A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme

To exploit the exponential decay found in numerical studies for the density matrix and its derivative with respect to nuclear displacements, we reformulate the coupled perturbed self-consistent field (CPSCF) equations and a quadratically convergent SCF (QCSCF) method for Hartree-Fock and density functional theory within a local density matrix-based scheme. Our D-CPSCF (density matrix-based CPSCF) and D-QCSCF schemes open the way for exploiting sparsity and to achieve asymptotically linear scaling of computational complexity with molecular size (M), in case of D-CPSCF for all O(M) derivative densities. Furthermore, these methods are even for small molecules strongly competitive to conventional algorithms.