The hydrogenic functions

A natural starting point for the discussion of radial basis functions is the one-electron hydrogenic system, for which the wave function can be written in a closed analytical form. Indeed, we would naively expect the simple hydrogenic eigenstates - the exact solutions for a one-electron atom - to represent an ideal set of functions in terms of which we may expand the more complicated orbitals of many-electron atoms. Somewhat surprisingly, perhaps, the hydrogenic orbitals have certain deficiencies when used as basis functions for many-electron atoms. Nevertheless, from a consideration of the hydrogenic wave functions, we should learn much about the desirable analytical properties of radial atomic basis functions. 

In atomic units, the Hamiltonian of a hydrogenic system with nuclear charge Z is given by 

The eigenstates of this operator consists of a set of bounded states and set of continuum states. In polar coordinates, the normalized hounded eigenfunctions may be written in the product form [9] 

2Zr/n and an exponential in —Zrin. The energy of the bounded hydrogenic state is given by 

The degeneracy of the hydrogenic states of different angular momenta is peculiar to the Coulomb potential and is lilted in a many-electron system. The degeneracy of the one-electron isotropic HO system is also different from that of the hydrogenic system see Section 6.6.1. 

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Certain catalyst manufacturers claims to have optimized the preparation (CoMo catalysts), the formulation or the promotion (aromatic saturation) of their catalysts to achieve an appropriate balance of the hydrogenation function to desulfurize the sterically hindered compounds and yield the 15 ppm S fuel. However, the actual trend is to use NiMo catalyst for the treatment of the more refractory compounds, below 200 ppm S [22],... 

The acid component of a hydrocracking catalyst can be an amorphous oxide, e.g., a silica-alumina ora zeolite, eg., USY. This component usually serves as a support for the metal compound responsible for the hydrogenation function. The metal compound can be a noble metal, e.g., Pt or Pd, or a mixture of sulfides, e.g., of Ni/Mo, NiAV, or Co/Mo. The relative amounts of the respective compounds have to be thoroughly balanced to achieve an optimum performance. 

Under the operating conditions, the reaction intermediates (w-hexenes and i-hexenes in n-hexane isomerization) are thermodynamically very adverse, hence appear only as traces in the products. These intermediates (which are generally olefinic) are highly reactive in acid catalysis, which explains that the rates of bifunctional catalysis transformations are relatively high. The activity, stability, and selectivity of bifunctional zeolite catalysts depend mainly on three parameters the zeolite pore structure, the balance between hydrogenating and acid functions, and their intimacy. In most of the commercial processes, the balance is in favor of the hydrogenation function, that is, the transformations are limited by the acid function. 

Unfortunately, the exponential radial dependence of the hydrogenic functions makes the evaluation of the necessary integrals exceedingly difficult and time consuming for general computation, and so another set of functions is now universally adopted. These are Cartesian Gaussian functions centered on nuclei. Thus, gj( 1) is a function centered on atom I ... 

In a hydrodesulfurization system, the hydrogenation function adds hydrogen to the tarry deposits. This reduces the concentration of coke precursors on the surface. There is, however, a slow accumulation of coke that reduces activity over a 1-2 year period. Refiners respond to this slow reduction in activity by raising the average temperature of the catalyst bed to maintain conversions. Eventually, however, an upper limit to the allowable temperature is reached and the catalyst must be removed and regenerated. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

The protodegermylation of adduct 25 to form 26 occurs under thermolysis. In the case of 27, hydrogermylation of the thiophene double bonds by action of excess of trichlorogermane occurs. The hydrogenative function of trichlorogermane was also marked in the reaction of alkoxybenzenes as shown below in Section VIII. 

A systematic study of the history of the hydrogenation function ip a series of fresh and deactivated commercial hydrocracking catalysts is reported using cyclohexene hydrogenation as a probe reaction. 

The variational function 0 can be provided with more parameters so that the energy can also be minimized with respect to these parameters, whereby a continually improved agreement with the correct energy can be obtained. The introduction of an effective nuclear charge % into the hydrogen functions instead of the nuclear charge i forms one of these extensions. 

A key question related to the design of HC catalysts is the importance of the proximity, or as it is often termed, the intimacy of the hydrogenation function to the acidic sites. Studies based on model HC catalyst systems (25) led to the development of intimacy criteria that need to be satisfied for sufficient rates of diffusion of olefinic intermediates between acidic and hydrogenation functions to achieve equilibria. Similarly, intimacy rules will apply to aromatic intermediates in order to minimise coke formation. Literature data on this aspect are rather scarce, however. 

It is, of course, one thing to want to emplace the hydrogenation function in a controllable manner, and another to be able to do it in practice. Industrial zeolite-Y based catalysts contain a binder, usually alumina or an ASA/alumina mixture. In this case one can direct Pd to the alumina phase by employing PdCL as the metal... 

When the hydrogenation function is embedded in the crystal voids of an MFI topology, the formation of trans-isomers is strongly reduced. After partial reduction of soy bean oil with such catalyst from an iodine value of 140 to 80, virtually no trans-isomers are obtained (56). This is the result of pore mouth catalysis combined with zeolite shape selectivity. Due to the bent character of the cts-isomer chains in triglycerides, trans-configured chains preferentially enter the pore mouths for hydrogenation. In this environment, metal-catalyzed cis-trans isomerization is restricted for steric reasons as multiple readsorption is minimal. 

It should be noted that H-mordenite itself is also able to isomerize alkanes such as pentane. However, the presence of platinum greatly improves its selectivity and stability, as shown in Fig. 2.8. Apparently the hydrogenation function prevents oligomerization etc. leading to coke formation. 

The isomerization of n-hexane at 250 °C, 26 bar. a H2/hydrocar-bon (HC) molar ratio of 22.4, and a WHSV of 3.3 g.g". h over fresh Pd-NiSMM led to a conversion of 56 %, almost without cracking. This isomerization activity was totally and irreversibly destroyed after injection of about 10 molecules pyridine per g Pd-NiSMM. Benzene hydrogenation over the poisoned catalyst (260 °C, 26 bar, H2/HC = 25) showed that the hydrogenation function of the catalyst was still active enough to hydrogenate benzene totally to cyclohexane, indicating that the metallic sites had only been partially poisoned, it at all. 

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