Linear scaling computation of the Fock matrix. V. Hierarchical cubature for numerical integration of the exchange-correlation matrix

Hierarchical cubature is a new method for achieving linear scaling computat ion of the exchange-correlation matrix central to Density Functional Theory . Hierarchical cubature combines a k-dimensional generalization of the bina ry search tree with adaptive numerical integration involving an entirely Ca rtesian grid. Hierarchical cubature overcomes strong variations in the elec tron density associated with nuclear cusps through multiresolution rather t han spherical-polar coordinate transformations. This unique Cartesian repre sentation allows use of the exact integration error during grid constructio n, supporting O(log N) range-queries that exploit locality of the Cartesian Gaussian based electron density. Convergence is controlled by tau (r), whi ch bounds the local integration error of the electron density. An early ons et of linear scaling is observed for RB3LYP/6-31G** calculations on water c lusters, commencing at (H2O)(30) and persisting with decreasing values of t au (r). Comparison with nuclear weight schemes suggests that the new method is competitive on the basis of grid points per atom. Systematic convergenc e of the RPBE0/6-31G** Ne-2 binding curve is demonstrated with respect to t au (r). (C) 2000 American Institute of Physics. [S0021-9606(00)32442-4].