Electron density fuzzy fragments

Transferability, adjustability, and additivity of fuzzy electron density fragments... 

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],... 

The approximate transferability of fuzzy fragment density matrices, and the associated technical, computational aspects of the idempotency constraints of assembled density matrices, as well as the conditions for adjustability and additivity of fragment density matrices are discussed in Section 4, whereas in Section 5, an algorithm for small deformations of electron densities are reviewed. The Summary in Section 6 is followed by an extensive list of relevant references. 

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. 

If the electron density partitioning results in subsystems without boundaries and with convergence properties which closely resemble the convergence properties of the complete system, then it is possible to avoid one of the conditions of the Holographic Electron Density Fragment Theorem , by generating fuzzy electron density fragments which do not have boundaries themselves, but then the actual subsystems considered cannot be confined to any finite domain D of the ordinary three-dimensional space E3. 

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. 

This, in turn, implies the exact additivity of the fuzzy electron density fragments p1 (r) as given by Equation (39). 

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. 

In structure determination from X-ray diffraction data, it sometimes happens that, on the Fourier maps, parts of the coming out structure are unclear. Fuzzy electron density maps may present problems in determining even the approximate positions of the respective fragments of the structure being analyzed. For example, the layered structure of the inclusion (intercalation) compound formed by Ni(NCS)2 (4-methylpyridine)4 (host) and methylcellosolve (guest) [1], The guest molecules are (Fig. 11.1) located on twofold crystal axes of unit cell symmetry and are orientationally disordered as shown in the picture. 

The Mulliken-Mezey Additive Fuzzy Electron Density Fragmentation Method... 

Linear Homotopies of Fuzzy Electron Density Fragments... 

In the general scheme described in subsequent sections, a functional group is regarded as a fuzzy body of electronic charge cloud, a fuzzy subset of the electronic charge density cloud of the complete molecule. In this context, a functional group is a special case of a fuzzy fragment of a molecular body, obtained by some subdivision... 

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. 

Based on the fragment density matrix Pk for the k-th fragment, the electron density of Mezey s fuzzy density fragment pk(r) is defined as... 

The fuzzy electron density fragment additivity rules (23) - (27) are exact at any given ab initio LCAO level, hence the reconstruction of the calculated electronic density p(r) of the given molecule from the corresponding fuzzy fragment electron densities pk(r) is also exact. 

The additive fuzzy electron density fragmentation scheme of Mezey is the basis of the Molecular Electron Density Lego Assembler (MEDLA) method [67,70-72], reviewed in section 4. of this report, where additional details and applications in local shape analysis are discussed. The MEDLA method was used for the generation of the first ab initio quality electron densities for macromolecules such as proteins [71,72] and other natural products such as taxol [66],... 

The exact additivity of the fragment density matrices Pk and the fuzzy fragment densities pk(r) defined according to Mezey s scheme (eqs. (23)-(27)) motivates the terminology additive, fuzzy electron density fragmentation method. The fuzzy electron density fragment additivity rule is exact at the given ab initio LCAO level. 

The linearity of density expressions (16) and (24) in the corresponding density matrices ensures exact additivity for the fuzzy electron density fragments, as described by eq. (27) ... 

The sum of all m fuzzy fragment densities pk(r) is, indeed, equal to the total electron density p(r) of the molecule. 

The application of the additive fuzzy electron density fragments for the building of electron densities of large molecules is called the Molecular Electron Density Lego Assembler method, or MEDLA method [5,37,66,67,70-72],... 

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