Sixth-order term of the gradient expansion of the kinetic-energy density functional

1 October 1981

journal article
research article
Published by American Physical Society (APS) in Physical Review A

Vol. 24 (4) , 1682-1688
https://doi.org/10.1103/physreva.24.1682

Abstract

A generalization of Hodges's method of obtaining the gradient expansion is derived and then employed to determine the sixth-order term. The formula for it is as follows:

T_{6} [ρ] = \frac{{(3 π^{2})}^{- 413}}{45 360 \int ρ^{\frac{- 1}{3}} [13 {(\frac{\nabla \nabla^{2} ρ}{ρ})}^{2} + \frac{2575}{144 {(\frac{\nabla^{2} ρ}{ρ})}^{3}} + \frac{249}{16 {(\frac{\nabla ρ}{ρ})}^{2} (\frac{\nabla^{4} ρ}{ρ})} + \frac{1499}{18 {(\frac{\nabla ρ}{ρ})}^{2} {(\frac{\nabla^{2} ρ}{ρ})}^{2}} - \frac{1307}{36 {(\frac{\nabla ρ}{ρ})}^{2} (\frac{\vec{\nabla} ρ \cdot \vec{\nabla} \nabla^{2} ρ}{ρ^{2}})} + \frac{343}{18 {(\frac{\vec{\nabla} ρ \cdot \vec{\nabla} \vec{\nabla} ρ}{ρ^{2}})}^{2} + \frac{8341}{72 (\frac{\nabla^{2} ρ}{ρ}) {(\frac{\nabla ρ}{ρ})}^{4}} - \frac{1 600 495}{2592 {(\frac{\nabla ρ}{ρ})}^{6}}] d^{3} r}}

. For atomic densities,

T_{6} [ρ]

is divergent near the nucleus and at large distances.

Keywords

KINETIC ENERGY

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