Ideal dipole approximation

In this still relatively simple molecule all pair interactions between the nine H-atoms at the carbons, the H at the hydroxyl, the four carbons and the oxygen, have to be accounted for. Treating the OH-group interaction also via a Lennard-Jones interaction plus an added (ideal) dipole contribution is already an approximation because intermolecular distances are too short to treat dipoles as ideal. In mathematical terms, the expressions derived for Debye- and Keesom-type interactions (1,4,4c) are only first approximations, the more so because the rotation of the dipole is restricted. In practice there is often no alternative than to make clever guesses about the various Uy r) functions. It is always possible to group some types of interaction together, to obtain more detailed expressions for yA. Such an equation for dumb-bell types of molecules have been given by Alejandre et al. ) and by Harris 2). 

The potentials for a uniform electric field corresponds to those of the electric dipole approximation, except that in the latter case the electric field is uniform only over the molecular volume and not over the entire space. Uniform fields extending over all space must be considered a mathematically convenient idealization they are allowed by Maxwell s equations, but not realizable experimentally. The same holds true for the source-free electromagnetic waves discussed above. It is also important to note that uniform fields do not form the static limit of an electromagnetic wave, in the limit % — 0 both the electric and magnetic fields go to zero. 

Verlet and Weis argued that one could expect to develop a satisfactory approximation scheme by retaining in the rotationally-invariant expansion of /i(12) only the terms that appear on a certain minimal basis, which consists of the terms that enter the MSA. [For a w(12) of ideal dipole form, these are just the 5, A, and D terms of (2.62), defined by (2.20)]. The linearization of (2.143) that will ensure the retention of these terms and introduce no others is given by... 

The polymers in Fig. 6.2 have centrosymmetric backbones in their ideal extended conformation. Dipole selection rules imply mutually exclusive linear absorption to odd-parity states such as F) in Eq. (29) and two-photon absorption (TPA) to even-parity states. We retain the Ag and Bu designation for the C2/, backbone of /-PA, but g and u indices apply equally to PDA, PPV, PT, and PS. NLO coefficients [86-89] are higher order perturbations in fi, with additional sums over states (SOS), polarizations, frequencies, and lifetimes. The transition dipoles (Ag /x 5u> control all NLO responses in the dipole approximation. To simplify notation, 2A and B refer to 2 Ag and I Bu below and correspond to excitation thresholds in Table 6.1. 

Carnie and Chan and Blum and Henderson have calculated the capacitance for an idealized model of an electrified interface using the mean spherical approximation (MSA). The interface is considered to consist of a solution of charged hard spheres in a solvent of hard spheres with embedded point dipoles, while the electrode is considered to be a uniformly charged hard wall whose dielectric constant is equal to that of the electrolyte (so that image forces need not be considered). 

While Onsager s formula has been widely used, there have also been numerous efforts to improve and generalize it. An obvious matter for concern is the cavity. The results are very sensitive to its size, since Eqs. (33) and (35) contain the radius raised to the third power. Within the spherical approximation, the radius can be obtained from the molar volume, as determined by some empirical means, for example from the density, the molar refraction, polarizability, gas viscosity, etc.90 However the volumes obtained by such methods can differ considerably. The shape of the cavity is also an important issue. Ideally, it should be that of the molecule, and the latter should completely fill the cavity. Even if the second condition is not satisfied, as by a point dipole, at least the shape of the cavity should be more realistic most molecules are not well represented by spheres. There was accordingly, already some time ago, considerable interest in progressing to more suitable cavities, such as spheroids91 92 and ellipsoids,93 using appropriate coordinate systems. Such shapes... 

In Section VILA a strongly idealized picture was described. The dielectric response of an oscillating nonrigid dipole was found in terms of collective vibrations of two charged particles. Now a more specific picture pertinent to an idealized water structure will be considered. Namely, we shall briefly consider thermal motions of a dipole as (i) pure rotations in Fig. 56b and (ii) pure translations in Fig. 58a. Item (i) presents the major interest for us, since we would like to roughly estimate on the basis of a molecular dynamics form of the absorption band stipulated by rotation of a dipole. Of course, even in terms of a simplified scheme, the internal rotations of a molecule should also be accompanied by its translations, so the Figs. 56a and 56b should somehow interfere. However, in Section IX.B.l we for simplicity will neglect this interference. This assumption approximately holds, since, as will be shown in Section IX.B.2, the mean frequencies of these two types of motion substantially differ. 

In essence, this difference between shell models and PPD models arises from the former s treatment of the induced dipole as a dipole of finite length. Polarization in physical atoms results in a dipole moment of a small, but finite, extent. Approximating this dipole moment as an idealized point dipole, as in the PPD models, is an attractive mathematical approximation and produces... 

One way to proceed Is to consider the dipole contribution as the leading term, dubbing all remaining parts "non-electrlc". Assuming the dipoles to be Ideal (also a debatable assumption on this scale) at least a first approximation is formulated ... 

The restriction to a single molecule in this idealized case guarantees that there is no macroscopic change in the surface LDOS upon chemisorption and that there is no dipole-dipole interaction in the infrared result. Experimentally, this is approximated by measuring the stretching frequency at a number of coverages 9 and extrapolating to 9 = 0. 

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