Levy constrained search formulation

The Levy constrained search formulation of the Hohenberg-Kohn theorem [18, 20, 21] now states that we can divide our search for the ground state energy into two steps We minimize EVo [n] first over all wave functions giving a certain density, and then over all densities ... 

Since we want to be able to do calculations on magnetic materials, we must expand the formalism to allow for spin-dependent external potentials. This is quite easily done with the use of the Levy constrained search formulation. 

The difficulty of proving this is that each spin component only determines the potential up to a constant, and that we only can get rid of one of the constants by changing energy scales in our potential [25]. This is fortunately enough not a big problem when doing actual calculations, since we then have v(r) and B(r) given [26]. It has been shown that the Levy constrained search formulation can be extended without problems to the spin-dependent case [27]. 

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... 

DFT can be developed from the constrained search formulation of Levy, [12] since we know that... 

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

M. Levy and J. P. Perdew, The constrained search formulation of density functional theo ry, in Density functional methods in physics, edited by R. M. Dreizler and J. da Providencia, pages 11-30, Plenum, 1985. 

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.)... 

An alternative proof of N-representability of such a class of electron densities was given by Harriman46. Eqs. 5-7 define a large class of electron densities relevant for chemical systems. Unless specified, the electron densities considered throughout the present review are TV-representable. The searching procedure of Eq. 4 is a key element of the Levy constrained-search. (LCS) formulation of density functional theory47. 

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. 

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