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Title: Linear response time-dependent density functional theory of the Hubbard dimer

Abstract

The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequency-dependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequency-dependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the ground-state exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. In conclusion, many exact results, limits, and expansions about those limits are given in the Appendices.

Authors:
 [1];  [1];  [2];  [3]
  1. Univ. de Oviedo, Oviedo (Spain); Consejo Superior de Investigaciones Cientificas (CSIC), Madrid (Spain). Nanomaterials and Nanotechnology Research Center
  2. City Univ. of New York (CUNY), NY (United States)
  3. Univ. of California, Irvine, CA (United States)
Publication Date:
Research Org.:
Univ. of California, Irvine, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); National Science Foundation (NSF)
OSTI Identifier:
1595120
Grant/Contract Number:  
[FG02-08ER46496; FIS2012-34858]
Resource Type:
Accepted Manuscript
Journal Name:
European Physical Journal. B, Condensed Matter and Complex Systems
Additional Journal Information:
[ Journal Volume: 91; Journal Issue: 7]; Journal ID: ISSN 1434-6028
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Carrascal, Diego J., Ferrer, Jaime, Maitra, Neepa, and Burke, Kieron. Linear response time-dependent density functional theory of the Hubbard dimer. United States: N. p., 2018. Web. doi:10.1140/epjb/e2018-90114-9.
Carrascal, Diego J., Ferrer, Jaime, Maitra, Neepa, & Burke, Kieron. Linear response time-dependent density functional theory of the Hubbard dimer. United States. doi:10.1140/epjb/e2018-90114-9.
Carrascal, Diego J., Ferrer, Jaime, Maitra, Neepa, and Burke, Kieron. Mon . "Linear response time-dependent density functional theory of the Hubbard dimer". United States. doi:10.1140/epjb/e2018-90114-9. https://www.osti.gov/servlets/purl/1595120.
@article{osti_1595120,
title = {Linear response time-dependent density functional theory of the Hubbard dimer},
author = {Carrascal, Diego J. and Ferrer, Jaime and Maitra, Neepa and Burke, Kieron},
abstractNote = {The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequency-dependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequency-dependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the ground-state exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. In conclusion, many exact results, limits, and expansions about those limits are given in the Appendices.},
doi = {10.1140/epjb/e2018-90114-9},
journal = {European Physical Journal. B, Condensed Matter and Complex Systems},
number = [7],
volume = [91],
place = {United States},
year = {2018},
month = {7}
}

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Cited by: 10 works
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Works referenced in this record:

Challenging adiabatic time-dependent density functional theory with a Hubbard dimer: the case of time-resolved long-range charge transfer
journal, January 2014

  • Fuks, Johanna I.; Maitra, Neepa T.
  • Phys. Chem. Chem. Phys., Vol. 16, Issue 28
  • DOI: 10.1039/C4CP00118D

Molecular excitation energies from time-dependent density functional theory
journal, April 2000


A long-range-corrected time-dependent density functional theory
journal, May 2004

  • Tawada, Yoshihiro; Tsuneda, Takao; Yanagisawa, Susumu
  • The Journal of Chemical Physics, Vol. 120, Issue 18
  • DOI: 10.1063/1.1688752

Asymptotic correction of the exchange–correlation kernel of time-dependent density functional theory for long-range charge-transfer excitations
journal, July 2004

  • Gritsenko, Oleg; Baerends, Evert Jan
  • The Journal of Chemical Physics, Vol. 121, Issue 2
  • DOI: 10.1063/1.1759320

Time-dependent exchange-correlation functional for a Hubbard dimer: Quantifying nonadiabatic effects
journal, December 2013


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


Theory of the Insulating State
journal, January 1964


Undoing static correlation: Long-range charge transfer in time-dependent density-functional theory
journal, June 2005

  • Maitra, Neepa T.
  • The Journal of Chemical Physics, Vol. 122, Issue 23
  • DOI: 10.1063/1.1924599

Landau parameters of almost-localized Fermi liquids
journal, June 1987


Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism
journal, May 1976


Relationship between long-range charge-transfer excitation energy error and integer discontinuity in Kohn–Sham theory
journal, December 2003

  • Tozer, David J.
  • The Journal of Chemical Physics, Vol. 119, Issue 24
  • DOI: 10.1063/1.1633756

Exchange-correlation kernel in time-dependent density functional theory
journal, October 2002


Time-dependent V-representability on lattice systems
journal, July 2008

  • Li, Yonghui; Ullrich, C. A.
  • The Journal of Chemical Physics, Vol. 129, Issue 4
  • DOI: 10.1063/1.2955733

Excitation energies of dissociating H2: A problematic case for the adiabatic approximation of time-dependent density functional theory
journal, November 2000

  • Gritsenko, O. V.; van Gisbergen, S. J. A.; Görling, A.
  • The Journal of Chemical Physics, Vol. 113, Issue 19
  • DOI: 10.1063/1.1318750

Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory
journal, July 1996


Double excitations within time-dependent density functional theory linear response
journal, April 2004

  • Maitra, Neepa T.; Zhang, Fan; Cave, Robert J.
  • The Journal of Chemical Physics, Vol. 120, Issue 13
  • DOI: 10.1063/1.1651060

The exchange-correlation energy of a metallic surface
journal, December 1975


Nonadiabatic Dynamics in Single-Electron Tunneling Devices with Time-Dependent Density-Functional Theory
journal, April 2018


Optical spectrum of a Hubbard chain
journal, September 1977


Exact time-dependent density functional theory for impurity models
journal, January 2013

  • Schmitteckert, Peter; Dzierzawa, Michael; Schwab, Peter
  • Physical Chemistry Chemical Physics, Vol. 15, Issue 15
  • DOI: 10.1039/c3cp44639e

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

Perspective: Fundamental aspects of time-dependent density functional theory
journal, June 2016

  • Maitra, Neepa T.
  • The Journal of Chemical Physics, Vol. 144, Issue 22
  • DOI: 10.1063/1.4953039

Assessment of dressed time-dependent density-functional theory for the low-lying valence states of 28 organic chromophores
journal, November 2011


Exact Kohn-Sham eigenstates versus quasiparticles in simple models of strongly correlated electrons
journal, January 2012


�ber die Zahl der Dispersionselektronen, die einem station�ren Zustand zugeordnet sind
journal, December 1925

  • Reiche, F.; Thomas, W.
  • Zeitschrift f�r Physik, Vol. 34, Issue 1
  • DOI: 10.1007/BF01328494

Time-dependent current density functional theory on a lattice
journal, January 2011


Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF
journal, February 2018

  • Li, Chen; Requist, Ryan; Gross, E. K. U.
  • The Journal of Chemical Physics, Vol. 148, Issue 8
  • DOI: 10.1063/1.5011663

Perspectives on double-excitations in TDDFT
journal, November 2011


Dynamical self-stabilization of the Mott insulator: Time evolution of the density and entanglement entropy of out-of-equilibrium cold fermion gases
journal, January 2011


Three-dimensional dynamics of a fermionic Mott wedding-cake in clean and disordered optical lattices
journal, September 2013

  • Kartsev, A.; Karlsson, D.; Privitera, A.
  • Scientific Reports, Vol. 3, Issue 1
  • DOI: 10.1038/srep02570

Dynamic polarizabilities and excitation spectra from a molecular implementation of time‐dependent density‐functional response theory: N 2 as a case study
journal, April 1996

  • Jamorski, Christine; Casida, Mark E.; Salahub, Dennis R.
  • The Journal of Chemical Physics, Vol. 104, Issue 13
  • DOI: 10.1063/1.471140

Dynamical Coulomb Blockade and the Derivative Discontinuity of Time-Dependent Density Functional Theory
journal, June 2010


Progress in Time-Dependent Density-Functional Theory
journal, May 2012


Bethe Ansatz Approach to the Kondo Effect within Density-Functional Theory
journal, February 2012


Analytic density functionals with initial-state dependence and memory
journal, August 2013


Adiabatic and local approximations for the Kohn-Sham potential in time-dependent Hubbard chains
journal, May 2014


On the determination of excitation energies using density functional theory
journal, January 2000

  • Tozer, David J.; Handy, Nicholas C.
  • Physical Chemistry Chemical Physics, Vol. 2, Issue 10
  • DOI: 10.1039/a910321j

Real-time, real-space implementation of the linear response time-dependent density-functional theory
journal, April 2006

  • Yabana, K.; Nakatsukasa, T.; Iwata, J. -I.
  • physica status solidi (b), Vol. 243, Issue 5
  • DOI: 10.1002/pssb.200642005

�ber die Gesamtst�rke der von einem Zustande ausgehenden Absorptionslinien
journal, December 1925


Density functionals and model Hamiltonians: Pillars of many-particle physics
journal, July 2013


A dressed TDDFT treatment of the 21Ag states of butadiene and hexatriene
journal, May 2004


Time-dependent density functional theory on a lattice
journal, September 2012


Generalized Kohn-Sham system in one-matrix functional theory
journal, June 2008


Frequency Dependence of the Exact Exchange-Correlation Kernel of Time-Dependent Density-Functional Theory
journal, February 2014


Correlated electrons in high-temperature superconductors
journal, July 1994


Inhomogeneous Electron Gas
journal, November 1964


The calculations of excited-state properties with Time-Dependent Density Functional Theory
journal, January 2013

  • Adamo, Carlo; Jacquemin, Denis
  • Chem. Soc. Rev., Vol. 42, Issue 3
  • DOI: 10.1039/C2CS35394F

On the mapping of time-dependent densities onto potentials in quantum mechanics
journal, January 2008

  • Baer, Roi
  • The Journal of Chemical Physics, Vol. 128, Issue 4
  • DOI: 10.1063/1.2822124

Charge transfer in time-dependent density-functional theory: Insights from the asymmetric Hubbard dimer
journal, June 2014


Extensive TD-DFT Benchmark: Singlet-Excited States of Organic Molecules
journal, August 2009

  • Jacquemin, Denis; Wathelet, Valérie; Perpète, Eric A.
  • Journal of Chemical Theory and Computation, Vol. 5, Issue 9
  • DOI: 10.1021/ct900298e

Application of the dressed time-dependent density functional theory for the excited states of linear polyenes
journal, April 2009

  • Mazur, Grzegorz; Włodarczyk, Radosław
  • Journal of Computational Chemistry, Vol. 30, Issue 5
  • DOI: 10.1002/jcc.21102

Twenty-five Years of High-Temperature Superconductivity – A Personal Review
journal, July 2013


Does density functional theory contribute to the understanding of excited states of unsaturated organic compounds?
journal, October 1999


Tuned Range-Separated Hybrids in Density Functional Theory
journal, March 2010


Dressed TDDFT study of low-lying electronic excited states in selected linear polyenes and diphenylopolyenes
journal, September 2010

  • Mazur, Grzegorz; Makowski, Marcin; Włodarczyk, Radosław
  • International Journal of Quantum Chemistry, Vol. 111, Issue 4
  • DOI: 10.1002/qua.22876

Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange
journal, August 2003

  • Dreuw, Andreas; Weisman, Jennifer L.; Head-Gordon, Martin
  • The Journal of Chemical Physics, Vol. 119, Issue 6
  • DOI: 10.1063/1.1590951

Doping a Mott insulator: Physics of high-temperature superconductivity
journal, January 2006

  • Lee, Patrick A.; Nagaosa, Naoto; Wen, Xiao-Gang
  • Reviews of Modern Physics, Vol. 78, Issue 1
  • DOI: 10.1103/RevModPhys.78.17

Adiabatic approximation in time-dependent reduced-density-matrix functional theory
journal, April 2010


Time-Dependent Density-Functional Theory and Strongly Correlated Systems: Insight from Numerical Studies
journal, October 2008


Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals
journal, April 2012

  • Kronik, Leeor; Stein, Tamar; Refaely-Abramson, Sivan
  • Journal of Chemical Theory and Computation, Vol. 8, Issue 5
  • DOI: 10.1021/ct2009363

Organic Electronic Materials: Recent Advances in the DFT Description of the Ground and Excited States Using Tuned Range-Separated Hybrid Functionals
journal, April 2014

  • Körzdörfer, Thomas; Brédas, Jean-Luc
  • Accounts of Chemical Research, Vol. 47, Issue 11
  • DOI: 10.1021/ar500021t

Resonant nonlinear polarizabilities in the time-dependent density functional theory
journal, November 2003

  • Tretiak, S.; Chernyak, V.
  • The Journal of Chemical Physics, Vol. 119, Issue 17
  • DOI: 10.1063/1.1614240

Towards a Description of the Kondo Effect Using Time-Dependent Density-Functional Theory
journal, November 2011


    Works referencing / citing this record:

    Application of the dressed time-dependent density functional theory for the excited states of linear polyenes
    journal, April 2009

    • Mazur, Grzegorz; Włodarczyk, Radosław
    • Journal of Computational Chemistry, Vol. 30, Issue 5
    • DOI: 10.1002/jcc.21102

    Real-time, real-space implementation of the linear response time-dependent density-functional theory
    journal, April 2006

    • Yabana, K.; Nakatsukasa, T.; Iwata, J. -I.
    • physica status solidi (b), Vol. 243, Issue 5
    • DOI: 10.1002/pssb.200642005

    Dressed TDDFT study of low-lying electronic excited states in selected linear polyenes and diphenylopolyenes
    journal, September 2010

    • Mazur, Grzegorz; Makowski, Marcin; Włodarczyk, Radosław
    • International Journal of Quantum Chemistry, Vol. 111, Issue 4
    • DOI: 10.1002/qua.22876

    The exchange-correlation energy of a metallic surface
    journal, December 1975


    Assessment of dressed time-dependent density-functional theory for the low-lying valence states of 28 organic chromophores
    journal, November 2011


    Perspectives on double-excitations in TDDFT
    journal, November 2011


    A dressed TDDFT treatment of the 21Ag states of butadiene and hexatriene
    journal, May 2004


    Density functionals and model Hamiltonians: Pillars of many-particle physics
    journal, July 2013


    Organic Electronic Materials: Recent Advances in the DFT Description of the Ground and Excited States Using Tuned Range-Separated Hybrid Functionals
    journal, April 2014

    • Körzdörfer, Thomas; Brédas, Jean-Luc
    • Accounts of Chemical Research, Vol. 47, Issue 11
    • DOI: 10.1021/ar500021t

    Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals
    journal, April 2012

    • Kronik, Leeor; Stein, Tamar; Refaely-Abramson, Sivan
    • Journal of Chemical Theory and Computation, Vol. 8, Issue 5
    • DOI: 10.1021/ct2009363

    Extensive TD-DFT Benchmark: Singlet-Excited States of Organic Molecules
    journal, August 2009

    • Jacquemin, Denis; Wathelet, Valérie; Perpète, Eric A.
    • Journal of Chemical Theory and Computation, Vol. 5, Issue 9
    • DOI: 10.1021/ct900298e

    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

    Three-dimensional dynamics of a fermionic Mott wedding-cake in clean and disordered optical lattices
    journal, September 2013

    • Kartsev, A.; Karlsson, D.; Privitera, A.
    • Scientific Reports, Vol. 3, Issue 1
    • DOI: 10.1038/srep02570

    On the determination of excitation energies using density functional theory
    journal, January 2000

    • Tozer, David J.; Handy, Nicholas C.
    • Physical Chemistry Chemical Physics, Vol. 2, Issue 10
    • DOI: 10.1039/a910321j

    Exact time-dependent density functional theory for impurity models
    journal, January 2013

    • Schmitteckert, Peter; Dzierzawa, Michael; Schwab, Peter
    • Physical Chemistry Chemical Physics, Vol. 15, Issue 15
    • DOI: 10.1039/c3cp44639e

    Excitation energies of dissociating H2: A problematic case for the adiabatic approximation of time-dependent density functional theory
    journal, November 2000

    • Gritsenko, O. V.; van Gisbergen, S. J. A.; Görling, A.
    • The Journal of Chemical Physics, Vol. 113, Issue 19
    • DOI: 10.1063/1.1318750

    Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange
    journal, August 2003

    • Dreuw, Andreas; Weisman, Jennifer L.; Head-Gordon, Martin
    • The Journal of Chemical Physics, Vol. 119, Issue 6
    • DOI: 10.1063/1.1590951

    Resonant nonlinear polarizabilities in the time-dependent density functional theory
    journal, November 2003

    • Tretiak, S.; Chernyak, V.
    • The Journal of Chemical Physics, Vol. 119, Issue 17
    • DOI: 10.1063/1.1614240

    Relationship between long-range charge-transfer excitation energy error and integer discontinuity in Kohn–Sham theory
    journal, December 2003

    • Tozer, David J.
    • The Journal of Chemical Physics, Vol. 119, Issue 24
    • DOI: 10.1063/1.1633756

    Double excitations within time-dependent density functional theory linear response
    journal, April 2004

    • Maitra, Neepa T.; Zhang, Fan; Cave, Robert J.
    • The Journal of Chemical Physics, Vol. 120, Issue 13
    • DOI: 10.1063/1.1651060

    A long-range-corrected time-dependent density functional theory
    journal, May 2004

    • Tawada, Yoshihiro; Tsuneda, Takao; Yanagisawa, Susumu
    • The Journal of Chemical Physics, Vol. 120, Issue 18
    • DOI: 10.1063/1.1688752

    Asymptotic correction of the exchange–correlation kernel of time-dependent density functional theory for long-range charge-transfer excitations
    journal, July 2004

    • Gritsenko, Oleg; Baerends, Evert Jan
    • The Journal of Chemical Physics, Vol. 121, Issue 2
    • DOI: 10.1063/1.1759320

    Undoing static correlation: Long-range charge transfer in time-dependent density-functional theory
    journal, June 2005

    • Maitra, Neepa T.
    • The Journal of Chemical Physics, Vol. 122, Issue 23
    • DOI: 10.1063/1.1924599

    On the mapping of time-dependent densities onto potentials in quantum mechanics
    journal, January 2008

    • Baer, Roi
    • The Journal of Chemical Physics, Vol. 128, Issue 4
    • DOI: 10.1063/1.2822124

    Time-dependent V-representability on lattice systems
    journal, July 2008

    • Li, Yonghui; Ullrich, C. A.
    • The Journal of Chemical Physics, Vol. 129, Issue 4
    • DOI: 10.1063/1.2955733

    Dynamic polarizabilities and excitation spectra from a molecular implementation of time‐dependent density‐functional response theory: N 2 as a case study
    journal, April 1996

    • Jamorski, Christine; Casida, Mark E.; Salahub, Dennis R.
    • The Journal of Chemical Physics, Vol. 104, Issue 13
    • DOI: 10.1063/1.471140

    Perspective: Fundamental aspects of time-dependent density functional theory
    journal, June 2016

    • Maitra, Neepa T.
    • The Journal of Chemical Physics, Vol. 144, Issue 22
    • DOI: 10.1063/1.4953039

    Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF
    journal, February 2018

    • Li, Chen; Requist, Ryan; Gross, E. K. U.
    • The Journal of Chemical Physics, Vol. 148, Issue 8
    • DOI: 10.1063/1.5011663

    Does density functional theory contribute to the understanding of excited states of unsaturated organic compounds?
    journal, October 1999


    Twenty-five Years of High-Temperature Superconductivity – A Personal Review
    journal, July 2013


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


    Progress in Time-Dependent Density-Functional Theory
    journal, May 2012


    Tuned Range-Separated Hybrids in Density Functional Theory
    journal, March 2010


    Dynamical self-stabilization of the Mott insulator: Time evolution of the density and entanglement entropy of out-of-equilibrium cold fermion gases
    journal, January 2011