Hardness qualitative concepts

DFT turns out to be well suited to quantitative expression of some of the qualitative concepts introduced in Section 1.2, such as electronegativity, hardness, and softness. The pnnciple of maximum hardness (p. 22) can be denved as a consequence of DFT, as can the concepts of hardness and softness. ... 

Hardness and softness as chemical concepts were presaged in the literature as early as 1952, in a paper by Mulliken [138], but did not become widely used till they were popularized by Pearson in 1963 [139]. In the simplest terms, the hardness of a species, atom, ion or molecule, is a qualitative indication of how polarizable it is, i.e. how much its electron cloud is distorted in an electric field. The adjectives hard and soft were said to have been suggested by D.H. Busch [140], but they appear in Mulliken s paper [138], p. 819, where they characterize the response to spatial separation of the energy of acid-base complexes. The analogy with the conventional use of these words to denote resistance to deformation by mechanical force is clear, and independent extension, by more than one chemist, to the concept of electronic resistance, is no surprise. The hard/soft concept proved useful, particularly in rationalizing acid-base chemistry [141]. Thus a proton, which cannot be distorted in an electric field since it has no electron cloud (we ignore the possibility of nuclear distortion) is a very hard acid, and tends to react with hard bases. Examples of soft bases are those in which sulfur electron pairs provide the basicity, since sulfur is a big fluffy atom, and such bases tend to react with soft acids. Perhaps because it was originally qualitative, the hard-soft acid-base (HSAB) idea met with skepticism from at least one quarter Dewar (of semiempirical fame) dismissed it as a mystical distinction between different kinds of acids and bases [142]. For a brief review of Pearson s contributions to the concept, which has been extended beyond strict conventional acid-base reactions, see [143],... 

Although the qualitative concepts such as electronegativity and hardness have been found to be useful in understanding various chemical reactions, they were not taken very seriously until recently because they did not have legitimate theoretical genesis. Rigorous quantitative definitions and methods for calculations [36-38] of electronegativity, hardness, and related quantities such as chemical potential, local hardness, softness, Fukui function, etc., have been provided within density functional... 

The qualitative concepts of hardness and softness were first introduced by Pearson [30-32,34], which later culminated in enunciation of the famous hard-soft acid-base principle. Quantification of these concepts had been in order and was accomplished within density functional theory by Parr and Pearson [52]. The energy stabilization due to soft-soft interaction can be expressed by rearranging Eq. (61) as [52] ... 

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. 

Thus, for a nomal charge-transfer interaction between a donor-acceptor couple like a Lewis acid-base pair, the chemical hardness (rj) of a given species signifies the measure of its reluctance towards further electron shift to the other species, thereby upholding its qualitative concept of compactness. 

Accurate equations of state for hydrocarbons are of considerable interest to the petroleum and natural gas industry, and this has fueled active research in this area. Early equations of state have used lattice or cell model descriptions. Although some of these approaches are in good agreement with experimental data, they contain adjustable parameters that cannot be determined a priori and the physical insights are clouded by several qualitative concepts such as free volume. The development of molecularly based equations of state has focused on predicting the volumetric properties of hard chain fluids. The reason for this is that the repulsive part of the potential is expected to dominate liquid structure and the effect of attractions can normally be treated as a perturbation as has been done successfully for simple liquids. 

Appealing and important as this concept of a molecule consisting of partially charged atoms has been for many decades for explaining chemical reactivity and discussing reaction mechanisms, chemists have only used it in a qualitative manner, as they can hardly attribute a quantitative value to such partial charges. Quantum mechanical methods (see Section 7.4) as well as empirical procedures (see... 

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

The strength of the complexation is a function of both the donor atom and the metal ion. The solvent medium is also an important factor because solvent molecules that are potential electron donors can compete for the Lewis acid. Qualitative predictions about the strength of donor-acceptor complexation can be made on the basis of the hard-soft-acid-base concept (see Section 1.2.3). The better matched the donor and acceptor, the stronger is the complexation. Scheme 4.3 gives an ordering of hardness and softness for some neutral and ionic Lewis acids and bases. 

Although sophisticated electronic structure methods may be able to accurately predict a molecular structure or the outcome of a chemical reaction, the results are often hard to rationalize. It therefore becomes difficult to apply the findings to other similar systems. Qualitative theories, on the other hand, are unable to provide accurate results, but they may be useful for gaining insight, e.g. why a certain reaction is favoured over another. They also provide a link to many concepts used by experimentalists. 

This concept was introduced qualitatively in the late 1950s and early 1960s by Pearson, in the framework of his classification of Lewis acids and bases, leading to the introduction of the hard and soft acids and bases (HSAB) principle [19-21]. This principle states that hard acids prefer to bond to hard bases and soft acids to soft bases. In many contributions, the factor of 1/2 is omitted. The inverse of the hardness was introduced as the softness S=l/rj [22]. A third quantity, which can be expressed as a derivative with respect to the number of electrons is the Fukui function, was introduced by Parr and Yang [23,24] ... 

Next, we shall describe why the magnitudes of the E and C numbers are not just quantitative manifestations of the HSAB concept, but give insight into intermolecular interactions which are absent in the qualitative soft-soft and hard-hard labeling of interactions. As can be seen from the data in Tables 3 and 4, each acid and base has both a C and an E number which could be thought to correspond to possessing properties of softness and hardness. If this were the case, ammonia, which Pearson labels hard, has a larger Cb value than benzene, which is labeled soft. 

It is possible to be consistent with our E and C equation and view intermolecular interactions in terms of concepts we could call hardness, softness and strength. However, in doing this, we will have to modify the qualitative ideas presented by Pearson (2) about what hardness and softness mean, vide infra. The approach involves converting the E and C equation to polar coordinates. Our acids and bases are represented as vectors in E and C space in Fig. 7. The dot product of these two vectors is given as... 

As we noted at the beginning of this book, the concept of hardness can be understood and defined in very different ways, and the later parts of the book have shown that according to the measurement method used, the results should be variously interpreted, if quantitatively and qualitatively comparable data are available. Day-to-day practice enforces an individual approach to hardness measurements in laboratories. 

The high affinity of metal for sulfur is clearly demonstrated by the enormous variety of metal sulfide minerals in nature. Qualitatively this might be understood using Pearson s concept of hard and soft acids and bases. It is better described, however, by the covalency of the metal-sulfur bond, particularly with the soft metals. 

A similar picture holds for other nucleophiles. As a consequence, there might seem little hope for a nucleophile-based reactivity relationship. Indeed this has been implicitly recognized in the popularity of Pearson s concept of hard and soft acids and bases, which provides a qualitative rationalization of, for example, the similar orders of reactivities of halide ions as both nucleophiles and leaving groups in (Sn2) substitution reactions, without attempting a quantitative analysis. Surprisingly, however, despite the failure of rate-equilibrium relationships, correlations between reactivities of nucleophiles, that is, comparisons of rates of reactions for one carbocation with those of another, are strikingly successful. In other words, correlations exist between rate constants and rate constants where correlations between rate and equilibrium constants fail. 

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... 

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