Taylor expansion function

As stated in the introduction, conceptual DFT is based on a series of reactivity descriptors mostly originating from a functional Taylor expansion of the E = E[N, v(r)] functional. These (<)nE/3NmSv(r)m ) quantities can be considered as response functions quantifying the response of a system for a given perturbation in N and/or v(r). In the case of molecular interactions (leading to a new constellation of covalent bonds or not), the perturbation is caused by the reaction partner. In Scheme 27.1 an overview of the interaction descriptors up to n 2 (for a more complex tabulation and discussion of descriptors up to n 3, see Refs. [11,12]) is given. 

Recently there has been a great deal of interest in nonlinear phenomena, both from a fundamental point of view, and for the development of new nonlinear optical and optoelectronic devices. Even in the optical case, the nonlinearity is usually engendered by a solid or molecular medium whose properties are typically determined by nonlinear response of an interacting many-electron system. To be able to predict these response properties we need an efficient description of exchange and correlation phenomena in many-electron systems which are not necessarily near to equilibrium. The objective of this chapter is to develop the basic formalism of time-dependent nonlinear response within density functional theory, i.e., the calculation of the higher-order terms of the functional Taylor expansion Eq. (143). In the following this will be done explicitly for the second- and third-order terms... 

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. 

Before turning to functional Taylor expansion, we note that many operations with ordinary derivatives can be extended to functional derivatives. We note, in particular, the chain rule of differentiation. 

We now consider the functional Taylor expansion. We start with a simple function of one variable f(x) for which the Taylor expansion about x = 0 is... 

We now view as a functional of rj which itself is a functional of In this way, we avoid the possibility of an infinite increment as in (D.4). Thus, the first-order functional Taylor expansion is... 

A central problem in the development of the theory is the formulation of a suitable expression for A[p] for the nonuniform system of interest. In treatments of SFE, two kinds of approach have been pursued. The first approach is to make a functional Taylor expansion of Aex[p] the excess... 

Prior to applications to SFE, WDAs were originally proposed to give a more realistic description of molecular packing effects in interfacial systems, and Evans [14] has given an interesting discussion of their relative suitability in the context of interfaces and SFE. WDAs are nonperturbative in character and so are not subject to the criticism leveled at the functional Taylor expansion approach. 

DFT studies of binary hard-sphere mixtures predate the simulation studies by several years. The earliest work was that of Haymet and his coworkers [221,222] using the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations. For example, it was Smithline and Haymet [221] who first analyzed the Hume-Rothery rule in the context of hard sphere mixture behavior and who first investigated the stability of substitutionally ordered solid solutions. The most accurate DFT results for hard-sphere mixtures have come from the WDA-based theories. In particular the results of Denton and Ashcroft [223] and those of Zeng and Oxtoby [224] give qualitatively correct behavior for hard spheres forming substitutionally disordered solid solutions. 

There are two approaches commonly used to derive an analytical connection between g(r) and u(r) the Percus-Yevick (PY) equation and the hypemetted chain (HNC) equation. Both are derived from attempts to form functional Taylor expansions of different correlation functions. These auxiliary correlation functions include ... 

Because of space limitations we considered only macroscopic kinetic systems, but the approach can be also applied to experiments of single-molecule kinetics [10]. The suggested approach is in an early stage of development. An important issue is extending the approach to nonneutral systems, for which the kinetic and transport properties of the labeled species are different from the corresponding properties of the unlabeled species. There are two different regimes (a) If the deviations from neutrality are small, the response can be represented by a functional Taylor expansion the terms of first order in the functional Taylor expansion correspond to the linear response law. (b) For large deviations, a phase linearization approach is more appropriate. 

We present here the operations of the functional derivative and functional Taylor expansion in a formal fashion, based on an analogy with the discrete case. For more details on the mathematical aspects, the reader is referred to Volterra (1931). 

The equilibrium density is a functional of the field n(i) that is, if we specify v[Pg.78]

Conversely, since v(r) is also a unique functional of p(i), a functional Taylor expansion of i (ri+s) and the use of the second Yvon equation (4.42) gives... 

In the Taylor series approximation.the free energy of the inhomogeneous phase (here, the crystal) is expanded in a functional Taylor expansion about a reference liquid density po(r). This expansion is subsequently truncated at second order to yield... 

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