Exploring Sparsity in Three-Dimensional Integration for Density-Functional Calculations

The cost of a density-functional calculation with three-dimensional integration remains to be order N3, although a large portion of the integration grid may have negligible effects on the generation of a matrix element, due to rapid decay of atom-centered basis functions with distance. This type of integration sparsity is exploited by prescreening for insignificant contributions based on a direct estimate of their magnitudes. The result is a substantial reduction in cost without sacrificing numerical precision. Timing on compact molecules shows that a near order N2 scaling with system size can be obtained for N≲300 basis functions. The overhead of prescreening is moderate and may be characterized by a break-even size N≈200.