Helium atom

A neutral helium atom contains two electrons bound to a nucleus of charge $+2e$. Even with a fixed nucleus and nonrelativistic kinetics, the repulsion $e^2/(4\pi {ϵ}_0{r}_{12})$ prevents exact separation into two hydrogenic one-electron problems. Its ground configuration is $1s^2$ with total spin zero, while distinct occupied orbitals allow both singlet helium states and triplet helium states.

The low $1s2s$ levels of a helium atom lack an allowed direct electric dipole decay to the ground $1s^2{}^1{S}_0$ level. The triplet is excluded by the spin selection rule, and the singlet-to-singlet $S$ transition is excluded by parity and orbital selection. Weaker radiative channels and collisions can still de-excite these states; metastable does not mean infinitely lived.

For fixed orthonormal orbitals $a,b$, symmetrizing or antisymmetrizing $a(1)b(2)$ gives electron-repulsion expectations $J_{ab}+K_{ab}$ and $J_{ab}-K_{ab}$. Antisymmetry of the complete electronic state therefore places the singlet above the triplet by $2K_{ab}$, with $K_{ab}$ the positive Coulomb exchange integral. Orbital relaxation and further correlation refine this comparison. Exchange splitting here is a consequence of Fermi statistics and the electrostatic interaction, not a separate classical spin-spin force.

A helium atom electronic state with total spin one. Its symmetric spin function requires an antisymmetric spatial wavefunction, excluding two electrons in the same spatial orbital. The three spin projections are magnetic substates of the same spin multiplet, not three separately distinguishable particles.

A helium atom electronic state with total spin zero. Its antisymmetric two-spin function requires a symmetric spatial wavefunction. Its approximate spectroscopic family is distinct from the triplet family under spin-conserving electric dipole transitions.

A bound-state configuration with one electron predominantly in $1s$ and the other in a different orbital $n\ell$. The two assignments of identical electrons to these orbitals must be combined with the appropriate exchange symmetry of two identical spin-one-half fermions. Symmetric spatial combinations accompany the singlet state; antisymmetric combinations accompany the triplet state.