Radial distribution function atom-specific

The MFA [1] introduces the perturbation due to the solvent effect in an averaged way. Specifically, the quantity that is introduced into the solute molecular Hamiltonian is the averaged value of the potential generated by the solvent in the volume occupied by the solute. In the past, this approximation has mainly been used with very simplified descriptions of the solvent, such as those provided by the dielectric continuum [2] or Langevin dipole models [3], A more detailed description of the solvent has been used by Ten-no et al. [4], who describe the solvent through atom-atom radial distribution functions obtained via an extended version of the interaction site method. Less attention has been paid, however, to the use of the MFA in conjunction with simulation calculations of liquids, although its theoretical bases are well known [5]. In this respect, we would refer to the papers of Sese and co-workers [6], where the solvent radial distribution functions obtained from MD [7] calculations and its perturbation are introduced a posteriori into the molecular Hamiltonian. 

The solvation structure around a molecule is commonly described by a pair correlation function (PCF) or radial distribution function g(r). This function represents the probability of finding a specific particle (atom) at a distance r from the atom being studied. Figure 4.32 shows the PCF of oxygen-oxygen and hydrogen-oxygen in liquid water. 

Consider now the mean oxygen density conditional on a specific alkyl configuration. Since that conditional mean oxygen density is less traditionally analyzed than the density profile shown in Fig. 1.9, we exploit another characterization tool, the proximal radial distribution (Ashbaugh and Paulaitis, 2001). Consider the volume that is the union of the volumes of spheres of radius r centered on each carbon atom see Fig. 1.10. The surface of that volume that is closer to atom i than to any other carbon atom has area fl, (r) with 0 < ff, (r) < 4tt. The proximal radial distribution function ( ) is defined as... 

The potential function used in MD calculations of liquid water is the so-called SPC potential—that is, a semiempirical potential yielding the almost correct radial distribution function for oxygen atoms at room temperatures. However, we emphasize that what we want to see is characters that are independent of detailed tunings of the system such as potential functions, bond lengths, bond angles, and even masses of atoms. The properties depending sensitively on them are attributed to system specificities and are outside our present concern. 

The radial distribution function in this form meets all the requirements for a 3D structure descriptor It is independent of the number of atoms, that is, the size of a molecule, it is unique regarding the three-dimensional arrangement of the atoms, and invariant against translation and rotation of the entire molecule. In addition, the RDF descriptors can be restricted to specific atom types or distance ranges to represent specific information in a certain three-dimensional structure space, for example, to describe sterical hindrance or structure/activity properties of a molecule. 

For an understanding of protein-solvent interactions it is necessary to explore the modifications of the dynamics and structure of the surrounding water induced by the presence of the biopolymer. The theoretical methods best suited for this purpose are conventional molecular dynamics with periodic boundary conditions and stochastic boundary molecular dynamics techniques, both of which treat the solvent explicitly (Chapt. IV.B and C). We focus on the results of simulations concerned with the dynamics and structure of water in the vicinity of a protein both on a global level (i.e., averages over all solvation sites) and on a local level (i.e., the solvent dynamics and structure in the neighborhood of specific protein atoms). The methods of analysis are analogous to those commonly employed in the determination of the structure and dynamics of water around small solute molecules.163 In particular, we make use of the conditional protein solute -water radial distribution function,... 

Radial distribution function Radial distribution function (RDF) is a term often utilized in analyzing the results of Monte Carlo or MD calculations. The RDF g(r) gives the probability of occurrence of an atom of type a at a distance r from an atom of type b. Peaks in the g(r) vs. r plots can be associated with solvation shells or specific neighbors and can be integrated to yield coordination numbers. 

It is to be stressed that this is a crystal-like theory. The detail of the band anisotropy will arise from the assumption of a specific crystal structure. Since the wave function we are considering is S-like, a band dispersion calculated for a f.c.c. crystal should not differ too much) at least near the band minimum) from that of a fluid if the effect of disorder is completely taken into account by the use of the radial distribution function in the calculation of the dipole moment induced on the atom at the center of the Wigner-Seitz cell. 

Another approach to help facilitate crystallisation is to bias the tr ectories of all the atoms moving in the (molten/amorphous) configuration to favour crystallisation. Specifically, order parameters (which can be bond distances, coordination numbers, nearest neighbour densities and radial distribution functions) may be used as a gauge of crystallinity and used to help drive the simulation (trajectory) to favour maximising the order parameter and, ultimately, induce crystallinity. 

The structure of chlc2 in liquid methanol, specifically the coordination of the chromophore metallic center to the methanol molecules can be discussed by representing the radial distribution function [RDF(r)] related to Mg-O interactions. A relevant issue concerns the coordination number of the metalic center. BOMD results for the structure indicate that the coordination number for Mg of chlc2 in liquid methanol is five [114] a feature that seems to be related with the displacement of the metallic center from the macrocycle plane. This is in contrast with the results from force field calculations that indicate a coordination number of six (four nitrogen atoms plus two oxygen atoms). Several works discussed the coordination of the central Mg atom of porph3rins and chlorophylls in different solvents and environments [112, 118, 134—138]. 

Neutron scattering methods provide the best experimental means cnr-rently available to probe the atomic strncture of aqueous solutions. It can be proved that a formal mathematical (Fourier transformation) link can be formed between the neutron scattering pattern obtained experimentally and the pair radial distribution functions Sotfi r) of pairs of atoms a and of the system. Knowledge of these functions, either individn-ally or as combinations [G (r)] specific to a particular atom (or ion), a,... 

It is worth noting that it is the radial distribution of core electrons in an atom, which is responsible for the reduction of the intensity when the diffraction angle increases. Thus, it is a specific feature observed in x-ray diffraction from ordered arrangements of atoms. If, for example, the diffraction of neutrons is of concern, they are scattered by nuclei, which may be considered as points. Hence, neutron scattering functions (factors) are independent of the diffraction angle and they remain constant for a given type of nuclei (also see Table 2.2). 

Interpreted in terms of the symmetrical form of the periodic table (Fig. 3), the quantum numbers that define the radial distances of r = n a specify the nodal surfaces of spherical waves that define the electronic shell structure. Knowing the number of electrons in each shell, the density at the crests of the spherical waves that represent periodic shells, i.e., at 1.5,3, etc. (a), can be calculated. This density distribution, shown in Fig. 7, decreases exponentially with Z and, like the TF central-field potential, is valid for all atoms and also requires characteristic scale factors to generate the density functions for specific atoms. The Bohr-Schrodinger... 

Unsatisfied by the low specificity of most of the derived residue-pair potentials, especially those obtained considering distances between C or atoms, some authors use a somewhat more detailed description of the protein conformation, in which each residue is represented not by one, but by two, interactions sites one for the backbone (Ca) and one representing the side-chain atoms most involved in the relevant interactions. When this is combined with the use of the radial pair distribution functions, as described above, residue-pair potentials displaying more specific hydrophilic interactions are obtained. 

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