CBS models

Geometry Method UHF/3-2IG(+) B3LYP /6-311G (2d,d,p) QCISD/6-311G(d,p) 

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The CBS models use the known asymptotic convergence of pair natural orbital expansions to extrapolate from calculations using a finite basis set to the estimated complete basis set limit. See Appendix A for more details on this technique. 

CBS models typically include a Hartree-Fock calculation with a very large basis set, an MP2 calculation with a medium-sized basis set (and this is also the level where the CBS extrapolation is performed), and one or more higher-level calculations with a medium-to-modest basis set. The following table outlines the components of the CBS-4 and CBS-Q model chemistries ... 

Remember that the CBS models begin with a large enough SCF calculation to obtain the desired level of accuracy (see Chapter 7) therefore, no explicit extrapolation of the SCF energy is included. CBS extrapolation involves computing the second-order and infinite-order corrections to the energy. 

CBS model chemistries make the correction resulting from these extrapolations to the second-order (MP2) correlation energy ... 

Chapter 7, High Accuracy Energy Models, describes several research procedures for predicting very accurate thermodynamic and energetic properties of systems, including Gl, G2, G2(MP2) and several Complete Basis Set (CBS) models. 

A comparison between the four CBS models is shown in Table 5.3. 

G2/CBS models, essentially all die correlation energy of the bond being broken must be recovered. This in mm necessitates large basis sets and sophisticated correlation methods. This is also the reason why ab initio energies are not converted into heats of formation, as is normally done for semi-empirical methods (eq. (3.89)), since the resulting values are poor unless a very high level of theory is employed. 

A comparison between the four CBS models is shown in Table 5.3. It should be noted that the G2-1 data set. with two exceptions fSOo and CO7), onlv >e(exp) - Z>e(ref)... 

G2/CBS models, essentially all die conelation energy of the bond being broken must be (due to a poor description of the electron-electron repulsion). In the lower half of the... 

The model that is formed by Eqs. (24) and (25), with the extra assumption that wall effects can be neglected (i.e., that the reactor is built up solely of central subchannel elementary cells such as the one depicted in Fig. 10), is called the catalyst bead (CB) model. Because of the assumption in the CB model that the value of the mass transfer coefficient at the flat ends of the particle equals the value at the cylindrical surface, it can be expected that this model overestimates the conversion obtained in a BSR, especially when the particle length-over-diameter ratio is much smaller than 0.5 and when the gap between consecutive particles on a string is small compared to the particle diameter. However, whether the CB model over- or underestimates the BSR performance also depends on the error made in the calculation of the particle effectiveness factor. 

In this equation, represents the effective lateral diffusion/dispersion constant for laminar flow a value on the order of is suitable, and for turbulent flow the molecular diffusion coefficient in this expression should be replaced by the turbulent diffusion coefficient. Based on these simple relations it can be calculated that under typical conditions, lateral reactant transport takes place only over distances of a few subchannels. In other words, if the gas velocity through the wall channels differs much from the velocity through the central subchannels, the nonuniform flow profile can have a significant effect on the overall reactant conversion. In these situations the CBS model can be expected to give a better estimate of the reactor performance than the CB model. 

The performance of lab-scale BSR modules in the SCR of NO can be predicted with an error of ca. 10% by four relatively simple mathematical models, whose parameters were determined via independent experiments. As expected, the LCF and the LCR model generally underpredict the conversion, whereas the CB model generally overestimates it. The CBS model gives the most accurate prediction of the NO conversion On the average, the predicted conversion deviates ca. 5% (relative) from the experimentally determined value. Such deviations can be attributed to stochastic variations in the experiments. 

Figure 19 Parity plots of the NO conversion in the BSR. (a) LCR model (b) LCF model (c) CB model (d) CBS model.

The use of the smaller basis for the QCISD(T) calculation means that the CBS-Q model is computationally faster than G2(MP2), but nevertheless gives slightly lower errors. A comparison among the four CBS models is shown in Table 5.6 (p. 218). 

Vj. and f similar to those in case of R-hel PA fibers. All the above arguments allow us to estimate the average size of conductive islands inside the R-hel PA nanofiber of the order 10 nm. The CB model predicts that the scaling exponent 1 is characteristic of ID system, while (1 2 is typical for 2D tunneling [87]. From this point of view the existence of two slopes in 7 vs. (V- Vj) / Vx plots at low temperature, namely 1.3 at low biases and 1.8-2.1 at high biases, can be attributed to the transition from ID to 2D tunneling with increase in excitation bias [85]. 

There are several other composite approaches, for example, complete basis set (CBS) models [56], focal-point analysis [57], multi-coefficient correlation methods [58], high-accuracy extrapolated ab initio thermo-chemistry (HEAT) [59] and the Weizmann-4 theory [60]. Hansen et al. [61] have employed the so-caUed MP2 DFT [193] scheme for analysing benzene ethylation over H-ZSM-5. Density functional calculations applying periodic boundary conditions [Perdew-Burke-Emzerhof (PBE) functional] were combined with MP2 energy calculations on a series of... 

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