Functional derivative definition

A consensus-derived definition and classification system for ARF has been proposed and is being validated (Fig. 75-1). Components of the system include both GFR and UOP plus two clinical outcomes. Definitions of risk of dysfunction, injury to and /ail ure of the kidney, loss of function, and end-stage kidney disease are included in the RIFLE acronym. 

To characterize a droplet size distribution, at least two parameters are typically necessary, i.e., a representative droplet diameter, (for example, mean droplet size) and a measure of droplet size range (for example, standard deviation or q). Many representative droplet diameters have been used in specifying distribution functions. The definitions of these diameters and the relevant relationships are summarized in Table 4.2. These relationships are derived on the basis of the Rosin-Rammler distribution function (Eq. 14), and the diameters are uniquely related to each other via the distribution parameter q in the Rosin-Rammler distribution function. Lefebvre 1 calculated the values of these diameters for q ranging from 1.2 to 4.0. The calculated results showed that Dpeak is always larger than SMD, and SMD is between 80% and 84% of Dpeak for many droplet generation processes for which 2left-hand side of Dpeak. The ratio MMD/SMD is... 

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. 

This puts a constraint on approximate functional derivatives of In general from the definition of it follows that... 

The strict mathematical definition of a functional derivative is slightly more subtle than the more familiar definition of a function s derivative, but conceptually you can think of this just as a regular derivative. The functional derivative is written using 8 rather than d to emphasize that it not quite identical to a normal derivative. 

The boundary conditions provide a tight coupling between the vorticity and stream-function fields. Also velocities still appear in the convective terms. Given the stream-function field, velocity is evaluated from the definition of stream function. That is, velocity is computed from stream-function derivatives. 

By this definition, an amide, RCONH2, but not a ketone, RCOCH3, is a functional derivative of a carboxylic acid. Several derivatives of carboxylic acids are given in Table 18-3, and methods for preparation of these derivatives are summarized in Tables 18-6 and 18-7 at the end of the chapter. 

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 local ground-state correlation potential is defined in RDFT as the functional derivative of Eq.(7) with respect to p. When infinitesimal variation of occupation numbers is allowed, a more practical definition follows from the fact that the unsymmetrical energy formuala used to construct Eq.(7) is itself a Landau functional of the occupation numbers [19]. Correlation energies of Landau quasiparticles, expressed as diagonal elements of a one-electron Hamiltonian matrix, are defined by differentiating with respect to occupation numbers to give... 

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). 

Consider the C-H bond in alkanes. Carbon is a more electronegative element than hydrogen. Consequently, the electron pair that forms this bond is shifted towards the carbon atom. In the extreme, an ionic representation of this bond can be given as pictured in 122 (Scheme 2.45). Within these conventions the carbon atom in an alkane can be approximated as a carbanion (oxidation level 0 by definition). Using this definition it becomes possible to apply oxidation-reduction terminology to the processes as if they occurred to ion pair 122. Thus, oxidation of 122 with the loss of one electron leads to the radical 123. With the loss of two electrons, the oxidation leads to carbocation 124. Similarly, the conversion of an alkane to an alcohol and the alcohol into an aldehyde and the aldehyde eventually to a carboxylic acid can unambiguously be classified as an oxidation sequence with the loss of two, four, and six electrons. The oxidation levels 1, 2, and 3 are ascribed respectively to these functional derivatives. The conversion of an alkane to an alkene or alkyne can be interpreted in an analogous fashion. 

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]... 

We present here a simplified definition of the operations of functional derivative and functional Taylor expansion. It is based on a formal generalization of the corresponding operations applied to functions of a finite number of independent variables. 

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. 

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... 

Let us now calculate the functional derivative 8(9/8v at a given potential v. According to our definition in the previous section we have to calculate the quantity... 

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