Subtractive scheme

In this application, the process analyzer is used in the vis-NIR spectral region to measure the clear top layer on a co-extruded polymer film. The bottom layer is pigmented to an opaque white color and its thickness cannot be determined by this method. Prior to the installation of the fiber-optic spectroscopy system, film samples were measured manually in the laboratory by a subtractive scheme. First, the total thickness of a sample was measured on a manual profilometer. The top layer of the polymer was removed with methylene chloride. The sample was then repositioned on the profilometer as closely as possible to the originally measured spot and the thickness of the second white layer was determined. The thickness of the top layer was then determined by difference. 

These results show that the minimal subtraction scheme eliminates ambiguities inherent in an extrapolation of e-expansion results to physical dimension d = 3. For the flow equations no extrapolation is necessary. Furthermore they are strictly independent of the parameter bu(z). 

Model A is the most straightforward implementation of a combined QM/MM model but in this type of QM/MM approach, the QM and MM regions do not interact in the quantum chemical calculation. In models of type A, the QM/MM energy of the whole system, totalQM/MM, is calculated in a simple subtractive scheme ... 

The representation (1.18) implies a subtraction scheme for calculating the finite part of the Wichmann-Kroll potential and the vacuum polarization charge density It was first considered by Wichmann and Kroll (1956). A detailed discussion of the evaluation of this contribution for high-Z nuclei of finite extent is presented in Soff and Mohr (1988) and Soff (1989). A special application of the computed vacuum polarization potential to muonic atoms has been presented in Schmidt et al. (1989). 

The subtraction scheme can be easily understood in the following way the whole system is described at the low level of theory (Efhigh level (E high w - E low lmel). The additive scheme is simply sum of low and high level descriptions of outer and inner parts, respectively, augmented by a cross-interaction term (E JJ ). 

A description of the partial-wave renormalization (PWR), used for calculating the first-order self energy and certain higher-order effects for the energy levels in highly cllarged ions, is presented. We put special emphasis on correction terms which have to be considered due to the use of the non-covariant subtraction scheme used in PWR. 

The U-, d-, and s-quark masses are estimates of so-called current-quark masses, in a mass-independent subtraction scheme such as MS at a scale /r 2 GeV. The c-and 6-quark masses are the running masses in the MS scheme. For the 6-quark we also quote the IS mass. These can be different from the heavy quark masses obtained in potential models. 

We adopt the minimal subtraction scheme where the D s are chosen to subtract the poles and only the poles. The calculation is order by order and so, to one loop order, one cannot determine D2, which involves (two loop term ). 

We show that the flow equation Eq. (74) is exact in the minimal subtraction scheme using dimensional regularization. More details may be found in Ref. [36,37]. 

Here we describe in more detail two effective schemes, that are used in the context of the model, considered in our review the massive and minimal subtraction schemes. 

Here, r is a rescaling parameter, which defines the scale of the external momenta in the minimal subtraction scheme. In the same way as in the Callan-Symanzik equation (77) the coefficients in (86) define the renormalization group functions ... 

Inspection of Eq. (3) shows that it would correspond to the total MM energy by changing the subscript in the first term. This leads to the subtractive scheme characteristic of the ONIOM method [5, 18-20] ... 

The difference to the subtractive scheme is that here a pure MM calculation is performed for only the outer region and the interaction between QM and MM regions is achieved by an explicit coupling term,... 

The result of the MM calculation for the QM part is subtracted to avoid double counting. In this straightforward realization [359, 360], the subtractive scheme is very simple to implement however, the coupling between the subsystems is handled entirely at the MM level, i.e. a polarization of the QM part by the MM part is not taken into account. Due to the simplicity of the approach, extensions to the combination of two QM approaches [361-363] or to the treatment of various layers are also possible. The latter approaches are known under the name ONIOM (our n-layered integrated MO and MM) [364-367]. Ryde and coworker also developed a subtractive scheme [32, 368, 369]. It is also used in Qpot developed by Sierka and Sauer [37, 39]. 

To avoid the negative effects of dangling bonds on the quantum mechanical solution for the internal part, terminating H atoms are added. They are also called link atoms, L. The L atoms and the internal part form the cluster, C = I -f L. The energy of the total system can still be approximated by the subtraction scheme ... 

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