Electronegativity

On the most basic level, electronegativity is determined by factors like the nuclear charge (the more protons an atom has, the more "pull" it will have on electrons) and the number and location of other electrons in the electron shells (the more electrons an atom has, the farther from the nucleus the valence electrons will be, and as a result, the less positive charge they will experience—both because of their increased distance from the nucleus and because the other electrons in the lower energy core orbitals will act to shield the valence electrons from the positively charged nucleus).

The term "electronegativity" was introduced by Jöns Jacob Berzelius in 1811, though the concept was known before that and was studied by many chemists including Avogadro. Despite its long history, an accurate scale of electronegativity was not developed until 1932, when Linus Pauling proposed an electronegativity scale that depends on bond energies, as a development of valence bond theory. It has been shown to correlate with several other chemical properties. Electronegativity cannot be directly measured and must be calculated from other atomic or molecular properties. Several methods of calculation have been proposed, and although there may be small differences in the numerical values of electronegativity, all methods show the same periodic trends between elements.

The most commonly used method of calculation is that originally proposed by Linus Pauling. This gives a dimensionless quantity, commonly referred to as the Pauling scale (χ_r), on a relative scale running from 0.79 to 3.98 (hydrogen = 2.20). When other methods of calculation are used, it is conventional (although not obligatory) to quote the results on a scale that covers the same range of numerical values: this is known as electronegativity in Pauling units.

As it is usually calculated, electronegativity is not a property of an atom alone, but rather a property of an atom in a molecule. Even so, the electronegativity of an atom is strongly correlated with the first ionization energy. The electronegativity is slightly negatively correlated (for smaller electronegativity values) and rather strongly positively correlated (for most and larger electronegativity values) with the electron affinity. It is to be expected that the electronegativity of an element will vary with its chemical environment, but it is usually considered to be a transferable property, that is to say, that similar values will be valid in a variety of situations.

By Pauling scale, caesium is the least electronegative element (0.79); fluorine is the most (3.98).

Methods of calculation

Pauling electronegativity

Pauling first proposed 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: $${\displaystyle |{\chi }_{\mathrm{A}}-{\chi }_{\mathrm{B}}|=(\mathrm{e}\mathrm{V}{)}^{-1/2}\sqrt{E_{\mathrm{d}}(\mathrm{A}\mathrm{B})-\frac{E_{\mathrm{d}}(\mathrm{A}\mathrm{A})+E_{\mathrm{d}}(\mathrm{B}\mathrm{B})}{2}}}$$ 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 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 at 0, then in his famous book at 2.1 to avoid negative electronegativity figures, later revised to 2.20 by Allred. 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 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, 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: $${\displaystyle E_{\mathrm{d}}(\mathrm{A}\mathrm{B})=\frac{E_{\mathrm{d}}(\mathrm{A}\mathrm{A})+E_{\mathrm{d}}(\mathrm{B}\mathrm{B})}{2}+({\chi }_{\mathrm{A}}-{\chi }_{\mathrm{B}}{)}^2\mathrm{e}\mathrm{V}}$$ or sometimes, a more accurate fit $${\displaystyle E_{\mathrm{d}}(\mathrm{A}\mathrm{B})=\sqrt{E_{\mathrm{d}}(\mathrm{A}\mathrm{A})E_{\mathrm{d}}(\mathrm{B}\mathrm{B})}+1.3({\chi }_{\mathrm{A}}-{\chi }_{\mathrm{B}}{)}^2\mathrm{e}\mathrm{V}}$$

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 approximately, 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 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.

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Periodic table of electronegativity by Pauling scale → Atomic radius decreases → Ionization energy increases → Electronegativity increases →

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.

Mulliken electronegativity

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: $${\displaystyle \chi =\frac{E_{\mathrm{i}}+E_{\mathrm{e}\mathrm{a}}}{2}}$$

As this definition is not dependent on an arbitrary relative scale, it has also been termed absolute electronegativity, 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, $${\displaystyle \chi =0.187(E_{\mathrm{i}}+E_{\mathrm{e}\mathrm{a}})+0.17}$$ and for energies in kilojoules per mole, $${\displaystyle \chi =(1.97\times {10}^{-3})(E_{\mathrm{i}}+E_{\mathrm{e}\mathrm{a}})+\mathrm{0.19.}}$$

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. 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., $${\displaystyle \mu (\mathrm{M}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{n})=-\chi (\mathrm{M}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{n})=-\frac{{\mathrm{E}}_{\mathrm{i}}+{\mathrm{E}}_{\mathrm{e}\mathrm{a}}}{2}}$$

Allred–Rochow electronegativity

A. Louis Allred and Eugene G. Rochow considered 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, $${\displaystyle \chi =3590\frac{Z_{\mathrm{e}\mathrm{f}\mathrm{f}}}{r_{\mathrm{c}\mathrm{o}\mathrm{v}}^2}+0.744}$$

Sanderson electronegativity equalization

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. With a knowledge of bond lengths, Sanderson's model allows the estimation of bond energies in a wide range of compounds. 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. This work underlies the concept of electronegativity equalization, which suggests that electrons distribute themselves around a molecule to minimize or equalize the Mulliken electronegativity. This behavior is analogous to the equalization of chemical potential in macroscopic thermodynamics.

Allen electronegativity

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,

$${\displaystyle \chi =\frac{n_{\mathrm{s}}{\epsilon }_{\mathrm{s}}+n_{\mathrm{p}}{\epsilon }_{\mathrm{p}}}{n_{\mathrm{s}}+n_{\mathrm{p}}}}$$

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.

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. These energies are obtained by averaging the energies of all spectroscopic terms, weighed according to the S, L and J quantum numbers. Unfortunately, for the transition metals and the lanthanides the spectroscopic energies of many terms are unknown since their energies overlap or have not been recorded and assigned yet. Therefore, Allen lists no electronegativities for the lanthanides and employed theoretical methods to estimate the electronegativities of the listed transition metals. The necessary data are available for most of the main group 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. 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 regarding their electronegativities calculated by the Allen method.

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