Fuzzy electronic densities

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

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). 

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. 

A variety of guests may be absorbed in organic zeolite and just very simple examples of the resulted structures are solved by X-ray diffraction. This is due to fuzzy electron density maps in the guest region when it is a chemical mixture or it is orientationally disordered. A rare example of the two-component guest zeolite structure is given below [11] (Fig. 11.12). 

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

Linear Homotopies of Fuzzy Electron Density Fragments... 

The above fuzzy electron density membership functions reflect the relative contributions of the fuzzy, three-dimensional charge clouds of the various molecular electron density distributions to the total electronic density of molecular family L. 

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

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],... 

With minor modifications, the fuzzy electron density membership function formalism of molecular families can also be applied to a family of functional groups within a molecule. Consider a molecule X and some electron density threshold a within the functional group range of density. Consider the functional groups appearing as separate density domains... 

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 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],... 

Even if each of the fuzzy electron density fragments pk(r) originate from a different molecule, a good approximation to the total electron density p(r) of the target molecule can be obtained using eq. (27) ... 

Density functional theory provides the means for defining fragment energies, based on the following, simple principle the fuzzy electron density fragment of a functional group, together with the associated set of nuclei, is treated as a complete... 

A quantum chemical approach is proposed for the representation of functional groups in chemistry. The approach is based on a simple density domain condition and on the additive, fuzzy electron density fragmenation method that also serves for the rapid caculation of ab initio quality electron densities of large molecules. Several aspects of the approach are described, including methods for similarity and complementarity analysis of functional groups. 

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