Distributed moment analysis

It has been recently shown [12] that the ELF topological analysis can also be used in the framework of a distributed moments analysis as was done for Atoms in Molecules (AIM) by Popelier and Bader [32, 33], That way, the Mo( 2) monopole term corresponds to the opposite of the population (denoted N) ... 

One solution to the volume problem was proposed using moment analysis. The steady-state volume of distribution (Vss) can be derived from the area under the curve (AUC) and the area under the first moment curve (AUMC). 

The scaling factor Sj can take any value between 0 and 1 and is applied to site j. The superscripts p and m indicate permanent and mutual induction, respectively. Equation (9-19) can be solved iteratively using similar procedures to those used to solve Eq. (9-3). The formal permanent moments can be calculated by subtracting induced moments from moments from ab initio calculations. For any conformation of a given compound the atomic multipoles can be determined from Distributed Multipole Analysis (DMA) [51]. 

The summation in Eq. (8.53) is slowly converging if a molecular charge distribution is represented by a single set of moments. However, the expression can be written as the summation over the distributed moments, centered at the nuclei j, which is precisely the information available from the multipole analysis ... 

In these equations kei is the elimination rate constant and AUMC is the area under the first moment curve. A treatment of the statistical moment analysis is of course beyond the scope of this chapter and those concepts may not be very intuitive, but AUMC could be thought of, in a simplified way, as a measure of the concentration-time average of the time-concentration profile and AUC as a measure of the concentration average of the profile. Their ratio would yield MRT, a measure of the time average of the profile termed in fact mean residence time. Or, in other words, the time-concentration profile can be considered a statistical distribution curve and the AUC and MRT represent the zero and first moment with the latter being calculated from the ratio of AUMC and AUC. 

Since the single-center multipole expansion of the interaction energy is divergent, one could use a kind of multicenter expansion. One can hope that the multipole expansion will provide better results if multipole moments and polarizabilities localized at various points of a molecule are used instead of global multipole moments and polarizabilities. This idea forms the basis of the so-called distributed multipole analysis of the electrostatic, induction, and dispersion interactions between molecules187 195. 

A moment analysis of Si MAS spectra of zeolites is shown to provide direct information on the number of second-nearest-neighbor Al-Al pairs. Monte Carlo computer calculations are described of randomized A1 distributions in zeolite frameworks, under restrictions of Loewenstein s and Dempsey s rules. The method is applied to a hypothetical square planar lattice which allows the various A1 distribution patterns to be visualized in simple displays, and to the zeolite X and Y framework. The results are compared with experimental data taken from the literature. 

Although it is possible in principle to derive the ground state g-factor value by measuring the Co/Ih ratio from the Method of Moments analysis, this is usually not practical, as the absorption bands lack a baseline-to-baseline band envelope or due to overlap with bands associated with other metal centers or chromophores. It should be noted that in some instances, excited state parameters also have to be carefully considered. For example, when ground state degeneracy (and the associated Boltzmaim distribution of population across... 

Farrar and co-workers recorded the photodissociation spectra of the isoelectronic Sr+-(NH3) system and observed similar large spectral shifts for the 5 P-5 S transition [25, 26]. They carried out a moment analysis of the absorption spectra and found a large increase in the electronic radial distribution in the ground state with... 

The results of Shankar and Lehnhoff [77] confirm the validity of the procedures currently used for the interpretation of HPLC data, which are based on moment analysis of the experimental data and on the additivity of the first and second moments of the distributions (see next section). These results also show that this approach is not valid for short systems. Multiple peaks or shoulders on a single band may appear, due to the interaction between transport and retention phenomena. Then the method of moment analysis is inadequate to characterize the... 

Summation of slices parallel to F2 of individual 2D peaks (primarily in crystalline materials) provides the complete assignment of x and rj by either simulating the extracted lineshapes or by other means such as moment analysis " and direct observation. Data from amorphous materials are harder to interpret since the projection of their 2D peaks on to the F2 dimension exhibits distributions rather than well-defined second-order quadrupolar MAS lineshapes. Yet, they can be analysed by spectral inversion, full 2D simulations or direct observation, as will be shown at the end of this section. 

Estimates of the dispersion coefficient and time lag can be obtained by integrating the experimental data to obtain the mean and variance of the residence time distribution. These values may be substituted into the moment expressions (Eqs. 30, 31) to obtain Kj and td. In this study, it was convenient to integrate the experimental data using the rectangular rule. The accuracy of the moment analysis can be improved by the use of optimum truncation points (33) or by the use of wei ted moments (32, 34). The disadvantage of these methods is that the expressions relating moments and model parameters become more complicated. 

Experiments were conducted at flowrates of 0.5,1.0,1.5, and 2.0 ml/min and applied voltages of 0, 100, 200, 300, and 400 V across the column. Moment analysis of the residence time distribution of the dyes was used to estimate the electrokinetic dispersion coefficients. The effects of the direction of the electric field and the packing size on Kj were also investigated. 

Some early attempts were made to determine anisotropies from second-moment analysis in which use is made of the anisotropic contribution to the observed second moment.5 6 Limitations of this method are the need for large shifts and fields and the often-required assumption of axial symmetry of the shielding tensor. In 1968 a method was reported that uses coherent averaging techniques7 to effectively narrow the dipolar-broadened lines of powdered solids. The observed spectrum is curve-fitted using a computer-broadened theoretical chemical shift distribution to give the principal components of the shielding tensor. 

Moment analysis provides the means to determine a model independent characteristic of the absorption rate or dissolution rate (Riegelman and Collier 1980). A single value characterising the rate of the entire dissolution or absorption process is obtained, which is called the mean absorption or dissolution time (MAT and MDT, respectively). These parameters can be determined without any assumptions regarding absorption or disposition pharmacokinetics, apart from the general prerequisites of linear pharmacokinetics and absence of intraindividual variability described above. MAT/MDT can be interpreted as the most probable time for a molecule to become absorbed/dissolved, based on a normal Gaussian distribution. 

Use of Eq. [23] automatically implies a power law distribution of pore sizes. A variety of other distribution functions, including the log-normal, incomplete Gamma, and Weibull distribution functions, have also been used to characterize natural pore-size distributions (e.g., Brutsaert, 1966). Furthermore, it is possible to parameterize the pore size distribution without resorting to a particular distribution function model using moment analysis (Brutsaert, 1966 Powers et al., 1992). 

The distributed multipole analysis method of Stone and co-workers is similar in concept but is based on nonredundant spherical harmonic representation of the multipoles (recall that whereas there are six second moments, only five are independent). He initially places numerous site multipoles at centers of orbital overlap. The individual monopoles are spread out along the molecular axis, and are thought to represent the distribution of charge the site dipoles are also spread out along the bond axis. This very detailed description is simplified into a three-site model, which includes a site in the F—H bond. However, the multipole expansion does not converge well, especially for the bond center site. 

Several papers have been written on the subject of analyzing the anisotropic distribution of v and j vectors of molecules produced by a photodissociation reaction [17,36-42]. The most satisfying scheme is the renormalized bipolar moment analysis proposed by Dixon [18]. The basic premise is to use bipolar harmonics as the basis set to describe the angular distribution of a single recoil velocity v and fragment angular momentum j ... 

Statistical moment analysis is a noncompartmental method, based on statistical moment theory, for calculation of the absorption, distribution, and elimination parameters of a drug. This approach to estimating pharmacokinetic parameters has gained considerable attention in recent years. 

One of the most useful properties of statistical moment analysis is that it permits estimation of the apparent volume of distribution that is independent of drug elimination (i.e. regardless of the model chosen to describe the concentration time data). 

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