The Quantum Mechanical Model of the Atom

Every model of the atom has been replaced by a better one: Dalton’s solid spheres, Thomson’s plum pudding, Rutherford’s nuclear atom, Bohr’s planetary orbits. The model chemists use today, the quantum mechanical model, emerged in the mid-1920s and has survived a century of increasingly demanding tests. It’s the model behind electron configurations, the periodic table, chemical bonding, spectroscopy and modern materials science. This article pulls together its key ideas and explains why it succeeds where earlier models failed.

How we got here

Year	Model	Key idea	Main limitation
1803–1808	Dalton	atoms are indivisible spheres	atoms have internal parts
1904	Thomson	electrons embedded in spread-out positive charge	the nucleus exists
1911	Rutherford	tiny dense nucleus with electrons around it	electrons should spiral inwards
1913	Bohr	electrons in fixed circular orbits with quantised energies	only works for one-electron atoms
1926 onwards	quantum mechanical	electrons as waves in orbitals; probability, not paths	still the accepted model

See the history of atomic models, Rutherford’s gold foil experiment and the Bohr model.

The key ideas

1. Electrons behave as waves

In 1924, Louis de Broglie proposed that electrons have wave properties, confirmed by electron diffraction in 1927. An electron bound to a nucleus behaves like a three-dimensional standing wave. See wave-particle duality.

2. The Schrödinger equation describes the electron

In 1926, Erwin Schrödinger wrote an equation whose solutions, called wavefunctions (ψ), describe the allowed states of electrons in atoms. Each solution has a definite energy. See the Schrödinger equation for chemists.

3. Energy is quantised

Only certain standing-wave patterns fit around a nucleus, so only certain energies are allowed. Unlike the Bohr model, where quantisation was assumed, here it follows naturally from the wave description.

4. Orbitals, not orbits

The allowed wavefunctions are orbitals: three-dimensional regions where the electron is likely to be found. Electrons don’t travel along paths. The square of the wavefunction, ψ², gives the probability of finding the electron at each point. See electron clouds and probability.

5. Uncertainty

The Heisenberg uncertainty principle means an electron can’t have a precise position and momentum at the same time, so definite orbits are impossible. See the Heisenberg uncertainty principle.

6. Four quantum numbers

Each electron is described by four quantum numbers:

• n (principal): shell, size and energy
• l (angular momentum): subshell and shape (s, p, d, f)
• mₗ (magnetic): orientation of the orbital
• mₛ (spin): +½ or −½

See quantum numbers explained and electron spin.

7. Filling rules

In atoms with many electrons, orbitals fill according to three rules:

• Aufbau: lowest energy first. See the Aufbau principle.
• Pauli exclusion: at most two electrons per orbital, with opposite spins. See the Pauli exclusion principle.
• Hund’s rule: fill equal-energy orbitals singly first, with parallel spins. See Hund’s rule.

What the model explains

The periodic table

The number of orbitals in each subshell (1, 3, 5, 7), each holding two electrons, gives subshell capacities of 2, 6, 10 and 14. These are exactly the widths of the s, p, d and f blocks of the periodic table. Elements in the same group have the same outer electron configuration, explaining their similar chemistry. See s, p, d and f blocks and why is the periodic table shaped like that?

Periodic trends

Shielding and penetration, described by the model, explain trends in atomic radius, ionisation energy and electronegativity. See effective nuclear charge and periodic trends explained.

Spectra of all elements

The model predicts the spectra of multi-electron atoms, including fine details (such as the splitting of lines) that the Bohr model couldn’t handle. See the hydrogen emission spectrum.

Chemical bonding and molecular shape

Covalent bonds form when orbitals overlap and electron waves reinforce between nuclei. The directional shapes of p and d orbitals, and hybrid orbitals formed from them, explain molecular shapes such as the tetrahedral geometry of methane. See VSEPR and molecular geometry.

Magnetism and colour

Unpaired electrons predicted by Hund’s rule explain paramagnetism; d-orbital energy splitting explains the colours of transition metal compounds. See paramagnetic vs diamagnetic and transition metals.

Exceptions and fine details

The model also explains unusual configurations such as chromium and copper, and relativistic effects that make gold yellow and mercury liquid. See electron configuration exceptions and why is gold yellow?

What it keeps from earlier models

Good new models don’t throw everything away:

• From Dalton: atoms of each element have characteristic masses; reactions rearrange atoms.
• From Rutherford: a tiny, dense, positive nucleus containing nearly all the mass.
• From Bohr: quantised energy levels, the principal quantum number n, and photons emitted or absorbed when electrons change levels.

What it discards is the idea of electrons as tiny planets following definite paths.

Describing the model in exams

A strong answer to “Describe the quantum mechanical model of the atom” might include:

1. A tiny, dense nucleus containing protons and neutrons.
2. Electrons described as waves, occupying orbitals rather than fixed orbits.
3. Orbitals as regions of high probability of finding an electron.
4. Energy levels (shells) divided into subshells (s, p, d, f) containing orbitals with characteristic shapes.
5. Each orbital holding up to two electrons with opposite spins.
6. The position and momentum of an electron can’t both be known precisely.

A worked illustration: oxygen

Here’s how the model describes a single oxygen atom (8 electrons):

• Nucleus: 8 protons and (usually) 8 neutrons, in a region a few femtometres across.
• Configuration: 1s² 2s² 2p⁴. The two 1s electrons form a compact spherical cloud close to the nucleus; the 2s electrons form a larger spherical cloud with one radial node; the four 2p electrons occupy three dumbbell-shaped orbitals at right angles.
• Spins: by Hund’s rule, two of the 2p electrons are paired and two are unpaired with parallel spins.
• Consequences: oxygen’s six outer electrons and two unpaired electrons explain why it typically forms two covalent bonds (as in water) or gains two electrons to form O²⁻, and its high effective nuclear charge explains why it’s the second most electronegative element.

Every one of these predictions comes from the same model, with no extra assumptions.

Is the model complete?

For chemistry, it’s extraordinarily successful. Its predictions, calculated with modern computers, match experimental measurements of energies, bond lengths and spectra to high precision. Physicists have extended it further: quantum electrodynamics accounts for tiny effects such as the Lamb shift in hydrogen’s energy levels, and relativistic quantum mechanics handles heavy atoms. But the core picture of orbitals, quantum numbers and probability remains the working model for all of chemistry.

Key takeaways

• The quantum mechanical model (1926 onwards) treats electrons as waves described by the Schrödinger equation.
• Electrons occupy orbitals, regions of probability, not fixed orbits; energy is quantised naturally.
• Four quantum numbers and three filling rules determine electron configurations.
• The model explains the periodic table, periodic trends, spectra, bonding, magnetism and colour.
• It keeps the nucleus from Rutherford and quantised energy from Bohr, but replaces orbits with orbitals.