Statistical analysis bond distances

In these models, the potential energy function is based on the molecular mechanics all-atom force field and includes the bond, angle, dihedral and non-bonded energy terms. The parameterization is based on the statistical analysis of sets of experimental structures. If a variable q describes a degree of freedom in the system (e.g., bond distances, angles, dihedrals) then, P(q), the probability distribution associated with this degree of freedom, is related to the potential of mean force, W(q), by the following equation... 

The valuable determination of r values by Bondi [51] relied on the closest non-bonded contacts in key selected compounds. With the volume of crystal structure data now available for interrogation through the Cambridge Structural Database [55], it is possible to use statistical analysis to extract nonbonded contact distances [56, 57]. Thus, to ascertain for atom type Y, it is possible to choose a probe atom Z and then extract all of the reliable crystal structures containing both Y and Z not mutually bonded, and then generate a... 

There are many other spectroscopic techniques, but let me mention just one more, the X-ray crystallography method, which yield complete 3D information on a molecule (Scheme 9.9b). The technique is based on the phenomenon that when X-rays hit a crystal, which is an ordered solid form made of many molecules, the material diffracts the X-rays in patterns that reflect the distances between the atoms in the molecule. As such, the X-ray diffraction pattern can be analyzed and can provide us with the bond distances and the angles in a molecule. The rather complex statistical mathematical analysis was developed by Herbert Aaron Hauptman and Jerome Karle, who were awarded the Nobel Prize for their work in 1985.In this manner, chemists, called X-ray crystallographers, are able today to determine the structure of any molecule that can be crystallized as an ordered solid (or half-ordered, as in the case of Dan Shechtman, who won the Nobel Prize in 2011 using X-ray crystallography and a related technique called transmission electron microscopy). This powerful method resulted in close to 30 Nobel Prizes. 

However, because of the weakness and nonspecific nature of vdW forces, nonbonded distances vary more widely than bond lengths. By a statistical analysis of numerous structures, Rowland and Taylor derived a novel system of vdW radii (Figure 3, Table 3). Atoms of Cl, Br, I, S, Se and Te, forming one covalent bond, show a highly anisotropic shape in intermolecular contacts (Figure 4), which can be described as an ellipsoid compressed along the bond direction, while harder N, O and F atoms remain essentially spherical. 

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