Fragment density

Fig. 4. Shapes of metal powder particles (a) spherical (b) rounded (c) angular (d) acicular (e) dendritic (f) kregular (g) porous and (h) fragmented. Density. The density of a metal powder particle is not necessarily identical to the density of the material from which it is produced because of...

All parameters of interest with respect to fragmentation will be discussed. The extent of damage or injury caused by these fragments is, however, not covered in this volume. (Parameters of the terminal phase include first, fragment density and velocity at impact, and second, resistance of people and structures to fragments.)... 

Approximate transferability of fuzzy density fragments is a key feature of the method, where the fuzzy fragments are custom-made in order to reproduce interfragment interactions. By increasing the size of the interaction shell about each fuzzy density fragment, the error of transferred fragment densities can be reduced below any positive threshold. One tool for this purpose is the Adjustable Density... 

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. 

Approximate transferability of fuzzy fragment density matrices... 

Transferred electron density fragments obtained by AFDF method can provide excellent approximations. One such approach, formulated in terms of transferability of fragment density matrices within the AFDF framework is a tool that has been suggested as an approach to macromolecular quantum chemistry [114, 115, 130, 142-146] and to a new density fitting algorithm in the crystallographic structure refinement process [161]. 

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 follows from the definition (35) of fragment density matrices P that implies exact additivity of these fragment density matrices, i.e., they add up to the density matrix P of the complete molecule,... 

With reference to the individual AO basis sets

fragment density matrices P t((p (Kt)) obtained from parent molecules Ms of nuclear configurations Kt, on the one hand, and the macromolecular AO basis set cp (K) of the macromolecular density matrix P (cp (K)) associated with the macromolecular nuclear configuration K, on the other hand, the following mutual compatibility conditions are assumed ... 

In order to fulfill compatibility condition (a), the local coordinate system of each parent molecule Mk can always be reoriented, resulting in a simple similarity transformation of the original fragment density matrix P (qS(Kk)) into a compatible fragment density matrix P (cp (K)),... 

The final macromolecular density matrix P(A") is rather sparse. The index relations described above help to identify the non-zero matrix elements of P(A"), and the actual computations can be restricted to those. Utilizing these restrictions and carrying out a finite number of steps only for the non-zero matrix elements of each fragment density matrix P (< Kk)), an iterative process is used for the assembly of the macromolecular density matrix P(AT) ... 

Small deformations of electron densities, adjustability and additivity conditions for fragment density matrices... 

The macromolecular density matrix built from such displaced local fragment density matrices does not necessarily fulfill the idempotency condition that is one condition involved in charge conservation. It is possible, however, to ensure idempotency for a macromolecular density matrix subject to small deformations of the nuclear arrangements by a relatively simple algorithm, based on the Lowdin transform-inverse Lowdin transform technique. 

L = Fragment length in direction of motion D Fragment density... 

From the perspective of MQS, this means that the similarity needs to be computed between fragments of a molecule. This requires methods to obtain a fragment density from a molecular electron density. Generally speaking, use is made of some operator wf acting on the molecular density pMol(r) to yield the fragment density as pt(r) ... 

If we are particularly interested in the nature of the bonding between the fragments of a molecular complex, a fragment deformation density may be calculated by subtracting fragment densities from the total distribution. In the case of a transition metal complex, the fragment may be a metal atom plus ligand, or just the density... 

A related theoretical approach to charge density transferability has been developed by Mezey and collaborators (Walker and Mezey 1993,1994). But rather than composing a molecule of standard pseudoatoms, the density of large molecules, including proteins, is constructed from the density of a number of standard theoretical fragments. The fragment densities are defined by the distribution... 

Olovsson and coworkers have pointed out that the superposition of the electron density of adjacent molecules in the experimental deformation density may lead to modification of the contours in the lone-pair region of the water molecules (Fernandes et al. 1990, McIntyre et al. 1990). To avoid this complication, it is preferable to partition the crystal density through the multipole analysis, after which comparisons can be based on individual molecule or fragment densities. 

Density of Fragments Determination. See Fragment Density Tests... 

Fragment Density Tests. See Vol 1, p XII and German Fragment Density Test in Vol 3,... 

Fragment Density Test. Same as Fragment Concentration Test... 

Fragment Density, Fragment Concentration- or Density of Splinters Tost(Splitterdicbteprobe) is described in PATR 2510(1958),P Ger 52... 

H. German Fragment Density Test (Splitter -dichteprobe in Ger). In this test, invented by Dr. G. Romer, the expl item to be tested is detonated in the middle of wooden boards, 2cm thick, surrounding it as shields... 

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