Local softness

Although we have concentrated in this chapter on the derivatives of the energy and density, there are other chemically meaningful concepts that can be derived from the ones presented here 144 161. Among these, the chemical softness, the inverse of the chemical hardness, and the local softness [47,48] have proven to be quite useful to explain intermolecular reactivity trends. 

Fukui Function and Local Softness as Reactivity Descriptors... 

In order to understand the detailed reaction mechanism such as the regio-selectivity, apart from the global properties, local reactivity parameters are necessary for differentiating the reactive behavior of atoms forming a molecule. The Fukui function [10] if) and local softness [11] t.v) are two of the most commonly used local reactivity parameters. 

The Fukui function is primarily associated with the response of the density function of a system to a change in number of electrons (N) under the constraint of a constant external potential [v(r)]. To probe the more global reactivity, indicators in the grand canonical ensemble are often obtained by replacing derivatives with respect to N, by derivatives with respect to the chemical potential /x. As a consequence, in the grand canonical ensemble, the local softness sir) replaces the Fukui function/(r). Both quantities are thus mutually related and can be written as follows ... 

The most useful and important application of Fukui function and local softness resides in the interpretation and thereby, prediction of reaction mechanism, especially in the site selectivity or regioselectivity. Since long FMO theory has generally been used to probe the regioselective nature of a reaction, in particular of organic compounds, but the DFT-based local reactivity parameters have emerged as... 

Evaluation of the only appropriate Fukui function is required for investigating an intramolecular reaction, as local softness is merely scaling of Fukui function (as shown in Equation 12.7), and does not alter the intramolecular reactivity trend. For this type, one needs to evaluate the proper Fukui functions (/+ or / ) for the different potential sites of the substrate. For example, the Fukui function values for the C and O atoms of H2CO, shown above, predicts that O atom should be the preferred site for an electrophilic attack, whereas C atom will be open to a nucleophilic attack. Atomic Fukui function for electrophilic attack (fc ) for the ring carbon atoms has been used to study the directing ability of substituents in electrophilic substitution reaction of monosubstituted benzene [23]. In some cases, it was shown that relative electrophilicity (f+/f ) or nucleophilicity (/ /f+) indices provide better intramolecular reactivity trend [23]. For example, basicity of substituted anilines could be explained successfully using relative nucleophilicity index ( / /f 1) [23]. Note however that these parameters are not able to differentiate the preferred site of protonation in benzene derivatives, determined from the absolute proton affinities [24],... 

Perhaps the most successful application of Fukui function and local softness is in the elucidation of the region-selective behavior of different types of pericyclic reactions including the 1,3-dipolar cycloadditions (13DC), Diels-Alder reactions, etc. These reactions can be represented as shown in Scheme 12.4. Considering the concerted approach of the two reactants A and B, there are two possible modes of addition as shown in Pathway-I and Pathway-II. 

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. 

These indices have been used to study the reactivity for a series of chlorobenzenes and a good correlation is observed, for example, between W and toxicity of chlorobenzene [41]. For a detail discussion of this concept and its applications, we refer the readers to a recent review [41,42]. For studying intramolecular reactivity, these philicity indices and local softness contain the same information as obtained from the Fukui functions, because they simply scale the Fukui functions. In some cases the relative electrophilicity and relative nucleophilicity may be used although they provide similar trends as s(r) and co(r) in most cases [43]. In the same vein, the spin-donicity and spin-philicity, which refer to the philicity of open-shell systems [44], could also be utilized to unravel the reactivity of high-spin species, such as the carbenes, nitrenes, and phosphinidenes [45]. 

Notice that from global softness, and hardness when chemical potential is given. The corresponding condensed-to-atom variants may be defined as... 

All of these regioselectivity indicators are called Fukui functions, in honor of Kenichi Fukui, who pioneered the analogous frontier orbital reactivity descriptors in the early 1950s [12-14]. The Fukui function and its twin, the local softness [15]. 

This chapter aims to present the fundamental formal and exact relations between polarizabilities and other DFT descriptors and is organized as follows. For pedagogical reasons, we present first the polarizability responses for simple models in Section 24.2. In particular, we introduce a new concept the dipole atomic hardnesses (Equation 24.20). The relationship between polarizability and chemical reactivity is described in Section 24.3. In this section, we clarify the relationship between the different Fukui functions and the polarizabilities, we introduce new concepts as, for instance, the polarization Fukui function, and the interacting Fukui function and their corresponding hardnesses. The formulation of the local softness for a fragment in a molecule and its relation to polarization is also reviewed in detail. Generalization of the polarizability and chemical responses to an arbitrary perturbation order is summarized in Section 24.4. 

The different Fukui functions are defined on the entire space and are normalized. Therefore, their average values decrease with the system size. In order to compare the properties of a molecular group in molecules of different sizes, one must compare instead, the corresponding local softnesses. For each Fukui function, there exists a local softness, v(r) defined simply by [13]... 

Global Softness and Local Softness (Side Chains) of Amino Acids Computed at MP2/6-311G(d,p) in Ref. [18]... 

Equation 24.104 can be used to compute ab initio the similarity and differences between the members of a molecular family. In the case of amino acids, for instance, [18], Sf is chosen as a constant fragment (the amino-carboxyl part) and S2 corresponds to the local softness of the side chain. In Table 24.1, we report the average value of local softness (Si) computed for each group of amino acids local softness only permits to clearly distinguish between the different types of amino acids which have all nearly the same global softness. Calculations of S and S2 for each of the twenty amino acids permits to prove the linear relationship (Equation 24.104) and deduces the values of Sf and <2°(1) [18]. 

Dispersion Interactions From Atoms-in-Molecules Polarizability to Local Softness.410... 

Hydrogen Bonding The Role of (Local) Softness in (Very) Strong Hydrogen Bonding... 

In Figure 27.3, the relationship between the hydrogen bond energy and the product of the local softness values of donor and acceptor atom (A and B) is given for a series of closely related O—H O types of bonds (NR2, N02 family, etc.). 

FIGURE 27.3 (a) / iiii (kcal/mol) vs. local softness of acceptor for NR2 family. (With... 

FIGURE 27.3 (continued) (c) Combined effect of the local softness values of the acceptor and donor atoms on hydrogen bonding energy. (Reprinted from Ozen, A.S., Aviyente, V., De Proft, F., and Geerlings, P., J. Phys. Chem., A110, 5860, 2006. With permission.)... 

Role of DFT Descriptors in the (Evaluation of) Dispersion Interaction From Local Polarizability to Local Softness... 

Both correlations indicate that polarizability at local level can be used to quantify the dispersion energy. In the next section, we take a further step within direction, moving to an atoms-in-molecules level, thereby linking local polarizability to local softness. 

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