Local alternatives to Becke weights for molecular quadrature

Most density functional theory (DFT) calculations employ Becke's methodology to partition the space in and around an n-atom molecule into n overlapping quasi-atomic regions. In this way, a single molecular integral for the energy (or other property) is reduced to a sum of n atomic integrals, each of which can then be approximated using a local grid in spherical polar coordinates. Unfortunately, the formal cost of Becke's partition is O(n2N), where N is the total number of grid points, and this can become a computational bottleneck when applied to large molecules. We propose two modifications of Becke's approach, each of which is formally exact and depends only on knowledge of the Delaunay neighbors of each nucleus. The two modifications can be used separately to produce schemes whose costs are O(nN) or together to produce a scheme whose cost is only O(N). Numerical tests for a variety of molecules and atomic grids suggest that the O(N) scheme should be implemented in all DFT software packages.