Constrained-search, Levy construction

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.)... 

The horizontal character of changes in the ground-state electronic structure is in contrast to a search for the equilibrium partitioning of the fixed molecular ground-state density p(r) into the subsystem densities p(r) = pa(r) = pa[p r] (a row vector), e.g., those of m constituent AIM or two reactants, which sum up to this given density, p(r) = PJr)- The latter is vertical in character, i.e., performed for the fixed molecular density [20-24,33], as is also the case in the Levy constrained search construction of F[p] [124]. As argued elsewhere [33], such a... 

Recently, a construction of such subsystem orbitals from the overlaping densities and Pg (Fig. lA) has been proposed (15), on the basis of the earlier developments (23-27) for the overall density p. Such density conserving orbitals can be used to describe the sybsystem wavefunctions that yield the prescribed densities of these complementary framgments of M., e.g., in the subsystem application (15) of the Levy (28) constrained search construction of the functional for the sum of the electronic kinetic and repulsion energies. 

In order to formally express these occupied orbitals as functionals of the relevant densities the constrained density/densities constructions of Levy [17] can be used. They involve searches over trial orbitals, defining the relevant Slater determinants for the subsystems or for M as whole, which conserve the prescribed density or densities. More specifically, the optimum orbitals of M as a whole can be expressed as functionals of the overall density, [p], through the familiar Levy construction of the density functional for the sum of the electronic kinetic and repulsion energies ... 

The particular choice of the equidensity orthmormal orbitals defining the Slater determinant that yields a prescribed electrcm density p(r) has been proposed by Harriman [18] on the basis of the pioneering works by Macke [19] and Gilbert [20]. Alternative constructions and extensions have also been suggested [21, 22]. in die density functional theory such A-electron wavefiinctions are mvolved in the formal density constrained search of Levy [17]. 

It has also been shown that the constrained-search construction of Levy can be used to determine orbitals defining the density functional for the nonadditive kinetic energy. It involves a search over the subsystem wave functions, which integrate to specified, overlapping densities of molecular fiagments. The trial wavefunctions of... 

We seek a density functional analog of (1.30). Instead of the original derivation of Hohenberg, Kohn and Sham [25,6], which was based upon reductio ad absurdum", we follow the constrained search approach of Levy [28], which is in some respects simpler and more constructive. 

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