We determine the energy density $\ensuremath{\xi}(3∕5)n{\ensuremath{\epsilon}}_{F}$ and the gradient correction $\ensuremath{\lambda}{\ensuremath{\hbar}}^{2}{(\mathbf{\ensuremath{\nabla}}n)}^{2}∕(8mn)$ of the extended Thomas-Fermi (ETF) density functional, where $n$ is the number density and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that $\ensuremath{\xi}=0.455$ and $\ensuremath{\lambda}=0.13$ give the best fit of the DMC data with an even number $N$ of particles. We also study the odd-even splitting $\ensuremath{\gamma}{N}^{1∕9}\ensuremath{\hbar}\ensuremath{\omega}$ of the ground-state energy for the unitary gas in a harmonic trap of frequency $\ensuremath{\omega}$ determining the constant $\ensuremath{\gamma}$. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.We determine the energy density $\ensuremath{\xi}(3∕5)n{\ensuremath{\epsilon}}_{F}$ and the gradient correction $\ensuremath{\lambda}{\ensuremath{\hbar}}^{2}{(\mathbf{\ensuremath{\nabla}}n)}^{2}∕(8mn)$ of the extended Thomas-Fermi (ETF) density functional, where $n$ is the number density and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that $\ensuremath{\xi}=0.455$ and $\ensuremath{\lambda}=0.13$ give the best fit of the DMC data with an even number $N$ of particles. We also study the odd-even splitting $\ensuremath{\gamma}{N}^{1∕9}\ensuremath{\hbar}\ensuremath{\omega}$ of the ground-state energy for the unitary gas in a harmonic trap of frequency $\ensuremath{\omega}$ determining the constant $\ensuremath{\gamma}$. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.

arXiv:0809.1820Physical Review ANov 18, 200885 citations

Density Functional Theory: Basic Results and Some Observations

Walter Kohn

Springer eBooksJan 1, 19855 citations

Generalized second-order Thomas-Fermi method for superfluid Fermi systems

Junchen Pei, Na Fei, Y. N. Zhang +1 more

Using the $\ensuremath{\hbar}$ expansion of the Green's function of the Hartree-Fock-Bogoliubov equation, we extend the second-order Thomas-Fermi approximation to generalized superfluid Fermi systems by including the density-dependent effective mass and the spin-orbit potential. We first implement and examine the full correction terms over different energy intervals of the quasiparticle spectra in calculations of finite nuclei. Final applications of this generalized Thomas-Fermi method are intended for various inhomogeneous superfluid Fermi systems.Using the $\ensuremath{\hbar}$ expansion of the Green's function of the Hartree-Fock-Bogoliubov equation, we extend the second-order Thomas-Fermi approximation to generalized superfluid Fermi systems by including the density-dependent effective mass and the spin-orbit potential. We first implement and examine the full correction terms over different energy intervals of the quasiparticle spectra in calculations of finite nuclei. Final applications of this generalized Thomas-Fermi method are intended for various inhomogeneous superfluid Fermi systems.

arXiv:1509.02616Physical Review CDec 18, 20156 citations

Some tests of an approximate density functional for the ground-state kinetic energy of a fermion system

Conyers Herring, Manoj Chopra

Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.

Physical Review AJan 1, 198854 citations

Modern Density Functional Theory

Viraht Sahni

Springer tracts in modern physicsJan 1, 202247 citations

n

is the number density and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that $\ensuremath{\xi}=0.455$ and $\ensuremath{\lambda}=0.13$ give the best fit of the DMC data with an even number

N

of particles. We also study the odd-even splitting $\ensuremath{\gamma}{N}^{1∕9}\ensuremath{\hbar}\ensuremath{\omega}$ of the ground-state energy for the unitary gas in a harmonic trap of frequency $\ensuremath{\omega}$ determining the constant $\ensuremath{\gamma}$. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.