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# Annals of Mathematical Sciences and Applications

## Volume 3 (2018)

### Number 2

### PEXSI-$\Sigma$: a Green’s function embedding method for Kohn–Sham density functional theory

Pages: 441 – 472

DOI: https://dx.doi.org/10.4310/AMSA.2018.v3.n2.a3

#### Authors

#### Abstract

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.

The work of X.L. was supported by the National Science Foundation under award DMS-1522617. The work of L.L. was partially supported by Laboratory Directed Research and Development (LDRD) funding from Berkeley Lab, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, the Alfred P. Sloan foundation, the DOE Scientific Discovery through the Advanced Computing (SciDAC) program and the DOE Center for Applied Mathematics for Energy Research Applications (CAMERA) program. The work of J.L. was supported in part by the National Science Foundation under awards DMS-1312659 and DMS-1454939.

Received 2 June 2016

Published 9 August 2018