Coulombic interactions energy models

We therefore learn that the short-range forces simultaneously affect both the pure coulombic interaction energy and the internal energies of the ions themselves, and reach the important conclusion that the nonideality of the solution is not sufficiently defined just by the presence of interactions (unless we also include the interactions of the electrons with the nuclei). That the polarizations should play a part in determining the distribution of ions is a fact which could not be deduced from the original Debye model. 

The band model employed in the last section did not take account of the Coulomb interaction energy between conduction electrons V(r) where r denotes the distance between two electrons. The long-range Coulomb interaction like... 

When the long-range Coulomb interaction is small, one can take account of only the on-site Coulomb interaction energy U between two electrons on a molecule. This electronic system is described in terms of the Hubbard model. Theoretical studies have shown that for UtW > 1 the electrons undergo the Mott transition which does not necessarily involve any structural changes. The electrons are localized with equal distance. They are apparently the same as the Wigner crystal described above. It is shown that the Mott transition is easy to occur when the charge density is l/molecule or I/site. 

We have developed a model to study the basic structural properties of solid C<,q. The model consists of two distinct types of intermolecular interactions. The dominant one is the van der Waals-type interactions between carbon atoms on different Cm molecules. A secondary short-range Coulomb interaction is modeled by a small charge transfer between the two types of bonds in the C60 molecule. In contrast to early calculations [6] which include the van der Waals interactions only, our model predicts correctly the observed cubic ground-state structure Pa3. Many structural properties calculated, such as the compressibility, cohesive energy, and specific heat, are in good agreement with experiments l7l. 

As long as we use the spherical ion model, the Coulomb interaction energy will obviously be at a minimum, and the molecule most stable, if the distance between the anions is as large as possible, i.e. if the molecule is linear. Would this stiU be so if we allow the ions to become polarized ... 

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

Imagine a model hydrogen molecule with non-interacting electrons, such that their Coulomb repulsion is zero. Each electron in our model still has kinetic energy and is still attracted to both nuclei, but the electron motions are completely independent of each other because the electron-electron interaction term is zero. We would, therefore, expect that the electronic wavefunction for the pair of electrons would be a product of the wavefunctions for two independent electrons in H2+ (Figure 4.1), which I will write X(rO and F(r2). Thus X(ri) and T(r2) are molecular orbitals which describe independently the two electrons in our non-interacting electron model. 

---