R-centroid

The unknown cubic and quartic terms, which are small, enter with a small weight and may safely be neglected in this case. Even the second term in this series amounts to less than 1% of the leading term (sometimes referred to as the r-centroid approximation), except at small separations where R is comparable to the collision diameter ( 5.71 bohr) and corrections amount to +2%. 

The significance of the. R-centroid is illustrated as follows. The electronic matrix element, He(R), may be expanded in a power series about an arbitrarily chosen internuclear distance, R (most usefully, R = Rc, the internuclear distance at which the two potential curves cross),... 

The i -centroid approximation has repeatedly been tested numerically. For perturbations between levels of potentials that intersect exactly once, the R centroid approximation can be regarded as more accurate than experimentally measurable matrix elements. 

The spectroscopic constants [1 ] and Morse potential functions for both states were used to calculate the Franck-Condon factors [8,9] and r-centroids [9,10] for the observed and a number of unobserved v v" transitions in the b X system. 

Franck-Condon Factors. r-Centroids. Transition Probabilities. 

Based on lifetime measurements in various A-state, v = 0 and 1, rotational levels (NH3 excitation by electron beam) [1], Franck-Condon (FC) factors r-centroids r, oscillator... 

Franck-Condon factors and r-centroids for c 11 (v = 0 to 3)<->a A (v" = 0 to 3) transitions of NH and ND were calculated using the own spectroscopic constants and literature data and assuming a Morse potential function for the molecules [11]. Based on lifetime measurements in the c-+a spectrum excited by electron impact on NH3 [23], Franck-Condon factors... 

Using the molecular constants from [3] and modified RKR potential energy functions for the b and d states, Franck-Condon factors and r-centroids for d —b transitions with v = 0 to 3 and v" = 0 to 3 were calculated [5]. 

As one should expect from the terminology, the earliest application of the q-centroid, strictly speaking, the r-centroid method was made in diatomic spectroscopy [131], although in a completely different physical sense. The q-centroid approximation presents a nontrivial generalization of the r-centroid approach of diatomic spectroscopy to the case of the nonradiative decay of polyatomic molecules. The importance of studying the q-dependence of the nonadiabatic coupling element was emphasized by Freed and Gelbart [132] and Freed and lin [27]. 

Fig. 3. Location of traverse sampling points, (a) Cross section of stack showing location of traverse points ( ) on perpendicular diameters. The circular cross section is divided into three equal areas at 0.5774 r, 0.8165 r, and r, where ris the radius. Sampling points are at the centroids of these areas at 0.38 r, 0.70 r, and 0.911 r. (b) Cross section of rectangular stack divided into 12 equal areas having traverse points ( ) at the centroid of each area.

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

Energy analyzers cannot be discussed without discussion of energy resolution, which is defined in two ways. Absolute resolution is defined as AE, the full width at half-maximum (FWHM) of a chosen peak. Relative resolution is defined as the ratio R of AE to the kinetic energy E of the peak energy position (usually its centroid), that is, R = AE/E. Thus absolute resolution is independent of peak position, but relative resolution can be specified only by reference to a particular kinetic energy. 

Irwan, R., Lane, R.G., 1999, Analysis of optimal centroid estimation applied to Shack-Hartmann sensing. Applied Optics 38, 6737 Kay, S.M., 1993, Fundamentals of statistical signal processing estimation theory, Prentice-Hall, New Jersey, pp 27-81... 

Figure 2. Negative FAB mass spectra of maitotoxin. The numbers denote the mass number at the centroid of each peak. A A survey scan at a low resolution (R=300). B. Resolution enhanced spectrum (R=3000) for ion clusters at around m/z 3300. C. Resolution enhanced spectrum (R=3000) for ion clusters at around miz 3400.

Where A F(z) is the free energy at z relative to that at the reactant state minimum zr, and the ensemble average < > is obtained by a quantum mechanical effective potential [15]. Note that the inherent nature of quantum mechanics is at odds with a potential of mean force as a function of a finite reaction coordinate. Nevertheless, the reaction coordinate function z[r] can be evaluated from the path centroids r, first recognized by Feynman and Flibbs as the most classical-like variable in quantum statistical mechanics and later explored by many researchers [14, 15]. 

Details of the derivation for Kleinert s variational perturbation (KP) theory can be found elsewhere [21], The nth-order KP approximation W, (r) to the centroid potential W (f) is given by... 

Etchebest, C., R. Lavery, and A. Pullman. 1982. The Calculations of Molecular Electrostatic Potential from a Multipole Expansion Based on Localized Orbitals and Developed at Their Centroids Accuracy and Applicability for Macromolecular Computations. Theor. Chim. Acta 62, 17. 

In addition, mention may be made of more recent electrostatic calculations due to Voitlander and his coworkers 110,177). Qualitatively the results are very similar to those of Robertson and McConnell 30), although the latter contain erroneous numerical values (177), but the choice of R as the metal to ring centroid distance, rather than the metal to carbon distance, leads to much better agreement with experiment. Thus, for Fe(Cp)2, the latter choice predicts 6 < n < o, but the former yields 5 o < ir, with a total splitting of some 18000 cm-1, in quite good agreement with experiment. 

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