  We present a spectrum-splitting approach to conduct all-electron Kohn-Sham
density functional theory (DFT) calculations by employing Fermi-operator
expansion of the Kohn-Sham Hamiltonian. The proposed approach splits the
subspace containing the occupied eigenspace into a core-subspace, spanned by
the core eigenfunctions, and its complement, the valence-subspace, and thereby
enables an efficient computation of the Fermi-operator expansion by reducing
the expansion to the valence-subspace projected Kohn-Sham Hamiltonian. The key
ideas used in our approach are: (i) employ Chebyshev filtering to compute a
subspace containing the occupied states followed by a localization procedure to
generate non-orthogonal localized functions spanning the Chebyshev-filtered
subspace; (ii) compute the Kohn-Sham Hamiltonian projected onto the
valence-subspace; (iii) employ Fermi-operator expansion in terms of the
valence-subspace projected Hamiltonian to compute the density matrix,
electron-density and band energy. We demonstrate the accuracy and performance
of the method on benchmark materials systems involving silicon nano-clusters up
to 1330 electrons, a single gold atom and a six-atom gold nano-cluster. The
benchmark studies on silicon nano-clusters revealed a staggering five-fold
reduction in the Fermi-operator expansion polynomial degree by using the
spectrum-splitting approach for accuracies in the ground-state energies of
$\sim 10^{-4} Ha/atom$ with respect to reference calculations. Further,
numerical investigations on gold suggest that spectrum splitting is
indispensable to achieve meaningful accuracies, while employing Fermi-operator
expansion.
