Crystal overlap effects

In general, overlap of incompletely filled p orbitals results in large deviations from pure ionic bonding, and covalent interactions result. Incompletely filled / orbitals are usually well shielded from the crystal field and behave as essentially spherical orbitals. Incompletely filled d orbitals, on the other hand, have a large effect on the energetics of transition metal compounds and here the so-called crystal field effects become important. 

In Eq. (10), E nt s(u) and Es(in) are the s=x,y,z components of the internal electric field and the field in the dielectric, respectively, and p u is the Boltzmann density matrix for the set of initial states m. The parameter tmn is a measure of the line-width. While small molecules, N<pure solid show well-defined lattice-vibrational spectra, arising from intermolecular vibrations in the crystal, overlap among the vastly larger number of normal modes for large, polymeric systems, produces broad bands, even in the crystalline state. When the polymeric molecule experiences the molecular interactions operative in aqueous solution, a second feature further broadens the vibrational bands, since the line-width parameters, xmn, Eq. (10), reflect the increased molecular collisional effects in solution, as compared to those in the solid. These general considerations are borne out by experiment. The low-frequency Raman spectrum of the amino acid cystine (94) shows a line at 8.7 cm- -, in the crystalline solid, with a half-width of several cm-- -. In contrast, a careful study of the low frequency Raman spectra of lysozyme (92) shows a broad band (half-width 10 cm- -) at 25 cm- -,... 

As already discussed above, the application of a cathodic overpotential step to a crystal face growing at steady state conditions leads to the formation of new growth pyramids at the emergence points of screw dislocations. The slope of these new pyramids is determined by the new final overpotential, rn, and is steeper than that of the pyramids growing at the initial overpotential, rji, before application of the potential step. If the new pyramids grow at rjf independently, i.e., without interaction or overlapping effects, they cover a part, Sex(0. of the crystal face surface given by [5.83]... 

It is also worthwhile to note that it is also possible to establish a close relation between the crystal field effects, covalent effects (overlap between the wave functions of an impurity ion and ligands) and electron-phonon interaction and JT effects [52-54]. It was shown in these works that it is possible to distinguish and analyze separately different contributions (arising from the point charge and exchange interactions) to the vibronic effects. 

The approximate total wave function W of the crystal is constructed as a simple Hartree product of the constituent wave functions, i.e. W is not antisymmetric with respect to the electron exchange between different motifs. Notice that due to the lack of exchange forces and overlap repulsion this wave function is not applicable at short intermolecular distances. Nevertheless, if we impose a reasonable separation of the motifs (e.g. based on experimental crystal structures), these effects can be neglected as far as we are interested in the Madelung-effect on the charge distributions. There are some recent attempts to take into account the exchange and overlap effects by appropriate pseudopotentials [162-164]. 

A considerable contribution to ab initio calculations is due to Newman and coworkers. More than twenty years ago, they evaluated the contribution of a variety of mechanisms to the one-ligand one-particle crystal field, namely overlap and exchange effects (Ellis and Newman 1967), exchange charge effects (Bishton et al. 1967), charge penetration (Ellis and Newman 1968), and ligand-ligand overlap effects (Curtis and Newman 1969). 

---