Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T15:25:01.752Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear Scaling Discontinuous Galerkin Density Matrix Minimization Method with Local Orbital Enriched Finite Element Basis: 1-D Lattice Model System

Published online by Cambridge University Press:  03 June 2015

Tiao Lu ,
Wei Cai ,
Jianguo Xin  and
Yinglong Guo
Tiao Lu*
Affiliation:
HEDPS & CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
Wei Cai*
Affiliation:
INS, Shanghai Jiaotong University, Shanghai 200240, P.R. China Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223-0001, USA
Jianguo Xin*
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223-0001, USA
Yinglong Guo*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
*
Corresponding author.Email:wcai@uncc.edu
Email:tlu@math.pku.edu.cn
Email:jxin@uncc.edu
Email:longlazio1900@gmail.com
Get access

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.

Keywords

35Q4065N3065Z0581Q05Density functional theorydensity matrix minimizationdiscontinuous Galerkin methodlinear scaling method
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 2 , August 2013 , pp. 276 - 300
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abgrall, R. and Shu, C.-W.Development of residual distribution schemes for the discontinuous Galerkin method: The scalar case with linear elements. Commun. Comput. Phys., 5(2-4):376–390, 2009.Google Scholar
[2]Akin, J.E.The generation of elements with singularities. Int. J. Numer. Meth. Engng., 10:1249–1259, 1976.Google Scholar
[3]Babusška, I. and Rosenzweig, B.A finite element scheme for domains with corners. Numer. Math., 20:1–21, 1972.Google Scholar
[4]Chelikowsky, J.R., Troullier, N., and Saad, Y.Finite difference-pseudopotential method: Electronic structure calculations without a basis. Phys. Rev. Lett., 72:1240–1243, 1994.CrossRefGoogle ScholarPubMed
[5]Cockburn, B. and Shu, C.-W.The local discontinuous galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis, 35(6):2440–2463, 1998.CrossRefGoogle Scholar
[6]Galli, Giulia. Linear scaling methods for electronic structure calculations and quantum molecular dynamics simulations. Current Opinion in Solid State and Materials Science, 1(6):864–874, 1996.CrossRefGoogle Scholar
[7]Galli, Giulia and Parrinello, Michele. Large scale electronic structure calculations. Phys. Rev. Lett., 69(24):3547–3550, Dec 1992.Google Scholar
[8]García-Cervera, C. J.A remark on an efficient real space method for orbital-free density-functional theory. Commun. Comput. Phys., 1(1):1–5, 2006.Google Scholar
[9]García-Cervera, C. J.An efficient real space method for orbital-free density-functional theory. Commun. Comput. Phys., 2(2):334–357, 2007.Google Scholar
[10]García-Cervera, C.J., Lu, J., Xuan, Y., and E, W.Linear-scaling subspace-iteration algorithm with optimally localized nonorthogonal wave functions for kohn-sham density functional theory. Phys. Rev. B, 79:115110, 2009.CrossRefGoogle Scholar
[11]Goedecker, S.Low complexity algorithms for electronic structure calculations. J. Comput. Phys., 118:261–268, 1995.CrossRefGoogle Scholar
[12]Goedecker, S.Linear scaling electronic structure methods. Rev. Mod. Phys., 71(4):1085–1123, Jul 1999.Google Scholar
[13]Goedecker, S. and Colombo, L.Efficient linear scaling algorithm for tight-binding molecular dynamics. Phys. Rev. Lett., 73:122–125, Jul 1994.CrossRefGoogle ScholarPubMed
[14]Hernaández, E. and Gillan, M. J.Self-consistent first-principles technique with linear scaling. Phys. Rev. B, 51(15):10157–10160, Apr 1995.CrossRefGoogle Scholar
[15]Kohn, W. and Sham, L. J.Self-consistent equations including exchange and correlation effects. Phys. Rev., 140:A1133–A1138, Nov 1965.CrossRefGoogle Scholar
[16]Li, X.-P., Nunes, R. W., and Vanderbilt, David. Density-matrix electronic-structure method with linear system-size scaling. Phys. Rev. B, 47(16):10891–10894, Apr 1993.Google Scholar
[17]Lin, L., Lu, J., Ying, L., and E, W.Adaptive local basis set for kohn-sham density functional theory in a discontinuous galerkin framework I: Total energy calculation. submitted to J. Comput. Phys., 2011.Google Scholar
[18]Lu, T. and Cai, W.A fourier spectral-discontinuous galerkin method for time-dependent 3-d schrödinger-poisson equations with discontinuous potentials. J. Comput. Appl. Math., 220:588–614, October 2008.Google Scholar
[19]Luenberger, D. and Ye, Y.Linear and Nonlinear Programming. Springer, third edition, 2008.CrossRefGoogle Scholar
[20]Martin, R.Electronic Structure: Basic Theory and Practical Methods. Cambridge Univ. Press, Cambridge, 2004.Google Scholar
[21]Mauri, F., Galli, G., and Car, R.Orbital formulation for electronic-structure calculations with linear system-size scaling. Phys. Rev. B, 47(15):9973–9976, Apr 1993.CrossRefGoogle ScholarPubMed
[22]Weeny, R. Mc. The density matrix in self-conssitent field theory. I. iterative constructon of the density matrix. Proc. Roy. Soc. Lond. A, 235:496–509, 1956.Google Scholar
[23]Slater, J.C.Wave functions in a periodic potential. Phys. Rev., 51:846–851, 1937.Google Scholar
[24]Troullier, N. and Martins, José Luriaas. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B, 43:1993–2006, Jan 1991.Google Scholar
[25]Wang, W. and Shu, C.-W.The wkb local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode. J. Scientific Computing, 40:360–374, 2009.Google Scholar
[26]Yang, W.Direct calculation of electron density in density-functional theory. Phys. Rev. Lett., 66:1438–1441, Mar 1991.CrossRefGoogle ScholarPubMed