Periodic Table from Quantum Mechanics

The periodic table is an empirical organization of the chemical elements by atomic number and recurring properties. Quantum mechanics explains why this recurrence is possible: a Coulomb-bound electron has discrete angular-momentum channels, electron spin doubles their one-particle capacity, fermionic antisymmetry limits their occupation, and electron–electron interactions turn those capacities into a nonhydrogenic sequence of valence structures.

That explanation is powerful but not a one-line derivation. Pauli exclusion does not determine which subshell fills next. The Madelung sequence is not an exact spectrum of the many-electron Hamiltonian. A ground-state configuration does not by itself determine chemistry. Periodicity emerges from several ingredients whose relative importance changes with nuclear charge, ionization state, and environment.

Required background. Configurations introduce occupation labels, Pauli antisymmetry supplies state counting, and the central-field approximation explains radial screening. These atomic ingredients identify scales that molecular bonding later reorganizes.

The Atomic Sequence

For a nucleus of charge $+Ze$ and $N$ electrons, the clamped-nucleus nonrelativistic Hamiltonian in atomic units is

$$H_Z^{(0)}(N) = \sum_{i=1}^{N} \left( -\frac{1}{2}\nabla_i^2 -\frac{Z}{r_i} \right) + \sum_{i<j}^{N} \frac{1}{r_{ij}}.$$

A neutral atom has $N=Z$. Moving one place to the right in the periodic table changes both the nuclear charge and the neutral-electron number:

$$(Z,N=Z) \longrightarrow (Z+1,N=Z+1).$$

The Hamiltonians of adjacent elements are therefore different many-body problems. Periodicity means that their low-energy valence sectors recur approximately after compact core structures have formed; it does not mean that the full Hamiltonian or wavefunction repeats.

For quantitative heavy-atom work one adds nuclear recoil, relativistic interactions, finite-nuclear-size effects, and radiative corrections:

$$H_Z = H_Z^{(0)} +H_{\mathrm{recoil}} +H_{\mathrm{rel}} +H_{\mathrm{QED}} +\cdots .$$

Chemical applications then add other nuclei, external fields, or an embedding environment. The boundary between an atomic trend and a chemical trend should always be stated.

Shells, Subshells, and Capacities

In a spherical one-electron reference potential, a spin-orbital can be labeled schematically by

$$\alpha=(n,\ell,m_\ell,m_s),$$

with

$$m_\ell=-\ell,\ldots,\ell, \qquad m_s=\pm\frac12.$$

For a fixed orbital angular momentum $\ell$, there are $2\ell+1$ magnetic orbitals and two spin projections. The nonrelativistic subshell capacity is therefore

$$g_\ell = 2(2\ell+1).$$

Subshell	$\ell$	Spatial orbitals	Spin-orbital capacity
$s$	$0$	$1$	$2$
$p$	$1$	$3$	$6$
$d$	$2$	$5$	$10$
$f$	$3$	$7$	$14$

These capacities explain the widths of the familiar blocks. They are exact state counts for the chosen central-field labels, even though the corresponding orbitals and their energies are approximate in a real atom.

Why a hydrogenic shell holds 2n²

For the Coulomb problem, all values

$$\ell=0,1,\ldots,n-1$$

occur at principal quantum number $n$. Including spin gives

$$\begin{aligned} g_n &= \sum_{\ell=0}^{n-1}2(2\ell+1)\\ &= 2n^2. \end{aligned}$$

This is a degeneracy count for an ideal hydrogenic shell. It is not the sequence of observed period lengths. Electron–electron interactions remove the accidental hydrogenic degeneracy between different $\ell$ values, and subshells with different $n$ then interleave in energy.

Nominal block sequence

The conventional long-form table can be summarized by the subshell families that dominate each period:

Period	Nominal sequence	Length
1	$1s$	$2$
2	$2s,\ 2p$	$8$
3	$3s,\ 3p$	$8$
4	$4s,\ 3d,\ 4p$	$18$
5	$5s,\ 4d,\ 5p$	$18$
6	$6s,\ 4f,\ 5d,\ 6p$	$32$
7	$7s,\ 5f,\ 6d,\ 7p$	$32$

This table is a structural map, not an assertion that every neutral ground configuration is obtained by rigidly filling orbitals in that printed order. Near-degenerate subshells rearrange, especially in the $d$ and $f$ regions.

Screening and Penetration

A central-field reference replaces the coupled electronic problem by radial equations of the form

$$-\frac12\frac{d^2P_{n\ell}}{dr^2} + \left[ \frac{\ell(\ell+1)}{2r^2} +V_{\mathrm{eff}}(r) \right]P_{n\ell} = \varepsilon_{n\ell}P_{n\ell}.$$

The effective potential includes nuclear attraction and an averaged response of the other electrons. It is useful to define a radius-dependent effective charge by

$$Z_{\mathrm{eff}}(r) \equiv -rV_{\mathrm{eff}}(r).$$

For a regular electronic density, the nuclear singularity dominates near the origin. For a neutral atom or positive ion with charge $q=Z-N\geq0$, one electron far outside the other $N-1$ electrons sees their residual core charge:

$$\begin{aligned} V_{\mathrm{eff}}(r) &\sim -\frac{Z}{r}, &&r\to0,\\ V_{\mathrm{eff}}(r) &\sim -\frac{q+1}{r}, &&r\to\infty. \end{aligned}$$

This is the physical residual-core asymptote. An arbitrary approximate mean-field potential need not reproduce it: a Hartree field containing all $N$ electrons includes self-interaction, and Hartree–Fock exchange is nonlocal. For a neutral atom the correct outer Coulomb tail is $-1/r$. For a singly negative ion the residual core is neutral, so the Coulomb coefficient vanishes and polarization and shorter-range physics determine binding.

A quoted scalar effective charge is an orbital- and model-dependent summary of the radial screening.

Penetration splits equal-n subshells

The centrifugal term suppresses high-$\ell$ amplitude near the nucleus:

$$V_{\mathrm{cent}}(r) = \frac{\ell(\ell+1)}{2r^2}.$$

At comparable principal scale, an $s$ orbital usually penetrates the core more strongly than a $p$ orbital, which penetrates more strongly than a $d$ or $f$ orbital. Greater penetration samples a less-screened nuclear attraction and tends to increase binding.

This qualitative ordering explains why $ns$ and $np$ are split in multi-electron atoms even though hydrogenic energies depend only on $n$. It also helps explain why an outer $ns$ family can begin a new period before the nominally lower-principal $(n-1)d$ family has filled.

Penetration is a radial statement, not a claim that one orbital lives entirely “inside” another. Atomic radial densities overlap.

How Periods and Blocks Emerge

In an interacting atom, an orbital energy carries model and state dependence:

$$\varepsilon_{n\ell} = \varepsilon_{n\ell} \bigl[ Z,N,\rho, \text{configuration}, \text{approximation} \bigr].$$

There is no fixed universal ladder that all neutral atoms traverse unchanged. As $Z$ and $N$ increase together:

1. compact occupied orbitals screen part of the nuclear charge;
2. penetrating valence orbitals respond strongly to the increased $Z$;
3. direct, exchange, and correlation energies change with occupation;
4. orbital relaxation moves the reference energies;
5. relativistic shifts grow and can reorder heavy-atom subshells.

The familiar filling sequence is therefore a successful pattern extracted from self-consistent atomic energetics and data, not an independent postulate of quantum mechanics.

The s and p blocks

Outside a compact core, the main-group patterns are approximately

$$\begin{gathered} \text{alkali-like:}\quad [\mathrm{core}]\,ns,\\ \text{alkaline earth:}\quad [\mathrm{core}]\,ns^2,\\ \text{p block:}\quad [\mathrm{core}]\,ns^2np^q,\\ q=1,\ldots,6. \end{gathered}$$

After an $np^6$ closure, the next low-energy valence electron often enters a more extended $(n+1)s$ orbital. This resets several observables: the first ionization energy drops, the characteristic radius grows, and a one-electron-outside-a-core structure reappears.

Lithium and sodium are not copies. Their valence orbitals have different radial scales, polarizabilities, excitation energies, and core responses. They nevertheless share a low-energy one-valence-electron architecture, which is why their chemistry has a recognizable family resemblance.

Helium is an instructive exception to any purely outer-configuration rule. Its closed $1s^2$ shell places it with the noble gases chemically even though the generic main-group closure is $ns^2np^6$.

The d and f blocks

Transition regions contain near-degenerate subshells with different radial character:

$$ns \quad\text{and}\quad (n-1)d,$$

or

$$ns,\quad (n-1)d,\quad (n-2)f.$$

Small changes in direct, exchange, correlation, or relativistic energy can then alter the leading neutral configuration. Ionization can reorder the same orbitals again; for many transition atoms, an $ns$ electron is removed before a $(n-1)d$ electron even though the neutral Aufbau mnemonic placed $ns$ first.

The block label identifies the broad valence sector being developed. It does not guarantee a pure configuration, a unique oxidation state, or a simple monotonic property trend.

Ionization Energy as an Observable Test

The first ionization energy is a total-energy difference between separately relaxed systems:

$$I_1(Z) = E_Z(N=Z-1) - E_Z(N=Z).$$

More generally, the $k$th ionization energy is

$$I_k(Z) = E_Z(Z-k) - E_Z(Z-k+1).$$

These equations define positive threshold energies when each $E_Z(N)$ is the ground-state energy of the indicated charge sector. They make clear why an ionization energy is not, in general, just “the energy of the last electron.”

For canonical spin-orbital Hartree–Fock, Koopmans’ frozen-orbital approximation gives

$$I_1 \approx -\varepsilon_{\mathrm{HOMO}}^{\mathrm{HF}}.$$

The difference from experiment includes orbital relaxation and correlation, together with relativistic, recoil, and radiative corrections at the precision where they matter. An approximate orbital eigenvalue and an observed removal threshold are different objects.

The measured sawtooth

First ionization energies for neutral atoms with $1\le Z\le20$, using NIST ASD 5.12 evaluated values. The broad rise across each main-group period and the sharp drops after He, Ne, and Ar reveal periodic shell closure. Local reversals are physical many-electron effects, not measurement noise.

Across the second and third periods, the first ionization energy broadly rises as increasing nuclear charge contracts and binds the valence region. At a noble-gas closure it is relatively large. Adding the next electron begins a more diffuse $s$ family, producing the sharp He-to-Li, Ne-to-Na, and Ar-to-K drops.

The pattern is periodic, not monotonic. Four useful local comparisons are shown with energies rounded to four significant figures:

Pair and first ionization energies	Main qualitative lesson
Be, B: $9.323,\ 8.298\ \mathrm{eV}$	The first $2p$ electron in B is easier to remove than a $2s$ electron from Be.
N, O: $14.53,\ 13.62\ \mathrm{eV}$	Interaction and term energies make removal from $2p^4$ easier than a smooth trend predicts.
Mg, Al: $7.646,\ 5.986\ \mathrm{eV}$	The $3s$-to-$3p$ change repeats the Be–B mechanism.
P, S: $10.49,\ 10.36\ \mathrm{eV}$	The half-filled and more-than-half-filled $3p$ sectors differ in exchange and relaxation energy.

Orbital penetration gives the first and third comparisons a useful starting intuition. The second and fourth require the many-electron terms of both the neutral atom and its ion. “Paired electrons repel” is too compressed to be a derivation: the measured threshold is a difference of two optimized correlated energies.

Successive ionization and core closure

Neutral sodium has the leading structure $[\mathrm{Ne}]\,3s$. NIST gives

$$\begin{aligned} I_1(\mathrm{Na}) &= 5.13907696\ \mathrm{eV},\\ I_2(\mathrm{Na}) &= 47.28636\ \mathrm{eV}. \end{aligned}$$

The large jump does not mean that a classical shell wall has been crossed. After the diffuse $3s$ electron is removed, the next threshold starts from the compact neon-like ion $\mathrm{Na}^+$. Successive ionization energies are therefore a direct empirical probe of changing charge sectors and core closure.

Effective Nuclear Charge and Broad Trends

A one-parameter estimate sometimes writes a valence binding scale as

$$E_{n\ell}^{\mathrm{eff}} \sim -\frac{Z_{\mathrm{eff}}^2}{2(n^*)^2}.$$

Here $n^*$ may include a quantum defect, and $Z_{\mathrm{eff}}$ is fitted or defined within a particular model. The expression is useful for trend reasoning but should not be inverted into a unique experimentally measured effective charge.

Across a main-group period

As $Z$ increases while electrons enter the same broad valence shell:

• core screening changes less rapidly than the nuclear charge;
• the valence region generally contracts;
• first ionization energies generally rise;
• static polarizabilities generally fall;
• electron-accepting and bond-polarizing tendencies often increase.

Every “generally” matters. Subshell changes, term structure, correlation, and relativistic effects create local reversals.

Down a group

Moving down a group introduces a valence orbital with a larger principal radial scale:

• the characteristic atomic size generally grows;
• the outer electron is usually more screened and easier to remove;
• excitation energies often decrease;
• the electron cloud is often more polarizable.

Core penetration and relativistic contraction can weaken or reverse a naive extrapolation. The heavy $6s$ and $7s$ sectors, for example, cannot be understood quantitatively by scaling a nonrelativistic hydrogen orbital.

From Atomic Trends to Chemical Trends

Chemistry compares energies of atoms, ions, molecules, solids, and their environments. An isolated neutral-atom configuration is useful input, but it is not the output of a bonding calculation.

Trend language	Quantum content and boundary
atomic size	Radial density and response of a specified state; no single radius operator equals every tabulated atomic radius.
ionization energy	A total-energy difference between charge sectors, not exactly one orbital eigenvalue.
electron affinity	An energy difference after attachment; some anions are weakly bound, metastable, or environment-stabilized.
electronegativity	A tendency inferred from molecular or thermochemical data; several inequivalent scales exist.
polarizability	An induced-dipole response through excited states; it is tensorial and frequency-dependent in general.
oxidation state	An electron-counting convention tied to bonding, not a direct isolated-atom observable.

For a nondegenerate isotropic atomic ground state, the static dipole polarizability has the spectral form

$$\alpha(0) = \frac{2}{3} \sum_{n\ne0} \frac{ \left| \langle n|\mathbf D|0\rangle \right|^2 }{ E_n-E_0 }.$$

The sum includes all dipole-coupled magnetic states and an integral over continuum states; $\mathbf D$ is the total electric dipole operator. Diffuse valence states and low excitation gaps tend to increase $\alpha(0)$. This helps connect the low first ionization energies of alkali atoms with their large response, but the relation is not an identity: transition matrix elements and the full excited spectrum also matter.

Why groups resemble one another

Elements in a group often present the same number and angular type of low-energy valence electrons outside a compact core. Their valence Hilbert spaces and dominant bonding channels then resemble one another.

The resemblance is approximate because the core changes:

• radial nodes and characteristic length scales differ;
• core polarization and core–valence correlation differ;
• accessible $d$ or $f$ channels can enter;
• spin–orbit coupling grows;
• oxidation and bonding energies need not scale together.

Group membership is therefore a highly successful structural classification, not a symmetry equivalence between different elements.

Transition and inner-transition trends

Partially filled $d$ and $f$ subshells support many Pauli-allowed terms, several oxidation sectors, and dense low-energy spectra. Their chemistry depends on a competition among:

$$\begin{gathered} \text{crystal or ligand fields},\\ \text{pairing and exchange},\\ \text{spin–orbit coupling},\\ \text{covalency and correlation}. \end{gathered}$$

The lanthanide contraction is associated with increasing nuclear charge across the $4f$ series combined with incomplete screening by the compact $4f$ electrons; relativistic effects refine the quantitative trend. It is not evidence that screening disappears.

Limits of periodic trends

Near-degenerate configurations can exchange order under small correlation, relaxation, or relativistic corrections. The characteristic relativistic parameter $Z\alpha$ grows with nuclear charge. Penetrating $s_{1/2}$ and $p_{1/2}$ spinors often contract, while changed screening can expand other orbitals. Heavy-element chemistry therefore requires a coupled relativistic calculation, not a uniform correction to a fixed filling diagram.

In molecules and solids, other nuclei break spherical symmetry and reorganize the valence states. Atomic group membership remains useful, but it does not determine a unique bond energy or oxidation state.

Exercises

Exercise 1: Why period 3 has eight elements

Hydrogenic counting assigns the $n=3$ shell a capacity of $18$. Explain why the third period nevertheless runs from Na to Ar and has length $8$.

Solution

The $n=3$ hydrogenic shell contains $3s$, $3p$, and $3d$, with capacities

$$2+6+10=18.$$

In neutral multi-electron atoms, screening and penetration remove the hydrogenic degeneracy. The $4s$ valence family becomes relevant before the $3d$ family is developed across the neutral sequence. Period 3 therefore consists nominally of

$$3s^2\,3p^6,$$

with length $8$. The $3d$ block appears between $4s$ and $4p$ in period 4. This is energetic interleaving, not a change in the state capacity of $3d$.

Exercise 2: Read the local ionization reversals

Use the table above to explain why $I_1(\mathrm B)<I_1(\mathrm{Be})$ and $I_1(\mathrm O)<I_1(\mathrm N)$ do not contradict the broad rise across period 2.

Solution

Boron begins the $2p$ subshell:

$$\mathrm{Be}:1s^2\,2s^2, \qquad \mathrm{B}:1s^2\,2s^2\,2p.$$

The $2p$ electron penetrates less strongly and is more weakly bound than a $2s$ electron in the neighboring system, so the first threshold drops at B.

Nitrogen and oxygen both ionize from $2p$, but their many-electron terms differ. Nitrogen’s half-filled $2p^3$ ground sector is strongly exchange stabilized. Removing one electron from oxygen’s $2p^4$ sector produces the favorable $2p^3$ ion. The exact comparison is still a difference of separately relaxed total energies; “pairing” is only shorthand for that term, exchange, and correlation balance.

Exercise 3: The sodium jump

Using the NIST values on this page, compute the ratio $I_2/I_1$ for sodium and explain its physical meaning.

Solution

The ratio is

$$\frac{I_2}{I_1} = \frac{47.28636}{5.13907696} \approx 9.20.$$

The first ionization removes the diffuse $3s$ valence electron. The second starts from the compact neon-like $\mathrm{Na}^+$ ion and removes a core electron. The large ratio is evidence for a change in electronic sector and shell closure, not a literal discontinuity in a classical radial shell.

Exercise 4: Similarity without identity

Lithium and sodium both have one $s$ electron outside a closed core. Give three reasons their atomic or chemical properties are not identical.

Solution

Any three of the following suffice:

• Li uses a $2s$ valence orbital, while Na uses $3s$; their radial nodes and sizes differ.
• The closed cores have different polarizabilities and core-excitation spectra.
• Their ionization and excitation energies differ.
• Core–valence correlation and exchange differ.
• Relativistic corrections, though modest here, are not identical.
• Molecular bond energies depend on partner atoms and geometry, not only the isolated neutral configuration.

The shared one-valence-electron structure explains family resemblance. It does not define a symmetry mapping one element’s Hamiltonian into the other’s.

References

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