Projection based embedding theory for solving Kohn-Sham density functional theory

TL;DR

This paper introduces a linear algebra perspective of projection based embedding theory (PET) for Kohn-Sham density functional theory, enabling reduced-cost calculations by confining computations to a system part while maintaining accuracy through a perturbation correction.

Contribution

It reformulates PET using linear algebra, develops a perturbation correction for the projector, and demonstrates improved accuracy in chemical system simulations.

Findings

PET can accurately compute ground state energies with reduced computational cost.

Perturbation correction enhances PET accuracy even under strong perturbations.

Numerical results show PET's effectiveness in small chemical systems.

Abstract

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.