The Polarizability Tensor

In the preceding sections the applied field was taken to be parallel to the principal axes of the ellipsoid. When the applied field E0 is arbitrarily directed, the induced dipole moment follows readily from superposition  

To evaluate (5.38) we need the components of p relative to the primed axes. Equation (5.37) can be written in matrix form 

In the interests of economy we shall write column vectors and matrices according to the following notational scheme  

In most experiments and observations we are confronted with a collection of very many particles unless special pains are taken to align the particles, or in the absence of a known alignment mechanism, we may reasonably assume that they are randomly oriented. Under these conditions the quantities of interest are the average cross sections (Cabs) and (Csca), which are independent of the polarization of the incident light provided that the particles are not intrinsically optically active. Let p(Cl) dti be the probability that one of the axes fixed relative to a particle, the x axis, say, lies within a solid angle dSl around the 

At optical frequencies, it is the dipole induced by the incident electric field that is normally important. The induced dipole is calculated from the following empirical relationship ] 

For an anisotropic material, the polarizability will be a tensor that will depend on the relative orientation of the principal axes of the material and the incident electric field. If the polarizability is biaxial and of the form, 

Applying rotation matrices defined in (2.67) to this tensor and assuming a uniaxial form (a2 = a3) gives 

Use of this form for the polarizability tensor in equation (4.10) leads to the result that the dipole will not be oriented parallel to the incident electric field, as it is when the polarizability is isotropic. Additionally, inserting the resulting expression for the dipole into equation (4.8) will produce a much more complex angular dependence for the scattered light electric field. 

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In a normal Hartree-Fock job, the hyperpolarizability tensor is given only in the archive entry, in the section beginning HyperPolar=. This tensor is also in lower tetrahedral order, but expressed in the input (Z-matrix) orientation. (This is also true of the polarizability tensor within the archive entry.)... 

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

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 polarizability tensor may therefore be defined by a set of nine components which reduce in number to six because the tensor is symmetric. The physical significance of molecular polarizability is often explained in terms of the polarizability ellipsoid which is defined by the equation ... 

In quantum theory as in classical theory the isotropic Raman spectrum is expressed in terms of the average value of the polarizibility tensor a(0) = (1/3) Sp a randomly changing in time due to collisions ... 

Usually, the individual components of the polarizability tensor 5 are not given, but only the average value of its diagonal elements which is defined as = 1/3 (a, + otyy + azz). 

The components of the translation and rotation vectors are given as Tx> Ty, T and RX Ry, Rz, respectively. The components of the polarizability tensor appear as linear combinations such as axx + (xyy> etc, that have the symmetry of the indicated irreducible representation. 

The dipole moment, p , induced on a site i is proportional to the electric field at that site, E . The proportionality constant is the polarizability tensor, a,. The dipole feels an electric field both from the permanent charges of the system and from the other induced dipoles. The expression for is... 

A stress-induced alignment can also be detected in Raman experiments. The sensitivity of a vibration to the polarization of the incident and scattered light in a Raman experiment is determined by the polarizability tensor for the vibration. Even in the absence of polarization information, IR absorption or Raman measurements made in the presence of stress can be used to detect a preferential alignment of a defect by the effect the alignment has on the relative intensities of the stress-split-components of a vibrational band. 

A computation of Raman intensities can be done precisely in the same way as for infrared intensities. One needs here, in addition to the wave functions of the initial and final state, the polarizability tensor a(r,0, < )). This is a symmetric tensor of rank 2 that in Cartesian coordinates can be written as... 

The second source is resonance Raman spectra. The so-called A term of the polarizability tensor for one mode in such a spectrum is given in a usual notation by (49) ... 

The methods described above are all based on the Born-Oppenheimer approximation. Therefore, they can be used to calculate polarizabilities of diatomic molecules for a given internuclear distance R. However, if one is interested in values of the polarizability tensors, and C", for a particular vibrational state /i )), one has to average the polarizability radial functions a(R) and C(R) with the vibrational wavefunction i.e., one has to... 

Table 3. HF dependence of the polarizability tensors (in atomic units) on the basis sets used in the calculations for three internuclear distances...

FIGURE 32. Diagonal elements of the polarizability tensors of ethylene and the aUyUc compounds (of. Table 7). Reprinted with permission from Reference 32. Copyright 1998 American Chemical Society... 

The polarizability tensor of a molecule related the components of the induced dipole moment of the molecule to the components of the electric field doing the inducing. It therefore has 9 components, axx, ctxy, etc., only 6 of which are independent. The theory of the Raman effect shows that a vibrational transition, from the totally symmetric ground state to an excited state of symmetry species F, will he Raman active if at least one of the following direct products contains the totally symmetric representation ... 

We noted in the preceding section that the polarizability of an ellipsoid is anisotropic the dipole moment induced by an applied uniform field is not, in general, parallel to that field. This anisotropy originates in the shape anisotropy of the ellipsoid. However, ellipsoids are not the only particles with an anisotropic polarizability in fact, all the expressions above for cross sections are valid regardless of the origin of the anisotropy provided that there exists a coordinate system in which the polarizability tensor is diagonal. 

It is not difficult to generalize the results of this section to an anisotropic ellipsoid the axes of which coincide with the principal axes of its permittivity tensor. The principal values of the polarizability tensor of such a particle are... 

For the P-phase of PVDF the fc-axis is a principal axis of the polarizability tensor of the repeat unit in the absence of an applied field, only the component of parallel to the (>-axis is nonzero. Equation (3) may thus be expressed in scalar form, where Ap is also directed along the b-axis. 

The theory of strain birefringence is elaborated in terms of the RIS model as applied to vinyl polymer chains. Additivity of the polarizability tensors for constituent groups is assumed. Stress-birefringence coefficients are calculated for PP and for PS. Statistical weight parameters which affect the Incidences of various rotational states are varied over ranges consistent with other evidence. The effects of these variations are explored in detail for isotactic and syndlotact/c chains. 

The stress-optical behaviour of an elastomeric network of PDET is measured over a wide range of elongation ratios and temperatures. Theoretical calculations are carried out with the RIS model. For Act, no reasonable modification of the conformational energies or contributions to the anisotropic part of the polarizability tensor would achieve agreement between theory and experiments. The discrepancy between theoretical and experimental results may be qualitatively explained by intermolecular interactions. Agreement between theory and experiment is only obtained assuming the unlikely value of about + 4.2 kJ mol-1 for E(on). 

Over the last years we have explored several advanced techniques for high-resolution rotational coherence spectroscopy (RCS [1]) in order to study the structures of molecules and clusters in the gas phase [2]. We have provided spectroscopic examples demonstrating (i) mass-selectivity (Fig. 1, [3]), (ii) that the rotational constants of the ground and electronic excited states can be obtained independently with high precision (lO MO"5, [4]), (iii) that the transition dipole moment alignment, (iv) centrifugal distortion constants, and (v) information on the polarizability tensor can be obtained (Fig.l, [5]). Here we review results pertaining to points (i), (ii), (iv) and (v) [2,3,5],... 

The method of fs DFWM spectroscopy and our theoretical model for the spectral simulation is discussed in our second contribution in this volume. The experimental setup has been detailed in a former publication [5], Here, we would like to highlight some special features of this technique with emphasis on the possibility to obtain, besides the rotational constants, centrifugal distortion constants (CDs) and information on the polarizability tensor (PT). 

The second important feature of fs DFWM spectroscopy is its dependence on the polarizability tensor. The first 50 ps of the experimental fs DFWM spectrum of formic acid vapor are plotted in Fig. 4a and a fitted simulation is given in Fig. 4b, [6]. From the resulting fitting parameters it is deduced that the the polarizability tensor has oblate symmetry . 

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