Levy’s constrained search

The paper of Parr and Bartolotti is prescient in many ways [1], It defines the shape function and describes its meaning. It notes the previously stated link to Levy s constrained search. It establishes the importance of the shape function in resolving ambiguous functional derivatives in the DFT approach to chemical reactivity—the subdiscipline of DFT that Parr has recently begun to call chemical DFT [6-9]. Indeed, until the recent resurgence of interest in the shape function, the Parr-Bartolotti paper was usually cited because of its elegant and incisive analysis of the electronic chemical potential [10],... 

Following Levy s constrained-search formulation [9] (see also [10]) we can perform the minimization in Eq. (9) in two steps, namely... 

The success of a determinantal approach, leading to one-electron equations in the HF approximation, served as inspiration for applying it to the exact GS problem. Stemming from the ideas of Slater [6], the method was formally completed in the work of Kohn and Sham (KS) [8], and is traditionally known as KS approach. We recall it now using again a Levy s constrained-search... 

Another route to construction of the approximate 1-RDM functional involves employment of expressions for E and D afforded by some size-consistent formalism of electronic structure theory. Mazziotti [42] proposed a geminal functional theory (GET) where an antisymmetric two-particle function (geminal) serves as the fundamental parameter. The one-matrix-geminal relationship allowed him to define a D-based theory from GET [43]. He generalized Levy s constrained search to optimize the universal functionals with respect to 2-RDMs rather than wavefunctions. 

We do not wish, however, to leave the reader with the impression that LS-DFT is either the only or the best constructive version of DFT. In the first place, let us remember that LS-DFT is closely related to Levy s constrained-search reformulation of DFT [48] which in the words of Cioslowski [19] "... yielded an implicit construction for the functional E[p]. In fact, severed attempts have been made [49] in the context the constrained-search formulation for the purpose of attaining an explicit construction for E p. In this vein, a vast array of known properties have been determined for the exact functionals [50-55] these properties, clearly, can be and have been incorporated into the construction of approximate functionals [56]. (In this respect, it is pertinent, however, to remark that the functionals obtained in LS-DFT satisfy by construction the requirements derived from ordinary scaling as the latter is a particular instance of local-scaling.)... 

Dunlap (1984) advocates a computationally simpler procedure that does not involve solving an energy expression involving more than one set of orbital occupations simultaneously. This method is conceptually similar to Levy s constrained search over single determinants to find the one yielding the lowest kinetic energy. In this case we search over all single determinants to find the... 

The gist of Cioslowski s work is to set up an energy functional that depends on the density, in the context of the constrained-search approach of Levy [84]. This functional is, therefore, defined by ... 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

Nb)i and Hi) ptflal being the difference of two functions integrable in the sense of Eq. 7, is also integrable in this sense. Therefore, ptpal is TV-representable and the Levy constrained search can be performed. Let s consider all possible sets of orthogonal functions such that ... 

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