The Pauli Principle, Atoms, and the Periodic Table

The exchange symmetry of the previous lesson fixes the form of every multi-electron state: antisymmetric under the interchange of any two electrons. Turning that constraint into the structure of atoms takes one construction, the Slater determinant, and one calculation, the splitting of the electron-electron repulsion into a direct and an exchange piece. Helium is the smallest atom where both electrons and their repulsion appear, and it displays the whole mechanism. Screening, the filling order of subshells, and Hund's rules then extend the same antisymmetry across the periodic table.

Slater determinants

A single electron in an atom occupies a spin-orbital, a product of a spatial orbital and a spin state,

where the composite label $\mu =(n,\ell ,m,m_s)$ collects every quantum number. An $N$-electron state must be totally antisymmetric in the $N$ coordinates ${\vec{x}}_1,\dots ,{\vec{x}}_N$. The antisymmetrized product of $N$ distinct spin-orbitals is the Slater determinant,

Rows are labelled by electrons, columns by spin-orbitals. Two determinant identities carry the physics.¹

• Antisymmetry. Interchanging two electrons swaps two rows, and a determinant changes sign under a row swap. The state is antisymmetric by construction.
• Exclusion. If two spin-orbitals coincide, two columns are equal and the determinant vanishes. No two electrons share all four quantum numbers.

For two electrons the determinant expands to the antisymmetric combination of the previous lesson,

Helium: the direct and exchange integrals

Helium has a nucleus of charge $Z=2$ and two electrons. Its Hamiltonian is

Drop ${\hat{V}}_{ee}$ and each electron sits in a hydrogenic potential with charge $Z=2$. The hydrogenic energies scale as $Z^2$, so the ground state, both electrons in the $1s$ orbital, has zeroth-order energy

The measured ground-state energy is $-79.0\text{eV}$ (the sum of the two ionization energies, $24.6+54.4\text{eV}$).² The $30\text{eV}$ discrepancy is the electron-electron repulsion, dropped in ${\hat{H}}_0$. Treat ${\hat{V}}_{ee}$ in first-order perturbation theory. For the ground configuration both electrons occupy $1s$, the spatial state is symmetric, and the spins form the singlet. The first-order energy shift is the expectation of ${\hat{V}}_{ee}$ in the $1s^2$ spatial state,

With $Z=2$ this is $34.0\text{eV}$, raising the estimate to $-108.8+34.0=-74.8\text{eV}$, within a few electron-volts of the measured $-79.0\text{eV}$. The remaining gap closes once the electrons are allowed to screen one another; the variational method with an effective charge $Z_{\text{eff}}=Z-\frac{5}{16}=1.69$ gives $-77.5\text{eV}$.³

Exchange splitting in the excited states

The instructive structure appears in the excited configurations $1sn\ell$, where one electron stays in $1s$ and the other occupies a higher orbital. Now the two spatial orbitals differ, ${\psi }_a={\psi }_{1s}$ and ${\psi }_b={\psi }_{n\ell }$, and the spatial state may be symmetric or antisymmetric,

Antisymmetry of the total state ties ${\psi }_{+}$ to the spin singlet ($S=0$, parahelium) and ${\psi }_{-}$ to the spin triplet ($S=1$, orthohelium). Evaluate $⟨{\hat{V}}_{ee}⟩$ in ${\psi }_{\pm }$. The cross terms of the symmetrized state generate two integrals,

with the direct (Coulomb) integral

and the exchange integral

$J$ is the classical electrostatic energy of the two charge clouds $|{\psi }_a{|}^2$ and $|{\psi }_b{|}^2$. $K$ has no classical analog: it comes from the interference of the two ways to assign electrons to orbitals, and it is positive for the Coulomb interaction. The singlet (parahelium) lies at $J+K$ and the triplet (orthohelium) at $J-K$, so

The triplet lies below the singlet by $2K$. Nothing magnetic entered the calculation; the interaction was pure electrostatic Coulomb. The triplet has an antisymmetric spatial state, its electrons avoid one another (the fermionic exchange correlation), their repulsion is reduced, and the energy drops. This is the physical content of the exchange integral: aligning the spins forces spatial antisymmetry, which lowers the Coulomb energy.⁴

Screening and effective charge

Beyond helium, an electron does not feel the bare nuclear charge $Z$. The inner electrons partly cancel it, so an outer electron sees a reduced effective charge $Z_{\text{eff}}<Z$. The cancellation depends on how far the electron penetrates toward the nucleus, and penetration depends on orbital angular momentum.

A low-$\ell$ orbital has a spatial density that reaches closer to the nucleus, past the screening cloud of inner electrons, and feels a larger $Z_{\text{eff}}$. A high-$\ell$ orbital is held out by the centrifugal barrier and is screened more completely. Within a shell of fixed $n$, energy therefore rises with $\ell$,

lifting the exact hydrogenic degeneracy in $\ell$. In hydrogen all $\ell$ at a given $n$ share an energy; screening in a multi-electron atom removes that accident.⁵

The aufbau order and electron configurations

Screening sets the order in which subshells fill. The empirical Madelung rule orders subshells by increasing $n+\ell$, and by increasing $n$ where $n+\ell$ ties. Filling in that order (the aufbau principle), placing electrons one at a time into the lowest available spin-orbital, generates the ground-state configuration of each element.

The rule is a good approximation, not a law: $4s$ fills before $3d$ because $4s$ ($n+\ell =4$) beats $3d$ ($n+\ell =5$), which puts potassium and calcium ahead of the transition metals. Half-filled and filled subshells carry extra stability, producing the well-known anomalies where the tabulated configuration departs from the naive filling.

Hund's rules

A partly filled subshell has many ways to distribute its electrons among the degenerate orbitals. Hund's rules select the ground-state term, and the first two are direct consequences of the exchange physics worked out for helium.

The first rule is the helium triplet generalized: aligning spins forces spatial antisymmetry, and the exchange integral pays for it. Applying the rules to the $2p$ series shows the pattern, with term symbols written ${}^{2S+1}{L}_J$.

Carbon ($2p^2$) puts both electrons in separate orbitals with parallel spins: $S=1$, and the largest compatible $L$ is $1$, giving a ${}^{3}P$ term; less than half filled, so $J=|L-S|=0$ and the ground term is ${}^3{P}_0$. Nitrogen ($2p^3$) is half-filled with all three spins parallel: $S=\frac{3}{2}$, and the only way to keep all $m_{\ell }$ distinct forces $L=0$, giving ${}^4{S}_{3/2}$. Oxygen ($2p^4$) has one doubly-occupied orbital: $S=1$, $L=1$, and now more than half filled, so $J=L+S=2$ and the term is ${}^3{P}_2$.

The shape of the periodic table

The blocks of the periodic table are the subshells being filled. The two columns on the left fill an $s$ subshell, the six on the right fill a $p$ subshell, the ten transition-metal columns fill a $d$ subshell, and the fourteen lanthanide and actinide columns fill an $f$ subshell. Column count is $2(2\ell +1)$, the number of spin-orbitals in the subshell: exclusion fixes the widths.

Two facts about ordinary matter descend directly from the antisymmetry of the electron wavefunction. Atoms have shell structure and chemistry because electrons cannot all fall into the $1s$ orbital; they stack into successive subshells, and the outermost, partly filled subshell sets an element's chemical behavior. And bulk matter resists compression because pressing atoms together forces their electrons toward common states that the exclusion principle forbids, producing the degeneracy pressure that holds up white dwarfs and neutron stars. Both are the Pauli principle, read at the scale of a single atom and at the scale of a star.