Statistical geometry correlations

Both pairs of lines are identical only when r = 1 in this limiting case, all four expressions in eqs. (30) and (31) are equal, and statistics has been replaced by simple geometry. If there is no correlation at all between the original values log k2 and log ki, i.e., ri2 = 0, apparent regression lines are obtained in the E versus log A plane (Figure 10) with the slopes (when Si = S2)... 

The application of the CNDO/2 method with standard geometry for various substituted 2-benzopyrylium cations results, as one could expect, in the preference for the resonance structures b-f with close values of charge densities on Ct and C3, and attempts to rationalize on the basis of these calculations yielded statistical agreement for correlations between change densities and H-NMR spectral data (90MI3) (cf. Section IV,A,2). 

The number of crystal structure data correlates with the number of statistical analyses of hydrogen-bond geometries. As the amount of crystal structure data increased, so did the efforts to use it as a basis for statistical analyses of the geometrical properties of hydrogen bonds. Because there was only a limited amount of neutron data available and the X-ray analyses did not have the accuracy to provide reliable hydrogen atom positions, most of the earlier analyses focused on the nonhydrogen donor-acceptor distances, i.e., O- O for O-H - O bonds and N 0 or N- -N for N—H 0 = C or N-H- -N bonds [33, 53, 381-384]. 

The intensities of the 83/2 and Pi/2 fluctuation bands in Figure 8 are in the ratio 2 to 1, virtually a statistical distribution however, the relative Intensity distributions for the Pi/2 and Pa/2 fluctuation bands differ notably. Much more pronounced differences are observed for the "hot bands" depicted in Figures 6 and 7 where the P3/2/ Pi/2 ratio varies with exciting frequency and in many instances approaches 8/1. This difference in intensity ratio must reflect very different geometries for the lower discrete states from which pumping occurs in bound-free transition. Again such a result would appear to correlate well with the theoretical analysis of the sodium trimer surface. Martin and Davidson predict that a linear symmetric conformation lies only 1050 cm above the ground... 

Modem quantum-chemical methods can, in principle, provide all properties of molecular systems. The achievable accuracy for a desired property of a given molecule is limited only by the available computational resources. In practice, this leads to restrictions on the size of the system From a handful of atoms for highly correlated methods to a few hundred atoms for direct Hartree-Fock (HF), density-functional (DFT) or semiempirical methods. For these systems, one can usually afford the few evaluations of the energy and its first one or two derivatives needed for optimisation of the molecular geometry. However, neither the affordable system size nor, in particular, the affordable number of configurations is sufficient to evaluate statistical-mechanical properties of such systems with any level of confidence. This makes quantum chemistry a useful tool for every molecular property that is sufficiently determined (i) at vacuum boundary conditions and (ii) at zero Kelvin. However, all effects from finite temperature, interactions with a condensed-phase environment, time-dependence and entropy are not accounted for. 

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