Screened hydrogenic model calculations

Here, emphasis is given to the application of few-state models in the description of the near-resonant vacancy exchange between inner shells. It is well known that the quantities relevant for inner-shell electrons may readily be scaled. Therefore, the attempt is made to apply as much as possible analytic functional forms to describe the characteristic quantities of the collision system. In particular, analytic model matrix elements derived from calculations with screened hydrogenic wave functions are applied. Hydrogenic wave functions are suitable for inner shells, since the electrons feel primarily the nuclear Coulomb field of the collision particles. Input for the analytic expressions is the standard information about atomic ionization potentials available in tabulated form. This procedure avoids a fresh numerical calculation for each new collision system. 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

Fig. 2.6 Comparison of the calculated structures for glycine in the gas-phase and in water (COSMO solvation model). Note that the central bond angle in the zwitterionic form 1 is distorted by the hydrogen bond length of 1.96A computed for this structure in the gas phase. When solvation effects are included in the calculation using COSMO, the electrostatic interaction is reduced in magnitude due to charge screening by water, and the bond angle distortion is no longer present.

Fig. 7.17 Plot of the calculated dielectric constant in silicon crystallites of different size. The broken curve corresponds to calculations based on the Penn model [Tsl], the dotted line corresponds to pseudopotential calculations [Wa5], while the full line is based on self-consistent linear screening calculation of hydrogenic impurities [AI4]. Redrawn from [AI4].

We have now collected almost all the pieces required for a first version of COSMO-RS, which starts from the QM/COSMO calculations for the components and ends with thermodynamic properties in the fluid phase. Although some refinements and generalizations to the theory will be added later, it is worthwhile to consider such a basic version of COSMO-RS because it is simpler to describe and to understand than the more elaborate complete version covered in chapter 7. In this model we make an assumption that all relevant interactions of the perfectly screened COSMO molecules can be expressed as local contact energies, and quantified by the local COSMO polarization charge densities a and a of the contacting surfaces. These have electrostatic misfit and hydrogen bond contributions as described in Eqs. (4.31) and (4.32) by a function for the surface-interaction energy density... 

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