Bond polarizability tensor

This effort has tried to relate the statistical segment anisotropy (b -to the anisotropy of the monomer unit (b - b2)m. The latter quantity is calculated from the assumption that bond polarizability tensors are additive. The number of monomer units in the statistical segment, Zq, may then be estimated from the relation... 

Each bond polarizability tensor is a 3 x3 square matrix which, in die case of an ordinaiy Raman experiment, has six independent elements. If, however, a special set of local bond Cartesian axes is chosen, the bond polarizability tensor can be presented in a simple diagonal form. Thus, the number of parameters characterizing electro-optica] properties of each molecular bond can be reduced to three. A local bond Cartesian system can be defined as follows (1) One of the coordinate axes is directed along the bond its unit vector is denoted by e lk) (2) The other two axes are chosen in such a w that they are perpendicular to the longitudinal vector and, at die same time, are at right angle one to another. Their particular orientation in space depends on the bond site symmetry. The respective unit vectors can be designated as e2W and e3(k). 

The indices 1, 2 and 3 correspond to the numbering of unit vectors as defined above. At diis stage important approximations are introduced. It is required diat the unit vector e Ik) lies always along the bond k and that the bond polarizability tensor preserves its simple representation when the bond participates in vibrational motion. These approximations are kept in order to reduce the number of intensity parameters. A generalized version of the valence-optical theory of Raman intensities which considers ofiT-diagonal elements of the bond polarizability tensor has been discussed by Rupprecht [299]. 

To each bond k of a molecule a local Cartesian coordinate system with z-axis directed along the bond direction is assigned. The other two bond coordinate axes are perpendicular to the z-axis and their particular orientation depends on the local symmetry of the bond. A bond polarizability tensor is presented in the form... 

In this equation ax(v) is the atomic polarizability tensor free from any rotational contribution. Its elements are, however, still interrelated through the dependency condition (9.84). The problem can be solved if a set of bond displacement coordinates [Eqs. (4.96) and (4.97)] instead of atomic displacement coordinates is used. A rotation-free bond polarizability tensor is defined as... 

An invariant of the bond polarizability tensor with respect to reorientations of the Cartesian reference frame is now defined as the trace of the product... 

Finally, die rotation-free bond polarizability tensor is evaluated with the aid of Eq. (9.101) (in units of 10 30 C.m/Y) ... 

Because an applied field in the y direction Ev can induce a dipole M with a component in the x direction Mx as well as the component in the y direction My, it is necessary that we specify the components of the polarizability tensor by two subscripts (Fig. 3). If the bond A—B of a diatomic molecule stretches during a vibrational mode, Mx and Mv will vary and therefore the corresponding polarizability tensor components will vary. 

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,... 

Polar bonds, solid-state materials with. 276-288 Polarizability tensor, 67 Polarization of ions, 129-134... 

These equations use Cartesian tensor notation in which a repeated Greek suffix denotes summation over the three components, and where ay7 is the third-rank antisymmetric unit tensor. For a molecule composed entirely of idealized axially symmetric bonds, for which [3 (G )2 = /3(A)2 and aG1 = 0 [13, 15], a simple bond polarizability theory shows that ROA is generated exclusively by anisotropic scattering, and the CID expressions then reduce to [13]... 

The authors used a [5 s4p 1 d/4s 1 p] basi s set in their calculations. In order to focus on the effects of the molecular interaction, they introduced the concept of a noninteracting dimer wherein the dimer wave function is a simple product (non-antisymmetrized) of the unperturbed monomer functions. The effects of the interaction are thus in evidence by comparison of the two columns in Table 3.31 from which it may be seen that the average polarizability is little affected, increasing from 16.48 to only 16.60. The anisotropy of the polarizability, however, as measured by y, undergoes a dramatic increase. Whereas the polarizability tensor is nearly spherical in the monomer, with all a.j values between 16.3 and 16.6, is increased up to 18 when the two molecules interact with one another. This increase is thus focused along the H-bond direction. 

The MM2 and MM3 force fields use a bond dipole description for Egi. The interaction created by the other atomic charges times the polarizability tensor. As each of the atoms... 

At the same time, the formally independent particle nature of DFT allows the application of standard interpretative tools developed for the HF approach. This is true not only for the standard MuUiken population analysis, but also for more sophisticated schemes, like the Natural Bond Orbital (NBO) analysis [9], the Atomic Polarizable Tensor population [10], or the Atom in Molecule (AIM) approach [11]. These tools allow the use of familiar and well known models to analyze the molecular wave function and to rationalize it in terms of classical chemical concepts. In short, DFT is providing very effective quantum... 

Computing Raman spectra is complicated by the fact that the polarizability of the system 9 is a tensor. Assuming that the polarizability can be represented as a sum of bond polarizabilities, one can write... 

The extension of the bond polarizability theory to ROA is based on the origin-dependence of G p and AaPr Thus using (2.6) the optical activity tensors of the molecule, written as sums of corresponding bond tensors, are... 

If a bond has a threefold for higher rotation axis, its polarizability tensor can be written as follows in terms of components referred to principal axes ... 

As demonstrated elsewhere45,5), if all the bonds in the molecule are axially-symmetric and achiral, the terms in the ROA expressions (4.8) involving intrinsic bond optical activity tensors and A vanish. For this particular case, the bond polarizability expressions (4.8) have been developed into a computational form that enables the ROA to be calculated for any normal model45) but because of their... 

Thus the polarizability tensor of the molecule is written as a sum of local atomic polarizabilities, each modified through dipolar interactions with the electric dipole moments on all the other atoms induced by the electric vector of the incident light wave. Similarly for the local atomic polarizabilities appearing in the origin-dependent parts of the optical activity tensors. But unlike the bond polarizability development, no allowance can be made for intrinsic local optical activity tensors Gj ap and Aj since these now pertain to spherical atoms. We refer to the original articles for the explicit Raman intensity and optical activity expressions generated by the atom dipole interaction theory. 

In addition to the Raman selection rules described above there are surface selection rules that apply for SERS because the process occurs close to metal surfaces [40—42]. The SERS surface selection rule predicts that the vibrational bands that have contributions from the Raman polarizability tensor component where z is the surface normal, will be most intense with weaker contributions from vibrational bands which have contributions from and o. This is essentially because tlic electric field of the exciting hght is enhanced in the direction of the surface normal (Figure 6.2). The surface selection rule for Raman spectroscopy is more complex than that for infrared spectroscopy. Modes with the bond axis paraUel to... 

McConnell and co-workers have demonstrated that information about the orientation of molecules in a condensed monolayer phase can be obtained by examining the polarization of the fluorescence. The underlying principle is shown in Fig. 6. The monolayer is assumed to be made up of domains of tilted amphiphiles and, for simplicity, the polarizability tensor of the fluorescent probe bonded to some of the molecules is taken to be perpendicular to the chain axis. If p-polarized light is directed to the surface at an angle with respect to the surface normal, the fluorescence excited in the domains will not be identical. The contrast difference between the domains will be reversed if the horizontal component of the direction of the... 

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