Intrinsic error calculations

Ans. The percentage of copper, 86%, is close and appears to be within the intrinsic error of measurement and calculation, but may not be conclusive. 

We have only concentrated on relativistic density functional calculations on heavy-element compounds. Calculations on compounds of light elements have been reported in Refs. [157-159] and primarily aimed at high numerical accuracy. However, for these compounds the intrinsic error of existing density functional treatments is probably much larger than the relativistic effects, so the purpose of such calculations remains unclear. 

Figure 5.6b The worksheet for the calculation of the two-electron repulsion term, equation 5.52. The values, calculated in column R, follow from the intrinsic Error function defined in the Engineering function library in EXCEL.

Some calculations can be carried out to give corrected depth profiles. Usually, this approach needs some modeling of the diffraction and refraction effects that arise in confocal Raman spectroscopy [41, 50]. Indeed, the depth profiles obtained in a conventional way contain, on the one hand, an intrinsic error due to the diffraction phenomena of the optical system, and on the other hand, an error due to the refractive index difference between the membrane (polymer sample) and its environment (bathing solutions or gas). 

For the other elements in the second row, there have been few direct calculations on the importance of 2s2p correlation. Possibly the best indication of the effect of inner-shell correlation comes from benchmark calculations determining the valence (complete) basis set limit. In Table 7 we show the intrinsic error computed at the CASSCF/ICMRCI level reported by Woon and Dunning for a series of second row molecules. The intrinsic error is the difference in the CBS limit at a specified level of correlation and experiment. It is the composite of a number of small effects including the error in the valence electron correlation treatment (n-particle error), any error in the CBS extrapolation, the relativistic correction, and the effect of inner-shell correlation as well as any error in... 

Before we consider the different properties analysed in the present chapter, it is appropriate to discuss in general terms the errors that arise in quantum-chemical calculations. Thus, in Section 15.2.1, we discuss the sources of errors in quantum-chemical calculations, introducing the concepts of apparent and intrinsic errors. Next, in Section 15.2.2, we review the measures of errors that we shall employ in our statistical investigations in this chapter. 

N-electron error is the difference between the result obtained with a given A -electron model and the result that would be obtained using the FCI wave function in the same basis set. As illustrated in Figure 15.1, we may think of the apparent error as the combined effect of the basis-set error and the iV-electron error. It should be noted that the A -electron error depends on the basis set employed in the calculation. Of interest is also the A-electron error in the limit of a complete basis, which we shall refer to as the intrinsic N-electron error or simply the intrinsic error. 

After the intrinsic and apparent errors of the standard models and basis sets have been established, this knowledge may be used in the design of balanced calculations, where the quality of the basis set matches the quality of the A -electron model. Consider the calculation depicted in Figure 15.2. From basis 2 to basis 3, a significant improvement in the calculated property is observed compared with the intrinsic error of the model beyond basis 4, the basis-set change is small compared with the intrinsic error. Thus, little improvement is observed beyond basis 4, which is said to represent a balanced level of description. Beyond this basis, the accuracy of the description cannot be substantially improved upon because the basis-set error is small compared with the intrinsic error of the wave-function model. The calculation is balanced in the sense that... 

The calculations presented in this section have shown that, whereas improvements in the correlation treatment in general increase the bond length, improvements in the one-electron treatment reduce it. The coupled-cluster hierarchy converges smoothly, with a reduction in the error by a factor of 3-5 at each level. Thus, whereas the intrinsic error (based on A and Aabs in Figure 15.3) at the Hartree-Fock level is about —3 pm, it is reduced to about —0.8 pm at the CCSD level and to about —0.2 pm at the CCSD(T) level. In contrast, the Mpller-Plesset hierarchy oscillates, with typical errors of about 0.5 pm at the MP2 level, 1.2 pm at the MP3 level and 0.4 pm at the MP4 level. Even for the small molecules in this study, the CISD model is inadequate, with an intrinsic error of —1.7 pm. Based on these observations, we may state that, in terms of computational accuracy and cost, the only successful models are Hartree-Fock, MP2, CCSD and CCSD(T). 

In Table 15.7, we have collected the equilibrium bond distances of the molecules in Table 15.1, calculated using the core-valence cc-pCVQZ basis set, correlating all the electrons in the system. For the CCSD(T) model, a comparison with experiment shows that, for 22 of the 29 bond lengths, the difference is less than or equal to 0.1 pm. This error is less than the intrinsic error of the CCSD(T) model and arises from a cancellation of errors. 

All values were calculated for intrinsic viscosity of gelatin B solutions to 37.4°C and compared against the value of Huggins, normally used as standard. It is noteworthy that each method has a relative error percentage and low for methods of more than four pair s values (Er% > 0.30). 

Having these severe approximations in mind, SCC-DFTB performs surprisingly well for many systems of interest, as discussed above. However, it has a lower overall accuracy than DFT or post HF methods. Therefore, applying it to new classes of systems should be only done after careful examination of its performance. This can be done e.g. by conducting reference calculations on smaller model systems with DFT or ab initio methods. A second source of errors is related to some intrinsic problems with the GGA functionals also used in popular DFT methods (SCC-DFTB uses the PBE functional), which are inherited in SCC-DFTB. This concerns the well known GGA problems in describing van der Waals interactions [32], extended conjugate n systems [45,46] or charge transfer excited states [47, 48],... 

G(X) and quantum yield of the product used for the calculation, respectively. Two possible sources of errors should be mentioned. One is the energy dependence of the photodecomposition. In order to obtain intrinsic yield should be measured at photon energies as close as possible to the absorption onset, or appropriate corrections should be used [154,155]. Both photon emission and photodecomposition studies show that the energy dependence is more severe for the smaller molecules than for the larger ones. Therefore, G(Si) for the larger molecules can be estimated with higher accuracy. The second source of error is that X may not solely form in Si molecule transformation. 

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