Distribution functions radial

Let us evaluate the different properties and applications of a molecular descriptor while keeping the aforementioned requirements for descriptors in mind. We will focus on a particular descriptor type the radial distribution function (RDF). RDF descriptors grew out of the research area of structure-spectrum correlations but are far more than simple alternative representations of molecules. The flexibility of these functions from a mathematical point of view allows applying them in several other contexts. This chapter will give a theoretical overview of RDF descriptors as well as their application for the characterization of molecules, in particular for similarity and diversity tasks. 

Radial distribution functions can be adapted quite flexibly to the desired representation of molecules. The RDF functions developed can be divided into several groups regarding the basic function type, the distance range of calculation, the type of distance information, the dimensionality, and the postprocessing steps. Most of the varieties of RDF descriptors introduced in this chapter can be combined arbitrarily to fit to the required task. As a consequence, more than 1,400 useful descriptors can be derived from radial functions [1], The molecules used for the calculation of descriptors are shown in the figures. 

The general RDF is an expression for the probability distribution of distances between each of which is measured between two points i and j, within a three-dimensional (3D) space of N points  

The exponential term leads to a Gaussian distribution around the distance r j with a half-peak width depending on the smoothing parameter B. 

The general RDF can be easily transformed to a basic molecular descriptor by applying it to the 3D coordinates of the atoms in a molecule. In molecular terms, is the distance between the atoms i and j in an Ai-atomic molecule. g r) is usually calculated for all unique pairs of atoms (denoted by i and j) in a certain distance range divided into eqnidistant intervals. Thus, the function g(r) is usually represented by its discrete form of an n-dimensional vector [ (rj), gir ),-, (rn)] calculated between and r . In this case, the RDF is considered as a molecular descriptor (RDF descriptor) for the three-dimensional arrangement of atoms in a molecule. 

How could the amorphous structure be characterized in a way that would allow comparison to experimentally measurable quantities In the case of crystals and quasicrystals we considered scattering experiments in order to determine the signature of the structure. We will do the same here. Suppose that radiation is incident on the amorphous solid in plane wave form with a wave-vector q. Since there is no periodic or regular structure of any form in the solid, we now have to treat each atom as a point from which the incident radiation is scattered. We consider that the detector is situated at a position R well outside the solid, and that the scattered radiation arrives at the detector with a wave-vector q in this case the directions of R and q must be the same. Due to the lack of order, we have to assume that the incident wave, which has an amplitude exp(iq r ) at the position r of an atom, is scattered into a spherical wave exp(i q R — r )/ R — r . We then have to sum the contributions of all these waves at the detector to find the total amplitude A(q, q R). This procedure gives 

The experimental signal is typically expressed in units of AoN/ Rp, which conveniently eUminates the factor in front of the integral in Eq. (12.20). The resulting quantity, called the structure factor S(k), does not depend on the direction of the scattering vector k, but only on its magnitude k = k, because the function g(r) has already been averaged over the directions of interatomic distances  

The structure factor is the quantity obtained directly by scattering experiments, which can then be used to deduce information about the structure of the amorphous solid, through g r). This is done by assuming a structure for the amorphous solid, calculating the g(r) for that structure, and comparing it with the experimentally determined g(r) extracted from S(k). This procedure illustrates the importance of good structural models for the amorphous structure. 

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Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

In general, it is diflfieult to quantify stnietural properties of disordered matter via experimental probes as with x-ray or neutron seattering. Sueh probes measure statistieally averaged properties like the pair-correlation function, also ealled the radial distribution function. The pair-eorrelation fiinetion measures the average distribution of atoms from a partieular site. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

Flere g(r) = G(r) + 1 is called a radial distribution function, since n g(r) is the conditional probability that a particle will be found at fif there is another at tire origin. For strongly interacting systems, one can also introduce the potential of the mean force w(r) tln-ough the relation g(r) = exp(-pm(r)). Both g(r) and w(r) are also functions of temperature T and density n... 

Figure A2.3.7 The radial distribution function g r) of a Lemiard-Jones fluid representing argon at T = 0.72 and p = 0.844 detennined by computer simulations using the Lemiard-Jones potential.

Microscopic theory yields an exact relation between the integral of the radial distribution function g(r) and the compressibility... 

Fig. 4. Radial distribution functions between the centre of a test cavity and the (jxygen atom of the surrounding water. The curves correspond to the different barrier heights for the softcore interaction illustrated in Fig. 3...

D MoRSE desaiptor, radial distribution function (RDF code), WHIM descriptors, GETAWAY descriptors,... 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

Figure 8-6. Comparison of the radial distribution function of the ctiair, boat, and twist conformations of cyclohexane (hydrogen atoms are not considered).

Topological descriptors and 3D descriptors calculated in distance space", such as 3D autocorrelation, surface autocorrelation, and radial distribution function... 

Models with 32 Radial Distribution Function Values and Eight Additional Descriptors... 

The compounds were described by a set of 32 radial distribution function (RDF) code values [27] representing the 3D structure of a molecule and eight additional descriptors. The 3D coordinates were obtained using the 3D structure generator GORINA [33]. 

The radial distribution Function (RDF) of an ensemble of N atoms can be interpreted as the probability distribution to find an atom in a spherical volume of... 

Figure 10.1-5. Predicted versus experimental solubility values of 496 compounds in the test set by a back-propagation neural network with 32 radial distribution function codes and eight additional descriptors.

A combination of physicochemical, topological, and geometric information is used to encode the environment of a proton, The geometric information is based on (local) proton radial distribution function (RDF) descriptors and characterizes the 3D environment of the proton. Counterpropagation neural networks established the relationship between protons and their h NMR chemical shifts (for details of neural networks, see Section 9,5). Four different types of protons were... 

The strncturcs in the database arc encoded using the radial distribution function (RDF) as a descriptor (cf Section 8,4,4). 

Fig. 6.2 Radial distribution function determined from a lOOps molecular dynamics simulation of liquid argon at a temperature of 100K and a density of 1.396gcm. ...

The radial distribution function can also be used to monitor the progress of the equilibration. This function is particularly useful for detecting the presence of two phases. Such a situation is characterised by a larger than expected first peak and by the fact that g r) does not decay towards a value of 1 at long distances. If two-phase behaviour is inappropriate then the simulation should probably be terminated and examined. If, however, a two-phase system is desired, then a long equilibration phase is usually required. 

The coarse-graining approach is commonly used for thermodynamic properties whereas the systematic or random sampling methods are appropriate for static structural properties such as the radial distribution function. 

FIGURE 2.2 Radial distribution functions for (a) a hard sphere fluid, (A) a real gas, (c) a liquid, (li) a crystal. 

Analyze the trajectories to obtain information about the system. This might be determined by computing radial distribution functions, dilfu-sion coefficients, vibrational motions, or any other property computable from this information. 

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

A very important aspect of both these methods is the means to obtain radial distribution functions. Radial distribution functions are the best description of liquid structure at the molecular level. This is because they reflect the statistical nature of liquids. Radial distribution functions also provide the interface between these simulations and statistical mechanics. 

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