Foundations — Hund's rule of maximum multiplicity

This page builds every word and symbol the parent note leans on, starting from a reader who has never seen an orbital diagram. Read top to bottom: each idea uses only the ones above it.

1. The atom as a set of "seats"

Look at the figure: three chairs drawn as boxes. That row of boxes is the picture we will reuse for the whole topic — every arrow we ever draw sits inside one of these boxes.

Why the topic needs it: Hund's rule never talks about single electrons floating alone; it always talks about how electrons are distributed across a row of chairs. Without the "chair" picture, phrases like "fill singly" have nothing to fill.

2. Degenerate — chairs that are equally comfy

• The three $p$ orbitals ($p_x,p_y,p_z$) of a shell are degenerate — three equally comfy chairs.
• The five $d$ orbitals are degenerate — five equally comfy chairs.

Why the topic needs it: "Maximum multiplicity" is the tie-breaker used only among degenerate orbitals. The word "degenerate" is the entry condition for the whole rule. Which set of chairs fills first (2p vs 3s...) is a different question, answered by the Aufbau Principle — not here.

3. Electron spin and the arrows ↑ ↓

The arrow is not literal spinning — it is a label with exactly two settings, like a coin that is either heads or tails.

Why the topic needs it: "All parallel spin" means all arrows pointing the same way (all ↑). "Paired" means ↑ and ↓ together in one chair. Every sentence of Hund's rule is about arrow directions, so we must have the arrow before anything else.

4. Why max 2 per chair, opposite arrows — Pauli

Why the topic needs it: This is the ceiling Hund works under. Pauli says "≤ 2 per chair, opposite when paired"; Hund then chooses how to arrange electrons under that ceiling. Two electrons can never be ↑↑ in the same chair — that possibility is forbidden, which is exactly why "parallel" forces electrons into separate chairs.

5. Counting: $n$, the number of unpaired electrons

Why the topic needs it: Every headline number in the parent — multiplicity, exchange pairs, magnetic moment — is a function of $n$ alone. Nail $n$ and the rest is arithmetic.

6. Total spin $S$ and the fraction $\frac{n}{2}$

Read this as: "line up all the lonely up-arrows, add half for each." Three lonely electrons $\Rightarrow S=\frac{3}{2}$.

Why the topic needs it: The parent's boxed formula, multiplicity $=2S+1$, is written in terms of $S$. We need $S$ as the bridge from the count $n$ to the multiplicity number.

7. Multiplicity and why "$2S+1$"

The figure shows the "spin ladder": for $S=1$, the ladder has rungs at $-1,0,+1$ — that's $2S+1=3$ rungs. Count the rungs, and that is the multiplicity.

Why "maximum multiplicity"? Since multiplicity $=n+1$, largest multiplicity = largest $n$ = most unpaired electrons. That is literally what Hund's rule chooses. The scary phrase reduces to "make $n$ as big as you can."

8. Choosing 2 from $n$ — the symbol $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$

Where it comes from (the picture): to make a pair, pick a first item ($n$ ways) then a second ($n-1$ ways) = $n(n-1)$ ordered picks; but "A then B" is the same pair as "B then A", so divide by 2.

Why the topic needs it: Each pair of same-spin electrons that can "swap" lowers energy by $K$ (see next). The number of such swappable pairs is exactly $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$, so this symbol is how the parent turns "spread out and parallel" into a stability number. Deep dive: Exchange Energy and Half-filled Stability.

9. Exchange integral $K$ (the stability unit)

Why the topic needs it: This is the quantitative reason parallel spins win — the deeper of the parent's two "why"s, and the reason $d^5$/$d^{10}$ are extra stable.

10. Magnetic moment $\mu$ (where $n$ shows up physically)

Why the topic needs it: It is the measurable consequence of Hund's arrangement — you can literally weigh unpaired electrons in a magnetic field.

How these feed the topic

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