Physics:Quantum Orbital hybridisation: Difference between revisions

From ScholarlyWiki
Jump to navigation Jump to search
← Older edit
imported>WikiHarold
No edit summary
Restore missing Quantum reference definitions
 
(6 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{Short description|Mixing (superposition) of atomic orbitals}}
{{Short description|Mixing (superposition) of atomic orbitals}}
In [[HandWiki:Chemistry|chemistry]], '''orbital hybridisation''' (or '''hybridization''') is the concept of mixing [[Physics:Atomic orbital|atomic orbital]]s to form new ''hybrid orbitals'' (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form [[Chemistry:Chemical bond|chemical bond]]s in [[Chemistry:Valence bond theory|valence bond theory]]. For example, in a carbon atom which forms four single bonds, the valence-shell s orbital combines with three valence-shell p orbitals to form four equivalent sp<sup>3</sup> mixtures in a [[Physics:Tetrahedral molecular geometry|tetrahedral]] arrangement around the carbon to bond to four different atoms. Hybrid orbitals are useful in the explanation of [[Physics:Molecular geometry|molecular geometry]] and atomic bonding properties and are symmetrically disposed in space. Usually hybrid orbitals are formed by mixing atomic orbitals of comparable energies.<ref>{{cite book |last1=Housecroft |first1=Catherine E. |last2=Sharpe |first2=Alan G. |title=Inorganic Chemistry |date=2005 |publisher=Pearson Prentice-Hal |isbn=0130-39913-2 |page=100 |edition=2nd}}</ref>
 
{{Quantum book backlink|Foundations}}
 
<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
 
<div style="width:280px;">
__TOC__
</div>
 
<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
In chemistry, '''orbital hybridisation''' (or '''hybridization''') is the concept of mixing atomic orbitals to form new ''hybrid orbitals'' (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds in valence bond theory. For example, in a carbon atom which forms four single bonds, the valence-shell s orbital combines with three valence-shell p orbitals to form four equivalent sp<sup>3</sup> mixtures in a tetrahedral arrangement around the carbon to bond to four different atoms. Hybrid orbitals are useful in the explanation of molecular geometry and atomic bonding properties and are symmetrically disposed in space. Usually hybrid orbitals are formed by mixing atomic orbitals of comparable energies.<ref>{{cite book |last1=Housecroft |first1=Catherine E. |last2=Sharpe |first2=Alan G. |title=Inorganic Chemistry |date=2005 |publisher=Pearson Prentice-Hal |isbn=0130-39913-2 |page=100 |edition=2nd}}</ref>
</div>
 
<div style="width:300px;">
[[File:AE4h.svg|thumb|280px|Orbital hybridisation in the Quantum Collection.]]
</div>
 
</div>


== History and uses ==
== History and uses ==


[[Chemist]] [[Biography:Linus Pauling|Linus Pauling]] first developed the hybridisation theory in 1931 to explain the structure of simple [[Physics:Molecule|molecule]]s such as [[Chemistry:Methane|methane]] (CH<sub>4</sub>) using [[Physics:Atomic orbital|atomic orbital]]s.<ref>{{citation
Chemist [[Biography:Linus Pauling|Linus Pauling]] first developed the hybridisation theory in 1931 to explain the structure of simple molecules such as methane (CH<sub>4</sub>) using atomic orbitals.<ref>{{citation
| last    = Pauling
| last    = Pauling
| first  = L.
| first  = L.
| year    = 1931
| year    = 1931
| title  = The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules
| title  = The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules
| journal = [[Organization:Journal of the American Chemical Society|Journal of the American Chemical Society]]
| journal = Journal of the American Chemical Society
| volume  = 53
| volume  = 53
| issue  = 4
| issue  = 4
Line 15: Line 32:
| doi    = 10.1021/ja01355a027
| doi    = 10.1021/ja01355a027
| bibcode = 1931JAChS..53.1367P
| bibcode = 1931JAChS..53.1367P
}}</ref> Pauling pointed out that a carbon atom forms four bonds by using one s and three p orbitals, so that "it might be inferred" that a carbon atom would form three bonds at right angles (using p orbitals) and a fourth weaker bond using the s orbital in some arbitrary direction. In reality, methane has four C–H bonds of equivalent strength. The angle between any two bonds is the [[Physics:Tetrahedral molecular geometry|tetrahedral bond angle]] of 109°28'<ref>{{cite journal|doi=10.1021/ed022p145|title=Valence Angle of the Tetrahedral Carbon Atom|first=W. E.|last=Brittin|journal=J. Chem. Educ.|date=1945|volume=22|issue=3|page=145|bibcode=1945JChEd..22..145B}}</ref> (around 109.5°). Pauling supposed that in the presence of four hydrogen atoms, the s and p orbitals form four equivalent combinations which he called ''hybrid'' orbitals. Each hybrid is denoted sp<sup>3</sup> to indicate its composition, and is directed along one of the four C–H bonds.<ref>L. Pauling ''The Nature of the Chemical Bond'' (3rd ed., Oxford University Press 1960) p.111–120.</ref> This concept was developed for such simple chemical systems, but the approach was later applied more widely, and today it is considered an effective [[Heuristic|heuristic]] for rationalizing the structures of organic compounds. It gives a simple orbital picture equivalent to Lewis structures.
}}</ref> Pauling pointed out that a carbon atom forms four bonds by using one s and three p orbitals, so that "it might be inferred" that a carbon atom would form three bonds at right angles (using p orbitals) and a fourth weaker bond using the s orbital in some arbitrary direction. In reality, methane has four C–H bonds of equivalent strength. The angle between any two bonds is the tetrahedral bond angle of 109°28'<ref>{{cite journal|doi=10.1021/ed022p145|title=Valence Angle of the Tetrahedral Carbon Atom|first=W. E.|last=Brittin|journal=J. Chem. Educ.|date=1945|volume=22|issue=3|page=145|bibcode=1945JChEd..22..145B}}</ref> (around 109.5°). Pauling supposed that in the presence of four hydrogen atoms, the s and p orbitals form four equivalent combinations which he called ''hybrid'' orbitals. Each hybrid is denoted sp<sup>3</sup> to indicate its composition, and is directed along one of the four C–H bonds.<ref>L. Pauling ''The Nature of the Chemical Bond'' (3rd ed., Oxford University Press 1960) p.111–120.</ref> This concept was developed for such simple chemical systems, but the approach was later applied more widely, and today it is considered an effective heuristic for rationalizing the structures of organic compounds. It gives a simple orbital picture equivalent to Lewis structures.


Hybridisation theory is an integral part of [[Chemistry:Organic chemistry|organic chemistry]], one of the most compelling examples being [[Chemistry:Baldwin's rules|Baldwin's rules]]. For drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons.<ref>{{Clayden|page=105}}</ref> Hybridisation theory explains bonding in alkenes<ref>''Organic Chemistry'', Third Edition Marye Anne Fox James K. Whitesell '''2003''' {{ISBN|978-0-7637-3586-9}}</ref> and methane.<ref>''Organic Chemistry'' 3rd Ed. '''2001''' Paula Yurkanis Bruice {{ISBN|978-0-130-17858-9}}</ref> The amount of p character or s character, which is decided mainly by orbital hybridisation, can be used to reliably predict molecular properties such as acidity or basicity.<ref>{{cite web
Hybridisation theory is an integral part of organic chemistry, one of the most compelling examples being Baldwin's rules. For drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons.<ref>{{cite book |last1=Clayden |first1=Jonathan |last2=Greeves |first2=Nick |last3=Warren |first3=Stuart |title=Organic Chemistry |edition=2nd |publisher=Oxford University Press |date=2012 |isbn=978-0-19-927029-3 |page=105}}</ref> Hybridisation theory explains bonding in alkenes<ref>''Organic Chemistry'', Third Edition Marye Anne Fox James K. Whitesell '''2003''' {{ISBN|978-0-7637-3586-9}}</ref> and methane.<ref>''Organic Chemistry'' 3rd Ed. '''2001''' Paula Yurkanis Bruice {{ISBN|978-0-130-17858-9}}</ref> The amount of p character or s character, which is decided mainly by orbital hybridisation, can be used to reliably predict molecular properties such as acidity or basicity.<ref>{{cite web
| title      = Acids and Bases
| title      = Acids and Bases
| url        = http://orgomadesimple.com/organic-chemistry-acids-and-bases/
| url        = http://orgomadesimple.com/organic-chemistry-acids-and-bases/
Line 28: Line 45:


== Overview ==
== Overview ==
Orbitals are a model representation of the behavior of electrons within molecules.  In the case of simple hybridization, this approximation is based on [[Physics:Atomic orbital|atomic orbital]]s, similar to those obtained for the hydrogen atom, the only neutral atom for which the [[Schrödinger equation]] can be solved exactly. In heavier atoms, such as carbon, nitrogen, and oxygen, the atomic orbitals used are the 2s and 2p orbitals, similar to excited state orbitals for hydrogen.
Orbitals are a model representation of the behavior of electrons within molecules.  In the case of simple hybridization, this approximation is based on atomic orbitals, similar to those obtained for the hydrogen atom, the only neutral atom for which the Schrödinger equation can be solved exactly. In heavier atoms, such as carbon, nitrogen, and oxygen, the atomic orbitals used are the 2s and 2p orbitals, similar to excited state orbitals for hydrogen.


Hybrid orbitals are assumed to be mixtures of atomic orbitals, superimposed on each other in various proportions. For example, in [[Chemistry:Methane|methane]], the C hybrid orbital which forms each [[Chemistry:Carbon|carbon]]–[[Software:Hydrogen|hydrogen]] bond consists of 25% s character and 75% p character and is thus described as sp<sup>3</sup> (read as ''s-p-three'') hybridised. [[Physics:Quantum mechanics|Quantum mechanics]] describes this hybrid as an sp<sup>3</sup> wavefunction of the form <math>N(s + \sqrt 3 p\sigma)</math>, where N is a normalisation constant (here 1/2) and pσ is a p orbital directed along the C-H axis to form a [[Chemistry:Sigma bond|sigma bond]]. The ratio of coefficients (denoted λ in general) is [[Square root of 3|<math>\color{blue}\sqrt 3</math>]] in this example. Since the [[Physics:Electron density|electron density]] associated with an orbital is proportional to the square of the wavefunction, the ratio of p-character to s-character is λ<sup>2</sup> = 3. The p character or the weight of the p component is N<sup>2</sup>λ<sup>2</sup> = 3/4.
Hybrid orbitals are assumed to be mixtures of atomic orbitals, superimposed on each other in various proportions. For example, in methane, the C hybrid orbital which forms each carbon–hydrogen bond consists of 25% s character and 75% p character and is thus described as sp<sup>3</sup> (read as ''s-p-three'') hybridised. [[Physics:Quantum mechanics|Quantum mechanics]] describes this hybrid as an sp<sup>3</sup> wavefunction of the form <math>N(s + \sqrt 3 p\sigma)</math>, where N is a normalisation constant (here 1/2) and pσ is a p orbital directed along the C-H axis to form a sigma bond. The ratio of coefficients (denoted λ in general) is <math>\color{blue}\sqrt 3</math> in this example. Since the electron density associated with an orbital is proportional to the square of the wavefunction, the ratio of p-character to s-character is λ<sup>2</sup> = 3. The p character or the weight of the p component is N<sup>2</sup>λ<sup>2</sup> = 3/4.


== Types ==
== Types ==
Line 37: Line 54:
{{anchor|sp3 hybrids}}
{{anchor|sp3 hybrids}}
[[File:AE4h.svg|thumb|150px|Four sp<sup>3</sup> orbitals.]]
[[File:AE4h.svg|thumb|150px|Four sp<sup>3</sup> orbitals.]]
Hybridisation describes the bonding of atoms from an atom's point of view. For a tetrahedrally coordinated carbon (e.g., [[Chemistry:Methane|methane]] CH<sub>4</sub>), the carbon should have 4 orbitals directed towards the 4 hydrogen atoms.
Hybridisation describes the bonding of atoms from an atom's point of view. For a tetrahedrally coordinated carbon (e.g., methane CH<sub>4</sub>), the carbon should have 4 orbitals directed towards the 4 hydrogen atoms.


Carbon's [[Physics:Ground state|ground state]] configuration is 1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>2</sup> or more easily read:
Carbon's ground state configuration is 1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>2</sup> or more easily read:
<!-- Don't use <math> here per WP:MOSCHEM "the change in size and font form is jarring to the reader" -->
<!-- Don't use <math> here per WP:MOSCHEM "the change in size and font form is jarring to the reader" -->
{| cellpadding=4px
{| cellpadding=4px
Line 52: Line 69:
  |}
  |}


This diagram suggests that the carbon atom could use its two singly occupied p-type orbitals to form two [[Chemistry:Covalent bond|covalent bond]]s with two hydrogen atoms in a [[Chemistry:Methylene (compound)|methylene]] (CH<sub>2</sub>) molecule, with a hypothetical bond angle of 90° corresponding to the angle between two p orbitals on the same atom. However the true H–C–H angle in singlet methylene is about 102°<ref name=shav>Isaiah Shavitt (1985), ''Geometry and singlet-triplet energy gap in methylene: A critical review of experimental and theoretical determinations''. Tetrahedron, volume 41, issue 8, page 1531 {{doi|10.1016/S0040-4020(01)96393-8}}</ref> which implies the presence of some orbital hybridisation.
This diagram suggests that the carbon atom could use its two singly occupied p-type orbitals to form two covalent bonds with two hydrogen atoms in a methylene (CH<sub>2</sub>) molecule, with a hypothetical bond angle of 90° corresponding to the angle between two p orbitals on the same atom. However the true H–C–H angle in singlet methylene is about 102°<ref name=shav>Isaiah Shavitt (1985), ''Geometry and singlet-triplet energy gap in methylene: A critical review of experimental and theoretical determinations''. Tetrahedron, volume 41, issue 8, page 1531 {{doi|10.1016/S0040-4020(01)96393-8}}</ref> which implies the presence of some orbital hybridisation.


The carbon atom can also bond to four hydrogen atoms in methane by an excitation (or promotion) of an electron from the doubly occupied 2s orbital to the empty 2p orbital, producing four singly occupied orbitals.
The carbon atom can also bond to four hydrogen atoms in methane by an excitation (or promotion) of an electron from the doubly occupied 2s orbital to the empty 2p orbital, producing four singly occupied orbitals.
Line 83: Line 100:
  |}
  |}


In CH<sub>4</sub>, four sp<sup>3</sup> hybrid orbitals are overlapped by the four [[Software:Hydrogen|hydrogen]]s' 1s orbitals, yielding four [[Chemistry:Sigma bond|σ (sigma) bonds]] (that is, four single covalent bonds) of equal length and strength.
In CH<sub>4</sub>, four sp<sup>3</sup> hybrid orbitals are overlapped by the four hydrogens' 1s orbitals, yielding four σ (sigma) bonds (that is, four single covalent bonds) of equal length and strength.


The following:
The following:
Line 94: Line 111:
[[File:Ethene-2D-flat.png|thumb|120px|Ethylene structure]]
[[File:Ethene-2D-flat.png|thumb|120px|Ethylene structure]]


Other carbon compounds and other molecules may be explained in a similar way. For example, [[Chemistry:Ethylene|ethylene]] (C<sub>2</sub>H<sub>4</sub>) has a double bond between the carbons. For this molecule, carbon sp<sup>2</sup> hybridises, because one [[Chemistry:Pi bond|π (pi) bond]] is required for the [[Chemistry:Double bond|double bond]] between the carbons and only three σ bonds are formed per carbon atom. In sp<sup>2</sup> hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals, usually denoted 2p<sub>x</sub> and 2p<sub>y</sub>. The third 2p orbital (2p<sub>z</sub>) remains unhybridised.
Other carbon compounds and other molecules may be explained in a similar way. For example, ethylene (C<sub>2</sub>H<sub>4</sub>) has a double bond between the carbons. For this molecule, carbon sp<sup>2</sup> hybridises, because one π (pi) bond is required for the double bond between the carbons and only three σ bonds are formed per carbon atom. In sp<sup>2</sup> hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals, usually denoted 2p<sub>x</sub> and 2p<sub>y</sub>. The third 2p orbital (2p<sub>z</sub>) remains unhybridised.
<!-- Don't use <math> here per WP:MOSCHEM "the change in size and font form is jarring to the reader" -->
<!-- Don't use <math> here per WP:MOSCHEM "the change in size and font form is jarring to the reader" -->
{| cellpadding=4px
{| cellpadding=4px
Line 113: Line 130:
=== sp ===
=== sp ===
[[File:AE2h.svg|thumb|150px|Two sp orbitals]]
[[File:AE2h.svg|thumb|150px|Two sp orbitals]]
The chemical bonding in compounds such as [[Chemistry:Alkyne|alkyne]]s with [[Chemistry:Triple bond|triple bond]]s is explained by sp hybridization. In this model, the 2s orbital is mixed with only one of the three p orbitals,
The chemical bonding in compounds such as alkynes with triple bonds is explained by sp hybridization. In this model, the 2s orbital is mixed with only one of the three p orbitals,


{| cellpadding=4px
{| cellpadding=4px
Line 126: Line 143:
  |}
  |}


resulting in two sp orbitals and two remaining p orbitals. The chemical bonding in [[Chemistry:Acetylene|acetylene]] (ethyne) (C<sub>2</sub>H<sub>2</sub>) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by p–p overlap. Each carbon also bonds to hydrogen in a σ s–sp overlap at 180° angles.
resulting in two sp orbitals and two remaining p orbitals. The chemical bonding in acetylene (ethyne) (C<sub>2</sub>H<sub>2</sub>) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by p–p overlap. Each carbon also bonds to hydrogen in a σ s–sp overlap at 180° angles.


== Molecule shape ==
== Molecule shape ==
[[File:Shapes of hybrid orbitals.svg|thumb|350px|Shapes of the different types of hybrid orbitals]]
[[File:Shapes of hybrid orbitals.svg|thumb|350px|Shapes of the different types of hybrid orbitals]]


Hybridisation helps to explain [[Physics:Molecular geometry|molecule shape]], since the angles between bonds are approximately equal to the angles between hybrid orbitals. This is in contrast to [[Physics:VSEPR theory|valence shell electron-pair repulsion (VSEPR) theory]], which can be used to predict molecular geometry based on empirical rules rather than on valence-bond or orbital theories.<ref>{{citation
Hybridisation helps to explain molecule shape, since the angles between bonds are approximately equal to the angles between hybrid orbitals. This is in contrast to valence shell electron-pair repulsion (VSEPR) theory, which can be used to predict molecular geometry based on empirical rules rather than on valence-bond or orbital theories.<ref>{{citation
| last    = Gillespie
| last    = Gillespie
| first  = R.J.
| first  = R.J.
Line 146: Line 163:


=== sp<sup>x</sup> hybridisation ===
=== sp<sup>x</sup> hybridisation ===
As the valence orbitals of main group elements are the one s and three p orbitals with the corresponding [[Octet rule|octet rule]], sp<sup>x</sup> hybridization is used to model the shape of these molecules.
As the valence orbitals of main group elements are the one s and three p orbitals with the corresponding octet rule, sp<sup>x</sup> hybridization is used to model the shape of these molecules.
{| class="wikitable"
{| class="wikitable"
! [[Physics:Coordination number|Coordination number]]
! Coordination number
! Shape
! Shape
! Hybridisation
! Hybridisation
Line 154: Line 171:
|- ----
|- ----
! 2
! 2
| [[Physics:Linear molecular geometry|Linear]]
| Linear
| sp hybridisation (180°)
| sp hybridisation (180°)
| [[Chemistry:Carbon dioxide|CO<sub>2</sub>]]
| CO<sub>2</sub>
|- ----
|- ----
! 3
! 3
| [[Physics:Trigonal planar molecular geometry|Trigonal planar]]
| Trigonal planar
| sp<sup>2</sup> hybridisation (120°)
| sp<sup>2</sup> hybridisation (120°)
| [[Chemistry:Boron trichloride|BCl<sub>3</sub>]]
| BCl<sub>3</sub>
|- ----
|- ----
! 4
! 4
| [[Physics:Tetrahedral molecular geometry|Tetrahedral]]
| Tetrahedral
| sp<sup>3</sup> hybridisation (109.5°)
| sp<sup>3</sup> hybridisation (109.5°)
| [[Chemistry:Carbon tetrachloride|CCl<sub>4</sub>]]
| CCl<sub>4</sub>
|- ----
|- ----
! Interorbital angles<ref name="NBO Perspective"/>
! Interorbital angles<ref name="NBO Perspective"/>
Line 173: Line 190:


=== sp<sup>x</sup>d<sup>y</sup> hybridisation ===
=== sp<sup>x</sup>d<sup>y</sup> hybridisation ===
As the valence orbitals of [[Chemistry:Transition metal|transition metal]]s are the five d, one s and three p orbitals with the corresponding [[18-electron rule|18-electron rule]], sp<sup>x</sup>d<sup>y</sup> hybridisation is used to model the shape of these molecules. These molecules tend to have multiple shapes corresponding to the same hybridization due to the different d-orbitals involved. A square planar complex has one unoccupied p-orbital and hence has 16 valence electrons.<ref name="Bayse Hall"/>
As the valence orbitals of transition metals are the five d, one s and three p orbitals with the corresponding 18-electron rule, sp<sup>x</sup>d<sup>y</sup> hybridisation is used to model the shape of these molecules. These molecules tend to have multiple shapes corresponding to the same hybridization due to the different d-orbitals involved. A square planar complex has one unoccupied p-orbital and hence has 16 valence electrons.<ref name="Bayse Hall"/>


{| class="wikitable"
{| class="wikitable"
! [[Physics:Coordination number|Coordination number]]
! Coordination number
! Shape
! Shape
! Hybridisation
! Hybridisation
Line 182: Line 199:
|- ----
|- ----
! 4
! 4
| [[Physics:Square planar molecular geometry|Square planar]]
| Square planar
| sp<sup>2</sup>d hybridisation
| sp<sup>2</sup>d hybridisation
| PtCl<sub>4</sub><sup>2−</sup>
| PtCl<sub>4</sub><sup>2−</sup>
|- ----
|- ----
! rowspan="2"| 5
! rowspan="2"| 5
| [[Physics:Trigonal bipyramidal molecular geometry|Trigonal bipyramidal]]
| Trigonal bipyramidal
| rowspan="2"| sp<sup>3</sup>d hybridisation
| rowspan="2"| sp<sup>3</sup>d hybridisation
| [[Chemistry:Iron pentacarbonyl|Fe(CO)<sub>5</sub>]]
| Fe(CO)<sub>5</sub>
|- ----
|- ----
| [[Physics:Square pyramidal molecular geometry|Square pyramidal]]
| Square pyramidal
| MnCl<sub>5</sub><sup>2−</sup>
| MnCl<sub>5</sub><sup>2−</sup>
|- ----
|- ----
! 6
! 6
| [[Physics:Octahedral molecular geometry|Octahedral]]
| Octahedral
| sp<sup>3</sup>d<sup>2</sup> hybridisation
| sp<sup>3</sup>d<sup>2</sup> hybridisation
| [[Chemistry:Molybdenum hexacarbonyl|Mo(CO)<sub>6</sub>]]
| Mo(CO)<sub>6</sub>
|- ----
|- ----
! rowspan="3"| 7
! rowspan="3"| 7
| [[Physics:Pentagonal bipyramidal molecular geometry|Pentagonal bipyramidal]]
| Pentagonal bipyramidal
| rowspan="3"| sp<sup>3</sup>d<sup>3</sup> hybridisation
| rowspan="3"| sp<sup>3</sup>d<sup>3</sup> hybridisation
| ZrF<sub>7</sub><sup>3−</sup>
| ZrF<sub>7</sub><sup>3−</sup>
|- ----
|- ----
| [[Physics:Capped octahedral molecular geometry|Capped octahedral]]
| Capped octahedral
| MoF<sub>7</sub><sup>−</sup>
| MoF<sub>7</sub><sup>−</sup>
|- ----
|- ----
| [[Physics:Capped trigonal prismatic molecular geometry|Capped trigonal prismatic]]
| Capped trigonal prismatic
| TaF<sub>7</sub><sup>2−</sup>
| TaF<sub>7</sub><sup>2−</sup>
|- ----
|- ----
! rowspan="3"| 8
! rowspan="3"| 8
| [[Physics:Square antiprismatic molecular geometry|Square antiprismatic]]
| Square antiprismatic
| rowspan="3"| sp<sup>3</sup>d<sup>4</sup> hybridisation
| rowspan="3"| sp<sup>3</sup>d<sup>4</sup> hybridisation
| ReF<sub>8</sub><sup>−</sup>
| ReF<sub>8</sub><sup>−</sup>
|- ----
|- ----
| [[Physics:Dodecahedral molecular geometry|Dodecahedral]]
| Dodecahedral
| Mo(CN)<sub>8</sub><sup>4−</sup>
| Mo(CN)<sub>8</sub><sup>4−</sup>
|- ----
|- ----
| [[Physics:Bicapped trigonal prismatic molecular geometry|Bicapped trigonal prismatic]]
| Bicapped trigonal prismatic
| ZrF<sub>8</sub><sup>4−</sup>
| ZrF<sub>8</sub><sup>4−</sup>
|- ----
|- ----
! rowspan="2"| 9
! rowspan="2"| 9
| [[Physics:Tricapped trigonal prismatic molecular geometry|Tricapped trigonal prismatic]]
| Tricapped trigonal prismatic
| rowspan="2"| sp<sup>3</sup>d<sup>5</sup> hybridisation
| rowspan="2"| sp<sup>3</sup>d<sup>5</sup> hybridisation
| ReH<sub>9</sub><sup>2−</sup>
| ReH<sub>9</sub><sup>2−</sup>
|- ----
|- ----
| [[Physics:Capped square antiprismatic molecular geometry|Capped square antiprismatic]]
| Capped square antiprismatic
|
|
|}
|}


=== sd<sup>x</sup> hybridisation ===
=== sd<sup>x</sup> hybridisation ===
In certain [[Chemistry:Transition metal|transition metal]] complexes with a low [[Chemistry:D electron count|d electron count]], the p-orbitals are unoccupied and sd<sup>x</sup> hybridisation is used to model the shape of these molecules.<ref name="NBO Perspective">{{cite book
In certain transition metal complexes with a low d electron count, the p-orbitals are unoccupied and sd<sup>x</sup> hybridisation is used to model the shape of these molecules.<ref name="NBO Perspective">{{cite book
| last1    = Weinhold
| last1    = Weinhold
| first1    = Frank
| first1    = Frank
Line 256: Line 273:
| title = Prediction of the Geometries of Simple Transition Metal Polyhydride Complexes by Symmetry Analysis
| title = Prediction of the Geometries of Simple Transition Metal Polyhydride Complexes by Symmetry Analysis
| first1= Craig |last1=Bayse |first2=Michael |last2=Hall
| first1= Craig |last1=Bayse |first2=Michael |last2=Hall
| journal = [[Organization:Journal of the American Chemical Society|J. Am. Chem. Soc.]]
| journal = J. Am. Chem. Soc.
| year = 1999
| year = 1999
| volume = 121
| volume = 121
Line 265: Line 282:


{| class="wikitable"
{| class="wikitable"
! [[Physics:Coordination number|Coordination number]]
! Coordination number
! Shape
! Shape
! Hybridisation
! Hybridisation
Line 271: Line 288:
|- ----
|- ----
! 3
! 3
| [[Physics:Trigonal pyramidal molecular geometry|Trigonal pyramidal]]
| Trigonal pyramidal
| sd<sup>2</sup> hybridisation (90°)
| sd<sup>2</sup> hybridisation (90°)
| [[Chemistry:Chromium trioxide|CrO<sub>3</sub>]]
| CrO<sub>3</sub>
|- ----
|- ----
! 4
! 4
| [[Physics:Tetrahedral molecular geometry|Tetrahedral]]
| Tetrahedral
| sd<sup>3</sup> hybridisation (70.5°, 109.5°)
| sd<sup>3</sup> hybridisation (70.5°, 109.5°)
| [[Chemistry:Titanium tetrachloride|TiCl<sub>4</sub>]]
| TiCl<sub>4</sub>
|- ----
|- ----
! 5
! 5
| [[Physics:Square pyramidal molecular geometry|Square pyramidal]]
| Square pyramidal
| sd<sup>4</sup> hybridisation (65.9°, 114.1°)
| sd<sup>4</sup> hybridisation (65.9°, 114.1°)
| Ta(CH<sub>3</sub>)<sub>5</sub>
| Ta(CH<sub>3</sub>)<sub>5</sub>
|- ----
|- ----
! 6
! 6
| ''C<sub>3v</sub>'' [[Physics:Trigonal prismatic molecular geometry|Trigonal prismatic]]
| ''C<sub>3v</sub>'' Trigonal prismatic
| sd<sup>5</sup> hybridisation (63.4°, 116.6°)
| sd<sup>5</sup> hybridisation (63.4°, 116.6°)
| [[Chemistry:Hexamethyltungsten|W(CH<sub>3</sub>)<sub>6</sub>]]
| W(CH<sub>3</sub>)<sub>6</sub>
|- ----
|- ----
! Interorbital angles<ref name="NBO Perspective"/>
! Interorbital angles<ref name="NBO Perspective"/>
Line 295: Line 312:


== Hypervalent molecules ==
== Hypervalent molecules ==
{{Main|Physics:Hypervalent molecule}}
=== Octet expansion ===
=== Octet expansion ===
In some general chemistry textbooks, hybridization is presented for main group coordination number 5 and above using an "expanded octet" scheme with d-orbitals first proposed by Pauling. However, such a scheme is now considered to be incorrect in light of computational chemistry calculations.
In some general chemistry textbooks, hybridization is presented for main group coordination number 5 and above using an "expanded octet" scheme with d-orbitals first proposed by Pauling. However, such a scheme is now considered to be incorrect in light of computational chemistry calculations.


{| class="wikitable"
{| class="wikitable"
! [[Physics:Coordination number|Coordination number]]
! Coordination number
! Molecular shape
! Molecular shape
! Hybridisation
! Hybridisation
Line 307: Line 322:
|- ----
|- ----
! 5
! 5
| [[Physics:Trigonal bipyramidal molecular geometry|Trigonal bipyramidal]]
| Trigonal bipyramidal
| sp<sup>3</sup>d hybridisation
| sp<sup>3</sup>d hybridisation
| [[Phosphorus pentafluoride|{{chem2|PF5}}]]
| {{chem2|PF5}}
|- ----
|- ----
! 6
! 6
| [[Physics:Octahedral molecular geometry|Octahedral]]
| Octahedral
| sp<sup>3</sup>d<sup>2</sup> hybridisation
| sp<sup>3</sup>d<sup>2</sup> hybridisation
| [[Sulfur hexafluoride|{{chem2|SF6}}]]
| {{chem2|SF6}}
|- ----
|- ----
! 7
! 7
| [[Physics:Pentagonal bipyramidal molecular geometry|Pentagonal bipyramidal]]
| Pentagonal bipyramidal
| sp<sup>3</sup>d<sup>3</sup> hybridisation
| sp<sup>3</sup>d<sup>3</sup> hybridisation
| [[Iodine heptafluoride|{{chem2|IF7}}]]
| {{chem2|IF7}}
|}
|}


In 1990, Eric Alfred Magnusson of the University of New South Wales published a paper definitively excluding the role of d-orbital hybridisation in bonding in hypervalent compounds of second-row ([[Chemistry:Period (periodic table)#Period 3|period 3]]) elements, ending a point of contention and confusion.  Part of the confusion originates from the fact that d-functions are essential in the basis sets used to describe these compounds (or else unreasonably high energies and distorted geometries result). Also, the contribution of the d-function to the molecular wavefunction is large.  These facts were incorrectly interpreted to mean that d-orbitals must be involved in bonding.<ref name="ReferenceA">{{cite journal | last1 = Magnusson | first1 = E. | year = 1990 | title = Hypercoordinate molecules of second-row elements: d functions or d orbitals? | journal = J. Am. Chem. Soc. | volume = 112 | issue = 22| pages = 7940–7951 | doi = 10.1021/ja00178a014 | bibcode = 1990JAChS.112.7940M }}</ref><ref>{{cite journal
In 1990, Eric Alfred Magnusson of the University of New South Wales published a paper definitively excluding the role of d-orbital hybridisation in bonding in hypervalent compounds of second-row (period 3) elements, ending a point of contention and confusion.  Part of the confusion originates from the fact that d-functions are essential in the basis sets used to describe these compounds (or else unreasonably high energies and distorted geometries result). Also, the contribution of the d-function to the molecular wavefunction is large.  These facts were incorrectly interpreted to mean that d-orbitals must be involved in bonding.<ref name="ReferenceA">{{cite journal | last1 = Magnusson | first1 = E. | year = 1990 | title = Hypercoordinate molecules of second-row elements: d functions or d orbitals? | journal = J. Am. Chem. Soc. | volume = 112 | issue = 22| pages = 7940–7951 | doi = 10.1021/ja00178a014 | bibcode = 1990JAChS.112.7940M }}</ref><ref>{{cite journal
| title  = Chemical Bonding to Hypercoordinate Second-Row Atoms: d Orbital Participation versus Democracy
| title  = Chemical Bonding to Hypercoordinate Second-Row Atoms: d Orbital Participation versus Democracy
| author1 = David L. Cooper
| author1 = David L. Cooper
Line 329: Line 344:
| author4 = Peter B. Karadakov
| author4 = Peter B. Karadakov
| author5 = Mario Raimondi
| author5 = Mario Raimondi
| journal = [[Organization:Journal of the American Chemical Society|Journal of the American Chemical Society]]
| journal = Journal of the American Chemical Society
| year    = 1994
| year    = 1994
| volume  = 116
| volume  = 116
Line 339: Line 354:


=== Resonance ===
=== Resonance ===
In light of [[Chemistry:Computational chemistry|computational chemistry]], a better treatment would be to invoke [[Chemistry:Sigma bond|sigma bond]] [[Resonance|resonance]] in addition to hybridisation, which implies that each resonance structure has its own hybridisation scheme. All resonance structures must obey the [[Octet rule|octet rule]].<ref name="hypervalent resonance">{{cite journal
In light of computational chemistry, a better treatment would be to invoke sigma bond resonance in addition to hybridisation, which implies that each resonance structure has its own hybridisation scheme. All resonance structures must obey the octet rule.<ref name="hypervalent resonance">{{cite journal
| title  = Increased valence (qualitative valence bond) descriptions of the electronic structures of electron-rich fluorine-containing molecules
| title  = Increased valence (qualitative valence bond) descriptions of the electronic structures of electron-rich fluorine-containing molecules
| author1 = Richard D. Harcourt
| author1 = Richard D. Harcourt
Line 353: Line 368:


{| class="wikitable"
{| class="wikitable"
! [[Physics:Coordination number|Coordination number]]
! Coordination number
! Resonance structures
! Resonance structures
|- ----
|- ----
! rowspan="2" | 5
! rowspan="2" | 5
| align="center" | [[Physics:Trigonal bipyramidal molecular geometry|Trigonal bipyramidal]]
| align="center" | Trigonal bipyramidal
|- ----
|- ----
| align="center" | 500px
| align="center" | 500px
|- ----
|- ----
! rowspan="2" | 6
! rowspan="2" | 6
| align="center" | [[Physics:Octahedral molecular geometry|Octahedral]]
| align="center" | Octahedral
|- ----
|- ----
| align="center" | 500px
| align="center" | 500px
|- ----
|- ----
! rowspan="2" | 7
! rowspan="2" | 7
| align="center" | [[Physics:Pentagonal bipyramidal molecular geometry|Pentagonal bipyramidal]]
| align="center" | Pentagonal bipyramidal
|- ----
|- ----
| align="center" | 500px
| align="center" | 500px
Line 382: Line 397:
Although ideal hybrid orbitals can be useful, in reality, most bonds require orbitals of intermediate character. This requires an extension to include flexible weightings of atomic orbitals of each type (s, p, d) and allows for a quantitative depiction of the bond formation when the molecular geometry deviates from ideal bond angles. The amount of p-character is not restricted to integer values; i.e., hybridizations like sp<sup>2.5</sup> are also readily described.
Although ideal hybrid orbitals can be useful, in reality, most bonds require orbitals of intermediate character. This requires an extension to include flexible weightings of atomic orbitals of each type (s, p, d) and allows for a quantitative depiction of the bond formation when the molecular geometry deviates from ideal bond angles. The amount of p-character is not restricted to integer values; i.e., hybridizations like sp<sup>2.5</sup> are also readily described.


The hybridization of bond orbitals is determined by [[Physics:Bent's rule|Bent's rule]]: "Atomic s character concentrates in orbitals directed towards electropositive substituents".
The hybridization of bond orbitals is determined by Bent's rule: "Atomic s character concentrates in orbitals directed towards electropositive substituents".


For molecules with lone pairs, the bonding orbitals are isovalent sp<sup>x</sup> hybrids. For example, the two bond-forming hybrid orbitals of oxygen in water can be described as sp<sup>4.0</sup> to give the interorbital angle of 104.5°.<ref>{{cite book
For molecules with lone pairs, the bonding orbitals are isovalent sp<sup>x</sup> hybrids. For example, the two bond-forming hybrid orbitals of oxygen in water can be described as sp<sup>4.0</sup> to give the interorbital angle of 104.5°.<ref>{{cite book
Line 412: Line 427:
| s2cid = 12677737
| s2cid = 12677737
| doi-access= free
| doi-access= free
}}</ref> The difference in extent of s and p orbitals increases further down a group.  The hybridisation of atoms in chemical bonds can be analysed by considering localised molecular orbitals, for example using natural localised molecular orbitals in a [[Chemistry:Natural bond orbital|natural bond orbital]] (NBO) scheme. In [[Chemistry:Methane|methane]], CH<sub>4</sub>, the calculated p/s ratio is approximately 3 consistent with "ideal" sp<sup>3</sup> hybridisation, whereas for [[Chemistry:Silane|silane]], SiH<sub>4</sub>, the p/s ratio is closer to 2. A similar trend is seen for the other 2p elements. Substitution of fluorine for hydrogen further decreases the p/s ratio.<ref name="The Chemical bond">{{cite book |last1        = Kaupp
}}</ref> The difference in extent of s and p orbitals increases further down a group.  The hybridisation of atoms in chemical bonds can be analysed by considering localised molecular orbitals, for example using natural localised molecular orbitals in a natural bond orbital (NBO) scheme. In methane, CH<sub>4</sub>, the calculated p/s ratio is approximately 3 consistent with "ideal" sp<sup>3</sup> hybridisation, whereas for silane, SiH<sub>4</sub>, the p/s ratio is closer to 2. A similar trend is seen for the other 2p elements. Substitution of fluorine for hydrogen further decreases the p/s ratio.<ref name="The Chemical bond">{{cite book |last1        = Kaupp
  |first1        = Martin
  |first1        = Martin
  |editor1-last  = Frenking
  |editor1-last  = Frenking
Line 457: Line 472:


== Photoelectron spectra ==
== Photoelectron spectra ==
One misconception concerning orbital hybridization is that it incorrectly predicts the [[Physics:Ultraviolet photoelectron spectroscopy|ultraviolet photoelectron spectra]] of many molecules. While this is true if [[Chemistry:Koopmans' theorem|Koopmans' theorem]] is applied to localized hybrids, quantum mechanics requires that the (in this case ionized) wavefunction obey the symmetry of the molecule which implies [[Resonance|resonance]] in valence bond theory. For example, in methane, the ionised states (CH<sub>4</sub><sup>+</sup>) can be constructed out of four resonance structures attributing the ejected electron to each of the four sp<sup>3</sup> orbitals. A linear combination of these four structures, conserving the number of structures, leads to a triply degenerate T<sub>2</sub> state and an A<sub>1</sub> state.<ref>{{cite journal
One misconception concerning orbital hybridization is that it incorrectly predicts the ultraviolet photoelectron spectra of many molecules. While this is true if Koopmans' theorem is applied to localized hybrids, quantum mechanics requires that the (in this case ionized) wavefunction obey the symmetry of the molecule which implies resonance in valence bond theory. For example, in methane, the ionised states (CH<sub>4</sub><sup>+</sup>) can be constructed out of four resonance structures attributing the ejected electron to each of the four sp<sup>3</sup> orbitals. A linear combination of these four structures, conserving the number of structures, leads to a triply degenerate T<sub>2</sub> state and an A<sub>1</sub> state.<ref>{{cite journal
| author1  = Andrei M. Tokmachev
| author1  = Andrei M. Tokmachev
| author2  = Andrei L. Tchougreeff
| author2  = Andrei L. Tchougreeff
Line 477: Line 492:
| pages    = 104–106
| pages    = 104–106
| isbn      = 978-0-470-03735-5
| isbn      = 978-0-470-03735-5
}}</ref> The difference in energy between each ionized state and the ground state would be [[Physics:Ionization energy|ionization energy]], which yields two values in agreement with experimental results.
}}</ref> The difference in energy between each ionized state and the ground state would be ionization energy, which yields two values in agreement with experimental results.


[[File:Methane Ionization.gif|none|Two distinct states for CH<sub>4</sub><sup>+</sup> exist (A<sub>1</sub> and T<sub>2</sub>), both of which result from the ionization of CH<sub>4</sub>. This gives rise to the two unique peaks on the photoelectron spectrum of methane.|frame]]
[[File:Methane Ionization.gif|none|Two distinct states for CH<sub>4</sub><sup>+</sup> exist (A<sub>1</sub> and T<sub>2</sub>), both of which result from the ionization of CH<sub>4</sub>. This gives rise to the two unique peaks on the photoelectron spectrum of methane.|frame]]


== Localised vs canonical molecular orbitals ==
== Localised vs canonical molecular orbitals ==
{{main|Localized molecular orbitals{{!}}Localised molecular orbitals|Natural bond orbital}}


Bonding orbitals formed from hybrid atomic orbitals may be considered as localised molecular orbitals, which can be formed from the delocalised orbitals of molecular orbital theory by an appropriate mathematical transformation. For molecules in the ground state, this transformation of the orbitals leaves the total many-electron wave function unchanged. The hybrid orbital description of the ground state is therefore ''equivalent'' to the delocalised orbital description for ground state total energy and electron density, as well as the molecular geometry that corresponds to the minimum total energy value.
Bonding orbitals formed from hybrid atomic orbitals may be considered as localised molecular orbitals, which can be formed from the delocalised orbitals of molecular orbital theory by an appropriate mathematical transformation. For molecules in the ground state, this transformation of the orbitals leaves the total many-electron wave function unchanged. The hybrid orbital description of the ground state is therefore ''equivalent'' to the delocalised orbital description for ground state total energy and electron density, as well as the molecular geometry that corresponds to the minimum total energy value.


=== Two localised representations ===
=== Two localised representations ===
{{Main|Chemistry:Sigma-pi and equivalent-orbital models}}
[[File:H2O lone pairs two descriptions.png|thumb|The symmetry-adapted and hybridised lone pairs of H<sub>2</sub>O]]
[[File:H2O lone pairs two descriptions.png|thumb|The symmetry-adapted and hybridised lone pairs of H<sub>2</sub>O]]
Molecules with multiple bonds or multiple lone pairs can have orbitals represented in terms of sigma and pi symmetry or equivalent orbitals. Different valence bond methods use either of the two representations, which have mathematically equivalent total many-electron [[Wave function|wave function]]s and are related by a [[Unitary transformation|unitary transformation]] of the set of occupied molecular orbitals.
Molecules with multiple bonds or multiple lone pairs can have orbitals represented in terms of sigma and pi symmetry or equivalent orbitals. Different valence bond methods use either of the two representations, which have mathematically equivalent total many-electron wave functions and are related by a unitary transformation of the set of occupied molecular orbitals.


For multiple bonds, the sigma-pi representation is the predominant one compared to the equivalent orbital ([[Chemistry:Bent bond|bent bond]]) representation. In contrast, for multiple lone pairs, most textbooks use the equivalent orbital representation. However, the sigma-pi representation is also used, such as by Weinhold and Landis within the context of [[Chemistry:Natural bond orbital|natural bond orbital]]s, a localised orbital theory containing modernised analogs of classical (valence bond/Lewis structure) bonding pairs and lone pairs.<ref>{{Cite book
For multiple bonds, the sigma-pi representation is the predominant one compared to the equivalent orbital (bent bond) representation. In contrast, for multiple lone pairs, most textbooks use the equivalent orbital representation. However, the sigma-pi representation is also used, such as by Weinhold and Landis within the context of natural bond orbitals, a localised orbital theory containing modernised analogs of classical (valence bond/Lewis structure) bonding pairs and lone pairs.<ref>{{Cite book
| title    = Discovering Chemistry with Natural Bond Orbitals
| title    = Discovering Chemistry with Natural Bond Orbitals
| last1      = Weinhold
| last1      = Weinhold
Line 506: Line 518:


== See also ==
== See also ==
* [[Physics:Crystal field theory|Crystal field theory]]
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
* [[Chemistry:Isovalent hybridization|Isovalent hybridisation]]
* [[Chemistry:Ligand field theory|Ligand field theory]]
* [[Physics:Linear combination of atomic orbitals|Linear combination of atomic orbitals]]
* MO diagrams
* [[Physics:VALBOND|VALBOND]]


== References ==
= References =
{{reflist}}
{{reflist}}


Line 522: Line 529:
* [http://college.hmco.com/chemistry/shared/media/zumdahl/dswmedia/undr_dcr/Ch14_u14a.dcr Understanding Concepts: Molecular Orbitals] {{Webarchive|url=https://archive.today/20130411024435/http://college.hmco.com/chemistry/shared/media/zumdahl/dswmedia/undr_dcr/Ch14_u14a.dcr |date=2013-04-11 }}
* [http://college.hmco.com/chemistry/shared/media/zumdahl/dswmedia/undr_dcr/Ch14_u14a.dcr Understanding Concepts: Molecular Orbitals] {{Webarchive|url=https://archive.today/20130411024435/http://college.hmco.com/chemistry/shared/media/zumdahl/dswmedia/undr_dcr/Ch14_u14a.dcr |date=2013-04-11 }}
* [https://www.grandinetti.org/orbital-hybridization General Chemistry tutorial on orbital hybridization]
* [https://www.grandinetti.org/orbital-hybridization General Chemistry tutorial on orbital hybridization]
{{Chemical bonding theory|state=expanded}}
{{Linus Pauling}}
[[Category:Chemical bonding]]
[[Category:Molecular geometry]]
[[Category:Stereochemistry]]
[[Category:Quantum chemistry]]


{{Sourceattribution|Orbital hybridisation}}
{{Sourceattribution|Orbital hybridisation}}

Latest revision as of 00:31, 24 May 2026


← Back to Foundations Book I

In chemistry, orbital hybridisation (or hybridization) is the concept of mixing atomic orbitals to form new hybrid orbitals (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds in valence bond theory. For example, in a carbon atom which forms four single bonds, the valence-shell s orbital combines with three valence-shell p orbitals to form four equivalent sp3 mixtures in a tetrahedral arrangement around the carbon to bond to four different atoms. Hybrid orbitals are useful in the explanation of molecular geometry and atomic bonding properties and are symmetrically disposed in space. Usually hybrid orbitals are formed by mixing atomic orbitals of comparable energies.[1]

Orbital hybridisation in the Quantum Collection.

History and uses

Chemist Linus Pauling first developed the hybridisation theory in 1931 to explain the structure of simple molecules such as methane (CH4) using atomic orbitals.[2] Pauling pointed out that a carbon atom forms four bonds by using one s and three p orbitals, so that "it might be inferred" that a carbon atom would form three bonds at right angles (using p orbitals) and a fourth weaker bond using the s orbital in some arbitrary direction. In reality, methane has four C–H bonds of equivalent strength. The angle between any two bonds is the tetrahedral bond angle of 109°28'[3] (around 109.5°). Pauling supposed that in the presence of four hydrogen atoms, the s and p orbitals form four equivalent combinations which he called hybrid orbitals. Each hybrid is denoted sp3 to indicate its composition, and is directed along one of the four C–H bonds.[4] This concept was developed for such simple chemical systems, but the approach was later applied more widely, and today it is considered an effective heuristic for rationalizing the structures of organic compounds. It gives a simple orbital picture equivalent to Lewis structures.

Hybridisation theory is an integral part of organic chemistry, one of the most compelling examples being Baldwin's rules. For drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons.[5] Hybridisation theory explains bonding in alkenes[6] and methane.[7] The amount of p character or s character, which is decided mainly by orbital hybridisation, can be used to reliably predict molecular properties such as acidity or basicity.[8]

Overview

Orbitals are a model representation of the behavior of electrons within molecules. In the case of simple hybridization, this approximation is based on atomic orbitals, similar to those obtained for the hydrogen atom, the only neutral atom for which the Schrödinger equation can be solved exactly. In heavier atoms, such as carbon, nitrogen, and oxygen, the atomic orbitals used are the 2s and 2p orbitals, similar to excited state orbitals for hydrogen.

Hybrid orbitals are assumed to be mixtures of atomic orbitals, superimposed on each other in various proportions. For example, in methane, the C hybrid orbital which forms each carbon–hydrogen bond consists of 25% s character and 75% p character and is thus described as sp3 (read as s-p-three) hybridised. Quantum mechanics describes this hybrid as an sp3 wavefunction of the form N(s+3pσ), where N is a normalisation constant (here 1/2) and pσ is a p orbital directed along the C-H axis to form a sigma bond. The ratio of coefficients (denoted λ in general) is 3 in this example. Since the electron density associated with an orbital is proportional to the square of the wavefunction, the ratio of p-character to s-character is λ2 = 3. The p character or the weight of the p component is N2λ2 = 3/4.

Types

sp3

Four sp3 orbitals.

Hybridisation describes the bonding of atoms from an atom's point of view. For a tetrahedrally coordinated carbon (e.g., methane CH4), the carbon should have 4 orbitals directed towards the 4 hydrogen atoms.

Carbon's ground state configuration is 1s2 2s2 2p2 or more easily read:

C ↑↓ ↑↓ ↑ ↑  
1s 2s 2p 2p 2p

This diagram suggests that the carbon atom could use its two singly occupied p-type orbitals to form two covalent bonds with two hydrogen atoms in a methylene (CH2) molecule, with a hypothetical bond angle of 90° corresponding to the angle between two p orbitals on the same atom. However the true H–C–H angle in singlet methylene is about 102°[9] which implies the presence of some orbital hybridisation.

The carbon atom can also bond to four hydrogen atoms in methane by an excitation (or promotion) of an electron from the doubly occupied 2s orbital to the empty 2p orbital, producing four singly occupied orbitals.

C* ↑↓ ↑ ↑ ↑ ↑
1s 2s 2p 2p 2p

The energy released by the formation of two additional bonds more than compensates for the excitation energy required, energetically favoring the formation of four C-H bonds.

According to quantum mechanics, the lowest energy is obtained if the four bonds are equivalent, which requires that they are formed from equivalent orbitals on the carbon. A set of four equivalent orbitals can be obtained that are linear combinations of the valence-shell (core orbitals are almost never involved in bonding) s and p wave functions,[10] which are the four sp3 hybrids.

C* ↑↓ ↑ ↑ ↑ ↑
1s sp3 sp3 sp3 sp3

In CH4, four sp3 hybrid orbitals are overlapped by the four hydrogens' 1s orbitals, yielding four σ (sigma) bonds (that is, four single covalent bonds) of equal length and strength.

The following:

A schematic presentation of hybrid orbitals overlapping hydrogen orbitals

translates into:

Methane's tetrahedral shape

sp2

Three sp2 orbitals.
Ethylene structure

Other carbon compounds and other molecules may be explained in a similar way. For example, ethylene (C2H4) has a double bond between the carbons. For this molecule, carbon sp2 hybridises, because one π (pi) bond is required for the double bond between the carbons and only three σ bonds are formed per carbon atom. In sp2 hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals, usually denoted 2px and 2py. The third 2p orbital (2pz) remains unhybridised.

C* ↑↓ ↑ ↑ ↑ ↑
1s sp2 sp2 sp2 2p

forming a total of three sp2 orbitals with one remaining p orbital. In ethylene, the two carbon atoms form a σ bond by overlapping one sp2 orbital from each carbon atom. The π bond between the carbon atoms perpendicular to the molecular plane is formed by 2p–2p overlap. Each carbon atom forms covalent C–H bonds with two hydrogens by s–sp2 overlap, all with 120° bond angles. The hydrogen–carbon bonds are all of equal strength and length, in agreement with experimental data.

sp

Two sp orbitals

The chemical bonding in compounds such as alkynes with triple bonds is explained by sp hybridization. In this model, the 2s orbital is mixed with only one of the three p orbitals,

C* ↑↓ ↑ ↑ ↑ ↑
1s sp sp 2p 2p

resulting in two sp orbitals and two remaining p orbitals. The chemical bonding in acetylene (ethyne) (C2H2) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by p–p overlap. Each carbon also bonds to hydrogen in a σ s–sp overlap at 180° angles.

Molecule shape

Shapes of the different types of hybrid orbitals

Hybridisation helps to explain molecule shape, since the angles between bonds are approximately equal to the angles between hybrid orbitals. This is in contrast to valence shell electron-pair repulsion (VSEPR) theory, which can be used to predict molecular geometry based on empirical rules rather than on valence-bond or orbital theories.[11]

spx hybridisation

As the valence orbitals of main group elements are the one s and three p orbitals with the corresponding octet rule, spx hybridization is used to model the shape of these molecules.

Coordination number Shape Hybridisation Examples
2 Linear sp hybridisation (180°) CO2
3 Trigonal planar sp2 hybridisation (120°) BCl3
4 Tetrahedral sp3 hybridisation (109.5°) CCl4
Interorbital angles[12] θ=arccos⁡(−1x)

spxdy hybridisation

As the valence orbitals of transition metals are the five d, one s and three p orbitals with the corresponding 18-electron rule, spxdy hybridisation is used to model the shape of these molecules. These molecules tend to have multiple shapes corresponding to the same hybridization due to the different d-orbitals involved. A square planar complex has one unoccupied p-orbital and hence has 16 valence electrons.[13]

Coordination number Shape Hybridisation Examples
4 Square planar sp2d hybridisation PtCl42−
5 Trigonal bipyramidal sp3d hybridisation Fe(CO)5
Square pyramidal MnCl52−
6 Octahedral sp3d2 hybridisation Mo(CO)6
7 Pentagonal bipyramidal sp3d3 hybridisation ZrF73−
Capped octahedral MoF7−
Capped trigonal prismatic TaF72−
8 Square antiprismatic sp3d4 hybridisation ReF8−
Dodecahedral Mo(CN)84−
Bicapped trigonal prismatic ZrF84−
9 Tricapped trigonal prismatic sp3d5 hybridisation ReH92−
Capped square antiprismatic

sdx hybridisation

In certain transition metal complexes with a low d electron count, the p-orbitals are unoccupied and sdx hybridisation is used to model the shape of these molecules.[12][14][13]

Coordination number Shape Hybridisation Examples
3 Trigonal pyramidal sd2 hybridisation (90°) CrO3
4 Tetrahedral sd3 hybridisation (70.5°, 109.5°) TiCl4
5 Square pyramidal sd4 hybridisation (65.9°, 114.1°) Ta(CH3)5
6 C3v Trigonal prismatic sd5 hybridisation (63.4°, 116.6°) W(CH3)6
Interorbital angles[12] θ=arccos⁡(±13(1−2x))

Hypervalent molecules

Octet expansion

In some general chemistry textbooks, hybridization is presented for main group coordination number 5 and above using an "expanded octet" scheme with d-orbitals first proposed by Pauling. However, such a scheme is now considered to be incorrect in light of computational chemistry calculations.

Coordination number Molecular shape Hybridisation Examples
5 Trigonal bipyramidal sp3d hybridisation PF
5
6 Octahedral sp3d2 hybridisation SF
6
7 Pentagonal bipyramidal sp3d3 hybridisation IF
7

In 1990, Eric Alfred Magnusson of the University of New South Wales published a paper definitively excluding the role of d-orbital hybridisation in bonding in hypervalent compounds of second-row (period 3) elements, ending a point of contention and confusion. Part of the confusion originates from the fact that d-functions are essential in the basis sets used to describe these compounds (or else unreasonably high energies and distorted geometries result). Also, the contribution of the d-function to the molecular wavefunction is large. These facts were incorrectly interpreted to mean that d-orbitals must be involved in bonding.[15][16]

Resonance

In light of computational chemistry, a better treatment would be to invoke sigma bond resonance in addition to hybridisation, which implies that each resonance structure has its own hybridisation scheme. All resonance structures must obey the octet rule.[17]

Coordination number Resonance structures
5 Trigonal bipyramidal
500px
6 Octahedral
500px
7 Pentagonal bipyramidal
500px

In computational VB theory

While the simple model of orbital hybridisation is commonly used to explain molecular shape, hybridisation is used differently when computed in modern valence bond programs. Specifically, hybridisation is not determined a priori but is instead variationally optimized to find the lowest energy solution and then reported. This means that all artificial constraints, specifically two constraints, on orbital hybridisation are lifted:
  • that hybridisation is restricted to integer values (isovalent hybridisation)
  • that hybrid orbitals are orthogonal to one another (hybridisation defects)

This means that in practice, hybrid orbitals do not conform to the simple ideas commonly taught and thus in scientific computational papers are simply referred to as spx, spxdy or sdx hybrids to express their nature instead of more specific integer values.

Isovalent hybridisation

Although ideal hybrid orbitals can be useful, in reality, most bonds require orbitals of intermediate character. This requires an extension to include flexible weightings of atomic orbitals of each type (s, p, d) and allows for a quantitative depiction of the bond formation when the molecular geometry deviates from ideal bond angles. The amount of p-character is not restricted to integer values; i.e., hybridizations like sp2.5 are also readily described.

The hybridization of bond orbitals is determined by Bent's rule: "Atomic s character concentrates in orbitals directed towards electropositive substituents".

For molecules with lone pairs, the bonding orbitals are isovalent spx hybrids. For example, the two bond-forming hybrid orbitals of oxygen in water can be described as sp4.0 to give the interorbital angle of 104.5°.[18] This means that they have 20% s character and 80% p character and does not imply that a hybrid orbital is formed from one s and four p orbitals on oxygen since the 2p subshell of oxygen only contains three p orbitals.

Hybridisation defects

Hybridisation of s and p orbitals to form effective spx hybrids requires that they have comparable radial extent. While 2p orbitals are on average less than 10% larger than 2s, in part attributable to the lack of a radial node in 2p orbitals, 3p orbitals which have one radial node, exceed the 3s orbitals by 20–33%.[19] The difference in extent of s and p orbitals increases further down a group. The hybridisation of atoms in chemical bonds can be analysed by considering localised molecular orbitals, for example using natural localised molecular orbitals in a natural bond orbital (NBO) scheme. In methane, CH4, the calculated p/s ratio is approximately 3 consistent with "ideal" sp3 hybridisation, whereas for silane, SiH4, the p/s ratio is closer to 2. A similar trend is seen for the other 2p elements. Substitution of fluorine for hydrogen further decreases the p/s ratio.[20] The 2p elements exhibit near ideal hybridisation with orthogonal hybrid orbitals. For heavier p block elements this assumption of orthogonality cannot be justified. These deviations from the ideal hybridisation were termed hybridisation defects by Kutzelnigg.[21]

However, computational VB groups such as Gerratt, Cooper and Raimondi (SCVB) as well as Shaik and Hiberty (VBSCF) go a step further to argue that even for model molecules such as methane, ethylene and acetylene, the hybrid orbitals are already defective and nonorthogonal, with hybridisations such as sp1.76 instead of sp3 for methane.[22]

Photoelectron spectra

One misconception concerning orbital hybridization is that it incorrectly predicts the ultraviolet photoelectron spectra of many molecules. While this is true if Koopmans' theorem is applied to localized hybrids, quantum mechanics requires that the (in this case ionized) wavefunction obey the symmetry of the molecule which implies resonance in valence bond theory. For example, in methane, the ionised states (CH4+) can be constructed out of four resonance structures attributing the ejected electron to each of the four sp3 orbitals. A linear combination of these four structures, conserving the number of structures, leads to a triply degenerate T2 state and an A1 state.[23][24] The difference in energy between each ionized state and the ground state would be ionization energy, which yields two values in agreement with experimental results.

Two distinct states for CH4+ exist (A1 and T2), both of which result from the ionization of CH4. This gives rise to the two unique peaks on the photoelectron spectrum of methane.

Localised vs canonical molecular orbitals

Bonding orbitals formed from hybrid atomic orbitals may be considered as localised molecular orbitals, which can be formed from the delocalised orbitals of molecular orbital theory by an appropriate mathematical transformation. For molecules in the ground state, this transformation of the orbitals leaves the total many-electron wave function unchanged. The hybrid orbital description of the ground state is therefore equivalent to the delocalised orbital description for ground state total energy and electron density, as well as the molecular geometry that corresponds to the minimum total energy value.

Two localised representations

The symmetry-adapted and hybridised lone pairs of H2O

Molecules with multiple bonds or multiple lone pairs can have orbitals represented in terms of sigma and pi symmetry or equivalent orbitals. Different valence bond methods use either of the two representations, which have mathematically equivalent total many-electron wave functions and are related by a unitary transformation of the set of occupied molecular orbitals.

For multiple bonds, the sigma-pi representation is the predominant one compared to the equivalent orbital (bent bond) representation. In contrast, for multiple lone pairs, most textbooks use the equivalent orbital representation. However, the sigma-pi representation is also used, such as by Weinhold and Landis within the context of natural bond orbitals, a localised orbital theory containing modernised analogs of classical (valence bond/Lewis structure) bonding pairs and lone pairs.[25] For the hydrogen fluoride molecule, for example, two F lone pairs are essentially unhybridised p orbitals, while the other is an spx hybrid orbital. An analogous consideration applies to water (one O lone pair is in a pure p orbital, another is in an spx hybrid orbital).

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ Housecroft, Catherine E.; Sharpe, Alan G. (2005). Inorganic Chemistry (2nd ed.). Pearson Prentice-Hal. p. 100. ISBN 0130-39913-2. 
  2. ↑ Pauling, L. (1931), "The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules", Journal of the American Chemical Society 53 (4): 1367–1400, doi:10.1021/ja01355a027, Bibcode: 1931JAChS..53.1367P 
  3. ↑ Brittin, W. E. (1945). "Valence Angle of the Tetrahedral Carbon Atom". J. Chem. Educ. 22 (3): 145. doi:10.1021/ed022p145. Bibcode: 1945JChEd..22..145B. 
  4. ↑ L. Pauling The Nature of the Chemical Bond (3rd ed., Oxford University Press 1960) p.111–120.
  5. ↑ Clayden, Jonathan; Greeves, Nick; Warren, Stuart (2012). Organic Chemistry (2nd ed.). Oxford University Press. p. 105. ISBN 978-0-19-927029-3. 
  6. ↑ Organic Chemistry, Third Edition Marye Anne Fox James K. Whitesell 2003 ISBN 978-0-7637-3586-9
  7. ↑ Organic Chemistry 3rd Ed. 2001 Paula Yurkanis Bruice ISBN 978-0-130-17858-9
  8. ↑ "Acids and Bases". http://orgomadesimple.com/organic-chemistry-acids-and-bases/. 
  9. ↑ Isaiah Shavitt (1985), Geometry and singlet-triplet energy gap in methylene: A critical review of experimental and theoretical determinations. Tetrahedron, volume 41, issue 8, page 1531 doi:10.1016/S0040-4020(01)96393-8
  10. ↑ McMurray, J. (1995). Chemistry Annotated Instructors Edition (4th ed.). Prentice Hall. p. 272. ISBN 978-0-131-40221-8
  11. ↑ Gillespie, R.J. (2004), "Teaching molecular geometry with the VSEPR model", Journal of Chemical Education 81 (3): 298–304, doi:10.1021/ed081p298, Bibcode: 2004JChEd..81..298G 
  12. ↑ 12.0 12.1 12.2 Weinhold, Frank; Landis, Clark R. (2005). Valency and bonding: A Natural Bond Orbital Donor-Acceptor Perspective. Cambridge: Cambridge University Press. pp. 367, 374–376, 381–383. ISBN 978-0-521-83128-4. 
  13. ↑ 13.0 13.1 Bayse, Craig; Hall, Michael (1999). "Prediction of the Geometries of Simple Transition Metal Polyhydride Complexes by Symmetry Analysis". J. Am. Chem. Soc. 121 (6): 1348–1358. doi:10.1021/ja981965+. Bibcode: 1999JAChS.121.1348B. 
  14. ↑ Kaupp, Martin (2001). ""Non-VSEPR" Structures and Bonding in d(0) Systems". Angew Chem Int Ed Engl 40 (1): 3534–3565. doi:10.1002/1521-3773(20011001)40:19<3534::AID-ANIE3534>3.0.CO;2-#. PMID 11592184. 
  15. ↑ Magnusson, E. (1990). "Hypercoordinate molecules of second-row elements: d functions or d orbitals?". J. Am. Chem. Soc. 112 (22): 7940–7951. doi:10.1021/ja00178a014. Bibcode: 1990JAChS.112.7940M. 
  16. ↑ David L. Cooper; Terry P. Cunningham; Joseph Gerratt; Peter B. Karadakov; Mario Raimondi (1994). "Chemical Bonding to Hypercoordinate Second-Row Atoms: d Orbital Participation versus Democracy". Journal of the American Chemical Society 116 (10): 4414–4426. doi:10.1021/ja00089a033. Bibcode: 1994JAChS.116.4414C. 
  17. ↑ Richard D. Harcourt; Thomas M. Klapötke (2003). "Increased valence (qualitative valence bond) descriptions of the electronic structures of electron-rich fluorine-containing molecules". Journal of Fluorine Chemistry 123 (1): 5–20. doi:10.1016/S0022-1139(03)00012-5. Bibcode: 2003JFluC.123....5H. 
  18. ↑ Frenking, Gernot; Shaik, Sason, eds (2014). "Chapter 3: The NBO View of Chemical Bonding". The Chemical Bond: Fundamental Aspects of Chemical Bonding. John Wiley & Sons. ISBN 978-3-527-66471-9. 
  19. ↑ Kaupp, Martin (2007). "The role of radial nodes of atomic orbitals for chemical bonding and the periodic table". Journal of Computational Chemistry 28 (1): 320–325. doi:10.1002/jcc.20522. ISSN 0192-8651. PMID 17143872. Bibcode: 2007JCoCh..28..320K. 
  20. ↑ Kaupp, Martin (2014). "Chapter 1: Chemical bonding of main group elements". in Frenking, Gernod; Shaik, Sason. The Chemical Bond: Chemical Bonding Across the Periodic Table. Wiley-VCH. ISBN 9781234567897. https://archive.org/details/guidetolcshinfor00doej. 
  21. ↑ Kutzelnigg, W. (August 1988). "Orthogonal and non-orthogonal hybrids". Journal of Molecular Structure: THEOCHEM 169: 403–419. doi:10.1016/0166-1280(88)80273-2. 
  22. ↑ Shaik, Sason; Danovich, David; Hiberty, Philippe (2022). "On the nature of the chemical bond in valence bond theory". The Journal of Chemical Physics 157 (9): 090901. doi:10.1063/5.0095953. PMID 36075734. Bibcode: 2022JChPh.157i0901S. https://cnrs.hal.science/hal-04310531v1/file/J.%20Chem.%20Phys.%20157%2C%20090901%20%282022%29.pdf. 
  23. ↑ Andrei M. Tokmachev; Andrei L. Tchougreeff; Igor A. Misurkin (2001). "Ionization potentials within semiempirical antisymmetrized product of strictly localized geminals approach". International Journal of Quantum Chemistry 85 (3): 109–117. doi:10.1002/qua.1060. 
  24. ↑ Sason S. Shaik; Phillipe C. Hiberty (2008). A Chemist's Guide to Valence Bond Theory. New Jersey: Wiley-Interscience. pp. 104–106. ISBN 978-0-470-03735-5. 
  25. ↑ Weinhold, Frank; Landis, Clark R. (2012). Discovering Chemistry with Natural Bond Orbitals. Hoboken, N.J.: Wiley. pp. 67–68. ISBN 978-1-118-11996-9. 

Source attribution: Orbital hybridisation

Retrieved from "https://scholarlywiki.org/index.php?title=Physics:Quantum_Orbital_hybridisation&oldid=10146"