Region Partitioning

FIGURE 23.1 Example of a QM/MM region partitioning for a S l reaction, (a) Entire molecule is shown with a dotted line denoting the QM region. (A) Molecule actually used for the QM calculation. 

Mezey, P. G. Catchment region partitioning of energy hypersurfaces,I., Theoret.chim.Acta(Berl.), 58 (1981), 390-330... 

Figure 2.17 Mixing in a crystallizer considering continuous phase (upper left-hand imaginary regions partitioned the others local size distribution of dispersion particle and mixedness).

Based on partition coefficient data, the hydrophilic solutes examined appear to permeate p-HEMA and p-HEMA with 1 mole % EGDMA via "bulk-like" water regions. Partition coefficients of steroids in p-HEMA are dominated by their high solubility of the steroids in the hydrophobic regions of the hydrogels or A domains, whereas, permeation is dominated by diffusion within "fluctuating pores."... 

AUKEMA, B.H., RICHARDS, G.R., KRAUTH, S.J., RAFFA, K.F., Species assemblage arriving at and emerging from trees colonized by Ips pini in the Great Lakes region Partitioning by time since colonization, season, and host species, Ann. Entomol. Soc. Am, 2004,97, 117-129. 

In order to compare two spectra visually, all points of a curve A are divided by the corresponding points of the curve B at the same wavelength region (partitive method, see Sec. 2.6.3.3). This has been carried out in Fig. 4-33. The spectra are taken from different concentrations of chymotrypsin. Theoretically, the division of both curves leads to a straight line, because this manipulation normalizes the data. In practice, small deviations caused by residual noise may occur [10, 29] (see Fig. 4-33, Ic to 4c). 

P. G. Mezey, Theor. Chim. Acta, 58, 309 (1981). Catchment Region Partitioning of Energy Hypersurfaces, 1. 

The D vibrational modes in the IS give a reactant region partition function on the form... 

These methods are based on homogeneity criteria that partitions an image into homogenous regions. This can be expressed by the following mathematical formalism. 

In the second picture, an interfacial layer or region persists over several molecular diameters due to a more slowly decaying interaction potential with the solid (note Section X-7C). This situation would then be more like the physical adsorption of vapors (see Chapter XVII), which become multilayer near the saturation vapor pressure (e.g.. Fig. X-15). Adsorption from solution, from this point of view, corresponds to a partition between bulk and interfacial phases here the Polanyi potential concept may be used (see Sections X-7C, XI-1 A, and XVII-7). 

This method, introduced originally in an analysis of nuclear resonance reactions, has been extensively developed [H, 16 and F7] over the past 20 years as a powerful ab initio calculational tool. It partitions configuration space into two regions by a sphere of radius r = a, where r is the scattered electron coordinate. 

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

The choice of where to locate the boundary between regions of the system is important. A number of studies have shown that very poor end results will be obtained if this is chosen improperly. There is no rigorous way to choose the best partitioning, but some general rules of thumb can be stated ... 

QM/MM methods do not allow for charge transfer between different regions of the system. Thus, partitioning should not divide sections expected to have a charge separation. 

In the combined QM-MM methodology the system being studies is partitioned into a quantum mechanical region and a molecular mechanical region (Fig. 1). The quantum... 

Figure 1 Schematic diagram depicting the partitioning of an enzymatic system into quantum and classical regions. The side chains of a tyrosine and valine are treated quantum mechanically, whereas the remainder of the enzyme and added solvent are treated with a classical force field.

Figure 2 A glutamate side chain partitioned into quantum and classical regions. The terminal CH2C02 group IS treated quantum mechanically, and the backbone atoms are treated with the molecular mechanics force field.

For a given Hamiltonian the calculation of the partition function can be done exactly in only few cases (some of them will be presented below). In general the calculation requires a scheme of approximations. Mean-field approximation (MFA) is a very popular approximation based on the steepest descent method [17,22]. In this case it is assumed that the main contribution to Z is due to fields which are localized in a small region of the functional space. More crudely, for each kind of particle only one field is... 

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