Local energy equation for two-electron atoms and relation between kinetic energy and electron densities

In early work, Dawson and March [J. Chem. Phys. {\bf 81}, 5850 (1984)] proposed a local energy method for treating both Hartree-Fock and correlated electron theory. Here, an exactly solvable model two-electron atom with pure harmonic interactions is treated in its ground state in the above context. A functional relation between the kinetic energy density $t(r)$ at the origin $r=0$ and the electron density $\rho (r)$ at the same point then emerges. The same approach is applied to the Hookean atom, in which the two electrons repel with Coulombic energy $e^2/r_{12}$, with $r_{12}$ the interelectronic separation, but are still harmonically confined. Again the kinetic energy density $t(r)$ is the focal point, but now generalization away from $r=0$ is also effected. Finally, brief comments are added about $He$-like atomic ions in the limit of large atomic number.