Interaction between electrons, effective

The interactions between electrons are inherently many-body forces. There are several methods in common use today which try to incorporate some, or all, of the many-body quantum mechanical effects. An important term is that of electronic exchange [57, 58]. Mathematically, when two particles in the many-body wavefunction are exchanged the wavefunction changes sign ... 

With the addition of a pseudopotential interaction between electrons and metal ions, the density-functional approach has been used82 to calculate the effect of the solvent of the electrolyte phase on the potential difference across the surface of a liquid metal. The solvent is modeled as a repulsive barrier or as a region of dielectric constant greater than unity or both. Assuming no specific adsorption, the metal is supposed to be in contact with a monolayer of water, modeled as a region of 3-A thickness (diameter of a water molecule) in which the dielectric constant is 6 (high-frequency value, appropriate for nonorientable dipoles). Beyond this monolayer, the dielectric constant is assumed to take on the bulk liquid value of 78, although the calculations showed that the dielectric constant outside of the monolayer had only a small effect on the electronic profile. 

Metal-insulator transitions in both crystalline and non-crystalline materials are often associated with the existence of magnetic moments. Moments on atoms in a solid are of course an effect of correlation, that is of interaction between electrons, and their full discussion is deferred until Chapter 3. But even within the approximation of non-interacting electrons in crystalline solids, metal-insulator transitions can occur. These will now be discussed. 

The electrons in a solid interact both with one another and with the lattice vibrations. A theme of this book is the effect of the interaction between electrons in inducing magnetic moments and metal-insulator transitions. Interaction with phonons also has an important effect, particularly in some transitional-metal oxides. In this chapter both kinds of interaction are introduced. 

We turn now to the interaction energy e2/r12 between electrons and consider first its effect on the Fermi surface. The theory outlined until this point has been based on the Hartree-Fock approximation in which each electron moves in the average field of all the other electrons. A striking feature of this theory is that all states are full up to a limiting value of the energy denoted by F and called the Fermi energy. This is true for non-crystalline as well as for crystalline solids for the latter, in addition, occupied states in fc-space are separated from unoccupied states by the "Fermi surface . Both of these features of the simple model, in which the interaction between electrons is neglected, are exact properties of the many-electron wave function the Fermi surface is a real physical quantity, which can be determined experimentally in several ways. 

As already discussed, there is an important case where resolution is determined by fundamental limitations of the electron optical system and not by electron scattering. This occurs with the high current shaped electron beams used in high throughput direct-write tools. The Coulomb interaction between electrons in these columns displaces the electrons from their intended trajectories and blurs the edges of the spot. As discussed above in connection to throughput, this effect, which is related to the Boersch effect (45) forces a compromise between throughput and resolution. 

To account for the interchannel coupling, or, which is the same, electron correlation in calculations of photoionization parameters, various many-body theories exist. In this paper, following Refs. [20,29,30,33], the focus is on results obtained in the framework of both the nonrelativistic random phase approximation with exchange (RPAE) [55] and its relativistic analogy the relativistic random phase approximation (RRPA) [56]. RPAE makes use of a nonrelativistic HF approximation as the zero-order approximation. RRPA is based upon the relativistic Dirac HF approximation as the zero-order basis, so that relativistic effects are included not as perturbations but explicitly. Both RPAE and RRPA implicitly sum up certain electron-electron perturbations, including the interelectron interaction between electrons from... 

Localized electrons in a partially filled shell carry a net spin (provided Hund s rule is obeyed), and interactions between these localized electrons and the collective electrons gives rise to a large effective field Hex acting on the collective electrons. Since the intraatomic exchange correlations minimize the electrostatic interactions between electrons of parallel spin, Hcx is directed parallel to the atomic moment due to localized electrons. Below a magnetic-ordering temperature, this internal field induces a contribution to the atomic moment from the collective electrons whether the localized electrons are ordered parallel or antiparallel. From equation 58, this contribution is... 

Finally, one should not overlook the possible role of correlation effects in atom-metal differences. In atoms the dominant contributions to the correlation energy arise from interaction between electrons of the same principal quantum number n, since these have the greatest overlap. As far as purely intraatomic electrons are concerned, the correlation terms differ very little between atom and metal correlation does, of course, affect interatomic screening of the final state vacancy in metals. 

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