Local-scaling transformation of the

The action of a local-scaling transformation of the density within the interacting Hohenberg-Kohn orbit carries the wavefunction 0 C,K C Cn into the transformed wavefunction [,WA l e c Cn- The same occurs within the noninteracting Hohenberg-Kohn orbital where the wavefunction s°lHK] g c SN... 

As discussed in Section 3.1.1, starting from an orbit generating wavefunction W(ri, , r,v) in position space, we may compute the optimal one-particle density Polt(r) = pii) jj(r) by optimizing the energy functional S[p x) j 1] subject to a normalization condition on the density. In other words, the optimal wavefunction , rV) within the position orbit is obtained by means of a local-scaling transformation of the orbit-generating wavefunction. 

By using local scaling transformation of the electron density, one can rewrite any kinetic energy functional as a one-point nonsymmetric WDA. Specifically, one has... 

Thus, for any p(r) e there exists a unique wavefunction generated by means of local-scaling transformation from the arbitrary generating wavefunction The set of all the wavefunctions thus generated, yielding densities p(f) in J g, is called an orbit and is denoted by... 

We describe in this Subsection the application of local-scaling transformations to the calculation of the energy for the lithium and beryllium atoms at the Hartree-Fock level [113]. (For other reformulations of the Hartree-Fock problem see [114] and referenres therein.) The procedure described here involves three parts. The first part is orbital transformation already discussed in Sect. 2.5. The second is intra-orbit optimization described in Sect. 4.3 and the third is inter-orbit optimization discussed in Sect. 4.6. 

Because a local-scaling transformation relates the densities p(r) and pg(r) by means of the following first-order differential equation [8] ... 

The above demonstrated possibility of obtaining numerical virtual orbitals indicate that the FD HF method can also be used as a solver of the Schrodinger equation for a one-electron diatomic system with an arbitrary potential. Thus, the scheme could be of interest to those who try to construct exchange-correlation potential functions or deal with local-scaling transformations within the functional density theory (32,33). 

Local-scaling transformations and the rigorous definition of the concept of orbit"... 

In order to obtain from [W] the fvmctional E[f] appearing in the variational prin-riple described by Eq. (24), we have to apply local-scaling transformations to the wavefunction M, or in view of Eq. (42), to the 1-matrix LA, to the 2-matrix Df and to the density pg. The pertinent expressions for these transformations are [21] ... 

By means of a local-scaling transformation carrying the initial density pn,g(r) into the object density p (r), we can obtain a transformation function fn(r) from which, in turn, we can generate the transformed wavefunctions... 

Because local-scaling transformations preserve the orthonormality of basis functions, condition (54) is immediately fulfilled. Hamiltonian orthogonality (Eq. (55)), however, is not satisfied. For this reason, one must solve the eigenvalue equation (51]... 

Clearly, we have in the present case that the union of all orbits exhausts the subclass Sn = The action of local-scaling transformations on the initial... 

Because the determinants appearing in the expansions of the configuration state functions" are constructed from a single-orbital set 4 k r, s) =1 where K > N, the effect of local-scaling transformations involves the replacement in each one of the single-Slater determinants of Eq. (87) of the initial orbitals by transformed orbitals belonging to the set ) = Thus, we are led to the set of... 

Further intra-orbit optimization becomes a rather delicate matter in view of the fact that one needs a very accurate representation of the one-partide density in order to reach the Hartree-Fock value. For this reason, we have performed local-scaling transformations from the optimal density pk(x) to the density pl(x) which comes from an approximate Hartree-Fock wavefunction whose energy is -14.572993 hartrees. The energy functional (Eq. (113)) reaches the value of -14.573 003 9 hartrees for pl(x) and a l, Pi. A closer approximation to the limiting Hartree-Fock value is attained when the transformation is carried out to the final density phf(x) of Boyd [92], In this case, the energy is lowered to —14.5730208 hartrees, a value that compares quite well with the limiting Hartree-Fock value of -14.573 02313 [90]. 

Carrying out a local-scaling transformation between the one-particle density pg(r) associated with and the fixed exact" density pci(r) of Esquivel and Bunge [93], using Eq. (110), we obtain a set of transformed orbitals from which we can generate a set formed by the following transformed single-Slater determinants ... 

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],... 

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...

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