Fuzzy density fragments approach

In more recent years, additional progress and new computational methodologies in macromolecular quantum chemistry have placed further emphasis on studies in transferability. Motivated by studies on molecular similarity [69-115] and electron density representations of molecular shapes [116-130], the transferability, adjustability, and additivity of local density fragments have been analyzed within the framework of an Additive Fuzzy Density Fragmentation (AFDF) approach [114, 131, 132], This AFDF approach, motivated by the early charge assignment approach of Mulliken [1, 2], is the basis of the first technique for the computation of ab initio quality electron densities of macromolecules such as proteins [133-141],... 

Mezey, P.G. (1998) A crystallographic structure refinement approach using ab initio quality additive fuzzy density fragments. In Advances in Molecular Structure Research, Hargittai, M. and Hargittai, I. (Eds.), JAI Press, New York. 

Using the additive, fuzzy density fragmentation (AFDF) approach of the macromolecular density pM(r)> the fuzzy fragment electron density... 

Electron density decreases exponentially with distance that suggests that an Additive Fuzzy Density Fragmentation (AFDF) approach can be used for both a fuzzy decomposition and construction of molecular electron densities. The simplest AFDF technique is the Mulliken-Mezey density matrix fragmentation [12,13], that is the basis of both the Molecular Electron Density Loge Assembler (MEDLA) [14-17] and the Adjustable Density Matrix Assembler (ADMA) [18-21] macromolecular quantum chemistry methods. 

A CRYSTALLOGRAPHIC STRUCTURE REFINEMENT APPROACH USING AB /N/T/O QUALITY ADDITIVE, FUZZY DENSITY FRAGMENTS... 

A simple, additive fragmentation approach to the molecular electronic density, proposed by the author, can be used for the construction of electronic densities and density-based shape representations for macromolecules. The simplest of these approaches is motivated by Mulliken s population analysis technique,and can be regarded as a natural generalization of Mulliken s approach a formal population analysis without integration. This method, the Mulliken-Mezey approach, is the simplest realization of a more general, additive fuzzy density fragmentation (AFDF) principle. ... 

The fundamental tool for the generation of an approximately transferable fuzzy electron density fragment is the additive fragment density matrix, denoted by Pf for an AFDF of serial index k. Within the framework of the usual SCF LCAO ab initio Hartree-Fock-Roothaan-Hall approach, this matrix P can be derived from a complete molecular density matrix P as follows. 

The computation of fuzzy electron density fragments and the construction of ab initio quality macromolecular electron densities is a novel approach to the quantum-chemical study and detailed modeling of large molecules. The additive ffizzy density fragmentation (AFDF) scheme of Mezey, described in a general form elsewhere, -is employed in the molecular electron density lego assembler (MEDLA) technique of Walker... 

If macromolecular density matrices are available, several new applications of quantum chemistry become possible. The ADMA technique provides the bridge between mainstream quantum chemistry and the additive fuzzy fragmentation approach. Most of the routine quantum chemical computational techniques, such as expectation value computations for various properties, become applicable to macromolecules if a macromolecular density matrbc can be computed. 

In many fuzzy electron density problems the density contributions of the functional groups and other moieties of the molecule X are also considered. In such cases, the membership function pFi (r) defined above is no longer appropriate. An alternative fuzzy set approach, that includes the effects of the electron density contributions of all other functional groups of the molecule, is based on Mezey s additive fuzzy density matrix fragmentation method [20,21],... 

The crystallographic structure refinement approach suggested is based on the viability of a fuzzy and additive density fragmentation technique. Ultimately, the fundamental quantum mechanical properties of density functionals justify the fuzzy fragmentation procedure, and the very same fundamental properties also suggest new approaches to local electron density shape analysis. These possibilities have relevance both to crystallographic structure refinement as well as to a density-based interpretation of the chemical properties of functional groups and other local... 

In fact, in a precise sense, no molecular fragment is rigorously transferable, although approximate transferability is an exceptionally useful and, if used judiciously, a valid approach within the limitations of the approximation. In particular, it is possible to define non-physical entities, such as fuzzy fragment electron densities, which do not exist as separate objects, yet they show much better transferability properties than actual, physically identifiable subsystems of well-defined, separate identity. This aspect of specially designed, custom- made , artificial subsystems of nearly exact additivity has been used to generate ab initio quality electron densities for proteins and other macromolecules. 

Whereas the first applications of the AFDF approach were based on a numerical combination of fuzzy fragment electron densities, each stored numerically as a set density values specified at a family of points in a three-dimensional grid, a more powerful approach is the generation of approximate macromolecular density matrices within the framework of the ADMA method [142-146]. A brief summary of the main steps in the ADMA method is given below. 

Some additional, quantum chemical and computational advantages of fuzzy fragments can be exploited in an approach designed to build electron densities of large molecules, partially motivated by an early approach of Christoffersen and Maggiora... 

In the following two sections two approaches will be discussed where molecular fragments are represented by fuzzy electron density models. 

Following the principles of the ZPA approach, these symmetry deficiency measures are generalizations of the folding-unfolding approach, equally applicable to crisp continuum sets and fuzzy sets, for example, to entire electron density distributions of molecules and various molecular fragments representing fuzzy functional groups. 

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