Obtaining a gradient-corrected kinetic-energy functional from the Perdew-Wang exchange functional

Lee, Lee, and Parr (LLP) have shown that the kinetic energy can be written in the same form as Becke’s exchange energy. This conjecture of LLP has been generalized to another exchange functional, namely, the Perdew-Wang exchange functional. As demonstrated by Lee and Parr, the exchange energy can be written K=πFFsΓ(r,s)drds with Γ(r,s)=‖γ(r,s)${\mathrm{\Vert }}^2$¯/${\mathit{n}}^2$(r), where ‖γ(r,s)${\mathrm{\Vert }}^2$¯ is the spherical average of ‖γ(r,s)${\mathrm{\Vert }}^2$. Using the generalization of LLP’s conjecture, it is demonstrated that Γ(r,s)= ${\mathit{e}}^{\mathrm{-}\mathit{s}2}$/β(r)+a[${\mathit{s}}^4$/${\mathrm{\beta }}_0^2$(r)]${\mathit{e}}^{\mathrm{-}\mathit{s}2}$/${\mathrm{\beta }}_0$(r), a=const, ${\mathrm{\beta }}_0$(r)=5[3${\mathrm{\pi }}^2$n(r)${]}^{\mathrm{-}2/3}$. At zeroth order, β(r)=${\mathrm{\beta }}_0$(r), the function Γ(r,s), gives exactly the modified Gaussian approximation proposed by Lee and Parr.