Gaussian contribution

Fig. 13 Isotopic line splitting of the V3 stretching vibration in single crystalline (see also Fig. 12(a)), after [108, 109], The origin of each absorption band is indicated by an isotopomer present in crystals of natural composition. While the absorption could be fitted by a Lorentzian band profile, the remaining peaks were dominated by the Gaussian contribution in the Voigt band shapes (solid lines below the spectrum). The sum result of fitting the isotopic absorption bands is inserted in the measured spectrum as a solid line...

It is noted that SOC is independent of Mc above values of Mc of about 5000 (even more). At lower values of 1 network chains are sufficiently short that non-gaussian contributions must be considered. 

The SAMI parametrization [74] further extends the number of two-electron integrals included in the treatment. They are calculated first for the AOs taken as in the STO-3G Gaussian basis set, but then scaled using the distance dependent functions containing adjustable parameters. The SAMI method has been parametrized for the elements H, Li, C, N, O, F, Si, P, S, Cl, Fe, Cu, Br, and I. Unfortunately, this parametrization was never thoroughly published and studied. The same applies to the PM5 method [75] which is implemented only by a commercial software, without adequate explanation.2 Further refinement of the system of correcting Gaussian contributions to the interatomic interaction functions has been proposed in [71],... 

Dq anisotropic Gaussian contribution from lattice distortions ... 

Tq isotropic Gaussian contribution from grain size ... 

Figure 3. The phase difference between each of the two Gaussian contributions to the excited state wave packet and the ground state wave packets plus fltj. The solid line gives the phase difference for the initially undistorted wave packet and the dashed line gives the phase difference for the distorted packet.

Figure 12. Width parameters for the initially undistorted Gaussian contribution to the excited state nuclear wave function.

It is recommended to associate to d(M) a probability P which will delineate the contour of the group as a function of x2(a ri). In fact, the Mahalonobis distance gives a default classification since it does not take into account the random errors of measurement and the non Gaussian contribution in the distribution of the data. 

In a typical first principles molecular dynamics simulation, a new Gaussian contribution is added to the potential V(Sa, t) every given time interval t amounting to few hundreds of ordinary molecular dynamics steps. In fact, in between a Gaussian and the next one, the system must be allowed to fully re-equilibrate and to explore exhaustively the available portion of phase space before blocking one part of it as the introduction of a new penalty function is designed to do. The schematic view of this approach is summarized by Fig. 2.7. 

Figure 9.9 shows an exemplary simulated IDF (black straight line) and the first three Gaussian contributions (light gray), (dark gray) and -2h (black) as dotted lines. 

The Thompson-Cox-Hastings function is often used to refine profiles with broad diffraction peaks because it is the more appropriate model for line-broadening analysis where the Lorentzian and Gaussian contributions for crystallite size and for microstrains are weighted. So in this case, the peak shape is simulated by the pseudo-Voigt function, which is a Unear combination of a Gaussian and a Lorentzian function (Table 8.5). 

The refinement of oxide as well as oxyfluoride diffractograms has shown that line broadening is not due to the Gaussian contribution of microstrains (DST tan0 = 0). The crystallite sizes vary from 16 to 7 nm and decrease when Ca/Ce atomic ratio raises but are of the same order of magnitude for oxides and oxyfluorides for the same Ca/Ce atomic ratio. The estimated standard deviations corrected with the Berar factor are given in parentheses. 

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