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In some sources, including my textbook it is cited that an atomic orbital is the 3 dimensional region where the probability of finding an electron is 90-95% . But, Other sources such as Wikipedia state otherwise. To quote Wikipedia, "In quantum mechanics, 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" With relevance to the fact that an orbital is always a wave function I agree that it can be regarded as a mathematical function, But what is this reference to "a pair of electrons"This has confused me.What is the the true definition of an orbital? I am new to this forum and Chemistry, please excuse me if there's some misconception. Thanks

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marked as duplicate by orthocresol, Jannis Andreska, Wildcat, Jan, Todd Minehardt Aug 21 '16 at 15:21

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    $\begingroup$ Graphical representations are usually drawn for the 95% probability integral, but the actual orbital is the function describing the distribution of said probabillity (not the integral!). Wikipedia is right, your textbook sloppily written. $\endgroup$ – Karl Jun 27 '16 at 18:45
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    $\begingroup$ As for the pair thing, read about spin, or rather don't, just take it for granted that two electrons may occupy the same orbital. Also, welcome to Chem.SE. $\endgroup$ – Ivan Neretin Jun 27 '16 at 18:59
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The mathematical definition is the most rigorous. The presence of the positively charged nucleus means that there is a pull on a negatively charged electron, which we describe with the potential energy. But electrons can only exist with certain discrete energy values. To do a full analysis you need to find the wavefunction that describes the electron. For a given electron, the wavefunction* at point (x, y, z) gives you the probability of finding the electron at the location; this is non-zero pretty much everywhere in space but is only significantly above zero in a very small region: this is what the first definition is getting at. Once you have the wavefunction you can calculate some properties, like the angular momentum. Since each electron has a discrete energy, each electron can be described by a set of four numbers, derived from the wavefunction: principle number (n), orbital angular momentum (l), magnetic angular momentum ($m_l$) and spin number ($m_s$). The spin number can be either +1/2 or -1/2 but we usually ignore it since normally that value doesn't affect the energy of the electron; this is the source of that "pair of electrons" business.

To summarize: An orbital is a specific state of an electron, characterized by its energy and various angular momenta and rigorously described by a mathematical function that satisfies Schroedinger's equation. This can be represented visually in many ways, such as showing the 95% probability solid, or as a line on a molecular-orbital diagram.

*Really this should be the wavefunction times its complex conjugate.

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