Polarizability derivatives with respect

Raman intensities are a function of the change in polarizability of a molecule. The polarizability, however, is a second derivative of the potential energy with respect to an external electric field which makes the Raman intensity dependent on a third-order derivative of the potential energy. For this reason, it is imperative to use an accurate analytical description of the polarizability to obtain a reasonable prediction of Raman intensity for a given vibrational frequency. Frisch et al. (1986) have derived the following expression for the polarizability derivative with respect to movement of atomic coordinates, x. 

The feature of the all considered experimental methods is that they allow us to define the values of the molecular polarizabilify only at the equilibrium position of molecular nuclei. To obtain the dependencies of molecular polarizabilities on the mutual location of nuclei in a molecule, the Raman effect can be used The line intensities of Raman spectra depend on the values of polarizability derivatives with respect to the nuclei displacements. The first works to define the polarizability derivatives of molecules have appeared immediately after the creation of the theory of Raman light scattering (Placzek theory of polarizability) [17]. However, the experimental technique of pre-laser period could not obtain the high-quality results. Some experimental results of this period are summarized in [18]. Currently, these data have only a historical interest. Now, laser technologies allow to increase the measurement accuracy and, as a result, significantly improve and revise the pre-laser data. Nevertheless, up to day the experimental data on flie polarizability derivatives of molecules are fragmentary and do not give the impression of systematic studies of the polarizability of molecules as a function of the nuclei coordinates, even for diatomic molecules [19-39]. 

The computation of Raman densities requires the derivatives of the polarizability tensor with respect to normal coordinates. Since we work with Cartesian representation, the first step in the calculation is to obtain the polarizability derivatives with respect to Cartesian 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]... 

Eqs. (9.24) and (9.25) can be expressed in tenns of bond polarizability derivatives with respect to internal coordinates. Thus, for a molecule with K bonds and M intemal coordinates we obtain [155]... 

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... 

The theoiy is applied in zero-order approximation. All polarizability derivatives are constrained equal to zero except bond polarizability derivatives with respect to stretching of the same bond, i.e. 3a0c)/SR 0 for R] = rjj only, where rjj is the stretching of the kd> bond. These restrictions result in considerable reduction of the number of reop, although their physical justification is questionable. 

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],... 

In these equations acq/ar (i = 1,2,3) denote bond polarizability derivatives with respect to the stretching of the same bond (the so-called zero-order reop) and and acq/d are derivatives with respect to the other internal coordinates the adjacent SO stretching and the OSO bending, respectively (first-order parameters). The term -0.121(aj-a2) last equation represents the rotational contribution to da JdSy Since the two S=0 bonds... 

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] ... 

The method is often used in zero order approximation. Atomic polarizability derivatives with respect to stretching of bonds containing atom i are retained only, i.e. do/dR 7 0 if either m or 1 equals i. These parameters are considered to a great extent independent of the atomic environment. 

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 ... 

In this expression 63x9 is a 3x9 null matrix containing the polarizability derivatives with respect to the three translational coordinates. The next three 3 3 blocks in the matrix comprise the polarizability derivatives with respect to the fiiree rotations p, Py and respectively, (e, = x, y, z) represent the equilibrium molecular polarizability tensor components and I is the principal inertia tensor of the molecule. 

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) ... 

Eq. (9.91) reveals the possibility of expressing molecular polarizability derivatives with respect to atomic Cartesian displacement coordinates in teims of electiYK>ptica] parameters. The validity of this relation will be checked in the case of SO2 molecule. 

Gaussian can also predict some other properties dependent on the second and h er derivatives of the energy, such as the polarizabilities and hyperpolarizabilities. These depend on the second derivative with respect to an electric field, and are included automatically in every Hartree-Fock frequency calculation. 

Many ab initio packages use the two key equations given above in order to calculate the polarizabilities and hyperpolarizabilities. If analytical gradients are available, as they are for many levels of theory, then the quantities are calculated from the first or second derivative (with respect to the electric field), as appropriate. If analytical formulae do not exist, then numerical methods are used. 

The intensities of Raman scattering depend on the square of the infinitesimal change of the polarizability a with respect to the normal coordinates, q. Since the polarizability itself is already the second derivative of the energy with respect to the electric field - see equa-... 

A. Partial Derivatives and Polarizability Coefficients Expansion of (8) yields a polynomial, the characteristic or secular polynomial, whose roots are determined by the values of the parameters , vw- The ground state energy (12) is likewise a function of the (a,j3) parameter values, as are all quantities such as AO coefficients in the MO s, charges q bond orders p t, etc. It is possible, therefore, to specify the h partial derivative with respect to any or at an arbitrary point defined by a set of values (a,j8) in the parameter space, and to make expansions such as... 

The intensity of Rayleigh scattering and the linear Raman effect is governed by the polarizability tensor apa of a molecule and its derivatives with respect to the normal coordinates. When the electric field of the exciting radiation is very high, further terms in the expression for the induced dipole moment 104)... 

In several cases, the polarizability distribution can be found by chemical intuition. For instance, in the case of naphthalene, which is made up of two identical fragments, the polarizability can be decomposed into two equivalent parts. Also, group or atom contributions can be deduced from a variety of schemes such as Stone s approach [74], the theory of atoms in molecules [75], the localization of molecular orbitals into chemical functions [76], atom/ bond additivity [77], the use of the acceleration gauge for the electric dipole operator [78], quantum mechanically determined induction energies [79], or calculated molecular quadrupole polarizabilities and their derivatives with respect to molecular deformations [80]. Several of these models consider charge... 

This section deals with analytical properties of the Hiickel molecular orbital theory and the associated isolated molecule method of predicting the active positions in a conjugated molecule. We shall deal with polarizability coefficients defined as certain partial derivatives with respect to the coulomb and resonance integrals described in Section III. The important derivatives are those relating to the total tt electron energy and to the charges q, fi ee valences and bond orders... 

The above analysis shows how the GAI is applied to the simplest polarizable interface in contact with a 11 electrolyte. Other more complicated situations have been analyzed for systems with more complex electrolytes and molecular solutes. More details can be found in reviews by Mohilner [1] and Parsons [G4]. The essential feature of these analyses is that an equation is derived which relates the change in interfacial tension to the change in the potential of the polarizable electrode with respect to that of a non-polarizable electrode, and to the chemical potentials of the components of the solution. 

The reason is simple the computation of the various (hyper)polarizability elements requires the knowledge of the one-electron density matrix derivatives with respect to the external field Cartesi ln components E , with v = x,y,z] in fact, by expanding the molecular dipole in terms of powers of the external field, we can derive ... 

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