Functional Derivatives and Differentiability in Density-Functional Theory

  • Ping Xiang
  • Yan Alexander WangEmail author
Conference paper
Part of the Progress in Theoretical Chemistry and Physics book series (PTCP, volume 31)


Based on Lindgren and Salomonson’s analysis on Fréchet differentiability [Phys Rev A 67:056501 (2003)], we showed a specific variational path along which the Fréchet derivative of the Levy-Lieb functional does not exist in the unnormalized density domain. This conclusion still holds even when the density is restricted within a normalized space. Furthermore, we extended our analysis to the Lieb functional and demonstrated that the Lieb functional is not Fréchet differentiable. Along our proposed variational path, the Gâteaux derivative of the Levy-Lieb functional or the Lieb functional takes a different form from the corresponding one along other more conventional variational paths. This fact prompted us to define a new class of unconventional density variations and inspired us to present a modified density variation domain to eliminate the problems associated with such unconventional density variations.


Density functional Density variation Functional differentiability Functional derivative 



Financial support for this project was provided by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.


  1. 1.
    Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford University Press, New YorkGoogle Scholar
  2. 2.
    Hohenberg P, Kohn W (1964) Phys Rev 136:B864CrossRefGoogle Scholar
  3. 3.
    Kohn W, Sham LJ (1965) Phys Rev 140:A1133CrossRefGoogle Scholar
  4. 4.
    Wang YA, Xiang P (2013) In: Wesolowski TA, Wang YA (eds) Recent advances in orbital-free density functional theory, Chap. 1. World Scientific, Singapore, pp 3–12Google Scholar
  5. 5.
    Lieb EH (1983) Int J Quantum Chem 24:243CrossRefGoogle Scholar
  6. 6.
    Englisch H, Englisch R (1983) Phys Stat Sol 123:711CrossRefGoogle Scholar
  7. 7.
    Englisch H, Englisch R (1984) Phys Stat Sol 124:373CrossRefGoogle Scholar
  8. 8.
    Lindgren I, Salomonson S (2003) Phys Rev A 67:056501CrossRefGoogle Scholar
  9. 9.
    Lindgren I, Salomonson S (2003) Adv Quantum Chem 43:95CrossRefGoogle Scholar
  10. 10.
    Lindgren I, Salomonson S (2004) Phys Rev A 70:032509CrossRefGoogle Scholar
  11. 11.
    Ekeland I, Temam R (1976) Convex analysis and variational problems. North-Holland, AmsterdamGoogle Scholar
  12. 12.
    Harris J, Jones RO (1974) J Phys F 4:1170CrossRefGoogle Scholar
  13. 13.
    Harris J (1984) Phys Rev A 29:1648CrossRefGoogle Scholar
  14. 14.
    Gunnarsson O, Lundqvist BI (1976) Phys Rev B 13:4274CrossRefGoogle Scholar
  15. 15.
    Langreth DC, Perdew JP (1980) Phys Rev B 21:5469CrossRefGoogle Scholar
  16. 16.
    Wang YA (1997) Phys Rev A 55:4589CrossRefGoogle Scholar
  17. 17.
    Wang YA (1997) Phys Rev A 56:1646CrossRefGoogle Scholar
  18. 18.
    Levy M (1979) Proc Natl Acad Sci USA 76:6062CrossRefPubMedGoogle Scholar
  19. 19.
    Nesbet RK (2001) Phys Rev A 65:010502CrossRefGoogle Scholar
  20. 20.
    Nesbet RK (2003) Adv Quantum Chem 43:1CrossRefGoogle Scholar
  21. 21.
    Dreizler RM, Gross EKU (1990) Density functional theory. Springer, BerlinCrossRefGoogle Scholar
  22. 22.
    Davidson ER (1976) Reduced density matrices in quantum chemistry. Academic, New YorkGoogle Scholar
  23. 23.
    Perdew JP, Levy M (1985) Phys Rev B 31:6264CrossRefGoogle Scholar
  24. 24.
    Englisch H, Englisch R (1983) Physica A 121:253CrossRefGoogle Scholar
  25. 25.
    Zhang YA, Wang YA (2009) Int J Quantum Chem 109:3199CrossRefGoogle Scholar
  26. 26.
    Milne RD (1980) Applied functional analysis: an introductory treatment. Pitman Publishing, UKGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of ChemistryUniversity of British ColumbiaVancouverCanada

Personalised recommendations