Constrained search method

Zhao, Q. and Parr, R.G. (1993). Constrained-search method to determine wave functions from electronic densities, J. Chem. Phys. 98, 543-548. 

Constrained Search Method for Constructing Kohn-Sham Potentials... 

Auxiliary basis functions for expanding exchange-correlation potentials are used in the Wu-Yang (WY) constrained search method. Excerpt from Teale et al. (2008)... 

In Chap. 4, the Kohn-Sham equation, which is the fundamental equation of DFT, and the Kohn-Sham method using this equation are described for the basic formalisms and application methods. This chapter first introduces the Thomas-Fermi method, which is conceptually the first DFT method. Then, the Hohenberg-Kohn theorem, which is the fundamental theorem of the Kohn-Sham method, is clarified in terms of its basics, problems, and solutions, including the constrained-search method. The Kohn-Sham method and its expansion to more general cases are explained on the basis of this theorem. This chapter also reviews the constrained-search-based method of exchange-correlation potentials from electron densities and... 

All the basis functions were given independently variable orbital exponents (CH4, 9 NH3,8 H20,7) and all exponents were optimised by the quadratically convergent direct search method of Fletcher (19). For comparison, the calculations were repeated with the GHOs constrained to have the symmetry of the molecule three independent variables for CH4 (1 sc, sp3, 1 sH) four for NH3 (1 sN, sp3, sp3, 1 sH) and four for H20 (1 sQ, sp3, sp3,1 sH). The most striking qualitative result is the confirmation of the results quoted earlier for H2 when the orbital exponents are all optimised, the GHO basis has the symmetry of the molecule there is no spatial symmetry dilemma.9)... 

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... 

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... 

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. 

The constrained optimization procedure, originally developed from the simplex method and first described by Box, is ideally suited to model refinement (.8). It is a search method that searches for the minimum of a multidimensional function within given intervals. It possesses all the advantages of search methods, among them that calculation of derivatives is not necessary, a test to assure the independence of variables can be omitted, and diverse variables can be easily included. These are exactly the requirements of model refinement where bond lengths, bond angles, torsion angles, and other parameters are used within experimentally defined limits. 

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]. 

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. 

Unrestricted searching is a relatively simple method of bounding the optimum for problems that are not constrained. In engineering design problems, it is almost always possible to state upper and lower bounds for every parameter, so unrestricted search methods are not widely used in design. 

Once the construction of an appropriately constrained cost function is complete, its minimization in the absolute sense becomes the target. Gradient search methods are patently unsuitable for the minimization of... 

Dunlap (1984) advocates a computationally simpler procedure that does not involve solving an energy expression involving more than one set of orbital occupations simultaneously. This method is conceptually similar to Levy s constrained search over single determinants to find the one yielding the lowest kinetic energy. In this case we search over all single determinants to find the... 

Equation 15 was used as a constraint with a value between 12 and 13 for Z (n-decane conversion), during optimization of the reaction variables, using a Non-linear Quasi-Newton search method with tangential extrapolation for estimates, forward differencing for estimation of partial derivatives, a tolerance of 0.05 and precision of 0.0005. The search was also constrained by boundary conditions 1 to -1 for the reaction variables x, and solved for maximization of Y . 

R. A. Dammkoehler, S. F. Karasek, E. F. B. Shands, and G. R. Marshall,/. Comput.-Aided Mol. Design, 3,3 (1989). Constrained Search of Conformational Hyperspace. This method is implemented in the RECEPTOR module for the SYBYL software package. Tripos Associates, St. Louis, MO. 

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