Semi-classical model and assessment of the two-electron atomic system energy in s orbitals

For physics and engineering undergraduate students, the semi-classical model formulation is a convenient approach to understanding electron behavior, which in turn facilitates the comprehension of more elaborate quantum mechanical formulations. In this line of thought, a two-electron atom semi-classical model was developed for the ground-state energy and excited states of He and ions. The two electrons are in a state occupying the same s orbital; thus, the average position distance was worked out from the classical orbital radius. Utilizing the Coulombic repulsion force between the electrons and the virial theorem, a relationship is established. Numerical results against optical spectroscopy yield an average energy relative error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>.</mml:mo> <mml:mn>23</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> . When compared against ab initio calculations, the average energy relative error is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> <mml:mo>.</mml:mo> <mml:mn>15</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> . Exact energy values of the system for helium using ICI, QD Monte Carlo, variational and CCSD methods assessed the model. These latter results comparison can serve the purpose of illustrating to the undergraduate student that semi-classical models, although not entirely accurate, can still provide a close agreement with the scientific fundamentals to comprehend the physics behind electrons’ behavior.