### Abstract

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 language | English |
---|---|

Article number | 285202 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 49 |

Issue number | 28 |

DOIs | |

Publication status | Published - 6 Jun 2016 |

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### Keywords

- 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

### Cite this

**On the divergence of gradient expansions for kinetic energy functionals in the potential functional theory.** / Sergeev, Alexey; Jovanovic, Raka; Kais, Sabre; Alharbi, Fahhad.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the divergence of gradient expansions for kinetic energy functionals in the potential functional theory

AU - Sergeev, Alexey

AU - Jovanovic, Raka

AU - Kais, Sabre

AU - Alharbi, Fahhad

PY - 2016/6/6

Y1 - 2016/6/6

N2 - 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.

AB - 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.

KW - Density functional theory

KW - Gradient expansion

KW - Padé approximants

KW - Potential functional theory

UR - http://www.scopus.com/inward/record.url?scp=84975046558&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975046558&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/49/28/285202

DO - 10.1088/1751-8113/49/28/285202

M3 - Article

VL - 49

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 28

M1 - 285202

ER -