Coulomb-type interactions

The idea behind this survey of nonbonded interactions is to get rid of them elegantly as explicit terms requiring separate calculations by means of Eq. (10.3). We shall examine to what extent nonbonded Coulomb-type interactions are at least approximately additive. The formulation of additivity is presented here for C H2 +2-2m hydrocarbons [208], where m is the number of six-membered cycles. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

Fig. 4.10. Electron momentum distributions for neon ( 75oi = 0.79 a.u. and /. 02 = 1.51 a.u.) subject to a linearly polarized monochromatic field with frequency ui = 0.057 a.u. and intensity I = 3.0 x 1014W/cm2, as functions of the electron momentum components parallel to the laser-field polarization. The left and the right panels correspond to the classical and to the quantum-mechanical model, respectively. The upper and lower panels have been computed for a contact and Coulomb-type interaction Vi2, respectively. In panels (a) and (d), and (h) and (e), the second electron is taken to be initially in a Is, and in a 2p state, respectively, whereas in panels (c) and (/) the spatial extension of the bound-state wave function has been neglected. The transverse momenta have been integrated over...

Table 2. The Coulomb-type interaction of the charge remaining on the ion at its optimum position with the charges in the cluster, when the ion is adsorbed at the hollow site (f/tot) and with the charges in the neutral Cun cluster (Co). The difference between the two quantities, U, is an indication of the interaction resulting from charge transfer from the ion to the cluster and the polarization of the cluster by the negative charge of the ion. All energies are given in kJmoff. ...

It should be mentioned that the well-known Coulomb-type interaction of a charged species above a metal surface with its image charge... 

Interactions of an ion M with ions N" at a distance beyond the borders of the WSC will decrease rapidly with increasing interatomic distance in covalent systems. This does not hold for ionic systems, since Coulomb-type interactions fall off slowly. If the ions N" are approximated as point charges with a net charge q, the Madelung potential due to these charges, which can be calculated by the Elwald summation technique as shown in [302], will modify the Fock matrix and the total energy in the following way... 

The main handicap of MD is the knowledge of the function [/( ). There are some systems where reliable approximations to the true (7( r, ) are available. This is, for example, the case of ionic oxides. (7( rJ) is in such a case made of coulombic (pairwise) interactions and short-range terms. A second example is a closed-shell molecular system. In this case the interaction potentials are separated into intraatomic and interatomic parts. A third type of physical system for which suitable approaches to [/( r, ) exist are the transition metals and their alloys. To this class of models belong the glue model and the embedded atom method. Systems where chemical bonds of molecules are broken or created are much more difficult to describe, since the only way to get a proper description of a reaction all the way between reactant and products would be to solve the quantum-mechanical problem at each step of the reaction. 

Abstract—A quantum mechanical treatment of the hydrogen bond by the method of valence structures is described. The interaction of the group A—K and the atom B is characterized by terms which include a donor-acceptor interaction between H and B, the decrease of repulsion between the non-bonded atoms as well as the Coulomb-type attraction. The treatment yields also the explanation of some spectroscopic phenomena of the hydrogen bond. 

In Fig. 4.8 the effect of the initial-state wave functions is explored, for the case where the crucial electron-electron interaction is the two-body Coulomb interaction (4.14a) and for the case where this interaction is the two-body contact interaction (4.14d), which is not restricted to the position of the ion. In both cases, the form factor includes the function (4.23), which favors momenta such that pi + p2 is large. This is clearly visible for the contact interaction (4.14d) and less so for the Coulomb interaction (4.14a) whose form factor also includes the factor (4.19), which favors pi = 0 (or p2 = 0)- We conclude that (i) the effect of the specific bound state of the second electron is marginal and (ii) that a pure two-body interaction, be it of Coulomb type as in (4.14a) or contact type as in (4.14d), yields a rather poor description of the data. A three-body effective interaction, which only acts if the second electron is positioned at the ion, provides superior results, notably the three-body contact interaction (4.14b), cf. the left-hand panel (d). This points to the significance of the interaction of the electrons with the ion, which so far has not been incorporated into the S-matrix theory beyond the very approximate description via effective three-body interactions such as (4.14b) or (4.14c). 

C. Hattig and B. A. Hess, Mol. Phys., 81, 813 (1994). Calculation of Orientation-Dependent Double-Tensor Moments for Coulomb-Type Intermolecular Interactions. 

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. 

It is known that two dominant inter-site interactions between the orbitals in solids are the superexchange-type and the cooperative Jahn-Teller type interactions. The former is attributed to the virtual exchange of an electron under the strong on-site Coulomb interaction [8-11], The explicit form of this interaction is given by... 

The ratio of Ni + to Ca in the aqueous phase is, in general, much less than unity. Thus, for the Ni + removal process to be selective (i.e., high R2Ni), the equilibrium constant, Ki, needs to be extremely high. Conventional cation exchangers with coulombic (electrostatic) type interaction are unable to attain such high selectivity (charge of calcium and nickel ion is the same). 

The exponent value of 0.6 in Jonscher regime is considered to arise by the ion-ion interactions, usually of the coulombic type. During the process of the hopping of the ions, even separate hopping events may have a broad distribution of relaxation times, and this effect can manifest as stretching of the relaxation times. Ngai s coupling model accounts for stretched exponential relaxation and considers it as a consequence of... 

We represent the cluster as composed of rigid molecules interacting through a sum of pairwise atom-atom potentials of Lennard-Jones and Coulomb types [37] ... 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

For molecules the evaluation of the Breit correction to the Coulomb-type electron-electron interaction operator becomes computationally highly demanding and cannot be routinely evaluated, not even on the Dirac-Fock level. To test the significance of the Breit interaction, the Gaunt term is evaluated as a first-order perturbation. It turned out that it can be neglected in most cases as can be seen from the DF 4- Bmag calculations cited in Table 2.1. 

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