View source for Covalent bond

{{Short description|Chemical bond by sharing of electron pairs}}
{{redirect|Covalent}}

[[File:Covalent bond hydrogen.svg|thumb|400px|A covalent bond forming H<sub>2</sub> (right) where two [[hydrogen atom]]s share the two [[electron]]s]]

A '''covalent bond''' is a [[chemical bond]] that involves the sharing of [[electrons]] to form [[electron pair]]s between [[atom]]s. These electron pairs are known as '''shared pairs''' or '''bonding pairs'''. The stable balance of attractive and repulsive forces between atoms, when they share [[electron]]s, is known as covalent bonding.<ref>{{cite book |last1=Whitten |first1=Kenneth W. |last2=Gailey |first2=Kenneth D. |last3=Davis |first3=Raymond E. |title=General Chemistry |date=1992 |publisher=Saunders College Publishing |isbn=0-03-072373-6 |page=264 |edition=4th |chapter=7-3 Formation of covalent bonds}}</ref> For many [[molecule]]s, the sharing of electrons allows each atom to attain the equivalent of a full valence shell, corresponding to a stable [[electronic configuration]]. In organic chemistry, covalent bonding is much more common than [[ionic bonding]], where atoms are separate and merely associated by [[electrostatic]] attraction.

Covalent bonding also includes many kinds of interactions, including [[sigma bond|σ-bonding]], [[pi bond|π-bonding]], <!-- do not change the following-->[[Metallic bonding|metal-to-metal bonding]],<!-- "nonmetal/nonmetal" is already covered as sigma/pi --> [[agostic interaction]]s, [[bent bond]]s, [[three-center two-electron bond]]s and [[three-center four-electron bond]]s.<ref>{{cite book| last = March| first = Jerry| title = Advanced Organic Chemistry: Reactions, Mechanisms, and Structure| url = https://archive.org/details/advancedorganicc04edmarc| url-access = registration| year = 1992| publisher = John Wiley & Sons| isbn = 0-471-60180-2 }}</ref><ref>{{cite book| author = Gary L. Miessler| author2 = Donald Arthur Tarr| title = Inorganic Chemistry| year = 2004| publisher = Prentice Hall| isbn = 0-13-035471-6| url-access = registration| url = https://archive.org/details/inorganicchemist03edmies}}</ref> The term "covalence" was introduced by Irving Langmuir in 1919, with Nevil Sidgwick using "co-valent link" in the 1920s. Merriam-Webster dates the specific phrase ''covalent bond'' to 1939,<ref>[[Merriam-Webster]] – Collegiate Dictionary (2000).</ref> recognizing its first known use. The prefix ''co-'' (jointly, partnered) indicates that "co-valent" bonds involve shared "[[valence (chemistry)|valence]]", as detailed in [[valence bond theory]].

In the molecule {{chem|H|2}}, the [[hydrogen]] atoms share the two electrons via covalent bonding.<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html |title=Chemical Bonds |publisher=Hyperphysics.phy-astr.gsu.edu |access-date=2013-06-09}}</ref> The covalent character of a bond, as opposed to a more ionic nature, is greatest between atoms that have similar [[electronegativity|electronegativities]], regardless of whether the elements are the same as each other. Covalent bonding that entails the sharing of electrons over more than two atoms is said to be [[delocalized electron|delocalized]].

== History ==
[[File:covalent.svg|thumb|200px|Early concepts in covalent bonding arose from this kind of image of the molecule of [[methane]]. Covalent bonding is implied in the [[Lewis structure]] by indicating electrons shared between atoms.]]

The term ''covalence'' in regard to bonding was first used in 1919 by [[Irving Langmuir]] in a ''[[Journal of the American Chemical Society]]'' article entitled "The Arrangement of Electrons in Atoms and Molecules". Langmuir wrote that "we shall denote by the term ''covalence'' the number of pairs of electrons that a given atom shares with its neighbors."<ref>{{cite journal|doi=10.1021/ja02227a002|volume=41|issue=6|pages=868–934|last=Langmuir|first=Irving|title=The Arrangement of Electrons in Atoms and Molecules|journal=Journal of the American Chemical Society|date=1919-06-01|url=https://zenodo.org/record/1429026}}</ref>

The idea of covalent bonding can be traced several years before 1919 to [[Gilbert N. Lewis]], who in 1916 described the sharing of electron pairs between atoms<ref>{{cite journal|doi=10.1021/ja02261a002|volume=38|issue=4|pages=762–785|last=Lewis|first=Gilbert N.|title=The atom and the molecule|journal=Journal of the American Chemical Society|date=1916-04-01|s2cid=95865413 |url=https://zenodo.org/record/1429068}}</ref> (and in 1926 he also coined the term "[[photon]]" for the smallest unit of radiant energy). He introduced the ''[[Lewis Structure|Lewis notation]]'' or ''electron dot notation'' or ''Lewis dot structure'', in which valence electrons (those in the outer shell) are represented as dots around the atomic symbols. Pairs of electrons located between atoms represent covalent bonds. Multiple pairs represent multiple bonds, such as [[double bond]]s and [[triple bond]]s. An alternative form of representation, not shown here, has bond-forming electron pairs represented as solid lines.<ref name=":0">{{Cite book|last=McMurry|first=John|title=Chemistry|publisher=Pearson|year=2016|isbn=978-0-321-94317-0|edition=7}}</ref>

Lewis proposed that an atom forms enough covalent bonds to form a full (or closed) outer electron shell. In the diagram of methane shown here, the carbon atom has a valence of four and is, therefore, surrounded by eight electrons (the [[octet rule]]), four from the carbon itself and four from the hydrogens bonded to it. Each hydrogen has a valence of one and is surrounded by two electrons (a duet rule) – its own one electron plus one from the carbon. The numbers of electrons correspond to full shells in the quantum theory of the atom; the outer shell of a carbon atom is the ''n''&nbsp;=&nbsp;2 shell, which can hold eight electrons, whereas the outer (and only) shell of a hydrogen atom is the ''n''&nbsp;=&nbsp;1 shell, which can hold only two.<ref name=":1">{{Cite book|last=Bruice|first=Paula|title=Organic Chemistry|publisher=Pearson|year=2016|isbn=978-0-13-404228-2|edition=8}}</ref>

While the idea of shared electron pairs provides an effective qualitative picture of covalent bonding, [[quantum mechanics]] is needed to understand the nature of these bonds and predict the structures and properties of simple molecules. [[Walter Heitler]] and [[Fritz London]] are credited with the first successful quantum mechanical explanation of a chemical bond ([[molecular hydrogen]]) in 1927.<ref name= London>{{cite journal|first1=W.|last1=Heitler|first2=F.|last2=London|title=Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik|trans-title=Interaction of neutral atoms and homeopolar bonds according to quantum mechanics|journal=Zeitschrift für Physik|volume=44|issue=6–7|pages=455–472|date=1927|doi=10.1007/bf01397394|bibcode=1927ZPhy...44..455H|s2cid=119739102}} English translation in {{cite book| last = Hettema| first = H.| title = Quantum Chemistry: Classic Scientific Papers| url = https://books.google.com/books?id=qsidHRJmUoIC| access-date = 2012-02-05| year = 2000| publisher = World Scientific| isbn = 978-981-02-2771-5| pages = 140}}</ref> Their work was based on the valence bond model, which assumes that a chemical bond is formed when there is good overlap between the [[atomic orbitals]] of participating atoms.

==Types of covalent bonds==
[[Atomic orbitals]] (except for s orbitals) have specific directional properties leading to different types of covalent bonds. [[Sigma bond|Sigma (σ) bond]]s are the strongest covalent bonds and are due to head-on overlapping of orbitals on two different atoms. A [[single bond]] is usually a σ bond. [[Pi bond|Pi (π) bonds]] are weaker and are due to lateral overlap between p (or d) orbitals. A [[double bond]] between two given atoms consists of one σ and one π bond, and a [[triple bond]] is one σ and two π bonds.<ref name=":0" />

Covalent bonds are also affected by the [[electronegativity]] of the connected atoms which determines the [[chemical polarity]] of the bond. Two atoms with equal electronegativity will make nonpolar covalent bonds such as H–H. An unequal relationship creates a polar covalent bond such as with H−Cl. However polarity also requires [[Geometry|geometric]] [[asymmetry]], or else [[dipole]]s may cancel out, resulting in a non-polar molecule.<ref name=":0" />

== Covalent structures ==
There are several types of structures for covalent substances, including individual molecules, [[molecular structures]], [[macromolecular]] structures and giant covalent structures. Individual molecules have strong bonds that hold the atoms together, but generally, there are negligible forces of attraction between molecules. Such covalent substances are usually gases, for example, [[HCl]], [[Sulfur dioxide|SO<sub>2</sub>]], [[Carbon dioxide|CO<sub>2</sub>]], and [[Methane|CH<sub>4</sub>]]. In molecular structures, there are weak forces of attraction. Such covalent substances are low-boiling-temperature liquids (such as [[ethanol]]), and low-melting-temperature solids (such as [[iodine]] and solid CO<sub>2</sub>). Macromolecular structures have large numbers of atoms linked by covalent bonds in chains, including synthetic polymers such as [[polyethylene]] and [[nylon]], and biopolymers such as [[protein]]s and [[starch]]. [[Network covalent bonding|Network covalent structures]] (or giant covalent structures) contain large numbers of atoms linked in sheets (such as [[graphite]]), or 3-dimensional structures (such as [[diamond]] and [[quartz]]). These substances have high melting and boiling points, are frequently brittle, and tend to have high electrical [[resistivity]]. Elements that have high [[electronegativity]], and the ability to form three or four electron pair bonds, often form such large macromolecular structures.<ref name="StranksEtAl1970">{{cite book |last1=Stranks |first1=D. R. |last2=Heffernan |first2=M. L. |last3=Lee Dow |first3=K. C. |last4=McTigue |first4=P. T. |last5=Withers |first5=G. R. A. |title=Chemistry: A structural view |year=1970 |publisher=Melbourne University Press |location=[[Carlton, Victoria|Carlton, Vic.]] |isbn=0-522-83988-6 |page=184}}</ref>

== One- and three-electron bonds ==
[[File:Graphical comparison of bonds.svg|200px|thumb|right|[[Lewis structure|Lewis]] and [[MO diagram]]s of an individual 2e<sup>−</sup> bond and 3e<sup>−</sup> bond]]
Bonds with one or three electrons can be found in [[radical (chemistry)|radical]] species, which have an odd number of electrons. The simplest example of a 1-electron bond is found in the [[dihydrogen cation]], {{chem|H|2|+}}. One-electron bonds often have about half the bond energy of a 2-electron bond, and are therefore called "half bonds". However, there are exceptions: in the case of [[dilithium]], the bond is actually stronger for the 1-electron {{chem|Li|2|+}} than for the 2-electron Li<sub>2</sub>. This exception can be explained in terms of [[Orbital hybridisation|hybridization]] and inner-shell effects.<ref>{{cite book | title=Valency and Bonding| publisher=Cambridge | year=2005 |pages=96–100 | last1=Weinhold|first1= F. |last2= Landis|first2= C. | isbn=0-521-83128-8}}</ref>

The simplest example of three-electron bonding can be found in the [[helium dimer]] cation, {{chem|He|2|+}}. It is considered a "half bond" because it consists of only one shared electron (rather than two);<ref>{{cite book |editor-last=Harcourt |editor-first=Richard D.|title=Bonding in Electron-Rich Molecules: Qualitative Valence-Bond Approach via Increased-Valence Structures |publisher=Springer |date=2015 |chapter=Chapter 2: Pauling "3-Electron Bonds", 4-Electron 3-Centre Bonding, and the Need for an "Increased-Valence" Theory|isbn=9783319166766}}</ref> in molecular orbital terms, the third electron is in an anti-bonding orbital which cancels out half of the bond formed by the other two electrons. Another example of a molecule containing a 3-electron bond, in addition to two 2-electron bonds, is [[nitric oxide]], NO. The oxygen molecule, O<sub>2</sub> can also be regarded as having two 3-electron bonds and one 2-electron bond, which accounts for its [[paramagnetism]] and its formal bond order of 2.<ref name="pauling">{{cite book|last=Pauling|first=L.|date=1960|title=The Nature of the Chemical Bond|url=https://archive.org/details/natureofchemical00paul|url-access=registration|publisher=Cornell University Press|pages=[https://archive.org/details/natureofchemical00paul/page/340 340–354]}}</ref> [[Chlorine dioxide]] and its heavier analogues [[bromine dioxide]] and [[Iodine oxide|iodine dioxide]] also contain three-electron bonds.

Molecules with odd-electron bonds are usually highly reactive. These types of bond are only stable between atoms with similar electronegativities.<ref name="pauling" />

[[Dioxygen]] is sometimes represented as obeying the octet rule with a double bond (O=O) containing two pairs of shared electrons.<ref>For example, ''General chemistry'' by R.H.Petrucci, W.S.Harwood and F.G.Herring (8th ed., Prentice-Hall 2002, {{ISBN|0-13-014329-4}}, p.395) writes the Lewis structure with a double bond, but adds a question mark with the explanation that there is some doubt about the validity of this structure because it fails to account for the observed paramagnetism.</ref> However, the ground state of this molecule is [[paramagnetic]], indicating the presence of unpaired electrons. Pauling proposed that this molecule actually contains two three-electron bonds and one normal covalent (two-electron) bond.<ref>L. Pauling ''The Nature of the Chemical Bond'' (3rd ed., Oxford University Press 1960) chapter 10.</ref> The octet on each atom then consists of two electrons from each three-electron bond, plus the two electrons of the covalent bond, plus one lone pair of non-bonding electrons. The bond order is 1+0.5+0.5=2.
{{multiple image | align = center | direction = horizontal | header = Modified Lewis structures with 3e bonds | width = 150 | image1 = Nitric oxide.svg | caption1  = Nitric oxide | image2 = Triplett-Sauerstoff.svg | caption2  = Dioxygen }}

== Resonance ==
{{Main|Resonance (chemistry)}}
There are situations whereby a single [[Lewis structure]] is insufficient to explain the electron configuration in a molecule and its resulting experimentally-determined properties, hence a superposition of structures is needed. The same two atoms in such molecules can be bonded differently in different Lewis structures (a single bond in one, a double bond in another, or even none at all), resulting in a non-integer [[bond order]]. The [[nitrate]] ion is one such example with three equivalent structures. The bond between the [[nitrogen]] and each oxygen is a double bond in one structure and a single bond in the other two, so that the average bond order for each N–O interaction is {{sfrac|2 + 1 + 1|3}} = {{sfrac|4|3}}.<ref name=":0" />

[[File:Nitrate-ion-resonance-2D.png|400px]]

=== Aromaticity ===
{{Main|Aromaticity}}

In [[organic chemistry]], when a molecule with a planar ring obeys [[Hückel's rule]], where the number of [[pi bond|π electrons]] fit the formula 4''n''&nbsp;+&nbsp;2 (where ''n'' is an integer), it attains extra stability and symmetry. In [[benzene]], the prototypical aromatic compound, there are 6 π bonding electrons (''n''&nbsp;=&nbsp;1, 4''n''&nbsp;+&nbsp;2&nbsp;=&nbsp;6). These occupy three delocalized π molecular orbitals ([[molecular orbital theory]]) or form conjugate π bonds in two resonance structures that linearly combine ([[valence bond theory]]), creating a regular [[hexagon]] exhibiting a greater stabilization than the hypothetical 1,3,5-cyclohexatriene.<ref name=":1" />

In the case of [[heterocyclic]] aromatics and substituted [[benzene]]s, the electronegativity differences between different parts of the ring may dominate the chemical behavior of aromatic ring bonds, which otherwise are equivalent.<ref name=":1" />

=== Hypervalence ===
{{Main|Hypervalent molecule}}
Certain molecules such as [[xenon difluoride]] and [[sulfur hexafluoride]] have higher coordination numbers than would be possible due to strictly covalent bonding according to the [[octet rule]]. This is explained by the [[three-center four-electron bond]] ("3c–4e") model which interprets the molecular wavefunction in terms of non-bonding [[HOMO/LUMO|highest occupied molecular orbital]]s in [[molecular orbital theory]] and [[resonance (chemistry)|resonance]] of sigma bonds in [[valence bond theory]].<ref>{{Cite book|last1=Weinhold|first1=F.|title=Valency and Bonding|last2=Landis|first2=C.|publisher=Cambridge University Press|year=2005|isbn=0521831288|location=|pages=275–306}}</ref>

=== Electron deficiency ===
{{Main|Electron deficiency}}
In [[three-center two-electron bond]]s ("3c–2e") three atoms share two electrons in bonding. This type of bonding occurs in [[boron hydrides]] such as [[diborane]] (B<sub>2</sub>H<sub>6</sub>), which are often described as electron deficient because there are not enough valence electrons to form localized (2-centre 2-electron) bonds joining all the atoms. However, the more modern description using 3c–2e bonds does provide enough bonding orbitals to connect all the atoms so that the molecules can instead be classified as electron-precise.

Each such bond (2 per molecule in diborane) contains a pair of electrons which connect the [[boron]] atoms to each other in a banana shape, with a proton (the nucleus of a hydrogen atom) in the middle of the bond, sharing electrons with both boron atoms. In certain [[cluster chemistry|cluster compounds]], so-called [[four-center two-electron bond]]s also have been postulated.<ref>{{cite journal |title= A new 4c–2e bond in {{chem|B|6|H|7|-}} |first1= K. |last1= Hofmann |first2= M. H. |last2= Prosenc |first3= B. R. |last3= Albert |journal= Chemical Communications |date= 2007 |volume= 2007 |issue= 29 |pages= 3097–3099 |doi= 10.1039/b704944g |pmid= 17639154 }}</ref>

== Quantum mechanical description ==
After the development of quantum mechanics, two basic theories were proposed to provide a quantum description of chemical bonding: [[valence bond theory|valence bond (VB) theory]] and [[molecular orbital theory|molecular orbital (MO) theory]]. A more recent quantum description<ref>{{cite journal|last1=Cammarata|first1=Antonio|last2=Rondinelli|first2=James M.|title=Covalent dependence of octahedral rotations in orthorhombic perovskite oxides|journal=Journal of Chemical Physics|date=21 September 2014|volume=141|issue=11|pages=114704|doi=10.1063/1.4895967|pmid=25240365|bibcode=2014JChPh.141k4704C}}</ref> is given in terms of atomic contributions to the electronic density of states.

=== Comparison of VB and MO theories ===
The two theories represent two ways to build up the [[electron configuration]] of the molecule.<ref name="Quanta">{{cite book | title=Quanta: A Handbook of Concepts| publisher=Oxford University Press | year=1974 |pages=147–148 | first=P. W.|last= Atkins | isbn=978-0-19-855493-6}}</ref> For valence bond theory, the atomic [[orbital hybridisation|hybrid orbitals]] are filled with electrons first to produce a fully bonded valence configuration, followed by performing a linear combination of contributing structures ([[resonance (chemistry)|resonance]]) if there are several of them. In contrast, for molecular orbital theory, a [[linear combination of atomic orbitals]] is performed first, followed by filling of the resulting [[molecular orbital]]s with electrons.<ref name=":0" />

The two approaches are regarded as complementary, and each provides its own insights into the problem of chemical bonding. As valence bond theory builds the molecular wavefunction out of localized bonds, it is more suited for the calculation of [[bond energy|bond energies]] and the understanding of [[reaction mechanism]]s. As molecular orbital theory builds the molecular wavefunction out of delocalized orbitals, it is more suited for the calculation of [[ionization energy|ionization energies]] and the understanding of [[Absorption spectroscopy|spectral absorption bands]].<ref>James D. Ingle Jr. and Stanley R. Crouch, ''Spectrochemical Analysis'', Prentice Hall, 1988, {{ISBN|0-13-826876-2}}</ref>

At the qualitative level, both theories contain incorrect predictions. Simple (Heitler–London) valence bond theory correctly predicts the dissociation of homonuclear diatomic molecules into separate atoms, while simple (Hartree–Fock) molecular orbital theory incorrectly predicts dissociation into a mixture of atoms and ions. On the other hand, simple molecular orbital theory correctly predicts [[Hückel's rule]] of aromaticity, while simple valence bond theory incorrectly predicts that cyclobutadiene has larger resonance energy than benzene.<ref name="Modern Physical Organic Chemistry">{{cite book|last=Anslyn|first=Eric V.|title=Modern Physical Organic Chemistry|year=2006|publisher=University Science Books|isbn=978-1-891389-31-3}}</ref>

Although the wavefunctions generated by both theories at the qualitative level do not agree and do not match the stabilization energy by experiment, they can be corrected by [[configuration interaction]].<ref name="Quanta"/> This is done by combining the valence bond covalent function with the functions describing all possible ionic structures or by combining the molecular orbital ground state function with the functions describing all possible excited states using unoccupied orbitals. It can then be seen that the simple molecular orbital approach overestimates the weight of the ionic structures while the simple valence bond approach neglects them. This can also be described as saying that the simple molecular orbital approach neglects [[electron correlation]] while the simple valence bond approach overestimates it.<ref name="Quanta"/>

Modern calculations in [[quantum chemistry]] usually start from (but ultimately go far beyond) a molecular orbital rather than a valence bond approach, not because of any intrinsic superiority in the former but rather because the MO approach is more readily adapted to numerical computations. Molecular orbitals are orthogonal, which significantly increases the feasibility and speed of computer calculations compared to nonorthogonal valence bond orbitals.

=== Covalency from atomic contribution to the electronic density of states ===
Evaluation of bond covalency is dependent on the [[basis set (chemistry)|basis set]] for approximate quantum-chemical methods such as COOP (crystal orbital overlap population),<ref>{{Cite journal|last1=Hughbanks|first1=Timothy|last2=Hoffmann|first2=Roald|date=2002-05-01|title=Chains of trans-edge-sharing molybdenum octahedra: metal-metal bonding in extended systems|journal=Journal of the American Chemical Society|volume=105|issue=11|pages=3528–3537|doi=10.1021/ja00349a027}}</ref> COHP (Crystal orbital Hamilton population),<ref>{{Cite journal|last1=Dronskowski|first1=Richard|last2=Bloechl|first2=Peter E.|date=2002-05-01|title=Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations|journal=The Journal of Physical Chemistry|volume=97|issue=33|pages=8617–8624|doi=10.1021/j100135a014}}</ref> and BCOOP (Balanced crystal orbital overlap population).<ref>{{Cite journal|last1=Grechnev|first1=Alexei|last2=Ahuja|first2=Rajeev|last3=Eriksson|first3=Olle|date=2003-01-01|title=Balanced crystal orbital overlap population—a tool for analysing chemical bonds in solids|journal=Journal of Physics: Condensed Matter|volume=15|issue=45|pages=7751|doi=10.1088/0953-8984/15/45/014|issn=0953-8984|bibcode=2003JPCM...15.7751G|s2cid=250757642 }}</ref> To overcome this issue, an alternative formulation of the bond covalency can be provided in this way.

The [[Center of mass|mass center]] {{tmath|cm(n,l,m_l,m_s)}} of an atomic orbital <math>| n,l,m_l,m_s \rangle ,</math> with [[quantum number]]s {{tmath|n,}} {{tmath|l,}} {{tmath|m_l,}} {{tmath|m_s,}} for atom A is defined as

:<math>cm^\mathrm{A}(n,l,m_l,m_s)=\frac{\int\limits_{E_0}\limits^{E_1} E g_{|n,l,m_l,m_s\rangle}^\mathrm{A}(E) dE}{\int\limits_{E_0}\limits^{E_1} g_{|n,l,m_l,m_s\rangle}^\mathrm{A} (E)dE}</math>

where <math>g_{|n,l,m_l,m_s\rangle}^\mathrm{A}(E)</math> is the contribution of the atomic orbital <math>|n,l,m_l,m_s \rangle</math> of the atom A to the total electronic density of states {{tmath|g(E)}} of the solid

:<math>g(E)=\sum_\mathrm{A}\sum_{n, l}\sum_{m_l, m_s}{g_{|n,l,m_l,m_s\rangle}^\mathrm{A}(E)}</math>

where the outer sum runs over all atoms A of the unit cell. The energy window {{tmath|[E_0, E_1]}} is chosen in such a way that it encompasses all of the relevant bands participating in the bond. If the range to select is unclear, it can be identified in practice by examining the molecular orbitals that describe the electron density along with the considered bond.

The relative position {{tmath|C_{n_\mathrm{A}l_\mathrm{A},n_\mathrm{B}l_\mathrm{B} } }} of the mass center of <math>| n_\mathrm{A},l_\mathrm{A}\rangle</math> levels of atom A with respect to the mass center of <math>| n_\mathrm{B},l_\mathrm{B}\rangle</math> levels of atom B is given as

:<math>C_{n_\mathrm{A}l_\mathrm{A},n_\mathrm{B}l_\mathrm{B}}=-\left|cm^\mathrm{A}(n_\mathrm{A},l_\mathrm{A})-cm^\mathrm{B}(n_\mathrm{B},l_\mathrm{B})\right|</math>

where the contributions of the magnetic and spin quantum numbers are summed. According to this definition, the relative position of the A levels with respect to the B levels is

:<math>C_\mathrm{A,B}=-\left|cm^\mathrm{A}-cm^\mathrm{B}\right|</math>

where, for simplicity, we may omit the dependence from the principal quantum number {{tmath|n}} in the notation referring to {{tmath|C_{n_\mathrm{A}l_\mathrm{A},n_\mathrm{B}l_\mathrm{B} }.}}

In this formalism, the greater the value of {{tmath|C_\mathrm{A,B},}} the higher the overlap of the selected atomic bands, and thus the electron density described by those orbitals gives a more covalent {{chem2|A\sB}} bond. The quantity {{tmath|C_\mathrm{A,B} }} is denoted as the ''covalency'' of the {{chem2|A\sB}} bond, which is specified in the same units of the energy {{tmath|E}}.

== Analogous effect in nuclear systems ==
An analogous effect to covalent binding is believed to occur in some nuclear systems, with the difference that the shared fermions are [[quarks]] rather than electrons.<ref>
{{cite journal
 |last1=Brodsky |first1=S. J.
 |title=Novel Features of Nuclear Chromodynamics |url=https://inspirehep.net/files/081b94935ae48c72c2d8d382b0d3da3c
 |journal=The European Physical Journal A
 |volume=53 |issue=3
 |year=2017
 |page=48
 |doi=10.1140/epja/i2017-12234-5
 |bibcode=2017EPJA...53...48B
 |osti=1341388
 |s2cid=126305939
 }}</ref> [[Particle physics|High energy]] [[proton]]-proton [[scattering]] [[Cross section (physics)|cross-section]] indicates that quark interchange of either u or d quarks is the dominant process of the [[nuclear force]] at short distance. In particular, it dominates over the [[Yukawa interaction]] where a [[meson]] is exchanged.<ref>
{{cite journal
 |last1=Brodsky |first1=S. J.
 |last2=Mueller |first2=A. H.
 |title=Using Nuclei to Probe Hadronization in QCD |url=https://inspirehep.net/files/265dea42f5f8eff75f4a59300d6f9157
 |journal=Physics Letters B
 |volume=206
 |year=1988
 |issue=4
 |page=685
 |doi=10.1016/0370-2693(88)90719-8
 |bibcode=1988PhLB..206..685B
 |osti=1448604
 }}</ref> Therefore, covalent binding by quark interchange is expected to be the dominating mechanism of nuclear binding at small distance when the bound [[hadrons]] have covalence quarks in common.<ref>
{{cite journal
 |last1=Bashkanova |first1=M.
 |last2=Brodsky |first2=S. J.
 |last3=Clement |first3=H.
 |title=Novel Six-Quark Hidden-Color Dibaryon States in QCD
 |journal=Physics Letters B
 |volume=727
 |year=2013
 |issue=4–5
 |page=438
 |doi=10.1016/j.physletb.2013.10.059
 |arxiv = 1308.6404
 |bibcode=2013PhLB..727..438B
 |s2cid=30153514
 }}</ref>

== See also ==
{{div col}}
* [[Bonding in solids]]
* [[Bond order]]
* [[Coordinate covalent bond]], also known as a dipolar bond or a dative covalent bond
* [[Covalent bond classification]] (or LXZ notation)
* [[Covalent radius]]
* [[Disulfide bond]]
* [[Orbital hybridisation|Hybridization]]
* [[Hydrogen bond]]
* [[Ionic bond]]
* [[Linear combination of atomic orbitals molecular orbital method|Linear combination of atomic orbitals]]
* [[Metallic bonding]]
* [[Noncovalent bonding]]
* [[Resonance (chemistry)]]
{{div col end}}

== References ==
{{Reflist|30em}}

== Sources ==
* {{cite web|url=https://www.chemguide.co.uk/atoms/bonding/covalent.html|title=Covalent bonding&nbsp;– Single bonds|publisher=chemguide|year=2000|access-date=2012-02-05}}
* {{cite web|url=https://www.chem.ox.ac.uk/vrchemistry/electronsandbonds/intro1.htm|title=Electron Sharing and Covalent Bonds|publisher=Department of Chemistry University of Oxford|access-date=2012-02-05}}
* {{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html#c5|title=Chemical Bonds|publisher=Department of Physics and Astronomy, Georgia State University|access-date=2012-02-05}}

== External links ==
* [http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_07.html Covalent Bonds and Molecular Structure] {{Webarchive|url=https://web.archive.org/web/20090210183035/http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_07.html |date=2009-02-10 }}
* [https://web.archive.org/web/20090430011156/http://www.chm.bris.ac.uk/pt/harvey/gcse/covalent.html Structure and Bonding in Chemistry—Covalent Bonds]

{{Chemical bonds}}
{{Chemical bonding theory}}
{{Organic chemistry}}

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[[Category:Chemical bonding]]