  Locality of compact one-electron orbitals expanded strictly in terms of local
subsets of basis functions can be exploited in density functional theory (DFT)
to achieve linear growth of computation time with systems size, crucial in
large-scale simulations. However, despite advantages of compact orbitals the
development of practical orbital-based linear-scaling DFT methods has long been
hindered because a compact representation of the electronic ground state is
difficult to find in a variational optimization procedure. In this work, we
show that the slow and unstable optimization of compact orbitals originates
from the nearly-invariant mixing of compact orbitals that are mostly but not
completely localized within the same subsets of basis functions. We also
construct an approximate Hessian that can be used to identify the problematic
nearly-invariant modes and obviate the variational optimization along them
without introducing significant errors into the computed energies. This enables
us to create a linear-scaling DFT method with a low computational overhead that
is demonstrated to be efficient and accurate in fixed-nuclei calculations and
molecular dynamics simulations of semiconductors and insulators.
