Electronic chemical potential energy

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. ... 

Though we and others (27-29) have demonstrated the utility and the improved sensitivity of the peroxyoxalate chemiluminescence method for analyte detection in RP-HPLC separations for appropriate substrates, a substantial area for Improvement and refinement of the technique remains. We have shown that the reactions of hydrogen peroxide and oxalate esters yield a very complex array of reactive intermediates, some of which activate the fluorophor to its fluorescent state. The mechanism for the ester reaction as well as the process for conversion of the chemical potential energy into electronic (excited state) energy remain to be detailed. Finally, the refinement of the technique for routine application of this sensitive method, including the optimization of the effi-ciencies for each of the contributing factors, is currently a major effort in the Center for Bioanalytical Research. 

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. 

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... 

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. 

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).

V x(x) the gradient of the chemical potential energy. However, again, it is, strictly speaking, just a mathematical substitution into a kinetic equation—it is a quasi thermodynamic potential, not an equilibrium thermodynamic potential. Equations (9a) and (9b) are often referred to as the drift and diffusion components, respectively, of the electron current. One can see immediately from Eqs. (6), (9a), and (9b) that WU(x) and V x(x) are independent forces in the photoconversion process and, therefore, that it is possible to drive a solar cell with either one, or both, of these forces. In fact, the different types of solar cell can be classified according to the relative importance of these two forces in the photoconversion process. 

Fig. 7.10 Change of energy (for F+, F and F ) as electrons are added to a species. The energies were calculated at the QCISD(T)/6-311+G level. The slope of the curve at any point (first derivative) is the electronic chemical potential, and the negative of the slope the electronegativity, of the species at that point. The curvature at any point (second derivative) is the hardness of the species). See too Table 7.12...

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