Constrained search

Dammkoehler R A, S F Karasek, E F B Shands and G R Marshall 1989. Constrained Search c Conformational Hyperspace. Journal of Computer-Aided Molecular Design 3 3-21. 

Similarly, the constrained-search scheme, even though being very elegant in appearance and strong in formal power, is only of theoretical value and offers no solution to practical considerations. Simply, the program indicated in Section 4.3 cannot be realized - how would we ever be able to search through all wave functions Since this is obviously impossible, setting up the functional F[p] = min (T IT + Vee I F) is impossible, too. A second... 

The variational principle is written in the form of a 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 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],... 

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... 

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... 

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

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 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 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. 

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 2-RDM formulation, Eq. (38), allows us to generalize the constrained search to approximately V-representable sets of 2-RDMs. In order to approximate the unknown functional Eee[V, D], we use here a reconstructive functional D[ D] that is, we express the elements D h in terms of the We neglect any explicit dependence of on the NOs themselves because the energy functional already has a strong dependence on the NOs via the one- and two-electron integrals. 

The interelectronic interactions W are defined using constrained search [21, 22] over all A-representable 2-RDMs that reduce to R g). Since the set of 2-RDMs in the definition of W contains the AGP 2-RDM of g, that set is not empty and W is well defined. Through this construction, E still follows the variational principle and coincides with the energy of a wavefunction ip, which reproduces R g) = D[ T ] and = W[g]. The latter is due to... 

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. 

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... 

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