Density functional theory: An introduction

American Journal of Physics

Volumn 68, Issue 1, 2000, Pages 69-79

  Argaman, Nathan ^a,b   Makov, Guy ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b NUCLEAR RESEARCH CENTER NEGEV   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034391541     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19375     Document Type: Article

References (64)

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3. W. Kohn and P. Vashishta, "General density functional theory," in Theory of the Inhomogeneous Electron Gas, edited by S. Lundqvist and N. H. March (Plenum, New York, 1983), pp. 79-147; see also additional chapters in this book.
4. The relationship between DFT and Legendre transforms was discussed mathematically by E. H. Lieb, "Density functionals for Coulomb systems." Int. J. Quantum Chem. 24, 243-277 (1983), and has occasionally been used in the literature - see, e.g., R. Fukuda, T. Kotani, Y. Suzuki, and S. Yokojima, "Density functional theory through Legendre transformation," Prog. Theor. Phys. 92, 833-862 (1994). However, we are unaware of any previous references which focus on the pedagogical value of this relationship.
5. The relationship between DFT and Legendre transforms was discussed mathematically by E. H. Lieb, "Density functionals for Coulomb systems." Int. J. Quantum Chem. 24, 243-277 (1983), and has occasionally been used in the literature - see, e.g., R. Fukuda, T. Kotani, Y. Suzuki, and S. Yokojima, "Density functional theory through Legendre transformation," Prog. Theor. Phys. 92, 833-862 (1994). However, we are unaware of any previous references which focus on the pedagogical value of this relationship.
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10. b)/T), and x ≠ y. This proof generalizes to the functional situation, where v(r) replaces μ; see the Appendix of M. Valiev and G. W. Fernando, "Generalized Kohn-Sham density functional theory via the effective action formalism," cond-mat/9702247. Note that the convexity or second derivative of Ω(μ) may vanish in extreme cases such as at the zero temperature limit (because there only terms with x = y remain) or in the thermodynamic limit, at the point of a first-order phase transition. Mathematical consistency may be restored by restricting attention to small but finite temperatures and large but finite volumes (cf. Ref. 18).
11. F(N) thus defined is not identical with that obtained by fixing the particle number - the equivalence of the canonical and grand-canonical ensembles is guaranteed only in the thermodynamic limit. However, we will later focus on the T→0 limit in which the equivalence is regained because the fluctuations of the particle number vanish.
12. k . The usual chain rules for derivatives follow, and can be used in deriving, e.g., the relationship n(r) = δΩ/δv (r). A more detailed introduction can be found in C. F. Stevens, "The six core theories of modern physics" (MIT Press, Cambridge, 1995), pp. 32-38, or in Ref. 1.
13. For alternatives, see R. F. Nalewajski, and R. G. Parr, "Legendre transforms and Maxwell relations in density functional theory," J. Chem. Phys. 77, 399-407 (1982) B. G. Baekelandt, A. Cedillo, and R. G. Parr, "Reactivity indices and fluctuation formulas in density functional theory: isomorphic ensembles and a new measure of local hardness," ibid. 103, 8548-8556 (1995).
14. For alternatives, see R. F. Nalewajski, and R. G. Parr, "Legendre transforms and Maxwell relations in density functional theory," J. Chem. Phys. 77, 399-407 (1982) B. G. Baekelandt, A. Cedillo, and R. G. Parr, "Reactivity indices and fluctuation formulas in density functional theory: isomorphic ensembles and a new measure of local hardness," ibid. 103, 8548-8556 (1995).
15. ψ→n<ψ|T̂+Û|ψ>, a definition which is valid for any reasonable n(r). Whereas the thermodynamic approach used here clarifies the special role of the density distribution n(r), in Levy's formulation one could equally well imagine other ways of constraining the search to other subspaces of the Hubert space.
16. ψ→n<ψ|T̂+Û|ψ>, a definition which is valid for any reasonable n(r). Whereas the thermodynamic approach used here clarifies the special role of the density distribution n(r), in Levy's formulation one could equally well imagine other ways of constraining the search to other subspaces of the Hubert space.
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18. P. Hohenberg and W. Kohn, "Inhomogeneous electron gas," Phys. Rev. 136, B864-867 (1964); the generalization to a finite-temperature grandcanonical ensemble was provided almost immediately by N. D. Mermin, "Thermal properties of the inhomogeneous electron gas," ibid. 137, A 1441-1443 (1965).
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20. J. D. van der Waals, Ph.D. thesis, University of Lieden (1873)
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23. We consider small but positive temperatures, T→0, rather than the situation at T=0, where the function N(μ) can take only integer values and becomes discontinuous [if one insists on working at T= 0, the functional derivative, e.g., in Eq. (21), must be redefined so that ∫ dr δn(r)=0]. For a discussion, see, e.g., J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, "Density-functional theory for fractional particle number: derivative discontinuities of the energy," Phys. Rev. Lett. 49, 1691-1694 (1982).
24. ni[n] is the kinetic energy of a system of noninteracting particles with the density n(r), rather than the kinetic energy of the interacting electron system.
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33. xc[n(r)].
34. eff (r) to vanish at infinity, one finds that Eq. (19) reduces to a statement of the equality of the chemical potentials for the interacting and the Kohn-Sham system.
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43. ↓).
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