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) $T_2\left[\rho \right]=\frac{1}{72}\int \frac{{\left|\overrightarrow{\nabla }\rho \right(\overrightarrow{\mathrm{r}}\left)\right|}^2}{\rho \left(\overrightarrow{\mathrm{r}}\right)}d\overrightarrow{\mathrm{r}}\ge \frac{{\pi }^{\frac{4}{3}}{2}^{\frac{2}{3}}}{24N^{\frac{2}{3}}}\int {\rho }^{\frac{5}{3}}\left(\overrightarrow{\mathrm{r}}\right)d\overrightarrow{\mathrm{r}}=\frac{10}{72}{\left[\frac{2}{3}\right]}^{\frac{2}{3}}\frac{T_0\left[\rho \right]}{N^{\frac{2}{3}}}$ and (ii) $\left|K_2\right[\rho \left]\right|=\int \frac{{\left|\overrightarrow{\nabla }\rho \right(\overrightarrow{\mathrm{r}}\left)\right|}^2}{{\rho }^{\frac{4}{3}}\left(\overrightarrow{\mathrm{r}}\right)}d\overrightarrow{\mathrm{r}}\ge \frac{27{\left(\frac{\pi }{2}\right)}^{\frac{4}{3}}}{N^{\frac{2}{3}}}\int {\rho }^{\frac{4}{3}}\left(\overrightarrow{\mathrm{r}}\right)d\overrightarrow{\mathrm{r}}=\frac{3\pi {\left(6\mathrm{}\pi \right)}^{\frac{2}{3}}}{N^{\frac{2}{3}}}\left|K_0\right[\rho \left]\right|$ have been derived [$N$ is the number of electrons and $\rho \left(\overrightarrow{\mathrm{r}}\right)$ 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.