Physics:Quantum sigma bond - Revision history

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	==External links==		==External links==
	*[http://goldbook.iupac.org/S05434.html IUPAC-definition]		*[http://goldbook.iupac.org/S05434.html IUPAC-definition]

	~~{{Chemical bonding theory}}~~

	{{DEFAULTSORT:Sigma Bond}}		{{DEFAULTSORT:Sigma Bond}}
	~~[[Category:Chemical bonding]]~~

	{{Sourceattribution\|Sigma bond}}		{{Sourceattribution\|Sigma bond}}

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	]]		]]

	In ~~[[HandWiki:Chemistry\|~~chemistry]], '''sigma bonds''' ('''σ bonds''') or '''sigma overlap''' are the strongest type of ~~[[Chemistry:Covalent\|~~covalent~~]] [[Chemistry:Chemical bond\|~~chemical bond]].<ref>{{cite book \|last1= Moore\|first1= John\|last2= Stanitski \|first2= Conrad L.\|last3= Jurs \|first3= Peter C.\|title= Principles of Chemistry: The Molecular Science\|url= https://books.google.com/books?id=ZOm8L9oCwLMC&q=sigma+bond+stronger+than+pi&pg=PA324\|isbn= 9780495390794\|date= 2009-01-21\|publisher= Cengage Learning}}</ref> They are formed by head-on overlapping between ~~[[Physics:Atomic orbital\|~~atomic ~~orbital]]s~~ along the internuclear axis. Sigma bonding is most simply defined for diatomic molecules using the language and tools of ~~[[Symmetry group\|~~symmetry ~~group]]s~~. In this formal approach, a σ-bond is symmetrical with respect to rotation about the bond axis. By this definition, common forms of sigma bonds are s+s, p<sub>z</sub>+p<sub>z</sub>, s+p<sub>z</sub> and d<sub>z<sup>2</sup></sub>+d<sub>z<sup>2</sup></sub> (where z is defined as the axis of the bond or the internuclear axis).<ref>{{cite book \|last1= Clayden\|first1= Jonathan\|last2= Greeves\|first2= Nick\|last3= Warren\|first3= Stuart\|title= Organic Chemistry \|edition= 2nd\|date=March 2012\|orig-year= 2002\|publisher= OUP Oxford\|location= Oxford\|isbn= 978-0199270293\|pages=101–136}}</ref>		In chemistry, '''sigma bonds''' ('''σ bonds''') or '''sigma overlap''' are the strongest type of covalent chemical bond.<ref>{{cite book \|last1= Moore\|first1= John\|last2= Stanitski \|first2= Conrad L.\|last3= Jurs \|first3= Peter C.\|title= Principles of Chemistry: The Molecular Science\|url= https://books.google.com/books?id=ZOm8L9oCwLMC&q=sigma+bond+stronger+than+pi&pg=PA324\|isbn= 9780495390794\|date= 2009-01-21\|publisher= Cengage Learning}}</ref> They are formed by head-on overlapping between atomic orbitals along the internuclear axis. Sigma bonding is most simply defined for diatomic molecules using the language and tools of symmetry groups. In this formal approach, a σ-bond is symmetrical with respect to rotation about the bond axis. By this definition, common forms of sigma bonds are s+s, p<sub>z</sub>+p<sub>z</sub>, s+p<sub>z</sub> and d<sub>z<sup>2</sup></sub>+d<sub>z<sup>2</sup></sub> (where z is defined as the axis of the bond or the internuclear axis).<ref>{{cite book \|last1= Clayden\|first1= Jonathan\|last2= Greeves\|first2= Nick\|last3= Warren\|first3= Stuart\|title= Organic Chemistry \|edition= 2nd\|date=March 2012\|orig-year= 2002\|publisher= OUP Oxford\|location= Oxford\|isbn= 978-0199270293\|pages=101–136}}</ref>
	Quantum theory also indicates that ~~[[Chemistry:Molecular orbital\|~~molecular ~~orbital]]s~~ (MO) of identical symmetry actually mix or ~~[[Physics:Orbital hybridisation\|~~''hybridize'']]. As a practical consequence of this mixing of diatomic molecules, the wavefunctions s+s and p<sub>z</sub>+p<sub>z</sub> molecular orbitals become blended. The extent of this mixing (or hybridization or blending) depends on the relative energies of the MOs of like symmetry.		Quantum theory also indicates that molecular orbitals (MO) of identical symmetry actually mix or ''hybridize''. As a practical consequence of this mixing of diatomic molecules, the wavefunctions s+s and p<sub>z</sub>+p<sub>z</sub> molecular orbitals become blended. The extent of this mixing (or hybridization or blending) depends on the relative energies of the MOs of like symmetry.

	[[File:Dihydrogen-LUMO-phase-3D-balls.png\|thumb\|right\|150px\|1sσ* antibonding molecular orbital in H<sub>2</sub> with nodal plane]]		[[File:Dihydrogen-LUMO-phase-3D-balls.png\|thumb\|right\|150px\|1sσ* antibonding molecular orbital in H<sub>2</sub> with nodal plane]]
	For homodiatomics (~~[[Chemistry:Homonuclear molecule\|~~homonuclear]] diatomic molecules), bonding σ orbitals have no nodal planes at which the wavefunction is zero, either between the bonded atoms or passing through the bonded atoms. The corresponding antibonding, or σ* orbital, is defined by the presence of one nodal plane between the two bonded atoms.		For homodiatomics (homonuclear diatomic molecules), bonding σ orbitals have no nodal planes at which the wavefunction is zero, either between the bonded atoms or passing through the bonded atoms. The corresponding antibonding, or σ* orbital, is defined by the presence of one nodal plane between the two bonded atoms.

	Sigma bonds are the strongest type of covalent bonds due to the direct overlap of orbitals, and the ~~[[Physics:Electron\|electron]]s~~ in these bonds are sometimes referred to as sigma electrons.<ref>{{cite book \|last1= Keeler\|first1= James\|last2= Wothers\|first2= Peter\|title= Chemical Structure and Reactivity \|edition= 1st\|date=May 2008\|publisher= OUP Oxford\|location= Oxford\|isbn= 978-0199289301\|pages=27–46}}</ref>		Sigma bonds are the strongest type of covalent bonds due to the direct overlap of orbitals, and the electrons in these bonds are sometimes referred to as sigma electrons.<ref>{{cite book \|last1= Keeler\|first1= James\|last2= Wothers\|first2= Peter\|title= Chemical Structure and Reactivity \|edition= 1st\|date=May 2008\|publisher= OUP Oxford\|location= Oxford\|isbn= 978-0199289301\|pages=27–46}}</ref>

	The symbol σ is the Greek letter sigma. When viewed down the bond axis, a σ MO has a ~~[[Circular symmetry\|~~circular symmetry]], hence resembling a similarly sounding "s" ~~[[Physics:Atomic orbital\|~~atomic orbital]].		The symbol σ is the Greek letter sigma. When viewed down the bond axis, a σ MO has a circular symmetry, hence resembling a similarly sounding "s" atomic orbital.

	Typically, a ~~[[Chemistry:Single bond\|~~single bond]] is a sigma bond while a multiple bond is composed of one sigma bond together with pi or other bonds. A ~~[[Chemistry:Double bond\|~~double bond]] has one sigma plus one ~~[[Chemistry:Pi bond\|~~pi bond]], and a ~~[[Chemistry:Triple bond\|~~triple bond]] has one sigma plus two pi bonds.		Typically, a single bond is a sigma bond while a multiple bond is composed of one sigma bond together with pi or other bonds. A double bond has one sigma plus one pi bond, and a triple bond has one sigma plus two pi bonds.
	{{clear\|right}}		{{clear\|right}}
	{\| align="right" width=278px style="text-align:center; margin:0 0 1ex 1em"		{\| align="right" width=278px style="text-align:center; margin:0 0 1ex 1em"
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	\|-		\|-
	\| colspan=2 style="font-size:87%" width=144px \|Symmetric (s–s and p–p)<br/>sigma bonds between atomic orbitals		\| colspan=2 style="font-size:87%" width=144px \|Symmetric (s–s and p–p)<br/>sigma bonds between atomic orbitals
	\| style="font-size:87%" \|A ~~[[Chemistry:Pi bond\|~~pi bond]],<br/>for comparison		\| style="font-size:87%" \|A pi bond,<br/>for comparison
	\|- style="font-size:6px; color:silver"		\|- style="font-size:6px; color:silver"
	\|——\|\|————————————————\|\|—————		\|——\|\|————————————————\|\|—————
	\|-		\|-
	\| colspan=2 \|128px <br />σ<sub style="font-size:87%">~~s–[[Physics:Orbital hybridisation\|hybrid]]~~</sub>		\| colspan=2 \|128px <br />σ<sub style="font-size:87%">s–hybrid</sub>
	\| 144px <br />σ<sub style="font-size:87%">s–p</sub>		\| 144px <br />σ<sub style="font-size:87%">s–p</sub>
	\|}		\|}
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	<div style="width:300px;">		<div style="width:300px;">
	[[File:Sigma bond.svg\|thumb\|280px\|σ bond between two atoms: localization of [[electron density]]		[[File:Sigma bond.svg\|thumb\|280px\|σ bond between two atoms: localization of electron density
	</div>		</div>

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	==Polyatomic molecules==		==Polyatomic molecules==
	Sigma bonds are obtained by head-on overlapping of atomic orbitals. The concept of sigma bonding is extended to describe bonding interactions involving overlap of a single lobe of one ~~[[Physics:Atomic~~ orbital~~\|orbital]]~~ with a single lobe of another. For example, ~~[[Chemistry:Propane\|~~propane]] is described as consisting of ten sigma bonds, one each for the two C−C bonds and one each for the eight C−H bonds.		Sigma bonds are obtained by head-on overlapping of atomic orbitals. The concept of sigma bonding is extended to describe bonding interactions involving overlap of a single lobe of one orbital with a single lobe of another. For example, propane is described as consisting of ten sigma bonds, one each for the two C−C bonds and one each for the eight C−H bonds.

	==Multiple-bonded complexes==		==Multiple-bonded complexes==
	Transition metal complexes that feature multiple bonds, such as the ~~[[Chemistry:Dihydrogen complex\|~~dihydrogen complex]], have sigma bonds between the multiple bonded atoms. These sigma bonds can be supplemented with other bonding interactions, such as ~~[[Chemistry:Pi backbonding\|~~π-back donation]], as in the case of W(CO)<sub>3</sub>(~~[[Chemistry:Tricyclohexylphosphine\|~~PCy<sub>3</sub>]])<sub>2</sub>(H<sub>2</sub>), and even δ-bonds, as in the case of ~~[[Chemistry:Chromium(II) acetate\|~~chromium(II) acetate]].<ref>{{cite journal\|last=Kubas\|first=Gregory\|title=Metal Dihydrogen and σ-Bond Complexes: Structure, Theory, and Reactivity\|journal=J. Am. Chem. Soc.\|year=2002\|volume=124\|issue=14\|pages=3799–3800\|doi=10.1021/ja0153417}}</ref>		Transition metal complexes that feature multiple bonds, such as the dihydrogen complex, have sigma bonds between the multiple bonded atoms. These sigma bonds can be supplemented with other bonding interactions, such as π-back donation, as in the case of W(CO)<sub>3</sub>(PCy<sub>3</sub>)<sub>2</sub>(H<sub>2</sub>), and even δ-bonds, as in the case of chromium(II) acetate.<ref>{{cite journal\|last=Kubas\|first=Gregory\|title=Metal Dihydrogen and σ-Bond Complexes: Structure, Theory, and Reactivity\|journal=J. Am. Chem. Soc.\|year=2002\|volume=124\|issue=14\|pages=3799–3800\|doi=10.1021/ja0153417}}</ref>

	== Organic molecules ==		== Organic molecules ==
	Organic molecules are often ~~[[Physics:Cyclic compound\|~~cyclic ~~compound]]s~~ containing one or more rings, such as ~~[[Chemistry:Benzene\|~~benzene]], and are often made up of many sigma bonds along with pi bonds. According to the '''sigma bond rule''', the number of sigma bonds in a molecule is equivalent to the number of atoms plus the number of rings minus one.		Organic molecules are often cyclic compounds containing one or more rings, such as benzene, and are often made up of many sigma bonds along with pi bonds. According to the '''sigma bond rule''', the number of sigma bonds in a molecule is equivalent to the number of atoms plus the number of rings minus one.

	:''N''<sub>σ</sub> = ''N''<sub>atoms</sub> + ''N''<sub>rings</sub> − 1		:''N''<sub>σ</sub> = ''N''<sub>atoms</sub> + ''N''<sub>rings</sub> − 1

	This rule is a special-case application of the [[Euler characteristic]] of the graph which represents the molecule.		This rule is a special-case application of the Euler characteristic of the graph which represents the molecule.

	A molecule with no rings can be represented as a ~~[[Tree (graph theory)\|~~tree]] with a number of bonds equal to the number of atoms minus one (as in dihydrogen, H<sub>2</sub>, with only one sigma bond, or ~~[[Chemistry:Ammonia\|~~ammonia]], NH<sub>3</sub>, with 3 sigma bonds). There is no more than 1 sigma bond between any two atoms.		A molecule with no rings can be represented as a tree with a number of bonds equal to the number of atoms minus one (as in dihydrogen, H<sub>2</sub>, with only one sigma bond, or ammonia, NH<sub>3</sub>, with 3 sigma bonds). There is no more than 1 sigma bond between any two atoms.

	Molecules with rings have additional sigma bonds, such as benzene rings, which have 6 C−C sigma bonds within the ring for 6 carbon atoms. The ~~[[Chemistry:Anthracene\|~~anthracene]] molecule, C<sub>14</sub>H<sub>10</sub>, has three rings so that the rule gives the number of sigma bonds as 24 + 3 − 1 = 26. In this case there are 16 C−C sigma bonds and 10 C−H bonds.		Molecules with rings have additional sigma bonds, such as benzene rings, which have 6 C−C sigma bonds within the ring for 6 carbon atoms. The anthracene molecule, C<sub>14</sub>H<sub>10</sub>, has three rings so that the rule gives the number of sigma bonds as 24 + 3 − 1 = 26. In this case there are 16 C−C sigma bonds and 10 C−H bonds.

	This rule fails in the case of molecules which, when drawn flat on paper, have a different number of rings than the molecule actually has - for example, ~~[[Chemistry:~~Buckminsterfullerene~~\|Buckminsterfullerene]]~~, C<sub>60</sub>, which has 32 rings, 60 atoms, and 90 sigma bonds, one for each pair of bonded atoms; however, 60 + 32 − 1 = 91, not 90. This is because the sigma rule is a special case of the [[Euler characteristic]], where each ring is considered a face, each sigma bond is an edge, and each atom is a vertex. Ordinarily, one extra face is assigned to the space not inside any ring, but ~~[[Truncated icosahedron#Truncated icosahedral graph\|~~when Buckminsterfullerene is drawn flat without any crossings]], one of the rings makes up the outer pentagon; the inside of that ring is the outside of the graph. This rule fails further when considering other shapes - toroidal fullerenes will obey the rule that the number of sigma bonds in a molecule is exactly the number of atoms plus the number of rings, as will nanotubes - which, when drawn flat as if looking through one from the end, will have a face in the middle, corresponding to the far end of the nanotube, which is not a ring, and a face corresponding to the outside.		This rule fails in the case of molecules which, when drawn flat on paper, have a different number of rings than the molecule actually has - for example, Buckminsterfullerene, C<sub>60</sub>, which has 32 rings, 60 atoms, and 90 sigma bonds, one for each pair of bonded atoms; however, 60 + 32 − 1 = 91, not 90. This is because the sigma rule is a special case of the Euler characteristic, where each ring is considered a face, each sigma bond is an edge, and each atom is a vertex. Ordinarily, one extra face is assigned to the space not inside any ring, but when Buckminsterfullerene is drawn flat without any crossings, one of the rings makes up the outer pentagon; the inside of that ring is the outside of the graph. This rule fails further when considering other shapes - toroidal fullerenes will obey the rule that the number of sigma bonds in a molecule is exactly the number of atoms plus the number of rings, as will nanotubes - which, when drawn flat as if looking through one from the end, will have a face in the middle, corresponding to the far end of the nanotube, which is not a ring, and a face corresponding to the outside.

	== See also ==		== See also ==

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	This rule fails in the case of molecules which, when drawn flat on paper, have a different number of rings than the molecule actually has - for example, [[Chemistry:Buckminsterfullerene\|Buckminsterfullerene]], C<sub>60</sub>, which has 32 rings, 60 atoms, and 90 sigma bonds, one for each pair of bonded atoms; however, 60 + 32 − 1 = 91, not 90. This is because the sigma rule is a special case of the [[Euler characteristic]], where each ring is considered a face, each sigma bond is an edge, and each atom is a vertex. Ordinarily, one extra face is assigned to the space not inside any ring, but [[Truncated icosahedron#Truncated icosahedral graph\|when Buckminsterfullerene is drawn flat without any crossings]], one of the rings makes up the outer pentagon; the inside of that ring is the outside of the graph. This rule fails further when considering other shapes - toroidal fullerenes will obey the rule that the number of sigma bonds in a molecule is exactly the number of atoms plus the number of rings, as will nanotubes - which, when drawn flat as if looking through one from the end, will have a face in the middle, corresponding to the far end of the nanotube, which is not a ring, and a face corresponding to the outside.		This rule fails in the case of molecules which, when drawn flat on paper, have a different number of rings than the molecule actually has - for example, [[Chemistry:Buckminsterfullerene\|Buckminsterfullerene]], C<sub>60</sub>, which has 32 rings, 60 atoms, and 90 sigma bonds, one for each pair of bonded atoms; however, 60 + 32 − 1 = 91, not 90. This is because the sigma rule is a special case of the [[Euler characteristic]], where each ring is considered a face, each sigma bond is an edge, and each atom is a vertex. Ordinarily, one extra face is assigned to the space not inside any ring, but [[Truncated icosahedron#Truncated icosahedral graph\|when Buckminsterfullerene is drawn flat without any crossings]], one of the rings makes up the outer pentagon; the inside of that ring is the outside of the graph. This rule fails further when considering other shapes - toroidal fullerenes will obey the rule that the number of sigma bonds in a molecule is exactly the number of atoms plus the number of rings, as will nanotubes - which, when drawn flat as if looking through one from the end, will have a face in the middle, corresponding to the far end of the nanotube, which is not a ring, and a face corresponding to the outside.

	==See also==		== See also ==
	~~<!-- *[[Chemistry~~:~~Pi bond~~\|~~Pi bond]]~~		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also/Matter}}
	*[[Chemistry:Delta bond\|~~Delta bond]] there is the navbox -->~~
	*Bond strength
	*[[Physics:~~Molecular geometry\|Molecular geometry]]~~
	~~<!-- *[[Chemistry:Covalent bond\|Covalent bond]]~~
	*Ionic bond there is the navbox -->

	==References==		==References==

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@@ Line 1: / Line 1: @@
 {{Short description|Covalent chemical bond}}
-[[File:Sigma bond.svg|thumb|right|250px|σ bond between two atoms: localization of [[electron density]]]]
+{{Quantum matter backlink|Molecules}}
 In [[HandWiki:Chemistry|chemistry]], '''sigma bonds''' ('''σ bonds''') or '''sigma overlap''' are the strongest type of [[Chemistry:Covalent|covalent]] [[Chemistry:Chemical bond|chemical bond]].<ref>{{cite book |last1= Moore|first1= John|last2= Stanitski |first2= Conrad L.|last3= Jurs |first3= Peter C.|title= Principles of Chemistry: The Molecular Science|url= https://books.google.com/books?id=ZOm8L9oCwLMC&q=sigma+bond+stronger+than+pi&pg=PA324|isbn= 9780495390794|date= 2009-01-21|publisher= Cengage Learning}}</ref> They are formed by head-on  overlapping between [[Physics:Atomic orbital|atomic orbital]]s along the internuclear axis. Sigma bonding is most simply defined for diatomic molecules using the language and tools of [[Symmetry group|symmetry group]]s. In this formal approach, a σ-bond is symmetrical with respect to rotation about the bond axis. By this definition, common forms of sigma bonds are s+s, p<sub>z</sub>+p<sub>z</sub>, s+p<sub>z</sub> and d<sub>z<sup>2</sup></sub>+d<sub>z<sup>2</sup></sub> (where z is defined as the axis of the bond or the internuclear axis).<ref>{{cite book |last1= Clayden|first1= Jonathan|last2= Greeves|first2= Nick|last3= Warren|first3= Stuart|title= Organic Chemistry |edition= 2nd|date=March 2012|orig-year= 2002|publisher= OUP Oxford|location= Oxford|isbn= 978-0199270293|pages=101–136}}</ref>
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   | 144px <br />σ<sub style="font-size:87%">s–p</sub>
   |}
 ==Polyatomic molecules==

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WikiHarold moved page Physics:Sigma bond to Physics:Quantum sigma bond without leaving a redirect

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imported>WikiHarold: WikiHarold moved page Physics:Sigma bond to Physics:Quantum sigma bond without leaving a redirect

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WikiHarold moved page Physics:Sigma bond to Physics:Quantum sigma bond without leaving a redirect

New page

{{Short description|Covalent chemical bond}}

[[File:Sigma bond.svg|thumb|right|250px|σ bond between two atoms: localization of [[electron density]]]]

In [[HandWiki:Chemistry|chemistry]], '''sigma bonds''' ('''σ bonds''') or '''sigma overlap''' are the strongest type of [[Chemistry:Covalent|covalent]] [[Chemistry:Chemical bond|chemical bond]].<ref>{{cite book |last1= Moore|first1= John|last2= Stanitski |first2= Conrad L.|last3= Jurs |first3= Peter C.|title= Principles of Chemistry: The Molecular Science|url= https://books.google.com/books?id=ZOm8L9oCwLMC&q=sigma+bond+stronger+than+pi&pg=PA324|isbn= 9780495390794|date= 2009-01-21|publisher= Cengage Learning}}</ref> They are formed by head-on overlapping between [[Physics:Atomic orbital|atomic orbital]]s along the internuclear axis. Sigma bonding is most simply defined for diatomic molecules using the language and tools of [[Symmetry group|symmetry group]]s. In this formal approach, a σ-bond is symmetrical with respect to rotation about the bond axis. By this definition, common forms of sigma bonds are s+s, pz+pz, s+pz and dz2+dz2 (where z is defined as the axis of the bond or the internuclear axis).<ref>{{cite book |last1= Clayden|first1= Jonathan|last2= Greeves|first2= Nick|last3= Warren|first3= Stuart|title= Organic Chemistry |edition= 2nd|date=March 2012|orig-year= 2002|publisher= OUP Oxford|location= Oxford|isbn= 978-0199270293|pages=101–136}}</ref>
Quantum theory also indicates that [[Chemistry:Molecular orbital|molecular orbital]]s (MO) of identical symmetry actually mix or [[Physics:Orbital hybridisation|''hybridize'']]. As a practical consequence of this mixing of diatomic molecules, the wavefunctions s+s and pz+pz molecular orbitals become blended. The extent of this mixing (or hybridization or blending) depends on the relative energies of the MOs of like symmetry.

[[File:Dihydrogen-LUMO-phase-3D-balls.png|thumb|right|150px|1sσ* antibonding molecular orbital in H2 with nodal plane]]
For homodiatomics ([[Chemistry:Homonuclear molecule|homonuclear]] diatomic molecules), bonding σ orbitals have no nodal planes at which the wavefunction is zero, either between the bonded atoms or passing through the bonded atoms. The corresponding antibonding, or σ* orbital, is defined by the presence of one nodal plane between the two bonded atoms.

Sigma bonds are the strongest type of covalent bonds due to the direct overlap of orbitals, and the [[Physics:Electron|electron]]s in these bonds are sometimes referred to as sigma electrons.<ref>{{cite book |last1= Keeler|first1= James|last2= Wothers|first2= Peter|title= Chemical Structure and Reactivity |edition= 1st|date=May 2008|publisher= OUP Oxford|location= Oxford|isbn= 978-0199289301|pages=27–46}}</ref>

The symbol σ is the Greek letter sigma. When viewed down the bond axis, a σ MO has a [[Circular symmetry|circular symmetry]], hence resembling a similarly sounding "s" [[Physics:Atomic orbital|atomic orbital]].

Typically, a [[Chemistry:Single bond|single bond]] is a sigma bond while a multiple bond is composed of one sigma bond together with pi or other bonds. A [[Chemistry:Double bond|double bond]] has one sigma plus one [[Chemistry:Pi bond|pi bond]], and a [[Chemistry:Triple bond|triple bond]] has one sigma plus two pi bonds.
{{clear|right}}
{| align="right" width=278px style="text-align:center; margin:0 0 1ex 1em"
|- style="font-size:6px; color:silver"
|——||————————————————||—————
|-
| valign=top style="font-size:87%" | Atomic

orbitals
| colspan=2 |250px
|-
| colspan=2 style="font-size:87%" width=144px |Symmetric (s–s and p–p) sigma bonds between atomic orbitals
| style="font-size:87%" |A [[Chemistry:Pi bond|pi bond]], for comparison
|- style="font-size:6px; color:silver"
|——||————————————————||—————
|-
| colspan=2 |128px σs–[[Physics:Orbital hybridisation|hybrid]]
| 144px σs–p
|}

==Polyatomic molecules==
Sigma bonds are obtained by head-on overlapping of atomic orbitals. The concept of sigma bonding is extended to describe bonding interactions involving overlap of a single lobe of one [[Physics:Atomic orbital|orbital]] with a single lobe of another. For example, [[Chemistry:Propane|propane]] is described as consisting of ten sigma bonds, one each for the two C−C bonds and one each for the eight C−H bonds.

==Multiple-bonded complexes==
Transition metal complexes that feature multiple bonds, such as the [[Chemistry:Dihydrogen complex|dihydrogen complex]], have sigma bonds between the multiple bonded atoms. These sigma bonds can be supplemented with other bonding interactions, such as [[Chemistry:Pi backbonding|π-back donation]], as in the case of W(CO)3([[Chemistry:Tricyclohexylphosphine|PCy3]])2(H2), and even δ-bonds, as in the case of [[Chemistry:Chromium(II) acetate|chromium(II) acetate]].<ref>{{cite journal|last=Kubas|first=Gregory|title=Metal Dihydrogen and σ-Bond Complexes: Structure, Theory, and Reactivity|journal=J. Am. Chem. Soc.|year=2002|volume=124|issue=14|pages=3799–3800|doi=10.1021/ja0153417}}</ref>

== Organic molecules ==
Organic molecules are often [[Physics:Cyclic compound|cyclic compound]]s containing one or more rings, such as [[Chemistry:Benzene|benzene]], and are often made up of many sigma bonds along with pi bonds. According to the '''sigma bond rule''', the number of sigma bonds in a molecule is equivalent to the number of atoms plus the number of rings minus one.

:''N''σ = ''N''atoms + ''N''rings − 1

This rule is a special-case application of the [[Euler characteristic]] of the graph which represents the molecule.

A molecule with no rings can be represented as a [[Tree (graph theory)|tree]] with a number of bonds equal to the number of atoms minus one (as in dihydrogen, H2, with only one sigma bond, or [[Chemistry:Ammonia|ammonia]], NH3, with 3 sigma bonds). There is no more than 1 sigma bond between any two atoms.

Molecules with rings have additional sigma bonds, such as benzene rings, which have 6 C−C sigma bonds within the ring for 6 carbon atoms. The [[Chemistry:Anthracene|anthracene]] molecule, C14H10, has three rings so that the rule gives the number of sigma bonds as 24 + 3 − 1 = 26. In this case there are 16 C−C sigma bonds and 10 C−H bonds.

This rule fails in the case of molecules which, when drawn flat on paper, have a different number of rings than the molecule actually has - for example, [[Chemistry:Buckminsterfullerene|Buckminsterfullerene]], C60, which has 32 rings, 60 atoms, and 90 sigma bonds, one for each pair of bonded atoms; however, 60 + 32 − 1 = 91, not 90. This is because the sigma rule is a special case of the [[Euler characteristic]], where each ring is considered a face, each sigma bond is an edge, and each atom is a vertex. Ordinarily, one extra face is assigned to the space not inside any ring, but [[Truncated icosahedron#Truncated icosahedral graph|when Buckminsterfullerene is drawn flat without any crossings]], one of the rings makes up the outer pentagon; the inside of that ring is the outside of the graph. This rule fails further when considering other shapes - toroidal fullerenes will obey the rule that the number of sigma bonds in a molecule is exactly the number of atoms plus the number of rings, as will nanotubes - which, when drawn flat as if looking through one from the end, will have a face in the middle, corresponding to the far end of the nanotube, which is not a ring, and a face corresponding to the outside.

==See also==

*Bond strength
*[[Physics:Molecular geometry|Molecular geometry]]


==References==
{{reflist}}

==External links==
*[http://goldbook.iupac.org/S05434.html IUPAC-definition]

{{Chemical bonding theory}}

{{DEFAULTSORT:Sigma Bond}}
[[Category:Chemical bonding]]

{{Sourceattribution|Sigma bond}}