Mapping between local potentials and ground state densities

Abstract

AbstractsThe mapping between the local one-body potential and the ground state density is discussed. The role of degenerate ground states, which form a set of measure zero in potential space, is shown to be amplified by the unitary transformation between degenerate densities, forming a space filling set in density space. A numerical experiment demonstrating these features is carried out. No indication of non-V representable densities is encountered.On discute la correspondance entre le potentiel local B un corps et la densit6 de 1'6tat fondamental. I1 est dtmontrt que l'importance des ttats fondamentaux dtgknkrks qui forment un ensemble de mesure ztro dans I'espace des potentiels, est amplifite par les transformations unitaires entre des densitts dtgtntrts que forment un ensemble complet dans l'espace des densitts. Une experience numtrique dtmontre ces aspects. Aucune indication de densitts non V-reprksentables a BtC trouv6e.Die Abbildung zwischen dem lokalen Einteilchenpotential und der Dichte des Grundzustands wird diskutiert. Es wird gezeigt dass die Rolle der entarteten Grundzustande, die eine Menge von Mass Null im Potentialraum bilden, von den unitaren Transformationen zwischen entarteten Dichten vergrossert wird, die eine Raumfiillende Menge im Dichteraum bilden. Ein numeiisches Experiment, das diese Tatsachen demonstriert, ist ausgefiihrt worden. Keine Andeutung von nicht V-darstellbaren Dichten wird gefunden.

Exact Density Functionals for Ground‐State Energies II. Details and Remarks

“…As is easily seen, in this case all E-V densities are PS-V representable. (This is no longer valid for the state space R-4) Thus we can confirm this conjecture from [25]. But for state spaces larger than C4 the fermion systems can possess a higher degeneracy and then, as mentioned above, Levy [12] and Lieb [7] showed that PS-V and E-V representability no longer coincide.…”

Energy Density Functional Theory of Many-Electron Systems

“…Hence, it can be regarded in the form (6.2-8) with 1/;i = 1l'(Xi), i = 1,2, ... , \I. In order to investigate the nature of mapping between ground-state d ~nsities and local external potentials, let us consider the three-dimensional model system of two noninteracting electrons with 1-1=4 as "the smallest system which is not obviously trivial" (Katriel et al, 1981b). (6.2-27), differs slightly, through the denominator, from that given by Eq.(6.2-5).…”

“…Katriel et al (1981b) Repeating this procedure many times, this three-dimensional array of the density space may be covered in such a way that some (accesible or A-type) regions are covered in an essentially uniform manner and the other (inaccesible or I-type) ones remain entirely forbidden (see Figure 6.2). In this way, one can choose the off-diagonal elements of the Ho-matrix once and for all for the whole Ho-familv of trial Hamiltonians.…”

“…from the density space (Katriel et al, 1981b). As an initial step, the trial V-matrix ''ir is chosen and the eigenvalue ground-state problem for the Hamiltonian Htr = Ho+ ' '',.t" is soh·ed; finally the trial density PtT is evaluated as the point in the density space with the coordinates pt r , p~r, and pt.;.…”

“…As an initial step, the trial V-matrix ''ir is chosen and the eigenvalue ground-state problem for the Hamiltonian Htr = Ho+ ' '',.t" is soh·ed; finally the trial density PtT is evaluated as the point in the density space with the coordinates pt r , p~r, and pt.;. Katriel et al (1981b) have shown that if Po belongs to the A-type region of the density space, the inverse problem has a solution whereas if Po is chosen from the I-type region, the optimization procedure fails to converge. Katriel et al (1981b) have shown that if Po belongs to the A-type region of the density space, the inverse problem has a solution whereas if Po is chosen from the I-type region, the optimization procedure fails to converge.…”