Molecular density, divisions

Should one additionally require, that the optimum subsystem densities reproduce a given molecular density, Y./3 Pp(r) = p(r), as is the case in the Hirshfeld division, one has to include the additional local constraint of the exhaustive partitioning ... 

Hence, one term each on the left- and right-hand sides of equation (9-29) is zero. Since the mass density of component i, p, and the molar density of component i, Q, are related by molecular weight, division by MW, produces the final form of the mass transfer equation for incompressible pseudo-binary mixtures with constant physical properties ... 

The division of the molecular volume into atomic basins follows from a deeper analysis based on the principle of stationary action. The shapes of the atomic basins, and the associated electron densities, in a functional group are very similar in different molecules. The local properties of the wave function are therefore transferable to a very good approximation, which rationalizes the basis for organic chemistry, that functional groups react similarly in different molecules. It may be shown that any observable... 

However, the division of the electron density at the iron nucleus into contributions arising from Is through 4s contributions can be done conveniently at the level of the canonical molecular orbitals. This arises because the iron Is, 2s, and 3s orbitals fall into an orbital energy range where they are well isolated and hence do not mix with any hgand orbital. Hence, the Is, 2s, and 3s contributions are well defined in this way. The 4s contribution then arises typically from several, if not many, molecular orbitals in the valence region that have contributions from the iron s-orbitals. Thus, the difference between the total electron density at the nucleus and... 

During the same period, commercialization of thermoplastic starch polymer blends was pursued by Novamont, a division of the Ferruzzi Group of Italy.162-172 Their products, marketed under the trade name Mater-Bi, are typically comprised of at least 60% starch or natural additive and hydrophilic, biodegradable synthetic polymers.64,165 It is stated that these blends form interpenetrated or semi-interpenetrated structures at the molecular level. Properties of typical commercial formulations have properties similar to those in the range of low- and high-density PE. Blends of Mater-Bi products with biodegradable polyesters have been claimed for use as water impervious films.173... 

To measure the volume of the Sephadex at equilibrium the equilibrated samples were contained in a calibrated tube (0.1 mL/division). An apparent volume was obtained from visual observation of the boundary defined by the layer to obtain the true volume an aliquot of the equilibrated solution phase was analysed for sodium by flame photometry. Most of the supernatant solution was then removed until 2.0 zt 0.025 mL of solution remained above the gel-defined boundary. Exactly 1.00 mL of water was added and the mixture was stirred sufficiently to assure a homogeneous aqueous phase. The solution phase was sampled immediately for sodium analysis with the flame photometer. From the observed dilution of the aqueous phase the true volume of the Sephadex gel at equilibrium was obtained. A correction for the matrix volume was based on the monomeric molecular weight of the Sephadex ( 220 25 capacity of 4.5 0.5 meq/dry g). In our samples, which contained about 88% water or 0.12 g acid/g sample, a volume of about 0.13/g is estimated for the matrix by assuming a density of approximately 0.9 for the dehydrated Sephadex. 

Hirshfeld division of the molecular ground-state density 151... 

HIRSHFELD DIVISION OF THE MOLECULAR GROUND-STATE DENSITY... 

Let us begin a survey of this development with a brief reminder of the Hirshfeld (H) partition scheme and its information-theoretic justification. The stockholder division [34] of the molecular electron density p(r). experimental or theoretical, into the AIM densities, p(r) = X< Pa(r)>... 

Postulate I. Among all possible partitions of the molecular ground-state density p into densities of molecular fragments [ p , for the fixed promolecule (free atom) reference, a natural choice in chemistry, there exists the equilibrium division into the stockholder fragments, that are characterized completely by p and by the reference densities alone. 

Historically, the field of electronic structure calculations has seen two largely independent lines of development on the one hand molecular quantum chemists have based most work on wavefunction techniques (the Hartree Fock [2, 3] and post-HF theories [4]) on the other, condensed-matter physicists had their reference method in Density Functional Theory (DFT) [5-7]. This formal division between molecular and solid-state communities has been due to the poor transferability of the standard computational methods between the two fields early DFT functionals underperform post-HF techniques in reproducing the known properties of small molecules... 

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