Linear-scaling implementation of exact exchange using localized   numerical orbitals and contraction reduction integrals
  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). 
Lionel A. Truflandier, Tsuyoshi Miyazaki and David R. Bowler
physics.chem-ph cond-mat.mtrl-sci physics.comp-ph