Energy derivatives, electron number chemical potential

Besides the already mentioned Fukui function, there are a couple of other commonly used concepts which can be connected with Density Functional Theory (Chapter 6). The electronic chemical potential p is given as the first derivative of the energy with respect to the number of electrons, which in a finite difference version is given as half the sum of the ionization potential and the electron affinity. Except for a difference in sign, this is exactly the Mulliken definition of electronegativity. ... 

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. 

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 genesis of chemical DFT can be traced back to the 1978 paper published by of Parr et al. [6]. That paper identified the electronic chemical potential as the derivative of the electronic energy with respect to the number of electrons at fixed molecular geometry ... 

In particular, is it possible to determine the chemical potential (which obviously depends on how the energy responds to variations in the number of electrons) from the variation of the electron density at fixed electron number Parr and Bartolotti show that this is not possible the derivatives in Equation 19.8 are equal to an arbitrary constant and thus ill defined. One has to remove the restriction on the functional derivative to determine the chemical potential. Therefore, the fluctuations of the electron density that are used in the variational method are insufficient to determine the chemical potential. 

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]... 

The effect of external field on reactivity descriptors has been of recent interest. Since the basic reactivity descriptors are derivatives of energy and electron density with respect to the number of electrons, the effect of external field on these descriptors can be understood by the perturbative analysis of energy and electron density with respect to number of electrons and external field. Such an analysis has been done by Senet [22] and Fuentealba [23]. Senet discussed perturbation of these quantities with respect to general local external potential. It can be shown that since p(r) = 8E/8vexl, Fukui function can be seen either as a derivative of chemical potential... 

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 methodology in this section concerns global properties, which can be written as first or second order derivatives of the energy with respect to the number of electrons N. In practice, these derivatives cannot be calculated analytically and their numerical calculation is performed using a finite difference approximation ("faute de mieux") very recently, a variational ansatz has been proposed [61]. For the chemical potential (or minus the electronegativity), the finite difference approximation becomes ... 

Some quantities in density-functional theory suggest analogies with chemical and thermodynamic concepts. For example, a chemical potential can be defined as the derivative of energy with respect to electron number... 

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 ... 

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... 

The electronic chemical potential (eqn (14)) has been identified as the negative of the electronegativity (x) and has been defined as the following first derivative of energy with respect to the number of electrons when the external potential u(r) remains unchanged, i.e. 

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