  Localized basis sets in the projector augmented wave formalism allow for
computationally efficient calculations within density functional theory (DFT).
However, achieving high numerical accuracy requires an extensive basis set,
which also poses a fundamental problem for the interpretation of the results.
We present a way to obtain a reduced basis set of atomic orbitals through the
subdiagonalization of each atomic block of the Hamiltonian. The resulting local
orbitals (LOs) inherit the information of the local crystal field. In the LO
basis, it becomes apparent that the Hamiltonian is nearly block-diagonal, and
we demonstrate that it is possible to keep only a subset of relevant LOs which
provide an accurate description of the physics around the Fermi level. This
reduces to some extent the redundancy of the original basis set, and at the
same time it allows one to perform post-processing of DFT calculations, ranging
from the interpretation of electron transport to extracting effective
tight-binding Hamiltonians, very efficiently and without sacrificing the
accuracy of the results.
