Levy constrained search formalism

When we proved the Hohenberg-Kohn theorem above, we made the assumption that the density is v-representable. By this is meant that the density is a density associated with the anti-symmetric ground state wave function and some potential v(r). Why is this important The reason is that we want to use the variational character of the energy functional  

Here tfo is the ground state wave function and Tno is any other wave function yielding the same density. We recognize this as the Hohenberg-Kohn functional, Fhk- It turns out that the ground state wave function of density n(r) can be defined as the wave function which yields n(r) and minimizes the Hohenberg-Kohn functional. 

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  

the minimization is over all n which are /V-representable. What this means is that n can be obtained from an anti-symmetric wave function, and that it 

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From a purist theoretical point of view, there is one further important result hidden in the Levy constrained-search strategy it provides a unique, albeit only formal, route to extract the ground state wave function F, from the ground state density p0. This is anything but a trivial problem, since there are many antisymmetric N-electron wave functions that yield... 

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 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. 

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]. 

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 ... 

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