Computational results

Instead of proceeding with a historical presentation, we discuss the computational results by methodology. Using HF/6-31G, the only feature located on the 2h sUce through the PES is a transition state with /ijg = 2.046 This geometry looks quite reasonable however, as is found with many other pericyclic reactions, the activation barrier is dramatically overestimated AH, = 55.0 kcal moE versus 

TABLE 4.6 Energies and for Transition States and Intermediates for the Cope 

A number of different CASSCF compulations have been reported, varying in the size of the active space and the basis set. ° Optimization of structures using CASSCF(6,6)/6-3 IG revealed both a diyl structure with / jg = 1.641 A and a transition state with / jg = 2.189A. The diyl structure is 1.9 kcal mol below the transition state. A transition state leading to the diyl was also located  

Why are the CASSCF computations not only quantitatively but also qualitatively incorrect Inclusion of dynamic correlation decreases the diradical connibution to the wavefunction, which is overestimated by CASSCF. This was a very surprising result at that time. The simple valence bond model described above would imply that the CASSCF approach should be satisfactory. The failure of CASSCF meant that much greater computational resources than anyone had expected would be needed to adequately describe even simple organic reactions, such as the Cope rearrangement. 

Since inclusion of dynamic electron correlation increases the importance of the aromatic contribution (Tgro ) at the expense of the diradical contributions, a single-reference post-HF method might be satisfactory. CCD and QCISD find a transition state with Rjg = 1.87 A. Both methods predict an activation barrier that is about 7 kcal moC too high.  

The model developed was based on the framework and model developed by Majozi and Zhu (2001) for the following reasons. Firstly, the models that exploit the structure of the SSN result in fewer binary variables than those derived from other mathematical methods, because the SSN only takes states into account while tasks are implicitly incorporated. Secondly, this model is based on the non-uniform discretization of the time horizon, thus resulting in fewer binary variables. Thirdly, the model is a MILP, thus solutions are globally optimal. 

Two distinctive models were developed in order to investigate the effectiveness of PIS operational philosophy. The first model is separated into two parts. The first part is used to determine the optimal throughput when there is zero intermediate storage available. Two situations were studied. Firstly the model was solved without the use of the PIS operational philosophy. Secondly, the model was solved with the PIS operational philosophy. In the simple example shown in this section a 50% increase in the throughput was achieved when the PIS operational philosophy was used. In both cases the models developed were a MILP, thus guaranteeing global optimality. 

The second part of the first model was used to determine the minimum amount of intermediate storage required to achieve the same throughput achieved when there is 

The second mathematical formulation presented, is a design model based on the PIS operational philosophy. This formulation is an MINLP model due to the capital cost objective function. The model is applied to a literature example and an improved design is achieved when compared to the flowsheet. The design model is then applied to an industrial case study from the phenols production facility to determine its effectiveness. The data for the case study are subject to a secrecy agreement and as such the names and details of the case study are altered. 

Unit Capacity Suitability Mean processing time (x) 

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Table 1. Computed results from the algorithm for sizing of the cracks from Fig.4a, Fig.4b, and Fig.4c and orientation angles d)=0°, 0=30° and 0=45°, and true values of the eorresponding crack parameters.

Ultrasonic pulses are generated within the sample on the transducer s resonance frequency, pass through sample and are received by the wideband receiver. After received pulse processing in the computer, results analysis is carried out. 

So called Ilydrogenic atomic orbitals (exact solutions for the hydrogen atom) h ave radial nodes (values of th e distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are n ot sensitive to these nodes and most simple calculation s use Slater atom ic orbitals ofthe form... 

More computational results are provided by Hite [75], including a second order reaction with different stoichiometry, and slab, as well as sphere geometry for the pellet. Overall it can be concluded that the approximate procedure of Hite and Jackson gives results of quite adequate... 

Var y the size of the increment in a in program statement 10. Tabulate the increment size, the computed result for a , and the calculated Wien constant. Comment on the relationship among the quantities tabulated. 

Computational results can be related to thermodynamics. The result of computations might be internal energies, free energies, and so on, depending on the computation done. Likewise, it is possible to compute various contributions to the entropy. One frustration is that computational software does not always make it obvious which energy is being listed due to the dilferences in terminology between computational chemistry and thermodynamics. Some of these differences will be noted at the appropriate point in this book. 

Because mesoscale methods are so new, it is very important to validate the results as much as possible. One of the best forms of validation is to compare the computational results to experimental results. Often, experimental results are not available for the system of interest, so an initial validation calculation is done for a similar system for which experimental results are available. Results may also be compared to any other applicable theoretical results. The researcher can verify that a sulficiently long simulation was run by seeing that the same end results are obtained after starting from several different initial configurations. 

CODESSA reads molecular structure files or output files created by other software packages as the starting point for QSAR analysis. It can import computational results from AMPAC, MOPAC, and Gaussian as well as structures in a number of common formats. 

This difference is shown in the next illustration which presents the qualitative form of a potential curve for a diatomic molecule for both a molecular mechanics method (like AMBER) or a semi-empirical method (like AMI). At large internuclear distances, the differences between the two methods are obvious. With AMI, the molecule properly dissociates into atoms, while the AMBERpoten-tial continues to rise. However, in explorations of the potential curve only around the minimum, results from the two methods might be rather similar. Indeed, it is quite possible that AMBER will give more accurate structural results than AMI. This is due to the closer link between experimental data and computed results of molecular mechanics calculations. 

In analyzing the computational results, the following quantities are very important ... 

Table 2-5 gives the input data and computer results. 

The computer results from Table 5-13 show the calculated compositions of benzene, diphenyl, triphenyl, and hydrogen. At a fixed feedrate, increasing V/F values correspond to movement through the plug flow reactor (i.e., increasing reactor volume). Thus, these results illustrate how the composition varies with position in the reactor. Here, the mole fraction of benzene decreases steadily as the reaction mixture progresses in the reactor, while the composition of diphenyl increases and reaches a maximum between 1,684 and 1,723 hr and thereafter decreases. This is often typical of an intermediate in consecutive reactions. 

Using the same values of the kinetic parameters as in Type 1, and given C o = 0-1 mo 1/1, it is possible to solve Equation 6-155 with Equations 6-127 and 6-128 simultaneously to determine the fractional conversion X. A computer program was developed to determine the fractional conversion for different values of (-iz) and a temperature range of 260-500 K. Eigure 6-30 shows the reaction profile from the computer results. 

Table 7-20 shows the computer results using an external heat exchanger involving a non-isothermal cooling medium. 

The quality of the computed results is only as good as the quality of the boundary conditions supplied by the user. 

Blast effects can be represented by a number of blast models. Generally, blast effects from vapor cloud explosions are directional. Such effects, however, cannot be modeled without conducting detailed numerical simulations of phenomena. If simplifying assumptions are made, that is, the idealized, symmetrical representation of blast effects, the computational burden is eased. An idealized gas-explosion blast model was generated by computation results are represented in Figure 4.24. Steady flame-speed gas explosions were numerically simulated with the BLAST-code (Van den Berg 1980), and their blast effects were calculated. 

It is important to realize that whenever qualitative or frontier molecular orbital theory is invoked, the description is within the orbital (Hartree-Fock or Density Functional) model for the electronic wave function. In other words, rationalizing a trend in computational results by qualitative MO theory is only valid if the effect is present at the HF or DFT level. If the majority of the variation is due to electron correlation, an explanation in terms of interacting orbitals is not appropriate. 

As was pointed out earlier (76AHCS1, p. 217), tautomeric equilibria for substituted isoindole-isoindolenine systems depend critically upon the substituents. Isoindole exists in the o-quinoid form 6. Computational results for the parent systems are given in Table III (99UP1). The results indicate that within the B3LYP functional only large basis sets provide reliable energy differences. 

The computational results (93JA2465) are consistent with the experimental findings of NMR spectroscopic studies (82JOC5132 97MRC35), which showed the presence of only the H tautomer of tetrazole in DMSO-dfi e = 49) solution. The content of H tautomer 27a in dioxane e = 22) at 30°C was estimated as 78% (82JST283) and 85% (75BSF1675) from its dipole moment 4.88 D and those of 1,5- and 2,5-dimethyltetrazoles as models for the H and 2H tautomers respectively. 

Computational results are reported for the isomerization of 1,4,5-trimethyl-imidazole (99MI233). They show that the isomerization occurs through the Dewar isomer arising from the excited singlet state. The formation of the triplet state is energetically favored however, the biradical intermediate cannot be produced because it has higher energy than the excited triplet state. 

Fio. 28. Phase diagram from computer results, (a) No two-liquid region, (b) Two-liquid region, (c) No vapor region. F stands for fluid. 

P(h) is just the barrier term referred to in Sect. 2, and has the form shown in Fig. 2.4. Also, S(h) takes the place of the driving force with the resultant growth rate the product of the two factors. Hence, a theoretical understanding of the factors leading to the computer results may be obtained. 

Data given in Tables 1-6 clearly show a significant dependence of P2 and p4 on amine concentration, that is, at least one of the apparent rate constants kj contains a concentration factor. Thus, according to the mathematical considerations outlined in the Analysis of Data Paragraph, both p2, P4 exponents and the derived variables -(P2 + p)4> P2 P4 ind Z (see Eqns. 8-12) are the combinations of the apparent rate constants (kj). To characterize these dependences, derived variables -(p2+p)4, P2 P4 and Z (Eqns. 8,11 and 12) were correlated with the amine concentration using a non-linear regression program to find the best fit. Computation resulted in a linear dependence for -(p2 + p)4 and Z, that is... 

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