Linear Scaling Discontinuous Galerkin Density Matrix Minimization Method with Local Orbital Enriched Finite Element Basis: 1-D Lattice Model System
Year: 2013
Communications in Computational Physics, Vol. 14 (2013), Iss. 2 : pp. 276–300
Abstract
In the first of a series of papers, we will study a discontinuous Galerkin (DG) framework for many electron quantum systems. The salient feature of this framework is the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteed piecewise polynomial basis in representing the Hamiltonian of the many body system. Such a flexibility is made possible by using the discontinuous Galerkin method to approximate the Hamiltonian matrix elements with proper constructions of numerical DG fluxes at the finite element interfaces. In this paper, we will apply the DG method to the density matrix minimization formulation, a popular approach in the density functional theory of many body Schrödinger equations. The density matrix minimization is to find the minima of the total energy, expressed as a functional of the density matrix ρ(r,r′), approximated by the proposed enriched basis, together with two constraints of idempotency and electric neutrality. The idempotency will be handled with the McWeeny's purification while the neutrality is enforced by imposing the number of electrons with a penalty method. A conjugate gradient method (a Polak-Ribiere variant) is used to solve the minimization problem. Finally, the linear-scaling algorithm and the advantage of using the local orbital enriched finite element basis in the DG approximations are verified by studying examples of one dimensional lattice model systems.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.290212.240812a
Communications in Computational Physics, Vol. 14 (2013), Iss. 2 : pp. 276–300
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 25