Introduction: The First Multi-Electron Atom
Helium, with two electrons, is the simplest atom that can't be solved exactly. It introduces exchange energy—a purely quantum effect with no classical analog.
The Hamiltonian
\[\hat{H} = -\frac{\hbar^2}{2m}(\nabla_1^2 + \nabla_2^2) - \frac{2e^2}{r_1} - \frac{2e^2}{r_2} + \frac{e^2}{r_{12}}\]The electron-electron repulsion \(e^2/r_{12}\) makes it non-separable.
Spatial and Spin Parts
Total wavefunction = spatial × spin, must be antisymmetric overall.
Parahelium (singlet): Antisymmetric spin, symmetric spatial
Orthohelium (triplet): Symmetric spin, antisymmetric spatial
Exchange Energy
Symmetric spatial wavefunctions allow electrons closer together → higher repulsion.
Antisymmetric spatial wavefunctions keep electrons apart → lower repulsion.
This energy difference is called exchange energy.
The Quantum Connection
Exchange energy is crucial for magnetism. Ferromagnetism arises when exchange favors parallel spins (triplet states). Helium shows the pattern: orthohelium (triplet) has lower energy than parahelium (singlet) for excited states, because antisymmetric spatial wavefunctions reduce electron-electron repulsion.