Inter-orbit optimization

In Fig. 7, we present a general scheme, comprising the intra- and inter-orbit optimizations appearing in the variational problem described by Eq. (138). We discuss this optimization process with reference to only three of the infinite number of orbits into which Hilbert space is decomposed. These orbits are... 

Fig. 7. Schematic representation of intra-orbit and inter-orbit optimizations...

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. 

The intra-orbit optimization process is carried out by varying the density parameters uj, 6,-, cj while keeping the wavefunction (or orbital) parameters fixed. Alternatively, inter-orbit optimization is accomplished by optimizing the wavefunction parameters for the energy functional... 

The resulting functional has been evaluated for the initial orbitals of the lithium and beryllium atoms given in Section 4.6. Using the same process of intra- and inter-orbit optimization carried out in Section 4.6, but substituting, in each step, the value for /(r) given by the Fade approximant obtained from Eq. (189), energy values were obtained for the Li and Be atoms that are indistingishable from those previously calculated with the exact values of/(r), namely, when Eq. (40) is solved. 

Moreover, the globed minimum can be reached by inter-orbit jumping . This may be accomplished by modifying the generating wavefunction at fixed final density. In the present case, this inter-orbit optimization is carried out by varying the parameters at fixed density. 

INTRA-ORBIT AND INTER-ORBIT OPTIMIZATION SCHEMES... 

Inter-orbit optimization of Cl wavefunctions via density-constrained variation... 

Inter-orbit optimization through the combined use of position and momentum energy functionals... 

Without resorting specifically to orbit-jumping", it is still possible to carry out inter-orbit optimization by combining intra-orbit optimizations performed in coordinate and momentum spaces [76]. In order to see how this can be achieved, consider the Fourier transformation of the orbit generating wavefunction M(ri, . rN) =... 

We base the present inter-orbit optimization scheme precisely on the fact that (pi, , pN) and (pi, , pV) are not necessarily equal. The following functional (constructed in analogy with Eq. (61))... 

In the spirit of the local-scaling version of density functional theory, we deal below with the calculation of the Hartree-Fock wavefunction and energy by means of a constrained variation at fixed density p(r) = phf( ) as well as by intra-orbit and inter-orbit optimizations. 

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