Frechet functional derivative

Assuming the existence of such a Frechet functional derivative [26,102] constitutes the locality hypothesis. If this hypothesis were valid, the OEL and Kohn-Sham equations would be equivalent, determining the same model or reference state. 

These conditions could be consistent if a Frechet functional derivative vf(r) = b/- /bp(r) exists for every well-defined density functional F[p], The locality hypothesis assumes this to be true. Various tests of this hypothesis are considered here. 

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. 

This defines a Gateaux functional derivative [26, 102], whose value depends on a direction in the function space, reducing to a Frechet derivative only if all e, are equal. Defining Tt = t + v, an explicit orbital index is not needed if Eq. (5.10) is interpreted to define a linear operator acting on orbital wave functions, TL — v = t. The elementary chain rule is valid when the functional... 

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. 

Because 6G is linear in its second argument, it must be expressible as a weighted integral of s(z) the weighting function, however, may still depend on the particular point (z) in Hilbert space where the Frechet differential is evaluated. Hence, the weighting function is given by a functional G [ ] of n z), which depends parametrically on x G [ ] is called the functional derivative of G[ ]. One has... 

The functional derivative is a particular important concept in DFT, and we shall here investigate that in some detail. Two forms of functional derivatives are here of interest, the Gateaux derivative and the Frechet derivative. 

Thus, the functional derivative depends on the way the density is changed, whereas the theory demands independence from the way the density is changed, that is, the so-caUed Frechet derivatives are needed. 

The Lagrange multiplier p. determined by normalization, is the chemical potential [232], such that pt = dE/dN when the indicated derivative is defined. This derivation requires the locality hypothesis, that a Frechet derivative of Fs p exists as a local function (r). 

If the hypothetical Frechet derivative vT(r) could be replaced by the operator t when acting on occupied orbital functions , there would be no contradiction. It... 

This proposition has been tested in the exact-exchange limit of the implied linear-response theory [329], The TDFT exchange response kernel disagrees qualitatively with the corresponding expression in Dirac s TDHF theory [79,289]. This can be taken as evidence that an exact local exchange potential does not exist in the form of a Frechet derivative of the exchange energy functional in TDFT theory. 

Frechet and coworkers recently described how living free radical polymerization can be used to make dendrigrafts. Either 2,2,6,6-tetramethylpiperidine oxide (TEMPO) modified polymerization or atom transfer radical polymerization (ATRP) can be used [96] (see Scheme 10). The method requires two alternating steps. In each polymerization step a copolymer is formed that contains some benzyl chloride functionality introduced by copolymerization with a small amount of p-(4-chloromethylbenzyloxymethyl) styrene. This unit is transformed into a TEMPO derivative. The TEMPO derivative initiates the polymerization of the next generation monomer or comonomer mixture. Alternatively, the chloromethyl groups on the polymer initiate an ATRP polymerization in the presence of CulCl or CuICl-4,4T dipyridyl complex. This was shown to be the case for styrene and n-butylmethacrylate. SEC shows clearly the increase in molecu-... 

These rod-like PAMAM dendrimers [83] are formally derived from the core functionalized aziridinyl macromonomer (dendron) A, just as Frechet et al. [104] have produced similar rod-like dendrimers by polymerization of the styrene functionalized monodendron (B) which was prepared via the convergent approach (see Fig. 21). 

Calculation of the first variation (Frechet derivative) of the vector wavefield We begin this section with an analysis of the equation for the vector wavefield variation. This equation can be derived by applying the perturbation operator to both sides of the vector Helmholtz equation (14.67), expressed in terms of the slowness function s(r). 

The theory also explains the value of 5. Unfortunately, that part of the story requires more sophisticated apparatus than we are prepared to discuss (operators in function space, Frechet derivatives, etc.). Instead we turn now to a concrete example of renormalization. The calculations are only approximate, but they can be done explicitly, using algebra instead of functional equations. 

For any orbital-functional model, an optimal effective (local) potential (OEP) can be constructed following a well-defined variational formalism [24,25]. If a Frechet derivative existed for the exchange-correlation energy E,lc for ground states, it would be obtained in an OEP calculation, while the minimum energy and corresponding reference state would coincide with OFT results. Thus numerically accurate OEP calculations test the locality hypothesis. 

When Exc p is specified, the relevant ground-state density for Hohenberg-Kohn theory is p0, computed using the equivalent orbital functional Exc in the OEL equahons, (Q — e,-)local potential w(r) in the corresponding KS equahons is determined by the KSC by minimizing T for p = p0. Assuming the locality hypothesis, that w — v is the Frechet derivative of the model ground-state functional h p — Ts[p, this implies that w = vh + vxc + v is a sum of local potentials. If i>xc in the OEL equahons was equivalent to a local potential vxc(r), the KS and OEL equations would produce the same model wave function. 

Values computed for He, Be, and Ne, respectively, are PT = 0.00000, 1.38904, 9.23714 (Hartree units). This confirms that a Frechet derivative of the functional Ts[p does not exist for more than two electrons. 

The function L(h) is the Frechet differential at the point x0 in the direction h, and the operator L( ) is sometimes termed the Frechet derivative at x0. Here, however, we shall reserve the term Frechet derivative to a function equivalent to the Gateaux derivative (32). Frechet differentiability is a stronger requirement than that of Gateaux differentiability, but—with the interpretations we make—the difference is quite subtle. 

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