Hund's Rules
Electrons are social creatures in theory but loners in practice. The Pauli exclusion principle forbids them from sharing quantum states, but Hund's rules describe how they actually distribute themselves among the available states when an atom's orbitals are being filled. Discovered by Friedrich Hund in 1925, these rules capture an empirical truth: electrons prefer to maximize their total spin and then minimize their total orbital angular momentum, subject to the constraints of quantum mechanics.
Rule One: Maximum multiplicity wins. When electrons fill degenerate orbitals — orbitals at the same energy level, like the three 2p orbitals — they spread out and align their spins parallel to one another before pairing up. This isn't just a suggestion. It's energetically favorable, and the reason is both electrostatic and quantum-mechanical.
Why Electrons Spread Out
Consider carbon, which has two electrons in its 2p subshell. The 2p subshell has three orbitals (corresponding to the magnetic quantum numbers -1, 0, and +1). Hund's first rule says: put one electron in each of two different orbitals, with both spins pointing the same way. Do not put both electrons in the same orbital with opposite spins.
Why? Two effects combine. First, electrons that occupy different orbitals spend more time far apart on average, reducing their electrostatic repulsion. Second, and more subtly, the exchange interaction — a purely quantum effect arising from the antisymmetry requirement on the fermionic wave function — lowers the energy when parallel-spin electrons occupy different orbitals. The antisymmetric spatial wave function associated with parallel spins has a node when the two electrons are at the same point in space. The electrons literally cannot be found at the same location simultaneously. This correlation reduces Coulomb repulsion without any classical force being invoked.
The Rules in Full
Rule One — Maximum S: Among all microstates arising from a given electron configuration, the term with the highest total spin S has the lowest energy. Electrons in degenerate orbitals prefer parallel spins.
Rule Two — Maximum L: For a given spin multiplicity, the term with the largest total orbital angular momentum L lies lowest. This is a consequence of the fact that larger L values tend to correlate with spatial arrangements that further separate the electrons.
Rule Three — J depends on filling: For a subshell that is less than half full, the level with the smallest total angular momentum J (where J = |L − S|) is the ground state. For a subshell more than half full, the level with the largest J (= L + S) is lowest. This rule involves spin-orbit coupling, which we will encounter separately.
Real-World Significance
Hund's rules explain why oxygen is paramagnetic. O₂ has two unpaired electrons, and those unpaired electrons give the molecule a magnetic moment. Liquid oxygen sticks between the poles of a magnet — a dramatic demonstration of Hund's first rule at work.
The rules also determine the magnetic properties of transition metal complexes, the colors of coordination compounds, and the structure of the periodic table's d-block. Iron, cobalt, and nickel owe their ferromagnetism to the way d electrons distribute themselves according to Hund's rules, creating large unpaired spin populations.
Hund's rules are simple in statement and profound in consequence. They reveal that nature's preference for order — spins aligned, energies minimized — is already baked into the quantum world long before we bring thermodynamics or external fields into the picture.