Coulomb potential, modified

In Eq. (154), we assume indeed that only the ions (Z 0) interact with each other and that the resulting interaction is simply the Coulomb potential modified by the zero-frequency dielectric constant e of the solvent. Of course, in an exact theory, we would have to take explicitly into account the interactions with the solvent, and the dielectric constant itself should come out of the calculation. The proper way of attacking this problem is based on the theory of the potential of average forces and is carefully analyzed in H. L. Friedman s monograph.11 However, the explicit calculations always become exceedingly complicated and, in one way or another, one always has to have recourse to an approximation of the type (154). It amounts to assuming ... 

If the potential V(r) is a pure Coulomb potential the asymptotic partial wave is given by the regular Coulomb function (4.64), apart from a constant phase factor. We strictly have no incident plane wave since the Coulomb potential modifies the wave function everywhere. We make the normalisation of the Coulomb distorted wave t/j,j(k,r) analogous to that of (4.83) by choosing the phase factor to be the Coulomb phase shift [Pg.95]

Besides its temperature dependence, hopping transport is also characterized by an electric field-dependent mobility. This dependence becomes measurable at high field (namely, for a field in excess of ca. 10d V/cm). Such a behavior was first reported in 1970 in polyvinylcarbazole (PVK) [48. The phenomenon was explained through a Poole-ITenkel mechanism [49], in which the Coulomb potential near a charged localized level is modified by the applied field in such a way that the tunnel transfer rale between sites increases. The general dependence of the mobility is then given by Eq. (14.69)... 

We discuss briefly the factors that determine the intensity of the scattered ions. During collision, a low energy ion does not penetrate the target atom as deeply as in RBS. As a consequence, the ion feels the attenuated repulsion by the positive nucleus of the target atom, because the electrons screen it. In fact, in a head-on collision with Cu, a He+ ion would need to have about 100 keV energy to penetrate within the inner electron shell (the K or Is shell). An approximately correct potential for the interaction is the following modified Coulomb potential [lj ... 

Because the diffusive flux is enhanced by this drift of a charge under the influence of the coulomb potential [as represented in eqn. (142)], the partially reflecting boundary condition (127) has to be modified to balance the rate of reaction of encounter pairs with the rate of formation of encounter pairs [eqn. (46)]. However, the rate of reaction of ion-pairs at encounter is usually extremely fast and the Smoluchowski condition, eqn. (5), is adequate. The initial and outer boundary conditions are the same as before [eqns. (131) and (128), respectively], representing on ion-pair absent until it is formed at time t0 and a negligibly small probability of finding the ion-pair with a separation r - ... 

VeQ is the electron-nuclear Coulomb potential at ligand Q, while / and K are modified Coulomb and exchange operators that act on spin-orbitals (x) as follows (c.f. 4-8),... 

This series arises naturally, when expressing the Coulomb potential of a charge separated by a distance s from the origin in terms of spherical coordinates. The positive powers result when r < s, while for r > s the potential is described by the negative powers. Similarly the solutions of the linearized Poisson-Boltzmann equation are generated by the analogous expansion of the shielded Coulomb potential exp[fix]/r of a non-centered point charge. Now the expansion for r > s involves the modified spherical Bessel-functions fo (x), while lor r < s the functions are the same as for the unshielded Coulomb potential,... 

In this case one may use the symmetry of the Coulomb potential and apply the comparison theorem for the ground state wavefunction ir r) and the "reflected" function external potential t/i(r) = U(err). As comparison theorem one finds ir r) > Hellmann-Feynman force is oriented into E+, that is, its n-projection is positive. Practically the same discussion was used in [34] to prove the monotone nature of the adiabatic potential for the ground state of the one-electron diatomic molecule (in the absence of the internuclear repulsion term). This statement is also easily modified for the Dirichlet boundary value problem for some region 2. One may formally require the external potential U to be infinite out of 2 (for the analysis of this statement see [35]). 

The specific structure of the states for Hp was described in detail in [79], where it is mentioned as a well-known physical effect. For example, it was noted in the theory of disordered semiconductors that a similar "ladder" structure of states is realized for the system where the Coulomb potential is modified within a sphere as a constant potential (see [86,87] for a qualitative discussion and analytical solution of the problem). For quantum chemistry, the situation is interesting, as was shown in a series of publications of Connerade, Dolmatov and others (see e.g. [19,88-91] note that the series of publications on confined many-electron systems by these authors is much wider). The picture described is realized to some extent for the effective potential of inner electrons in multi-electron atoms, as it is defined by orbital densities with a number of maximal points. The existence of a number of extrema generates a system of the type described above [89]. This situation was modeled and described for the one-electron atom in [88] it is similar to that one described in Sections 5.2 and 5.3. 

The other principal radiative correction is the vacuum polarization (Fig. 3 b)). It describes the interaction of a fermion with virtual electron-positron pairs which can be thought present in the vacuum for short times without violating the energy-time uncertainty relation. If external fields are present, these virtual pairs are influenced and act like a polarizable medium. Therefore the Coulomb interaction of the nucleus with the electrons is modified which leads to an energy shift compared to the pure Coulomb potential energy eigenvalue. 

Moreover, the evaluation of the Coulomb potential, for the full range of inter-peak distances, might become protracted, because of the many pairs to be considered. It is thus necessary to limit the extent of the calculation, particularly for macromolecules. We modified the Coulomb expression with a force-switching function dedicated to the Coulomb interaction, as established by Field [73] ... 

We now calculate the ground-state energy of an electron gas interacting with the lattice charges through the modified Coulomb potential (131). The calculations follow the same steps as for the case of point charges. We shall therefore merely state the points at which differences must be introduced. This, of course, only happens in the contributions where the lattice plays some role, i.e., the Madelung term and the polarization term. 

Fig. 10. Plot of e versus rf for the modified Coulomb potential. Experimental data for lithium, sodium, and potassium.

One thing that is clear, however, is that the crudest model of positive point charges is definitely inadequate for actual metals. The modified Coulomb potential, however, gives some hope of being able to represent the real situation roughly thus, a given metal in this picture should be characterized by two parameters ... 

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