Models of Chemical Bonding and “Empirical” Methods

Abstract

This chapter defines the different atomic radii and their use to predict a bond length. The valence-shell electron-pair repulsion (VSEPR) model and the ligand close-packing (LCP) model are reviewed. Finally, different empirical correlations are reported.

8.6 Appendix: Electronegativity (χ)

8.1.1 8.6.1 Introduction

Electronegativity is a measure of the relative ability of an atom in a molecule to attract electrons to itself. It is generally a dimensionless parameter. It is an essential property of the atoms in a molecule, and several correlations have been shown between electronegativity and molecular properties; see Sect. 8.5.3.1. In particular, electronegativity helps characterize the bonding between the atoms, e.g., the bond polarity. The difficulty is that electronegativity is defined artificially using an arbitrary scale, and there are several different definitions.

8.1.2 8.6.2 Pauling Scale (Pauling 1932, 1960)

Pauling assumed that if two diatomic homonuclear molecules AA and BB interact to form diatomic heteronuclear molecules AB, the bond energy of AB should be the average of the two homonuclear bond energies of AA and BB, provided that the electrons are shared evenly. However, the observed heteronuclear bond energy is always found larger than the average. Pauling attributed this increase to the difference of electronegativity of the two atoms. He originally defined the difference in electronegativity between the atoms A and B as

$$\left| {\chi_{\text{A}} - \chi_{\text{B}} } \right| = \frac{1}{{\sqrt {\text{eV}} }}\sqrt {E({\text{AB}}) - \frac{{E({\text{AA}}) + E({\text{BB}})}}{2}}$$

(8.20)

where the dissociation energies E are expressed in eV.

Latter, he replaced the arithmetic mean by a geometric mean, which gives better results. The new definition is

$$\left| {\chi_{\text{A}} - \chi_{\text{B}} } \right| = 0.102\sqrt {E({\text{AB}}) - \sqrt {E({\text{AA}}) \times E({\text{BB}})} }$$

(8.21)

the dissociation energies E being expressed in kJ mol⁻¹.

This equation only gives differences; an origin is needed. It was assumed that the most electronegative element, fluorine, had the value 3.98.

8.1.3 8.6.3 Allred-Rochow Scale (Allred and Rochow 1958)

The electronegativity is evaluated from the Coulombic force of attraction between the effective nuclear charge and that of an outer electron. Its expression is

$$\chi = 3590\frac{{Z_{\text{eff}} }}{{r_{\text{cov}}^{2} }} + 0.744$$

(8.22)

The effective nuclear charge, Z_eff, can be estimated using Slater’s rules and r_cov is the covalent radius expressed in pm.

8.1.4 8.6.4 Other Scales

Mulliken (1934) defined the electronegativity as the average of the energy required to remove an electron from an atom, i.e., the ionization energy E_i, and the energy released by gain of one electron, i.e., the electron affinity E_ea. Its expression is

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

(8.23)

This electronegativity has the same unit as E_i + E_ca, usually eV. However, it is usual to transform these absolute values into values comparable to the Pauling values,

$$\chi = 1.97 \times 10^{ - 3} \left( {E_{i} + E_{\text{ca}} } \right) + 0.19$$

(8.24)

where the energies are expressed in expressed in kJ mol⁻¹.

Sanderson (1983a, b) has proposed a method of calculation based on the reciprocal of the atomic volume.

Some data for the different scales are given in Table 8.9.

Table 8.9 Table of electronegativities^a

8.1.5 8.6.5 Group Electronegativity

It is also possible to associate an electronegativity to a functional group (such as OH and CH₃). The group electronegativities are derived either by experimental methods or computational methods. The problem is that the different methods give values that are not always compatible. For a review, see Bratsch (1985).

8.1.6 8.6.6 Electronegativity Equalization (Sanderson 1983a, b)

In the following, it is assumed that Sanderson’s electronegativiy scale is used. It is given in Table 8.9.

The electronegativity is assumed to be a property of the atom. Actually, it depends on the atomic structure when atoms combine. Consider a diatomic molecule AB. When the bond is formed, the initially more electronegative atom acquires more than half share of the bonding electrons, i.e., it gains a partial negative charge, whereas the other atom gets a partial positive charge. The effect of the partial negative charge is to diminish the effective nuclear charge, to increase the atomic radius, and to diminish the electronegativity. For the atom with a partial positive charge, the contrary happens. The consequence of these electron transfers is that the electronegativities are equalized throughout the molecule, and the electronegativity of the bonded atoms, χ_M is the geometric mean of electronegativities χ_I of all component atoms

$$\chi_{\text{M}} = \left( {\prod\limits_{i = 1}^{n} {\chi_{i} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}}}$$

(8.25)

where n is the number of atoms of the molecule.

8.1.7 8.6.7 Partial Atomic Charge

A partial charge is a non-integer charge value on an atom (in elementary charge unit) due to the asymmetric distribution of electrons in chemical bonds. They are used in molecular mechanics to compute the electrostatic interaction energy (see (2.47) in Sect. 2.16.2). They are also useful for a qualitative understanding of the structure: It is the goal of this chapter. Finally, because chemical reactions often occur by attack on some reagent on the more positive or more negative site in a molecule, it is interesting to have reliable predictions of atom charges. The difficulty is that assigning charges to individual atoms is arbitrary and various methods have been proposed. Among the orbital-based charges, there is the AIM method (see Sect. 2.18) and the natural bond orbital (NBO) charges (Reed et al. 1985). A more recent method, charge model 5 (CM5), is described by Marenich et al. (2012).

The electronegativity equalization, (7.5), permits to estimate the partial charge δ_i on atom i

$$\delta_{i} = \frac{{\chi_{\text{M}} - \chi_{i} }}{{\Delta \chi_{i} }}$$

(8.26)

∆χ_i is the charge that the atom i would have undergone if it had acquired a unit charge. It may be estimated with the following equation

$$\Delta \chi_{i} = 1.57\sqrt {\chi_{i} }$$

(8.27)

The weak point of this method is that it assigns the same partial charge to each atom of the same kind. For instance, in CH₃OH, all the H have the same charge.

Fortunately, Rappé and Goddard (1991) proposed a more sophisticated version. The charge of an atom A may be written

$$E_{\text{A}} (Q) = E_{{{\text{A}}0}} + \left( {\frac{\partial E}{\partial Q}} \right)_{{{\text{A}}0}} Q_{\text{A}} + \frac{1}{2}\left( {\frac{{\partial^{2} E}}{{\partial Q^{2} }}} \right)_{{{\text{A}}0}} Q_{\text{A}}^{2} + \cdots$$

(8.28)

For the neutral atom (Q_A = 0): E_A(0) = E_A0

Limiting the develoment to second order gives for the cation (Q_A = + 1)

$$E_{\text{A}} ( + 1) = E_{{{\text{A}}0}} + \left( {\frac{\partial E}{\partial Q}} \right)_{{{\text{A}}0}} + \frac{1}{2}\left( {\frac{{\partial^{2} E}}{{\partial Q^{2} }}} \right)_{{{\text{A}}0}} = E_{\text{i}}$$

(8.29)

and for the anion (Q_A = −1)

$$E_{\text{A}} ( - 1) = E_{{{\text{A}}0}} - \left( {\frac{\partial E}{\partial Q}} \right)_{{{\text{A}}0}} + \frac{1}{2}\left( {\frac{{\partial^{2} E}}{{\partial Q^{2} }}} \right)_{{{\text{A}}0}} = - E_{\text{ca}}$$

(8.30)

Combining (8.10) and (8.11) gives

$$\left( {\frac{\partial E}{\partial Q}} \right)_{{{\text{A}}0}} = \frac{1}{2}\left( {E_{i} + E_{\text{ca}} } \right) = \chi_{\text{A}}^{0}$$

(8.31)

where χ_A is the electronegativity as defined by Mulliken, (8.4) and

$$\left( {\frac{{\partial^{2} E}}{{\partial Q^{2} }}} \right)_{{{\text{A}}0}} = E_{i} - E_{\text{ca}} = J_{\text{AA}}^{0} = 2\eta_{\text{A}}^{ 0}$$

(8.32)

J_AA is called idempotential and η_A is the atomic hardness. Using (8.12) and (8.13), (8.9) may be rewritten

$$E_{\text{A}} (Q) = E_{{{\text{A}}0}} + \chi_{\text{A}}^{O} Q_{\text{A}} + \frac{1}{2}J_{\text{AA}}^{0} Q_{\text{A}}^{2}$$

(8.33)

$\chi_{\text{A}}^{0}$ and $J_{\text{AA}}^{0}$ can be derived from atomic data, see Table 1 of Rappé and Goddard (1991). To calculate the charge distribution, it is necessary to evaluate the interatomic electrostatic energy, $\sum\limits_{{\text{A}} < {\text{B}}} {Q_{\text{A}} Q_{\text{B}} } J_{\text{AB}}$ where J_AB is the Coulomb interaction which is inversely proportional to R_AB, the distance between A and B. The total electrostatic energy is

$$E(Q_{1} \ldots Q_{N} ) = \sum\limits_{\text{A}} {\left( {E_{{{\text{A}}0}} + \chi_{\text{A}}^{0} Q_{\text{A}} + \frac{1}{2}J_{\text{AA}}^{0} Q_{\text{A}}^{2} } \right)} + \sum\limits_{{\text{A}} < {\text{B}}} {J_{\text{AB}} Q_{\text{A}} Q_{\text{B}} }$$

(8.34)

where N is the number of atoms.

Deriving with respect to Q_A gives

$$\chi_{\text{A}} (Q_{1} \ldots Q_{N} ) = \frac{\partial E}{{\partial Q_{\text{A}} }} = \chi_{\text{A}}^{0} + \sum\limits_{\text{B}} {J_{\text{AB}} } Q_{\text{B}}$$

(8.35)

At equilibrium, we should have

$$\chi_{ 1} = \chi_{ 2} = \cdots = \chi_{N}$$

(8.36)

Adding the condition of total charge

$$Q_{\text{tot}} = \sum\limits_{i = 1}^{N} {Q_{i} }$$

(8.37)

gives a system of N simultaneous equations that can be solved to obtain the charges.

There is another easy way to calculate reasonable partial charges using a method developed by Allen (1989).

$$\begin{aligned} Q_{\text{A}} & = {\text{group}}\,{\text{number}}\,{\text{of}}\,A - {\text{number}}\,{\text{of}}\,{\text{lone}}\,{\text{pair}}\,{\text{electrons}}\,{\text{on}}\,A \\ & \quad - (\chi_{\text{A}} /\sum \chi ) \times \left( {{\text{number}}\,{\text{of}}\,{\text{bonding}}\,{\text{electrons}}\,{\text{shared}}\,{\text{by}}\,A} \right) \\ \end{aligned}$$

(8.38)

where ∑χ is the sum of the electronegativities of atom A and the atoms to which A is bonded. In this equation, the choice of the electronegativity scale (either Pauling or Allred-Rochow) is not important).

Table 8.10 gives the partial atomic charges for formaldehyde. The agreement between the different methods is only qualitative.

8.10 Partial atomic charges for formaldehyde, H₂C=O, using different methods