On the divergence of gradient expansions for kinetic energy functionals in the potential functional theory

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We consider the density of a fermionic system as a functional of the potential, in one-dimensional case, where it is approximated by the Thomas-Fermi term plus semiclassical corrections through the gradient expansion. We compare this asymptotic series with the exact answer for the case of the harmonic oscillator and the Morse potential. It is found that the leading (Thomas-Fermi) term is in agreement with the exact density, but the subdominant term does not agree in terms of the asymptotic behavior because of the presence of oscillations in the exact density, but their absence in the gradient expansion. However, after regularization of the density by convolution with a Gaussian, the agreement can be established even in the subdominant term. Moreover, it is found that the expansion is always divergent, and its terms grow proportionally to the factorial function of the order, similar to the well-known divergence of perturbation series in field theory and the quantum anharmonic oscillator. Padé-Hermite approximants allow summation of the series, and one of the branches of the approximants agrees with the density.

Original languageEnglish
Article number285202
JournalJournal of Physics A: Mathematical and Theoretical
Issue number28
Publication statusPublished - 6 Jun 2016



  • Density functional theory
  • Gradient expansion
  • Padé approximants
  • Potential functional theory

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Modelling and Simulation
  • Statistics and Probability

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