2 dimensional analysis molecular geometry

Information on those features of the three-dimensional structure of a molecule (small or large) that are important with respect to their chemical or biochemical reactivities can often be obtained by comparisons of analogous molecular geometries in various different crystal structures. Effective techniques for comparing three-dimensional structures are therefore crucial to these types of analyses. If the structure of, for example, the citrate ion is compared as it occurs in several different crystal structures, some measure will be obtained of the possible conformational variability of this ion, as shown in Figure 16.1. Each such conformation determined from a crystal structure analysis represents a low-energy form of the ion. 

The CMF approach can offer an alternative solution to this problem. Instead of using discrete sets of representative conformations, one can consider for each molecule an infinite number of conformations organized into a continuous manifold, so-called corrformational space . This provides the ability to apply functional data analysis not only to molecular fields but also to molecular geometry in a consistent way. Such corrformational space can be described by means of some probability density function pdf) in 3N-dimensional Euclidean space, where N is the number of atoms in the molecule under study. Having applied several approximations from the arsenal of statistical physics, one can obtain the following expression for calculating atomic kernels instead of Eq. (13.6) ... 

The purpose of X-ray analysis of crystal stmctures is to provide information on the positions of the individual atoms of a molecule, their interatomic distances, bond angles, and other features of molecular geometry such as the planarity of a particular group of atoms, the angles between the planes, and torsion angles around the bonds. The resulting three-dimensional representation of the atomic contents, rapidly determined by modern computerized techniques, of the crystal establishes the complete conformational stfucture and geometrical details hitherto unknown. 

One-dimensional quadrupole echo NMR lineshape analysis of powder samples is particularly informative when fast, discrete jumps occur between sites of well-defined geometry as, for example, in a phenyl group undergoing two-site exchange. In this case, the characteristic Pake-pattern is transformed into an axially asymmetric lineshape with an apparent asymmetry parameter r] 9 0 (see Equation (6.2.3)) [1-8]. The asymmetric lineshapes, shown on the left in Fig. 6.2.2, can be derived by considering how the individual components of the principal EFG tensor become averaged by the discrete jumps. Within the molecular frame, and in units of as defined by Equation (6.2.2), the static axially symmetric tensor consists of the components = 1, = — 1/2, and V y = — 112. This traceless tensor satisfies the... 

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