Linear-scaling implementation of exact exchange using the density matrix: method and accuracy

We present enhancements to the computational efficiency of exact exchange calculations using the density matrix and local support functions. We introduce a numerical method which avoids the explicit calculation the four-center two-electron repulsion integrals and reduces the prefactor scaling by a factor N, where N is the number of atoms within the range of the exact exchange Hamiltonian. This approach is based on a contraction-reduction scheme, and takes advantage of the discretization space which enables the direct summation over the support functions in a localized space. Using the sparsity property of the density matrix, the scaling of the prefactor can be further reduced to reach asymptotically O(N).