Multielectron atoms — spin, Pauli, and the periodic table

The electron is a tiny magnet with two faces — how spin and the Pauli exclusion principle force Slater determinants on us and write the periodic table for free.

Opening

In chapter 2 we saw that a single electron in a hydrogen atom lives in an orbital labelled by three quantum numbers $(n, \ell, m_\ell)$. This chapter adds one decisive piece to that picture — spin. The moment spin enters, an orbital can host two electrons rather than one, and that ceiling of “two” is enforced — in a way that refuses to factor apart — by the Pauli exclusion principle, expressed most cleanly as the vanishing of a determinant. By the end of the chapter you should be able to reconstruct, rather than memorise, the electron configurations from helium to uranium, and to say in one line which quantum number each axis of the periodic table is a picture of.

Main 1 — Spin: the second face the electron carries around

Besides its spatial position, every electron carries one unit of an intrinsic angular momentum called $\vec S$. Quantum mechanics fixes its magnitude to

$$|\vec S| = \hbar \sqrt{s(s+1)}, \qquad s = \tfrac{1}{2},$$

where $\hbar$ (h-bar, the reduced Planck constant) sets the unit of every angular momentum in quantum theory. The crucial point is that $s = 1/2$ is nailed down for an electron — the electron is a “half-integer spin” particle, a fermion.

Spin’s other degree of freedom is its projection onto a chosen axis (call it $z$):

$$S_z = m_s \hbar, \qquad m_s = \pm \tfrac{1}{2}.$$

So the spin of an electron takes only two values: “up” ($m_s = +1/2$, written $\uparrow$) and “down” ($m_s = -1/2$, written $\downarrow$). The consequence: each spatial orbital comes automatically paired with two spin-orbitals — one with spin up, one with spin down.

A single electron is now fully specified by four quantum numbers, $(n, \ell, m_\ell, m_s)$, with $n = 1, 2, 3, \dots$, $\ell = 0, 1, \dots, n-1$, $m_\ell = -\ell, \dots, +\ell$, and $m_s = \pm 1/2$. Following chemists’ convention we call the $\ell = 0, 1, 2, 3$ orbitals $s, p, d, f$ respectively. (That "$s$" is not the spin $s$; the letter is a 19th-century spectroscopy fossil — sharp — and the collision of names is purely an accident.)

Main 2 — Pauli exclusion and Slater determinants

Because electrons are fermions, swapping the labels of two electrons must multiply the total wavefunction by $-1$ — that is, the wavefunction must be antisymmetric:

$$\Psi(1, 2) = - \Psi(2, 1).$$

The labels “1, 2” stand jointly for the spatial coordinates and spin of each electron. This is not a convention dressed up as a rule — it is an experimentally verified fact about nature, and every half-integer-spin particle obeys it.

Suppose two electrons occupy distinct spin-orbitals $\phi_a$ and $\phi_b$. The simplest wavefunction that satisfies the antisymmetry condition is

$$\Psi(1, 2) = \frac{1}{\sqrt{2}}\left[\phi_a(1)\phi_b(2) - \phi_a(2)\phi_b(1)\right] = \frac{1}{\sqrt{2}}\det\!\begin{pmatrix}\phi_a(1) & \phi_b(1) \\ \phi_a(2) & \phi_b(2)\end{pmatrix}.$$

The right-hand side is the determinant of a 2×2 matrix; a multielectron wavefunction written this way is called a Slater determinant. A determinant changes sign when two rows are swapped and vanishes when two columns coincide, and these two properties immediately deliver:

• Swapping the electron labels 1 and 2 (= swapping rows) flips the sign of $\Psi$ — antisymmetry is built in automatically.
• If the two spin-orbitals coincide, $\phi_a = \phi_b$, the two columns of the matrix coincide and $\Psi \equiv 0$ — no two electrons may share a single spin-orbital. This is the Pauli exclusion principle in its strongest form.

Slater determinants generalise to $N$ electrons: the antisymmetric $N$-electron wavefunction is the determinant of the $N \times N$ matrix $\phi_i(j)$ normalised by $\sqrt{N!}$. This form returns in earnest in chapter 12 when we set up molecular orbital theory.

Main 3 — Reading the periodic table: Aufbau and Hund’s rule

With these two tools we now build the periodic table. The Aufbau (build-up) principle is almost embarrassingly simple — fill the lowest-energy spin-orbital first. Each spatial orbital has room for two electrons (one with each $m_s$); when it is full move on. The average filling order seen in multielectron atoms is

$$1s \to 2s \to 2p \to 3s \to 3p \to 4s \to 3d \to 4p \to 5s \to 4d \to 5p \to \cdots$$

with capacities $2, 6, 10, 14$ for the $s, p, d, f$ subshells respectively (which is just $2(2\ell+1)$). Two observations to fix in mind:

1. The $s, p, d, f$ blocks match the vertical groupings of the periodic table directly — 2 columns of $s$-block, 6 of $p$-block, 10 of $d$-block, 14 of $f$-block.
2. The slightly surprising fact that $4s$ fills before $3d$ is exactly what makes the first row of transition metals interesting. The subtle chemistry of transition metals begins right at that crossover.

One more rule completes the construction: Hund’s rule — when several orbitals of equal energy are available (e.g. the three $2p$ orbitals $p_x, p_y, p_z$), electrons first spread out one to each orbital with parallel spins before any pairing begins. That is why carbon (C, six electrons) sits in the ground state $1s^2 2s^2 2p^2$ with its two $2p$ electrons in two different orbitals and spins aligned, rather than paired in a single orbital. This single fact sets the stage for carbon chemistry — and therefore, essentially, all of organic chemistry.

In Python

# Build the helium ground-state two-electron Slater determinant by hand
# and verify antisymmetry under electron-label exchange.
# Spatial 1s orbital phi(r) = (1/sqrt(pi)) exp(-r), with Bohr radius a_0 = 1.
import numpy as np

def phi_1s(r):
    return np.exp(-r) / np.sqrt(np.pi)

def alpha(spin):           # 1 only on spin "up"
    return 1.0 if spin == "up" else 0.0

def beta(spin):            # 1 only on spin "down"
    return 1.0 if spin == "down" else 0.0

def chi_up(r, spin):       # 1s with spin up (spin-orbital)
    return phi_1s(r) * alpha(spin)

def chi_dn(r, spin):       # 1s with spin down
    return phi_1s(r) * beta(spin)

def slater(e1, e2):
    r1, s1 = e1
    r2, s2 = e2
    M = np.array([[chi_up(r1, s1), chi_dn(r1, s1)],
                  [chi_up(r2, s2), chi_dn(r2, s2)]])
    return np.linalg.det(M) / np.sqrt(2)

e1 = (0.5, "up")
e2 = (1.2, "down")
psi_12 = slater(e1, e2)
psi_21 = slater(e2, e1)   # swap the electron labels
print(f"Psi(1, 2) = {psi_12: .6f}")
print(f"Psi(2, 1) = {psi_21: .6f}   (should be the negative)")
print(f"sum       = {psi_12 + psi_21: .2e}  (≈ 0)")

If $\Psi(1,2)$ and $\Psi(2,1)$ come out as exact negatives of each other and their sum is zero to machine precision, you have touched with your own hands the fact that a Slater determinant automatically enforces fermionic antisymmetry. Rewrite the snippet so that both electrons share the same spin-orbital (say both $\chi_\uparrow$) and the determinant collapses to zero — the Pauli exclusion principle falling out as a single line of code.

To the next chapter

Chapter 4: the quantum nature of light moves the stage from electrons sitting in spin-orbitals to electrons jumping between them, absorbing or emitting light as they go. Which combinations of the quantum numbers $(n, \ell, m_\ell, m_s)$ make a transition allowed — the so-called selection rules — will be the opening sentence of spectroscopy.