Polyelectronic systems

The hydrogen atom and the molecular ion ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{+}}$ are the only two systems whose eigenstates are known exactly. For all other systems, we obtain Hamiltonians containing electrostatic electron-electron interaction terms that make it impossible to solve Schrödinger's equation analytically. Such a phenomenon is very common in physics, as static systems of interacting particles generally admit no simple solutions due to inter-particle interaction terms.

We must therefore abandon the usual mathematical arsenal of eigenfunctions...