Atomic orbital

Atomic orbitals are mathematical functions used to describe the wavelike behaviour and spatial distribution of electrons in atoms. Within quantum mechanics, an orbital represents a region around the nucleus where the probability of finding an electron is highest. Each orbital is characterised by a set of quantum numbers defining its energy, shape, and orientation, forming the basis of the modern electron-cloud model of atomic structure.

Quantum Mechanical Foundations

Electrons in atoms behave as both particles and waves. Instead of travelling in fixed paths like planets around a star, they exist as standing wave patterns defined by solutions to the Schrödinger equation. These wave functions provide a probability distribution rather than a precise trajectory, meaning that an electron’s charge appears spread out over space. When more than one electron occupies an atom, each electron state is a superposition of many possible configurations, though orbitals offer a practical means of visualising the underlying structure.
Each orbital is described by three principal quantum numbers:

• n (principal quantum number) – indicates the energy level and approximate size of the orbital.
• l (angular momentum quantum number) – determines the shape of the orbital (0 = s, 1 = p, 2 = d, 3 = f, etc.).
• mₗ (magnetic quantum number) – specifies the orientation of the orbital in space.

A fourth quantum number, mₛ (spin quantum number), distinguishes the two spin states of electrons occupying the same orbital. No more than two electrons can occupy a single orbital due to the Pauli exclusion principle.

Shapes and Types of Orbitals

The familiar labels s, p, d, and f arise from early spectroscopic descriptions—sharp, principal, diffuse, and fundamental. Their shapes stem from the angular dependence of the solutions to the Schrödinger equation:

• s orbitals (l = 0): spherical distributions.
• p orbitals (l = 1): dumbbell-like shapes oriented along x, y, and z axes.
• d orbitals (l = 2): more complex, cloverleaf or elongated structures.
• f orbitals (l = 3): intricate multi-lobed shapes.

Higher orbitals, labelled g, h, i, and k (excluding j), follow in alphabetical order.
Orbitals with definite magnetic quantum numbers are often expressed in complex form, but real-valued orbitals can be formed as linear combinations of these and are commonly represented using harmonic polynomials such as xy or xz, reflecting their angular features. These real orbitals provide the visual shapes typically found in textbooks.

Electron Configurations and the Periodic Table

Orbital structures underpin the periodic table. The s, p, d, and f blocks reflect the maximum number of electrons—2, 6, 10, and 14—accommodated by each set of orbitals. Electrons occupy orbitals in order of increasing energy, described by the Aufbau principle, though this ordering becomes less tidy for transition metals and ions where subshell energies converge.
Electron configurations, such as 1s² 2s² 2p⁶ for neon, indicate the number of electrons in each subshell. These notations describe dominant contributions within the more complex wave functions that arise when multiple electrons interact.

Wavelike and Particlelike Characteristics

Electrons in orbitals exhibit:

• Wavelike behaviour – They form standing-wave patterns and occupy regions of space rather than discrete points. The lowest-energy orbital resembles a fundamental frequency, while higher states resemble harmonics.
• Particlelike behaviour – Electrons remain countable particles with discrete charge and spin. They transition between orbitals by absorbing or emitting quantised amounts of energy, usually in the form of photons.

The dual nature of electrons means that orbital diagrams show mathematical idealisations (eigenstates). In reality an electron may exist in superpositions of many such states, giving rise to highly flexible and dynamic electron distributions.

Formal Quantum Description

In full quantum mechanical terms, atomic orbitals are one-electron functions used to approximate the multi-electron wave function of an atom. The exact state of an atom is an eigenstate of its Hamiltonian and depends on the coordinates of all electrons simultaneously. Approximations such as the Hartree–Fock method or configuration interaction express the many-electron state as linear combinations of Slater determinants built from individual orbitals.
Orbitals thus serve as practical approximations that allow physicists and chemists to model atomic behaviour, visualise electron transitions, and predict atomic spectra. Spectral lines correspond to transitions between states characterised by particular combinations of orbitals, summarised in term symbols such as ¹S₀ or ³P₂.

Hydrogenlike and Approximate Orbitals

The simplest orbitals are the hydrogenlike orbitals—exact analytic solutions to the Schrödinger equation for a single electron bound to a nucleus. For atoms with more electrons, these hydrogenlike functions serve as starting points for constructing approximate orbitals through mathematical methods. Their form depends on the coordinate system:

• Spherical coordinates are ideal for atoms, where wave functions separate neatly into radial and angular parts.
• Cartesian coordinates are often used in molecular contexts, particularly when constructing polyatomic molecular orbitals.

Atomic orbitals therefore bridge the gap between an exact quantum description of electrons and the simplified models needed in chemistry and materials science.

Role in Chemical Bonding and Atomic Structure

Orbitals are fundamental building blocks for understanding how atoms interact. They guide predictions about bonding patterns, ionisation energies, atomic size, and chemical reactivity. Orbital overlaps form the basis of covalent bonding, while differences in energy and occupancy determine ionic and metallic interactions.