Chemical hardness defined

The electronic chemical hardness, defined in eqn (23), gives a measure of the chemical reactivity of a given compound. Also the chemical potential,... 

Physical hardness can be defined to be proportional, and sometimes equal, to the chemical hardness (Parr and Yang, 1989). The relationship between the two types of hardness depends on the type of chemical bonding. For simple metals, where the bonding is nonlocal, the bulk modulus is proportional to the chemical hardness density. The same is true for non-local ionic bonding. However, for covalent crystals, where the bonding is local, the bulk moduli may be less appropriate measures of stability than the octahedral shear moduli. In this case, it is also found that the indentation hardness—and therefore the Mohs scratch hardness—are monotonic functions of the chemical hardness density. 

Figure 16.1 The chemical hardness of an atom, molecule, or ion is defined to be half. The value of the energy gap between the bonding orbitals (HOMO—highest orbitals occupied by electrons), and the anti-bonding orbitals (LUMO—lowest orbitals unoccupied by electrons). The zero level is the vacumn level, so I is the ionization energy, and A is the electron affinity, (a) For hard molecules the gap is large (b) it is small for soft molecules. The solid circles represent valence electrons. Adapted from Atkins (1991).

We discussed mainly some of the possible applications of Fukui function and local softness in this chapter, and described some practical protocols one needs to follow when applying these parameters to a particular problem. We have avoided the deeper but related discussion about the theoretical development for DFT-based descriptors in recent years. Fukui function and chemical hardness can rigorously be defined through the fundamental variational principle of DFT [37,38]. In this section, we wish to briefly mention some related reactivity concepts, known as electrophilicity index (W), spin-philicity, and spin-donicity. 

FIGURE 21.2 Profiles of (a) dipole moment (in D), (b) chemical hardness (in kcal/mol), and (c) CO and CS bond electronic populations for the reaction shown in Equation 21.9. Vertical dashed lines indicate the limits of the reaction regions defined in the text. 

In a second approach of the reactivity, one fragment A is represented by its electronic density and the other, B, by some reactivity probe of A. In the usual approach, which permits to define chemical hardness, softness, Fukui functions, etc., the probe is simply a change in the total number of electrons of A. [5,6,8] More realistic probes are an electrostatic potential cf>, a pseudopotential (as in Equation 24.102), or an electric field E. For instance, let us consider a homogeneous electric field E applied to a fragment A. How does this field modify the intermolecular forces in A Again, the Hellman-Feynman theorem [22,23] tells us that for an instantaneous nuclear configuration, the force on each atom changes by... 

As mentioned in [Section 24.1], and as already demonstrated in Equation 24.39, the Fukui functions as well as the chemical hardness of an isolated system can be properly defined without invoking any change in its electron number. We define a new Fukui function called polarization Fukui function, which very much resembles the original formulation of the Fukui function but with a different physical interpretation. Because of space limitation, only a brief presentation is given here. More details will appear in a forthcoming work [33]. One assumes a potential variation <5wext(r), which induces a deformation of the density 8p(r). A normalized polarization Fukui function is defined by... 

Although we will not discuss the finite difference approximations E1 (r) further in this section, it is useful to note that these functions can be defined also as potential derivatives of a gap and a chemical potential. The gap is the usual definition of the chemical hardness [8,9],... 

Chemical hardness and softness are much newer ideas than electronegativity, and they were quantified only fairly recently. Parr and Pearson (1983) proposed to identify the curvature (i.e. the second derivative) of the E versus N graph (e.g. Fig. 7.10) with hardness, rj [151]. This accords with the qualitative idea of hardness as resistance to deformation, which itself accommodates the concept of a hard molecule as resisting polarization - not being readily deformed in an electric field if we choose to define hardness as the curvature of the E versus N graph, then... 

The HSAB principle can be considered as a condensed statement of a very large amount of experimental information, but cannot be labelled a law, since a quantitative definition of the intuitive concepts of chemical hardness (T ) and softness (S) was lacking. This problem was solved when the hardness found an exact, and also an operational, definition in the framework of the Density Functional Theory (DFT) by Parr and co-workers [2], In this context, the hardness is defined as the second order derivative of energy with respect to the number of electrons and has the meaning of resistance to change in the number of electrons. The softness is the inverse of the hardness [3]. Moreover, these quantities are defined in their local version [4, 5] as response functions [6] and have found a wide application in the chemical reactivity theory [7],... 

The second derivative of the energy with respect to the number of electrons is called absolute hardness t] (or chemical hardness) [Parr and Pearson, 1983], which for a molecule with Nd electrons is defined as ... 

Starting from (2), it is possible to define a local (regional) counterpart for the co quantity as follows use the inverse relationship between chemical hardness and the global softness S = I/1722 and the additivity rule for S, namely S = to rewrite (2). 

Based on the above, it is not surprising that the structure gap has received so much attention it is a reality. One major reason that the structure gap has proved so hard to bridge lies in the difficulty in preparing structurally and chemically well-defined model catalysts that can be used to elucidate the above and related questions. A second reason is the increased complexity of possible surface reactions when more elaborate surface geometries are considered. 

There are still more reasons to believe that rj, as defined in Equation (2.12), is indeed what is meant by chemical hardness. To understand this, it is necessary to see whether the chemical concepts derived by DFT are compatible with molecular orbital (MO) theory.This theory is certainly the most widely used by chemists and is very successful in many areas. It is almost universally applied to explain structure and bonding, visible-UV spectra, chemical reactivity and detailed mechanisms of chemical reactions. 

Apart from the radical cases, it would appear that Figure 2.2 offers a most graphic and concise way of defining what is meant by chemical hardness ... 

Both derivatives appearing on the right hand side of Eq. (4) are chemically significant. The chemical hardness (q) is defined [6J as... 

The necessity for a quantitative definition for chemical hardness was addressed by Parr and Pearson [6], who defined the hardness as... 

Define electronegativity and chemical hardness of atoms and give approximations of them using the optical properties of the atoms. 

Chemical hardness represents the resistance of the system to exchange electrons with a specified or unspecified environment. Chemical softness S is simply defined as the inverse of chemical hardness S = l/rf. 

The quantity rj N is the equivalent of the chemical hardness rj in spin-free conceptual DFT, except for the fact that the derivative has an extra Ns constraint it has been defined as the smallest gap between the frontier Kohn-Sham orbitals, rjss can be termed the spin-hardness. The hardnesses in the [No[,Np,Voi,vp representation are given by refs. 66 and 69... 

Once defined the global measure characterizing molecular systems, i.e., the absolute electronegativity, there is realized that the chemical hardness is not constant over the entire molecule. Therefore, also the local sizes can be defined, which have been called as Fukui functions by Parr and Yang (1984) ... 

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