The text book ground state electron configuration of the Helium atom (with two electrons and a kernel with two protons and two neutrons) is described as a 1s2 state meaning that two spherically symmetric electrons with opposite spin have identical exponentially decreasing spatial wave functions and thus form the innermost K-shell, so to speak on top of each other without separation.
However, the energy of such configurations ranges from - 2.75 to - 2.84 (a. u.) depending on the rate of exponential decay, while the observed energy is - 2.903, and so the stated 1s2 ground state is not the true physical ground state.
To handle this contradiction, the 1s2 state is subjected to various perturbations of which those of Hylleraas from 1929 with a dependence on the interelectronic distance give the correct energy by allowing the complete wave function depending on 6 spatial dimensions to account for electron separation. The energy of the perturbed wave function with separated electrons is thus correct, but its form displaying the true electron configuration of Helium is kept as a secret to the text book reader to which only the non-true 1s2 state is presented.
The successs of the Hylleraas wave function results from allowing the two electrons to be separated and thus decrease the inter electron energy. In the simplest such model the two electrons are in opposite position at a distance of d = 1/2 (a.u.) for which configuration the total energy E can be computed as follows (as in the Rutherford model displayed above):
- E = - 2 x 2/d + 2 x 1/2 x 1/ d^2 + 1/2d = - 8 + 4 + 1 = - 3
More generally, this simple model gives the following (surprisingly good) approximations of the ground state energies of the following 2 electron ions of n-electron atoms according to the formula
- E = - 2 x n/d + 2 x 1/2 x 1/ d^2 + 1/2d with d = 1/n:
- Li+ : - 7.5 (- 7.3)
- Be++ : - 14 (- 13.7)
- B+++ : - 22.5 ( - 22.2)
- C4+ : - 33 ( - 32.8)
- N5+ : - 45.5 (- 45.2)
- 06+ : - 60 (- 59.8)
- F7+ : -76.5 (- 76.3)
- Ne8+ : -95 (- 94.9)