Density functional theory constrained-search

In the context of density functional theory (DFT), the shape function can be considered to be the fundamental variable in the Levy-constrained search [5],... 

The basic quantity in density functional theory is the energy functional which within constrained search [24, 25] is defined as... 

In the second place, a quite useful characteristic of LS-DFT is that it renders possible to transform an arbitrary wavefunction, say, the Hartree-Fock single Slater determinant into a locally-scaled one associated with a given one-particle density such as the exact one. Thus, one can easily generate a locally-scaled Hartree-Fock wavefunction that yields the exact p. In this sense, one finds much common ground between LS-DFT and those constructive realizations of the constrained-search approach which reformulate the Hartree-Fock method as well as with those developments which pose the optimized potential method as a particular instance of density functional theory [42,43,57-61]. 

M. Levy and J. P. Perdew, The constrained search formulation of density functional theory, in Density Functional Methods in Physics (Dreizler, R. M. and Providencia, J. da, eds) New York Plenum, pp. 11-30 (1985). 

Levy, M., Perdew, J. The Constrained Search Formulation of Density Functional Theory, In Dreizler, R. M., da Providencia, J. (eds.) (1985). Density Functional Methods in Phywzcs, Plenum Press, New York, AM7D/1S7 Senes B.-Physics 123,11-31. 

The original density functional theory (DFT), based on Hohenberg-Kohn theorems [1], Kohn-Sham equations [2] and the Levy constrained search formulation [3], is a rigorous approach for determining the ground-state density and ground-state energy for any A/ -electron system. Here the electron number... 

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]. 

To complete the definition of the functional derivatives and of the chemical potential /x, we extend the constrained search from wavefunctions to ensembles [49,50]. An ensemble or mixed state is a set of wavefunctions or pure states and their respective probabilities. By including wavefunctions with different electron numbers in the same ensemble, we can develop a density functional theory for non-integer particle number. Fractional particle mun-bers can arise in an open system that shares electrons with its environment, and in which the electron number fluctuates between integers. 

Levy M (1983) The constrained search approach, mappings to extianal potentials, and virial-like theorems for electron-density and one-matrix eneigy-fimctional theories. In Keller J, Gazquez JL (eds) Density functional theory, vol 187, Lecture Notes in Physics Springer, Heidelberg, pp 9-35... 

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. 

Both formal analysis and computational developments associated with DFT can be carried over intact to nDFT. For example, the exact two-particle ground-state density, no(x), can be determined through a constrained search [34] for that many-particle, properly symmetrized or antisymmetrized wave function, with symmetry imposed with respect to ordinary particles, which yields n0 and also minimizes the many-particle energy, T + Vpp, where Vpp denotes the interparticle interaction in two-particle space. Essentially any method developed within a single-particle application of DFT for the study of electronic structure can, with appropriate technical modifications, be extended to two-, or rc-particle states. The use of multiple-scattering theory to calculate fully correlated two-particle densities in solids will be given in a future publication. 

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