Matrix distance dependence

It is inherently surprising that geminal interactions are typically weaker than vicinal interactions, because the former involve orbitals that are in closer spatial proximity. The reasons for this counterintuitive distance dependence can be seen by decomposing the geminal Fock-matrix element into individual atomic hybrid contributions. 

Points with a constant Euclidean distance from a reference point (like the center) are located on a hypersphere (in two dimensions on a circle) points with a constant Mahalanobis distance to the center are located on a hyperellipsoid (in two dimensions on an ellipse) that envelops the cluster of object points (Figure 2.11). That means the Mahalanobis distance depends on the direction. Mahalanobis distances are used in classification methods, by measuring the distances of an unknown object to prototypes (centers, centroids) of object classes (Chapter 5). Problematic with the Mahalanobis distance is the need of the inverse of the covariance matrix which cannot be calculated with highly correlating variables. A similar approach without this drawback is the classification method SIMCA based on PC A (Section 5.3.1, Brereton 2006 Eriksson et al. 2006). 

Both the electronic couphng matrix element and the outer-sphere component of the nuclear reorientation parameter are thought to vary with donor-acceptor separation and orientation [29, 49]. It has been shown in studies of Os and Ru-ammines bridged by polyproline spacers that the distance dependence of X can be greater than that of [50]. Dielectric continuum models of solvent reorganization predict that Xg will increase with... 

Suppose we approach M atoms of an element having an unfilled outer shell, disposed in a lattice. The point may be made clear if we suppose to approach hydrogen atoms (outer shell 1 s ). Equations (11) and (12) would predict the broadening of the electron state in a half-filled s-band, which should therefore allow metallic behaviour. Apparently, this would happen for any inter-atomic distance a and, therefore also at infinite distance. What would change is, of course, the bandwidth, which is determined by matrix elements dependent on the interatomic distance a but the metallic behaviour, depending essentially on the fact that the electrons have available energy states within the band, should occur also at distances where the atoms may well be supposed to be isolated. 

The pre-exponential factor A in Eq. 1 is a weak function of the temperature and the reorganization energy, and strongly dependent upon the electronic coupling matrix element V. In the simplest case, V may be assumed to be exponentially dependent upon the through-space donor-acceptor separation r. This yields a distance dependence for electron transfer of ... 

Note that all results (the scores t, the scores distances, the orthogonal distances, and the scatter matrix x) depend on the number of components k. But to simplify the notations, we do not explicitly add a subscript k. 

Until recently very little quantitative experimental data concerning the distance dependence of electron-transfer rates were available. From experiments on electron transfer between statistically distributed donor and acceptor species in an inert glassy matrix it had been concluded (Miller et al., 1984) that the rate falls of sharply with increasing donor-acceptor separation and that at a given edge-to-edge separation Re (in A) the fastest rate (k in s ) achievable under optimally exothermic conditions would be given by the exponential expression eq. (1) ... 

The metal-ligand matrix elements, Hml and Hlm, are distance dependent and can be expanded in terms of symmetry-adapted nuclear displacements, Qryim). For a JT crystal, in this expansion the linear term is similar to the JT Hamiltonian, as (1)... 

Main Distance-Dependent Matrix 4D Molecular Similarity Analysis... 

Then, the Main Distance-Dependent Matrix (MDDM), for each pair of Interaction Pharmacophore Elements (IPEs) is estimated. The IPEs are specific and independent groups representing molecule functionality (Table Numerical Entriesl). The seventh IPE type (HS) encodes information about the overall molecular shape since all the nonhydrogen atoms are considered. From the MDDM estimated for the HS-HS pair, a similarity measure with respect to the whole molecule is obtained. 

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