Dipole length formula

More commonly used (cf. Example 1.6) is the related dipole length formula... 

The use of the dipole length formula is based on the evaluation of the transition dipole moment... 

Dipole length formula, 23, S6-Slt Dipole moment, 465. See also Electric dipole moment Magnetic dipole moment excipicx, 282 excited-slate. 47-48. 132 induced.130 permanent, 130-31 Dipole strength, IS8 Dipole velocity formula, 2, 56-57 Direct reaction. See Reaction 1,4-Disilabenzene, 105-7 Disproportionation, 228-29, 380, 390, 433 Disrotatory. See Elecirocyclic reaction ... 

Some difficulties in comparing the experimental kinetic data with the outer-sphere reorganization energy calculated from the Marcus formula (28) result from several assumptions made in this theory. The reactant was assumed to have a spherical shape with a symmetric charge distribution. No field penetration into the metal was considered. Also, the spatial dispersion of the dielectric permittivity of the medium was not taken into account. In fact, the positions and orientations of dipoles around a given ion are correlated with each other therefore the reorientation of one dipole, under the influence of the external field, changes to some extent the reorientation of other dipoles within the distance defined by the correlation length. 

To apply the formula to our case of a vibrating particle, note that this represents an oscillating electric dipole of moment p = es., where 5 is the displacement of the electron at any instant from its position of equilibrium. Then the energy radiated per second in a specified direction is given by the Poynting vector, the length of which is... 

Formula 9.2 is value when the distance between the dipole and an observation point is significantly greater than the characteristic length Xq, or measurements are performed at times exceeding ro(n 1). In the latter case eq. 9.21 can be presented as ... 

We will consider the behavior of the field at the low-frequency part of the spectrum when the skin depth in both media exceeds the distance from the dipole to the interface and the probe length. In deriving asymptotic formulae we will use the approach described in chapter four, namely the interval of integration is presented as a sum of two parts, the internal part where 0 < mL < moL < 1, and the external one as m > mo. Within the external interval radicals mi and m2 can be expanded in a series by powers of k /m and k /m. For this reason the integral at the external interval is presented as a series of terms having even powers of k. Within the internal interval exponents can be expanded... 

The differences in the bond lengths, especially between the bonds N/C-2 and N/C-4 indicate that the delocalization of the r-electrons is affected by the heteroatoms. As in the case of furan, the structural formula with two r-bonds is a good representation of the electronic structure of the molecule. The ionization energy of oxazole is 9.83 eV and its dipole moment is 1.5 D. 

The measured values of coherence lengths / i, l, and the corresponding cubic non-linear susceptibilities Fn, Fj., F,j for all the compounds investigated are shown in Table. The effective cubic polarizabilities > f, yf, y and y,j were calculated using the isotropic (Vuks) form for the local electric field. The value of (the projection of the vectorial part of the quadratic hyperpolarizability on the direction of the molecular dipole moment jl) was calculated from the formula ... 

In the above formulation Ry is the distance between centroids, while the overall distance dependence is still of the 12-6 type. Both well depth and equilibrium separation depend on the distance and relative orientation of the two ellipsoids, and two extra parameters, p. and v, add flexibility to the model by exponential scaling of the implied dot products between orientation vectors. All in all, a GB potential depends on four parameters, the length/breadth ratios for and a, plus p, and v. Figure 13.8 shows the nice result an orientation dependence of the potential. In addition, the ellipsoids or the discoids can be made electrically active by imposing some central multipoles along the molecular envelope the calculation of the multipole energy then is easily accomplished by standard electrostatic formulas (see Section 4.2 and ref. [30]). For two dipoles di and dj at a distance R, for example, the total simulation energy of the liquid crystalline sample then becomes ... 

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