Editing Atomic orbital (section)

== Shapes of orbitals ==
[[File:Atomic-orbital-cloud_n6_l0_m0.png|thumb|Transparent cloud view of a computed 6s {{math|1=(''n'' = 6, ''ℓ'' = 0, ''m'' = 0)}} hydrogen orbital. The s orbitals, though spherically symmetric, have radially placed wave-nodes for {{math|''n'' > 1}}. Only s orbitals invariably have a center anti-node; the other types never do.]]

Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space. Instead the diagrams are approximate representations of boundary or [[isosurface|contour surfaces]] where the probability density {{math|{{!}} ψ(''r'', ''θ'', ''φ'') {{!}}<sup>2</sup>}} has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although {{math|{{!}} ''ψ'' {{!}}<sup>2</sup>}} as the square of an [[absolute value]] is everywhere non-negative, the sign of the [[wave function]] {{math|ψ(''r'', ''θ'', ''φ'')}} is often indicated in each subregion of the orbital picture.

Sometimes the {{math|ψ}} function is graphed to show its phases, rather than {{math|{{!}} ψ(''r'', ''θ'', ''φ'') {{!}}<sup>2</sup>}} which shows probability density but has no phase (which is lost when taking absolute value, since {{math|ψ(''r'', ''θ'', ''φ'')}} is a [[complex number]]). {{math|{{abs|ψ(''r'', ''θ'', ''φ'')}}<sup>2</sup>}} orbital graphs tend to have less spherical, thinner lobes than {{math|ψ(''r'', ''θ'', ''φ'')}} graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, to show wave function phase, shows mostly {{math|ψ(''r'', ''θ'', ''φ'')}} graphs.

The lobes can be seen as [[standing wave]] [[wave interference|interference]] patterns between the two counter-rotating, ring-resonant [[traveling wave]] {{mvar|m}} and {{math|−''m''}} modes; the projection of the orbital onto the xy plane has a resonant {{mvar|m}} wavelength around the circumference. Although rarely shown, the traveling wave solutions can be seen as rotating banded tori; the bands represent phase information. For each {{mvar|m}} there are two standing wave solutions {{math|⟨''m''⟩ + ⟨−''m''⟩}} and {{math|⟨''m''⟩ − ⟨−''m''⟩}}. If {{math|1=''m'' = 0}}, the orbital is vertical, counter rotating information is unknown, and the orbital is ''z''-axis symmetric. If {{math|1=''ℓ'' = 0}} there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric.

''[[Node (physics)|Nodal]] planes'' and ''nodal spheres'' are surfaces on which the probability density vanishes. The number of nodal surfaces is controlled by the quantum numbers {{mvar|n}} and {{mvar|ℓ}}. An orbital with azimuthal quantum number {{mvar|ℓ}} has {{mvar|ℓ}} radial nodal planes passing through the origin. For example, the s orbitals ({{math|1=''ℓ'' = 0}}) are spherically symmetric and have no nodal planes, whereas the p orbitals ({{math|1=''ℓ'' = 1}}) have a single nodal plane between the lobes. The number of nodal spheres equals {{mvar|n−ℓ−1}}, consistent with the restriction {{mvar|ℓ ≤ n−1}} on the quantum numbers. The principal quantum number controls the total number of nodal surfaces which is {{mvar|n−1}}.<ref>{{Cite book |last=Stanley Raimes |url=http://archive.org/details/thewavemechanicsofelectronsinmetalsraimes |title=The Wave Mechanics Of Electrons In Metals |date=1963 |publisher=North-Holland Publishing Company - Amsterdam |pages=39}}</ref> Loosely speaking, {{mvar|n}} is energy, {{mvar|ℓ}} is analogous to [[orbital eccentricity|eccentricity]], and {{mvar|m}} is orientation.

In general, {{mvar|n}} determines size and energy of the orbital for a given nucleus; as {{mvar|n}} increases, the size of the orbital increases. The higher nuclear charge {{mvar|Z}} of heavier elements causes their orbitals to contract by comparison to lighter ones, so that the size of the atom remains very roughly constant, even as the number of electrons increases.
[[File:Sr core-electron orbitals for Wiki.jpg|thumb|Experimentally imaged 1s and 2p core-electron orbitals of Sr, including the effects of atomic thermal vibration and excitation broadening, retrieved from energy dispersive x-ray spectroscopy (EDX) in scanning transmission electron microscopy (STEM).<ref name="Jeong 165140">{{Cite journal|last1=Jeong|first1=Jong Seok|last2=Odlyzko|first2=Michael L.|last3=Xu|first3=Peng|last4=Jalan|first4=Bharat|last5=Mkhoyan|first5=K. Andre|date=26 April 2016|title=Probing core-electron orbitals by scanning transmission electron microscopy and measuring the delocalization of core-level excitations|journal=Physical Review B|volume=93|issue=16|pages=165140|doi=10.1103/PhysRevB.93.165140|bibcode = 2016PhRvB..93p5140J |doi-access=free}}</ref>]]
Also in general terms, {{mvar|ℓ}} determines an orbital's shape, and {{mvar|m<sub>ℓ</sub>}} its orientation. However, since some orbitals are described by equations in [[complex number]]s, the shape sometimes depends on {{mvar|m<sub>ℓ</sub>}} also. Together, the whole set of orbitals for a given {{mvar|ℓ}} and {{mvar|n}} fill space as symmetrically as possible, though with increasingly complex sets of lobes and nodes.

The single s orbitals (<math>\ell=0</math>) are shaped like spheres. For {{math|1=''n'' = 1}} it is roughly a [[ball (mathematics)|solid ball]] (densest at center and fades outward exponentially), but for {{math|1=''n'' ≥ 2}}, each single s orbital is made of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well). See illustration of a cross-section of these nested shells, at right. The s orbitals for all {{mvar|n}} numbers are the only orbitals with an anti-node (a region of high wave function density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have angular momentum, and thus avoid the nucleus (having a wave node ''at'' the nucleus). Recently, there has been an effort to experimentally image the 1s and 2p orbitals in a SrTiO<sub>3</sub> crystal using scanning transmission electron microscopy with energy dispersive x-ray spectroscopy.<ref name="Jeong 165140" /> Because the imaging was conducted using an electron beam, Coulombic beam-orbital interaction that is often termed as the impact parameter effect is included in the outcome (see the figure at right).

The shapes of p, d and f orbitals are described verbally here and shown graphically in the ''Orbitals table'' below. The three p orbitals for {{math|1=''n'' = 2}} have the form of two [[ellipsoid]]s with a [[point of tangency]] at the [[atomic nucleus|nucleus]] (the two-lobed shape is sometimes referred to as a "[[dumbbell]]"—there are two lobes pointing in opposite directions from each other). The three p orbitals in each [[Electron shell|shell]] are oriented at right angles to each other, as determined by their respective linear combination of values of&nbsp;{{mvar|m<sub>ℓ</sub>}}. The overall result is a lobe pointing along each direction of the primary axes.

Four of the five d orbitals for {{math|1=''n'' = 3}} look similar, each with four pear-shaped lobes, each lobe tangent at right angles to two others, and the centers of all four lying in one plane. Three of these planes are the xy-, xz-, and yz-planes—the lobes are between the pairs of primary axes—and the fourth has the center along the x and y axes themselves. The fifth and final d orbital consists of three regions of high probability density: a [[torus]] in between two pear-shaped regions placed symmetrically on its z axis. The overall total of 18 directional lobes point in every primary axis direction and between every pair.

There are seven f orbitals, each with shapes more complex than those of the d orbitals.

Additionally, as is the case with the s orbitals, individual p, d, f and g orbitals with {{mvar|n}} values higher than the lowest possible value, exhibit an additional radial node structure which is reminiscent of harmonic waves of the same type, as compared with the lowest (or fundamental) mode of the wave. As with s orbitals, this phenomenon provides p, d, f, and g orbitals at the next higher possible value of {{mvar|n}} (for example, 3p orbitals vs. the fundamental 2p), an additional node in each lobe. Still higher values of {{mvar|n}} further increase the number of radial nodes, for each type of orbital.

The shapes of atomic orbitals in one-electron atom are related to 3-dimensional [[spherical harmonics]]. These shapes are not unique, and any linear combination is valid, like a transformation to [[cubic harmonic]]s, in fact it is possible to generate sets where all the d's are the same shape, just like the {{math|p<sub>''x''</sub>, p<sub>''y''</sub>,}} and {{math|p<sub>''z''</sub>}} are the same shape.<ref>{{cite journal| doi = 10.1021/ed045p45 | title = The five equivalent d orbitals | year = 1968 | last1 = Powell | first1 = Richard E. | journal = Journal of Chemical Education | volume = 45| issue = 1 | page = 45|bibcode = 1968JChEd..45...45P }}</ref><ref>{{cite journal| doi =10.1063/1.1750628 | title =Directed Valence | year =1940 | last1 =Kimball | first1 =George E. | journal =The Journal of Chemical Physics | volume =8| issue =2 | page =188|bibcode = 1940JChPh...8..188K }}</ref>

[[File:Schrodinger model of the atom.svg|thumb|The 1s, 2s, and 2p orbitals of a sodium atom]]
Although individual orbitals are most often shown independent of each other, the orbitals coexist around the nucleus at the same time. Also, in 1927, [[Albrecht Unsöld]] proved that if one sums the electron density of all orbitals of a particular azimuthal quantum number  {{mvar|ℓ}} of the same shell {{mvar|n}} (e.g., all three 2p orbitals, or all five 3d orbitals) where each orbital is occupied by an electron or each is occupied by an electron pair, then all angular dependence disappears; that is, the resulting total density of all the atomic orbitals in that subshell (those with the same  {{mvar|ℓ}}) is spherical. This is known as [[Spherical harmonics#Addition theorem|Unsöld's theorem]].

=== Orbitals table ===
This table shows the real hydrogen-like wave functions for all atomic orbitals up to 7s, and therefore covers the occupied orbitals in the ground state of all elements in the periodic table up to [[radium]] and some beyond. "ψ" graphs are shown with '''−''' and '''+''' [[wave function]] phases shown in two different colors (arbitrarily red and blue). The {{math|p<sub>''z''</sub>}} orbital is the same as the {{math|p<sub>0</sub>}} orbital, but the {{math|p<sub>''x''</sub>}} and {{math|p<sub>''y''</sub>}} are formed by taking linear combinations of the {{math|p<sub>+1</sub>}} and {{math|p<sub>−1</sub>}} orbitals (which is why they are listed under the {{math|1=''m'' = ±1}} label). Also, the {{math|p<sub>+1</sub>}} and {{math|p<sub>−1</sub>}} are not the same shape as the {{math|p<sub>0</sub>}}, since they are pure [[spherical harmonics]].

{| class="wikitable"
|-
!
! s ({{math|1=''ℓ'' = 0}})
! colspan="3" |p ({{math|1=''ℓ'' = 1}})
! colspan="5" |d ({{math|1=''ℓ'' = 2}})
! colspan="7" |f ({{math|1=''ℓ'' = 3}})
|-
!
! {{math|1=''m'' = 0}}
! {{math|1=''m'' = 0}}
! colspan="2" |{{math|1=''m'' = ±1}}
! {{math|1=''m'' = 0}}
! colspan="2" |{{math|1=''m'' = ±1}}
! colspan="2" |{{math|1=''m'' = ±2}}
! {{math|1=''m'' = 0}}
! colspan="2" |{{math|1=''m'' = ±1}}
! colspan="2" |{{math|1=''m'' = ±2}}
! colspan="2" |{{math|1=''m'' = ±3}}
|-
!
! s
! p<sub>''z''</sub>
! p<sub>''x''</sub>
! p<sub>''y''</sub>
! d<sub>''z''<sup>2</sup></sub>
! d<sub>''xz''</sub>
! d<sub>''yz''</sub>
! d<sub>''xy''</sub>
! d<sub>''x''<sup>2</sup>−''y''<sup>2</sup></sub>
! f<sub>''z''<sup>3</sup></sub>
! f<sub>''xz''<sup>2</sup></sub>
! f<sub>''yz''<sup>2</sup></sub>
! f<sub>''xyz''</sub>
! f<sub>''z''(''x''<sup>2</sup>−''y''<sup>2</sup>)</sub>
! f<sub>''x''(''x''<sup>2</sup>−3''y''<sup>2</sup>)</sub>
! f<sub>''y''(3''x''<sup>2</sup>−''y''<sup>2</sup>)</sub>
|-
!{{math|1=''n'' = 1}}
| [[File:S1M0.png|50px]]
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!{{math|1=''n'' = 2}}
| [[File:S2M0.png|50px]]
| [[File:P2M0.png|50px]]
| [[File:Px orbital.png|50px]]
| [[File:Py orbital.png|50px]]
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!{{math|1=''n'' = 3}}
| [[File:S3M0.png|50px]]
| [[File:P3M0.png|50px]]
| [[File:P3x.png|50px]]
| [[File:P3y.png|50px]]
| [[File:D3M0.png|50px]]
| [[File:Dxz orbital.png|50px]]
| [[File:Dyz orbital.png|50px]]
| [[File:Dxy orbital.png|50px]]
| [[File:Dx2-y2 orbital.png|50px]]
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!{{math|1=''n'' = 4}}
| [[File:S4M0.png|50px]]
| [[File:P4M0.png|50px]]
| [[File:P4M1.png|50px]]
| [[File:P4M-1.png|50px]]
| [[File:D4M0.png|50px]]
| [[File:D4xz.png|50px]]
| [[File:D4yz2.png|50px]]
| [[File:D4xy.png|50px]]
| [[File:D4x2-y2.png|50px]]
| [[File:F4M0.png|50px]]
| [[File:Fxz2 orbital.png|50px]]
| [[File:Fyz2 orbital.png|50px]]
| [[File:Fxyz orbital.png|50px]]
| [[File:Fz(x2-y2) orbital.png|50px]]
| [[File:Fx(x2-3y2) orbital.png|50px]]
| [[File:Fy(3x2-y2) orbital.png|50px]]
|-
!{{math|1=''n'' = 5}}
| [[File:S5M0.png|50px]]
| [[File:P5M0.png|50px]]
| [[File:P5M1.png|50px]]
| [[File:P5y.png|50px]]
| [[File:D5M0.png|50px]]
| [[File:D5xz.png|50px]]
| [[File:D5yz.png|50px]]
| [[File:D5xy.png|50px]]
| [[File:D5x2-y2.png|50px]]
| '''. . .'''
| '''. . .'''
| '''. . .'''
| '''. . .'''
| '''. . .'''
| '''. . .'''
| '''. . .'''
|-
!{{math|1=''n'' = 6}}
| [[File:S6M0.png|50px]]
| [[File:P6M0.png|50px]]
| [[File:P6x.png|50px]]
| [[File:P6y.png|50px]]
| '''. . .''' ‡
| '''. . .''' ‡
| '''. . .''' ‡
| '''. . .''' ‡
| '''. . .''' ‡
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
|-
!{{math|1=''n'' = 7}}
| [[File:S7M0.png|50px]]
| '''. . .''' †
| '''. . .''' †
| '''. . .''' †
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
| '''. . .''' *
|}

<nowiki>*</nowiki> ''No elements with 6f, 7d or 7f electrons have been discovered yet.''

† ''Elements with 7p electrons have been discovered, but their [[electronic configuration]]s are only predicted – save the exceptional [[lawrencium|Lr]], which fills 7p<sup>1</sup> instead of 6d<sup>1</sup>.''

‡ ''For the elements whose highest occupied orbital is a 6d orbital, only some electronic configurations have been confirmed.'' ([[Meitnerium|Mt]], [[Darmstadtium|Ds]], [[Roentgenium|Rg]] and [[Copernicium|Cn]] are still missing).

These are the real-valued orbitals commonly used in chemistry. Only the <math>m = 0</math> orbitals where are eigenstates of the orbital angular momentum operator, <math>\hat L_z</math>. The columns with <math>m = \pm 1, \pm 2,\cdots</math> are combinations of two eigenstates. See [[:File:Atomic orbitals spdf m-eigenstates and superpositions.png|comparison in the following picture]]: [[File:Atomic orbitals spdf m-eigenstates and superpositions.png|thumb|Atomic orbitals spdf m-eigenstates and superpositions]]

=== Qualitative understanding of shapes ===
The shapes of atomic orbitals can be qualitatively understood by considering the analogous case of [[Vibrations of a circular drum|standing waves on a circular drum]].<ref>{{cite journal| last1=Cazenave, Lions|first1=T., P.|title=Orbital stability of standing waves for some nonlinear Schrödinger equations| journal=Communications in Mathematical Physics|year=1982| volume=85|issue=4 | pages= 549–561|doi = 10.1007/BF01403504 |bibcode = 1982CMaPh..85..549C| last2=Lions| first2=P. L.|s2cid=120472894}}</ref> To see the analogy, the mean vibrational displacement of each bit of drum membrane from the equilibrium point over many cycles (a measure of average drum membrane velocity and momentum at that point) must be considered relative to that point's distance from the center of the drum head. If this displacement is taken as being analogous to the probability of finding an electron at a given distance from the nucleus, then it will be seen that the many modes of the vibrating disk form patterns that trace the various shapes of atomic orbitals. The basic reason for this correspondence lies in the fact that the distribution of kinetic energy and momentum in a matter-wave is predictive of where the particle associated with the wave will be. That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the [[Uncertainty principle|Heisenberg uncertainty principle]] for details of the mechanism).

This relationship means that certain key features can be observed in both drum membrane modes and atomic orbitals. For example, in all of the modes analogous to '''s'''&nbsp;orbitals (the top row in the animated illustration below), it can be seen that the very center of the drum membrane vibrates most strongly, corresponding to the [[antinode]] in all '''s'''&nbsp;orbitals in an atom. This antinode means the electron is most likely to be at the physical position of the nucleus (which it passes straight through without scattering or striking it), since it is moving (on average) most rapidly at that point, giving it maximal momentum.

A mental "planetary orbit" picture closest to the behavior of electrons in '''s'''&nbsp;orbitals, all of which have no angular momentum, might perhaps be that of a [[Keplerian orbit]] with the [[orbital eccentricity]] of 1 but a finite major axis, not physically possible (because [[particle]]s were to collide), but can be imagined as a [[limit (mathematics)|limit]] of orbits with equal major axes but increasing eccentricity.<!-- could somebody make an illustration? -->

Below, a number of drum membrane vibration modes and the respective wave functions of the hydrogen atom are shown. A correspondence can be considered where the wave functions of a vibrating drum head are for a two-coordinate system {{math|ψ(''r'', ''θ'')}} and the wave functions for a vibrating sphere are three-coordinate {{math|ψ(''r'', ''θ'', ''φ'')}}.

<gallery mode="nolines" perrow="3" widths="200px" caption="s-type drum modes and wave functions">
File:Drum vibration mode01.gif|Drum mode <math>u_{01}</math>
File:Drum vibration mode02.gif|Drum mode <math>u_{02}</math>
File:Drum vibration mode03.gif|Drum mode <math>u_{03}</math>
File:Phi 1s.gif|Wave function of 1s orbital (real part, 2D-cut, <math>r_\mathrm{max}=2 a_0</math>)
File:Phi 2s.gif|Wave function of 2s orbital (real part, 2D-cut, <math>r_\mathrm{max}=10 a_0</math>)
File:Phi 3s.gif|Wave function of 3s orbital (real part, 2D-cut, <math>r_\mathrm{max}=20 a_0</math>)
</gallery>

None of the other sets of modes in a drum membrane have a central antinode, and in all of them the center of the drum does not move. These correspond to a node at the nucleus for all non-'''s''' orbitals in an atom. These orbitals all have some angular momentum, and in the planetary model, they correspond to particles in orbit with eccentricity less than 1.0, so that they do not pass straight through the center of the primary body, but keep somewhat away from it.

In addition, the drum modes analogous to '''p''' and '''d''' modes in an atom show spatial irregularity along the different radial directions from the center of the drum, whereas all of the modes analogous to '''s'''&nbsp;modes are perfectly symmetrical in radial direction. The non-radial-symmetry properties of non-'''s''' orbitals are necessary to localize a particle with angular momentum and a wave nature in an orbital where it must tend to stay away from the central attraction force, since any particle localized at the point of central attraction could have no angular momentum. For these modes, waves in the drum head tend to avoid the central point. Such features again emphasize that the shapes of atomic orbitals are a direct consequence of the wave nature of electrons.

<gallery mode="nolines" perrow="3" widths="200px" caption="p-type drum modes and wave functions">
File:Drum vibration mode11.gif|Drum mode <math>u_{11}</math>
File:Drum vibration mode12.gif|Drum mode <math>u_{12}</math>
File:Drum vibration mode13.gif|Drum mode <math>u_{13}</math>
File:Phi 2p.gif|Wave function of 2p orbital (real part, 2D-cut, <math>r_\mathrm{max}=10 a_0</math>)
File:Phi 3p.gif|Wave function of 3p orbital (real part, 2D-cut, <math>r_\mathrm{max}=20 a_0</math>)
File:Phi 4p.gif|Wave function of 4p orbital (real part, 2D-cut, <math>r_\mathrm{max}=25 a_0</math>)
</gallery>

<gallery mode="nolines" perrow="3" widths="200px" caption="d-type drum modes">
File:Drum vibration mode21.gif|Drum mode <math>u_{21}</math> 
File:Drum vibration mode22.gif|Drum mode <math>u_{22}</math>
File:Drum vibration mode23.gif|Drum mode <math>u_{23}</math>
</gallery>