  Quantum embedding theories are playing an increasingly important role in
bridging different levels of approximation to the many body Schr\"odinger
equation in physics, chemistry and materials science. In this paper, we present
a linear algebra perspective of the recently developed projection based
embedding theory (PET) [Manby et al, J. Chem. Theory Comput. 8, 2564, 2012],
restricted to the context of Kohn-Sham density functional theory. By
partitioning the global degrees of freedom into a `system' part and a `bath'
part, and by choosing a proper projector from the bath, PET is an in principle
exact formulation to confine the calculation to the system part only, and hence
can be carried out with reduced computational cost. Viewed from the perspective
of the domain decomposition method, one particularly interesting feature of PET
is that it does not enforce a boundary condition explicitly, and remains
applicable even when the discretized Hamiltonian matrix is dense, such as in
the context of the planewave discretization. In practice, the accuracy of PET
depends on the accuracy of the projector for the bath. Based on the linear
algebra reformulation, we develop a first order perturbation correction to the
projector from the bath to improve its accuracy. Numerical results for real
chemical systems indicate that with a proper choice of reference system, the
perturbatively corrected PET can be sufficiently accurate even when strong
perturbation is applied to very small systems, such as the computation of the
ground state energy of a SiH$_3$F molecule, using a SiH$_4$ molecule as the
reference system.

Computational Physics;Numerical Analysis