The Functional Derivative

How does one minimize or, for that matter, finds other extrema, of a functional Referring to the functional as F(y(x)) the question is how are total changes of y for the entire range of x (from a to b) going to change F. Thus the entire range of x is to be considered and if one were to select probe values of x for this, one would add these up so that 

Since a limit is taken to obtain dF (lim e 0) the higher terms of G may be ignored. The functional dG/dy is referred to as the functional derivative of F and is simply represented by the notation dFIdy rather than using a new letter. One important property of a functional derivative is obtained from the mathematics involved with Euler-LaGrange relationships. If F is of the form 

By setting SF/ = 0 one should obtain the extrema for the variation of F as a function of y as evaluated for the entire range from a to b. As with functions, whether a particular extremum is a local minimum, maximum or a (vertical) inflection point may be determined by the second and third derivatives. 

The extension of the functional to higher dimensions follows the same principles. For n classical particles, one can construct a functional describing the positions and velocities of aU the particles, in which case there would be 6n dimensions. 

A well-known relationship in statistics is if two sets of observations are independent of each other then the variances are additive. However, if they are not, then there is what is referred to as a correlation between the observations. In terms of probability this can be expressed as follows. 

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To determine the position of the tricritical point and the structure of the ordered phases stable below the bifurcation we analyze the asymptotic form of Qeff for e 0. At local minima the functional derivative of Qeff with respect to all the OPs vanishes. From this condition and from (45), (58), (47), and (64) we find that at the metastable states... 

When the solid phase 0+ at x = -f oo coexists with the gas phase 0 at X = -oo, the stationary profile of the phase field is determined so as to minimize the free energy functional F (56). The functional derivative gives... 

A series of benzimidazole and benzimidazolone derivatives from the Janssen laboratories has provided an unusually large number of biologically active compounds, particularly in the area of the central nervous system. Reaction of imidazolone itself with isopropenyl acetate leads to the singly protected imidazolone derivative 51. Alkylation of this with 3-chloro-l-bromopropane affords the functionalized derivative Use of this... 

In hydrates with their open structure the relative contribution of second and third neighbor solvent molecules to w(r) is only of the order of i of that in the much denser face-centered cubic lattice. It is therefore a better approximation to neglect second and third neighbors altogether than to use the functions derived by Wen tor f et al.u for the face-centered cubic lattice including contributions due to second and third shell neighbors. 

Note in particular that the exchange-correlation functional that emCTges here does not involve the kinetic energy. From the perspective of the DFT literature, (3.16) is a formulation of the Hohenberg-Kohn functional that is constructed to ensure that the functional derivatives required for variational minimization actually exist. We return to these issues in Sect. 3.3. Also note that in the time-dependent case the external potential V(r, )is often considered to be explicitly... 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

While the behavior of the exchange-correlation potential Vxc (recall from equation (5-16) that the exchange-correlation potential Vxc is defined as the functional derivative of the... 

Note that the operator fKS differs from the Fock operator f that we introduced in Section 1.3 in connection with the Hartree-Fock scheme only in the way the exchange and correlation potentials are treated. In the former, the non-classical contributions are expressed via the - in its exact form unknown - exchange-correlation potential Vxc, the functional derivative of Exc with respect to the charge density. In the latter, correlation is neglected... 

First, the self-energy operator is replaced by a local exchange-correlation potential, which is given by the functional derivative of the exchange-correlation energy with respect to the electron density ... 

The density fluctuations in the second term are approximated by atomic charge monopoles Aqa, which allows to approximate the functional derivatives by an analytical function yap (for a more detailed discussion, see Refs. [22,40,41]). 

The functional derivative in Eq. (60) represents deterministic relaxation of the system toward a minimum value of the free-energy functional E[< )(r, f)], which is usually taken to have the form of the coarse-grained Landau-Ginzburg free energy... 

To initiate a DF calculation with E% A a guesstimate of p is used to find the functional derivative... 

It is however possible to obtain a physically meaningful representation of 0(r) for cations, in the context of density functional theory. The basic expression here is the fundamental stationary principle of DFT, which relates the electronic chemical potential ju, with the electrostatic potential and the functional derivatives of the kinetic and exchange-correlation contributions [20] ... 

In the Kohn-Sham equation above, the Coulomb potential and the XC potential are obtained from their energy counterparts by taking the functional derivative of the latter with respect to the density. Thus... 

Further analysis from the minimum action principle shows that the exchange (xc) potential is then the functional derivative of that quantity in terms of the density ... 

In particular, is it possible to determine the chemical potential (which obviously depends on how the energy responds to variations in the number of electrons) from the variation of the electron density at fixed electron number Parr and Bartolotti show that this is not possible the derivatives in Equation 19.8 are equal to an arbitrary constant and thus ill defined. One has to remove the restriction on the functional derivative to determine the chemical potential. Therefore, the fluctuations of the electron density that are used in the variational method are insufficient to determine the chemical potential. 

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. 

A different test, one less satisfactory because the standard of comparison is simulated water not real water, is obtained by examining the functions hon(R) and dd(-R) predicted. The functions derived from the Narten model are shown in Fig. 54 they should be compared with those for simulated water, displayed in Figs. 27 and 28. Just as for the function hoo R), the curves for simulated water are narrower and higher than those in Fig. 54. In all other respects the agreement between the two sets of functions is excellent. It now remains to be shown that a full calculation, based on precisely the random network model proposed will reproduce the data as well as this (sensibly equivalent ) model. 

In this work, the electronic kinetic energy is expressed in terms of the potential energy and derivatives of the potential energy with respect to nuclear coordinates, by use of the virial theorem (5-5). Thus, the results are valid for ail bound electronic states. However, the functional derived for E does not obey a variational principle with respect to (Pg ( )), even though in... 

The Dirac and the L6vy-Leblond equations establish relationships between the large and the small components of the wavefunctions. If these relationships are to be fulfilled by the functions derived from a variational procedure, the basis sets for the large and for the small components have to be constructed accordingly. In particular, the relation... 

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