Relationships between the terms in the gradient expansion: Kinetic and exchange energy functionals

Rigorous lower bounds for all bound-state systems, for the first-gradient corrections to the kinetic and exchange energy functionals, viz., (i) ${T}₂[{ρ}]=1/72{∫}{{|{{→}}{{∇}}{ρ}({{→}}{r})|}²}{{ρ}({{→}}{r})}d{{→}}{r}{≥}{{{π}}4/3{2}2/3}{24{N}2/3}{∫}{{ρ}}5/3({{→}}{r})d{{→}}{r}=10/72{[2/3]}2/3{{T}₀[{ρ}]}{{N}2/3}$ and (ii) $|{K}₂[{ρ}]|={∫}{{|{{→}}{{∇}}{ρ}({{→}}{r})|}²}{{{ρ}}4/3({{→}}{r})}d{{→}}{r}{≥}{27{({{π}}{2})}4/3}{{N}2/3}{∫}{{ρ}}4/3({{→}}{r})d{{→}}{r}={3{π}{(6{π})}2/3}{{N}2/3}|{K}₀[{ρ}]|$ have been derived [$N$ is the number of electrons and ${ρ}({{→}}{r})$ is the electron density]. Numerical investigations on these bounds employing Hartree-Fock atomic-electron densities have been carried out. An empirical relationship between the Hartree-Fock ${T}₂$ and ${T}₀$ for neutral atoms has also been presented.