Hund's rule of maximum multiplicity

The Core Statement

WHAT it decides: the arrangement of electrons within a subshell, not which subshell fills first (that's the Aufbau principle).

WHY it works — two physical reasons:

Reduced electron–electron repulsion. Two electrons in different orbitals are spatially farther apart than two in the same orbital, so they repel less.
Exchange energy (the deeper reason). Electrons of the same spin in different orbitals are indistinguishable and can "exchange". Each such swappable pair lowers the energy by an amount $K$ (the exchange integral). More parallel-spin electrons ⇒ more exchange pairs ⇒ more stability.

HOW we use it: fill singly, spins parallel, then pair.

Deriving "Multiplicity" from Scratch

So Hund's rule = choose the electron configuration with the largest $n$ (most unpaired, all parallel).

Exchange Energy — the "WHY", made quantitative

HOW to count exchange pairs: if there are $n$ parallel-spin electrons, the number of distinct pairs that can exchange is $\text{pairs} = \binom{n}{2} = \frac{n(n-1)}{2}.$

Why this step? Any 2 out of the $n$ same-spin electrons form one exchange pair; choosing 2 from $n$ is $\binom{n}{2}$ .

Total exchange stabilization $= -K \cdot \dfrac{n(n-1)}{2}$ .

Figure — Hund's rule of maximum multiplicity

Common Mistakes (Steel-manned)

Active Recall

Recall Quick self-test (hide answers, forecast first)

Multiplicity of $2p^3$ ? → $4$ (n=3).
Multiplicity of $2p^4$ ? → $3$ (n=2).
Exchange pairs in $d^5$ ? → $\binom{5}{2}=10$ .
Why is Cr $3d^5 4s^1$ ? → Half-filled $d^5$ maximizes exchange energy/multiplicity.

Recall Feynman: explain to a 12-year-old

You've got a row of identical empty chairs (orbitals) all comfy in the same way. New kids (electrons) walk in. Instead of two kids squeezing onto one chair right away, each kid takes their own chair first, and they all face the same way (same spin). Kids sitting alone and facing the same direction are happiest — squeezing together only starts once every chair already has someone. That "everyone gets their own seat, same direction" rule is Hund's rule.

Flashcards

Hund's rule states electrons in degenerate orbitals do what before pairing?

Occupy each orbital singly with parallel spins.

Define spin multiplicity in terms of unpaired electrons n.

Multiplicity

=2S+1=n+1

, where

S=n/2

Why does maximizing unpaired parallel-spin electrons lower energy?

Less e–e repulsion (different orbitals) + more exchange energy stabilization.

Number of exchange pairs for n parallel-spin electrons?

\binom{n}{2}=n(n-1)/2

Multiplicity of a nitrogen atom (2p³)?

4 (three unpaired electrons).

Multiplicity of oxygen atom (2p⁴)?

3 (two unpaired electrons).

Which principle sets order between subshells (not Hund)?

Aufbau principle (n+l rule).

Why is Cr [Ar]3d⁵4s¹?

A half-filled 3d⁵ gives maximum multiplicity/exchange energy stability.

Magnetic moment formula from unpaired electrons?

\mu=\sqrt{n(n+2)}\,\mu_B

Multiplicity of a fully filled subshell like 2p⁶?

1 (n=0 unpaired).

Connections

Aufbau Principle — decides which subshell fills first.
Pauli Exclusion Principle — caps 2 electrons/orbital with opposite spins.
Electron Spin Quantum Number — origin of $m_s=\pm\tfrac12$ and total spin $S$ .
Exchange Energy and Half-filled Stability — the deep "why" behind Cr, Cu, Mn.
Paramagnetism and Diamagnetism — unpaired electrons ⇒ paramagnetic.
Magnetic Moment of Atoms — $\mu=\sqrt{n(n+2)}$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Hund ka rule bolta hai ki jab electrons ko degenerate orbitals (same energy wale, jaise teen $p$ orbitals ya paanch $d$ orbitals) me daalte ho, to pehle har orbital me ek-ek electron jaata hai, aur sabka spin same direction me (sab ↑). Pairing (do electron ek orbital me, ↑↓) tabhi shuru hoti hai jab har orbital me ek electron aa chuka ho. Simple funda: "bus seat rule" — pehle sab khaali seats bharo, phir doubling karo.

Aisa kyun? Do reasons. Ek, agar do electron alag orbitals me hain to woh door hote hain, isliye repulsion kam hoti hai. Do (deep reason), same-spin electrons ke beech exchange energy milti hai — jitne zyada parallel-spin electrons, utne zyada exchange pairs $\binom{n}{2}$ , utni zyada stability. Isi wajah se half-filled ( $p^3, d^5$ ) aur fully-filled ( $d^{10}$ ) configurations extra stable hote hain, aur Cr ka config $3d^5 4s^1$ ho jaata hai.

Multiplicity ka matlab: unpaired electrons $n$ ho to $S=n/2$ aur multiplicity $=2S+1=n+1$ . Toh "maximum multiplicity" ka seedha matlab hai "maximum unpaired electrons". Jaise nitrogen $2p^3$ me 3 unpaired ⇒ multiplicity 4; oxygen $2p^4$ me 2 unpaired ⇒ multiplicity 3.

Yaad rakho: Hund yeh nahi batata ki kaunsa subshell pehle bharega (woh Aufbau ka kaam hai), aur na yeh Pauli hai (jo bolta hai ek orbital me max 2, opposite spin). Hund sirf ek degenerate set ke andar electrons kaise arrange honge, yeh decide karta hai. Exam me magnetism ke questions ( $\mu=\sqrt{n(n+2)}$ ) directly is rule se aate hain.

The Core Statement

Deriving "Multiplicity" from Scratch

Exchange Energy — the "WHY", made quantitative

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Quantum Atomic Structure

Connections