Dipole interaction potential

Themiodynamic stability requires a repulsive core m the interatomic potential of atoms and molecules, which is a manifestation of the Pauli exclusion principle operating at short distances. This means that the Coulomb and dipole interaction potentials between charged and uncharged real atoms or molecules must be supplemented by a hard core or other repulsive interactions. Examples are as follows. 

Fig. 14.5 Dipole-dipole interaction potential for (a) the case in which v E, (b) the case in which v LE with 0 = 0 and = jt/2 (from ref. 14).

Fig. 8. Schematic representation of the hydrogen-bond potential. The solidline (dipole-dipole interaction potential) and the dot (position of minimum, and energy at the minimum) are the only features that are known. The dashed line is an empirical potential constructed to fit tiho-kfiown data (Poland and Scheraga, 1967).

As an ion and a dipole are separated by the distance r and the intermolecular distance is much larger than the dipole distance, the ion-dipole interaction potential is given by ... 

Assume that the magnetic moments ijl and in Eq (15.19) are both directed along the z-axis. Show that the angular dependence of the dipole-interaction potential energy then contains the factor (3 cos 6 — ). Show that this quantity vanishes when averaged over all orientations. 

Difficulties arise in the band structure treatment for quasilinear periodic chains because the scalar dipole interaction potential is neither periodic nor bounded. These difficulties are overcome in the approach presented in [115] by using the time-dependent vector potential, A, instead of the scalar potential. In that formulation the momentum operator p is replaced by tt =p + (e/c A while the corresponding quasi-momentum Ic becomes k = lc + (e/c)A. Then, a proper treatment of the time-dependence of k, leads to the time-dependent self-consistent field Hartree-Fock (TDHF) equation [115] ... 

For this case fli2 corresponds to the quasistatic dipole-dipole interaction potential. 

By resolution of both light-absorption and CD bands of ARN and FAD and by the use of both experimental and theoretical values for transition moments, conformations of the relevant molecules were deduced. However, because of difficulties in accurate assignment and determination of the various transition moments and because of the limitations inherent in calculations using the coupled oscillator model and dipole-dipole interaction potentials, only a qualitative picture of the conformations involved could be expected (12). Generally, reciprocal relations between known transitions were taken as evidence for juxtaposition of the corresponding chromophores (12). The optical data obtained for coenzymes or their model compounds may be utilized in favorable cases for estimating structural features in the bound state, in which the interactions with the protein environment will also have to be evaluated. 

Example 4.1 (Dipolar Molecular Liquid) The Stockmayer model for a molecular liquid with permanent dipoles [388] treats this as a system of rigid bodies (spheres) with dipole-dipole interaction potential we already encountered this in the exercises of Chap. 1. From a computational and modelling perspective, there is no real reason to use spheres (and perhaps good reasons not to) for example they might be taken to be ellipsoidal. Let us assume that they are identical ellipsoids. The Hamiltonian for a more general dipolar system can be written... 

Where Krot represents the rotational kinetic energy expressed in terms of the body-fixed angular momentum vectors, Atrans is the translational kinetic energy, U is the potential energy which is a function of all the positions and dipole orientations. The vectors iik can be taken to be unit vectors reflecting the dipole orientation, so they are subject to the constraint /tjt = 1. The total dipole-dipole interaction potential then takes the form... 

The linear terms in the expansion Eq. (1.39) do not contribute to the flexoelectric coefficients because the dipole-dipole interaction potential is odd both in di and d2 and hence the corresponding contributions vanish after averaging over the orientation of the molecular axes. Thus it is necessary to take into account the quadratic terms in the expansion of the direct correlation function. Then the contribution from the dipole-dipole correlations to the flexocoefficients can be written in the form ... 

Example 1.1 Dipole-Dipole Interaction Potential A dipole consists of two point charges at a fixed distance apart. A dipole-dipole interaction system is illustrated in Fig. 1.7. For convenience ofnotation, we label the charge species as a, 6, c, and d. The potential energy of interaction between the dipoles is the sum of the coulombic interactions between the point charges, i.e.. 

A further simplified form of the dipole-dipole interaction potential is often used for dipole-dipole separation distances that are large compared to 6 = b, the dipole length. Such a situation exists for molecules in the vapor state. Let yac = jrac and expand jac about 6 = 0 in a Taylor series, i.e.. 

All of these approaches employ the classical capture approximation and use a charge-dipole interaction potential of the form... 

A FIGURE 10.9 Parameters appearing in the derivation of the dipole-dipole interaction potential. 

The introduction of an >-substituent (CN, Cl, or OH) into a primary n-alkyl chloride considerably enhances the rate of 5 n2 chloride exchange in the gas phase. Reactivity trends suggest that the acceleration is due primarily to through-space solvation of the transition state, especially charge-dipole interactions. Potential-energy surfaces are discussed. In further work by the same group, the translational energy dependence of the rate constants of several gas-phase 5 n2 and carbonyl addition-elimination reactions has been measured by FT-ICR spectroscopy. The results were interpreted by RRKM calculations. 

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