Polarizability tensor invariants

The dependence of the polarizability tensor invariants of the van der Waals X2-Y complexes on the frequency of the electromagnetic field and on the complex... 

Polarizability tensor invariants a R) and y R) for the family of the most stable configurations of the CH4-N2 complex can be also presented in the form of Taylor series in the vicinity of R = 6.84 ao. 

These expressions allow estimating the derivatives of the polarizability tensor invariants of the complex for the analysis of scattering processes in methane-nitrogen gas media. Particularly, it is seen that the first derivatives da R)/dR = 0.172 a.u. and 9y (7 )/dR = —0.018 a.u. are of different sign at 7 e and considerably differ in their absolute value. 

As it was noted in Chaps. 3 and 4 the dipole moment modulus and the polarizability tensor invariants practically do not change for the family of the most stable configurations [52]. We can expect, that the first-hyperpolarizability tensor invariants which are important for description of interaction-induced hyper-Rayleigh scattering, also change weakly for all most stable configurations. 

The anisotropy of the polarizability is described by the tensor invariant y2, which determines the depolarization ratio. This invariant y2 is averaged over all configurations of the chain molecule treated in the RIS approximation, interdependenca of rotations about neighboring bonds is taken into account. 

To see why this is the case, we first consider the portion of the response that arises from llsm. According to Equation (10), we can express (nsm(t) nsm(0)> in terms of derivatives of llsm with respect to the molecular coordinates. Since in the absence of intermolecular interactions the polarizability tensor of an individual molecule is translationally invariant, FIsm is sensitive only to orientational motions. Since the trace is a linear function of the elements of n, the trace of the derivative of a tensor is equal to the derivative of the trace of a tensor. Note, however, that the trace of a tensor is rotationally invariant. Thus, the trace of any derivative of with respect to an orientational coordinate must be zero. As a result, nsm cannot contribute to isotropic scattering, either on its own or in combination with flDID. On the other hand, although the anisotropy is also rotationally invariant, it is not a linear function of the elements of 11. The anisotropy of the derivative of a tensor therefore need not be zero, and nsm can contribute to anisotropic scattering. 

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. 

The terms isotropic and anisotropic are meant to refer to the invariants of the polarizability tensor, not to scattering per se we prefer generally terms such as trace and anisotropy scattering. 

The Gas Phase Polarizability Anisotropy. Murphy50 has measured the depolarization ratio for Rayleigh scattering, pR, and analysed the intensity distribution in the rotational Raman spectrum of the vapour at 514.5 nm. The ratio R20 of the invariants of the a,-,aA/ tensor can be determined by fitting the rotational Raman distribution, and a is known (from the Zeiss-Meath formula). Knowledge of the three quantities, a, pR and R2o, allows the polarizability anisotropy, Aa, and the three principal values of the tensor to be calculated. The polarizability anisotropy invariant is numerically equal to the quantity,... 

As the molecule rotates, the vector u reorients and the tensor a changes in time. We note that because aSap is independent of u, this part of the polarizability tensor does not change as the molecule rotates. aSap is said to be rotationally invariant. Pap on the other hand depends on u, so that this part of the polarizability changes as the molecule rotates. 

The invariants a[ and y/ represent combinations between the six independent components Socjj/SQi (j, k = x, y, z) of the polarizability tensor aq [Eqs. (8.42) and (8.43)]. These quantities can be determined from experiment for very few small and highly symmetric molecules only. Typically, the number of intensity parameters to be evaluated exceeds by far the number of experimental observations. The XY2 bent... 

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... 

The Raman (Eq. 2) and ROA (Eq. 3) invariants require the evaluation of the first-order derivatives of three polarizability tensors (Bo/O/f, ), (6A/6R ), and (0G /07 ia)- A review by Buckingham [34] defines all these polarizabilities. The dipole strength and rotational strength entering into the IR (Eq. 4) and VCD (Eq. 5) intensities require the calculation of the atomic polar tensors (APTs, Pm = and of the atomic axial tensors (AATs,... 

For some systems of simple symmetry, as linear molecules, the invariants of dipole (hyper)polarizability tensors, mean values, and anisotropies are easily defined and commonly used. Most of them are measurable quantities or can be deduced from experimental observations. 

To study the orientation of the sample, the aforementioned experiments must be related to the molecular polarizabilities. By using upper-case letters for the laboratory coordinate system and lower-case letters for the molecule-fixed coordinate system, the derived polarizability tensor can be found. For excitation polarized along F, the scattered radiation, which is polarized along F, will have an intensity that is proportional to [ a )FF f- F and F may be either vertical or horizontal). The appropriate relationships have been determined [5,6]. In an isotropic system, there are orientationally invariant terms, S and y, that are defined as... 

The form of Lx and V, Equation (2.144), makes it evident that the invariants If< of products of transition tensors can be written in the frame of the polarizability theory as sums over mono- and dinuclear terms ... 

Since the Raman techniques described are sensitive to (fl(t) - fi (0)>, we are interested in the properties of products of two elements of II, subject to the above constraints. Furthermore, it is necessarily true that any tensor elements of n that are related by reversed indices are identical, e.g., nxy = Flyx. This leaves us only two independent elements of R<3) to consider. It is conventional to express these independent elements in terms of rotationally invariant features of n One of these invariants (usually denoted a) is given by one third of the trace of FI (24). Since this invariant measures the average polarizability of the system, it is known as the isotropic component of n. The other invariant is generally denoted ft, and in the principal axis system of FI it is given by (24)... 

In general, the polarizability is a tensor whose invariants, trace and anisotropy, give rise to polarized and fully depolarized light scattering, respectively. Collision-induced light scattering is caused by the excess polarizability of a collisional pair (or a larger complex of atoms or molecules) that arises from the intermolecular interactions. In Section I.l, we are concerned with the definition, measurement, and computation of interaction-induced polarizabilities and their invariants. 

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