  In this paper, we propose a new Green's function embedding method called
PEXSI-$\Sigma$ for describing complex systems within the Kohn-Sham density
functional theory (KSDFT) framework, after revisiting the physics literature of
Green's function embedding methods from a numerical linear algebra perspective.
The PEXSI-$\Sigma$ method approximates the density matrix using a set of nearly
optimally chosen Green's functions evaluated at complex frequencies. For each
Green's function, the complex boundary conditions are described by a self
energy matrix $\Sigma$ constructed from a physical reference Green's function,
which can be computed relatively easily. In the linear regime, such treatment
of the boundary condition can be numerically exact. The support of the $\Sigma$
matrix is restricted to degrees of freedom near the boundary of computational
domain, and can be interpreted as a frequency dependent surface potential. This
makes it possible to perform KSDFT calculations with $\mathcal{O}(N^2)$
computational complexity, where $N$ is the number of atoms within the
computational domain. Green's function embedding methods are also naturally
compatible with atomistic Green's function methods for relaxing the atomic
configuration outside the computational domain. As a proof of concept, we
demonstrate the accuracy of the PEXSI-$\Sigma$ method for graphene with
divacancy and dislocation dipole type of defects using the DFTB+ software
package.
