One electron in φ·α
↑
φ
There is no occupancy conflict.
Page 01 · occupancy before hybridization
Pauli does not forbid mixing orbitals.
It forbids something more specific: placing two electrons in the same spin-orbital. That distinction lets us talk about orbital superposition without confusing it with electron filling.
One electron: superposition is ordinary quantum mechanics
Let χ₁, χ₂, … be orthonormal one-electron orbitals. A normalized linear combination is another perfectly valid one-electron orbital. The coefficients determine its shape, phase and nodal structure.
φ(r) = Σᵢ cᵢ χᵢ(r)
Σᵢ |cᵢ|² = 1
Nothing in Pauli's principle singles out “pure” 2s or 2p as more allowed than a normalized mixture of them.
One-electron stateschematic
↑
φ
φ = cos θ · 2s + sin θ · 2pz
Still one orbital. Still one electron.
The exclusion rule
The object Pauli constrains is the spin-orbital
A spin-orbital combines a spatial orbital φ(r) with a spin state α or β. Electrons are fermions, so the total many-electron wavefunction must change sign when any two electrons are exchanged.
Ψ(...,xᵢ,...,xⱼ,...) = −Ψ(...,xⱼ,...,xᵢ,...)
Two opposite spins in φ
↑↓
φ
φ·α and φ·β are different spin-orbitals.
The same spin-orbital twice
↑↑×
φ
The antisymmetrized two-electron state is identically zero.
Interactive schematic
Mix the orbital; Pauli's statement does not change
A subtler point
Different-looking orbitals can represent the same occupied subspace
Suppose φ₁ and φ₂ are two occupied orthonormal orbitals. A unitary rotation can produce φ′₁ and φ′₂ that look different while spanning exactly the same two-dimensional occupied space.
[φ′₁] [ cos θ sin θ][φ₁]
[φ′₂] = [−sin θ cos θ][φ₂]
⇔
span{φ₁,φ₂}
=
span{φ′₁,φ′₂}
This is why orbital pictures are useful but not unique. The many-electron state depends on the occupied space and its
antisymmetry, not on one privileged drawing of individual orbitals.
Connection to the familiar filling diagrams
Orbital boxes are a bookkeeping device
Aufbau and Hund's rule help decide how electrons are distributed among available orbitals in simple atomic models. Pauli supplies the occupancy constraint for each orbital. For a p subshell, three spatial orbitals are available, so the subshell can hold up to six electrons—not two.
Example: p³ and p⁶three spatial p orbitals
↑
pₓ
↑
pᵧ
↑
p_z
→
↑↓
pₓ
↑↓
pᵧ
↑↓
p_z
Schematic inspiration: 88Guru · Electron Configuration.
The illustrations and wording on this page are original; the Pauli statement is given at the orbital/spin-orbital level.