Isotropic polarizabilities

The Slater-Kirkwood equation (Eq. 39) was selected with N = 4 for carbon and N = 1 for hydrogen. The success of the equivalent calculation for the intermolecular interaction of CH4 molecules was mentioned in the previous section. Atoms, rather than bonds, were chosen as the basis for the calculation because the location of the atom centers is unambiguous and the approximation of isotropic polarizability is better for an atom than for a bond. Possible deviations from isotropic polarizability are discussed in Section V. Ketelaar19 gives for the atomic polarizabilities of hydrogen and carbon a = 0.42 and 0.93x 10-24 cm3, respectively. The resulting equation for the London energy is... 

The development of the methods described in Section 9.2 was an important step in modeling polarization because it led to accurate calculations of molecular polarizability tensors. The most serious issue with those methods is known as the polarization catastrophe since they are unable to reproduce the substantial decrease of the total dipole moment at distances close to contact as obtained from ab initio calculations. As noted by Applequist et al. [49], and Thole [50], a property of the unmodified point dipole is that it may originate infinite polarization by the cooperative interaction of the two induced dipoles in the direction of the line connecting the two. The mathematical origins of such singularities are made more evident by considering a simple system consisting of two atoms (A and B) with isotropic polarizabilities, aA and c b. The molecular polarizability, has two components, one parallel and one perpendicular to the bond axis between A and B,... 

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.

The bond additivity approximation (BAA) appears to work for polymers dissolved in isotropically polarizable nonpolar solvents. However in the gas phase, BAA has been shown to be incorrect by Ward and coworkers (11). It has been speculated that the solvent provides a symmetrical environment in which local electric fields at a given bond caused by adjoining bonds, are cancelled by fields due to solvent molecules. Thus assuming the correctness of the RIS and BAA models, the configurational average over all internal degrees of freedom r is given by... 

To answer this question, let us first consider a neutral molecule that is usually said to be polar if it possesses a dipole moment (the term dipolar would be more appropriate)1 . In solution, the solute-solvent interactions result not only from the permanent dipole moments of solute or solvent molecules, but also from their polarizabilities. Let us recall that the polarizability a of a spherical molecule is defined by means of the dipole m = E induced by an external electric field E in its own direction. Figure 7.1 shows the four major dielectric interactions (dipole-dipole, solute dipole-solvent polarizability, solute polarizability-solvent dipole, polarizability-polarizability). Analytical expressions of the corresponding energy terms can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres (Suppan, 1990). These four non-specific dielectric in-... 

If solvent (or environment) relaxation is complete, equations for the dipole-dipole interaction solvatochromic shifts can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres and within the assumption of equal dipole moments in Franck-Condon and relaxed states. The solvatochromic shifts (expressed in wavenumbers) are then given by Eqs (7.3) and (7.4) for absorption and emission, respectively ... 

Another problem comes in examining the polarizability. In the physical picture, the spherically symmetric molecule, just like an atom, has isotropic polarizability. In the chemical picture, for a diatomic molecule we have two unique polarizabilities (1) and in the internal coordinate system or (2) dzz = 5 (o xc + (isotropic polarizability) and Aa = — [polar-... 

Molar Kerr constants mK and dipole moments squared of polytoxyethylene giycoils (POEG) and polyjoxyethylene dimethyl ether)s (POEDE) are reported in the isotropically polarizable solvents carbon tetrachloride, cyclohexane, and dioxane. Data for mK/x for POEG appear to reach an asymptotical value, Calculations of mK/x and /x based on the RIS model show good agreement with the experimental results. 

Figure 9. Top An isotropic atom/molecule and crystal with isotropic polarizabilities will give rise to a spherical wave surface and index ellipsoid. Middle Another isotropic atom/molecule and crystal with larger isotropic polarizabilities will give rise to a smaller spherical wave surface and a larger spherical index ellipsoid. Bottom An anisotropic atom/molecule and crystal with anisotropic polarizabilities will give rise to an ordinary spherical wave surface and an ellipsoidal extraordinary wave surface. The index ellipsoid will have major and minor axes.

Christopher J. Cramer and their co-workers during the last decade [61,100, 55, 56], In SMx, terms responsible for cavity foimation. dispersion, solvent structure and local field polarization are present [51,57], The solvation energy is obtained via the usual approximation that the solute, treated at the quantum mechanical level, is immersed in an isotropic polarizable continuum representing the solvent. Therefore the standard free energy of the solute in solution can be expressed as ... 

The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44], Although these results overestimate the dispersion energy, the correct l/d3 dependence was recovered (d is the metal-molecule distance). Later studies [45 17] extended the work of Lennard-Jones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule-metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability a embedded in a dielectric medium with permittivity esol (note that no cavity is built around the molecule) reads ... 

Table 1 Molecular static isotropic polarizabilities a obtained with the LDA, BP and SAOP potentials and with the accurate vxc (see [17])...

The solvent is described as an isotropic polarizable dielectric medium at equilibrium at a given pressure and temperature. 

Even for the case of isotropic polarizabilities, where ap = aq = a, it follows from Eq. (3-36) that the total polarizability will be anisotropic. If we want to define effective polarizabilities from Eq. (3-37) for the members, we must (arbitrarily ) distribute the interaction term. For ap = aq equipartitioning could work, leading to local anisotropy with a// (local)>a and a (local)[Pg.53]

Figure 6 Difference in isotropic polarizability between homologous alkanes. All polarizabilities reported in atomic units.

In Figure 6, we have plotted the differential methylene polarizability (the isotropic polarizability per methylene unit) of the n-alkanes. The most notable feature of this plot is the increase in the isotropic polarizability with the number of carbon atoms. From the decomposition of the polarizability along the principal directions in Figure 7 (with the Cartesian directions defined as for Figure 3), we note that the differential polarizability increases only in the x direction—that is, in the component parallel to the carbon chain. In the other two directions, the polarizability converges smoothly except for weak oscillations in the in-plane ayy component, oppositely directed to those in the other in-plane component axx. These oscillations, which are absent in the isotropic polarizability, may be linked to the alter-... 

Depolarized scattering occurs because of various forms of particle anisotropy. Distinct classes of depolarizing scatterers include nonspherical particles with uniform isotropic (scalar) polarizabilities (sometimes called form anisotropy), inhomogeneous particles with nonuniform distributions of isotropic polarizability, and particles with anisotropic (tensor) polarizabilities. For each of these classes, the intensity of depolarized light scattered by a particle will change as the particle translates, rotates, or manifests internal rearrangement of its scattering elements. DDLS can provide information on the dynamics of each of these processes. 

For molecules with anisotropic polarizabilities, where the assumption Piseries solution in the form times powra s of a r gM The contribution to Be of the lead term in the series proportional to vanishes on averaging over all orientations if the intomolecular potential is a function of fu only. For isotropically polarizable molecules the lead term also vanishes, leaving the series equivalent to the second term on the right side of equation 3a). The lead term in this series, (4Treo) 2<4 i2, is exactly what is obtained by setting the denominator of the second term in equation (23a) equal to one. 

As examples of molecules having, in a linear approximation, isotropic polarizability, we adduce ones of tetrahedral symmetry such as CH, CO, and the like. Here, neither a permanent dipole nor quadrupole is available, and the lowest non-zero moment is an octupole having the non-zero components Qxis The electric field of such octupoles, given by equation (48e), induces a dipole in another molecule, and we have by (204) in the approximation of pairwise correlations ... 

Solution of a polar liquid in a solvent with isotropic molecules. Assume the molecules of the solvent to be isotropically polarizable, and those of the solute to be dipolar. Equation (227) now yields ... 

If the solvent with isotropically polarizable molecules contains in solution a liquid whose molecules are quadrupolar, we obtain ... 

On taking into consideration the electric dipoles as isotropically polarizable, we get in place of Van Vleck s formula (260) the relation ... 

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