Leading corrections to local approximations II (with turning points)

Abstract

Quantum corrections to Thomas-Fermi (TF) theory are investigated for noninteracting one-dimensional fermions with known uniform semiclassical approximations to the density and kinetic energy. Their structure is analyzed, and contributions from distinct phase space regions (classically-allowed versus forbidden at the Fermi energy) are derived analytically. Universal formulas are derived for both particle numbers and energy components in each region. For example, in the semiclassical limit, exactly 1/(6\pi3^{1/2}) of a particle leaks into the evanescent region beyond a turning point. The correct normalization of semiclassical densities is proven analytically in the semiclassical limit. Energies and densities are tested numerically in a variety of one-dimensional potentials, especially in the limit where TF theory becomes exact. The subtle relation between the pointwise accuracy of the semiclassical approximation and integrated expectation values is explored. The limitations of the semiclassical formulas are also investigated when the potential varies too rapidly. The approximations are shown to work for multiple wells, except right at the mid-phase point of the evanescent regions. The implications for density functional approximations are discussed.

References in corpus (7)

• DFT: A Theory Full of Holes?
• The Importance of being consistent
• Exact condition on the Kohn-Sham kinetic energy, and modern parametrization of the Thomas-Fermi density
• Electronic structure via potential functional approximations
• Leading corrections to local approximations
• Corrections to Thomas-Fermi densities at turning points and beyond
• Quantum oscillations in the kinetic energy density: Gradient corrections from the Airy gas

Cited by in corpus (1)

• Jellium-with-gap model applied to semilocal kinetic functionals