Energy derivatives, electron number 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. 

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. 

A rigorous quantum-mechanical calculation of some of the energy derivatives is unique to DFT alone [52]. The first and second derivatives with respect to the number of electrons, 0E/6N and 0 E/0N, recognised respectively as measures of chemical electronegativity [60] and hardness [61,62], are amenable to a rigorous calculation [52, 55,63,64]. For a system of N electrons characterised by an external potential v(f) (arising, for example, from the nuclei in an atom, molecule or cluster), the energy density functional can be expressed as... 

In the same spirit, considering the transfer of an electron from an atom X to another identical atom X, the associated energy quantity is the energy difference, (7x - Ax), which should be a measure of the ease or difficulty of charge fluctuation between two identical atoms. This quantity, therefore, is interpreted as the measure of hardness/softness of an atomic species. The quantitative definition of the chemical hardness parameter q has thus been made fl = (7 - A)/2, which is interpreted " to represent (again, within a finite difference approximation) the second derivative of the energy quantity with respect to the electron number N, viz., q = (l/2)(3 W3N ). 

Parr and Pearson (1983) defined the hardness of a molecule as a function of its total energy E and its electron number N. The first partial derivative of E with respect to N is the Mulliken electronegativity x -... 

The second derivative of the energy with respect to the number of electrons is the hardness r) (the inverse quantity is called the softness), which again may be approximated in term of the ionization potential and electron affinity. 

In the preceding sections, we have analyzed the derivatives of the energy and of the density with respect to the number of electrons. The former is identified with the concepts of chemical potential (electronegativity) and hardness and measure the... 

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]... 

Parr and Pearson have introduced the absolute hardness as the second derivative of the energy EA with respect to the number of electrons NA at constant external potential [18]... 

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],... 

The local hardness 17 (r) can be interpreted as the mixed derivative of the energy, comprising differentiation with respect to the global number of electrons and electron density,... 

From the theoretical point of view, the electrophilicity concept has been recently discussed in terms of global reactivity indexes defined for the ground states of atoms and molecules by Roy et al.18 19. In the context of the conceptual density functional theory (DFT), a global electrophilicity index defined in terms of the electronic chemical potential and the global hardness was proposed by Maynard et al.20 in their study of reactivity of the HIV-1 nucleocapsid protein p7 zinc finger domains. Recently, Parr, Szentp ly and Liu proposed a formal derivation of the electrophilicity, co, from a second-order energy expression developed in terms of the variation in the number of electrons.21... 

Within density functional theory, the chemical potential and the hardness t] become partial derivatives of the system s energy E expressed as a functional of an external potential V(r), i.e., the nuclear conformation, and a function of the number of electrons N ... 

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