An atomic orbital is a wavefunction describing the spatial state of an electron in an atom. In quantum mechanics, it specifies probability amplitudes rather than a definite path around the nucleus. Atomic orbitals are exact stationary-state solutions for idealized one-electron atoms and serve as components of approximate descriptions of many-electron atoms. The term also commonly refers to diagrams depicting the associated spatial probability density, although an orbital is mathematically a function, not a bounded region. (goldbook.iupac.org)
Mathematical description
For a one-electron atom, stationary orbitals satisfy the time-independent Schrödinger equation,
[ \hat H\psi=E\psi, ]
where (\hat H) is the Hamiltonian operator and (E) is the state's energy. According to the Born rule, the probability density is (|\psi(\mathbf r)|^2). The probability of finding the electron within a specified volume is the integral of this density over that volume. A normalized orbital has total probability one. Consequently, a picture of an orbital should not be interpreted as a solid object or an electron's trajectory. (openstax.org)
For hydrogen and hydrogen-like ions, spherical symmetry permits separation into radial and angular factors:
[ \psi_{n\ell m_\ell}(r,\theta,\phi) =R_{n\ell}(r)Y_\ell^{m_\ell}(\theta,\phi). ]
The radial factor describes dependence on distance from the nucleus; the angular factor is a spherical harmonic. This separation provides the mathematical foundation for familiar orbital shapes and their classification. (ocw.mit.edu)
Quantum numbers and classification
In the nonrelativistic, central-field description, three quantum numbers label a spatial orbital:
- Principal quantum number, (n): a positive integer identifying the shell.
- Orbital angular momentum quantum number, (\ell): an integer from (0) to (n-1), identifying the subshell.
- Magnetic quantum number, (m_\ell): an integer from (-\ell) to (+\ell), specifying the projection of orbital angular momentum along a chosen axis.
The letters (s), (p), (d), and (f) correspond respectively to (\ell=0,1,2,3). Thus, “3p” identifies the (n=3,\ell=1) subshell, not one uniquely specified orbital. Each subshell contains (2\ell+1) spatial orbitals: one (s), three (p), five (d), or seven (f). (openstax.org)
An electron additionally has spin angular momentum, with spin projection (m_s=+\tfrac12) or (-\tfrac12). Spin is not part of the spatial orbital itself. The Pauli exclusion principle allows at most two electrons to occupy the same spatial orbital, provided their spin projections differ. Accordingly, the maximum capacities of (s), (p), (d), and (f) subshells are 2, 6, 10, and 14 electrons. (openstax.org)
Shapes, phases, and nodes
The probability density of an (s) orbital is spherically symmetric. Familiar real (p) orbitals have two lobes separated by a nodal plane. Most conventional real (d) orbital drawings show four lobes, while (d_{z^2}) has two axial lobes and a toroidal region. These drawings generally display selected probability-density surfaces rather than sharp physical boundaries; bound orbital functions extend beyond the depicted surfaces. (openstax.org)
A node is a location where the wavefunction vanishes. In the conventional real hydrogenic orbital representation, an orbital has (n-\ell-1) radial nodes and (\ell) angular nodes. For example, (2s) has one spherical radial node, whereas (2p) has one angular nodal plane and no radial node. Different colors on orbital lobes usually indicate opposite signs of a real wavefunction, not opposite electric charges. (ocw.mit.edu)
The commonly drawn (p_x) and (p_y) orbitals are linear combinations of complex orbitals with (m_\ell=\pm1). They are useful directional representations, but neither individually has a definite angular-momentum projection along the (z)-axis. (ocw.mit.edu)
Energies and electron configurations
In the ideal nonrelativistic Coulomb model of a hydrogen-like atom, orbital energy depends only on (n) and is proportional to (-Z^2/n^2), where (Z) is the atomic number. Orbitals sharing (n) are therefore degenerate: they have equal energy despite different shapes. An external magnetic field can remove some degeneracies. (openstax.org)
In many-electron atoms, electron–electron repulsion, shielding, and penetration into the inner electron distribution alter subshell energies. Orbital occupation is expressed through an electron configuration, such as (1s^22s^22p^6) for neon. Ground-state configurations reflect energy ordering, the exclusion principle, and Hund's rules. The familiar filling sequence is useful but has exceptions; orbital energies also change with occupation and ionization. Shell structure underlies recurring properties in the periodic table. (openstax.org)
Many-electron descriptions and chemical bonding
An interacting many-electron atom has a wavefunction depending jointly on all electron coordinates. Individual orbitals are therefore not separate exact wavefunctions for distinguishable electrons. Methods such as the Hartree–Fock method approximate this collective problem using one-electron orbitals and an average treatment of interactions; more elaborate calculations account for additional electron correlation. (ocw.mit.edu)
Atomic orbitals also supply the language and mathematical building blocks for describing chemical bonds. In valence bond theory, orbital overlap supports localized bonding descriptions. Orbital hybridization forms directional combinations of orbitals on the same atom, such as the four tetrahedrally directed (sp^3) hybrids. In molecular orbital theory, combinations of functions centered on different atoms construct orbitals extending across a molecule. Their relative phases help determine whether combinations are bonding or antibonding. (openstax.org)