Electronic chemical hardness

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. 

The electronic chemical hardness, r is the curvature of the U versus N curve. Thus it is the second derivative of U with respect to N ... 

The electronic chemical hardness, defined in eqn (23), gives a measure of the chemical reactivity of a given compound. Also the chemical potential,... 

Figure 16.1 The chemical hardness of an atom, molecule, or ion is defined to be half. The value of the energy gap between the bonding orbitals (HOMO—highest orbitals occupied by electrons), and the anti-bonding orbitals (LUMO—lowest orbitals unoccupied by electrons). The zero level is the vacumn level, so I is the ionization energy, and A is the electron affinity, (a) For hard molecules the gap is large (b) it is small for soft molecules. The solid circles represent valence electrons. Adapted from Atkins (1991).

Chemical hardness measures the resistance to a chemical change. This may be seen by considering a reaction between two molecules C and D in which electrons are transferred from D to C. Initially the two chemical potentials are ... 

To get an approximate expression for the chemical hardness, start with an expression for the electronic chemical potential. Let a hypothetical atom have an energy, UQ. Subtract one electron from it. This costs I = ionization energy. Alternatively, add one electron to it. This yields A = electron affinity. The derivative = electronic chemical potential = p = AU/AN = (I + A)/2. The hardness is the derivative of the chemical potential = r = Ap/AN = (I - A)/2. 

One more step provides an operational definition. The HOMO level lies, I = ionization energy, below the vacuum level, while the LUMO level lies, A = electron affinity, below it. Thus, the chemical hardness lies midway in the gap and usually is given in units of eV. 

The bonding in solids is similar to that in molecules except that the gap in the bonding energy spectrum is the minimum energy band gap. By analogy with molecules, the chemical hardness for covalent solids equals half the band gap. For metals there is no gap, but in the special case of the alkali metals, the electron affinity is very small, so the hardness is half the ionization energy. 

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). 

Another property that is related to chemical hardness is polarizability (Pearson, 1997). Polarizability, a, has the dimensions of volume polarizability (Brinck, Murray, and Politzer, 1993). It requires that an electron be excited from the valence to the conduction band (i.e., across the band gap) in order to change the symmetry of the wave function(s) from spherical to uniaxial. An approximate expression for the polarizability is a = p (N/A2) where p is a constant, N is the number of participating electrons, and A is the excitation gap (Atkins, 1983). The constant, p = (qh)/(2n 2m) with q = electron charge, m = electron mass, and h = Planck s constant. Then, if N = 1, (1/a) is proportional to A2, and elastic shear stiffness is proportional to (1/a). 

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. 

The next step is the identification of the concept of chemical hardness, 17, with the second derivative of the energy with respect to the number of electrons, formulated by Parr and Pearson [14]... 

Two quantities derived from DFT are the electronic chemical potential /r and the chemical hardness 17 [2]. The definitions of these quantities are... 

The variable hardness in this work is the local hardness as given by the basic theory [2]. The electronic chemical potential in this work is a property if a given molecule (arrangement of nuclei) is also of the approximate wave function used to describe it. This does not represent an equilibrium system. The variation of the chemical potential is a consequence. 

FIGURE 21.2 Profiles of (a) dipole moment (in D), (b) chemical hardness (in kcal/mol), and (c) CO and CS bond electronic populations for the reaction shown in Equation 21.9. Vertical dashed lines indicate the limits of the reaction regions defined in the text. 

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... 

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... 

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... 

The global reactivity descriptors, such as chemical potential and chemical hardness, are the derivative of energy with respect to the number of electrons. The formal expressions for chemical potential (p.) and chemical hardness (rj) are [1,11]... 

Chemical hardness Electronic Fukui function./ /) Linear response function... 

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 natural way to approximate the chemical potential and chemical hardness in DFT is to evaluate them directly from the calculated ionization energy and electron... 

STRUTINSKY S SHELL-CORRECTION METHOD IN THE EXTENDED KOHN-SHAM SCHEME APPLICATION TO THE IONIZATION POTENTIAL, ELECTRON AFFINITY, ELECTRONEGATIVITY AND CHEMICAL HARDNESS OF ATOMS... 

Abstract. Calculations of the first-order shell corrections of the ionization potential, 6il, electron affinity, 5 A, electronegativity, ix, and chemical hardness. Sir] are performed for elements from B to Ca, using the previously described Strutinsky averaging procedure in the frame of the extended Kohn-Sham scheme. A good agreement with the experimental results is obtained, and the discrepancies appearing are discussed in terms of the approximations made. 

In the next section we shall recall the definitions of the chemical concepts relevant to this paper in the framework of DFT. In Section 3 we briefly review Strutinsky s averaging procedure and its formulation in the extended Kohn-Sham (EKS) scheme. The following section is devoted to the presentation and discussion of our results for the residual, shell-structure part of the ionization potential, electron affinity, electronegativity, and chemical hardness for the series of atoms from B to Ca. The last section will present some conclusions. 

---