More than One Electron Atoms

Author

Daniel Fischer

Beyond Hydrogen

In previous modules, we focused on the hydrogen atom, whose single electron allows for an exact analytical solution of the Schrödinger equation. Real atoms, however, contain two or more electrons, and their mutual Coulomb interaction makes the problem far more complex — and much richer in physics.

In this chapter, we will build on the concepts developed for hydrogen and extend them to multi-electron atoms. We will see how the additional electron–electron interactions lead to energy shifts and complex coupling schemes that determine the organization of the periodic table and the spectra of real elements.

Scope of this Section

In this module, we explore the physics of multi-electron systems. Our main goals are to:

Describe how the Coulomb repulsion between electrons modifies atomic energies and wavefunctions.
Explain the helium atom as the simplest example of a two-electron system.
Understand the concept of configuration interaction and Hartree–Fock theory as approximations to many-electron systems.
Connect atomic shell structure to the organization of the periodic table.
Understand how angular momenta of multiple electrons combine (\(LS\) and \(jj\) coupling).
Recognize how excited states and spectral lines emerge from these multi-electron configurations.

Structure of this Module

Helium (Basics) – The simplest multi-electron atom and the first example of electron–electron interaction.
Two-Electron States in Helium – Symmetry, spin, and exchange effects in the helium atom.
Calculations for Multielectron Atoms – Approximation methods, including the Hartree and Hartree–Fock approaches.
Shell Model and Periodic Table – How atomic structure leads to the periodic organization of the elements.
Angular Momentum Coupling in Many-Electron Atoms – \(LS\) and \(jj\) coupling, term symbols, and selection rules.
Excited States – Understanding optical spectra and transitions in complex atoms.

Key idea:
Each step beyond hydrogen introduces a new layer of complexity — but also deeper insight into how quantum mechanics explains chemical behavior and atomic spectra.