Coulomb Interaction / Potential

In summary, the movement of a high-energy electron in a solid may be described by a set of three Equations (1), (4) and (6). From these equations we may conclude that for high-energy electron diffraction the problem of multiple elastic and inelastic scattering by a solid is entirely determined by two functions, i.e. (1) the Coulomb interaction potential averaged over the motion of the crystal particles (V(r)> and (2) the mixed dynamic form factor S(r, r, E) of inelastic excitations of the solid. 

The structures of ionic solids may be accounted for quite accurately by the use of a coulombic interaction potential between neighbouring ion pairs together with a suitable ion-core repulsion. 

The Boltzmann equation works reasonably well when electrons collide mainly with neutral species. Electron-electron or electron-ion collisions involve coulombic interactions that have a longer range than that of electron-neutral species interactions. Coulombic-interaction potentials vary inversely with separation, but electron-neutral species interaction potentials vary inversely with the fifth or sixth power of separation. 

For charged sites, it is the asymptotics of the site-site Coulomb interaction potential, and the constraint (25) ensures the condition of local electroneutrality is obeyed. 

The key is that a single-center expansion of the transition density, implicit in a multipolar expansion of the Coulombic interaction potential, cannot capture the complicated spatial patterns of phased electron density that arise because molecules have shape. The reason is obvious if one considers that, according to the LCAO method, the basis set for calculating molecular wavefunctions is the set of atomic orbital basis functions localized at atomic centers a set of basis functions localized at one point in a molecule is unsatisfactory. 

The configuration coordinates of electrons (p) and nuclei (R) in the new frame are related to the laboratory one by rk = u + pk, Qk. = u + Rk, symbolically written as r =u+p, Q=u+R, and T=(p,R). Ke represents the electrons kinetic energy operators Vee (p), VeN(p, R) and Vnn(R) are the standard Coulomb interaction potentials they are invariant to origin translation. The vector u is just a vector in real space R3. Kn is the kinetic energy operator of the nuclei, and in this work the electronic Hamiltonian He(r Z) includes all Coulomb interactions. This Hamiltonian would represent a general electronic system submitted to arbitrary sources of external Coulomb potential. 

To summarize, we have presented a review on the renormalization group theory in the reduced many-body density matrix basis (DMRG method), and we have applied it to conjugated organic systems, with both short range and long range Coulomb interaction potentials. We... 

Here, r and R denote the electronic and nuclear coordinates, respectively, and pj and Pk are the operators of their conjugate momenta. K (r R) is the Coulombic interaction potential among electrons and nuclei. 

The values of the four interaction terms (surface contact, side chain inter-molecular contacts, relative solvation energy and Coulombic interaction potential) were computed for the 200 best matching geometries generated by the BoGlE module, for each of the 15 docking cases. 

Here r and R denote collectively the electronic and nuclear coordinates, respectively, Te is the kinetic-energy operator of the electrons, and V(r, R) represents the Coulombic interaction potential of all particles. The Born-Oppenheimer adiabatic electronic states depend parametrically on the nuclear coordinates R. The Vn(R) define the adiabatic electronic potential-energy surfaces. 

If the electron transfer in the A. B pair is very fast and irreversible, the reaction is limited by the frequency of diffiisional collisions, which depends on the viscosity ri, radius r (r = re), and the Coulomb interaction potential, which is determined by the sign and value of charges za and % (see Chapter 5)... 

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