Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-06T02:42:25.465Z Has data issue: false hasContentIssue false
  • English
  • Français

On the “Preconditioning” Function Used in Planewave DFT Calculations and its Generalization

Part of: Basic linear algebra General and miscellaneous specific topics Graph theory Numerical linear algebra

Published online by Cambridge University Press:  03 July 2015

Yunkai Zhou ,
James R. Chelikowsky ,
Xingyu Gao  and
Aihui Zhou
Yunkai Zhou*
Affiliation:
Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA
James R. Chelikowsky
Affiliation:
Center for Computational Materials, Institute for Computational Engineering and Science, and Departments of Physics and Chemical Engineering, University of Texas, Austin, TX 78712, USA
Xingyu Gao
Affiliation:
HPCC, Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China
Aihui Zhou
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
*Corresponding author. Email addresses: [email protected] (Y. Zhou), [email protected] (J. R. Chelikowsky), [email protected] (X. Gao), [email protected] (A. Zhou)
Get access

Abstract

The Teter, Payne, and Allan “preconditioning” function plays a significant role in planewave DFT calculations. This function is often called the TPA preconditioner. We present a detailed study of this “preconditioning” function. We develop a general formula that can readily generate a class of “preconditioning” functions. These functions have higher order approximation accuracy and fulfill the two essential “preconditioning” purposes as required in planewave DFT calculations. Our general class of functions are expected to have applications in other areas.

Keywords

Density functional theoryplanewavepreconditioning functioneigenvalue problem

MSC classification

Secondary: 00A69: General applied mathematics 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 15A18: Eigenvalues, singular values, and eigenvectors 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 167 - 179
Copyright
Copyright © Global-Science Press 2015 

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]Banerjee, A. S., Elliott, R. S., and James, R. D.. A spectral scheme for Kohn-Sham density functional theory of clusters. ArXiv e-prints, arXiv:1404.3773, 2014.Google Scholar
[2]Benzi, M.. Preconditioning techniques for large linear systems: A survey. J. Comput. Phys., 182(2):418–477, 2002.Google Scholar
[3]Engel, B. and Dreizler, R.M.. Density Functional Theory: An Advanced Course. Theoretical and Mathematical Physics. Springer, 2011.CrossRefGoogle Scholar
[4]Giannozzi, P.et al.. QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials. Journal of Physics: Condensed Matter, 21(39):395502 (19pp), 2009.Google Scholar
[5]Gonze, X.et al.. ABINIT: First-principles approach of materials and nanosystem properties. Computer Phys. Commun., 180:2582–2615, 2009.Google Scholar
[6]Hohenberg, P. and Kohn, W.. Inhomogeneous electron gas. Phys. Rev., 136:B864–871, 1964.Google Scholar
[7]Kaxiras, E.. Atomic and Electronic Structure of Solids. Cambridge University Press, 2003.Google Scholar
[8]Kohanoff, J.. Electronic Structure Calculations for Solids and Molecules: Theory and Computational Methods. Cambridge Univ. Press, 2006.CrossRefGoogle Scholar
[9]Kohn, W. and Sham, L. J.. Self-consistent equations including exchange and correlation effects. Phys. Rev., 140:A1133–1138, 1965.Google Scholar
[10]Kresse, G. and Furthmü ller, J.. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B, 54(16):11169–11186, 1996.Google Scholar
[11]Levitt, A. and Torrent, M.. Parallel eigensolvers in plane-wave density functional theory. Comp. Phys. Comm., 187:98–105, 2015.Google Scholar
[12]Martin, R. M.. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, 2004.Google Scholar
[13]Motamarri, P. and Gavini, V.. A subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization. ArXiv e-prints, arXiv:1406.2600, 2014.Google Scholar
[14]Saad, Y.. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, second edition, 2003.CrossRefGoogle Scholar
[15]Sholl, D. and Steckel, J. A.. Density Functional Theory: A Practical Introduction. Wiley-Interscience, 2009.CrossRefGoogle Scholar
[16]Teter, M. P., Payne, M. C., and Allan, D. C.. Solution of Schrödinger’s equation for large systems. Phys. Rev. B, 40:12255–12263, 1989.Google Scholar
[17]ABINIT webpage. http://www.abinit.org/.Google Scholar
[18]Quantum ESPRESSO webpage. http://www.quantum-espresso.org/.Google Scholar
[19]VASP webpage. http://cms.mpi.univie.ac.at/vasp/.Google Scholar
[20]Wilkinson, J. H.. The Algebraic Eigenvalue Problem. Oxford University Press, 1965.Google Scholar
[21]Yang, C., Meza, J. C., Lee, B., and Wang, L.-W.. KSSOLV — a MATLAB Toolbox for Solving the Kohn-Sham Equations. ACM Trans. Math. Softw., 36(2):10:1–35, 2009.Google Scholar
[22]Zhou, Y.. A block Chebyshev-Davidson method with inner-outer restart for large eigenvalue problems. J. Comput. Phys., 229(24):9188–9200, 2010.Google Scholar
[23]Zhou, Y. and Li, R.-C.. On the essence of “pre-conditioned” eigen-algorithms. (to be submitted).Google Scholar
[24]Zhou, Y. and Saad, Y.. A Chebyshev-Davidson algorithm for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl., 29(3):954–971, 2007.Google Scholar
[25]Zhou, Y., Saad, Y., Tiago, M. L., and Chelikowsky, J. R.. Parallel self-consistent-field calculations using Chebyshev-filtered subspace acceleration. Phys. Rev. E, 74(6):066704, 2006.Google Scholar