Induced dipole empirical models

Collision-induced dipoles manifest themselves mainly in collision-induced spectra, in the spectra and the properties of van der Waals molecules, and in certain virial dielectric properties. Dipole moments of a number of van der Waals complexes have been measured directly by molecular beam deflection and other techniques. Empirical models of induced dipole moments have been obtained from such measurements that are consistent with spectral moments, spectral line shapes, virial coefficients, etc. We will briefly review the methods and results obtained. 

Spectroscopic measurement. Specifically, if the induced dipole moment and interaction potential are known as functions of the intermolecular separation, molecular orientations, vibrational excitations, etc., an absorption spectrum can in principle be computed potential and dipole surface determine the spectra. With some caution, one may also turn this argument around and argue that the knowledge of the spectra and the interaction potential defines an induced dipole function. While direct inversion procedures for the purpose may be possible, none are presently known and the empirical induced dipole models usually assume an analytical function like Eqs. 4.1 and 4.3, or combinations of Eqs. 4.1 through 4.3, with parameters po, J o, <32, etc., to be chosen such that certain measured spectral moments or profiles are reproduced computationally. 

Such procedures work well if the appropriate functional form of the induced dipole is chosen, if the interaction potential is known, and if high-quality measured spectra are available over a wide range of frequencies, intensities and temperatures - conditions which are rarely met to the degree desirable. Nevertheless, for a few simple systems like rare-gas pairs, highly polarizable molecular gases, etc., more or less satisfactory empirical dipole models were thus obtained. 

If one now chooses the appropriate induced dipole model, Eqs. 4.1 through 4.3, or a suitable combination of these, with N parameters po, >7, R0,. .., and one has at least N theoretical moment expressions available, an empirical dipole moment may be obtained which satisfies the conditions exactly, or in a least-mean-squares fashion [317, 38]. We note that a formula was given elsewhere that permits the determination of the range parameter, 1/a, directly from a ratio of first and zeroth moments it was used to determine a number of range parameters from a wide selection of measured moments [189]. In early work, an empirical relationship between the range parameter and the root, a, of the potential is assumed, like 1/a 0.11 a. That relationship is, however, generally not consistent with recent data believed to-be reliable. 

Table 4.1. Empirical induced dipole models for the translational band Eq. 4.30, with n = 1 where B(n) 0.

Besides the asymptotic terms above, a number of near-range (exponential) induced dipole components do in general exist. For some empirical models, such near-range terms were added for better fits of measured spectra. 

Early numerical estimates of ternary moments [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in the fundamental band, which we will consider in Chapter 6, and hard-sphere interaction potentials. While the main conclusions are at least qualitatively supported by more detailed calculations, significant quantitative differences are observed that are related to three improvements that have been possible in recent work [296] improved interaction potentials the quantum corrections of the distribution functions and new, accurate induced dipole functions. The force effect is by no means always positive, nor is it always stronger than the cancellation effect. 

Early work. Previous numerical estimates [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in... 

Previous work of the kind was generally based on empirical induced dipole models whose parameters were adjusted to fit measured spectra. For molecular systems like hydrogen pairs, empirical dipole models are highly simplified, for example, by either suppressing the anisotropic overlap terms, the AL = 21 components, in favor of an overlap term in the... 

The profiles of the rototranslational absorption of CH4-CH4 in the far infrared have been reported [56] see Fig. 3.22 for an example. The treatment of the spectra is based on the multipolar induction model and an advanced isotropic potential empirical overlap-induced dipole components have also been included for fitting the experimental data at several temperatures (126 through 300 K). At the lower temperatures, satisfactory fits of the measurements are possible. The analysis seems to suggest that at temperatures near room temperature a significant rotation-induced distortion of the tetrahedral frames occurs which affects the properties of the individual molecules (multipole strengths, molecular symmetry, polarizabilities, and perhaps the interaction). 

The interaction between adsorbed particles was also taken into account in terms of some models of induced inhomogeneity (see the above representation), e.g. in de Boer s dipole-dipole interaction model [64], but compared with the lattice gas model, they must be treated as semi-empirical. A semi-empirical model for the collective interaction of adsorbed particles with catalyst surface was also suggested by Snagovskii and Ostrovskii [37]. 

The effect of induced dipoles in the medium adds an extra term to the molecular Hamilton operator. = -r R (16.49) where r is the dipole moment operator (i.e. the position vector). R is proportional to the molecular dipole moment, with the proportional constant depending on the radius of the originally implemented for semi-empirical methods, but has recently also been used in connection with ab initio methods." Two other widely available method, the AMl-SMx and PM3-SMX models have atomic parameters for fitting the cavity/dispersion energy (eq. (16.43)), and are specifically parameterized in connection with AMI and PM3 (Section 3.10.2). The generalized Bom model has also been used in connection with force field methods in the Generalized Bom/Surface Area (GB/SA) model. In this case the Coulomb interactions between the partial charges (eq. (2.19)) are combined... 

Higher-order multipole moments enhance the forces between particles at short distances and their neglect is extremely questionable, especially if fine effects are looked at, as for instance the ground-state properties of close-packed lattice structures [244,246-251] or the viscosity To go beyond the point dipole approximation Klingenberg and co-workers [ 173,252] developed an empirical force expression for the interaction between two dielectric spheres in a uniform external field from the munerical solution of Laplace s equation [253]. Recently, Yu and co-workers [254,255] proposed a computationally efficient (approximate) dipole-induced-dipole model based on a multiple image method which accounts partially for multipolar interactions. 

Since these parameters are based upon microscopic model processes (both chemical and physical), they act as probes of solute solvent interactions such as hydrogen-bonding, charge-dipole, dipole-dipole, and dipole-induced dipole. These empirical parameters have found extensive usage(7), and new empirical parameters continue to... 

The derivation of the potential energy for London dispersion forces is quite involved, and usually such interactions are not quantitatively modeled by equations of the sort we have been presenting here. Typically, the empirically derived Lennard-Jones "6-12" potential discussed in Chapter 2 or a related function is used. To a first approximation, as with the dipole-induced-dipole, the energy of interaction can be considered to drop off with an dependence. 

Polymers can have dipoles in the monomeric unit that can be decomposed in two different components parallel or perpendicular to the chain backbone. The dipole moment parallel to the chain backbone giving rise to an "end-to-end" net polarization vector will induce the so-called dielectric normal mode dielectric relaxation that can be studied using theoretical models. The dipole moment perpendicular to the chain backbone will lead to the segmental a-relaxation that can only be described using empirical models, since no definitive theoritical framework exists for this universal process. 

Fig. 2.2 Self-Consistent Reaction Field (SCRF) model for the inclusion of solvent effects in semi-empirical calculations. The solvent is represented as an isotropic, polarizable continuum of macroscopic dielectric e. The solute occupies a spherical cavity of radius ru, and has a dipole moment of p,o. The molecular dipole induces an opposing dipole in the solvent medium, the magnitude of which is dependent on e.

A. Rare-gas Compounds.—The recent preparation of a number of rare-gas compounds has led to a significant number of calculations on these molecules, particularly on diatomic species. The repulsive interaction and dispersion interaction between like rare-gas atoms has been dealt with above. Several years ago Matcha and Nesbet284and Gilbert and Wahl276 presented calculations on the species NeHe, ArHe, andNeAr. The interaction of two dissimilar rare-gas diatoms gives rise to a dipole moment, which leads to far-i.r. collision-induced spectra of rare-gas mixtures. For this reason, the dipole moments of such species are of interest, but dispersion contributions are not described in the HF approximation and have to be computed by an empirical expression ft — B exp (—Rjp). Much more work is needed on this problem. Wahl and co-workers have reported the numerical results of OVC calculations on HeNe,240 but no detailed discussion was given. In cases like these, the HF model should yield reliable interaction potentials, since A and B are both closed systems. 

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