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We present enhancements to the computational efficiency of exact exchangecalculations using the density matrix and local support functions. We introducea numerical method which avoids the explicit calculation the four-centertwo-electron repulsion integrals and reduces the prefactor scaling by a factorN, where N is the number of atoms within the range of the exact exchangeHamiltonian. This approach is based on a contraction-reduction scheme, andtakes advantage of the discretization space which enables the direct summationover the support functions in a localized space. Using the sparsity property ofthe density matrix, the scaling of the prefactor can be further reduced toreach asymptotically O(N).

Linear-scaling implementation of exact exchange using localized numerical orbitals and contraction reduction integrals