Atomic orbitals

From wavefunction to probability

The wavefunction ψ can be positive, negative or zero; |ψ|² is proportional to probability density. The sign of ψ is not an electrical charge. It is a phase of the wavefunction, and phase becomes chemically important when orbitals combine to make bonds.

Because a quantum state extends through space, an atom has no sharp electron-cloud edge. Orbital pictures choose a surface that encloses some convenient fraction of the total probability—often about 90%—so the shape can be drawn.

The nucleus lies at the center. An s orbital is spherically symmetric. A p subshell contains three mutually perpendicular orbitals. Five d orbitals and seven f orbitals have more complex angular patterns.

Shells, subshells and orbital labels

An orbital label such as 3p contains two pieces of information. The principal quantum number n = 3 identifies the shell; p means angular-momentum quantum number ℓ = 1. For a given ℓ there are 2ℓ + 1 orientations: one s orbital, three p orbitals, five d orbitals and seven f orbitals.

Nodes, penetration and energy

Nodes are not empty shells inserted by hand; they follow from the wavefunction. The total number of nodes is n − 1. Of these, ℓ are angular nodes and n − ℓ − 1 are radial nodes.

In a hydrogen atom, orbitals with the same n have the same energy. In multi-electron atoms that degeneracy is broken because electrons repel one another and different orbital shapes penetrate the inner electron cloud differently. An s electron can have substantial probability close to the nucleus, so s orbitals often experience a larger effective nuclear charge than p orbitals in the same shell.

This penetration difference helps explain why 2s lies below 2p in multi-electron atoms and why orbital energies cannot be inferred from n alone.

Interpretation and limits of orbital pictures

Orbital shapes help explain directionality in bonding, hybridization models, molecular geometry and selection rules in spectroscopy. The phase pattern of p and d orbitals determines whether overlap between neighboring atoms is constructive or destructive.

But an orbital drawing is not a snapshot of one electron’s path. Quantum mechanics predicts probability distributions and measurable energies; it does not assign a definite classical trajectory. Nor should colored lobes be interpreted as positive and negative charge. Different colors usually indicate opposite wavefunction phase.

Experiment connects the model to reality through spectra, ionization energies, scattering and electron-density measurements. Orbitals are theoretical objects, but they make testable predictions.

Probability density and radial probability are different graphs

For an s orbital, |ψ|² is largest near the nucleus, but a thin spherical shell near r = 0 contains very little volume. The radial probability distribution includes the spherical volume factor 4πr², so its most probable radius is not necessarily where |ψ|² itself is largest. This distinction matters whenever a graph is labeled “probability versus distance.”

For hydrogen 1s, the electron density is greatest at the nucleus while the radial probability peaks at the Bohr radius. Neither statement means the electron follows a circular orbit of that radius.

Hydrogen orbitals versus orbitals in many-electron atoms

Hydrogenic orbitals are exact solutions of the one-electron Coulomb problem. Real multi-electron atoms require approximations because each electron also repels all the others. Chemists still use atomic orbitals because self-consistent orbital models capture the dominant shell and subshell structure extremely well.

In hydrogen, 2s and 2p have the same energy. In a multi-electron atom, 2s penetrates the inner region more strongly and is stabilized relative to 2p. This is why the simple hydrogen energy formula depending only on n cannot be transferred unchanged to carbon, oxygen or iron.

Real orbitals, complex orbitals and orientation

The familiar p_x, p_y and p_z drawings are convenient real combinations of states with definite magnetic quantum number. Quantum mechanics also allows complex combinations. The choice of axes can rotate the drawings without changing the physics of an isolated atom.

What becomes physically meaningful in a molecule or crystal is the relation between the orbital and its environment: an internuclear axis, ligand arrangement, electric field or crystal symmetry can select preferred orientations and split formerly degenerate orbital energies.

Orbital size is a distribution, not one radius

An orbital does not possess a single geometric radius. Chemists may discuss the most probable radius, an expectation value of r, or a surface enclosing a chosen percentage of probability. Those definitions answer different questions and should not be confused with tabulated atomic radii derived from bonds or crystals.

This is why an orbital can have a small probability close to the nucleus and still extend far outward with a long tail. Penetration refers to the inner part of that distribution, while overall size concerns the full spatial spread.

Exercises

3p nodes

For a 3p orbital, find the number of angular and radial nodes.

Subshell capacity

How many orbitals and electrons can a d subshell contain?

Phase

Do the + and − lobes drawn on a p orbital represent positive and negative electric charge?