  We present a systematic hierarchy of approximations for {\it local}
exact-decoupling of four-component quantum chemical Hamiltonians based on the
Dirac equation. Our ansatz reaches beyond the trivial local approximation that
is based on a unitary transformation of only the atomic block-diagonal part of
the Hamiltonian. Systematically, off-diagonal Hamiltonian matrix blocks can be
subjected to a unitary transformation to yield relativistically corrected
matrix elements. The full hierarchy is investigated with respect to the
accuracy reached for the electronic energy and molecular properties on a
balanced test molecule set that comprises molecules with heavy elements in
different bonding situations. Our atomic (local) assembly of the unitary
transformation needed for exact decoupling provides an excellent local
approximation for any relativistic exact-decoupling approach. Its order-$N^2$
scaling can be further reduced to linear scaling by employing the
neighboring-atomic-blocks approximation. Therefore, it is an efficient
relativistic method perfectly well suited for relativistic calculations on
large molecules. If a large molecule contains many light atoms (typically
hydrogen atoms), the computational costs can be further reduced by employing a
well-defined non-relativistic approximation for these light atoms without
significant loss of accuracy.
