Coulomb damping

Stiffness and energy dissipative characteristics of the soil as often represented by secant shear (or Young s) modulus and viscous or Coulomb damping. 

Eq. (56) indicates that an ellipse will be obtained if x is plotted against y. In other words, in viscous damping an elliptic hysteresis loop will be obtained. The energy dissipated per volume per cycle due to Coulomb damping in an oscillatory motion is given ... 

Functional fonns based on the above ideas are used in the FIFD [127] and Tang-Toeimies models [129], where the repulsion tenn is obtained by fitting to Flartree-Fock calculations, and in the XC model [92] where the repulsion is modelled by an ab initio Coulomb tenn and a semi-empirical exchange-repulsion tenn Cunent versions of all these models employ an individually damped dispersion series for the attractive... 

While nonbonded atom pairs will typically not come within 1A of each other, it is possible for covalently bound pairs, either directly bounds, as in 1-2 pairs, or at the vertices of an angle, as in 1-3 pairs. Accordingly it may be considered desirable to omit the 1-2 and 1-3 dipole-dipole interactions as is commonly performed on additive force fields for the Coulombic and van der Waals terms. However, it has been shown that inclusion of the 1-2 and 1-3 dipole-dipole interactions is required to achieve anistropic molecular polarizabilites when using isotropic atomic polariz-abilites [50], For example, in a Drude model of benzene in which isotropic polarization was included on the carbons only inclusion of the 1-2 and 1-3 dipole-dipole interactions along with the appropriate damping of those interactions allowed for reproduction of the anisotropic molecular polarizability of the molecule [64], Thus, it may be considered desirable to include these short range interactions in a polarizable force field. 

Thus, the ionic coulomb potential is damped exponentially within a Thomas-Fermi screening length = 1 /ktf. It follows from eqs (2.41) and (6.10) that... 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

The approximate potentials for the description of ionic systems consist of three terms, a screened Coulomb term, exp(-ar)/r, a repulsive term, (rn, n = 12, usually), and an attractive dispersion term (r6). If the damping factor in the Coulomb-term is not present, i.e., a = 0, the energy function is evaluated using the summation method suggested by deLeeuw65. ... 

AB2 Systems. Since layered structures are important variants in AB2 systems (e.g. the Cdl2 and CdCb families of structures), it was deemed necessary to always use a Coulomb potential without a damping term. The method chosen to perform the full Coulomb sum was the one suggested by de Leeuw66. Again, the resulting structure candidates in general did not depend on the exact values of the effective potential. The most common structure types found are summarized in table 3. We note that these include many commonly found structures like anatas,... 

In the EH-CSD approach it is not convenient to decouple electrostatic terms into rigid Coulombic and polarization contributions the effective Hamiltonian leads to compute these two terms together. Exchange repulsive terms are hardly computed when the second partner of the interaction is a liquid they may be obtained with delicate simulation procedures, and it is convenient to decouple them into two contributions, namely the work spent to form a cavity of a suitable shape and an additional repulsion contribution. Dispersion contributions may be kept we shall examine this term in more detail later. Charge-transfer contributions are damped in liquids their inclusion could introduce additional problems in the definition of Vjnt via continuous solvent distributions. It is advisable to neglect them, as it is done in the interaction potentials used in simulations with the present approach it is possible to describe the charge transfer effect by enlarging the solute M —> M-Sn. 

Note that the main dissimilarities seen in Fig. 5 are related with the Independence of dispersions and damping coefficients of optic-like mode at small k domain. For example, for LiF the characteristic frequency of these excitations decreases when k increases. An inverse situation is observed for KrAr. One may suppose that such a dissimilarity is caused for finite k mainly by an antiferromagnetic type of interactions in a mixture of charged particles, whereas the long-range characte of Coulombic potential becomes a crucial fac-... 

Figure 3.17 Time evolution of the ISe ISh exciton population as a function of the relaxation times and Coulomb coupling for the single and biexcitons. The rise time of the biexciton (formation time) and the presence of strong or damped quantum beating depends upon the relative values of the Wc, yi, and j2 as shown above. Source Shabaev et al. (2006).

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