Polarizability, Hardness, and Softness

Hardness increases with electronegativity and with positive charge. Thus, for the halogens the order is E CF Br F, and for second-row anions, E HO HjN- H3C. Eor cations, hardness decreases with size and increases with positive charge, so that H+ Li+ Na+ K+. The proton, lacking any electrons, is infinitely hard. In solution it does not exist as an independent entity but contributes to the hardness of some protonated species. Metal ion hardness increases with oxidation state as the electron cloud contracts with the removal of each successive electron. All these as 

The HSAB theory provides a useful precept for understanding Lewis acid-base interactions in that hard acids prefer hard bases and soft acids prefer soft bases. The principle can be applied to chemical equilibria in the form of the principle of maximum hardnesswhich states that molecules arrange themselves so as to 

Description of Molecular Structure Using Valence Bond Concepts 

The hard-hard interactions are dominated by electrostatic attraction, whereas soft-soft interactions are dominated by mutual polarization. Electronegativity and hardness determine the extent of electron transfer between two molecular fragments in a reaction. This can be approximated numerically by the expression 

Pearson, J. Am. Chem. Soc., 85, 3533 (1963) T. L. Ho, Hard and Soft Acids and Bases in Organic Chemistry, Academic Press, New York, 1977 W. B. Jensen, The Lewis Acid-Base Concept, Wiley-Interscience, New York, 1980, Chap. 8. 

Another property that is closely related to electronegativity and position in the periodic table is polarizability. Polarizability is related to the size of atoms and ions and the 

Qualitative relationship between orbital energies, hardness, and softness 

Numerical measures of polarizability analogous to electronegativity have been defined. Hardness t] is numerically expressed as 

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Some nucleophilic tendencies towards carbon are shown in Table 21. They vary by a factor of only ca. 10 within any one solvent, from the least reactive (2,4-dinitrophenoxide in MeOH) to the most reactive (e.g. CeHsS" in MeOH). If, as shown, solvation effects can produce changes of 10 in nucleophilic tendencies then it is clearly pointless, unless solvent is specified and its effect taken into account, to discuss rate data, for reactions in methanol and in other protic solvents, in terms of intrinsic properties of the nucleophile, such as structure, charge type, polarizability, hardness and softness, size, a-e fects, ability to adjust valence shells to transition state requirements, bond strength, and so on. Solvation of the nucleophile is a major factor in determining nucleophilic tendencies. 

Section 2.4 Polarizability Hard and Soft Acid-Base Theory... 

The concepts of electronegativity, polarizability, hardness, and softness are all interrelated. For the kind of qualitative applications we make in discussing reactivity, the concept that initial interactions between reacting molecules can be dominated by either partial electron transfer and bond formation (soft reactants) or by electrostatic interaction (hard reactants) is an important generalization. 

The qualitative ideas of valence bond (VB) theory provide a basis for understanding the relationships between structure and reactivity. Molecular orbital (MO) theory offers insight into the origin of the stability associated with delocalization and also the importance of symmetry. As a central premise of density functional theory (DFT) is that the electron density distribution determines molecular properties, there has be an effort to apply DFT to numerical evaluation of the qualitative concepts such as electronegativity, polarizability, hardness, and softness. The sections that follow explore the relationship of these concepts to the description of electron density provided by DFT. 

Bases of low polarizabiUty such as fluoride and the oxygen donors are termed hard bases. The corresponding class a cations are called hard acids the class b acids and the polarizable bases are termed soft acids and soft bases, respectively. The general rule that hard prefers hard and soft prefers soft prevails. A classification is given in Table 3. Whereas the divisions are arbitrary, the trends are important. Attempts to provide quantitative gradations of "hardness and softness" have appeared (14). Another generaUty is the usual increase in stabiUty constants for divalent 3t5 ions that occurs across the row of the Periodic Table through copper and then decreases for zinc (15). 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

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). 

Polarizabilities are responses to a potential (the gradient of which is a field). On the contrary, Fukui functions, chemical hardness and softness are responses to a transfer or removal of an integer number of electrons. Both responses are DFT descriptors but the responses which involve a change in the number of... 

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. 

The extent to which an atom or molecule s charge distribution is affected by an external electric field E (which could be due to an approaching reactant) is governed, to first order, by its polarizability a. It was really a to which Pearson was referring in his hard and soft acid-base theory, which rationalizes a large number of chemical reactions. The terms hard and soft refer, respectively, to low and high polarizability. 

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. 

A more refined but still debated in the literature notion is Pearson s Hard and Soft Acids and Bases (HSAB) principle [9,41], which quantifies energy changes to second order according to which hard (soft) acids (electron pair acceptors) prefer to interact with hard (soft) bases (electron pair donors). Soft likes soft relates to covalent bonds being facilitated by high polarizabilities, while hard likes hard relates to a creation of predominantly electrostatic interactions. 

The HSAB (hard and soft acids and base) principle is that hard acids prefer to interact with a hard base, and soft acids with soft bases. Hard bases are not polarizable, and inclnde those with 0-donor atoms. Soft bases are more polarizable, and inclnde S-donor bases. Solvent hardness/softness can be assessed by comparing the Gibbs free energy of transfer of a soft cation like Ag from hard water to the solvent with the Gibbs free energy of transfer of similarly sized hard cations like Na and K. Table 3.9 shows some solvents listed in increasing softness. ... 

The Lewis acid/base complex is formed via an overlap between a doubly occupied orbital of the donor D and vacant orbital of the acceptor A. This acid/base approach was extended by Pearson who divided Lewis acids and bases into two groups, hard and soft, according to their electronegativity and polarizability (principle of hard and soft acids and bases (HSAB concept). Hard acids (e.g., H, Lf, Na, BF3, AICI3, hydrogen-bond donors HX) and hard bases (e.g., F", CL, HO, RO, H2O, ROH, R2O,... 

S, I) are more polarizable (they are further away from the nucleus), and since larger soft species have, in general, lower solvation energies, they are better nucleophiles as compared to smaller hard species. (For definition of the term hard and soft nucleophiles, see Box 13.1.) Thus, we can qualitatively understand why, for example, nucleophilicity increases from F to Cl to Br to V (Table 13.3), and why HS- is a stronger nucleophile than OH". 

As noted above, the hardness or softness of an acidic or basic site is not an inherent property of the particular atom at that site, but can be influenced by the substituent atoms The addition of soft, polarizable substituents can soften an otherwise hard center and the presence of electron-withdrawing substituents can reduce the softness of a site. The acidic boron atom is borderline between hard and soft. Addition of three hard, electronegative fluorine atoms hardens the boron and makes it a hard Lewis acid. Conversely, addition of three soft, electropositive hydrogens54 softens the boron and makes it a soft Lewis acid. Examples of the difference in hardness of these two boron acids are... 

There is no obvious reason why oA should express Pearson s scale of softness. Nevertheless, it is evident that it succeeds much better than for instance the molar polarizabilities a given in Table 2. The border-line cases between hard and soft central atoms have aA around 3 eV, whereas typical hard behaviour is found when oA is below 2 eV. A mild criticism is that oA has a tendency to increase more with the oxidation number z than appropriate for the chemical softness, producing aA =... 

One of the most useful tools for predicting the outcome of chemical reactions is the principle of hard and soft acids and bases (HSAB), formulated by Pearson in 1963 [13-15]. This prindple states that hard acids will react preferentially with hard bases, and soft acids with soft bases, hard and soft referring to sparsely or highly polarizable reactants. A selection of hard and soft Lewis acids and bases is given in Table 1.1. 

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