Polarizability of an Isolated Molecule

Polarizability of an Isolated Molecule.—In Cartesian tensor notation, the components of the molecular dipole moment p, induced by an electric field E, can be written as ... 

It is important to emphasize that the SOR is not the inevitable consequence of fundamental physical principles rather, it is a very plausible hypothesis, which has extensive experimental support for polymer solutions and melts. In other words, there is no reason to assume that the SOR is valid under all possible flow conditions or for all possible polymer liquids. Some situations under which the SOR is expected to fail are mentioned in the next section. Many constitutive relations for solutions and melts predict that the SOR will hold, but even this apparent generality is somewhat misleading. The derivation of an SOR starts at a measurable molecular property, the optical polarizability of an isolated molecule a, and leads to a macroscopic refractive index tensor n, in a nontrivial way several substantial assumptions are necessary. Most rheological models (for flexible chains) that proceed to an SOR assume the derivation of Kuhn and Gritn (1942) for the polarizability anisotropy of a Gaussian subchain and thus in a sense make the same assumptions for the optical half of the SOR (Larson, 1988). Therefore differences between constitutive relations and their predictions for an SOR usually stem from differences in the calculation of t. 

From the point of view of phase transferability, or, more generally transferability to thermodynamic conditions different from that at which parametrization was carried out, RWKl and RWK2 perform better than TIP4P and SPC. The latter, for example, are certainly unable to satisfactory describe the low density vapor, as their dipole moment (2.18 and 2.27 D) [67] is much larger than that of an isolated molecule (1.86 D) just to take into account many-body effects (mainly polarizability) required for a good description of the condensed phase. 

For the vibration of CF4, the Raman polarizability tensor of an isolated molecule is isotropic, and, consequently, monomer polarizabilities of Eq. (2) do not contribute to Vj depolarized Raman spectral wings. On the other hand, the Raman incremental pair polarizability tensor for the normal vibration V that takes place in Eq. (2) is [78]... 

As we already discussed in Section 2.5.3 for excess polarizabilities of molecules dissolved in a solvent, the London dispersion interactions between molecules in a solvent medium may be very different from those of isolated molecules in free space. The intrinsic permanent dipole moment, p, and polarizability of an isolated gas molecule, a, may be different in the liquid state or when dissolved in a medium, and this can only be determined by experiment. 

The existence of these inequalities is less surprising when it is remembered that the early stages of the localization process described above correspond precisely to polarization of the isolated molecule, and that the subsequent changes in levels and orbitals, as discussed in Section III, follow essentially by an analytic continuation. It follows that predictions of the sequence of active centres, in an even alternant hydrocarbon, based on localization energies must agree with those based on polarizabilities. This... 

By expanding / so1 as a function of the dipole of the isolated molecule and the polarizability a of the molecule, it is possible to obtain an expression for ffJJP /dQ as a function of e, the solute refractive index n, the solution refractive index ns and a [17,18]. Note that the Buckingham approach accounts for nonequilibrium solvent effects (see below), described in terms of the optical dielectric constant eopt. A comparison between PCM calculated IR intensities and classical equations is reported in ref. [8],... 

An analogous relation holds for /3 (—w w,0). The solute-solvent interactions are treated here on the level of dipolar reaction fields. Such effects are increasingly taken into account in quantum chemical calculations on linear and non-linear optical properties of molecules (Karelson and Zerner, 1992 Willets and Rice, 1993 Bishop, 1994a Di Bella etai, 1994 Tomasi and Persico, 1994). Therefore, the solute polarizabilities represent a natural level at which theoretical and experimental results should be compared. Expressions (90)-(92) allow us to identify the major terms that are missing when results of NLO solution experiments are compared with theoretical calculations for molecules in the gas phase. It turns out that the contributions induced by the reaction field are substantial and often even larger than those of the isolated molecule (Mikkelsen et al., 1993). 

Perhaps more challenging to resolve than the choice of electron correlation treatment for weak interaction problems is the basis-set selection. Partly, this reflects the fact that basis sets have traditionally been devised for describing chemical bonding, not for the subtle juxtaposition of effects in weak interaction. They have to do both for weak interaction potential evaluations. This issue in basis-set selection, the adequacy of the basis to describe the electronic structure effects that comprise the interaction, can impose more stringent requirements than for describing an isolated molecule. Consider polarization as a contributor to interaction. Since the adequate determination of molecular polarizabilities requires basis sets augmented with diffuse and higher-/ basis functions [37], an... 

It has been demonstrated that the properties of multiple, interconnected intermolecular bonds are different from those of isolated bonds. This effect is called cooperativity and its origin is the polarization that the formation of an intermolecular bond induces in the electron density of the interacting molecules. When a molecule makes more than one bond, the second one is made with the polarized molecule. The relevance of the effect is clearly shown in Table 1.2.4 for the O-H O bond in water aggregates. But such a polarization effect is expected to be present in all kinds of intermolecular bonds, and will be proportional to the electronic polarizability of the molecule. Experimentally, the existence of polarization effects can be demonstrated by comparing the formation energy of an isolated water dimer (-5.44 kcal mol-1 [45], that is, 2.72 kcal mol-1 per water molecule) with the formation energy per water molecule in ice at 0 K (-11.3 kcal mol-1 [46]). 

The above expressions were derived for the polarizabilities of molecules in free space or in a dilute gas (mostly air). However, we often encounter molecules interacting in a liquid solvent medium, which reduces the interaction pair potential by around e, or more the extent of this reduction depends on several factors. First of all, the intrinsic polarizability and dipole moment of an isolated gas molecule may be different when it is itself in the liquid state, or alternatively when dissolved in a solvent medium. This is because of the difference in interaction strength and also the separation distance between molecules. Thus, the polarizability values are best determined by experiment. Second, a dissolved molecule can only move by displacing an equal volume of solvent from its path. If the molecule has the same polarizability as the solvent molecules, that is if no electric held is reflected by the molecule, it is invisible in the solvent medium and does not experience any induction force. Thus, the polarizability of the molecule, a, must represent the excess or effective polarizability of a molecule over that of the solvent. Landau and Lifshitz applied a continuum approach and modeled a molecule, i, as a dielectric sphere of radius, ah having... 

As an example of an anisotropic crystal, let us consider a molecular crystal with a > 1 molecules per unit cell. Let us investigate the optical properties of the crystal in the frequency range u> io where io is the transition frequency to that of a nondegenerate excited state of the isolated molecule. The polarizability of the molecule in the vicinity of the dipole-allowed transition can be presented by the relation... 

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