The bond polarizability model

In accordance with Placzek s theory (1934) we can write the real part of the complex transition polarizability as the dynamic vibrational polarizability operator (which is a function of a static configuration Q of nuclei) acting on the vibrational state functions and 

Both the initial vibrational state and the final vibrational state belong to the electronic ground state. 

Similarly to the transition polarizability the vibrational transition optical activity tensors are written as  

We can now expand the operators in a Taylor series in the normal coordinates and replace the p normal vibrational coordinates Qp by the local internal vibrational coordinates Sq  

Looking at the expressions for A (Eqs. 6.3-23 to 6.3-26) we see that we need five different products of the tensors and to calculate the optical activity of 

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The Raman line shape is calculated with the bond polarizability model as described above. The unpolarized Raman line shape computed from the sum of the VV and VH line shapes is shown in Fig. 3. One again sees fair agreement between theory and experiment, with excellent peak position and evidence of a... 

As can be seen from the precedmg sections, the evaluation of reop for a given molecule is quite a cumbersome and tedious process. In order to simplify the calculation inocedure, Montero and Del Rio have put forward a compact formulation of die valence-optical dieory of Raman intensities known as bond polarizability model [296,297]. The bond polarizability model has been applied in extracting Raman intensity parameters for a number of molecules [263,302-311]. 

In the bond polarizability model the changes of bond lengdi and bond direction during vibrational distortions are described by introducing a displacement vector 1. It is defined in terms of a set of polar coordinates, namely ... 

In zero-order approximation of the bond polarizability model the following three types of electro-optical parameters representing derivatives of the bond polarizability with respect to bond displacement coordinates are defined [296,297]... 

In this section the predictive power of the bond polarizability model is tested in calculating Raman intensities of propyne by transferring parameters from other molecules. 

Calculated spectral characteristics for propyne obtained by transferring bond polarizability parameters are compared with those evaluated by RHF/6-3 lG(d,p) ab initio MO calculations [311] in Table 9.10 and Fig. 9.4. It is seen that the predicted spectrum obtained in applying the bond polarizability model is in better agreement with experiment than the ab initio estimated spectrum. More advanced quantum mechanical computations are, evidently, needed to satisfactorily calculate the Raman intensities of propyne. The computations employing transferable sets of polarizability derivatives are simple and give good results. 

Raman intensities of the molecular vibrations as well as of their crystal components have been calculated by means of a bond polarizibility model based on two different intramolecular force fields ([87], the UBFF after Scott et al. [78] and the GVFF after Eysel [83]). Vibrational spectra have also been calculated using velocity autocorrelation functions in MD simulations with respect to the symmetry of intramolecular vibrations [82]. 

Interpretation of the v(CO) Raman intensities of a variety of carbonyls using a bond polarizability model... 

Fortunately, later calculations reviewed by Birshtein, Volkenstein, Gotlib and Ptitsyn (750) showed that the deviation of ZQ from Z is largest just for the free rotating chain, when compared with a great number of more realistic models. The conclusion was drawn by these authors that, for practical calculation purposes, Zg and Z0 can be equated. This means that, in principle, the optical anisotropy (oq — oq) of the random link, as used in previous sections, can be calculated with the aid of the experimentally accessible s (see below), when the bond polarizabilities are taken from the literature (757). In this way (oq — oq) is calculated as the anisotropy of a stretched piece of chain, containing s monomer units. This latter number is obtained by combining the first... 

In the Raman case, three distinct general computational thedries have been proposed the bond polarizability theory, the atom dipole interaction theory and localized molecular orbital theories. In the first and third of these the normal modes of vibration, and hence the vibrational quantum states, must embrace a chiral nuclear framework. They are therefore analogous to the inherently chiral chromo-phore model of electronic optical activity in which the electronic states are delo-... 

One advantage of the bond polarizability theory is that, since it is based on a decomposition of the molecule into bonds or groups that can support local internal vibrational coordinates, it can be applied to idealized normal modes containing just a few internal coordinates and so can provide conceptual models of the generation of ROA by some simple chiral structures. Indeed, as mentioned above, the bond polarizability theory actually developed out of a synthesis of the two-group model and the inertial model, both of which have been applied in detail to a number of simple chiral structures 3 5). 

A number of transition metals also show electronically driven asymmetries in their coordination environments. Typical of these are and U which form one or two strong bonds to O to give the vanadyl (VO) " and uranyl (U02) complexes. Cu typically shows an elongation of two trans bonds in its octahedral coordination sphere [46]. Cu and Hg ", though not strictly transition metals, tend to form two strong colinear bonds. Other distortions tend to be weaker and are only expressed when other factors favor a distorted crystal structure. The distortion of the Ti environment in BaTiOi is primarily a steric effect (see Section 10.6.2), but the polarizability of the Tr cation probably contributes. Similar effects are found for other cations with a d° configuration [47]. Most of these effects are not well understood in detail but their presence must be taken into account when using the bond valence model. 

The total potential energy of the Drude polarizable model contains the terms representative of the interaction with the static electric field, interaction with other dipoles and the self-energy associated with the Drude oscillators, in addition to the standard contributions representing bonding terms (bonds, angles, dihedrals, etc.) and intermolecular interactions represented by Lennard-Jones (Lj) "6-12" term ... 

The exploitation of localized orbitals for dispersion energy calculations has already been proposed since the early works on local correlation methods [41 5]. In classical and semiclassical models most often the atoms are selected as force centers only a few works exploit the advantages related to the use of two-center localized orbitals and lone pairs. A notable exception is the recent work of Silvestrelh and coworkers [46-50], who adapted the Tkatchenko-Scheffler model [16] for maximally localized Wannier functions (MLWF), which are essentially Boys localized orbitals for solids. It is worthwhile to mention that one of the very first use of the bond polarizabilities as interacting units for the description of London dispersion forces has been suggested as early as in 1969 by Claverie and Rein [51] see also [52],... 

The application of bond polarizability model at zero-order level, the results obtained are satisfactory. It can be concluded that the set of polarizability parameters transferred from ethane and acetylene is close to the actual set characterizing Raman intensities of propyne. Minor refinement of parameter values is required to obtain a very good fit between observed and calculated spectral curves. 

Epoi also relies on a local picture as it uses polarizabilities distributed at the Boys LMOs centroids [44] on bonds and lone pairs using a method due to Garmer et al. [35], In this framework, polarizabilities are distributed within a molecular fragment an therefore, the induced dipoles do not need to interact together (like in the Appleq-uist model) within a molecule as their value is only influenced by the electric fields from the others interacting molecules. 

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,... 

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