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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.^{[1]} 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.^{[2]} 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.

The idea that electrons might revolve around a compact nucleus with definite angular momentum was convincingly argued in 1913 by Niels Bohr,^{[4]} and the Japanese physicist Hantaro Nagaoka published an orbit-based hypothesis for electronic behavior as early as 1904.^{[5]} 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.^{[6]}

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.^{[7]} 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: **s**harp, **p**rincipal, **d**iffuse, and **f**undamental, the rest being named in alphabetical order (omitting j).^{[8]}^{[9]}

The most widely accepted model currently describes electrons using four quantum numbers, [[Math:c|n \ ]], [[Math:c|\ell \ ]], [[Math:c|m_\ell \ ]], and [[Math:c|m_s\,\!]]. It is also the common nomenclature in the classical description of nuclear particle states (e.g., proton and neutrons.)

- The first, [[Math:c|n \ ]], describes the electron shell, or energy level.
- The value of [[Math:c|n \ ]] ranges from 1 to "n", where "n" is the shell containing the outermost electron of that atom.

- The second, [[Math:c|\ell \ ]], describes the electron shell (0 = s orbital, 1 = p orbital, 2 = d orbital, 3 = f orbital, etc.).
- The value of [[Math:c|\ell \ ]] ranges from [[Math:c|0]] to [[Math:c|\ n-1]]. This is because the first p orbital (
*l*=1) appears in the second electron shell (*n*=2), the first d orbital (*l*=2) appears in the third shell (*n*=3), and so on. A quantum number beginning in 3,0,... describes an electron in the s orbital of the third electron shell of an atom.

- The value of [[Math:c|\ell \ ]] ranges from [[Math:c|0]] to [[Math:c|\ n-1]]. This is because the first p orbital (
- The third, [[Math:c|m_\ell \ ]], describes the specific orbital (or "cloud") within that subshell.*
- The values of [[Math:c|m_\ell \ ]] range from [[Math:c|-\ell \ ]] to [[Math:c| \ell \ ]].

- The fourth, [[Math:c|m_s\,\!]], describes the spin of the electron within that orbital.*
- Because an orbital never contains more than two electrons, [[Math:c|m_s\,\!]] will be either -1/2 or +1/2, corresponding with "spin" and "opposite spin".

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.

**Acknowledgements**

This web page reuses material from Wikipedia page http://en.wikipedia.org/wiki/Atomic_orbital + http://en.wikipedia.org/wiki/Quantum_number under the rights of *CC-BY-SA* license. As a result, the content of this page is and will stay available under the rights of this license regardless of restrictions that apply to other pages of this website.

^{1 }Milton Orchin,Roger S. Macomber, Allan Pinhas, and R. Marshall Wilson(2005)"Atomic Orbital Theory"^{2 }Daintith, J.. Oxford Dictionary of Chemistry^{3 }The Feynman Lectures on Physics -The Definitive Edition, Vol 1 lect 6 pg 11. Feynman, Richard; Leighton; Sands. (2006) Addison Wesley ISBN 0-8053-9046-4^{4 }B.Niels (1913).*On the Constitution of Atoms and Molecules*. Philosophical Magazine**1**476.^{5 }H.Nagaoka (1904).*Kinetics of a System of Particles illustrating the Line and the Band Spectrum and the Phenomena of Radioactivity*. Philosophical Magazine 445–455.^{6 }B.Bryson. A Short History of Nearly Everything^{7 }M.Robert. (1932).*Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations*. Phys. Rev.**1**49–71.^{8 }D.Griffiths. Introduction to Quantum Mechanics^{9 }I.Levine. Quantum Chemistry

Orbital applet by Carlo Barraco, Todd Fuller