Abstract
Density functional theory (DFT) has become a basic tool for the study of electronic structure of matter, in which the Hohenberg–Kohn theorem plays a fundamental role in the development of DFT. In this paper, we present a simple, selfcontained and mathematically rigorous proof using the Fundamental Theorem of Algebra. We also show the Hohenberg–Kohn theorem for systems with some more general external potentials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ayers P.W.: Density per particle as a descriptor of Coulombic systems. Proc. Nat. Acad. Sci. USA 97, 1959–1964 (2000)
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. 124, 054101–054107 (2006)
Ayers P.W., Levy M.: Time-independent (static) density-functional theories for pure excited states: extensions and unification. Phys. Rev. A 80, 012508–012523 (2009)
Ayers P.W., Liu S.: Necessary and sufficient conditions for the N-representability of density functionals. Phys. Rev. A 75, 022514–022527 (2007)
Cohen A.j., Mori-Sánchez P., Yang W.: Challenges for density functional theory. Chem. Rev. 112, 289–320 (2012)
Eschrig H.: The Fundamentals of Density Functional Theory. Eagle, Leipzig (2003)
Fournais S., Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Østergard Sørensen T.: Analyticity of the density of electronic wavefunctions. Ark. Mat. 42, 143–415 (2004)
Fournais S., Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Østergard Sørensen T.: Analytic structure of many-body Coulombic many-electron wave functions. Commun. Math. Phys. 289, 291–310 (2009)
Gross, E.K.U., Dreizler, R.M. (eds.), Density Functional Theory. Plenum Press, New York (1995)
Hadjisavvas N., Theophilou A.: Rigorous formulation of the Kohn-Sham theory. Phy. Rev. A 30, 2183–2186 (1984)
Hohenberg P., Kohn W.: Inhomogeneous gas. Phys. Rev. 136, B864–B871 (1964)
Jensen F.: Introduction to Computational Chemistry. Wiley, New York (1999)
Kato T.: On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. 10, 151–177 (1957)
Kohn W., Sham L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. A 140, 4743–4754 (1965)
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. 103, 818–823 (2005)
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. 106, 1795–1798 (2006)
Lammert P.E.: Differentiability of Lieb functional in electronic density functional theory. Int. J. Quantum Chem. 107, 1943–1953 (2007)
Lammert P.E.: Well-behaved coarse-grained model of density theory. Phys. Rev. A 82, 012109–012230 (2010)
Levy M.: University variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solutions of thev-representability problems. Proc. Nat. Acad. Sci. USA 76, 6062–6065 (1979)
Levy M.: Electron densities in search of Hamiltonians. Phys. Rev. A 26, 1200–1208 (1982)
Levy M.: On the simple constrained-search reformulation of the Hohenberg-Kohn theorem to include degeneracies and more (1964–1979). Int. J. Quantum Chem. 110, 3140–3144 (2010)
Lieb E.H.: Density functionals for Coulomb systems. Int. J. Quantum Chem. 24, 243–277 (1983)
Martin R.: Electronic Structure: basic theory and practical methods. Cambridge University Press, London (2004)
Mezey P.G.: The holographic electron density theorem and quantum similarity measures. Mol. Phys. 96, 169–178 (1999)
Parr R.G., Yang W.T.: Density-Functional theory of atoms and molecules. Oxford University Press, Oxford (1989)
Pino R., Bokanowski O., Ludeña E.V., Boada R.L.: A re-statement of the Hohenberg-Kohn theorem and its extension to finite subspaces. Theor. Chem. Account 118, 557–561 (2007)
Smithies F.: A forgotten paper on the fundamental theorem of algebra. Notes Rec. R. Soc. Lond. 54, 333–341 (2000)
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. 107, 762–763 (2007)
Leeuwen R.: Density functional approach to the many-body problem: key concepts and exact functionals. Adv. Quantum Chem. 43, 25–94 (2003)
Yang W.T., Ayers P.W., Wu Q.: Potential functionals: dual to density functionals and solution to the v-representability problem. Phys. Rev. Lett. 92, 146404–146407 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the National Science Foundation of China under grants 10871198 and 10971059, the Funds for Creative Research Groups of China under grant 11021101, the National Basic Research Program of China under grant 2011CB309703, and the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.
Rights and permissions
About this article
Cite this article
Zhou, A. Hohenberg–Kohn theorem for Coulomb type systems and its generalization. J Math Chem 50, 2746–2754 (2012). https://doi.org/10.1007/s10910-012-0061-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-012-0061-3
Keywords
- Coulomb system
- Density functional theory
- Electronic structure
- Fundamental Theorem of Algebra
- Hohenberg–Kohn