Two-Electron Atoms

Phase 2: Building Complexity

Syllabus Goal

Bridge between hydrogen and complex atoms. Role of Pauli exclusion principle.

Helium Atom

Hamiltonian includes electron-electron repulsion term which makes it unsolvable analytically: $$ H = -\frac{\hbar^2}{2m} (\nabla_1^2 + \nabla_2^2) - \frac{Ze^2}{r_1} - \frac{Ze^2}{r_2} + \frac{e^2}{|\mathbf{r}_1 - \mathbf{r}_2|} $$

Pauli Exclusion Principle

Wave function must be antisymmetric under exchange of fermions. $$ \Psi(1,2) = -\Psi(2,1) $$

• Parahelium: Symmetric space (Singlet spin).
• Orthohelium: Antisymmetric space (Triplet spin).