Local-scaling transformations, for

We have reviewed here the implementation of the inverse method for going from densities to potentials, based on local-scaling transformations. For completeness, let us mention, however, that several other methods have also been advanced to deal with this inverse problem [101-111]. Consider the decomposition of into orbits Such orbits are characterized by the fact that... 

There is still a very important area of contact between LS-DFT and other approaches having to do with the direct evaluation of the Kohn-Sham potentials [6,62-66] from known densities. In this respect, the work, of Zhao, Morrison and Parr [66] is of particular interest as it provides an alternative to local-scaling transformations for a fixed-density variation of the kinetic energy functional. 

Several methods have been advanced in the literature for the purpose of going from densities to potentials [78-88]. We review here the use of local-scaling transformations for carrying out the density-constrained kinetic energy minimization of a non-interacting system, as through the solution of this problem one can obtain exact Kohn-Sham orbitals and potentials. 

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] ... 

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]]...

Local-Scaling Transformation Version of Density Functional Theory Talbe 3. Orbital parameters for function... 

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... 

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]. 

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... 

Let us finish this Section by discussing the relationship between the Kohn-Sham-like equations advanced above and the actual Kohn-Sham equations. From the perspective of local-scaling transformations, we can analyze this relationship as follows. First of all, we assume that, for an interacting system, we are able to select an orbit-generating wavefunction belonging to the... 

Let us note, however, that in spite of the fact that the orbit jumping optimization is carried out at fixed density, the resulting wavefunction, epi[p%) = is not necessarily associated with the fixed density p ]],(r). For this reason, it is then possible to apply a local-scaling transformation to it and produce an optimized wavefunction which, at the same time, is associated with the fixed density. We denote this wavefunction by Moreover, we can... 

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. 

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. 

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... 

Improvement of Correlated Wave Functions for Helium by Means of Local-Scaling Transformations... 

In the present review of LS-DFT the constructive nature of this approach leading to the actual formulation of density functionals has been emphasized. We have shown explicit expressions for the kinetic and exchange energy functionals and have compared the former to a number of the usual representations advanced in conventional DFT. We have further analyzed the correlation problem from the point of view of local-scaling transformations and have made some... 

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). 

Let us go back to Fig. 1 and consider the orbits" Ol for = 1, , HK, . These orbits" or columns appearing in Fig. 1 are made up of wavefunctions belonging to Hilbert space (at this point we just assume that they exist a formal definition is given in Section 2.6). We assume, furthermore, that these orbits" are endowed with the following characteristic no two wavefunctions belonging to the same orbit" can have the same density, i.e., there is a one to one correspondence between p r) e Af and W g (this fact is proven in Section 2.6, using local-scaling transformations). We assume, moreover, that the union of all orbits" exhausts Hilbert space. Clearly, in terms of these orbits", the variational principle can be reformulated as follows [21] ... 

Combining the above equation with Eq. (30), we obtain the following first-order differential equation for the evolution of the one-particle density as a result of the application of local-scaling transformations ... 

The uniqueness of the local-scaling transformation guarantees that within an orbit G[ c CN there exists a one to one correspondence between one particle densities p r ) e Mb and /V-particle wavefunctions W df C Cn- This very important result is fundamental for obtaining the explidt expression for the energy density functional within an orbit. This is discussed in Section 2.8. 

Let us now prove that the set Av of u-representable one-particle densities is a subset of M. For this purpose, let us use local-scaling transformations. However, as these transformations can only be rigorously applied to densities belonging to Mb,... 

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] ... 

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]... 

Let us now indicate how local-scaling transformations can be used in order to carry out the constrained minimization of the kinetic energy functional [85-88]. The strategy that we have adopted is first to select a Slater determinant such as the one appearing in Eq. (73), as the trial wavefunction which generates the particular orbit d c Sn- For the case of atoms, the one-particle orbitals ( >g,i r) from which this Slater determinant is constructed are explicitly given by g,nim(r) = Rg,ni r)Yiimi(0,), where the subindex i has been replaced by the quantum numbers n, /, m. The radial functions are expanded as... 

Calculation of the energy and wavefunction for the beryllium atom at the Hartree-Fock level by in the context of local-scaling transformations... 

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]. 

In order to illustrate how intra-orbit optimization of the energy may be accomplished by non-variational methods, let us consider some of the entries in Table 7. Let us assume that the orbit-generating wavefunction for orbit is W, which, according to Eq. (86) has the expansion coefficients (7 and yields the density pg(x). For the primitive orbital set A, the energy associated with this wavefunction is -14.538796 hartrees. Now, any displacement within orbit O must be accomplished by means of a local-scaling transformation. Consider that we carry out such a transformation between densities pg(x) and phf(x) and that by solving Eq. (37) we obtain the transformation function f(r). By means of Eq. (110), we can then transform the initial set A into a locally-scaled one from which the new wavefunction M HF can be constructed. Notice that because local-scaling transformations act only on the orbitals, the transformed wavefunction conserves the... 

Considering that the optimal energy for an untransformed function is—14.599936 hartrees (first entry for set C in Table 7), we see that local-scaling transformations have a considerable effect on these configuration interaction wavefunctions. Since the best locally-scaled energy is -14.612495 hartrees (last entry for set C in Table 7), we observe that these transformations improve the energy by -0.012 559 hartrees. 

---