Fast Hartree-Fock theory using local density fitting approximations

Density fitting approximations are applied to generate the Fock matrix in Hartree-Fock calculations. By localizing the orbitals in each iteration and performing separate fits for each orbital the O(N-4) scaling of the computational effort for the exchange can be reduced to O(N). We also use the Poisson method to replace almost all Coulomb integrals with simple overlaps, an efficient alternative to diagonalization, and dual basis sets such that the Hartree-Fock calculation is performed in a smaller basis than the subsequent treatment of electron correlation. The accuracy and efficiency of the method is demonstrated in calculations with almost 4000 basis functions. The errors introduced by the local approximations on HF and MP2 energies are small compared to those that arise from the density fitting, and the fitting errors themselves (typically 1-10 microhartree per atom) are very small compared, for example, to the effect of basis set variations.