Chemical distance space

Following the argument by Bunde and Drager [62], front propagation is well defined in the chemical distance space. The front has to propagate with constant velocity, i.e., the front position in the chemical distance space behaves like / t. Euclidean and chemical distances are related by (6.1). The front position in real... 

Here y is the average and cr is the standard deviation of the Euclidean distances of the k nearest neighbors of each compound in the training set in the chemical descriptor space, and Z is an empirical parameter to control the significance level, with the default value of 0.5. If the distance from an external compound to its nearest neighbor in the training set is above Dc, we label its prediction unreliable. 

Here, L2 represents the chemical property space corresponding to the VCL, d(x, rf) represents an appropriate distance metric between the lead compound ry and the compound , p xf is the relative weight that can be attached to compound (if all compounds are of equal importance, then the weights p(xy) = for each i), and Mis typically much smaller than N. That is, this problem seeks a subset of M lead compounds ry in a descriptor space such that the average distance of a compound from its nearest lead compound is minimized. Alternatively, this problem can also be formulated as finding an optimal partition of the descriptor space co into M clusters Ay and assigning to each cluster a lead compound rj such that the following cost function is minimized ... 

In Euclidean cZ-dimensional spaces Flory exponent depends only on d. A good (although not exact) estimation was given by Flory formula, Eq. (9) [7, 8], It is well known [47] that the critical phenomena depend by the decisive mode on various ftactal characteristics of basic stmcture. It becomes obvious, that excepting the ftactal (Hausdorflf) dimension Dj. physical phenomena on ft actals depend on many other dimensions, including skeleton fractal dimension [48], dimension of minimum (or chemical) distance [29] and so on. It also becomes clear, that regular random walks on fractals have anomalous fractal dimension [49] and that the vibrational excitations spectrum is characterized by spectral (fiac-ton) dimension d=2D d [41, 50],... 

Rupp M, Schneider P, Schneider G. Distance phenomena in high-dimensional chemical descriptor spaces Consequences for similarity-based approaches. J Comput Chem 2009 30 2285-2296. 

Excluded volume interactions act between all monomers irre-speaive of their chemical distance along the chain. At short distances, they are typically repulsive, modeling the fact that two objeas cannot occupy the same space. This sometimes can be followed by a longer range attrartion. The most popular potential to model excluded volume interactions is the Letmard-Jones (LJ) potential ... 

In special cases (as in colloidal solutions) some particles can be considered as essential and other particles as irrelevant , but in most cases the essential space will itself consist of collective degrees of freedom. A reaction coordinate for a chemical reaction is an example where not a particle, but some function of the distance between atoms is considered. In a simulation of the permeability of a lipid bilayer membrane for water [132] the reaction coordinate was taken as the distance, in the direction perpendicular to the bilayer, between the center of mass of a water molecule and the center of mass of the rest of the system. In proteins (see below) a few collective degrees of freedom involving all atoms of the molecule, describe almost all the... 

Spanned by tbc atoms 4, 2, and 1, and 2, 1, and 3 (tlic ry-planc), Except of the first three atoms, each atom is described by a set of three internal coordinates a distance from a previously defined atom, the bond angle formed by the atom with two previous atoms, and the torsion angle of the atom with three previous atoms. A total of 3/V - 6 internal coordinates, where N is the number of atoms in the molecule, is required to represent a chemical structure properly in 3D space. The number (,3N - 6) of internal coordinates also corresponds to the number of degrees of freedom of the molecule. 

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

Table 12.7 Selected sources of spacing distances with hazardous chemicals (see Bibliography) ...

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