Science China Mathematics

, Volume 62, Issue 1, pp 63–68 | Cite as

A mathematical aspect of Hohenberg-Kohn theorem

  • Aihui ZhouEmail author


The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of matter. In this article, we study the Hohenberg-Kohn theorem for a class of external potentials based on a unique continuation principle.


density functional theory electronic structure unique continuation principle Hohenberg-Kohn theorem 





This work was supported by National Natural Science Foundation of China (Grant Nos. 91730302, 9133202 and 11671389), and the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences (Grant No. QYZDJ-SSW-SYS010). The author thanks Mr. Bing Yang for the useful discussion on the unique continuation property and the anonymous referees for their careful reviews and helpful suggestions that improved the presentation of this paper.


  1. 1.
    Adams R A, Fournier J J F. Sobolev Spaces, 2nd ed. Amsterdam: Academic Press, 2003zbMATHGoogle Scholar
  2. 2.
    Ayers P W, Golden S, Levy M. Generalizations of the Hohenberg-Kohn theorem, I: Legendre transform constructions of variational principles for density matrices and electron distrbution functions. J Chem Phys, 2006, 124: 054101CrossRefGoogle Scholar
  3. 3.
    Boosting materials modelling. Nature Materials, 2016, 15: 365, nmat4619.pdfCrossRefGoogle Scholar
  4. 4.
    Eschrig H. The Fundamentals of Density Functional Theory. Leipzig: Eagle, 2003zbMATHGoogle Scholar
  5. 5.
    Fournais S, Hoffmann-Ostenhof M, Hoffmann-Ostenhof T, et al. Positivity of the spherically averaged atomic oneelectron density. Math Z, 2008, 259: 123–130MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hadjisavvas N, Theophilou A. Rigorous formulation of the Kohn-Sham theory. Phys Rev A (3), 1984, 30: 2183–2186MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hohenberg P, Kohn W. The inhomogeneous electron gas. Phys Rev, 1964, 136: 864–871MathSciNetCrossRefGoogle Scholar
  8. 8.
    Jerison D, Kenig C E. Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann of Math (2), 1985, 121: 463–494MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kohn W, Sham L J. Self-consistent equations including exchange and correlation effects. Phys Rev A, 1965, 140: 4743–4754MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kryachko E S. On the original proof by reductio ad absurdum of the Hohenberg-Kohn theorem for many-electron Coulomb systems. Int J Quantum Chem, 2005, 103: 818–823CrossRefGoogle Scholar
  11. 11.
    Kryachko E S. On the proof by reductio ad absurdum of the Hohenberg-Kohn theorem for ensembles of fractionally occupied states of Coulomb systems. Int J Quantum Chem, 2006, 106: 1795–1798CrossRefGoogle Scholar
  12. 12.
    Kryachko E S, Ludeñ E V. Density functional theory: Foundations reviewed. Phys Rep, 2014, 544: 123–239MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kvaal S, Helgaker T. Ground-state densities from the Rayleigh-Ritz variation principle and from density functional theory. J Chem Phys, 2015, 143: 184106CrossRefGoogle Scholar
  14. 14.
    Lammert P E. Differentiability of Lieb functional in electronic density functional theory. Int J Quantum Chem, 2007, 107: 1943–1953CrossRefGoogle Scholar
  15. 15.
    Levy M. University variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solutions of the v-representability problems. Proc Natl Acad Sci USA, 1979, 76: 6062–6065CrossRefGoogle Scholar
  16. 16.
    Levy M. Electron densities in search of Hamiltonians. Phys Rev A, 1982, 26: 1200–1208Google Scholar
  17. 17.
    Lieb E H. Density functionals for Coulomb systems. Int J Quantum Chem, 1983, 24: 243–277CrossRefGoogle Scholar
  18. 18.
    Martin R. Electronic Structure: Basic Theory and Practical Methods. London: Cambridge University Press, 2004CrossRefzbMATHGoogle Scholar
  19. 19.
    Parr R G, Yang W T. Density-Functional Theory of Atoms and Molecules. Oxford: Oxford University Press, 1989Google Scholar
  20. 20.
    Pino R, Bokanowski O, Ludexxxxxa E V, et al. A re-statement of the Hohenberg-Kohn theorem and its extension to finite subspaces. Theor Chem Account, 2007, 118: 557–561CrossRefGoogle Scholar
  21. 21.
    Reed M, Simon B. Methods of Modern Mathematical Physics IV: Analysis of Operators. San Diego: Academic Press, 1978zbMATHGoogle Scholar
  22. 22.
    Redner S. Citation statistics from 110 years of physical review. Physics Today, 2005, 58: 49–54CrossRefGoogle Scholar
  23. 23.
    Schechter M, Simon B. Unique continuation for Schrodinger operators with unbounded potentials. J Math Anal Appl, 1980, 77: 482–492MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Szczepanik W, Dulak M, Wesolowski T A. Comment on "On the original proof by reductio ad absurdum of the Hohenberg-Kohn theorem for many-electron Coulomb systems". Int J Quantum Chem, 2007, 107: 762–763CrossRefGoogle Scholar
  25. 25.
    van Noorden R, Maher B, Nuzzo R. The top 100 papers. Nature, 2014, 514: 550–553CrossRefGoogle Scholar
  26. 26.
    Wolff T H. Recent work on sharp estimates in second-order elliptic unique continuation problems. J Geom Anal, 1993, 3: 621–650MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhou A. Hohenberg-Kohn theorem for Coulomb type systems and its generalization. J Math Chem, 2012, 50: 2746–2754MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhou A. Some open mathematical problems in electronic structure models and calculations (in Chinese). Sci Sin Math, 2015, 45: 929–938Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

Personalised recommendations