More Advanced Calculations

Why stop with a single hydrogenic Is orbital on either centre A little thought shows that the 2s and 2po- orbitals might also make a contribution to the bonding, and so we could write 

The best orbitals for this simple system are those which exploit its symmetry. James and Coolidge (1935) found that a good approximation to the lOg orbital was 

LCAO with best exponent of 1.238 Finkelstein and Horowitz (1928) 2.354 106.8 

LCAO with optimized exponents. Is and 2pff Dickinson (1933) 2.716 106.1 

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A more advanced calculation shows that the potential (defined by the data points) has a minimum at 127.34pm with a corresponding energy of t460.244 0222 h-... 

The question remains whether the nodal planes, essential for the qualitative analysis, remain in the more advanced calculations of wavefunctions. To test this point, the electronic localization function (ELF) as implemented by B. Silvi and A. Savin [24] is used. In Figure 3 we summarize the results. 

Simple and more advanced calculations of lattice energies many applications. 

We shall compare the potential curves obtained with the two different methods. Second order perturbation theory (CASPT2) has been used to estimate the remaining correlation effects in the FCI calculation with the smaller number of orbitals. This approach will be described in detail below. The spectroscopic constants are presented in Table 5-2. As can be seen, the two results are almost identical. The results are obviously far from experiment because of the small basis set used but that is not relevant to the present discussion. With the smaller number of orbitals we can now perform much more advanced calculations using larger basis sets and approach the experimental values. As an illustration, such a result is also given in the table. 

Another standing topic during the last two decades has been to evaluate the electronic structure of solids, surfaces and adsorbates on surfaces. This can be done using standard band structure methods [107] or in more recent years slab codes for studies of surfaces. An alternative and very popular approach has been to model the infinite solid or surface with a finite cluster, where the choice of the form and size of the cluster has been determined by the local geometry. These clusters have in more advanced calculations been embedded in some type of external potential as discussed above. It should be noted that these types of cluster have in general quite different geometries compared with... 

Our ability to understand the structure and properties of water in all its forms has been dramatically enhanced by the use of computer simulation. Early studies of the liquid used simple representations of the potential surface. These were often three or four point-charge distributions, adjusted to fit dipole and quadrupole moments, embedded in a simple spherical nonelectrostatic interaction. The simulations used classical Monte Carlo (MC) or molecular dynamic (MD) calculations, and the water molecules were assumed to be rigid. Recently, more advanced calculations have been based on quantum simulations, the introduction of intramolecular degrees of freedom, and accurate potential surfaces. As one side benefit... 

The activation energy of the insertion of coordinated ethylene estimated by the ab initio method was found to be 15 kcal/mol Despite the application of a more advanced calculation technique these results are less compatible with the experimental data on solid titanium chloride-based catalysts, when the activation energy of the propagation step is 3-6 kcal/mol (Table 10). Probably, this incompatability is due to the model used in ref. which describes the AC as a bimetallic complex CljTiCHj with A1(CH3)3. However, it is important to note that the calculations performed by means of the nonempirical method confirm the concept implying that in the active center the alkyl group occupies an intermediate position between the octahedral sites and that in olefin coordination the AC structure is reconstructed. 

The extent of sulphur s d orbital contribution to the jr system in addition to that of one p lone electron pair was greatly overemphasized in the early HMO treatment 23> of thiophene 6. But more advanced calculations show that... 

The result of a Flartree-Fock calculation is a set of K molecular orbitals, where K is the number of basis functions in the calculation. The N electrons are then fed into these orbitals in accordance with the Aufbau principle, two electrons per orbital, starting with the lowest-energy orbitals. The remaining orbitals do not contain any electrons these are known as the virtual orbitals. Alternative electronic configurations can be generated by exciting electrons from the occupied orbitals to the virtual orbitals these excited configurations are used in more advanced calculations that will be discussed in Chapter 3. 

In the next chapter results of accurate quantum-chemical calculations on transition metal surfaces will be presented. Here we will develop a method that enables an estimate of elementary quantum-chemical features, without the need for extensive quantum-chemical calculations. The result is of limited and only qualitative value and we will verify it in the next chapter with the aid of more sophisticated procedures. The method is of use to us because it permits the development and illustration of concepts that will facilitate our analysis of more advanced calculations. In this section we will analyze the electronic structure of a face-centered cubic (f.cx.) transition metal. We will use the method to discuss the valence electron structure of the (lll)i (100) and (110) surfaces of f.c.c. crystals. Specifically we are interested to know whether subbands can be identified, whose bandwidths can be estimated by means of the Bethe lattice approximation and how their respective local densities of states behave as a function of valence electron band filling. 

Extension material to support this unit is available on our website. More advanced calculations involving equilibrium constants, and an introduction to free energy changes (AG) are to be found in Appendix 15 on the website. 

Of course, we cannot expect the description to reflect all the details of the charge distribution in the butadiene molecule, but one may expect this approach to be able to reflect at least some rough features of the yr electron distribution. If the results of more advanced calculations contradicted the rough particle-in-box results, then we should take a closer look at them and... 

The cycloheptatrienyl carbocation (20) is also a six n electron system, and all six electrons can go into bonding orbitals. Its delocalization energy is calculated to be 2.99)8 (Figure 4.26), which is close to the value of 50 kcal/mol of delocalization stabilization calculated by other methods. Therefore, the cycloheptatrienyl cation is an especially stable carbocation, although it is still a cation and is certainly not as stable as benzene. On the other hand, the cycloheptatrienyl anion is a 4n Ji system and thus is predicted to be antiaromatic by HMO theory. More advanced calculations suggest that any energy consequences of electron delocalization in the anion must be very small. 

Another useful calculation is the remaining operating time until a battery is empty. A simple estimate can be calculated using the remaining capacity of the battery (derived from SOC and last full capacity) divided by the present discharge current. More advanced calculations can use an average or predictive discharge current or compensate for temperature, just to list a few examples. The typical calculated data values commonly used in laptop computers was standardized in the Smart Battery Data Specification, Revision 1.1, published in 1998 [12]. 

Theory, which explicitly includes chromophore-chromophore electrostatic interactions, certainly makes a number of predictions that can be readily verified. The predicted maxima in graphs of the electro-optic coefficient versus number density indicate that there is an ideal size for spherically shaped chromophores characterized by specific dipole moments, polarizabilities, and ionization potentials. More advanced calculations suggest that prolate ellipsoidal shapes are the least favorable for optimizing optical nonlinearily. Both spherical and oblate ellipsoidal shapes offer advantages. Finally, calculations suggest that to understand fully the observed poling behavior, including temporal behavior, molecular dynamics must be taken into account. 

Thermal combustion reactions are very fast. The sub-stoichiometric combustion of methane is a complex process with many radical reactions [466]. The reaction pattern depends on the residence time/temperature distribution. Hence, it is important to couple the kinetic models with CFD simulations by post processing [466] or by direct coupling in more advanced calculations. 

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