Origin of the Deviations

In order to study the origin of the deviations observed, we first consider the statistical convergence of the QCL data. As a representative example. Fig. 14 shows the absolute error of the adiabatic population as a function of the number of iterations N—that is, the number of initially starting random walkers. The data clearly reveal the well-known 1/Vn convergence expected for Monte Carlo sampling. We also note the occurrence of the sign problem mentioned above. It manifests itself in the fact that the number of iterations increases almost exponentially with propagation time While at time t = 10 fs only 200 iterations are sufficient to obtain an accuracy of 2%, one needs N = 10 000 at t = 50 fs. 

Once this procedure is complete, the user has a permanent record of Raman shift accuracy, absolute instrument response, and instrument response function for a given experimental session. Over time, these periodic records provide an indication of instrumental drift and performance changes. In addition, they may be checked at a future date if a particular sample appears to deviate from expected behavior. If the instrument was at fault, the daily qualification spectra may provide insight into the origin of the deviation. 

It is not easy to locate the origin of the deviations between the various sets of thermodynamic data listed in Table 1. Rau et al. mentioned that the enthalpies of all reactions except for the formation of S2 could be varied slightly without much effect on the total sulfur pressure which, on the other hand, was taken as the final proof of the consistency of the data set. In addition, the entropy of Sg used by these authors (430 J mol at 298 K) does... 

From the data in Table 8, spin-orbit effects on bond lengths, harmonic vibrational frequencies and dissociation energies for (113)F were obtained, and listed in Table 15. The spin-orbit effects fi-om the one- and two-component REP results deviate somewhat from those from DK and DC based all-electron results. The spin-orbit effects evaluated by Seth et al. [78] from ARPP-SOCI calculations are also in overall better agreement with those from RECP than those from DK/DC results. The origin of the deviation is not clear, but spin-orbit effects are qualitatively similar enough to make the discussion of spin-orbit effects based on the RECP results valid. The variations in spin-orbit effects for R and (Og, obtained at various levels of theory by Seth et al, are uniformly smaller in comparison with other results both for (113)F and (113)H, which is probably due to... 

The origin of the deviation from the cross relation for the reactions of [V (0)2(bpy)2] or [Ru (0)(bpy)2(py)] with xanthene is not known. Three other HAT reactions of xanthene do not show this large deviation (Table 1.4), so the deviations are not likely to be due to an erroneous xanthene selfexchange rate constant [taken as that of 9,10-dihydroanthracene (DHA, Table 1.1)]. Tunneling could be important in reactions of Ru (0)(bpy)2(py), as the reaction with DHA has > 35. Interestingly, the cross relation... 

Even if the results obtained for model (calibration) samples deviate fromLambert-Beer s law, the observed relationship between the A value and the concentration of the analyte may still be suitable for a fit-for-purpose practical quantitative analysis. In such a case, it is useful to express the observed results with a nonlinear relationship between the Aj value and the concentration of the target substance. This nonlinear relationship can be used for quantitative analysis, regardless of the origin of the deviation from Lambert-Beer s law. In a limited range of the concentration of the analyte, the nonhnear relationship may sometimes be approximated by a linear relationship, which is often easier to use practically. 

These considerations constitute the basic support of hybrid QM/MM methods. In fact MM potentials are able to describe the most important factors influencing the electronic structure of a given group, when expressed at the QM level, and also to describe the steric factors which are often at the origin of the deviations with respect to the norm in the reactivity of chemical functions. 

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