Polarizability derivatives with respect symmetry coordinates

The transfrumation of vibrational intensities in Raman spectra into molecular parameto s involves sevoal calculation stages. An essential initial step is the reduction of intensity data to polarizability derivatives with respect to symmetiy vibrational coordinates. As pointed out in previous ciutyters, the inverse electro-optical problem of vibrational intensities can be performed with success only for molecules possessing sufficient symmetry. The transformation between da/dQ and do/dSj derivatives is carried out widi die aid of the normal coordinate transformation matrix Lg according to the expression ... 

Polarizability derivatives with respect to symmetry coordinates obtained from Eqs. (9.1) and (9.2) are not always purely intramolecular quantities since contributions from the compensatory molecular rotation accompanying some vibrations may be present. Such contributions arise in the cases of non-totally symmetric modes of molecules having a non-spherical polarizability ellipsoid. Polarizability derivatives corrected for contributions from molecular rotation can be obtained according to the relation... 

Symmetry coordinates and rotational correction terms to polarizability derivatives with respect to symmetry coordinates for acetonitrile (Reprinted from Ref [2S ] widi permission of John Wiley Sons, Ltd., Copyright (1993] John Wil Sons, Ltd.)... 

Rotatiooal comectirai terms are mass-depmident quantities and are, therefore, different for different isotopes with identical symmetry. This affects die values of the respective polarizability derivatives with respect to symmetiy coordinates which will also vary in the respective series. In such a case it is convenient to use the polarizability derivatives of a given (reference) molecule from the series as a standard as proposed by Escribano, et al. [71]. Thus, the intensity analysis is carried out in a uniform way. The polarizability derivatives of each molecule i from die series are related to diose of die reference species through the equation [71]... 

It is more convenient to use polarizability derivatives with respect to symmetry coordinates instead of derivatives with respect to normal coordinates. Recalling that internal vibrational coordinates are connected with symmetry coordinates through the orthogonal U matrix Eq. (2.6)] and using more compact notation, we can write... 

In this section the general equations (9.33) and (9.34) of VOTR are applied in interpreting Raman intensities of SO2. As was pointed out in Section 8.11, the Raman intensity experiment for bent XY2 molecules is not favorable in deteimining a complete set of molecular polarizability derivatives with respect to normal coordinates. That is why, in order to overcome the indeterminacy problem, a set of polarizability derivatives, with respect to symmetry coordinates for SO2 evaluated by means of ab initio MO calculations, is used [301]. The do/dSj derivatives forming the ag matrix are computed by applying the numerical differentiation approach. Other entries needed in solving the problem are taken from the same source [301]. Structural parameters for the sulfur dioxide molecule are given in Table 9.5. The Cartesian reference system and definition of internal coordinates and unit bond vectors are shown in Fig. 9.2. The a tensor employed in the calculations is as follows (in units of or rad" ) [301],... 

The general formula of the bond polarizability model relates molecular polarizability derivatives with respect to symmetry vibrational coordinates with the set of electro-optical parameters [296,297] ... 

In these equations oq and as are the matrices containing molecular polarizability derivatives with respect to normal and symmetry vibrational coordinates, respectively [Eqs. (8.41) and (9.3)], and Op is an array comprising polarizability derivatives with respect to molecular translations and rotations. The matrix product ag Bg represents die so-called vibrational atomic polarizability tensor Vq accounting for the changes in molecular polarizability with molecular vibrations. The Vq tensor for the entire molecule can be expressed as a juxtaposhion of individual atomic tensors ... 

It can be seen fi-om Eqs. (9.86) and (9.87), diat polarizability derivatives with respect to symmetry coordinates belonging to non-totally symmetric modes are different for the two isotopic species. This is expected since the do/dSj derivatives are mass-dependent quantities. The Op tensor is evaluated from Eq. (9.81) ... 

For single crystal samples, the crystal-fixed coordinate system is identical to the laboratory system. Therefore the intensities of the Raman scattered light in the various polarization components can be obtained directly in terms of the polarizability component derivatives from Table II of Brith, Ron, and Schnepp (1969) (reproduced from Wilson, Decius and Cross, 1955), If, however, a randomly oriented powder is investigated, the results must be averaged over orientations, a procedure also described in these two references. If the derivatives have been calculated with respect to the symmetry coordinates S, the derivatives with respect to the normal eoordinates are given in analogy to (4.10). Finally, the observed intensity is obtained by substitution in the equation as follows (Woodward, 1967) ... 

Raman and hyper-Raman tensors for Lane class 3m Here, the Raman and the hyper-Raman tensors for trigonal lattices shall be given. The property tensor for the Raman effect is the derivative of the polarizability with respect to the normal coordinate and is a polar second rank i-tensor which is symmetric if the resonant Raman effect or a degenerate ground state are excluded while the corresponding tensor for the hyper-Raman effect is a polar i-tensor of rank three the internal symmetry of which is dependent on the experimental conditions. 

The heavy-isotope approach to evaluate rotational contributions to polarizability derivatives [288] will be illustrated with calculations on a series of molecules consisting of acetonitrile (C3V synunetiy), dichloromethane (C2v symmetry) and acetone (C2v symmetry). Structural parameters and polarizability tensors employed in die calculations are surtunarized in Table 9.1. Since the axes of the Cartesian reference systems (Fig. 9.1) are chosen to coincide with the respective inertial axes, the static polarizability tensors acquire sirtqile diagonal form. The symmetry coordinates corresponding to vibrations which may crmtain contributions from compensatory molecular rotation for the three molecules are given in Tables 9.2, 9.3 and 9.4, respectively. The following heavy isotopes are employed ... 

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