Chemical potential, electronic

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. ... 

Molecular Orbital Electronegativity as Electron Chemical Potential in Semiempirical SCF Schemes... 

MOLECULAR ORBITAL ELECTRONEGATIVITY AS ELECTRON CHEMICAL POTENTIAL... 

We are now ready for computing the electron chemical potential within the u> scheme. Since ours is a Htickel-like scheme, the total energy Etot is the sum of the orbital energies multiplied by the pertinent occupations, and therefore... 

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. 

Experimental data as well as density functional theory show that the ground-state properties of solids depend primarily on the densities of the valence electrons. Therefore, pE may be considered to be the electronic chemical potential (Pearson, 1997). Since pE denotes the energy per mole of... 

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. 

It is however possible to obtain a physically meaningful representation of 0(r) for cations, in the context of density functional theory. The basic expression here is the fundamental stationary principle of DFT, which relates the electronic chemical potential ju, with the electrostatic potential and the functional derivatives of the kinetic and exchange-correlation contributions [20] ... 

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. 

Extension of this method for correcting the energies of approximate wave functions to systems containing more electrons and orbitals would be very useful. But difficulties quickly arise. The interelectronic effects become complicated because of exchange and correlation. More importantly, in DFT, it is only the highest occupied orbital whose energy is equal to the electronic chemical potential. This potential is valid for the total electron density. 

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 ... 

The electronic chemical potential is constant for a system in its electronic ground state, which led Parr et al. to associate the chemical potential with minus one times the electronegativity, since the electronegativity is also equalized in the ground state [7]. This equalization of the chemical potential also suggests that electronic structure theory can be expressed in a way that resembles classical thermodynamics. Ergo, Parr et al. wrote the total differential of the energy as... 

The paper of Parr and Bartolotti is prescient in many ways [1], It defines the shape function and describes its meaning. It notes the previously stated link to Levy s constrained search. It establishes the importance of the shape function in resolving ambiguous functional derivatives in the DFT approach to chemical reactivity—the subdiscipline of DFT that Parr has recently begun to call chemical DFT [6-9]. Indeed, until the recent resurgence of interest in the shape function, the Parr-Bartolotti paper was usually cited because of its elegant and incisive analysis of the electronic chemical potential [10],... 

The purely electronic derivatives, calculated for the rigid molecular geometry, determine the system electronic chemical potential... 

To neutralize the electrical charge in the homogeneous dense u, d quark matter, roughly speaking, twice as many d quarks as u quarks are needed, i.e., rid — 2nu, where nv,d are the number densities for u and d quarks. This induces a mismatch between the Fermi surfaces of pairing quarks, i.e., pd — Hu = 10 25n, where pe is the electron chemical potential. 

We will demonstrate below that the Ginzburg number Gi = AT/Tc), which determines the broadness of the energy region near the critical temperature, where fluctuations essentially contribute, is Gi A(Tc/iiq)4 with A 500 in our case. To compare, for clean metals A 100, p,q — fi,., the latter is the electron chemical potential. Thus Gi 1, if Tc is rather high, Tc (f -t- )p,q, and we expect a broad region of temperatures, where fluctuation effects might be important. 

We consider stellar matter in the quark core of compact stars consisting of electrons in chemical equilibrium with u and d quarks. Hence Pd = pu + Pe, where pe is the electron chemical potential. The thermodynamic potential of such matter is... 

Fig. 7.15 Band filling in an intercalation model according to Friedel s (1954) notion of screening. The upper panel shows the position of the bands at various degrees of filling the lower panel shows the corresponding values of the electron chemical potential (Fermi energy).

Global hardness. The partial charge concept helps the chemist to visualize within a molecule or a network, how the electronic density changes as a function of the spatial location of the various atomic constituents. Another useful information for chemical reactivity would be to visualize where the electronic chemical potential variation should be the largest or the lowest. Such a parameter is called a frontier index and may be defined in density-functional theory as f = dQHOMO/LUMO N (16). Point-charge approximation of this relation shows that each atom of a chemical compound should have a frontier index fj such that (17) ... 

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