Eigen equations

In all cases the key to assigning mechanism is the ability to detect and measure the equilibrium constant K. The equilibrium constant Kos can be estimated through the Fuoss-Eigen equation,10 as shown in equation 1.22. Usually, Kos is ignored in the case of L = solvent. 

Here =K n and =g n. Equation 6 reduces to an eigen equation by renormalizing the eigenvector ... 

The complete solution W(x, t) is then expressed through the sum over the particular solutions W (x, f), over the set of eigenvalues n. After the separation of Eq. (43) through division by (0Dlt aTn)(pn, one arrives at the two eigen-equations... 

Bronsted plots with some scatter should always be examined carefully because they might conform with the Eigen equation. A classic case is the acid-catalysed reaction of aniline with HNCO which can be fitted to a shallow Bronsted line (slope = —0.19)[9] however, the data are also consistent with an Eigen dependence (Fig. 11.5B) arising from a mechanism involving trapping of an addition intermediate by proton transfer (Scheme 11.11). 

The Eq. (4-9) is an eigen equation and gives multiple excited states by single diagonalization. The different SAC-CI solutions are therefore orthogonal to each other. 

The data in Table 4 were obtained by Bednar and Jencks for the protonation of CN" by a series of general acids (AH, Equation 32). Graph logA k = overall second-order rate constant) versus pA fit the data to the Eigen Equation log = log/ , - log(l + 10 "" ) (assuming k x = k and comment on the results. There is no need to correct the data statistically. 

The fit to an Eigen equation of the proton transfer step to the carbon base (CN") indicates that the 2 step is not rate limiting and the cyanide ion behaves as if it were a heteroatom base. The anomalously large values of the rate constants for oxonium and hydroxide ions are consistent with their faster diffusion rates resulting from Grotthus translational-type effects. 

The data for the acid-catalysed formation of carbinolamine from methoxyamine and 4-methoxybenzaldehyde fit the Eigen equation and follow a plot similar to that shown in Figure 20. The parameters are ... 

The fit to the Eigen equation is consistent with a mechanism involving a stepwise proton transfer to the oxyanion of the zwitterion (Scheme 35). The parameter pA is close to pA, calculated for the protonation of the zwitterion (8.97). 

The refractive indices of vacuum deposited thin films of the studied catenanes and rotaxanes were determined by the m-lines technique. It consists on finding solutions of an eigen equation for modes propagating in thin film. For a planar waveguide. 

The equilibrium constant X, 2 for formation of the ion-pair (sometimes referred to, less specifically, as Kq ) can be derived from the Fuoss—Eigen equation [59, 60]... 

First, let s introduce the key idea of the proposed approach with a simple ideal example. Consider the generalized eigen equation for the mth mode of the structure ... 

This prior PDF is based on choosing a Gaussian PDF as a probability model for the eigen equation errors, where the prior covariance matrix Y,eq controls the size of these equation errors. The uncertainty in the equation errors for each mode are modeled as independent and identically distributed, so ... 

The prior PDF p(k, 0, C) implies that, given a class of dynamical models and before using the dynamic test data, the most probable values of X and 0 are those that minimize the Euclidean norm (2-norm) of the error in the eigen equation for the dynamical model. This implies that the prior most probable values of X and 0 are the squared modal frequencies and mode shapes of a dynamical model, but these values are never explicitly required. This prior PDF will have multiple peaks because there is no implied ordering of the modes here. 

Similarly, since the eigen-equation of the non-perturbed valence state by means of its energy is accommodated with the creation-annihilation quantum rules ... 

When passing to the energy related quantities, similarly, one employs the ionization and affinity states (1.30) and (1.31) into the perturbed energy (1.32) that, at its turn, needs the pre-evaluation of the expressions calling the eigen-equation (1.29)... 

For better understanding how the semi-empirical methods were introduced, worth reconsidering the molecular eigen-equation (1.29) under the so called independent-electron problem... 

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