The effects of the Pauli exclusion principle in determining the ionization energies of the helium atom and helium-like ions

Abstract

For helium and helium-like ions, we have examined the differences between the values of the ionization energies as calculated from the Bohr theory and those measured in experiments. We find that these differences vary linearly with the atomic number of the system. Using this result, we show how the Bohr model for single-electron systems may be modified in a simple way so that it may be applied to two-electron systems as well.

Few-electron atoms with linear Bohr–Sommerfeld electron paths

“…No, our purpose is to illustrate how concepts from the old theory can be applied, the results thus obtained and the shortcomings of the same, all of which can be contrasted with the modern quantum-mechanical analysis. In this sense, the objective of this paper is very similar to that of Deeney and O'Leary [14] who attribute the main shortcoming of the Bohr atom to its neglect of the Pauli principle. In this paper, we will lend support to this contention quantitatively.…”

“…The ground-state energy of a two-electron atomic system within the original Bohr picture where the two electrons share the same circular orbit but displaced along it by a phase difference of π (which is the phase difference which minimizes the interelectron repulsion) can easily be shown (see e.g. reference [14]) to be E = −(Z − 1/4) 2 /2 which differs from equation ( 11) (vide infra) merely by a constant term. This result corresponds to the situation of mechanical equilibrium between the two electrons in the shared circular orbit and does not require any further approximations within the model assumptions.…”

“…Naively, one might expect that electron correlation will decrease the ground-state energy further, on account of the Coulombic repulsion between electrons. However, as pointed out by Deeney and O'Leary [14] (among others), the Pauli principle leads to an apparent attraction between the ground-state electrons (sometimes phrased in terms of a 'Pauli potential' within density-funtional theory, but commonly called 'exchange' within wavefunction theory). Inclusion of the Pauli principle will in this case increase the calculated energy.…”

Calculating the ground state energy of hydrogen molecules and helium hydride ions using Bohr’s quantum theory

“…Thus the existence of α in equation (1) will reduce the repulsive electrical force between two hydrogen nuclei. According to Deeney's calculation [16], the value of α can be taken as 0.129.…”

Lateral Position Uncertainty of Electrons in Bohr Hydrogen-like Atoms: An Implication of Heisenberg Uncertainty Principle

“…Deeney et al pointed out differences between the values calculated from the Bohr Theory and those found by experiments according to atomic number Z. Author has mentioned a need for additional mechanisms, whose effects should be added to those already present in the Bohr Theory in order to account reductions in the observed ionization energies[11]. Later,Kuo presented analyses for the uncertainties in measurements of the radial position r D , radial momentum p D , relative dispersion of radial position, r r / D and the product of both uncertainties, p r D D in a non-relativistic hydrogen-like atoms depending on the quantum numbers [12].…”