Editing Atomic orbital (section)

== Quantum numbers ==
{{Main|Quantum number}}

Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers. These quantum numbers occur only in certain combinations of values, and their physical interpretation changes depending on whether [[real number|real]] or [[complex number|complex]] versions of the atomic orbitals are employed.

=== Complex orbitals ===
[[File:Electronic_levels.svg|thumb|450px|alt=Electronic levels|Energetic levels and sublevels of polyelectronic atoms]]
In physics, the most common orbital descriptions are based on the solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure [[spherical harmonic]]. The quantum numbers, together with the rules governing their possible values, are as follows:

The [[principal quantum number]] {{mvar|n}} describes the energy of the electron and is always a [[positive integer]]. In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of ''n''; these orbitals together are sometimes called ''[[electron shells]]''.

The [[azimuthal quantum number]] {{mvar|ℓ}} describes the orbital angular momentum of each electron and is a non-negative integer. Within a shell where {{mvar|n}} is some integer {{math|''n''<sub>0</sub>}}, {{mvar|ℓ}} ranges across all (integer) values satisfying the relation <math>0 \le \ell \le n_0-1</math>. For instance, the {{math|1=''n'' = 1}}&nbsp;shell has only orbitals with <math>\ell=0</math>, and the {{math|1=''n'' = 2}}&nbsp;shell has only orbitals with <math>\ell=0</math>, and <math>\ell=1</math>. The set of orbitals associated with a particular value of&nbsp;{{mvar|ℓ}} are sometimes collectively called a ''subshell''.

The [[magnetic quantum number]], <math>m_\ell</math>, describes the projection of the orbital angular momentum along a chosen axis. It determines the magnitude of the current circulating around that axis and the orbital contribution to the [[electron magnetic moment|magnetic moment of an electron]] via the [[Magnetic moment#Ampèrian loop model|Ampèrian loop]] model.<ref>{{Cite book |last=Greiner |first=Walter |url=http://archive.org/details/quantummechanics0001grei |title=Quantum mechanics : Introduction |date=1994 |publisher=Springer-Verlag |isbn=978-0-387-58080-7 |edition=2nd corrected |location=New York, Berlin, Heidelberg |pages=163}}</ref> Within a subshell <math>\ell</math>, <math>m_\ell</math> obtains the integer values in the range <math>-\ell \le m_\ell \le \ell </math>.

The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of <math>m_\ell</math> available in that subshell. Empty cells represent subshells that do not exist.

{| class="wikitable"
|-
!
! {{math|1=''ℓ'' = 0 (s)}}
! {{math|1=''ℓ'' = 1 (p)}}
! {{math|1=''ℓ'' = 2 (d)}}
! {{math|1=''ℓ'' = 3 (f)}}
! {{math|1=''ℓ'' = 4 (g)}}
! ...
|-
! {{math|1=''n'' = 1}}
| <math>m_\ell=0</math>
| || || || || ...
|-
! {{math|1=''n'' = 2}}
| 0 || −1, 0, 1
| || || || ...
|-
! {{math|1=''n'' = 3}}
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2
| || || ...
|-
! {{math|1=''n'' = 4}}
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2 || −3, −2, −1, 0, 1, 2, 3
| || ...
|-
! {{math|1=''n'' = 5}}
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2 || −3, −2, −1, 0, 1, 2, 3 || −4, −3, −2, −1, 0, 1, 2, 3, 4
| ...
|-
! ...
| ... || ... || ... || ... || ... || ...
|}

Subshells are usually identified by their <math>n</math>- and <math>\ell</math>-values. <math>n</math> is represented by its numerical value, but <math>\ell</math> is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with <math>n=2</math> and <math>\ell=0</math> as a '2s subshell'.

Each electron also has angular momentum in the form of [[Spin (physics)|quantum mechanical spin]] given by spin ''s'' = {{sfrac|1|2}}. Its projection along a specified axis is given by the [[spin magnetic quantum number]], ''m<sub>s</sub>'', which can be +{{sfrac|1|2}} or −{{sfrac|1|2}}. These values are also called "spin up" or "spin down" respectively.

The [[Pauli exclusion principle]] states that no two electrons in an atom can have the same values of all four quantum numbers. If there are two electrons in an orbital with given values for three quantum numbers, ({{mvar|n}}, {{mvar|ℓ}}, {{mvar|m}}), these two electrons must differ in their spin projection ''m<sub>s</sub>''.

The above conventions imply a preferred axis (for example, the ''z'' direction in Cartesian coordinates), and they also imply a preferred direction along this preferred axis. Otherwise there would be no sense in distinguishing {{math|1=''m'' = +1}} from {{math|1=''m'' = −1}}. As such, the model is most useful when applied to physical systems that share these symmetries. The [[Stern–Gerlach experiment]]{{Emdash}}where an atom is exposed to a magnetic field{{Emdash}}provides one such example.<ref>{{cite journal|last1=Gerlach |first1=W. |last2=Stern |first2=O. |title=Das magnetische Moment des Silberatoms|journal=[[Zeitschrift für Physik]]|volume=9|issue=1 |pages=353–355|year=1922|doi=10.1007/BF01326984|bibcode = 1922ZPhy....9..353G |s2cid=126109346 }}</ref>

=== Real orbitals ===
[[Image:Orbital p1-px animation.gif|thumb|220px|Animation of continuously varying superpositions between the {{math|p{{sub|1}}}} and the {{math|p{{sub|''x''}}}} orbitals. This animation does not use the Condon–Shortley phase convention.]]

Instead of the complex orbitals described above, it is common, especially in the chemistry literature, to use ''real'' atomic orbitals. These real orbitals arise from simple linear combinations of complex orbitals. Using the [[Spherical harmonics#Condon–Shortley phase|Condon–Shortley phase convention]], real orbitals are related to complex orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics. Letting <math>\psi_{n,\ell, m}</math> denote a complex orbital with quantum numbers {{mvar|n}}, {{mvar|&ell;}}, and {{mvar|m}}, the real orbitals <math>\psi_{n, \ell, m}^{\text{real}}</math> may be defined by<ref>{{cite book |last1=Thaller |first1=Bernd |title=Advanced visual quantum mechanics |date=2004 |publisher=Springer/TELOS |location=New York |isbn=978-0387207773}}</ref>

<math display="block">\begin{align}
  \psi_{n,\ell, m}^{\text{real}} &= \begin{cases}
    \sqrt{2} (-1)^m \text{Im}\left\{\psi_{n,\ell,|m|}\right\} &\text{ for } m<0 \\[2pt]
\psi_{n,\ell,|m|} &\text{ for } m=0\\[2pt]
    \sqrt{2} (-1)^m \text{Re}\left\{\psi_{n,\ell,|m|}\right\} &\text{ for } m>0
  \end{cases} \\[4pt]
  &= \begin{cases}
    \frac{i}{\sqrt{2}}\left(\psi_{n,\ell, -|m|} - (-1)^m \psi_{n,\ell, |m|}\right) & \text{ for } m<0 \\[2pt]
    \psi_{n, \ell, |m|}& \text{ for } m=0 \\[4pt]
\frac{1}{\sqrt{2}}\left(\psi_{n,\ell, -|m|} + (-1)^m \psi_{n,\ell, |m|}\right) & \text{ for } m>0
  \end{cases}
\end{align}</math>

If <math>\psi_{n,\ell, m}(r, \theta, \phi) = R_{nl}(r) Y_{\ell}^m(\theta, \phi)</math>, with <math>R_{nl}(r)</math> the radial part of the orbital, this definition is equivalent to <math>\psi_{n,\ell, m}^{\text{real}}(r, \theta, \phi) = R_{nl}(r) Y_{\ell m}(\theta, \phi)</math> where <math>Y_{\ell m}</math> is the real spherical harmonic related to either the real or imaginary part of the complex spherical harmonic <math>Y_{\ell}^m</math>.

Real spherical harmonics are physically relevant when an atom is embedded in a crystalline solid, in which case there are multiple preferred symmetry axes but no single preferred direction.{{citation needed|date=February 2022}} Real atomic orbitals are also more frequently encountered in introductory chemistry textbooks and shown in common orbital visualizations.<ref>{{cite book |title=General chemistry: principles and modern applications. |author1=Petrucci, Ralph |author2=Herring, F. |author3=Madura, Jeffry |author4=Bissonnette, Carey |date=2016 |edition=11th |publisher=Prentice Hall |location=[Place of publication not identified] |isbn=978-0133897319}}</ref> In real hydrogen-like orbitals, quantum numbers {{mvar|n}} and {{mvar|&ell;}} have the same interpretation and significance as their complex counterparts, but {{mvar|m}} is no longer a good quantum number (but its absolute value is).

Some real orbitals are given specific names beyond the simple <math>\psi_{n, \ell, m}</math> designation. Orbitals with quantum number {{math|1=''&ell;'' = 0, 1, 2, 3, 4, 5, 6...}} are called {{math|s, p, d, f, g, h, i, ...}} orbitals. With this one can already assign names to complex orbitals such as <math>2\text{p}_{\pm 1} = \psi_{2, 1, \pm 1}</math>; the first symbol is the {{mvar|n}} quantum number, the second character is the symbol for that particular {{mvar|&ell;}} quantum number and the subscript is the {{mvar|m}} quantum number.

As an example of how the full orbital names are generated for real orbitals, one may calculate <math>\psi_{n, 1, \pm 1}^{\text{real}}</math>. From the [[table of spherical harmonics]], <math display="inline">\psi_{n, 1, \pm1} = R_{n, 1}Y_1^{\pm 1} = \mp R_{n, 1} \sqrt{3/8\pi} \cdot (x\pm i y)/r</math> with <math display=inline>r = \sqrt{x^2+y^2+z^2}</math>. Then

<math display=block>\begin{align}
  \psi_{n, 1, +1}^\text{real} &= R_{n, 1} \sqrt{\frac{3}{4\pi}} \cdot \frac{x}{r}\\
  \psi_{n, 1, -1}^\text{real} &= R_{n, 1} \sqrt{\frac{3}{4\pi}} \cdot \frac{y}{r}
\end{align}</math>

Likewise <math display="inline">\psi_{n, 1, 0} = R_{n, 1} \sqrt{3/4\pi} \cdot z/r</math>. As a more complicated example:

<math display=block>
\psi_{n, 3, +1}^\text{real} = R_{n, 3} \frac{1}{4} \sqrt{\frac{21}{2\pi}} \cdot \frac{x\cdot (5z^2 - r^2)}{r^3}
</math>

In all these cases we generate a Cartesian label for the orbital by examining, and abbreviating, the polynomial in {{math|''x'', ''y'', ''z''}} appearing in the numerator. We ignore any terms in the {{math|''z'', ''r''}} polynomial except for the term with the highest exponent in {{mvar|z}}.
We then use the abbreviated polynomial as a subscript label for the atomic state, using the same nomenclature as above to indicate the <math>n</math> and <math>\ell</math> quantum numbers.{{cn|date=March 2024}}

<math display=block>
\begin{align}
  \psi_{n, 1, -1}^\text{real} &= n\text{p}_y = \frac{i}{\sqrt{2}} \left(n\text{p}_{-1} + n\text{p}_{+1}\right)\\
  \psi_{n, 1, 0}^\text{real} &= n\text{p}_z = 2\text{p}_0\\
  \psi_{n, 1, +1}^\text{real} &= n\text{p}_x = \frac{1}{\sqrt{2}} \left(n\text{p}_{-1} - n\text{p}_{+1}\right)\\
  \psi_{n, 3, +1}^\text{real} &= nf_{xz^2} = \frac{1}{\sqrt{2}} \left(nf_{-1} - nf_{+1}\right)
\end{align}
</math>

The expression above all use the [[Spherical harmonics#Condon–Shortley phase|Condon–Shortley phase convention]] which is favored by quantum physicists.<ref>{{cite book| last=Messiah|first=Albert|title=Quantum mechanics : two volumes bound as one| year=1999|publisher=Dover|location=Mineola, NY| isbn=978-0-486-40924-5|edition=Two vol. bound as one, unabridged reprint}}</ref><ref>{{cite book|author1=Claude Cohen-Tannoudji |author2=Bernard Diu |author3=Franck Laloë |translator=from the French by Susan Reid Hemley|title=Quantum mechanics |year=1996 |publisher=Wiley-Interscience |isbn=978-0-471-56952-7 |display-authors=etal}}</ref> Other conventions exist for the phase of the spherical harmonics.<ref name=Levine7ed>{{cite book| last=Levine|first=Ira|title=Quantum Chemistry| edition=7th| year=2014| publisher=Pearson Education| isbn=978-0-321-80345-0| pages=141–2}}</ref><ref>{{cite journal|author1=Blanco, Miguel A. |author2=Flórez, M. |author3=Bermejo, M. |date= December 1997|title=Evaluation of the rotation matrices in the basis of real spherical harmonics|journal=Journal of Molecular Structure: THEOCHEM |volume=419 |issue=1–3|pages=19–27|doi=10.1016/S0166-1280(97)00185-1}}</ref> Under these different conventions the <math>\text{p}_x</math> and <math>\text{p}_y</math> orbitals may appear, for example, as the sum and difference of <math>\text{p}_{+1}</math> and <math>\text{p}_{-1}</math>, contrary to what is shown above.

Below is a list of these Cartesian polynomial names for the atomic orbitals.<ref>{{cite book |title=General chemistry : principles and modern applications. |date=2016 |publisher=Prentice Hall |location=[Place of publication not identified] |isbn=978-0133897319}}</ref><ref>{{cite journal |last1=Friedman |title=The shapes of the f orbitals |journal=J. Chem. Educ. |year=1964 |volume=41 |issue=7 |page=354 |doi=10.1021/ed041p354 |bibcode=1964JChEd..41..354F }}</ref> There does not seem to be reference in the literature as to how to abbreviate the long Cartesian spherical harmonic polynomials for <math>\ell>3</math> so there does not seem be consensus on the naming of <math>g</math> orbitals or higher according to this nomenclature.{{cn|date=March 2024}}
{| class="wikitable"
|-
!
! <math>\psi_{m=-3}+\psi_{m=+3}</math>
! <math>\psi_{m=-2}+\psi_{m=+2}</math>
! <math>\psi_{m=-1}+\psi_{m=+1}</math>
! <math>\psi_{m=0}</math>
! <math>\psi_{m=-1}-\psi_{m=+1}</math>
! <math>\psi_{m=-2}-\psi_{m=+2}</math>
! <math>\psi_{m=-3}-\psi_{m=+3}</math>
|-
!  <math>\ell=0</math>
|| || || || <math>\text{s}</math> || || ||
|-
!  <math>\ell=1</math>
|| || || <math>\text{p}_y</math> || <math>\text{p}_z</math> || <math>\text{p}_x</math> || ||
|-
!  <math>\ell=2</math>
|| || <math>\text{d}_{x^2-y^2}</math>|| <math>\text{d}_{yz}</math> || <math>\text{d}_{z^2}</math> || <math>\text{d}_{xz}</math> || <math>\text{d}_{xy
}</math>||
|-
!  <math>\ell=3</math>
|| <math>\text{f}_{y(3x^2-y^2)}</math>|| <math>\text{f}_{z(x^2-y^2)}</math>|| <math>\text{f}_{yz^2}</math> || <math>\text{f}_{z^3}</math> || <math>\text{f}_{xz^2}</math> || <math>\text{f}_{xyz}</math>|| <math>\text{f}_{x(x^2-3y^2)}</math>
|}