Principal component analysis geometry

A distance geometry calculation consists of two major parts. In the first, the distances are checked for consistency, using a set of inequalities that distances have to satisfy (this part is called bound smoothing ) in the second, distances are chosen randomly within these bounds, and the so-called metric matrix (Mij) is calculated. Embedding then converts this matrix to three-dimensional coordinates, using methods akin to principal component analysis [40]. 

Ab initio electron correlated calculations of the equilibrium geometries, dipole moments, and static dipole polarizabilities were reported for oxadiazoles <1996JPC8752>. The various measures of delocalization in the five-membered heteroaromatic compounds were obtained from MO calculations at the HF/6-31G level and the application of natural bond orbital analysis and natural resonance theory. The hydrogen transfer and aromatic energies of these compounds were also calculated. These were compared to the relative ranking of aromaticity reported by J. P. Bean from a principal component analysis of other measures of aromaticity <1998JOC2497>. 

If the origin ( 0 ) is chosen at the centroid of the atoms, then it can be shown that distances from this point can be computed from the interatomic distances alone. A fundamental theorem of distance geometry states that a set of distances can correspond to a three-dimensional object only if the metric matrix g is rank three, i.e., if it has toee positive and N — 3 zero eigenvalues. This is not a trivial theorem, but it may be made plausible by thinking of the eigenanalysis as a principal component analysis all of the distance properties of the molecule should be describable in terms of three components , which would be the x, y and z coordinates. If we denote the eigenvector matrix as w and the eigenvalues A., the metric matrix can be written in two ways ... 

Murray-Rust, P Motherwell, S. Computer retrieval and analysis of molecular geometry. 1. General principles and methods, Acta Cryst. 1978, B34, 2518-2526. A rough application of principal components, or factor analysis, is a shirt that has just one size parameter (S, M, F, XF), instead of a specification of waistline, chest width, arm, or leg lengths, etc., in the assumption that these body parameters are correlated. 

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... 

Thus, if the analytical stress distribution is known, the loading factor for each principal stress can be calculated. In this process, compressive stresses are usually neglected. In many realistic component geometries, the stresses need to be determined by numerical techniques, such as finite-element analysis. From Eq. (9.1) with o in=0, the survival probability for a single element R j under the action of a single principal stress can be written as... 

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