Hund's rules
/ HOONT /
A single electron configuration such as carbon's 2p^2 does not fix one energy — the two p-electrons can share their spins and orbital motion in several distinct ways, each a different atomic term with its own energy. Hund's rules are three quick prescriptions that tell you, without diagonalizing anything, which of those arrangements is the ground state — the lowest-energy term of the isolated atom.
Applied in strict order of priority: (1) maximize the total spin S, i.e. pick the largest spin multiplicity 2S+1; (2) among the terms tied on S, maximize the total orbital angular momentum L; (3) among those tied on both S and L, take the total angular momentum J = |L - S| if the open subshell is less than half filled, and J = L + S if it is more than half filled (a half-filled subshell has L = 0, so J = S). Rule 1 is exchange physics: electrons with parallel spins are forced apart by antisymmetry, lowering their mutual Coulomb repulsion. Rule 2 is a milder version of the same avoidance for orbital motion. Rule 3 is the sign of the spin-orbit interaction, which flips between the first and second half of a shell.
You meet Hund's rules whenever you assign a ground-state term symbol, explain why oxygen (2p^4) is a spin-triplet paramagnet, or predict the magnetic moment of a rare-earth ion. The honest caveats: they are reliable only for the ground term of an isolated atom in the LS-coupling regime (light-to-moderate atoms), rule 3 assumes LS coupling and fails for heavy atoms where spin-orbit rivals the electron-electron term splitting, and the rules say nothing trustworthy about the ordering of excited terms.
Carbon 2p^2: rule 1 gives S = 1 (a triplet), rule 2 gives L = 1 (a P term), so the term is ^3P; the subshell is less than half filled, so rule 3 selects J = |1 - 1| = 0. The ground term is ^3P_0, and the excited ^1D_2 and ^1S_0 terms lie above it.
The three rules pick the single ground term out of the multiplet a configuration generates.
Hund's rules order the ground term reliably but should not be trusted to order excited terms — the second and third rules in particular can invert for excited configurations.