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Title: Real time time-dependent density functional theory using higher order finite-element methods

Abstract

We present a computationally efficient approach to solve the time-dependent Kohn-Sham equations in real-time using higher-order finite-element spatial discretization, applicable to both pseudopotential and all-electron calculations. To this end, we develop an a priori mesh adaption technique, based on the semi-discrete (discrete in space but continuous in time) error estimate on the time-dependent Kohn-Sham orbitals, to construct an efficient finite-element discretization. Subsequently, we obtain the full-discrete error estimate to guide our choice of the time-step. We employ spectral finite-elements along with special reduced order quadrature to render the overlap matrix diagonal, thereby simplifying the inversion of the overlap matrix that features in the evaluation of the discrete time-evolution operator. We use the second-order Magnus operator as the time-evolution operator, wherein the action of the discrete Magnus operator, expressed as exponential of a matrix, on the Kohn-Sham orbitals is obtained efficiently through an adaptive Lanczos iteration. We observe close to optimal rates of convergence of the dipole moment with respect to spatial and temporal discretization, for both pseudopotential and all-electron calculations. We demonstrate a staggering 100-fold reduction in the computational time afforded by higher-order finite-elements over linear finite-elements, for both pseudopotential and all-electron calculations. Further, for similar level of accuracy, wemore » obtain significant computational savings by our approach as compared to state-of-the-art finite-difference methods. We also demonstrate the competence of higher-order finite-elements for all-electron benchmark systems. Lastly, we observe good parallel scalability of the proposed method on many hundreds of processors.« less

Authors:
 [1];  [1]
  1. Univ. of Michigan, Ann Arbor, MI (United States)
Publication Date:
Research Org.:
Univ. of Michigan, Ann Arbor, MI (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF); US Army Research Office (ARO)
OSTI Identifier:
1564082
Alternate Identifier(s):
OSTI ID: 1656745
Grant/Contract Number:  
SC0017380; ACI-1053575; AC02-05CH11231; W911NF-15-1-0158
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review B
Additional Journal Information:
Journal Volume: 100; Journal Issue: 11; Journal ID: ISSN 2469-9950
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
36 MATERIALS SCIENCE; Electronic structure; Ab initio calculations; Finite-element method; Time-dependent DFT; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Kanungo, Bikash, and Gavini, Vikram. Real time time-dependent density functional theory using higher order finite-element methods. United States: N. p., 2019. Web. doi:10.1103/PhysRevB.100.115148.
Kanungo, Bikash, & Gavini, Vikram. Real time time-dependent density functional theory using higher order finite-element methods. United States. doi:10.1103/PhysRevB.100.115148.
Kanungo, Bikash, and Gavini, Vikram. Mon . "Real time time-dependent density functional theory using higher order finite-element methods". United States. doi:10.1103/PhysRevB.100.115148. https://www.osti.gov/servlets/purl/1564082.
@article{osti_1564082,
title = {Real time time-dependent density functional theory using higher order finite-element methods},
author = {Kanungo, Bikash and Gavini, Vikram},
abstractNote = {We present a computationally efficient approach to solve the time-dependent Kohn-Sham equations in real-time using higher-order finite-element spatial discretization, applicable to both pseudopotential and all-electron calculations. To this end, we develop an a priori mesh adaption technique, based on the semi-discrete (discrete in space but continuous in time) error estimate on the time-dependent Kohn-Sham orbitals, to construct an efficient finite-element discretization. Subsequently, we obtain the full-discrete error estimate to guide our choice of the time-step. We employ spectral finite-elements along with special reduced order quadrature to render the overlap matrix diagonal, thereby simplifying the inversion of the overlap matrix that features in the evaluation of the discrete time-evolution operator. We use the second-order Magnus operator as the time-evolution operator, wherein the action of the discrete Magnus operator, expressed as exponential of a matrix, on the Kohn-Sham orbitals is obtained efficiently through an adaptive Lanczos iteration. We observe close to optimal rates of convergence of the dipole moment with respect to spatial and temporal discretization, for both pseudopotential and all-electron calculations. We demonstrate a staggering 100-fold reduction in the computational time afforded by higher-order finite-elements over linear finite-elements, for both pseudopotential and all-electron calculations. Further, for similar level of accuracy, we obtain significant computational savings by our approach as compared to state-of-the-art finite-difference methods. We also demonstrate the competence of higher-order finite-elements for all-electron benchmark systems. Lastly, we observe good parallel scalability of the proposed method on many hundreds of processors.},
doi = {10.1103/PhysRevB.100.115148},
journal = {Physical Review B},
number = 11,
volume = 100,
place = {United States},
year = {2019},
month = {9}
}

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Works referenced in this record:

Real-time time-dependent density functional theory approach for frequency-dependent nonlinear optical response in photonic molecules
journal, October 2007

  • Takimoto, Y.; Vila, F. D.; Rehr, J. J.
  • The Journal of Chemical Physics, Vol. 127, Issue 15
  • DOI: 10.1063/1.2790014

Memory in Time-Dependent Density Functional Theory
journal, June 2002


Time-dependent local-density approximation in real time
journal, August 1996


Finite element method for solving Kohn–Sham equations based on self-adaptive tetrahedral mesh
journal, July 2008


Modeling Fast Electron Dynamics with Real-Time Time-Dependent Density Functional Theory: Application to Small Molecules and Chromophores
journal, April 2011

  • Lopata, Kenneth; Govind, Niranjan
  • Journal of Chemical Theory and Computation, Vol. 7, Issue 5
  • DOI: 10.1021/ct200137z

Partition of unity finite element method for quantum mechanical materials calculations
journal, February 2017


Architecture of Qbox: A scalable first-principles molecular dynamics code
journal, January 2008

  • Gygi, F.
  • IBM Journal of Research and Development, Vol. 52, Issue 1.2
  • DOI: 10.1147/rd.521.0137

Daubechies wavelets as a basis set for density functional pseudopotential calculations
journal, July 2008

  • Genovese, Luigi; Neelov, Alexey; Goedecker, Stefan
  • The Journal of Chemical Physics, Vol. 129, Issue 1
  • DOI: 10.1063/1.2949547

Finite-element approach to band-structure analysis
journal, May 1986


Local density-functional theory of frequency-dependent linear response
journal, December 1985


Error estimation and adaptive meshing in strongly nonlinear dynamic problems
journal, April 1999


Finite-element method for electronic structure
journal, March 1989

  • White, Steven R.; Wilkins, John W.; Teter, Michael P.
  • Physical Review B, Vol. 39, Issue 9
  • DOI: 10.1103/PhysRevB.39.5819

Adaptive Finite Element Approximations for Kohn--Sham Models
journal, January 2014

  • Chen, Huajie; Dai, Xiaoying; Gong, Xingao
  • Multiscale Modeling & Simulation, Vol. 12, Issue 4
  • DOI: 10.1137/130916096

Numerical analysis of finite dimensional approximations of Kohn–Sham models
journal, September 2011

  • Chen, Huajie; Gong, Xingao; He, Lianhua
  • Advances in Computational Mathematics, Vol. 38, Issue 2
  • DOI: 10.1007/s10444-011-9235-y

Oscillator strength distribution of in the time-dependent density functional theory
journal, November 2009


Gaussian finite-element mixed-basis method for electronic structure calculations
journal, January 2005


Plane-wave pseudopotential implementation of explicit integrators for time-dependent Kohn-Sham equations in large-scale simulations
journal, December 2012

  • Schleife, André; Draeger, Erik W.; Kanai, Yosuke
  • The Journal of Chemical Physics, Vol. 137, Issue 22
  • DOI: 10.1063/1.4758792

octopus: a tool for the application of time-dependent density functional theory
journal, September 2006

  • Castro, Alberto; Appel, Heiko; Oliveira, Micael
  • physica status solidi (b), Vol. 243, Issue 11
  • DOI: 10.1002/pssb.200642067

An h-adaptive finite element solver for the calculations of the electronic structures
journal, May 2012


Center of Mass and Relative Motion in Time Dependent Density Functional Theory
journal, April 1995


Efficacious Form for Model Pseudopotentials
journal, May 1982


Time-dependent quantum transport: An exact formulation based on TDDFT
journal, July 2004


Finite element approach for density functional theory calculations on locally-refined meshes
journal, May 2007

  • Fattebert, J. -L.; Hornung, R. D.; Wissink, A. M.
  • Journal of Computational Physics, Vol. 223, Issue 2
  • DOI: 10.1016/j.jcp.2006.10.013

A Kohn–Sham equation solver based on hexahedral finite elements
journal, April 2012


Configurational forces in electronic structure calculations using Kohn-Sham density functional theory
journal, April 2018


Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory
journal, March 2009

  • Stein, Tamar; Kronik, Leeor; Baer, Roi
  • Journal of the American Chemical Society, Vol. 131, Issue 8
  • DOI: 10.1021/ja8087482

Demonstration of initial-state dependence in time-dependent density-functional theory
journal, March 2001


DFT-FE – A massively parallel adaptive finite-element code for large-scale density functional theory calculations
journal, January 2020


The Magnus expansion and some of its applications
journal, January 2009


NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations
journal, September 2010

  • Valiev, M.; Bylaska, E. J.; Govind, N.
  • Computer Physics Communications, Vol. 181, Issue 9, p. 1477-1489
  • DOI: 10.1016/j.cpc.2010.04.018

Adaptive Finite Element Method for Solving the Exact Kohn−Sham Equation of Density Functional Theory
journal, March 2009

  • Bylaska, Eric J.; Holst, Michael; Weare, John H.
  • Journal of Chemical Theory and Computation, Vol. 5, Issue 4
  • DOI: 10.1021/ct800350j

Real-space grid implementation of the projector augmented wave method
journal, January 2005


Large-Scale Electronic-Structure Calculations Based on the Adaptive Finite-Element Method
journal, November 1998

  • Tsuchida, Eiji; Tsukada, Masaru
  • Journal of the Physical Society of Japan, Vol. 67, Issue 11
  • DOI: 10.1143/JPSJ.67.3844

All-electron Kohn–Sham density functional theory on hierarchic finite element spaces
journal, October 2013


Self-Consistent Equations Including Exchange and Correlation Effects
journal, November 1965


Efficient pseudopotentials for plane-wave calculations
journal, January 1991


Time-dependent Density Functional Results for the Dynamic Hyperpolarizability of C 60
journal, April 1997

  • van Gisbergen, S. J. A.; Snijders, J. G.; Baerends, E. J.
  • Physical Review Letters, Vol. 78, Issue 16
  • DOI: 10.1103/PhysRevLett.78.3097

Higher-order adaptive finite-element methods for orbital-free density functional theory
journal, August 2012

  • Motamarri, Phani; Iyer, Mrinal; Knap, Jaroslaw
  • Journal of Computational Physics, Vol. 231, Issue 20
  • DOI: 10.1016/j.jcp.2012.04.036

Non-periodic finite-element formulation of Kohn–Sham density functional theory
journal, February 2010

  • Suryanarayana, Phanish; Gavini, Vikram; Blesgen, Thomas
  • Journal of the Mechanics and Physics of Solids, Vol. 58, Issue 2
  • DOI: 10.1016/j.jmps.2009.10.002

Exponential Integrators for Large Systems of Differential Equations
journal, September 1998

  • Hochbruck, Marlis; Lubich, Christian; Selhofer, Hubert
  • SIAM Journal on Scientific Computing, Vol. 19, Issue 5
  • DOI: 10.1137/S1064827595295337

Excitation Energies from Time-Dependent Density-Functional Theory
journal, February 1996


Ab initio simulations of sodium using time-dependent density-functional theory
journal, November 1992


All-electron time-dependent density functional theory with finite elements: Time-propagation approach
journal, October 2011

  • Lehtovaara, Lauri; Havu, Ville; Puska, Martti
  • The Journal of Chemical Physics, Vol. 135, Issue 15
  • DOI: 10.1063/1.3651239

Simulating molecular conductance using real-time density functional theory
journal, October 2006


Excitations in Time-Dependent Density-Functional Theory
journal, January 2003


Time-dependent density-functional theory in the projector augmented-wave method
journal, June 2008

  • Walter, Michael; Häkkinen, Hannu; Lehtovaara, Lauri
  • The Journal of Chemical Physics, Vol. 128, Issue 24
  • DOI: 10.1063/1.2943138

Self-consistent-field calculations using Chebyshev-filtered subspace iteration
journal, November 2006

  • Zhou, Yunkai; Saad, Yousef; Tiago, Murilo L.
  • Journal of Computational Physics, Vol. 219, Issue 1
  • DOI: 10.1016/j.jcp.2006.03.017

The SIESTA method for ab initio order- N materials simulation
journal, March 2002

  • Soler, José M.; Artacho, Emilio; Gale, Julian D.
  • Journal of Physics: Condensed Matter, Vol. 14, Issue 11
  • DOI: 10.1088/0953-8984/14/11/302

Density-functional approach to nonlinear-response coefficients of solids
journal, June 1989


Finite-element methods in electronic-structure theory
journal, March 2001


Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach
journal, May 1999


Density-Functional Theory for Time-Dependent Systems
journal, March 1984


Time-dependent electron localization function
journal, January 2005


Higher-order adaptive finite-element methods for Kohn–Sham density functional theory
journal, November 2013


Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration
journal, December 2006


Large-scale all-electron density functional theory calculations using an enriched finite-element basis
journal, January 2017


Spectrum-splitting approach for Fermi-operator expansion in all-electron Kohn-Sham DFT calculations
journal, January 2017


A method for ab initio nonlinear electron-density evolution
journal, February 2001

  • Baer, Roi; Gould, Recca
  • The Journal of Chemical Physics, Vol. 114, Issue 8
  • DOI: 10.1063/1.1342761

All-electron density functional theory and time-dependent density functional theory with high-order finite elements
journal, January 2009

  • Lehtovaara, Lauri; Havu, Ville; Puska, Martti
  • The Journal of Chemical Physics, Vol. 131, Issue 5
  • DOI: 10.1063/1.3176508

Ground State of the Electron Gas by a Stochastic Method
journal, August 1980


Wavelet-based linear-response time-dependent density-functional theory
journal, June 2012


Finite element methods in ab initio electronic structure calculations
journal, April 2005

  • Pask, J. E.; Sterne, P. A.
  • Modelling and Simulation in Materials Science and Engineering, Vol. 13, Issue 3
  • DOI: 10.1088/0965-0393/13/3/R01

Propagators for the time-dependent Kohn–Sham equations
journal, August 2004

  • Castro, Alberto; Marques, Miguel A. L.; Rubio, Angel
  • The Journal of Chemical Physics, Vol. 121, Issue 8
  • DOI: 10.1063/1.1774980

Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds: EFFICIENT INTEGRATION OF FUNCTIONS WITH SHARP GRADIENTS AND CUSPS
journal, June 2012

  • Mousavi, S. E.; Pask, J. E.; Sukumar, N.
  • International Journal for Numerical Methods in Engineering, Vol. 91, Issue 4
  • DOI: 10.1002/nme.4267

Intense few-cycle laser fields: Frontiers of nonlinear optics
journal, April 2000


Real-time adaptive finite element solution of time-dependent Kohn–Sham equation
journal, January 2015


Time-dependent quantum transport: A practical scheme using density functional theory
journal, July 2005


On the Non‐Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals
journal, March 1950

  • Löwdin, Per‐Olov
  • The Journal of Chemical Physics, Vol. 18, Issue 3
  • DOI: 10.1063/1.1747632

On Magnus Integrators for Time-Dependent Schrödinger Equations
journal, January 2003