Editing Electronegativity (section)

==Methods of calculation==

===Pauling electronegativity===
{{Anchor|Pauling electronegativity}}

[[Linus Pauling|Pauling]] first proposed<ref name="paulingJACS"/> the concept of electronegativity in 1932 to explain why the [[covalent bond]] between two different atoms (A–B) is stronger than the average of the A–A and the B–B bonds. According to [[valence bond theory]], of which Pauling was a notable proponent, this "additional stabilization" of the [[heteronuclear]] bond is due to the contribution of [[Ionic bond|ionic]]  [[Resonance (chemistry)|canonical form]]s to the bonding.

The difference in electronegativity between atoms A and B is given by:
<math display="block">|\chi_{\rm A} - \chi_{\rm B}| = ({\rm eV})^{-1/2} \sqrt{E_{\rm d}({\rm AB}) - \frac{E_{\rm d}({\rm AA}) + E_{\rm d}({\rm BB})} 2}</math>
where the [[Bond dissociation energy|dissociation energies]], ''E''<sub>d</sub>, of the A–B, A–A and B–B bonds are expressed in [[electronvolt]]s, the factor (eV)<sup>−{{frac|1|2}}</sup> being included to ensure a dimensionless result. Hence, the difference in Pauling electronegativity between hydrogen and [[bromine]] is 0.73 (dissociation energies: H–Br, 3.79&nbsp;eV; H–H, 4.52&nbsp;eV; Br–Br 2.00&nbsp;eV)

As only differences in electronegativity are defined, it is necessary to choose an arbitrary reference point in order to construct a scale. Hydrogen was chosen as the reference, as it forms covalent bonds with a large variety of elements: its electronegativity was fixed first<ref name="paulingJACS"/> at 2.1, later revised<ref name="Allred">{{cite journal
 |author= Allred, A. L.
 |year= 1961
 |journal= Journal of Inorganic and Nuclear Chemistry
 |volume= 17
 |issue= 3–4
 |pages= 215–221
 |title= Electronegativity values from thermochemical data
 |doi= 10.1016/0022-1902(61)80142-5}}</ref> to 2.20. It is also necessary to decide which of the two elements is the more electronegative (equivalent to choosing one of the two possible signs for the square root). This is usually done using "chemical intuition": in the above example, [[hydrogen bromide]] dissolves in water to form H<sup>+</sup> and Br<sup>−</sup> ions, so it may be assumed that bromine is more electronegative than hydrogen. However, in principle, since the same electronegativities should be obtained for any two bonding compounds, the data are in fact overdetermined, and the signs are unique once a reference point has been fixed (usually, for H or F).

To calculate Pauling electronegativity for an element, it is necessary to have data on the dissociation energies of at least two types of covalent bonds formed by that element. A. L. Allred updated Pauling's original values in 1961 to take account of the greater availability of thermodynamic data,<ref name="Allred"/> and it is these "revised Pauling" values of the electronegativity that are most often used.

The essential point of Pauling electronegativity is that there is an underlying, quite accurate, semi-empirical formula for dissociation energies, namely:
<math display="block">E_{\rm d}({\rm AB}) = \frac{E_{\rm d}({\rm AA}) + E_{\rm d}({\rm BB})} 2 + (\chi_{\rm A} - \chi_{\rm B})^2 {\rm eV}</math>
or sometimes, a more accurate fit
<math display="block">E_{\rm d}({\rm AB}) =\sqrt{E_{\rm d}({\rm AA}) E_{\rm d}({\rm BB})}+1.3(\chi_{\rm A} - \chi_{\rm B})^2 {\rm eV}</math>

These are approximate equations but they hold with good accuracy. Pauling obtained the first equation by noting that a bond can be approximately represented as a quantum mechanical superposition of a covalent bond and two ionic bond-states. The covalent energy of a bond is approximate, by quantum mechanical calculations, the [[geometric mean]] of the two energies of covalent bonds of the same molecules, and there is additional energy that comes from ionic factors, i.e. polar character of the bond.

The geometric mean is approximately equal to the [[arithmetic mean]]—which is applied in the first formula above—when the energies are of a similar value, e.g., except for the highly electropositive elements, where there is a larger difference of two dissociation energies; the geometric mean is more accurate and almost always gives positive excess energy, due to ionic bonding.  The square root of this excess energy, Pauling notes, is approximately additive, and hence one can introduce the electronegativity. Thus, it is these semi-empirical formulas for bond energy that underlie the concept of Pauling electronegativity.

The formulas are approximate, but this rough approximation is in fact relatively good and gives the right intuition, with the notion of the polarity of the bond and some theoretical grounding in quantum mechanics. The electronegativities are then determined to best fit the data.

In more complex compounds, there is an additional error since electronegativity depends on the molecular environment of an atom. Also, the energy estimate can be only used for single, not for multiple bonds. The [[standard enthalpy of formation|enthalpy of formation]] of a molecule containing only single bonds can subsequently be estimated based on an electronegativity table, and it depends on the constituents and the sum of squares of differences of electronegativities of all pairs of bonded atoms. Such a formula for estimating energy typically has a relative error on the order of 10% but can be used to get a rough qualitative idea and understanding of a molecule.

{{Periodic table (electronegativities)}}

===Mulliken electronegativity===
{{Anchor|Mulliken electronegativity}}

[[Image:Pauling and Mullikan electronegativities.png|thumb|upright=1.35|The correlation between Mulliken electronegativities (''x''-axis, in kJ/mol) and Pauling electronegativities (''y''-axis).]]
[[Robert S. Mulliken]] proposed that the [[arithmetic mean]] of the first [[ionization energy]] (E<sub>i</sub>) and the [[electron affinity]] (E<sub>ea</sub>) should be a measure of the tendency of an atom to attract electrons:<ref>{{cite journal |author = Mulliken, R. S.|year =1934 |journal = [[Journal of Chemical Physics]] |volume = 2 |title = A New Electroaffinity Scale; Together with Data on Valence States and on Valence Ionization Potentials and Electron Affinities |doi = 10.1063/1.1749394 |pages = 782–793 |issue = 11|bibcode = 1934JChPh...2..782M }}</ref><ref>{{cite journal |author= Mulliken, R. S. |year=1935 |title = Electronic Structures of Molecules XI. Electroaffinity, Molecular Orbitals and Dipole Moments |journal = [[Journal of Chemical Physics|J. Chem. Phys.]] |volume = 3 |doi = 10.1063/1.1749731 |pages = 573–585
|issue = 9|bibcode = 1935JChPh...3..573M }}</ref>
<math display="block">\chi = \frac{E_{\rm i} + E_{\rm ea}} 2 </math>

As this definition is not dependent on an arbitrary relative scale, it has also been termed '''absolute electronegativity''',<ref>{{cite journal |author=Pearson, R. G. |year=1985 |title=Absolute electronegativity and absolute hardness of Lewis acids and bases |journal=[[Journal of the American Chemical Society|J. Am. Chem. Soc.]] |volume=107 |issue=24 |pages=6801–6806 |doi=10.1021/ja00310a009}}</ref> with the units of [[Joule per mole|kilojoules per mole]] or [[electronvolt]]s. However, it is more usual to use a linear transformation to transform these absolute values into values that resemble the more familiar Pauling values. For ionization energies and electron affinities in electronvolts,<ref>{{cite book |last1=Huheey |first1=J.E. |last2=Keiter |first2=E.A. |last3=Keiter |first3=R.L. |date=December 1, 2008 |orig-year=1978 |chapter=17 |editor1-last=Kauffman |editor1-first=G.B. |title=Inorganic Chemistry: Principles of Structure and Reactivity |url=https://www.pdfdrive.com/inorganic-chemistry-principles-of-structure-and-reactivity-e175855674.html |type=digitalized |language=en |edition=3rd |location=New York |publication-date=1900 |page=167 |doi=10.1021/ed050pA379.1 |isbn=9780060429874 |oclc=770736023 |id= inorganicchemist00huhe_0 |archive-url=https://web.archive.org/web/20190908204147/https://www.pdfdrive.com/inorganic-chemistry-principles-of-structure-and-reactivity-e175855674.html |archive-date=September 8, 2019 |access-date=December 15, 2020 |via=Oxford University Press}} [https://archive.org/details/inorganicchemist00huhe_0 Alt URL]</ref>
<math display="block">\chi = 0.187(E_{\rm i} + E_{\rm ea}) + 0.17 \,</math>
and for energies in kilojoules per mole,<ref>This second relation has been recalculated using the best values of the first ionization energies and electron affinities available in 2006.</ref>
<math display="block">\chi = (1.97\times 10^{-3})(E_{\rm i} + E_{\rm ea}) + 0.19.</math>

The Mulliken electronegativity can only be calculated for an element whose electron affinity is known. [[Electron affinity (data page)|Measured values are available]] for 72 elements, while approximate values have been [[Electron affinity (data page)|estimated or calculated]] for the remaining elements.

The Mulliken electronegativity of an atom is sometimes said to be the negative of the [[chemical potential]].<ref>{{cite journal |last1=Franco-Pérez |first1=Marco |last2=Gázquez |first2=José L. |title=Electronegativities of Pauling and Mulliken in Density Functional Theory |journal=Journal of Physical Chemistry A |date=31 October 2019 |volume=123 |issue=46 |pages=10065–10071 |doi=10.1021/acs.jpca.9b07468 |pmid=31670960 |bibcode=2019JPCA..12310065F |s2cid=207814569 }}</ref>  By inserting the energetic definitions of the ionization potential and electron affinity into the Mulliken electronegativity, it is possible to show that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons., i.e.,
<math display="block">\mu(\rm Mulliken) = -\chi(\rm Mulliken) = {}-\frac{E_{\rm i} + E_{\rm ea}} 2 </math>

===Allred–Rochow electronegativity===
{{Anchor|Allred–Rochow electronegativity}}

[[Image:Pauling and Allred-Rochow electronegativities.png|thumb|upright=1.35|The correlation between Allred–Rochow electronegativities (''x''-axis, in Å<sup>−2</sup>) and Pauling electronegativities (''y''-axis).]]
[[Albert L. Allred|A. Louis Allred]] and [[Eugene G. Rochow]] considered<ref>{{cite journal|author1=Allred, A. L. |author2=Rochow, E. G. |year=1958| journal=Journal of Inorganic and Nuclear Chemistry|volume=5|pages=264–268|doi=10.1016/0022-1902(58)80003-2|title=A scale of electronegativity based on electrostatic force|issue=4}}</ref> that electronegativity should be related to the charge experienced by an electron on the "surface" of an atom: The higher the charge per unit area of atomic surface the greater the tendency of that atom to attract electrons. The [[effective nuclear charge]], ''Z''<sub>eff</sub>, experienced by [[valence electron]]s can be estimated using [[Slater's rules]], while the surface area of an atom in a molecule can be taken to be proportional to the square of the [[covalent radius]], ''r''<sub>cov</sub>. When ''r''<sub>cov</sub> is expressed in [[picometre]]s,<ref>{{cite book |last1=Housecroft |first1=C.E. |last2=Sharpe |first2=A.G. |title=Inorganic Chemistry |url=https://www.pearson.com/us/higher-education/program/Housecroft-Inorganic-Chemistry-5th-Edition/PGM2178749.html |type=eBook |language=en |volume=3 |edition=15th |location=Switzerland |publisher=Pearson Prentice-Hall |publication-date=November 1, 1993 |page=38 |doi=10.1021/ed070pA304.1|isbn=9780273742753 |via=University of Basel}} [https://archive.org/details/Inorganic_Chemistry_4th_edition_by_Catherine_Housecroft_Alan_G._Sharpe Alt URL]</ref>
<math display="block">\chi = 3590{{Z_{\rm eff}}\over{r^2_{\rm cov}}} + 0.744</math>

===Sanderson electronegativity equalization===
[[Image:Pauling and Sanderson electronegativities.png|thumb|upright=1.35|The correlation between Sanderson electronegativities (''x''-axis, arbitrary units) and Pauling electronegativities (''y''-axis).]]
[[Robert Thomas Sanderson|R.T. Sanderson]] has also noted the relationship between Mulliken electronegativity and atomic size, and has proposed a method of calculation based on the reciprocal of the atomic volume.<ref>{{cite journal|author=Sanderson, R. T. |year=1983 |title=Electronegativity and bond energy| journal=Journal of the American Chemical Society|volume=105|pages=2259–2261|doi=10.1021/ja00346a026 |issue=8}}</ref> With a knowledge of bond lengths, Sanderson's model allows the estimation of bond energies in a wide range of compounds.<ref>{{cite book|author=Sanderson, R. T.|year=1983|title=Polar Covalence|location=New York|publisher=Academic Press|isbn=978-0-12-618080-0|url-access=registration|url=https://archive.org/details/polarcovalence0000sand}}</ref> Sanderson's model has also been used to calculate molecular geometry, ''s''-electron energy, [[NMR]] spin-spin coupling constants and other parameters for organic compounds.<ref>{{cite journal |last1=Zefirov |first1=N. S. |first2=M. A. |last2=Kirpichenok |first3=F. F. |last3=Izmailov |first4=M. I. |last4=Trofimov |title=Calculation schemes for atomic electronegativities in molecular graphs within the framework of Sanderson principle |journal=[[Doklady Akademii Nauk SSSR]] |year=1987 |volume=296 |pages=883–887}}</ref><ref>{{cite journal |doi=10.1007/s11172-006-0105-6|title=Application of the electronegativity indices of organic molecules to tasks of chemical informatics |year=2005|author=Trofimov, M. I.|journal=Russian Chemical Bulletin|volume=54|pages=2235–2246|last2=Smolenskii|first2=E. A.|issue=9|s2cid=98716956 }}</ref> This work underlies the concept of '''electronegativity equalization''', which suggests that electrons distribute themselves around a molecule to minimize or to equalize the Mulliken electronegativity.<ref name= Lipkowitz>

{{cite book |title=Reviews in computational chemistry |author1=SW Rick |author2=SJ Stuart |chapter=Electronegativity equalization models |editor1=Kenny B. Lipkowitz |editor2=Donald B. Boyd |chapter-url=https://books.google.com/books?id=IqWXSLz6QE8C&pg=PA106 |page=106 |isbn=978-0-471-21576-9 |year=2002 |publisher=Wiley}}</ref> This behavior is analogous to the equalization of chemical potential in macroscopic thermodynamics.<ref name=Parr>{{cite book |title=Density-functional theory of atoms and molecules |author1=Robert G. Parr |author2=Weitao Yang |url=https://books.google.com/books?id=mGOpScSIwU4C&pg=PA91 |page=91 |isbn=978-0-19-509276-9 |year=1994 |publisher=Oxford University Press}}</ref>

===Allen electronegativity===
[[Image:Pauling and Allen electronegativities.png|thumb|upright=1.35|The correlation between Allen electronegativities (''x''-axis, in kJ/mol) and Pauling electronegativities (''y''-axis).]]
Perhaps the simplest definition of electronegativity is that of Leland C. Allen, who has proposed that it is related to the average energy of the [[valence electron]]s in a free atom,<ref>{{cite journal |doi=10.1021/ja00207a003 |title=Electronegativity is the average one-electron energy of the valence-shell electrons in ground-state free atoms|year=1989|author=Allen, Leland C.|journal=Journal of the American Chemical Society |volume=111|pages=9003–9014 |issue=25}}</ref><ref>{{cite journal|doi=10.1021/ja992866e|title=Configuration Energies of the Main Group Elements|year=2000|author=Mann, Joseph B. |author2=Meek, Terry L. |author3=Allen, Leland C. |journal=Journal of the American Chemical Society |volume=122 |pages=2780–2783|issue=12}}</ref><ref>{{cite journal|doi=10.1021/ja9928677|title=Configuration energies of the d-block elements |year=2000|author=Mann, Joseph B. |author2=Meek, Terry L. |author3=Knight, Eugene T. |author4=Capitani, Joseph F. |author5=Allen, Leland C. |journal=Journal of the American Chemical Society|volume=122|pages=5132–5137|issue=21}}</ref>

<math display="block">\chi = {n_{\rm s}\varepsilon_{\rm s} + n_{\rm p}\varepsilon_{\rm p} \over n_{\rm s} + n_{\rm p}}</math>

where ''ε''<sub>s,p</sub> are the one-electron energies of s- and p-electrons in the free atom and ''n''<sub>s,p</sub> are the number of s- and p-electrons in the valence shell.

The one-electron energies can be determined directly from [[Spectroscopy|spectroscopic data]], and so electronegativities calculated by this method are sometimes referred to as '''spectroscopic electronegativities'''. The necessary data are available for almost all elements, and this method allows the estimation of electronegativities for elements that cannot be treated by the other methods, e.g. [[francium]], which has an Allen electronegativity of 0.67.<ref name="Fr">The widely quoted Pauling electronegativity of 0.7 for francium is an extrapolated value of uncertain provenance. The Allen electronegativity of caesium is 0.66.</ref> However, it is not clear what should be considered to be valence electrons for the d- and f-block elements, which leads to an ambiguity for their electronegativities calculated by the Allen method.

On this scale, [[neon]] has the highest electronegativity of all elements, followed by [[fluorine]], [[helium]], and [[oxygen]].

{{periodic table (electronegativity by Allen scale)}}