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

Abstract

Rigorous lower bounds for all bound-state systems, for the first-gradient corrections to the kinetic and exchange energy functionals, viz., (i) ${T}_{2}[\ensuremath{\rho}]=\frac{1}{72}\ensuremath{\int}\frac{{|\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})|}^{2}}{\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}\ensuremath{\ge}\frac{{\ensuremath{\pi}}^{\frac{4}{3}}{2}^{\frac{2}{3}}}{24{N}^{\frac{2}{3}}}\ensuremath{\int}{\ensuremath{\rho}}^{\frac{5}{3}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}=\frac{10}{72}{\left[\frac{2}{3}\right]}^{\frac{2}{3}}\frac{{T}_{0}[\ensuremath{\rho}]}{{N}^{\frac{2}{3}}}$ and (ii) $|{K}_{2}[\ensuremath{\rho}]|=\ensuremath{\int}\frac{{|\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})|}^{2}}{{\ensuremath{\rho}}^{\frac{4}{3}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}\ensuremath{\ge}\frac{27{(\frac{\ensuremath{\pi}}{2})}^{\frac{4}{3}}}{{N}^{\frac{2}{3}}}\ensuremath{\int}{\ensuremath{\rho}}^{\frac{4}{3}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}=\frac{3\ensuremath{\pi}{(6\ensuremath{\pi})}^{\frac{2}{3}}}{{N}^{\frac{2}{3}}}|{K}_{0}[\ensuremath{\rho}]|$ have been derived [$N$ is the number of electrons and $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ 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.