.