Density functional derivatives definitions

The mathematical issues relevant to the definition of density functional derivatives can be considered in the simple model of noninteracting electrons. As in the KSC [4], this singles out the kinetic energy. The /V-electron Hamiltonian operator is H = T + V. Orbital functional derivatives determine the noninteracting OEL equations... 

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. ... 

General properties and definitions of polarizabilities can be introduced without invoking the complete DFT formalism by considering first an elementary model the dipole of an isolated, spherical atom induced by a uniform electric field. The variation of the electronic density is represented by a simple scalar the induced atomic dipole moment. This coarse-grained (CG) model of the electronic density permits to derive a useful explicit energy functional where the functional derivatives are formulated in terms of polarizabilities and dipole hardnesses. 

The theorem holds if the exchange-correlation potential VXc equals the functional derivative of the exchange-correlation energy /iXc with respect to the electron density p - an operational definition, which is intrinsic to DFT. 

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],... 

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). 

All this is in contradiction to the density functional definition of electronegativity as presented originally by Parr[36], where it is also inferred that the electronegativity for all the (natural) orbitals of the system should be the same. The discrepancy between these two different derivations and points of view[29,36] is still to be resolved. In addition, the global definition of electronegativity as promulgated within density functional theory, does not appear to be particularly useful, as it yields the same... 

In principle, testing the accuracy of a given approximation to T ad[pA, Pb would require exact reference data for T ad[pA, Pb - In general, such reference data for a given pair of electron densities pA and ps can be obtained by means of the Levy constrained-search procedure and the definition of T ad[pA,Pb]- In the embedding potential of Eq. 53, however, not T ad[pA, Pb] but its functional derivative is used. It is, therefore, useful to start with the analysis of the accuracy of approximations to STrd[pA,ps]... 

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of averaged quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

Alternatively, it might be that any well-defined density functional necessarily has a Frechet functional derivative, so that the locality property is inherent in the definition vF (r) = 8F/8p [18,19] and can be assumed without detailed proof. The mathematical object so defined must be proven to exist if this definition is to have any meaning. Counterexamples show that a local functional derivative does not exist in cases for which it can be tested. Either the theory must be abandoned or the definition must be generalized. 

Then we refer to (SF[p]/Sp(r))p=ft as the functional derivative at the density p0. There are different definitions of this concept, as we shall discuss in the next section. If the functional has an extremum (maximum or minimum) at the density p0, then the functional derivative will vanish in that point. 

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