Molecular polarizability tensor

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,... 

It was proven [96] that the molecular polarizability can be written as a sum of intrinsic atomic polarizabilities of the atoms in the molecule and a charge delocalization term. Thus, the xy element of the molecular polarizability tensor of molecule A can be decomposed as... 

A simple calculation for urea by Spackman is instructive. Urea crystallizes in an acentric space group (it is a well-known nonlinear optical material), in which the symmetry axes of the molecules coincide with the two-fold axes of the space group. All molecules are lined up parallel to the tetragonal c axis. If the electric field is given by E, and the principal element of the diagonalized molecular polarizability tensor along the c axis by oc , the induced moment along the polar c axis is... 

Therefore, consider a crystal with two molecules per cell, equivalent under translational symmetry operations, as in the anthracene crystal. The molecular polarizability tensors of the two host molecules, in positions 1 and 2, for a molecular transition are given the form... 

The induced component arises from the interaction of the molecular polarizability tensors, 7t, with electric fields. The induced dipole moment on molecule i is given by ... 

We mention that, by contrast, the more familiar Raman spectra arising from the permanent polarizabilities of the individual (noninteracting) molecules of the complex are not considered a part of the supermolecular spectra, or of CILS. In ordinary Raman spectroscopy of rarefied gases the invariants of the permanent molecular polarizability tensor are conveniently considered to be not affected by intermolecular interactions, an approximation that is often justified because induced spectral components are usually much weaker than ordinary allowed Raman bands. 

Miller, K.J. (1990b). Calculation of the Molecular Polarizability Tensor. J.Am.Chem.Soc., 112, 8543-8551. 

In a dense medium, collisional effects as well as time and space fluctuations of multipolar molecular fields will in general lead to changes in the molecular polarizability tensor. Then, according to Eq. (1), the pair polarizability tensor of two molecules A and B, Aap, acquires an incremental collision-induced... 

There have been a number of reports of analogous surface selection rules for Raman spectra [27-29]. However, for SERS, the situation is complicated by the essential roughness of the metal surface and the mixture of enhancement mechanisms, in addition to the facts that the Raman effect depends upon the molecular polarizability tensor and the excitation frequencies are typically high enough to reduce the metal conductivity to levels where finite parallel, as well as perpendicular, electric vectors are established at the surface. An excellent recent review of this subject has been written by Creighton [30]. Suffice it to say here that surface selection rules evidently do exist for Raman spectroscopy but they are more complicated than the rule for infrared and EELS. 

The correlation function involves the elements aaf of the molecular polarizability tensor in the laboratory fixed coordinate system. The aif change with time because of molecular reorientation. Note that the only q dependence on the right-hand side of Eq. (7.1.3) is in the translational factor Fs(q, t). The (0)try (/)> is purely local in character and hence does not depend on q. In the remaining sections of this chapter we evaluate this correlation function for various combinations of molecular symmetries and models of reorientation in fluids. 

The evolution of the orientational distribution results from the orientational hole burning, orientational redistribution, and rotational diffusion. If we assume that the trans and cis molecules are anisometrically shaped and that they may be characterized by uniaxial molecular polarizability tensors (cigar-shaped molecules), then the evolution of the angular distribution has the general form... 

An additional extension of the Drude model includes terms to improve the treatment of the orientation of molecular polarizabilities. Traditionally, in non-polarizable models electrostatic interactions, as well as LJ terms, between atoms bonded to each other or separated by two covalent bonds, 1,2 and 1,3 pairs, respectively, are ignored. Similarly, in the Drude model the interactions between core charges as well as those between the Drude oscillators and core charges are excluded for 1-2 and 1-3 pairs. However, the ability to preserve Coulomb interactions between Drude oscillators (i.e., dipole-dipole interactions) for the 1,2 and 1,3 atom pairs is important for accurate reproduction of the molecular polarizability tensor. At the same time, the use of point charges for these interactions is problematic as their spatial separation is small enough that the Coulombic approximation fails. To overcome this, the electrostatic shielding treatment proposed by Thole [30] is applied, in which the Coulomb interactions between charges / and j are modulated by a factor, S,y, as follows ... 

Table 8 also includes the calculated principal components of the molecular polarizability tensors denotes the component parallel to the carbon chain, while denotes the perpendicular components, Ij in the plane containing the carbon nuclei, o perpendicular to this plane as well. The parallel component is the largest in each case and it exhibits the largest homologous increase, a result of the contribution to this component from the transfer of charge over the length of the molecule between the terminal methyl groups. 

Here, N stands for the molecular packing density, h = 3el(2e+ 1) is the cavity field factor, e = (ej + 2e )/3 is the averaged dielectric constant, F is the Onsager reaction field, < //> and are the principal elements of the molecular polarizability tensor. 

We may obtain working selection rules for vibrational Raman transitions by expanding the molecular polarizability tensor in the normal coordinates Qi... 

If the molecular polarizability tensor is not spherical, then the factorization in equation 17c is not possible without an additional assumption. Owing to the molecular rotation (t)is now a function of time as well as the phase factor. If iSfe assume,however, that the orientation of the molecules is independent of their position, then the factorization can be performed without the appearance of cross terms and the CF separates into a transla-lational and a rotational component as follows ... 

Let US Start examining the first contribution to the induced dipole liM in Eq. (3.1) which is linear in the inducing electric field E. The linear molecular polarizability tensor in the molecular coordinate system can be written, within the oriented gas model, as ... 

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