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Title: On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals

Abstract

In this paper, we provide a rigorous proof that the Hartree Fock energy, as a function of the fractional electron number, E(N), is piecewise concave. Moreover, for semi-local density functionals, we show that the piecewise convexity of the E(N) curve, as stated in the literature, is not generally true for all fractions. By an analysis based on exchange-only local density approximation and careful examination of the E(N) curve, we find for some systems, there exists a very small concave region, corresponding to adding a small fraction of electrons to the integer system, while the remaining E(N) curve is convex. Several numerical examples are provided as verification. Although the E(N) curve is not convex everywhere in these systems, the previous conclusions on the consequence of the delocalization error in the commonly used density functional approximations, in particular, the underestimation of ionization potential, and the overestimation of electron affinity, and other related issues, remain unchanged. Finally, this suggests that instead of using the term convexity, a modified and more rigorous description for the delocalization error is that the E(N) curve lies below the straight line segment across the neighboring integer points for these approximate functionals.

Authors:
ORCiD logo [1];  [2]
+ Show Author Affiliations
  1. Duke Univ., Durham, NC (United States). Dept. of Chemistry
  2. Duke Univ., Durham, NC (United States). Dept. of Chemistry; South China Normal Univ., Guangzhou (China). Key Lab. of Theoretical Chemistry of Environment. School of Chemistry and Environment
Publication Date:
Research Org.:
Energy Frontier Research Centers (EFRC) (United States). Center for Complex Materials from First Principles (CCM); Duke Univ., Durham, NC (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF)
OSTI Identifier:
1465680
Alternate Identifier(s):
OSTI ID: 1361757
Grant/Contract Number:  
SC0012575; CHE-13-62927
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 146; Journal Issue: 7; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; protons; Hilbert space; electron affinities; band gap; ground states; local density approximations; electron densities of states; many electron systems

Citation Formats

Li, Chen, and Yang, Weitao. On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals. United States: N. p., 2017. Web. doi:10.1063/1.4974988.
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Li, Chen, & Yang, Weitao. On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals. United States. https://doi.org/10.1063/1.4974988
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Li, Chen, and Yang, Weitao. Tue . "On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals". United States. https://doi.org/10.1063/1.4974988. https://www.osti.gov/servlets/purl/1465680.
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@article{osti_1465680,
title = {On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals},
author = {Li, Chen and Yang, Weitao},
abstractNote = {In this paper, we provide a rigorous proof that the Hartree Fock energy, as a function of the fractional electron number, E(N), is piecewise concave. Moreover, for semi-local density functionals, we show that the piecewise convexity of the E(N) curve, as stated in the literature, is not generally true for all fractions. By an analysis based on exchange-only local density approximation and careful examination of the E(N) curve, we find for some systems, there exists a very small concave region, corresponding to adding a small fraction of electrons to the integer system, while the remaining E(N) curve is convex. Several numerical examples are provided as verification. Although the E(N) curve is not convex everywhere in these systems, the previous conclusions on the consequence of the delocalization error in the commonly used density functional approximations, in particular, the underestimation of ionization potential, and the overestimation of electron affinity, and other related issues, remain unchanged. Finally, this suggests that instead of using the term convexity, a modified and more rigorous description for the delocalization error is that the E(N) curve lies below the straight line segment across the neighboring integer points for these approximate functionals.},
doi = {10.1063/1.4974988},
journal = {Journal of Chemical Physics},
number = 7,
volume = 146,
place = {United States},
year = {Tue Feb 21 00:00:00 EST 2017},
month = {Tue Feb 21 00:00:00 EST 2017}
}
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https://doi.org/10.1063/1.4974988
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Works referenced in this record:

Density‐functional thermochemistry. III. The role of exact exchange
journal, April 1993

  • Becke, Axel D.
  • The Journal of Chemical Physics, Vol. 98, Issue 7, p. 5648-5652
  • DOI: 10.1063/1.464913

Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy
journal, December 1982

Failure of the random-phase-approximation correlation energy
journal, April 2012

Exchange and correlation in open systems of fluctuating electron number
journal, October 2007

Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals
journal, November 2006

  • Ruzsinszky, Adrienn; Perdew, John P.; Csonka, Gábor I.
  • The Journal of Chemical Physics, Vol. 125, Issue 19
  • DOI: 10.1063/1.2387954

A nonempirical scaling correction approach for density functional methods involving substantial amount of Hartree–Fock exchange
journal, May 2013

  • Zheng, Xiao; Zhou, Ting; Yang, Weitao
  • The Journal of Chemical Physics, Vol. 138, Issue 17
  • DOI: 10.1063/1.4801922

Orbital relaxation effects on Kohn–Sham frontier orbital energies in density functional theory
journal, April 2015

  • Zhang, DaDi; Zheng, Xiao; Li, Chen
  • The Journal of Chemical Physics, Vol. 142, Issue 15
  • DOI: 10.1063/1.4918347

Analytical evaluation of Fukui functions and real-space linear response function
journal, April 2012

  • Yang, Weitao; Cohen, Aron J.; De Proft, Frank
  • The Journal of Chemical Physics, Vol. 136, Issue 14
  • DOI: 10.1063/1.3701562

Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
journal, January 1988

Density-functional exchange-energy approximation with correct asymptotic behavior
journal, September 1988

Exchange-correlation energy from pairing matrix fluctuation and the particle-particle random-phase approximation
journal, September 2013

Challenges for Density Functional Theory
journal, December 2011

  • Cohen, Aron J.; Mori-Sánchez, Paula; Yang, Weitao
  • Chemical Reviews, Vol. 112, Issue 1
  • DOI: 10.1021/cr200107z

Tests of functionals for systems with fractional electron number
journal, April 2007

  • Vydrov, Oleg A.; Scuseria, Gustavo E.; Perdew, John P.
  • The Journal of Chemical Physics, Vol. 126, Issue 15
  • DOI: 10.1063/1.2723119

Density functionals for coulomb systems
journal, September 1983

  • Lieb, Elliott H.
  • International Journal of Quantum Chemistry, Vol. 24, Issue 3
  • DOI: 10.1002/qua.560240302

A challenge for density functionals: Self-interaction error increases for systems with a noninteger number of electrons
journal, August 1998

  • Zhang, Yingkai; Yang, Weitao
  • The Journal of Chemical Physics, Vol. 109, Issue 7
  • DOI: 10.1063/1.476859

Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis
journal, August 1980

  • Vosko, S. H.; Wilk, L.; Nusair, M.
  • Canadian Journal of Physics, Vol. 58, Issue 8
  • DOI: 10.1139/p80-159

Perspective on "Density-functional theory for fractional particle number: derivative discontinuities of the energy"
journal, February 2000

  • Zhang, Yingkai; Yang, Weitao
  • Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), Vol. 103, Issue 3-4
  • DOI: 10.1007/s002149900021

Perspective: Fifty years of density-functional theory in chemical physics
journal, May 2014

  • Becke, Axel D.
  • The Journal of Chemical Physics, Vol. 140, Issue 18
  • DOI: 10.1063/1.4869598

Many-electron self-interaction error in approximate density functionals
journal, November 2006

  • Mori-Sánchez, Paula; Cohen, Aron J.; Yang, Weitao
  • The Journal of Chemical Physics, Vol. 125, Issue 20
  • DOI: 10.1063/1.2403848

Insights into Current Limitations of Density Functional Theory
journal, August 2008

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    Works referencing / citing this record:

    On extending Kohn-Sham density functionals to systems with fractional number of electrons
    journal, June 2017

    • Li, Chen; Lu, Jianfeng; Yang, Weitao
    • The Journal of Chemical Physics, Vol. 146, Issue 21
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    • Gould, Tim; Kronik, Leeor; Pittalis, Stefano
    • The Journal of Chemical Physics, Vol. 148, Issue 17
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    Density functional approximations for orbital energies and total energies of molecules and solids
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    • Baerends, Evert Jan
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    journal, September 2017

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    • DOI: 10.1093/nsr/nwx111
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