Based on the VB approximation, the bond strengths can be considered mainly about the potential energy: the nuclear charge Z felt by valence electrons at the covalent boundary. And the term of Schrödinger’s Wave Equation incorporating bond strength is the potential energy Ze²ψ/r. (2.1.1) − h 2 ∇ 2 ψ / 8 π 2 m − Z e 2 ψ / 4 π ɛ 0 r = E ψ

Ionic bonds are omnidirectional. The nuclear charge Z possesses the power of ionizing radiation and radiates positive charge in all directions. Therefore, the nuclear charge Z is directly proportional to the bond strengths but has the delocalized ionic nature.

The covalent radius, r_c is the other important part of the potential here. It is a distance from nucleus to a charge density wherein the bonding atoms are aligned and localized at very specific bond lengths, and it is inversely proportional to the bond strengths. In the case of the hydrogen atom, as Equation (2.1.1) shows, as the electron approaches the nucleus, the potential energy dives down toward minus-infinity, in order for the total energy E to remain constant, and its kinetic energy shoots up toward positive-infinity. So a compromise is reached in which theory tells us that the fall potential energy is just twice the kinetic energy, and the electron dances at an average distance that corresponds to the Bohr radius [25]. The calculation of the H₂ molecule by Heitler and London showed that as the inter-nuclear distance decreased, the potential energy associated with interactions between nucleus and electrons dropped very markedly until a minimum was reached, and then (owing to the greater effect of internuclear repulsion at much smaller internuclear distances) to rise sharply [26]. The minimum corresponded fairly closely to the experimentally determined value of R, 0.74 Å (covalent radius = 0.37 Å, see below 4.2). Therefore, the covalent radius has the harmoniously localized covalent nature.

As atoms and molecules have the dual nature, ionocovalency, the bond strengths can be considered as a combination of the ionic and the covalent functions [10]. So the ionic function I can be considered as a function of the nuclear charge Z: I(Z); and the covalent function C can be considered as a function of the covalent radius r_c: C(r_c⁻¹), wherein the reciprocal of r_c can be defined as atomic covalency.

Ionocovalently, the bond strengths and electronegativity, therefore, can be accounted for on the basis of the dual nature of bonds, and functionally defined as ionocovalency, a product of the ionic and the covalent functions: (2.1.2) I C = I ( Z * ) C ( r c − 1 )

And so the bond strengths, the potential energy and electronegativity have the IC framework of ionocovalency, which shows an effective ionocovalent potential, the attraction power that should be Pauling postulated.

According to the Bohr energy model (2.1.3) E = − Z 2 m e 4 / 8 n 2 h 2 ɛ 0 2 = − R Z 2 / n 2we have derived the effective nuclear charge Z* from ionization energy and the effective principal quantum number n* [8,9]: (2.1.4) Z * = n * ( I z / R ) ½where I_z is the ultimate IE. R is the Rydberg constant, R = 2π²μ⁴2e⁴/h² = 13.6 eV, h is Planck’s constant. Substituting Equation (2.1.4) into (2.1.2), we can naturally correlate the bond properties to the quantum-mechanics, and get the IC model: (2.1.5) I ( I z ) C ( n * r c − 1 ) = n * ( I a v / R ) ½ r c − 1where the effective principal quantum number n* is related to the electron energy, distribution and the distance from the nucleus. Hence, n* can be considered as an energy of ionic function and also as a spatial distance of covalent function. The n*r_c⁻¹, which is related to the spatial overlap, can be defined as spatial-covalency of covalent function (r_c⁻¹ is a linear- or σ-covalency).

As an application of ionocovalency, by Plotting Pauling values of X_p against n*(I_av/R)^½r_c⁻¹, we obtain a new IC-potential electronegativity (2.1.6) X I C = 0.412 n * ( I a v / R ) ½ r c − 1 + 0.387

Based on the above IC model, our previous electronegativity scale X_z [8,9] can be accounted for IC-force: (2.1.7) I C = I ( V ) C ( r − 1 ) = I ( Z * ) C ( r − 2 ) = n * ( I z / R ) ½ r c − 2 (2.1.8) X z = 0.241 n * ( I z / R ) ½ r c − 2 + 0.775

In Equation 2.1.5 and 2.1.6, I_av is ionicity (the average IE of valence shell electrons) determined by the following IC orbital hybrid bonding procedures. The diagrams are adapted from those in the excellent article by Blaber [27].

(1) Ionization promotion: As a valence shell fills, the successive increased ionization energy of an electron would provide the promotion energy for hybrid orbital formation e.g., consider gaseous molecules of BeF₂. The fluorine atom has the electron configuration: 1s²2s²2p⁵. The beryllium atom has the electron configuration: 1s²2s².

In the ground state, there are no unpaired electrons. However, the beryllium atom could obtain an unpaired electron by promoting an electron from the 2s orbital to the 2p orbital by its lower first IE (9.32 eV). The beryllium atom can now forms two covalent bonds with fluorine atoms:

(2) Ionicity hybridization: We can combine functions for the 2s and 2p electrons to produce a “hybrid” orbital for both electrons. The charge transfer and charge distribution would occur from the higher energy level of the 2p orbital to the lower energy level of the 2s orbital to form an energy-lower and identical 2sp hybrid orbital. The ideas developed are adequate for calculation of the charge density identically distributed. The IEs of the 2s and 2p electrons can be averaged to result in a hybridizing ionicity, I_av: (2.1.9) I a v = n − 1 Σ i = 1 n I iwhere I_i is the IE of single electron of valence shell, n is number of valence shell electrons.

By Equation (2.1.9), we get an average IE: I_av = 13.76 eV from the first IE (9.32 eV) and second IE (18.2 eV):

first: Be[He](2s)² → Be⁺[He] (2s)¹ − 9.32 eV

(3) Ionocovalency: by localizing the hybridized ionicity at the covalent boundary r_c to form an ionocovalent bond of 2sp hybrid orbitals:

Hence, for beryllium, IC = I(I_av)C(n*r_c⁻¹) = n*(I_av/R)^½r_c⁻¹ = 1.99(13.76/13.6)^½ (0.970)⁻¹ = 2.064 And X_IC = 0.412 n*(I_av/R)^½ r_c⁻¹ + 0.387 = 0.412*1.99(13.76/13.6)^½(0.970) ⁻¹ + 0.387 = 1.237.

Similarly, an s orbital can also mix with all three p orbitals in the same subshell. For CH₄ by Equation (2.1.9), we get ionicity: I_av = 37.015 eV from the first IE (11.26 eV), second IE (24.40 eV), third IE (47.90 eV) and fourth IE (64.50 eV)

of the all 2s and 2p electrons and form 2sp³ hybrid orbitals:

The IE is taken from Mackay et al. [28], the covalent radius r_c, is taken from Pauling [29], Batsanov [30], Cordero et al. [31], Rappe et al. [32] and the effective principal quantum number n* is from our previous work [8,9]. The ionocovalency scale, IC, calculated from Equation (2.1.5) is listed in Chart 1 and the electronegativity scale, X_IC calculated from Equation (2.1.6) are listed in Table 1.

Based on the IC model, ionocovalency can be finally defined as “the effective potential caused by the ionicity on a bonding pair of electrons at the localized covalent boundary in different valence hybrid orbital states, forming an ionocovalent charge density”.

Natural Values: Chart 1 shows that IC has a same value range as that of its atomic core charge. This phenomenon is particularly noticeable for the top period elements which have not yet experienced much shielding effects. The atomic core charge is an effective nuclear charge and is always markedly less than the actual nuclear charge Z, but, as the shielding is not perfect, the core charge increases as Z increases, but more slowly. In the IC model, the effective nuclear charge is determined by natural IE, not from the calculation by shielding constants.

Ionocovalent Continuum: Chart 1 and Table 1 show that the ionocovalency IC and the electronegativity X_IC exhibit evidently the ionocovalency character. The greater the IC and the X_IC, the more covalent and the less ionic the cation is, and vice versa. Generally, across the period, the more right-hand-side an element is, the more covalent it is. And the more down-ward an element is, the more ionic it is. The IC and X_IC increase from the lower left to the upper right of the Periodic Table across the s and p blocks, and decrease down most columns. Trends parallel periodic trends in IE.

The ionocovalency is a continuum which is certainly an improvement over the old ionic-versus-covalent dichotomy. In so-called ‘pure’ ionic bonding, an electron is transferred completely from one atom to another, once this transfer is complete, the IC potential will act to try to pull this electron back to its parent ion. If IC partially succeeds then there will be some electron density, n*(I_av/R)^½r_c⁻¹, in the region in between the ions, which is the situation in a covalent bond. Thus ‘pure’ ionic and ‘pure’ covalent bonds could be seen as two extremes of an IC continuum. And so a covalent scale has the ionic degree.

Energy-lowered Hybrid Bonding: The ionocovalent bonding procedures runs a charge promotion, charge distribution and energy-lowered hybrid sequence. Using Equation 2.1.5. and 2.1.6., we get the energy-lowered hybrid IC and X_IC values which are evidently lower than that of unhybridized values. For example, for carbon we have the energy-lowered 2sp³ hybrid IC = 4.320 and X_IC = 2.167. In the unhybridized situation, IC would be as higher as 5.702 and all unhybridized conventional electronegativities are as higher as than 2.5 for carbon.

Pauling’s scale is estimated from the bond dissociation energies of two atoms, hydrogen and chlorine, and then arbitrarily extended to all elements not based on the quantitative configuration energy data [3].

Hydrogen has the lowest energy, E = −13.6 eV. Batsanov has an experimental covalent radius of 0.37 Å [30] equal to the Heitler-London’s half H-H value R (R = 0.74 Å) [26]. Based on these spectroscopic data and the IC model (Equation 2.1.5), we reach its IC value of 2.297 eV and its electronegativity, X_IC of 1.333 (Table 2 and Figure 1) that is not that conventionally high as 2.2.

The result is strongly supported by the point-charge distribution in hydrides (Table 3) proposed by Mo [33], which shows that in the typical ionic LiH (lithium monohydride) hydrogen gains point charge of 0.783, in the weak ionic BeH (beryllium monohydride) gains only 0.044, but in the covalent BH (boron monohydride) starts to loss point charge. That means that the ionocovalency and the electronegativity of hydrogen are smaller than that of boron. And moreover, the data of electric dipole moments for AlH (aluminum monohydride) (Table 4), proposed by the National Institute of Standards and Technology [34], shows that the aluminum end of the dipole is negative. That means that the ionocovalency and the electronegativity of hydrogen are smaller than that of aluminum.

Hydrogen has one valency orbital and a single electron. It can be an anion to form an ionic bond by gaining another electron and it can be a cation to form a covalent bond by sharing another electron. Therefore, the IC and the X_IC values of hydrogen happen to lie on the border between the weaker ionic beryllium and the weaker covalent boron (Chart 1 and Table 1). We can assign the IC value of hydrogen (2.297) as a standard to estimate the ionocovalent character of the cations. The cations with IC values smaller than that of hydrogen we call the ionic cations and those with IC values greater than that of hydrogen we call the covalent cations. The cations with IC values greater than that of beryllium (2.064) and smaller than that of boron (3.291) we might call borderline cations. The greater the IC than that of hydrogen, the more covalent and the less ionic the caton is, and vice versa.

Chart 2 and Chart 3 show that the IC and the X_IC scales of the top period rationalize an interesting empirical observation of a similar situation that exists for the pairs of elements. The first element in a given family of the periodic chart tends to resemble the second element in the family to the right as indicated below:

The reason for this relationship is that the pairs of element have approximately similar ionocovalency, IC = I(I_av)C(n*r_c⁻¹), due to the approximately similar ionic function I(I_av) (Chart 4) and covalent function C(n*r_c⁻¹) (Chart 5). Because of the electron configuration nature of the elements, the downwards vertical trend in decreasing covalency r_c⁻¹ is the opposite of the downwards vertical trend in increasing principal quantum number n*, and in this section of the Periodic Table the two opposing trends in C(n*r_c⁻¹) approximately cancel each other, resulting in similar values of ionicity, I(I_av).

There are some arguments about the values of electronegativities of carbon, sulfur, selenium, tellurium, iodine and hydrogen [22]. Chart 1 shows IC values in the order: S e 2 + ( 3.146 ) > S 2 + ( 3.121 ) > C 2 + ( 2.998 ) > T e 2 + ( 2.832 ) > I + ( 2.530 ) > H + ( 2.297 )

The results are consistent with the observations that hydrides H₂Se, H₂S, H₂C, H₂Te and HI form H₃O⁺ ions in water [35].

As Thomas reviewed, the electronegativity of carbon and sulfur in most of the scale are almost identical. The key point, however, so far as their role as poisons is concerned, is that they differ markedly in the distance at which they sit on the nickel overlayers [36]. The calculations for these locations show that sulfur is very much stronger than carbon as a poison.

The results are also consistent with the experiment data of the dipole moment, which indicates that the electron clouds on the C-S and C-I bond in the molecules CS₂ and CI₄ are close to the sulfur end and the iodine end, respectively [37]. From IC model data (Chart 4), we can see that S⁶⁺ has a greater ionicity than that of C⁴⁺: I_av (S⁶⁺ = 46.077, C⁴⁺ = 37.015), although they have the close spatial covalency, n*r_c⁻¹ (C⁴⁺ = 2.618, S⁶⁺ = 2.805) (Chart 5).

As Chart 1 and Table 1 show, the IC and X_IC run the same uneven trend that is expected to decrease down a group with the features in terms of As³⁺(IC = 3.364, X_IC = 1.773) and Bi³⁺(IC = 3.162, X_IC = 1.690) both having higher than expected values, comparing those of P³⁺(IC = 3.286, X_IC = 1.741) and Sb³⁺(IC = 3.036, X_IC = 1.638). This apparent anomaly derives from the filling of the 3d row prior to gallium and the 4f row prior to thallium, both of which lead to higher effective nuclear charges (Table 1) than the previous periods as a result of inefficient screening of the nuclear charge by the 3d and 4f electrons, respectively.

The IC values with a trend of decreasing at d⁵ and d¹⁰ agree well with the variation in values of E⁰ (M²⁺/M) [38] as a function of dⁿ configuration for the first row of transition metals; the d⁰ corresponds to M = Ca (Table 5).

The IC model, based on the VB approximation’s intuitive appeal and determined by covalent radius and ionization energy, is in accord with the relativistic effects with which contributions to the unusual chemistry of the heavier elements are two principal consequences. First, the s orbitals become more stable. Second, d and f orbitals expand and their energies are less.

For the inert pair effect in Tl(I), Pb(II), and Bi(III), the relativistic effects can give a qualitative verbalization: “The s orbitals of the heavier elements become more stable than otherwise expected” [39]. In the IC model, as Table 6 shows, the effect is attributable to the fact that the bond property in this case is controlled by the ionic function I(I_z, I_av). They are more stable in ionic compounds than in the entirely covalent form. Their IEs for forming higher covalent bonds are too much higher to form a stable hybridizing ionicity I_av:

According to the relativistic effects: “The s orbitals of the heavier elements become more stable than otherwise expected” [39], we can only give a qualitative overview on the color of gold and silver, but this has nothing to do with the color of copper.

In the IC model, the phenomenon of “the Color of Copper, Silver and Gold” is attributable to the fact that their bond structure is controlled by their ionocovalency dual properties and we can get a satisfactory explanation by the Dual method.

As Table 7 shows, with the increase of the contraction of the s orbital, the outer d orbitals expand and their ionicities I_av are decreased from Cu³⁺ via Ag³⁺ to Au³⁺, however, with the increase of their effective principle quantum number n* the covalency r_c⁻¹ decreases from Cu³⁺ to Ag³⁺ but increases from Ag³⁺ to Au³⁺, which causes the spatial covalency n*r_c⁻¹ of Cu³⁺ higher than that of Ag³⁺ but close to Au³⁺, leading to same trend in ionocovalency.

This quantitative trend in their structure and energy nicely reflect their character of color. Copper and gold are the only two elemental metals with a natural color other than gray or silver, which depends on their ionic delocalizing property of “electron sea” that is capable of absorbing and re-emitting photons over a wide range of frequencies. Copper in its liquefied state, a pure copper surface without ambient light, appears somewhat greenish, a characteristic shared with gold.

Villesuzanne et al. proposed the study: “New considerations on the role of covalency in ferroelectric niobates and tantalites” [23]. Here, covalency means the amount of mixing of oxygen 2p and metal d orbitals to form valence bands; it is evaluated quantitatively through the computation of the crystal orbital overlap population (COOP). The energies of Ta 5d and Nb 4d atomic orbitals are the same in EHTB parameters. The bond lengths are equal too, as found experimentally. The difference in COOP’s occurs because of larger radial extension of Ta 5d compared to Nb 4d orbitals, leading to a greater overlap with oxygen 2p orbitals. The fact that Ta⁵⁺-O bonds are more covalent than Nb⁵⁺-O bonds is due to a larger radial expansion of Ta 5d orbitals. This effect is not accounted for in Pauling electronegativity scales [3], which give information on the energy difference between valence orbitals, not on their spatial overlap. The arguments led to the opposite assumption of reference [24] concerning the covalency of Ta⁵⁺-O and Nb⁵⁺-O bonds from Pauling electronegativity X_p: Ta(1.5) < Nb(1.6).

In their later paper, they proposed that the explicit calculation of the electronic structure—COOP’s in particular—gives a larger covalency for Ta⁵⁺-O bonds than for Nb⁵⁺-O bonds. This result is retrieved in the Allred and Rochow scale [7] and in Zhang electronegativity scales for ions [9]. The results can be fairly well accounted in IC model: the energies of Ta 5d and Nb 4d atomic orbitals are the same in EHTB parameters due to having similar atomic ionicity I_av of 24.89 and 27.02, respectively. The bond lengths are equal due to having similar linear covalency r_c⁻¹ of 0.745 and 0.745, respectively. The big difference is the spatial covalency, n*r_c⁻¹, in I(I_av)C(n*r_c⁻¹) = n*(I_av/R)^½r_c⁻¹. The Ta 5d orbitals, compared to Nb 4d orbitals, involve greater spatial covalency, n*r_c⁻¹, (Ta⁵⁺ = 3.246, Nb⁵⁺ = 2.869), leading to a greater overlap with oxygen 2p orbitals and a greater IC: Ta⁵⁺ (4.393) > Nb⁵⁺ (4.043) and X_IC: Ta⁵⁺(2.197) > Nb⁵⁺(2.053).

As the IC model, n*(I_av/R)^½r_c⁻¹, is defined as ionocovalent density of the effective nuclear charges at covalent boundary, it is strongly related with the Mössbauer parameters δ and Δ [40,41]. The value of the isomer shift, δ, depends particularly on the density of s electrons at the nucleus. Therefore, in iron-57 an increase in electron density causes a negative isomer shift; since d electrons tend to shield the nucleus slightly from the s electrons, the value of δ falls as the number of d electrons in the iron atom falls. Mean values of δ [42], Z* and IC for some oxidation states of iron are shown in Table 8 and Figure 2.

Fajans suggested the rules to estimate the extent to which a cation could polarize an anion and thus induce covalent character. This Fajans phenomenon happens to be the IC-potential, the ionocovalency, the effective ionic potential (or the effective polarizing power), n*(I_av/R)^½r_c⁻¹.

The simple form of the ionic potential considered the valence charge of the ion with respect to its size. The valence charge is numerically equal to the number of valence electrons of the ion. In some cases we may consider the effective nuclear charge Z*. For two ions of the same actual nuclear charge, Hg²⁺ and Ca²⁺, the Hg²⁺ has the higher effective nuclear charge Z* (4.490) and the IC (3.118), it is considerably more polarizing and its compounds are considerably more covalent than those of Ca²⁺ which has the smaller effective nuclear charge Z* (2.807) and the IC (1.617). So we have their related melting points HgCl₂ = 276 and CaCl₂ = 772. Comparisons of more compounds are listed in Table 9 (below).

Table 9 shows that for the covalent bonding, the increased covalent bonding resulting from increasing the ionicity I_av, the σ⁻ covalency r_c⁻¹ or the spatial covalency n*r_c⁻¹ can lower the transition temperatures. The melting points decrease with increasing the covalency r_c⁻¹ and the spatial covalency n*r_c⁻¹.

However, for ionic bonding (see 4.5.), the ionic compounds are characterized by very strong IC potentials holding the ions together. Increasing the ionic function I(Z*, I_av) tends to increase the lattice energy of a crystal. For compounds which are predominantly ionic, increased ionic function I(Z*, I_av) or covalent function C(r_c⁻¹, n*r_c⁻¹) will result in increased melting points.

The IC model Equation 2.1.5 is correlated with the electrostatic energy of a cation in Born-Landé equation of the lattice energy (4.5.1) U = − Z 2 e 2 A N / 4 π ɛ 0 r ( 1 − n − 1 )

The both equations reveal how ionic bond strengths vary with the cation ionic charges and inversely with the distance between ions in the lattice. The IC gives a reasonable correlation to the lattice energy as shown in Table 10 and Table 11.

As we have described [8–10], the stability of a metal complex (the strength of metal-ligand bond) should be a function of the electron-attraction power of the metal. The IC value agrees fairly well with the lattice energy and the crystal field stabilization energy (CFES):

Table 11 shows that the IC values correlate with the lattice energies derived from Born-Haber cycle data for MCl₂ where M is a first row d-block metal; the point for d⁰ corresponds to CaCl₂. (Data are not available for scandium where the stable oxidation state is +3) [38].

Crystal field stabilization energy (CFES) causes to d level an effect upon thermodynamic stability of complex ions. Owing to the overall contraction in size on traversing a period from left to right, there is an increase in CFES from Ca²⁺ to Zn²⁺ and same trend occurred in IC and X_IC, For weak-field ligands (Table 12), the formation constants log β [38] follow the uneven trend and the IC and X_IC run in same way: Mn²⁺ < Fe²⁺< Co²⁺ < Ni²⁺ < Cu²⁺ < Zn²⁺.

In Table 13 an interpretation for an interesting comparison can be made between the predominantly ionic species CsF and BaF₂ and the more covalent species KBr and CaBr₂. For ionic species, the bond strengths are controlled by the ionic function I(I_av). Doubling the I_av from 3.890 to 7.605 in the highly ionic fluorides produces the expected increase in lattice energy and correspondingly doubles the transition temperatures (from 684 °C to 1,28 °C). For covalent bonding, the covalent function C(n*r_c⁻¹) is controlling factor. The little change in covalency r_c⁻¹: 0.513, 0.576 and spatial covalency n*r_c⁻¹: 1.769, 1.987 produces the expected little change in transition temperatures (from 730 °C to 765 °C), despite the doubling of I_av from 4.340 to 9.345.

Table 14 shows why electronegativity of sodium is higher than that of lithium. When we correlate IC or X_IC with lattice energies, there is the “inverted” sodium-lithium electronegativity [8,9,45]; that Li⁺ has unexpectedly low values of IC and X_IC. However, we can ask I(I_av, Z*) and C(n*, r_c⁻¹) to dialectically explain it: after 1st filling of p orbitals, Na⁺ reaches a much higher effective nuclear charge Z*(1.777) than that of Li⁺(1.253). The spatial covalency n*r_c⁻¹(Na⁺ = 1.838, Li⁺ = 1.624) does not cancel the higher effective nuclear charge Z* anymore. Dialectically, however, Li⁺ still has higher ionicity, I_av(5.390) and covalency r_c⁻¹(0.816) than that of Na⁺(I_av = 5.140, r_c⁻¹ = 0.636) although covalency is not so important in lattice energy.

Changzheng et al. [46] presented an effective synthetic protocol to produce high quality InN nanocrystals using indium iodide (InI₃). There has been a question: “Is it possible for high-quality InN to be synthesized from indium halides?” The positive answer has been found in their work using InI₃. Concerning the four kinds of indium halides, InF₃, InCl₃, InBr₃, and InI₃, InI₃ has the strongest covalent ability. As is known, when two atoms form a chemical bond, the greater the difference between the electronegativity values for the two atoms, the more ionic the chemical bond between them [8–10].

According to the IC model, in the effective polarizing power, n*(I_av/R)^½r_c⁻¹, both the effective principle quantum number, n*, and the covalent radius, r_c, for halogens are increased in the order: F < Cl < Br < I (Table 1). The polarizability of the anion will be related to its “softness”; that is, to the deformability of its electron cloud. Both increasing n* and r_c will cause this cloud to be less under the influence of the nuclear charge of the anion and more easily influenced by the charge on the cation. So concerning the four kinds of indium halides, InI₃ is more covalent than the other three. And it is possible for high-quality InN to be synthesized from indium iodide (InI₃).

Comparison of melting points for the anion pairs KF/KBr and CaCl₂/CaBr₂ from Table 9 can be treated in same way. KBr and CaBr₂ are more covalent and have lower melting points than KF and CaCl₂ respectively (see 4.4.).