Chemical hardness density

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

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. 

Equation 16.12 expresses a relation between q and B.This is not a universal relation, but it does apply to the sp-bonded elements of the first four columns of the Periodic Table. Using chemical hardness values given by Parr and Yang (1989), and atomic volumes from Kittel (1996), it has been shown that the bulk moduli of the Group I, II, III, and IV elements are proportional to the chemical hardness density (CH/atomic volume) (Gilman, 1997). The correlation lines pass nearly through the coordinate origins with correlation coefficients, r = 0.999. Thus physical hardness is proportional to chemical hardness (Pearson, 2004). 

The Group IV elements also show a linear correlation of their octahedral shear moduli, C44(lll) with chemical hardness density (Eg/2Vm).This modulus is for for shear strains on the (111) planes. It is a measure of the shear stiffnesses of the covalent bonds. The (111) planes lie normal to the bonds that connect the atoms in the diamond (or zinc blende) structure. In terms of the three standard moduli for cubic symmetry (Cn, Q2, and C44), the octahedral shear modulus is given by C44(lll) = 3CV1 + [4C44/(Cn - Ci2)]. Since the (111) planes have three-fold symmetry, they have only one shear modulus. The bonds across the octahedral planes have high resistance to shear which probably results from electron correlation in the bonds (Gilman, 2002). 

It is shown that the stabilities of solids can be related to Parr s physical hardness parameter for solids, and that this is proportional to Pearson s chemical hardness parameter for molecules. For sp-bonded metals, the bulk moduli correlate with the chemical hardness density (CffD), and for covalently bonded crystals, the octahedral shear moduli correlate with CHD. By analogy with molecules, the chemical hardness is related to the gap in the spectrum of bonding energies. This is verified for the Group IV elements and the isoelec-tronic III-V compounds. Since polarization requires excitation of the valence electrons, polarizability is related to band-gaps, and thence to chemical hardness and elastic moduli. Another measure of stability is indentation hardness, and it is shown that this correlates linearly with reciprocal polarizability. Finally, it is shown that theoretical values of critical transformation pressures correlate linearly with indentation hardness numbers, so the latter are a good measure of phase stability. 

Chattaraj PK, Parr RG (1993) Density Functional Theory of Chemical Hardness. 80 11-26 Cheh AM, Neilands JP (1976) The j -Aminoevulinate Dehydratases Molecular and Environmental Properties. 29 123-169 Chimiak A, Neilands JB (1984) Lysine Analogues of Siderophores. 58 89-96 Christensen JJ, see Izatt RM (1973) 16 161-189... 

Wood is prized for its physical properties, such as strength, compressibility, hardness, density, color, or grain pattern. Chemists classify physical and chemical properties as either intensive or extensive. All chemical properties are intensive, but physical properties can be either. Density is an important physical property of matter that is often used for identifying substances. By determining the density of a piece of wood, you can identify the specific sample. 

Since chemical hardness is related to the gaps in the bonding energy spectra of covalent molecules and solids, the band gap density (Eg/Vm) may be substituted for it. When the shear moduli of the III-V compound crystals (isoelec-tronic with the Group IV elements) are plotted versus the gap density there is again a simple linear correlation. 

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. 

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... 

Chattaraj, P. K. and R. G. Parr. 1993. Density functional theory of chemical hardness. Struct. Bond. 80 11-25. 

This chapter will be concerned with computing the three response functions discussed above—the chemical potential, the chemical hardness, and the Fukui function—as reliably as possible for a neutral molecule in the gas phase. This involves the evaluation of the derivative of the energy and electron density with respect to the number of electrons. 

The analytical techniques discussed previously can be used to study the EPDM network as such or its formation in time as well as to determine relationships between the network structure and the properties of the vulcanisates. In a preliminary approach some typical vulcanised EPDM properties, i.e., hardness, tensile strength, elongation at break and tear strength, have been plotted as a function of chemical crosslink density (Figure 6.6). The latter is either determined directly via 1H NMR relaxation time measurements or calculated from the FT-Raman ENB conversion (Table 6.3). It is concluded that for these unfilled, sulfur-vulcanised, amorphous EPDM, the chemical crosslink density is the main parameter determining the vulcanisate properties. It is beyond the purpose of this review to discuss these relationships in a more detailed and theoretical way. 

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],... 

Although many polymer properties are greatly influenced by molecular weight, some other important properties are not. For example, chain length does not affect a polymer s resistance to chemical attack. Physical properties such as color, refractive index, hardness, density, and electrical conductivity are also not greatly influenced by molecular weight. 

Chattaraj, P. K., Parr, R. G. Density Functional Theory of Chemical Hardness. Vol. 80,... 

The positive sign of chemical hardness is a consequence of the concavity of the energy with respect to the number of electrons. The Fukui function, /(r), appears in both equations since it represents the sensitivity of the chemical potential to the changes in the external potential and that of the density with respect to the number of electrons,6... 

Many approaches already do this, for example, by incorporating known chemical constraints, densities or hard-sphere repulsions. Many of the emerging methods described below have this flavor, and as time goes on our ability to complex our data and our modeling approaches will only increase. 

Nevertheless, it is a great advantage in science to have quantitative definitions so that one can measure what one is speaking about, and express it in numbers. Fortunately this is what has happened to chemical hardness. The means by which this has come about lies in density functional theory. This will be the topic of the next chapter. 

There are several cogent reasons to include a chapter on the solid state in a treatise devoted to chemical hardness, and other concepts, derived from density functional theory. One is that DFT has been the theoretical method of choice in dealing with solid-state problems for a number of years. 

A second reason to consider solids here is that for this state the concept of physical hardness is important. Physical hardness is the resistance to a change in volume or shape of a solid object, produced by mechanical forces. Remembering that chemical hardness is subject to a restriction of constant nuclear positions, we see that physical hardness has the effect of removing this restriction. Nuclear positions must change, and this must be accompanied by a change in the electron density. 

The first five chapters of this volume introduced the subject of chemical hardness, that is, the resistance to changes in the electron density function of a chemical system. The nuclei were supposedly held fixed in position. In spite of this limitation, a number of applications of chemical hardness to a better understanding of bonding energies, rates of reaction and structures, were given. 

Chattaraj PK, Parr RG (1993) Density of Functional Theory of Chemical Hardness. 80 11-26... 

It has been discussed in Section 2 that concepts such as electronegativity and hardness could explain important aspects of chemical reactions and could be related to different physico-chemical properties. Density functional theory has been found to provide a rigorous theoretical background for electronegativity, hardness, and related concepts. 

Thus, one can see that within the framework provided by density functional theory, the basic equations for the description of a chemical event, Eqs. (4) and (7), may be expressed in terms of basic variables such as the chemical potential (electronegativity), the chemical hardness and the fukui function (frontier orbitals). In fact, through this approach one may introduce a coherent quantitative language of hardness and softness functions which are nonlocal, local, and global [29]. The global softness is given by... 

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