Atomic orbitals

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.

Hydrogen orbitals. N, L and M are the first quantum numbers (the forth number has no effect on the cloud shape)
Despite the obvious analogy to planets revolving around the Sun, electrons cannot be described as solid particles and so atomic orbitals rarely, if ever, resemble a planet's elliptical path. A more accurate analogy might be that of a large and often oddly-shaped atmosphere (the electron), distributed around a relatively tiny planet (the atomic nucleus). Atomic orbitals exactly describe the shape of this atmosphere only when a single electron is present in an atom. When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection (sometimes termed the atom’s “electron cloud” [3]) 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,[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: sharp, principal, diffuse, and fundamental, the rest being named in alphabetical order (omitting j).[8][9]

The quantum numbers

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 first, , describes the electron shell, or energy level.
    • The value of ranges from 1 to "n", where "n" is the shell containing the outermost electron of that atom.
  • The second, , describes the electron shell (0 = s orbital, 1 = p orbital, 2 = d orbital, 3 = f orbital, etc.).
    • The value of ranges from to . 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 third, , describes the specific orbital (or "cloud") within that subshell.*
    • The values of range from to .
  • The fourth, , describes the spin of the electron within that orbital.*
    • Because an orbital never contains more than two electrons, will be either -1/2 or +1/2, corresponding with "spin" and "opposite spin".

Hydrogen orbitals

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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  1. 1 Milton Orchin,Roger S. Macomber, Allan Pinhas, and R. Marshall Wilson(2005)"Atomic Orbital Theory"
  2. 2 Daintith, J.. Oxford Dictionary of Chemistry
  3. 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. 4 B.Niels (1913). On the Constitution of Atoms and Molecules. Philosophical Magazine 1 476.
  5. 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. 6 B.Bryson. A Short History of Nearly Everything
  7. 7 M.Robert. (1932). Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations. Phys. Rev. 1 49–71.
  8. 8 D.Griffiths. Introduction to Quantum Mechanics
  9. 9  I.Levine. Quantum Chemistry


Orbital applet by Carlo Barraco, Todd Fuller