Kohn-Sham chemical potential

All these functional derivatives are well defined and do not involve any actual derivative relative to the electron number. It is remarkable that the derivatives of the Kohn-Sham chemical potential /rs gives the so-called radical Fukui function [8] either in a frozen orbital approximation or by including the relaxation of the KS band structure. On the other hand, the derivative of the Kohn-Sham HOMO-FUMO gap (defined here as a positive quantity) is the so-called nonlinear Fukui function fir) [26,32,50] also called Fukui difference [51]. 

Higher-order derivatives with respect to external potential define xi(r, r1), Xi(r, r1, r"), etc., and their response with N define j(r, rJ), g2(r, r, / ), etc. This chain of derivatives is diagrammatically depicted in Figure 25.1 [22]. Thus, an exact one-electron formulation of all chemical responses (linear and nonlinear hardness, FF) in terms of Kohn-Sham orbital of the unperturbed system was derived [22b]. 

In fact, the true form of the exchange-correlation functional whose existence is guaranteed by the Hohenberg-Kohn theorem is simply not known. Fortunately, there is one case where this functional can be derived exactly the uniform electron gas. In this situation, the electron density is constant at all points in space that is, n(r) = constant. This situation may appear to be of limited value in any real material since it is variations in electron density that define chemical bonds and generally make materials interesting. But the uniform electron gas provides a practical way to actually use the Kohn-Sham equations. To do this, we set the exchange-correlation potential at each position to be the known exchange-correlation potential from the uniform electron gas at the electron density observed at that position ... 

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. 

E. J. Baerends, O. V. Gritsenko, and R. van Leeuwen, in Chemical Applications of Density Functional Theory, B. B. Laird, R. B. Ross, and T. Ziegler, Eds., American Chemical Society, Washington, DC, 1996, pp. 20-41. Effective One-Electron Potential in the Kohn-Sham Molecular Orbital Theory. 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

On the other hand, there is a discrepancy between the predictions of the Kohn-Sham theory of Sect. 2 and of the direct ensemble approach of the present section for the electrophilic and nucleophilic Fukui functions and for the chemical potential. Direct use of the ensemble of Eqs. (28) yields results for f (r), Eq. (34a), and for n, Eq. (30a), which are independent of JT in the range... 

Instead of supposing there to be a single Kohn-Sham potential, one can think of it as a vector in Fock space. For each sheet ft = N of the latter, there is a component vKS(r,N) and a corresponding set of Kohn-Sham equations. Density functional theory and Kohn-Sham theory hold separately on each sheet. Ensemble-average properties are then composed of weighted contributions from each sheet, computable sheet by sheet via the techniques of DFT and the KS equations. Nevertheless, though completely valid, this procedure would yield for the reactivity indices f(r), s(r), and S the results already obtained directly from Eqs. (28). We are left without proper definitions of chemical-reactivity indices for systems with discrete spectra at T = 0 [43]. 

The electron-transfer reactivities are defined as derivatives of the electron-density p(r) with respect to total electron number Jf, Ufr), or chemical potential p, s r). The treatment of JT as a continuous variable [8-12] is justified by reference to the ensemble formulation of density-functional theory [8,18] and, in consequence, of the Kohn-Sham theory. We show in Sect. 4, in previously unpublished work [42], that this ensemble formulation yields either vanishing or infinite local and global softnesses for localized systems with... 

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