Electronegativity

Pauling first proposed[3] 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 canonical forms to the bonding.

The difference in electronegativity between atoms A and B is given by:

$${\displaystyle |\chi _{\rm {A}}-\chi _{\rm {B}}|=({\rm {eV}})^{-1/2}{\sqrt {E_{\rm {d}}({\rm {AB}})-{\frac {E_{\rm {d}}({\rm {AA}})+E_{\rm {d}}({\rm {BB}})}{2}}}}}$$

where the dissociation energies, E_d, of the A–B, A–A and B–B bonds are expressed in electronvolts, the factor (eV)^−1⁄2 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 eV; H–H, 4.52 eV; Br–Br 2.00 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[3] at 2.1, later revised[8] 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⁺ and Br⁻ 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,[8] 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:

$${\displaystyle E_{\rm {d}}({\rm {AB}})={\frac {E_{\rm {d}}({\rm {AA}})+E_{\rm {d}}({\rm {BB}})}{2}}+(\chi _{\rm {A}}-\chi _{\rm {B}})^{2}{\rm {eV}}}$$

or sometimes, a more accurate fit

$${\displaystyle E_{\rm {d}}({\rm {AB}})={\sqrt {E_{\rm {d}}({\rm {AA}})E_{\rm {d}}({\rm {BB}})}}+1.3(\chi _{\rm {A}}-\chi _{\rm {B}})^{2}{\rm {eV}}}$$

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 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 of electronegativity by Pauling scale

→ Atomic radius decreases → Ionization energy increases → Electronegativity increases →

	1	2		3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Group →
↓ Period
1	H 2.20																		He
2	Li 0.98	Be 1.57												B 2.04	C 2.55	N 3.04	O 3.44	F 3.98	Ne
3	Na 0.93	Mg 1.31												Al 1.61	Si 1.90	P 2.19	S 2.58	Cl 3.16	Ar
4	K 0.82	Ca 1.00		Sc 1.36	Ti 1.54	V 1.63	Cr 1.66	Mn 1.55	Fe 1.83	Co 1.88	Ni 1.91	Cu 1.90	Zn 1.65	Ga 1.81	Ge 2.01	As 2.18	Se 2.55	Br 2.96	Kr 3.00
5	Rb 0.82	Sr 0.95		Y 1.22	Zr 1.33	Nb 1.6	Mo 2.16	Tc 1.9	Ru 2.2	Rh 2.28	Pd 2.20	Ag 1.93	Cd 1.69	In 1.78	Sn 1.96	Sb 2.05	Te 2.1	I 2.66	Xe 2.60
6	Cs 0.79	Ba 0.89		Lu 1.27	Hf 1.3	Ta 1.5	W 2.36	Re 1.9	Os 2.2	Ir 2.20	Pt 2.28	Au 2.54	Hg 2.00	Tl 1.62	Pb 1.87	Bi 2.02	Po 2.0	At 2.2	Rn 2.2
7	Fr >0.79[en 1]	Ra 0.9		Lr 1.3[en 2]	Rf	Db	Sg	Bh	Hs	Mt	Ds	Rg	Cn	Nh	Fl	Mc	Lv	Ts	Og
				La 1.1	Ce 1.12	Pr 1.13	Nd 1.14	Pm –	Sm 1.17	Eu –	Gd 1.2	Tb 1.1	Dy 1.22	Ho 1.23	Er 1.24	Tm 1.25	Yb –
				Ac 1.1	Th 1.3	Pa 1.5	U 1.38	Np 1.36	Pu 1.28	Am 1.13	Cm 1.28	Bk 1.3	Cf 1.3	Es 1.3	Fm 1.3	Md 1.3	No 1.3

See also: Electronegativities of the elements (data page)
There are no reliable sources for Pm, Eu and Yb other than the range of 1.1–1.2; see Pauling, Linus (1960). The Nature of the Chemical Bond. 3rd ed., Cornell University Press, p. 93.

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_i) and the electron affinity (E_ea) should be a measure of the tendency of an atom to attract electrons:[9][10]

$${\displaystyle \chi ={\frac {E_{\rm {i}}+E_{\rm {ea}}}{2}}}$$

As this definition is not dependent on an arbitrary relative scale, it has also been termed absolute electronegativity,[11] with the units of kilojoules per mole or electronvolts. 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,[12]

$${\displaystyle \chi =0.187(E_{\rm {i}}+E_{\rm {ea}})+0.17\,}$$

and for energies in kilojoules per mole,[13]

$${\displaystyle \chi =(1.97\times 10^{-3})(E_{\rm {i}}+E_{\rm {ea}})+0.19.}$$

The Mulliken electronegativity can only be calculated for an element whose electron affinity is known. Measured values are available for 72 elements, while approximate values have been estimated or calculated for the remaining elements.

The Mulliken electronegativity of an atom is sometimes said to be the negative of the chemical potential.[14] 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.,

$${\displaystyle \mu ({\rm {Mulliken)=-\chi ({\rm {Mulliken)={}-{\frac {E_{\rm {i}}+E_{\rm {ea}}}{2}}}}}}}$$

The correlation between Allred–Rochow electronegativities (x-axis, in Å⁻²) and Pauling electronegativities (y-axis).

A. Louis Allred and Eugene G. Rochow considered[15] 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_eff, experienced by valence electrons 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_cov. When r_cov is expressed in picometres,[16]

$${\displaystyle \chi =3590{{Z_{\rm {eff}}} \over {r_{\rm {cov}}^{2}}}+0.744}$$

The correlation between Sanderson electronegativities (x-axis, arbitrary units) and Pauling electronegativities (y-axis).

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.[17] With a knowledge of bond lengths, Sanderson's model allows the estimation of bond energies in a wide range of compounds.[18] 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.[19][20] 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.[21] This behavior is analogous to the equalization of chemical potential in macroscopic thermodynamics.[22]

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 electrons in a free atom,[23][24][25]

$${\displaystyle \chi ={n_{\rm {s}}\varepsilon _{\rm {s}}+n_{\rm {p}}\varepsilon _{\rm {p}} \over n_{\rm {s}}+n_{\rm {p}}}}$$

where ε_s,p are the one-electron energies of s- and p-electrons in the free atom and n_s,p are the number of s- and p-electrons in the valence shell. It is usual to apply a scaling factor, 1.75×10⁻³ for energies expressed in kilojoules per mole or 0.169 for energies measured in electronvolts, to give values that are numerically similar to Pauling electronegativities.

The one-electron energies can be determined directly from 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.[26] 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.

The variation of Pauling electronegativity (y-axis) as one descends the main groups of the periodic table from the second period to the sixth period

In general, electronegativity increases on passing from left to right along a period and decreases on descending a group. Hence, fluorine is the most electronegative of the elements (not counting noble gases), whereas caesium is the least electronegative, at least of those elements for which substantial data is available.[26] This would lead one to believe that caesium fluoride is the compound whose bonding features the most ionic character.

There are some exceptions to this general rule. Gallium and germanium have higher electronegativities than aluminium and silicon, respectively, because of the d-block contraction. Elements of the fourth period immediately after the first row of the transition metals have unusually small atomic radii because the 3d-electrons are not effective at shielding the increased nuclear charge, and smaller atomic size correlates with higher electronegativity (see Allred-Rochow electronegativity and Sanderson electronegativity above). The anomalously high electronegativity of lead, in particular when compared to thallium and bismuth, is an artifact of electronegativity varying with oxidation state: its electronegativity conforms better to trends if it is quoted for the +2 state with a Pauling value of 1.87 instead of the +4 state.

In inorganic chemistry, it is common to consider a single value of electronegativity to be valid for most "normal" situations. While this approach has the advantage of simplicity, it is clear that the electronegativity of an element is not an invariable atomic property and, in particular, increases with the oxidation state of the element.

Allred used the Pauling method to calculate separate electronegativities for different oxidation states of the handful of elements (including tin and lead) for which sufficient data were available.[8] However, for most elements, there are not enough different covalent compounds for which bond dissociation energies are known to make this approach feasible. This is particularly true of the transition elements, where quoted electronegativity values are usually, of necessity, averages over several different oxidation states and where trends in electronegativity are harder to see as a result.

The chemical effects of this increase in electronegativity can be seen both in the structures of oxides and halides and in the acidity of oxides and oxoacids. Hence CrO₃ and Mn₂O₇ are acidic oxides with low melting points, while Cr₂O₃ is amphoteric and Mn₂O₃ is a completely basic oxide.

The effect can also be clearly seen in the dissociation constants pK_a of the oxoacids of chlorine. The effect is much larger than could be explained by the negative charge being shared among a larger number of oxygen atoms, which would lead to a difference in pK_a of log₁₀(1⁄4) = –0.6 between hypochlorous acid and perchloric acid. As the oxidation state of the central chlorine atom increases, more electron density is drawn from the oxygen atoms onto the chlorine, diminishing the partial negative charge of individual oxygen atoms. At the same time, the positive partial charge on the hydrogen increases with a higher oxidation state. This explains the observed increased acidity with an increasing oxidation state in the oxoacids of chlorine.

The electronegativity of an atom changes depending on the hybridization of the orbital employed in bonding. Electrons in s orbitals are held more tightly than electrons in p orbitals. Hence, a bond to an atom that employs an sp^x hybrid orbital for bonding will be more heavily polarized to that atom when the hybrid orbital has more s character. That is, when electronegativities are compared for different hybridization schemes of a given element, the order χ(sp³) < χ(sp²) < χ(sp) holds (the trend should apply to non-integer hybridization indices as well). While this holds true in principle for any main-group element, values for the hybridization-specific electronegativity are most frequently cited for carbon. In organic chemistry, these electronegativities are frequently invoked to predict or rationalize bond polarities in organic compounds containing double and triple bonds to carbon.