Physics:Quantum bond order: Difference between revisions
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[[File: | [[File:Quantum_bond_order_concept_map.svg|thumb|280px|bond order in the Quantum Collection.]] | ||
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==Examples== | ==Examples== | ||
The bond order itself is the number of | The bond order itself is the number of electron pairs (covalent bonds) between two atoms.<ref>[https://goldbook.iupac.org/terms/view/B00705 IUPAC Gold Book: Bond number]</ref> For example, in diatomic nitrogen N≡N, the bond order between the two nitrogen atoms is 3 (triple bond). In acetylene H–C≡C–H, the bond order between the two carbon atoms is also 3, and the C–H bond order is 1 (single bond). In carbon monoxide, {{chem2|-C\tO+}}, the bond order between carbon and oxygen is 3. In thiazyl trifluoride {{chem2|N\tSF3}}, the bond order between sulfur and nitrogen is 3, and between sulfur and fluorine is 1. In diatomic oxygen O=O the bond order is 2 (double bond). In ethylene {{chem2|H2C\dCH2}} the bond order between the two carbon atoms is also 2. The bond order between carbon and oxygen in carbon dioxide O=C=O is also 2. In phosgene {{chem2|O\dCCl2}}, the bond order between carbon and oxygen is 2, and between carbon and chlorine is 1. | ||
In some molecules, bond orders can be 4 ( | In some molecules, bond orders can be 4 (quadruple bond), 5 (quintuple bond) or even 6 (sextuple bond). For example, potassium octachlorodimolybdate salt ({{chem2|K4[Mo2Cl8]}}) contains the {{chem2|[Cl4Mo\qMoCl4](4–)}} anion, in which the two Mo atoms are linked to each other by a bond with order of 4. Each Mo atom is linked to four {{chem2|Cl−}} ligands by a bond with order of 1. The compound (terphenyl)–CrCr–(terphenyl) contains two chromium atoms linked to each other by a bond with order of 5, and each chromium atom is linked to one terphenyl ligand by a single bond. A bond of order 6 is detected in ditungsten molecules {{chem2|W2}}, which exist only in a gaseous phase. | ||
===Non-integer bond orders=== | ===Non-integer bond orders=== | ||
In molecules which have | In molecules which have resonance or nonclassical bonding, bond order may not be an integer. In benzene, the delocalized molecular orbitals contain 6 pi electrons over six carbons, essentially yielding half a pi bond together with the sigma bond for each pair of carbon atoms, giving a calculated bond order of 1.5 (one and a half bond). Furthermore, bond orders of 1.1 (eleven tenths bond), 4/3 (or 1.333333..., four thirds bond) or 0.5 (half bond), for example, can occur in some molecules and essentially refer to bond strength relative to bonds with order 1. In the nitrate anion ({{chem2|NO3−}}), the bond order for each bond between nitrogen and oxygen is 4/3 (or 1.333333...). Bonding in dihydrogen cation {{chem2|H2+}} can be described as a covalent one-electron bond, thus the bonding between the two hydrogen atoms has bond order of 0.5.<ref>{{Cite book|author=Clark R. Landis|author2=Frank Weinhold |title=Valency and bonding: a natural bond orbital donor-acceptor perspective |publisher=Cambridge University Press |location=Cambridge, UK |year=2005 |pages=91–92 |isbn=978-0-521-83128-4 }}</ref> | ||
==Bond order in molecular orbital theory== | ==Bond order in molecular orbital theory== | ||
In | In molecular orbital theory, bond order is defined as half the difference between the number of bonding electrons and the number of antibonding electrons as per the equation below.<ref>{{cite book | author1 = Jonathan Clayden | last2 = Greeves | first2 = Nick | author3 = Stuart Warren | title = Organic Chemistry | edition = 2nd | publisher = Oxford University Press | date = 2012 | isbn = 978-0-19-927029-3 | page = 91}}</ref><ref>{{cite book | title = Inorganic Chemistry | last1 = Housecroft | first1 = C. E. | last2 = Sharpe | first2 = A. G. | year = 2012 | publisher = Prentice Hall | edition = 4th | isbn = 978-0-273-74275-3 | pages = 35–37}}</ref> This often but not always yields similar results for bonds near their equilibrium lengths, but it does not work for stretched bonds.<ref name = Manz2017>{{cite journal | doi = 10.1039/c7ra07400j | journal = RSC Adv. | title = Introducing DDEC6 atomic population analysis: part 3. Comprehensive method to compute bond orders | author1 = T. A. Manz | year = 2017 | volume = 7 | issue = 72 | pages = 45552–45581| bibcode = 2017RSCAd...745552M |doi-access = free}}</ref> Bond order is also an index of bond strength and is also used extensively in valence bond theory. | ||
:''bond order'' = {{sfrac|''number of bonding electrons'' - ''number of antibonding electrons''|2}} | :''bond order'' = {{sfrac|''number of bonding electrons'' - ''number of antibonding electrons''|2}} | ||
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Generally, the higher the bond order, the stronger the bond. Bond orders of one-half may be stable, as shown by the stability of {{chem2|H2+}} (bond length 106 pm, bond energy 269 kJ/mol) and {{chem2|He2+}} (bond length 108 pm, bond energy 251 kJ/mol).<ref>Bruce Averill and Patricia Eldredge, ''Chemistry: Principles, Patterns, and Applications'' (Pearson/Prentice Hall, 2007), 409.</ref> | Generally, the higher the bond order, the stronger the bond. Bond orders of one-half may be stable, as shown by the stability of {{chem2|H2+}} (bond length 106 pm, bond energy 269 kJ/mol) and {{chem2|He2+}} (bond length 108 pm, bond energy 251 kJ/mol).<ref>Bruce Averill and Patricia Eldredge, ''Chemistry: Principles, Patterns, and Applications'' (Pearson/Prentice Hall, 2007), 409.</ref> | ||
Hückel molecular orbital theory offers another approach for defining bond orders based on molecular orbital coefficients, for planar molecules with delocalized π bonding. The theory divides bonding into a sigma framework and a pi system. The π-bond order between atoms ''r'' and ''s'' derived from Hückel theory was defined by [[Biography:Charles Coulson|Charles Coulson]] by using the orbital coefficients of the Hückel MOs:<ref>{{cite book |last1=Levine |first1=Ira N. |title=Quantum Chemistry |date=1991 |publisher=Prentice-Hall |isbn=0-205-12770-3 |page=567 |edition=4th}}</ref><ref>{{cite journal |last1=Coulson |first1=Charles Alfred |title=The electronic structure of some polyenes and aromatic molecules. VII. Bonds of fractional order by the molecular orbital method |journal=Proceedings of the Royal Society A |date=7 February 1939 |volume=169 |issue=938 |pages=413–428 |doi=10.1098/rspa.1939.0006 |bibcode=1939RSPSA.169..413C |doi-access=free }}</ref> | Hückel molecular orbital theory offers another approach for defining bond orders based on molecular orbital coefficients, for planar molecules with delocalized π bonding. The theory divides bonding into a sigma framework and a pi system. The π-bond order between atoms ''r'' and ''s'' derived from Hückel theory was defined by [[Biography:Charles Coulson|Charles Coulson]] by using the orbital coefficients of the Hückel MOs:<ref>{{cite book |last1=Levine |first1=Ira N. |title=Quantum Chemistry |date=1991 |publisher=Prentice-Hall |isbn=0-205-12770-3 |page=567 |edition=4th}}</ref><ref>{{cite journal |last1=Coulson |first1=Charles Alfred |title=The electronic structure of some polyenes and aromatic molecules. VII. Bonds of fractional order by the molecular orbital method |journal=Proceedings of the Royal Society A |date=7 February 1939 |volume=169 |issue=938 |pages=413–428 |doi=10.1098/rspa.1939.0006 |bibcode=1939RSPSA.169..413C |doi-access=free }}</ref> | ||
:<math>p_{rs} = \sum_i n_ic_{ri}c_{si}</math>, | :<math>p_{rs} = \sum_i n_ic_{ri}c_{si}</math>, | ||
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Here the sum extends over π molecular orbitals only, and ''n<sub>i</sub>'' is the number of electrons occupying orbital ''i'' with coefficients ''c<sub>ri</sub>'' and ''c<sub>si</sub>'' on atoms ''r'' and ''s'' respectively. Assuming a bond order contribution of 1 from the sigma component this gives a total bond order (σ + π) of 5/3 = 1.67 for benzene, rather than the commonly cited bond order of 1.5, showing some degree of ambiguity in how the concept of bond order is defined. | Here the sum extends over π molecular orbitals only, and ''n<sub>i</sub>'' is the number of electrons occupying orbital ''i'' with coefficients ''c<sub>ri</sub>'' and ''c<sub>si</sub>'' on atoms ''r'' and ''s'' respectively. Assuming a bond order contribution of 1 from the sigma component this gives a total bond order (σ + π) of 5/3 = 1.67 for benzene, rather than the commonly cited bond order of 1.5, showing some degree of ambiguity in how the concept of bond order is defined. | ||
For more elaborate forms of molecular orbital theory involving larger | For more elaborate forms of molecular orbital theory involving larger basis sets, still other definitions have been proposed.<ref>{{cite journal |last1=Sannigrahi |first1=A. B. |last2=Kar |first2=Tapas |title=Molecular orbital theory of bond order and valency |journal=Journal of Chemical Education |date=August 1988 |volume=65 |issue=8 |pages=674–676 |doi=10.1021/ed065p674 |bibcode=1988JChEd..65..674S |url=https://pubs.acs.org/doi/abs/10.1021/ed065p674 |access-date=5 December 2020}}</ref> A standard quantum mechanical definition for bond order has been debated for a long time.<ref>IUPAC Gold Book [http://goldbook.iupac.org/BT07005.html ''bond order'']</ref> A comprehensive method to compute bond orders from quantum chemistry calculations was published in 2017.<ref name = Manz2017 /> | ||
==Other definitions== | ==Other definitions== | ||
The bond order concept is used in | The bond order concept is used in molecular dynamics and bond order potentials. The magnitude of the bond order is associated with the bond length. According to Linus Pauling in 1947, the bond order between atoms ''i'' and ''j'' is experimentally described as | ||
:<math>s_{ij} = \exp{\left[\frac{d_{1} - d_{ij}}{b}\right]}</math> | :<math>s_{ij} = \exp{\left[\frac{d_{1} - d_{ij}}{b}\right]}</math> | ||
where ''d''<sub>1</sub> is the single bond length, ''d<sub>ij</sub>'' is the bond length experimentally measured, and ''b'' is a constant, depending on the atoms. Pauling suggested a value of 0.353 | where ''d''<sub>1</sub> is the single bond length, ''d<sub>ij</sub>'' is the bond length experimentally measured, and ''b'' is a constant, depending on the atoms. Pauling suggested a value of 0.353 Å for ''b'', for carbon-carbon bonds in the original equation:<ref>{{cite journal | last = Pauling | first = Linus | title = Atomic Radii and Interatomic Distances in Metals | journal = Journal of the American Chemical Society | volume = 69 | issue =3 | pages = 542–553 | date = March 1, 1947 | doi=10.1021/ja01195a024}}</ref> | ||
:<math>d_{1} - d_{ij} = 0.353~\text{ln}(s_{ij})</math> | :<math>d_{1} - d_{ij} = 0.353~\text{ln}(s_{ij})</math> | ||
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The value of the constant ''b'' depends on the atoms. This definition of bond order is somewhat ''ad hoc'' and only easy to apply for diatomic molecules. | The value of the constant ''b'' depends on the atoms. This definition of bond order is somewhat ''ad hoc'' and only easy to apply for diatomic molecules. | ||
= References = | |||
{{Reflist}} | {{Reflist}} | ||
{{Sourceattribution|Bond order}} | {{Sourceattribution|Bond order}} | ||
Latest revision as of 00:31, 24 May 2026
Bond order is a Book II topic in the Quantum Collection. It is a measure of the effective number of chemical bonds between two atoms, often related to the occupation of bonding and antibonding molecular orbitals. In quantum chemistry it helps explain bond length, bond strength, vibrational frequency, and molecular stability. Bond order can be described in valence bond theory, molecular orbital theory, and computational electronic-structure methods. The concept links electron sharing, orbital overlap, resonance, and the quantum structure of molecules.
Examples
The bond order itself is the number of electron pairs (covalent bonds) between two atoms.[1] For example, in diatomic nitrogen N≡N, the bond order between the two nitrogen atoms is 3 (triple bond). In acetylene H–C≡C–H, the bond order between the two carbon atoms is also 3, and the C–H bond order is 1 (single bond). In carbon monoxide, −
C≡O+
, the bond order between carbon and oxygen is 3. In thiazyl trifluoride N≡SF
3, the bond order between sulfur and nitrogen is 3, and between sulfur and fluorine is 1. In diatomic oxygen O=O the bond order is 2 (double bond). In ethylene H
2C=CH
2 the bond order between the two carbon atoms is also 2. The bond order between carbon and oxygen in carbon dioxide O=C=O is also 2. In phosgene O=CCl
2, the bond order between carbon and oxygen is 2, and between carbon and chlorine is 1.
In some molecules, bond orders can be 4 (quadruple bond), 5 (quintuple bond) or even 6 (sextuple bond). For example, potassium octachlorodimolybdate salt (K
4[Mo
2Cl
8]) contains the [Cl
4Mo≣MoCl
4]4− anion, in which the two Mo atoms are linked to each other by a bond with order of 4. Each Mo atom is linked to four Cl−
ligands by a bond with order of 1. The compound (terphenyl)–CrCr–(terphenyl) contains two chromium atoms linked to each other by a bond with order of 5, and each chromium atom is linked to one terphenyl ligand by a single bond. A bond of order 6 is detected in ditungsten molecules W
2, which exist only in a gaseous phase.
Non-integer bond orders
In molecules which have resonance or nonclassical bonding, bond order may not be an integer. In benzene, the delocalized molecular orbitals contain 6 pi electrons over six carbons, essentially yielding half a pi bond together with the sigma bond for each pair of carbon atoms, giving a calculated bond order of 1.5 (one and a half bond). Furthermore, bond orders of 1.1 (eleven tenths bond), 4/3 (or 1.333333..., four thirds bond) or 0.5 (half bond), for example, can occur in some molecules and essentially refer to bond strength relative to bonds with order 1. In the nitrate anion (NO−
3), the bond order for each bond between nitrogen and oxygen is 4/3 (or 1.333333...). Bonding in dihydrogen cation H+
2 can be described as a covalent one-electron bond, thus the bonding between the two hydrogen atoms has bond order of 0.5.[2]
Bond order in molecular orbital theory
In molecular orbital theory, bond order is defined as half the difference between the number of bonding electrons and the number of antibonding electrons as per the equation below.[3][4] This often but not always yields similar results for bonds near their equilibrium lengths, but it does not work for stretched bonds.[5] Bond order is also an index of bond strength and is also used extensively in valence bond theory.
- bond order =
Generally, the higher the bond order, the stronger the bond. Bond orders of one-half may be stable, as shown by the stability of H+
2 (bond length 106 pm, bond energy 269 kJ/mol) and He+
2 (bond length 108 pm, bond energy 251 kJ/mol).[6]
Hückel molecular orbital theory offers another approach for defining bond orders based on molecular orbital coefficients, for planar molecules with delocalized π bonding. The theory divides bonding into a sigma framework and a pi system. The π-bond order between atoms r and s derived from Hückel theory was defined by Charles Coulson by using the orbital coefficients of the Hückel MOs:[7][8]
- ,
Here the sum extends over π molecular orbitals only, and ni is the number of electrons occupying orbital i with coefficients cri and csi on atoms r and s respectively. Assuming a bond order contribution of 1 from the sigma component this gives a total bond order (σ + π) of 5/3 = 1.67 for benzene, rather than the commonly cited bond order of 1.5, showing some degree of ambiguity in how the concept of bond order is defined.
For more elaborate forms of molecular orbital theory involving larger basis sets, still other definitions have been proposed.[9] A standard quantum mechanical definition for bond order has been debated for a long time.[10] A comprehensive method to compute bond orders from quantum chemistry calculations was published in 2017.[5]
Other definitions
The bond order concept is used in molecular dynamics and bond order potentials. The magnitude of the bond order is associated with the bond length. According to Linus Pauling in 1947, the bond order between atoms i and j is experimentally described as
where d1 is the single bond length, dij is the bond length experimentally measured, and b is a constant, depending on the atoms. Pauling suggested a value of 0.353 Å for b, for carbon-carbon bonds in the original equation:[11]
The value of the constant b depends on the atoms. This definition of bond order is somewhat ad hoc and only easy to apply for diatomic molecules.
References
- ↑ IUPAC Gold Book: Bond number
- ↑ Clark R. Landis; Frank Weinhold (2005). Valency and bonding: a natural bond orbital donor-acceptor perspective. Cambridge, UK: Cambridge University Press. pp. 91–92. ISBN 978-0-521-83128-4.
- ↑ Jonathan Clayden; Greeves, Nick; Stuart Warren (2012). Organic Chemistry (2nd ed.). Oxford University Press. p. 91. ISBN 978-0-19-927029-3.
- ↑ Housecroft, C. E.; Sharpe, A. G. (2012). Inorganic Chemistry (4th ed.). Prentice Hall. pp. 35–37. ISBN 978-0-273-74275-3.
- ↑ 5.0 5.1 T. A. Manz (2017). "Introducing DDEC6 atomic population analysis: part 3. Comprehensive method to compute bond orders". RSC Adv. 7 (72): 45552–45581. doi:10.1039/c7ra07400j. Bibcode: 2017RSCAd...745552M.
- ↑ Bruce Averill and Patricia Eldredge, Chemistry: Principles, Patterns, and Applications (Pearson/Prentice Hall, 2007), 409.
- ↑ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice-Hall. p. 567. ISBN 0-205-12770-3.
- ↑ Coulson, Charles Alfred (7 February 1939). "The electronic structure of some polyenes and aromatic molecules. VII. Bonds of fractional order by the molecular orbital method". Proceedings of the Royal Society A 169 (938): 413–428. doi:10.1098/rspa.1939.0006. Bibcode: 1939RSPSA.169..413C.
- ↑ Sannigrahi, A. B.; Kar, Tapas (August 1988). "Molecular orbital theory of bond order and valency". Journal of Chemical Education 65 (8): 674–676. doi:10.1021/ed065p674. Bibcode: 1988JChEd..65..674S. https://pubs.acs.org/doi/abs/10.1021/ed065p674. Retrieved 5 December 2020.
- ↑ IUPAC Gold Book bond order
- ↑ Pauling, Linus (March 1, 1947). "Atomic Radii and Interatomic Distances in Metals". Journal of the American Chemical Society 69 (3): 542–553. doi:10.1021/ja01195a024.
Source attribution: Bond order