CLS models

Low-pressure steam-generating Cl models are particularly sensitive to changes in water chemistry, and their rough waterside surfaces makes them susceptible to deposition, and thus to the risk of overheating and subsequent cracking. Typically, they are quite small boilers, most being less than 200 hp. 

In ab initio methods (which, by definiton, should not contain empirical parameters), the dynamic correlation energy must be recovered by a true extension of the (single configuration or small Cl) model. This can be done by using a very large basis of configurations, but there are more economical methods based on many-body perturbation theory which allow one to circumvent the expensive (and often impracticable) large variational Cl calculation. Due to their importance in calculations of polyene radical ion excited states, these will be briefly described in Section 4. 

The application of CL model and initial information allowed the researchers to map the critical loads of various heavy metals for different ecosystems. 

A CLS model can be built using either experimentally measured spectra or estimated spectra to populate K. Use of the latter option requires that a series of standards be obtained for which the known concenna-tions (C) of all components are known. The spectra of such standards (Xsta) could then be used to estimate K by least squares ... 

The same calibration fit metrics, RMSEE and r, can be used to express the fit of a CLS model. However, one must first calculate the model-estimated component concennations for the calibration standards ... 

From the technical viewpoint, the matrix inversion (C C) in Equation 12.36 can be very unstable if any two of the analyte concentrations in the calibration standards happen to be highly correlated to one another. This translates to the need for careful experimental design in the preparation of calibration standards for CLS modeling, which is particularly challenging because multiple constituents must be considered. In addi-... 

Although the CLS model can be considered rather rigid and limited in its scope of application, its advantages can be considerable in cases where it is applicable. Furthermore, recent work [47-51] has shown that extensions of the CLS model can reduce some of this rigidity , thus enabling the power of direct calibration methods to be applied to a wider scope of practical applications. Such extensions of CLS will be discussed in the following section. 

Appropriately, extended mixture models involve an extension of the CLS model ... 

The coefficient of determination, R, of the Cl model is 0.98. The maximum error for Cl prediction is below 7%. Figure 12.9 shows Cl estimated from H-NMR spectra versus that calculated from its original formula. 

There are twa steps to validating an ICLS model. The first is to verify that the estimated puse spectra are reasonable the CLS model assumptions are then validated. Sewral diagnostic tools for validating the pure spectra are discussed, and a summary is found at the end of the section in Table 5.8. The primary use of these diagaostic tools is to investigate whether the estimated pure spectra are reasonaWc. 

Once the f e spectra have been estimated and validated, the CLS model is validated foil tving the approach discussed for DCLS (Section 5.2.1.1). This can be done rith the calibration set of mixtures or with an additional validation set. The validation of the pure spectra already gives some confidence in the model because violations of the CLS assumptions will be identified when reason e pure spectra are not obtained. However, the DCLS validation tools found IE Table 5.1 should still be used. For this example, this second step in tlie model validation did not reveal any problems with e model (results not shovia). See Example 2 for discnission on the complete ICLS model validation. 

Assume also that a validation sample has been collected with concentrations for Sj, Sy, and ij of 1, 3, and 2 respectively (c = [1 3 2]). Assuming linear additivity holds, the resulting response vector for this mixture sample is r= (12 8 10]. When validating tire models using this sample, the known information is the measured spectrum, r= [12 8 10], and the component amcentrations for the known analytes c = [1 3]- The steps for validating the CLS model arc shown in Figure 5.62 and include (a) formulating... 

A commonly perceived weakness of PCR/PLS is that it usually takes many samples to comstruct rhe model. This can be tnie if relying on natural designs (see Chapter 2 and Appendix A). It is tnie that, in general, more samples are required to build a PLS model than a CLS model (see Section 5-2). However, this is because the inverse models are typically correcting for effects that cannot be modeled using the classical methods. The perception that hundreds of samples are always required to build inverse models is simply not true. One rule of thumb is that there should be at least three times as many samples as factors. 

The conclusion from the validation of Example 1 is that the CLS model assumptions are valid. The measures of performance for the DCLS model are as follows (see Table 5.1 for a description of these figures of merit) ... 

Equation (3.17) is the fundamental equation of the CLS model that allows calibration and prediction. The calibration step consists of calculating S, which is the matrix of coefficients that will allow the quantification of future samples. S is found by entering the spectra and the known concentration of a set of calibration samples in eqn (3.17). These calibration samples, which can be either pure standards or mixtures of the analytes, must contain in total all the analytes that will be found in future samples to be predicted. Then, eqn (3.17) for I calibration samples becomes... 

The multivariate quantitative spectroscopic analysis of samples with complex matrices can be performed using inverse calibration methods, such as ILS, PCR and PLS. The term "inverse" means that the concentration of the analyte of interest is modelled as a function of the instrumental measurements, using an empirical relationship with no theoretical foundation (as the Lambert Bouguer-Beer s law was for the methods explained in the paragraphs above). Therefore, we can formulate our calibration like eqn (3.3) and, in contrast to the CLS model, it can be calculated without knowing the concentrations of all the constituents in the calibration set. The calibration step requires only the instrumental response and the reference value of the property of interest e.g. concentration) in the calibration samples. An important advantage of this approach is that unknown interferents may be present in the calibration samples. For this reason, inverse models are more suited than CLS for complex samples. 

A last note refers to the design of the calibration set. Although we do not need to know either the spectra or the concentration of the interferences, we must be sure that the calibration samples contain the analytes and interferences which might contribute to the response of the unknown samples. In this way, the calculated regression coefficients can remove the contribution of the interferences in the predictions. If the instrumental measurement on the unknown sample contains signals from non-modelled interferences, biased predictions are likely to be obtained (as in the CLS model). 

C. Ab Initio Cl Models of Simple Double, Charged, and Dative 7r-Bonds... 

A CLS model can be implemented using either measured pure component spectra (K) or estimated pure component spectra (K). For most process analytical applications, the second option is most recommended because it is rare that high-quality pure component spectra of all analytes can be physically obtained. Furthermore, even if they could be obtained, they might not sufficiently represent the spectra of the pure components in the process mixture. However, the use of estimated pure component spectra requires that a series of standards of known concentration (C) for all analytes be analyzed by a reference analytical method, thus generating a series of standard spectra (Xsta). If one can manage to do this, the estimated pure component spectra (K) can be calculated ... 

The estimated pure component spectra K (or the measured pure component spectra, K, if they are sufficiently relevant) are the parameters for the CLS model. Once they are determined, the concentrations for all analytes in a future sample (cp) can be estimated from the spectrum of that sample (xp) ... 

When applying the Cl model to the ground state of helium, the 2p2 S basis function makes a significant contribution, and the expansion... 

Using a four-phase model consisting of ambient/simple grade/film/ substrate, we fit the data to obtain the dispersion of optical constants for each films in the range of 1.55-6.53 eV. The Cauchy model was used as a model for the substrate and fixed during the fitting. The Cody-Lorentz (CL) model [14] was used as a model for the film. 

In the range of 1.55-6.35 eV, the absorption coefficient of SI was represented by one CL model. Based on the assumptions that the Hf-N bonds and Hf-O bonds independently form the parabolic band structure, and that total absorption coefficient is the sum of the absorption coefficients related to the Hf-N bonds and the Hf-O bonds two CL models were used as models for S2 and S3 films. Due to the degradation of Hf-O-N thin films, the simple grade model was applied after the fitting of films. 

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