Functions, scaling transformations

Assignments I, J XO H A, P FNZ indices data vector scaling factor for ordinate intermediate results function to transform %-probability values to NPS-scale x is the independent variable in the polynomial. (See Section 5.1.1.)... 

The first is to normalize the data, making them suitable for analysis by our most common parametric techniques such as analysis of variance ANOYA. A simple test of whether a selected transformation will yield a distribution of data which satisfies the underlying assumptions for ANOYA is to plot the cumulative distribution of samples on probability paper (that is a commercially available paper which has the probability function scale as one axis). One can then alter the scale of the second axis (that is, the axis other than the one which is on a probability scale) from linear to any other (logarithmic, reciprocal, square root, etc.) and see if a previously curved line indicating a skewed distribution becomes linear to indicate normality. The slope of the transformed line gives us an estimate of the standard deviation. If... 

Ludena, E. V., L6pez-Boada Local-Scaling Transformation Version of Density Functional Theory Generation of Density Functionals. 180, 169-224 (1996). 

Local-Scaling Transformation Version of Density Functional Theory ... 

Local-scaling transformations, or point transformations, are generalizations of the well-known scaling transformations. The latter have been widely used in many domains of the physical sciences. Scaling transformations carry a vector into /( ) = Xr, where k is just a constant. In the case of local-scaling transformations, A is a function (i.e., k = A(r)). Notice that the transformed vector/(r) 6 conserves the same direction as the original one and is given by /(r) = k(f)r. In terms of the operator/associated with this transformations, we can relate F and J(F) by ... 

The full-fledged introduction of local-scaling transformations into density functional theory took place in the works of Kryachko, Petkov and Stoitsov [28-30, 32, 34], and of Kryachko and Ludena [1, 20, 31, 33, 35-37],... 

Let us consider now the application of local-scaling transformations to sets of single-particle functions or orbitals. As it was shown in Sect. 2.1, a set of plane waves gives rise to the transformed orbitals described by Eq. (2). In particular, the application of this transformation to one-dimensional plane-waves leads to Harriman s equidensity orbitals [27], which are given by ... 

Fig. 5. Graph of the iocai-scaling transformation function/(r) and related quantities...

Table I. Selected values of the Raffenetti-Hartree-Fock orbitals Isg and 2shf for Be, of their locally-scaled transformed functions IsJ, and 2sgr and of their differenees di, = Ish, - Isgj,) and = (2siif — 2sgp). [Reproduced with permission from Table I Ludeiia et al. [Ill]]...

The explicit construction to which Cioslowski refers is that provided by the density-driven approach, advanced in 1988. But, already in 1986, an alternative way for carrying out this explicit construction had been set forward by Kryachko, Petkov and Stoitsov [28]. This new approach - based on localscaling transformations - was further developed by these same authors [29, 30, 32, 34], by Kryachko and Ludena [1, 20, 31, 33, 35-37], and by Koga [51]. In this Section we show that Cioslowski s density-driven method corresponds to a finite basis representation of the local-scaling transformation version of density functional theory [38]. 

It is instructive, however, in order to establish the connection between the usual methods in quantum chemistry - based on molecular orbitals - and the local-scaling transformation version of density functional theory, to discuss Cioslowski s work in some detail. 

It is clear, therefore, that Cioslowski s approach based on density-driven orbitals [74, 75, 77], corresponds to a finite orbital representation of the local-scaling transformation version of density functional theory [38]. 

The Construction of Approximate Energy Density Functionals by Means of Local-Scaling Transformations... 

In Table VIII, we present the local-scaling- transformation-energy results for lithium and beryllium and compare them with results obtained with other methods. It is worth mentioning that the Hartree-Fock results for these atoms are a first instance of atxurate energy values obtained within the context of a formalism based on density functional theory. 

Considering a particular excited state n, we can obtain a local-scaling transformation function / (r) connecting the initial density p ,g(r) with the object density p (r), and generate the transformed wavefunctions ... 

We review in this Section some recent work by Ludena, Lopez-Boada and Pino [113] on the construction of energy functionals that depend explicitly upon the one-particle density, but which are generated in the context of the local-scaling-transformation version of density functional theory. This work does not consider the general case involving exchange and correlation, but restricts itself to the exchange-only Hartree-Fock approximation. 

The way in which local-scaling transformations have been used for the minimization of the kinetic energy functional is as follows [108-111], An arbitrary Slater determinant is selected to be the orbit-generating... 

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