Intuition The one core ideaElectrons that share the same "spin" and live in the same subshell can secretly swap places , and every possible swap makes the atom a little more stable. So an atom sometimes rearranges its electrons — even breaking the usual filling order — just to maximize the number of these swaps, which happens exactly at half-filled and fully-filled subshells.
This page assumes you have seen NONE of the notation in the parent note. We build every symbol from the ground up, in an order where each idea rests on the one before it. When you finish, re-reading the parent topic should feel like reading plain English.
Definition Nucleus and electronsEvery atom has a tiny, heavy, positively charged centre called the nucleus , surrounded by light, negatively charged particles called electrons . Opposite charges attract, so the nucleus pulls the electrons inward; like charges repel, so the electrons push each other apart.
Picture a bee-swarm around a lamp: the lamp (nucleus) pulls, the bees (electrons) jostle. The topic is ultimately about how the bees arrange themselves to be as calm (low-energy) as possible.
Intuition Why "energy" is the refereeNature prefers the arrangement with the lowest total energy . "Stable" just means "low energy." Every rule in this chapter is really answering: which arrangement of electrons has the lowest energy?
Electrons don't float randomly; each one has an address made of layers.
Definition Shell → subshell → orbital
A shell is a main energy level, labelled by a whole number called the principal quantum number n = 1 , 2 , 3 , … n n
A subshell is a room-type on that floor, labelled by a letter: s , p , d , f
An orbital is a single "seat" — a region where up to two electrons can sit.
Definition The letter is really a number: the quantum numberl azimuthal quantum number l s ↔ l = 0 , p ↔ l = 1 , d ↔ l = 2 , f ↔ l = 3 l the letter and the number are the same fact — d is l = 2
The letter tells you how many orbitals (seats) the subshell has:
Subshell
l Number of orbitals
Max electrons (2 ×
s 0
1
2
p 1
3
6
d 2
5
10
f 3
7
14
Look at the figure: each box is one orbital (seat). The p d f one electron per box and "full" means two per box .
3 d 5 ( shell number ) ( subshell letter ) ( how many electrons ) 3 d 5 d 5 electrons. Since d one per box = half-filled.
Worked example Reading the notation
2 p 3 p half-filled .2 p 6 fully-filled .3 d 10 fully-filled .
This is why the parent note keeps saying p 3 , d 5 , f 7 p 6 , d 10 , f 14 one-per-box and two-per-box .
Every electron carries a built-in property called spin , which comes in exactly two flavours. We draw them as arrows: ↑ ("spin up") and ↓ ("spin down"). Think of it as each electron holding a little compass needle that can only point up or down — nothing in between.
Why do we need spin? Because two rules of the atom depend entirely on it:
Definition Pauli Exclusion Principle (the seating rule)Two electrons in the same orbital (same box) must have opposite spins — one ↑ and one ↓. See Pauli Exclusion Principle . This is why each box holds at most two electrons: once a box has an ↑ and a ↓, there is no third arrow left to add.
Definition Parallel spinsTwo electrons have parallel spins if their arrows point the same way (both ↑, or both ↓). This word "parallel" is the heartbeat of the whole topic — the stability bonus comes only from parallel-spin electrons.
Now: given several boxes and several electrons, how do we fill them ?
Definition Hund's Rule of Maximum MultiplicityWhen filling orbitals of the same subshell , electrons first go one to a box, all with parallel spin , before any box gets a second electron. See Hund's Rule of Maximum Multiplicity .
Definition What "multiplicity" meansMultiplicity is a score for how many spins point the same way. If S + 2 1 − 2 1 unpaired ), the multiplicity is 2 S + 1 more parallel unpaired spins → bigger multiplicity . "Maximum multiplicity" therefore just means "make as many spins parallel as the boxes allow." For 3 d 5 S = 2 5 = 2 ( 2 5 ) + 1 = 6 d
Intuition Why one-per-box first?Two electrons crammed in the same box repel strongly (they're close). Spreading them into separate boxes keeps them apart — cheaper. And keeping their spins parallel unlocks the exchange bonus (next section). So 3 d 5 ↑↑↑↑↑ — five boxes, one parallel electron each. That is why half-filled subshells are special: they are the largest set of all-parallel electrons a subshell can hold.
The parent note counts "exchange pairs" using a symbol that looks scary but isn't.
Common mistake A quick warning about the letter usedUp to now n shell number (principal quantum number). In this section only, we borrow the general letter n "how many parallel electrons we are counting" — a different quantity. To avoid confusion we write the count as q ( 2 q ) n q
Definition The combination symbol( 2 q ) ( 2 q ) q unordered pair from q same pair as picking B then A.
Worked example Counting by hand to trust the formula5 electrons labelled 1,2,3,4,5. List the pairs: (1,2)(1,3)(1,4)(1,5)(2,3)(2,4)(2,5)(3,4)(3,5)(4,5) → 10 pairs. Formula: 2 5 × 4 = 10 2 4 × 3 = 6 10 6
s 1 s one box , so its "half-filled" state is s 1 q = 1 ( 2 1 ) = 2 1 ⋅ 0 = 0 A lone electron has nobody to swap with, so it earns no exchange stabilization at all. This is why the special "half-filled" story is really about p 3 , d 5 , f 7 not about s 1 s
Now the payoff. When two electrons have parallel spin , quantum mechanics says they are indistinguishable and can swap places without creating a new state — and each such possible swap lowers the energy .
Definition The exchange integralK K overlap . Bigger overlap → bigger K
K Why it feels equal: the tidy formula multiplies by one K The fix: K which two orbitals overlap. A 3 d 3 d K 3 d 4 s K K ij i j K approximation — good for intuition, not exact.
Intuition Reading the sum symbol∑ ∑ i < j what to add: one term − K ij ( 2 q ) − K
To explain why Cr and Cu are "anomalies," you first need the normal order — and we can draw it rather than just name it.
Definition Aufbau PrincipleElectrons fill the lowest-energy subshell first, building up one at a time (see Aufbau Principle ). The question is: which subshell is lowest?
( n + l ) n l s = 0 , p = 1 , d = 2 , f = 3
Compute n + l Smaller n + l
If two subshells tie on n + l smaller n fills first.
This is exactly the (n+l) Rule and Orbital Energies ; the figure below builds its ladder step by step.
Worked example The key near-tie:4 s 3 d
4 s n + l = 4 + 0 = 4 3 d n + l = 3 + 2 = 5
Since 4 4 s normally fills before 3 d close their energies sit on the ladder — this near-tie is the whole reason a small exchange bonus can flip the order for Cr and Cu , letting an electron slip into 3 d 3 d 5 3 d 10
Definition Ionization energyThe energy needed to rip one electron off an atom. See Ionization Energy Trends . If an atom is extra-stable (half/full subshell), it holds its electrons tighter , so its ionization energy is unusually high (e.g. nitrogen's 2 p 3
Nucleus pulls, electrons repel
Shells subshells orbitals
Hunds rule max multiplicity
Half filled and full subshells
Normal filling 4s before 3d
Cover the right side; can you answer before revealing?
What the exponent in 3 d 5 The number of electrons in that subshell (here 5 electrons in the d
How many orbitals (boxes) each subshell has s = 1 , p = 3 , d = 5 , f = 7
The two quantum numbers used on this page n l s = 0 , p = 1 , d = 2 , f = 3
What "half-filled" means in box language Exactly one electron per box, all parallel spins (p 3 , d 5 , f 7
Why s 1 A lone s ( 2 1 ) = 0
What "multiplicity" means The score 2 S + 1
What "parallel spins" means Two electrons whose spin arrows point the same way (both ↑ or both ↓).
The Pauli seating rule A single orbital holds at most two electrons, and they must have opposite spins.
What Hund's rule tells you to do Fill each box singly with parallel spins before pairing any box.
The meaning of ( 2 q ) The number of unordered pairs you can pick from q 2 q ( q − 1 )
Why ( 2 5 ) = 10 ( 2 4 ) = 6 The difference of 4 is the extra 3 d 3 d 3 d 5 4 s 1
What K Energy saved per possible swap of a parallel-spin pair; it depends on how much the two orbitals overlap.
Why K 3 d , 3 d > K 3 d , 4 s 3 d 3 d 3 d 4 s
How the ( n + l ) 4 s 3 d 4 s n + l = 4 3 d n + l = 5 4 s
