Polarizability and hardness

POLARIZABILITY AND HARD AND SOFT ACID-BASE THEORY... 

Pearson and Edwards noted that some substrates in Sn2 reactions seem more susceptible to the Bronsted basicity of a nucleophile, while other substrates seem more susceptible to its polarizability. Pearson proposed that bases be divided into two categories soft (polarizable) and hard (nonpolarizable). On the basis of equilibrium data, Pearson concluded that H, Li, Na, Mg, and Ca are hard acids, while Cu, Ag, Hg, I, Br, I2, and Br2 are soft. Pearson noted that hard acids show greater association with hard bases, while soft acids prefer soft bases. One explanation for this behavior is the ionic/covalent bond description, in which it is assumed that hard acids and hard bases are attracted primarily through ionic interactions, while soft acids and soft bases associate through covalent bonds. In addition, the two... 

The proton is the most typical class (a) ion. Pearson termed Lewis acids of class (a) character as hard acids and those of class (b) character as soft acids. In classifying Lewis acids the criterion of Ahrland, Chatt and Davies o as used whenever possible, but other rules were introduced. Soft acids are expected to combine with soft bases and hard acids prefer to combine with hard acids. Soft bases are polarizable (high polarizability) and hard bases are non-polarizable. 

Pearson s classification of ligands and metals as soft acids and bases can readily be applied to metal 71-complexes. Here the term soft means polarizable and hard indicates nonpolarizable. According to the general guideline,... 

Sicilia, E., Russo, N., and Mineva, T. 2001. Correlation between energy, polarizability, and hardness profiles in the isomerization reaction of HNO and CINO. J. Phys. Chem. A 105 442--f50. 

The i5p-titanium(IV) atom is hard, ie, not very polarizable, and can be expected to form its most stable complexes with hard ligands, eg, fluoride, chloride, oxygen, and nitrogen. Soft or relatively polarizable ligands containing second- and third-row elements or multiple bonds should give less stable complexes. The stabihty depends on the coordination number of titanium, on whether the ligand is mono- or polydentate, and on the mechanism of the reaction used to measure stabihty. 

Hard Bases. The donor atoms are of high electronegativity and low polarizability and are hard to oxidize. They hold their valence electrons tightly. 

Hard Acids. The acceptor atoms are small, have high positive charge, and do not contain unshared pairs in their valence shells. They have low polarizability and high electronegativity. 

The first task is to decide whether the members of a given group are Lewis acids or bases. Then evaluate the relative softness and hardness based on polarizability, taking into account correlations with electronegativity, size, and charge. Refer to the periodic table in assessing the trends. 

Pearson (1966) defines a soft base as one in which the donor atom is of high polarizability and low electronegativity and is easily oxidized or associated with empty, low-lying orbitals . A hard base has opposite properties. The donor atom is of low polarizability and high electronegativity, is hard to reduce, and is associated with empty orbitals of high energy. ... 

Another feature of the metal ions that are typically involved in cementitious bonding in AB cements is that most of them fall into the category of hard in Pearson s Hard and Soft Acids and Bases scheme (Pearson, 1963). The underlying principle of this classification is that bases may be divided into two categories, namely those that are polarizable or soft, and those that are non-polarizable or hard. Lewis acids too may be essentially divided into hard and soft, depending on polarizability. From these classifications emerges the useful generalization that hard acids prefer to associate with hayd bases and soft acids prefer to associate with soft bases (see Section 2.3.7). 

Hard acids (HA) have low polarizability and small dimensions, higher oxidation numbers and the hardness increases with increasing oxidation number they bind bases primarily through ionic bonds. Typical examples are H+, Na+, Hg2+, Ca2+, Sn2+, V02+, V022+, (CH3)2Sn2+, A1(CH3)3,I7+, I5+, Cl7+, co2, R3C+, so3. 

Hard bases (HB) have low polarizability and strong Br0nsted basicity in the V, VI and VII groups of the periodic table, the first atom is always the hardest and the hardness decreases with increasing atomic number in the group. Typical examples are H20, OH-, F-, O2-, CH3COO, S042-, Cl-,... 

The polarizable fluctuating charge model in CHARMM results from the work of Patel, Brooks and co-workers [92, 214], The water model is based on the TIP4P-FQ model of Rick, Stuart and Berne [17], In the development of the force field the electronegativities and hardnesses were treated as empirical parameters and do not have any association with experimental or QM values, for example, from ionization energies and electron affinities of single atoms. 

Since hardness and the shear modulus are usually proportional, the factors that determine the shear moduli need to be understood. The shear moduli are functions of the local polarizability and this depends on the valence electron density, as well as the energy needed to promote a valence electron to its first excited state. The latter depends on the strength of the chemical bond between two atoms. This will be discussed in more detail in Chapter 3. 

The connection between hardness and a measure of the ease with which a crystal can change its shape is its reciprocal polarizability (as shown above). The softer the crystal, the greater its polarizability. The hardnesses have the dimensions of pressure, and the polarizabilities are derived from the dielectric constants through the Clausius-Mosottii equation. That is ... 

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. 

General properties and definitions of polarizabilities can be introduced without invoking the complete DFT formalism by considering first an elementary model the dipole of an isolated, spherical atom induced by a uniform electric field. The variation of the electronic density is represented by a simple scalar the induced atomic dipole moment. This coarse-grained (CG) model of the electronic density permits to derive a useful explicit energy functional where the functional derivatives are formulated in terms of polarizabilities and dipole hardnesses. 

The concept of dipole hardness permit to explore the relation between polarizability and reactivity from first principles. The physical idea is that an atom is more reactive if it is less stable relative to a perturbation (here the external electric field). The atomic stability is measured by the amount of energy we need to induce a dipole. For very small dipoles, this energy is quadratic (first term in Equation 24.19). There is no linear term in Equation 24.19 because the energy is minimum relative to the dipole in the ground state (variational principle). The curvature hi of E(p) is a first measure of the stability and is equal exactly to the inverse of the polarizability. Within the quadratic approximation of E(p), one deduces that a low polarizable atom is expected to be more stable or less reactive as it does in practice. But if the dipole is larger, it might be useful to consider the next perturbation order ... 

As we have seen, an atom under pressure changes its electron structure drastically and consequently, its chemical reactivity is also modified. In this direction we can use the significant chemical concepts such as the electronegativity and hardness, which have foundations in the density functional theory [9]. The intuition tells us that the polarizability of an atom must be reduced when it is confined, because the electron density has less possibility to be extended. Furthermore, it is known that the polarizability is related directly with the softness of a system [14], Thus, we expect atoms to be harder than usual when they are confined by rigid walls. Estimates of the electronegativity, x and die hardness, tj, can be obtained from [9]... 

For instance, in structure 12-e, the C-X and C-0 dipole moments are additive, leading to a destabilization of the molecule by increasing the energy. In structure 12-a, offset of the C-X and C-0 dipole moments minimizes electrostatic interactions, thus leading to a more stable conformation. This electrostatic model was supported by the observed increase of the percentage of the equatorial conformation of 2-methoxy tetrahydropyran (14) when moving from a non-polar to a polar solvent (Table 3).12 In this model, the polar groups are not polarizable and lead to dipole/dipole (hard/hard) interactions. 

Acceptors may be considered as either hard"or soft". Hard acceptors, such as the proton or alkali metal ions are hardly polarizable and tend to react preferentially with light donor atoms14, ls ... 

According to the hard and soft acids and bases (HSAB) principle, developed by Pearson in 1963232,233, Lewis acids and Lewis bases are divided into two groups hard and soft. Pearson correlated the hardness of acids and bases with their polarizability, whereby soft acids and bases are large and easily polarizable, and vice versa. A selected list of Lewis acids ordered according to their hardness in aqueous solution is presented in Table 18. The HSAB principle predicts strong association of like partners. Hard acid-soft base complexes mainly result from electrostatic interactions, while soft acid-soft base complexes are dominated by covalent interactions. 

Cadmium(II) and zinc(II) systems other than cyanides Among the i acceptors of the zinc group, the softness rapidly decreases from the markedly soft Hg2+ to the mildly soft Cd2+ and to the distinctly hard Zn2+. As mentioned above, only very soft ligands such as CN are coordinated to Cd2+ or Zn + by bonds which are essentially covalent. Nevertheless, covalent bonding is still important for the formation of the Cd2+ halide complexes. This is evident from the fact that the values of AHn become more exothermic as the halide becomes larger and consequently more polarizable and susceptible to covalent bonding. This trend results in the (6) or soft sequence for the halide systems of... 

Later on, Pearson [75] introduced the concept of hard and soft acid and bases (HSABs) hard acids (defined as small-sized, highly positively charged, and not easily polarizable electron acceptor) prefer to associate with hard bases (i.e., substances that hold their electrons tightly as a consequence of large electronegativities, low polarizabilities, and difficnlty of oxidation of their donor atoms) and soft acids prefer to associate with soft bases, giving thermodynamically more stable complexes. According to this theory, the proton is a hard acid, whereas metal cations may have different hardnesses. 

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