Orbital local-scaling transformation

Orbital local-scaling transformation approach fermionic systems in the ground state 45... 

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]]...

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... 

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 . 

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]. 

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... 

The uniqueness of the local-scaling transformation guarantees that within an orbit there exists a one-to-one correspondence between one particle... 

Hohenberg-Kohn orbit. Clearly, within the application of local-scaling transformations to any initial wavefunction leads to the exact ground-state wavefunction as well as to the exact ground-state density. 

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. 

Calculation of Kohn-Sham Orbitals and Potentials by Local-Scaling Transformations... 

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... 

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... 

Consider a local-scaling transformation that carries the vector r into the vector (r)f. Applying this transformation to the arbitrary generating atomic orbital set g,i r) = Rg,niii r)Yiiirni.(9,4>) iLi one obtains the following set of transformed orbitals p,i r) = f p,r>iii(r)Viiim( (0, where the... 

The wavefunction w(n, r2, ri2) has been recently described [47]. The results are presented in Table 3. As it can be seen, in all cases the energy is improved by a local-scaling transformation. Also, note that as discussed above, a transformation to Pexact does not necessarily lead to the lowest energy (it does not yield a minimum in an arbitrary orbit). 

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] ... 

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

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. 

In Sections 3.1.5 and 3.1.6 we discuss two important alternative methods to intraorbit optimization. Both are based on the use of local-scaling transformations in order to produce sets of transformed orbitals which are then directly employed in the calculation of the total energy. In the non-variational case, we deal with arbitrary orbitals which are locally-scaled in order to yield the Hartree-Fock one-particle density, which we assume to be known beforehand. In the second method, the final density is optimized by energy minimization. But as in the previous case, locally-scaled transformed orbitals are used in the energy calculation. 

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... 

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... 

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... 

---