Density functional theory definition

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 aim of this chapter is to discuss chemical reactivity and its application in the real world. Chemical reactivity is an established methodology within the realm of density functional theory (DFT). It is an activity index to propose intra- and intermolecular reactivities in materials using DFT within the domain of hard soft acid base (HS AB) principle. This chapter will address the key features of reactivity index, the definition, a short background followed by the aspects, which were developed within the reactivity domain. Finally, some examples mainly to design new materials related to key industrial issues using chemical reactivity index will be described. I wish to show that a simple theory can be state of the art to design new futuristic materials of interest to satisfy industrial needs. 

More insight into these processes is obtained by studying the particle number dependent properties of density functionals. This of course requires a suitable definition of these density functionals for fractional particle number. The most natural one is to consider an ensemble of states with different particle number (such an ensemble is for instance obtained by taking a zero temperature limit of temperature dependent density functional theory [84]). We consider a system of N + co electrons where N is an integer and 0 < m < 1. For the corresponding electron density we then have... 

These studies, which employed density functional theory (DFT) methods (B3LYP/LANL2DZ/Gaussian 98) proposed that the reactions of all alkali metal phenoxides with C02 followed a similar ground mechanism that comprised three intermediates and three transition states. In step 1, C02 must first be activated by an alkali metal phenoxide. In the case of the sodium phenoxide [24a], C02 can only attack at the polarized O-Na bond to form a Ph0Na/C02 complex as the first intermediate (structure 4). The calculation definitely rules out a direct C-C bond formation at the aromatic ring. 

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],... 

Density functional theory thus offers precise definitions of previously poorly defined chemical quantities, enabling their first principles calculation. This part of DFT is termed by Parr as conceptual DFT [6]. 

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]. 

We have now seen that the effort of Parr and collaborators [8-12] to put Fukui s frontier-orbital concept of chemical reactivity on sound footing in density-functional theory through the definition of the Fukui function and the local and global softness works only for extended systems. This restriction to extended systems raises a sixth issue. In both the local softness and the Fukui function, Eqs. (54) and (53a), the orbitals at the chemical potential represent both the LUMO and the HOMO in the Fukui sense. However, there is a continuum of unoccupied KS states above the chemical potential accessible even to weak chemical perturbations any linear combination of which could in principle be selected as the LUMO, and similarly for states below fi and the HOMO. This ambiguity in the frontier-orbital concept obviously applies as well to localized systems when there is more than one KS state significantly affected by a chemical perturbation. 

Notice that other definitions of chemical potential may sometimes appear in literature, particularly in the density functional theory (where the electronic chemical potential is considered as the functional derivative of the density functional with respect to the electron density), and also in the description of relativistic systems in theoretical physics (see [v, vi] and references cited). 

HJ point out that in the detailed work on H2 by Kolos and Wolniewicz,114 the first excited state 3 2 was found to have a very weak minimum at a large separation. This binding presumably arises from a van der Waals force which is not included in the density functional theory when a local approximation to exchange and correlation is employed. Nevertheless, as HJ point out, their study of the corresponding state of the dimers Li2-Cs2 revealed a weak, but definite maximum in each case. Rough estimates of binding energy and equilibrium separation are shown in Table 16. It is, of course, possible that these results are a consequence of the local spin-density approximation, so that further work will... 

To go into this idea quantitatively, we need definitions of hardness and softness, and a rank order for acids and bases on a scale of hardness. This has been done in two ways one based on molecular orbital theory, and the other on density functional theory. 

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