An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. These functions may serve as three-dimensional graph of an electron’s likely location. The term may thus refer directly to the physical region defined by the function where the electron is likely to be. Specifically, atomic orbitals are the possible quantum states of an individual electron in the collection of electrons around a single atom, as described by the orbital function.) tends toward a generally spherical zone of probability describing where the atom’s electrons will be found.
The idea that electrons might revolve around a compact nucleus with definite angular momentum was convincingly argued in 1913 by Niels Bohr, and the Japanese physicist Hantaro Nagaoka published an orbit-based hypothesis for electronic behavior as early as 1904. However, it was not until 1926 that the solution of the Schrödinger equation for electron-waves in atoms provided the functions for the modern orbitals.
Because of the difference from classical mechanical orbits, the term "orbit" for electrons in atoms, has been replaced with the term orbital—a term first coined by chemist Robert Mulliken in 1932. Atomic orbitals are typically described as “hydrogen-like” (meaning one-electron) wave functions over space, categorized by n, l, and m quantum numbers, which correspond to the electrons' energy, angular momentum, and an angular momentum direction, respectively. Each orbital is defined by a different set of quantum numbers and contains a maximum of two electrons. The simple names s orbital, p orbital, d orbital and f orbital refer to orbitals with angular momentum quantum number l = 0, 1, 2 and 3 respectively. These names indicate the orbital shape and are used to describe the electron configurations as shown on the right. They are derived from the characteristics of their spectroscopic lines: sharp, principal, diffuse, and fundamental, the rest being named in alphabetical order (omitting j).
The most widely accepted model currently describes electrons using four quantum numbers, , , , and . It is also the common nomenclature in the classical description of nuclear particle states (e.g., proton and neutrons.)
The simplest atomic orbitals are those that occur in an atom with a single electron, such as the hydrogen atom. They can be obtained analytically (see applet). An atom of any other element ionized down to a single electron is very similar to hydrogen, and the orbitals take the same form.
For atoms with two or more electrons, the governing equations can only be solved only approximately.
A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: N, L, and M. The rules restricting the values of the quantum numbers, and their energies, explain the electron configuration of the atoms and the periodic table.
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Orbital applet by Carlo Barraco, Todd Fuller