Local-scaling transformations

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 ... 

In order to see that they correspond to the general transformations studied by Moser [58], consider the effect of applying a local-scaling transformation, denoted by the operator f, to each of the coordinates appearing in the wavefunc-tion i(Fi,. ..,F v) 6 ifjv. Hence, the resulting wavefunction 2(Fi,...,Fjv) e is given 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 now show that when we apply a local-scaling transformation to a closed contour Si(r C) of p, (defined by pi( ) — C = 0), we obtain a closed... 

Fig. 3. Schematic representation of the transformation of a closed density contour curve of pj (dark contour on the left-hand side) into that of p2 (dark contour right) by local-scaling transformations...

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 ... 

We illustrate here a specific example of the application of local-scaling transformations to atomic orbitals [111]. Consider the i is(r) and / 2s( ) orbitals of the Raffenetti type for the beryllium atom [71] ... 

The density associated with the Hartree-Fock-Raffenetti wavefunction is denoted by puVif)- We take this to be the initial density in our local-scaling transformation, i.e., pi r) = puVir)- We take as the final density, that associated with the 650-term Cl wavefunction of Esquivel and Bunge [73], which we call P2ir) = pair). These two densities are practically about the same, as can be seen clearly in Fig. 4, where we have also plotted their difference. The transformed radial orbitals are given by ... 

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]]...

Thus, for example, from wavefunction we can generate an orbit which contains - among the infinite number of wavefunctions obtained through the application of local-scaling transformation, the particular wavefunctions and The important aspect of orbits is that the uniqueness of the... 

Equation (99) is completely equivalent to Eq. (39). Thus, the transformations discussed here and the local-scaling transformations coincide in this spherically symmetric case. It is clear however, that the procedure outlined in this Section is much more flexible than the one presented in Sect. 2.2. 

Just as it was done in position space, we can introduce local-scaling transformations in momentum space, in view of the fact that the momentum densities satisfy the following conditions ... 

Thus, we can consider the following local-scaling transformation in momentum space ... 

It follows from the above considerations that local-scaling transformations can be advanced in momentum space on an equal footing with those in position space. In particular, wavefunctions in momentum space can be transformed so as to generate new wavefunctions that have the property of belonging to an orbit . 

Local-scaling transformations have been employed [39] in order to obtain a one-particle density in position space from the one-particle density in momentum space, and vice versa. This problem arises from example when we have a y(p), obtained from experimental Compton profiles, and wish to calculate the corresponding p(r) [98]. 

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... 

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