Correlated energy from radial density-energy relations

Here, we demonstrate that the radial distribution function can be mapped into a radial density-energy space and the relationship between the radial density and radial energy is linear for the ground and excited states of helium-like systems; the gradient of the resulting straight line delivers the energy of the state considered. To utilize this finding, a simple analytical expression for the total energy in terms of the density at the most probable nucleus-electron distance of the systems considered is derived using a fitting procedure.

Automated summary

We find that the radial distribution function can be transformed into a radial density-energy space, and there is a linear relationship between the radial density and radial energy for the ground and excited states of helium-like systems. The gradient of the resulting straight line provides the energy of the considered state. Based on this finding, we derive a simple analytical expression for the total energy in terms of the density at the most probable nucleus-electron distance of the systems considered using a fitting procedure.