Short-ranged potential

Flere u. j(r,T,P) is the short-range potential for ions, and e is the dielectric constant of the solvent. The solvent averaged potentials are thus actually free energies that are fimctions of temperature and pressure. The... 

This expression contains the contribution of the short-range potential included earlier in, so that the excess free energy, to this level of approximation, is... 

Figure 12.2. A schematic inlerionic short-range potential function, after Stoneham el al. (1996).

Till this point an electric coupling with the wall is assumed. More generally, we may assume the existence of a short-range potential located at the wall. The following form will be used ... 

This functional form is continuous and there are no large forces at long range, but there is damage done to the short range potential which should probably be reparameterized to compensate for this. 

A special case of pairwise additivity (for sufficiently short-range potentials) is the limit of pair ( bond ) transferability or constancy from one environment to another. As mentioned above, such transferability is found to hold rather well for covalent bonds, but the corresponding transferability of H-bonds cannot be assumed without further justification. Such H-bond-additivity or -transferability assumptions often... 

Another short-range force that occurs in a solid is the weak van der Waals attraction between electron orbitals. There are a number of expressions for the short-range potential that takes both these factors into account. One commonly used expression is the Buckingham potential ... 

The first two terms describe each subsystem dressed with the interaction of each other. It is now apparent that exchange forces between the two subsystems have to be included in order to get the total force acting on the nuclei. This latter force is usually mimicked with a repulsive short range potential. A pseudo potential method can also be used in a microscopic approach to the surrounding medium effects [104,105],... 

Assume for simplicity that there is just one ionic species in the system with big ions, whose finite radius r0 should be taken into consideration. Denote their concentration by C. Assume further that besides the electric force —zVip there is another short-range potential force — Vip acting on the ion that prevents its approaching another ion closer than 2ro- To account for this assume for the potential ip a steep smooth monotonic dependence on C of the sort... 

Sometimes in the literature, a resonance refers to the passage of the phase shift across 7t/2, but that does not necessarily mean steep slope of 5(E), a large time delay, or formation of a QBS. On the other hand, a large time delay is not necessarily associated with the formation of a QBS. For example, Wigner s threshold law for a short-range potential states that the s-wave phase shift 50(E) is proportional to E1/2 at low energies E [33]. This leads to the time delay At a E 1/2, which becomes extremely large at very low E (and is positive if the interaction is attractive), in spite of its irrelevance to any QBS [34], Note... 

In the beginning, let us examine for simplicity the one electron center-donor. The interaction of the donor with tunneling electron can be described by the sum of the short-range potential t/ (r) and Coulomb attractive potential Vo(r) — — jr where r is the distance from the donor center (here and below in this chapter it is used the atomic units). It means that far from the donor, the donor-positive ion interacts with the electron, so the donor is neutral in the initial state. The total donor potential is... 

Let the electron-acceptor interaction be described by the short-range potential UsA(f—fA) where rA is the center of acceptor coordinate. It is supposed that the wave function of the electron bound state on the acceptor with energy E, TG(F — ta, E), is an exact one, i.e. it is considered not only UA(f—rA) but the donor-electron interaction also. Then, the exact value of the matrix element of the electron donor-acceptor transfer is equal [1] ... 

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