Numerical test of the sixth-order gradient expansion for the kinetic energy:Application to the monovacancy in jellium

Abstract

Although the density-gradient expansion for the noninteracting fermion kinetic energy is known through sixth order, numerical tests and applications of it have so far only been made through fourth order because its sixth-order term (${\mathrm{T}}_{6}$) diverges in the exponentially decaying tail of the electron density for a finite system. We show that ${\mathrm{T}}_{6}$ provides a useful correction to the density functional ${\mathrm{T}}_{0}$+${\mathrm{T}}_{2}$+${\mathrm{T}}_{4}$ for a problem in which the density n(r) is everywhere bounded from below: the formation energy of a monovacancy in jellium. Thus the sixth-order expansion may be useful in condensed-matter physics or perhaps more generally via convergent resummation. We also suggest that the most important terms of the series may be those involving |\ensuremath{\nabla}n${\mathrm{|}}^{2}$ and ${\mathrm{\ensuremath{\nabla}}}^{2}$n, but no other derivative.