Numerical methods for the inverse problem of density functional theory
Abstract
Here, the inverse problem of Kohn–Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems.
- Authors:
-
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Purdue Univ., West Lafayette, IN (United States)
- Publication Date:
- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Org.:
- NSF Funded from Graduate School; USDOE
- OSTI Identifier:
- 1421863
- Report Number(s):
- SAND-2017-7018J
Journal ID: ISSN 0020-7608; 655000
- Grant/Contract Number:
- AC04-94AL85000
- Resource Type:
- Accepted Manuscript
- Journal Name:
- International Journal of Quantum Chemistry
- Additional Journal Information:
- Journal Volume: 118; Journal Issue: 1; Journal ID: ISSN 0020-7608
- Publisher:
- Wiley
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; density functional theory; inverse problems; PDE-constrained optimization
Citation Formats
Jensen, Daniel S., and Wasserman, Adam. Numerical methods for the inverse problem of density functional theory. United States: N. p., 2017.
Web. doi:10.1002/qua.25425.
Jensen, Daniel S., & Wasserman, Adam. Numerical methods for the inverse problem of density functional theory. United States. https://doi.org/10.1002/qua.25425
Jensen, Daniel S., and Wasserman, Adam. Mon .
"Numerical methods for the inverse problem of density functional theory". United States. https://doi.org/10.1002/qua.25425. https://www.osti.gov/servlets/purl/1421863.
@article{osti_1421863,
title = {Numerical methods for the inverse problem of density functional theory},
author = {Jensen, Daniel S. and Wasserman, Adam},
abstractNote = {Here, the inverse problem of Kohn–Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems.},
doi = {10.1002/qua.25425},
journal = {International Journal of Quantum Chemistry},
number = 1,
volume = 118,
place = {United States},
year = {Mon Jul 17 00:00:00 EDT 2017},
month = {Mon Jul 17 00:00:00 EDT 2017}
}
Web of Science
Works referenced in this record:
Libxc: A library of exchange and correlation functionals for density functional theory
journal, October 2012
- Marques, Miguel A. L.; Oliveira, Micael J. T.; Burnus, Tobias
- Computer Physics Communications, Vol. 183, Issue 10
PDE-constrained optimization with error estimation and control
journal, April 2014
- Hicken, J. E.; Alonso, J. J.
- Journal of Computational Physics, Vol. 263
Density-Based Partitioning Methods for Ground-State Molecular Calculations
journal, July 2014
- Nafziger, Jonathan; Wasserman, Adam
- The Journal of Physical Chemistry A, Vol. 118, Issue 36
Generation of finite difference formulas on arbitrarily spaced grids
journal, January 1988
- Fornberg, Bengt
- Mathematics of Computation, Vol. 51, Issue 184
The Morse oscillator in position space, momentum space, and phase space
journal, April 1988
- Dahl, Jens Peder; Springborg, Michael
- The Journal of Chemical Physics, Vol. 88, Issue 7
Exchange-correlation potential with correct asymptotic behavior
journal, April 1994
- van Leeuwen, R.; Baerends, E. J.
- Physical Review A, Vol. 49, Issue 4
Boundaryless beam propagation
journal, January 1996
- Ladouceur, François
- Optics Letters, Vol. 21, Issue 1
ImprovedSCF convergence acceleration
journal, January 1982
- Pulay, P.
- Journal of Computational Chemistry, Vol. 3, Issue 4
Reference electronic structure calculations in one dimension
journal, January 2012
- Wagner, Lucas O.; Stoudenmire, E. M.; Burke, Kieron
- Physical Chemistry Chemical Physics, Vol. 14, Issue 24
Determination of Kohn–Sham effective potentials from electron densities using the differential virial theorem
journal, October 2012
- Ryabinkin, Ilya G.; Staroverov, Viktor N.
- The Journal of Chemical Physics, Vol. 137, Issue 16
Variational finite-difference representation of the kinetic energy operator
journal, October 2001
- Maragakis, P.; Soler, José; Kaxiras, Efthimios
- Physical Review B, Vol. 64, Issue 19
Kohn-Sham calculations with the exact functional
journal, July 2014
- Wagner, Lucas O.; Baker, Thomas E.; Stoudenmire, E. M.
- Physical Review B, Vol. 90, Issue 4
Dual consistency and functional accuracy: a finite-difference perspective
journal, January 2014
- Hicken, J. E.; Zingg, D. W.
- Journal of Computational Physics, Vol. 256
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
Numerical Methods for Evaluating the Derivatives of Eigenvalues and Eigenvectors
journal, June 1975
- Rudisill, Carl S.; Chu, Yee-Yeen
- AIAA Journal, Vol. 13, Issue 6
DFT in a nutshell
journal, July 2012
- Burke, Kieron; Wagner, Lucas O.
- International Journal of Quantum Chemistry, Vol. 113, Issue 2
An efficient method for finding the minimum of a function of several variables without calculating derivatives
journal, February 1964
- Powell, M. J. D.
- The Computer Journal, Vol. 7, Issue 2
Controlling the Dynamics of Many-Electron Systems from First Principles: A Combination of Optimal Control and Time-Dependent Density-Functional Theory
journal, October 2012
- Castro, A.; Werschnik, J.; Gross, E. K. U.
- Physical Review Letters, Vol. 109, Issue 15
A direct optimization method for calculating density functionals and exchange–correlation potentials from electron densities
journal, January 2003
- Wu, Qin; Yang, Weitao
- The Journal of Chemical Physics, Vol. 118, Issue 6
One-dimensional mimicking of electronic structure: The case for exponentials
journal, June 2015
- Baker, Thomas E.; Stoudenmire, E. Miles; Wagner, Lucas O.
- Physical Review B, Vol. 91, Issue 23
Fragment-based treatment of delocalization and static correlation errors in density-functional theory
journal, December 2015
- Nafziger, Jonathan; Wasserman, Adam
- The Journal of Chemical Physics, Vol. 143, Issue 23
A survey of the parallel performance and accuracy of Poisson solvers for electronic structure calculations
journal, November 2013
- García-Risueño, Pablo; Alberdi-Rodriguez, Joseba; Oliveira, Micael J. T.
- Journal of Computational Chemistry, Vol. 35, Issue 6
Newton-Type Minimization via the Lanczos Method
journal, August 1984
- Nash, Stephen G.
- SIAM Journal on Numerical Analysis, Vol. 21, Issue 4
Demonstration of initial-state dependence in time-dependent density-functional theory
journal, March 2001
- Maitra, Neepa T.; Burke, Kieron
- Physical Review A, Vol. 63, Issue 4
An Introduction to the Adjoint Approach to Design
journal, December 2000
- Giles, Michael B.; Pierce, Niles A.
- Flow, Turbulence and Combustion, Vol. 65, Issue 3/4, p. 393-415
From electron densities to Kohn-Sham kinetic energies, orbital energies, exchange-correlation potentials, and exchange-correlation energies
journal, September 1994
- Zhao, Qingsheng; Morrison, Robert C.; Parr, Robert G.
- Physical Review A, Vol. 50, Issue 3
Computational chemistry from the perspective of numerical analysis
journal, April 2005
- Le Bris, Claude
- Acta Numerica, Vol. 14
Computational Methods for Inverse Problems
book, January 2002
- Vogel, Curtis R.
- Frontiers in Applied Mathematics
Optimized effective potential made simple: Orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potential
journal, July 2003
- Kümmel, Stephan; Perdew, John P.
- Physical Review B, Vol. 68, Issue 3
Optimized effective potentials from electron densities in finite basis sets
journal, November 2007
- Bulat, Felipe A.; Heaton-Burgess, Tim; Cohen, Aron J.
- The Journal of Chemical Physics, Vol. 127, Issue 17
Gradient Type Optimization Methods For Electronic Structure Calculations
journal, January 2014
- Zhang, Xin; Zhu, Jinwei; Wen, Zaiwen
- SIAM Journal on Scientific Computing, Vol. 36, Issue 3
Density functional approach to quantum lattice systems
journal, February 1985
- Chayes, J. T.; Chayes, L.; Ruskai, Mary Beth
- Journal of Statistical Physics, Vol. 38, Issue 3-4
Smoothing and Differentiation of Data by Simplified Least Squares Procedures.
journal, July 1964
- Savitzky, Abraham.; Golay, M. J. E.
- Analytical Chemistry, Vol. 36, Issue 8
Numerical density-to-potential inversions in time-dependent density functional theory
journal, January 2016
- Jensen, Daniel S.; Wasserman, Adam
- Physical Chemistry Chemical Physics, Vol. 18, Issue 31
Analysis of Costate Discretizations in Parameter Estimation for Linear Evolution Equations
journal, January 1995
- Vogel, C. R.; Wade, J. G.
- SIAM Journal on Control and Optimization, Vol. 33, Issue 1
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming
journal, April 2005
- Wächter, Andreas; Biegler, Lorenz T.
- Mathematical Programming, Vol. 106, Issue 1
PDE-constrained optimization with error estimation and control
conference, November 2012
- Hicken, Jason; Alonso, Juan
- 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
A survey of the parallel performance and the accuracy of Poisson solvers for electronic structure calculations
preprint, January 2012
- García-Risueño, Pablo; Alberdi-Rodriguez, Joseba; Oliveira, Micael J. T.
- arXiv
Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems
text, January 2015
- Andrade, Xavier; Strubbe, David A.; De Giovannini, Umberto
- arXiv
One Dimensional Mimicking of Electronic Structure: The Case for Exponentials
text, January 2015
- Baker, Thomas E.; Stoudenmire, E. Miles; Wagner, Lucas O.
- arXiv
Variational finite-difference representation of the kinetic energy operator
text, January 2001
- Maragakis, P.; Soler, Jose M.; Kaxiras, Efthimios
- arXiv
Works referencing / citing this record:
Inverting the external‐potential‐to‐electron density map in systems of electrons by minimization of a least‐squares fit functional for analytic potentials
journal, September 2019
- Pérez‐Jordá, José M.
- International Journal of Quantum Chemistry, Vol. 120, Issue 1
Exact exchange-correlation potentials from ground-state electron densities
journal, October 2019
- Kanungo, Bikash; Zimmerman, Paul M.; Gavini, Vikram
- Nature Communications, Vol. 10, Issue 1
Non-additive non-interacting kinetic energy of rare gas dimers
journal, March 2018
- Jiang, Kaili; Nafziger, Jonathan; Wasserman, Adam
- The Journal of Chemical Physics, Vol. 148, Issue 10
Accurate embedding through potential reconstruction: A comparison of different strategies
journal, August 2018
- Schnieders, David; Neugebauer, Johannes
- The Journal of Chemical Physics, Vol. 149, Issue 5
Constructing a non-additive non-interacting kinetic energy functional approximation for covalent bonds from exact conditions
journal, October 2018
- Jiang, Kaili; Nafziger, Jonathan; Wasserman, Adam
- The Journal of Chemical Physics, Vol. 149, Issue 16
Adiabatic connection in density functional theory in two-dimensions: A semi-analytic wavefunction based study for two-electron atomic systems
journal, November 2019
- Singh, Rabeet; Patra, Bikash; Patra, Abhilash
- The Journal of Chemical Physics, Vol. 151, Issue 20
Description of intermolecular charge transfer with subsystem density-functional theory
journal, October 2019
- Schulz, Anika; Jacob, Christoph R.
- The Journal of Chemical Physics, Vol. 151, Issue 13
Universal nature of different methods of obtaining the exact Kohn–Sham exchange-correlation potential for a given density
journal, March 2019
- Kumar, Ashish; Singh, Rabeet; Harbola, Manoj K.
- Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 52, Issue 7
Improvement of functionals in density functional theory by the inverse Kohn–Sham method and density functional perturbation theory
journal, November 2019
- Naito, Tomoya; Ohashi, Daisuke; Liang, Haozhao
- Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 52, Issue 24
Exchange-correlation magnetic fields in spin-density-functional theory
journal, September 2019
- Pluhar, Edward A.; Ullrich, Carsten A.
- Physical Review B, Vol. 100, Issue 12
First step in the nuclear inverse Kohn-Sham problem: From densities to potentials
journal, February 2020
- Accorto, G.; Brandolini, P.; Marino, F.
- Physical Review C, Vol. 101, Issue 2
Numerical construction of the density-potential mapping
journal, October 2018
- Nielsen, Soeren E. B.; Ruggenthaler, Michael; van Leeuwen, Robert
- The European Physical Journal B, Vol. 91, Issue 10
Non-additive Non-interacting Kinetic Energy of Rare Gas Dimers
text, January 2017
- Jiang, Kaili; Nafziger, Jonathan; Wasserman, Adam
- arXiv
Constructing a Non-additive Non-interacting Kinetic Energy Functional Approximation for Covalent Bonds from Exact Conditions
text, January 2018
- Jiang, Kaili; Nafziger, Jonathan; Wasserman, Adam
- arXiv
Exchange-correlation magnetic fields in spin-density-functional theory
text, January 2019
- Pluhar, Edward A.; Ullrich, Carsten A.
- arXiv
Kinetic-Energy Density-Functional Theory on a Lattice
journal, June 2018
- Theophilou, Iris; Buchholz, Florian; Eich, F. G.
- Journal of Chemical Theory and Computation, Vol. 14, Issue 8
Self-Consistent Density-Functional Embedding: A Novel Approach for Density-Functional Approximations
journal, September 2019
- Mordovina, Uliana; Reinhard, Teresa E.; Theophilou, Iris
- Journal of Chemical Theory and Computation, Vol. 15, Issue 10
Exact exchange-correlation potentials from ground-state electron densities
journal, October 2019
- Kanungo, Bikash; Zimmerman, Paul M.; Gavini, Vikram
- Nature Communications, Vol. 10, Issue 1