The Periodic Table from Quantum Mechanics

Imagine building an atom one electron at a time, keeping the nucleus at charge $Z$ fixed. A single electron would settle into the hydrogenic $1s$ orbital with energy $-Z^2(13.6\,\text{eV})$. If electrons ignored one another and ignored exclusion, every atom would be that — all $Z$ electrons crushed into $1s$, no shells, no chemistry, no you. Two quantum facts rescue the universe from that dead simplicity. The first is exclusion. Because electrons are identical fermions, no two can share all four labels $(n,\ell,m_\ell,m_s)$, so the $1s$ orbital fills after two electrons and the next must climb to $n=2$. Stacking electrons into successively higher orbitals — the building-up procedure — is what makes atoms grow in layers. The second is repulsion. Each electron does not see the bare nucleus; it sees the nucleus screened by the smeared-out charge of the electrons closer in. The result is that an electron in a many-electron atom moves, to good approximation, in a central potential set by an effective charge $Z_{\text{eff}}<Z$. Screening does something the bare hydrogen atom never does: it splits the $\ell$-degeneracy. In hydrogen, $2s$ and $2p$ have identical energy. In a real atom, the $2s$ orbital is more penetrating — its wavefunction has amplitude close to the nucleus, inside the screening cloud, where it feels nearly the full $Z$ — so it binds more tightly than $2p$. Order the subshells by how deeply they penetrate and you recover the empirical filling order. The single number that captures all of this is $Z_{\text{eff}}$, and its climb across a period is the story of the whole table.

The sawtooth is the periodic table read straight off an instrument. Each period climbs from an alkali metal (one loosely held valence electron, easy to ionize) up to a noble gas (a saturated shell, extremely hard to ionize), then crashes as the next shell begins. The gross tooth spacing is set by the shell capacities; the small dips within each rise — beryllium above boron, nitrogen above oxygen — are the fine print that only subshell closure and exchange stability explain. Both the coarse teeth and the fine dips are quantum mechanics.

The gross architecture is exclusion arithmetic. A subshell with orbital quantum number $\ell$ contains $2\ell+1$ values of $m_\ell$, and each takes two spins, so it holds $2(2\ell+1)$ electrons: $s\to2$, $p\to6$, $d\to10$, $f\to14$. A full shell of principal number $n$ sums these:

where: the inner sum $\sum_{\ell=0}^{n-1}(2\ell+1)=n^2$ is the arithmetic series $1+3+5+\cdots+(2n-1)$. The physical read: exclusion turns counting orbitals into counting electrons, and the shell capacities $2n^2 = 2,8,18,32$ are the raw material of the table. This capacity law is owned by Pauli Exclusion Principle (which states "shell capacities $2(2\ell+1)$ and $2n^2$ giving 2,8,18,32"); we take it and organize it. The periods are not, however, simply the shells — if they were, the fourth row would have $2n^2=32$ elements, but it has 18. The period boundaries are set by energy, through the Madelung filling order, which is owned by Aufbau Principle and Electron Configurations: fill orbitals by increasing $n+\ell$, breaking ties toward smaller $n$. Cutting the Madelung sequence at each closed $s$–$p$ block gives the actual periods.

Now the owned physics. Exclusion and Madelung give the skeleton; electron–electron repulsion gives the ordering and the chemistry. The many-electron Hamiltonian

is not separable — the last sum couples every pair. The central-field approximation replaces it with each electron moving in an averaged potential: the bare nucleus shielded by the smeared charge of the others, so electron $i$ feels an effective charge $Z_{\text{eff}} = Z - \sigma$, where $\sigma$ is the shielding constant. The screened hydrogenic energy is then

where: $\sigma$ is the total shielding seen by the electron; the $Z\to Z_{\text{eff}}$ replacement inherits the hydrogenic $1/n^2$ and $a\to a_0/Z_{\text{eff}}$ scaling (the bare-charge scaling $a\to a_0/Z$, $E\to Z^2 E_h$ is owned by Bohr Radius and Atomic Scales). The physical read: because low-$\ell$ orbitals penetrate the screening cloud, they feel a larger $Z_{\text{eff}}$ and bind more tightly, which is why $2s$ lies below $2p$ and $4s$ below $3d$ — the very ordering that Madelung encodes empirically. Slater's rules (1930) turn $\sigma$ into an explicit recipe. For an electron in an $ns$ or $np$ orbital: each other electron in the same $n$ shell contributes $0.35$; each electron in the $n{-}1$ shell contributes $0.85$; each deeper electron contributes $1.00$. (For $d$ and $f$ electrons the same-shell rule is $0.35$ but all inner electrons contribute $1.00$.) These three numbers, crude as they are, capture the essential trend.

Monotone $Z_{\text{eff}}$ predicts a monotone rise in ionization energy across a period. The data has two conspicuous dips, and they are the periodic table's most instructive fine print. Beryllium ($1s^22s^2$) ionizes at $9.32$ eV but boron ($1s^22s^22p^1$) at only $8.30$ eV — a drop as $Z$ increases. Nitrogen ($2p^3$) ionizes at $14.53$ eV but oxygen ($2p^4$) at $13.62$ eV — another drop. Neither is explained by screening; both are subshell effects. The boron dip is subshell closure: boron's ionized electron is the first $2p$, which lies above the closed $2s^2$ subshell and is therefore easier to remove than beryllium's $2s$ electron. The oxygen dip is exchange stability: nitrogen's $2p^3$ has one electron in each of the three $p$ orbitals with parallel spins — a half-filled subshell that Hund's first rule makes especially stable — while oxygen must double up one $p$ orbital, paying a pairing penalty. These half-filled and filled-subshell stabilizations are owned by Hund's Rules; the periodic table is where they show up as observable kinks.

Example 1 — Slater's $Z_{\text{eff}}$ for a fluorine $2p$ electron, and the trend it sets. Compute the effective charge seen by a valence electron in fluorine ($Z=9$, configuration $1s^22s^22p^5$) and place it in the across-period trend.

Step 1. Enumerate the shielding electrons. Fluorine's $2p$ electron shares the $n=2$ shell with six others ($2s^2$ and the other four $2p$), and sits above two $1s$ electrons in the $n{-}1$ shell.

Step 2. Apply Slater's rules. Same-shell contribution: $6\times0.35=2.10$. Inner-shell contribution: $2\times0.85=1.70$. Total shielding $\sigma=3.80$, so $Z_{\text{eff}} = 9 - 3.80 = \mathbf{5.20}$.

Step 3. Boundary check — the trend. This sits between oxygen's $Z_{\text{eff}}=4.55$ and neon's $5.85$, on a monotone climb $3.25\to3.90\to4.55\to5.20\to5.85$ from carbon to neon. Each step adds a proton shielded by only $0.35$, so $Z_{\text{eff}}$ rises by $0.65$ per element. The prediction — atoms get harder to ionize and physically smaller left-to-right across period 2 — matches the measured ionization climb ($11.26\to14.53\to13.62\to17.42\to21.56$ eV) and radius contraction, dips aside.

Example 2 — the oxygen dip, and why it needs Hund. Explain why $I_1(\text{O})=13.62$ eV is lower than $I_1(\text{N})=14.53$ eV despite oxygen's larger $Z_{\text{eff}}$.

Step 1. Screening alone predicts the wrong sign. Oxygen's $Z_{\text{eff}}=4.55$ exceeds nitrogen's $3.90$, so a pure central-field model predicts oxygen should be harder to ionize — the opposite of what is measured.

Step 2. Read the configurations. Nitrogen is $2p^3$: one electron in each of the three $p$ orbitals, all spins parallel — a half-filled subshell whose extra stability comes from exchange energy, owned by Hund's Rules. Oxygen is $2p^4$: it must place a fourth electron into an already-occupied $p$ orbital, paying a pairing penalty and losing the half-filled stability.

Step 3. Boundary check. The electron removed from oxygen is precisely that paired one, so it comes off more easily than a nitrogen $p$ electron — the $0.91$ eV dip. The screening model gives the trend; exchange stability gives the exception. The periodic table is the ledger where both are read.

The quantum periodic table is not a finished chapter — its edges are active physics. The heaviest elements, synthesized one atom at a time (oganesson, $Z=118$, completed period 7 in 2016), test whether relativistic effects distort the expected chemistry: in superheavy atoms the inner $s$ electrons move at a sizable fraction of $c$, contract, and change valence behavior, so element 118 may not be a well-behaved noble gas. Relativistic quantum chemistry — screening plus Dirac dynamics — is what predicts the color of gold and the liquidity of mercury from first principles. On the computational side, density functional theory and coupled-cluster methods have turned the qualitative screening picture of this article into routine predictions of ionization energies, molecular geometries, and reaction barriers to chemical precision.

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §5.2 · Bransden & Joachain, Physics of Atoms and Molecules, 2nd ed., Ch. 8 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 7 · Slater, Phys. Rev. 36, 57 (1930) · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Complement A$_{\text{XIV}}$.

Provenance. The quantum-number scaffold, capacities $2(2\ell+1)$ and $2n^2$, and the Madelung $n+\ell$ ordering are standard (Griffiths §5.2). Noble-gas atomic numbers $2,10,18,36,54,86$ are exact cumulative period sums, re-summed in the verification script; the period lengths $2,8,8,18,18,32,32$ are their successive differences. Slater's screening constants $(0.35, 0.85, 1.00)$ follow Slater (1930); the worked $Z_{\text{eff}}$ values for C/N/O/F/Ne ($3.25/3.90/4.55/5.20/5.85$) are recomputed, neon matching the standard $5.85$. First ionization energies are NIST values (Li $5.39$, Be $9.32$, B $8.30$, C $11.26$, N $14.53$, O $13.62$, F $17.42$, Ne $21.56$ eV). Editorial note: the crude $E=13.6\,Z_{\text{eff}}^2/n^2$ estimate overshoots badly in absolute magnitude ($116$ eV for neon vs the measured $21.56$ eV); we surface that gap as a teaching warning — the rule predicts trends, not energies. Shell capacities are owned by Pauli Exclusion Principle, the Madelung order by Aufbau Principle and Electron Configurations, half-filled-shell stability by Hund's Rules, and the $Z$-scaling of orbital sizes and energies by Bohr Radius and Atomic Scales.

Builds on: Pauli Exclusion Principle · Slater Determinants · Bosons and Fermions · The Helium Atom · Hund's Rules Leads to: Hund's Rules · Exchange Forces and Exchange Energy · Singlet and Triplet States in Two-Electron Systems