The Constrained-Search Approach

In this section we introduce a different way of looking at the variational search connected to the Hohenberg-Kohn treatment. Recall the variational principle, equation (1-13) as introduced in Chapter 1 

In words, we search over all allowed, antisymmetric N-electron wave functions and the one that yields the lowest expectation value of the Hamilton operator (i. e. the energy) is the ground state wave function. 

In order to connect this variational principle to density functional theory we perform the search defined in equation (4-13) in two separate steps first, we search over the subset of all the infinitely many antisymmetric wave functions Ex that upon quadrature yield a particular density px (under the constraint that the density integrates to the correct number of electrons). The result of this search is the wave function T xin that yields the lowest 

The energy due to the external potential is determined simply by the density and is therefore independent of the wave function generating that density. Hence, it is the same for all wave functions integrating to a particular density and we can separate it from the kinetic and electron-electron repulsion contributions 

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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 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]. 

B. The Constrained Search Approach First choose a fixed electron density p(r) and consider the family,... 

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. 

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... 

A very different approach has been followed by Zhao et al. [54-57] who based their method on the constrained search definition of the Kohn-Sham kinetic energy. It follows from this definition that, from all Slater determinants which yield a given density, the Kohn-Sham determinant will minimize the kinetic energy. Suppose we have an exact density po- If one minimizes the Kohn-Sham kinetic energy under the constraint... 

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],... 

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... 

Constrained Search and the related approach of SCAMPI both integrate the conformational analysis closely with the pharmacophore discovery. This has the advantage that the sampling of conformational space can be more focused on key regions. With both Catalyst and DANTE, conformational analysis was explicitly kept separate, in the latter to allow one to take advantage of any innovations in conformational analysis tools. And, indeed, there continues to be a steady flow of new approaches in conformational analysis—pharmacophore discovery is critically dependent on high-quality exhaustive conformational analysis. Based on our experience thus far, we cannot conclude that either approach is superior (integrated vs. external). Furthermore, a consensus has not yet been reached on the optimal manner to perform conformational search as needed by pharmacophore discovery. This will continue to be a fruitful area of research. 

This statement of the theorem of Hohenberg and Kohn is the justification of many of the modern density functional approaches to chemistry. Here we shall outline the generalized and very concise constrained search proof as described by Levy [45-48]. This proof, as we shall see, lends itself to further generalizations that are of special advantage in the case of large molecules, such as proteins. 

The most important distinction between DEDA and other wave function-based EDA approaches [4-15] lies in the calculation of the frozen density energy. We have explained above how DEDA uses constrained search to variationally calculate the energy of the frozen density state where fragments densities are superimposed without distortions. This approach not only yields an optimal... 

This taboo search approach is then realized by solving the constrained minimization problems ... 

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... 

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