Side quest · Aufbau, Pascal and Fibonacci

Why an orbital-filling toy model can land on Fibonacci numbers.

Aufbau gives us an ordered list of increasingly costly one-electron states. Fibonacci enters only after we add a separate combinatorial rule: among a line of ordered positions, no two occupied positions may sit next to one another.

Step 1 · physical ordering idea

Aufbau: lower-energy states are filled before higher-energy ones

In the familiar atomic picture, electrons are placed into low-energy orbitals first, subject also to Pauli and Hund. The common Madelung \(n+\ell\) order is a useful mnemonic, not an exact universal energy theorem: orbital energies depend on the atom and on occupancy, and well-known configuration exceptions occur.

typical order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < ...

Important separation. Real Aufbau filling does not say “occupied orbitals may not be adjacent.” That rule is introduced below only to create the Fibonacci counting problem.

Step 2 · binary abstraction

Turn an ordered list into 0/1 slots

Let \(x_i=1\) mean position \(i\) is selected and \(x_i=0\) mean it is not.

x = (x₁,x₂,...,x_N), xᵢ ∈ {0,1}

Now impose the extra rule

xᵢ xᵢ₊₁ = 0

so consecutive 1s are forbidden.

Interactive

Choose N

N = 5

Step 3 · exact counting

Exactly k occupied positions

Choose k non-adjacent slots

C(N−k+1, k)

Compress away the mandatory gaps between occupied sites.

Sum over k

Σₖ C(N−k+1,k)

The largest possible k is \(\lceil N/2\rceil\).

Fibonacci identity

Σₖ C(N−k+1,k) = F(N+2)

Convention: \(F_0=0,\ F_1=1\).

k occupied	count	formula
Total

Step 4 · Pascal’s triangle

The coefficients lie on a shallow diagonal

For fixed \(N\), the coefficients \(\binom{N-k+1}{k}\) are entries of Pascal’s triangle. Their shallow-diagonal sum is \(F_{N+2}\).

Why Fibonacci appears

The recurrence is even simpler

Let \(A_N\) be the number of legal strings of length \(N\). If the first slot is 0, there are \(A_{N-1}\) possibilities. If it is 1, the next slot must be 0, leaving \(A_{N-2}\) possibilities.

A_N = A_(N−1) + A_(N−2), A_0 = 1, A_1 = 2

Hence \(A_N=F_{N+2}\).

Keep the categories separate

Physics retained

an approximate energy ordering of atomic orbitals
Pauli and Hund constrain real electron filling
Aufbau organizes low-to-high energy occupation

Extra combinatorial rule

reduce each ordered position to a binary choice
forbid adjacent occupied positions
that nearest-neighbour exclusion creates Fibonacci counting

So the statement is not “Aufbau generates Fibonacci numbers.” The correct statement is: an ordered orbital list, combined with a nearest-neighbour exclusion rule, has the same counting structure as Fibonacci strings.

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