Tensor polarizability

The dipole polarizability tensor characterizes the lowest-order dipole moment induced by a unifonu field. The a tensor is syimnetric and has no more than six independent components, less if tire molecule has some synnnetry. The scalar or mean dipole polarizability... 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in temis of an ensemble average over the microscopic nonlmear polarizability tensors of the individual constituents. 

If we neglect pure dephasing, the general tensor element of the third order hyperpolarizability relates to those of the first order polarizability tensor according to... 

Flere, the linear polarizability, a (oip 2), corresponds to the doorway stage of the 4WM process while to the window stage. We also see the (complex) Raman resonant energy denominator exposed. Of the tliree energy denominator factors required at third order, the remaining two appear, one each, m the two Imear polarizability tensor elements. 

Lee S Y 1983 Placzek-type polarizability tensors for Raman and resonance Raman scattering J. Chem. Phys. 78 723-34... 

In the final column of the character table are given the assignments to symmetry species and These are the components of the symmetric polarizability tensor... 

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.)... 

Gelbart (1974) has reviewed a number of theories of the origins of the depolarized spectrum. One of the simplest models is the isolated binary collision (IBC) model of McTague and Bimbaum (1968). All effects due to the interaction of three or more particles are ignored, and the scattering is due only to diatomic collision processes. In the case that the interacting particles A and B are atoms or highly symmetrical molecules then there are only two unique components of the pair polarizability tensor, and attention focuses on the anisotropy and the incremental mean pair polarizability... 

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 ... 

Ratio of principal components of differential polarizability tensor... 

For simplicity, consider a single unit of structure with a cylindrical differential polarizability tensor. The scattered intensity Is is then... 

The Raman measurements provide values directly for P)mn, the coefficients of the Legendre expansion related to coordinates axes chosen with respect to the principal axes of the differential polarizability tensor, hence the superscript r. The coefficients Pimn for the orientation of the units of structure must then be obtained by further calculation from the P)mn. 

For a cylindrical differential polarizability tensor we can write ax = a2 and r = v.3l[Pg.91]

It can readily be appreciated that in the absence of any knowledge regarding the differential polarizability tensor, it is a difficult exercise to obtain precise information from the Raman measurements. However, if r is known, Equations (19) are six linear simultaneous Equations in the six quantities (ot2I0N0) 1, P)0o> P220) PIoo> P420 and... 

Die-drawing 143 Die-drawn fibers 144 Differential polarizability tensor 91 Differential scanning calorimetry 63, 165 Diffusion 189... 

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. 

Electronic polarizability is often included in force fields via the use of induced dipoles. Assuming that hyperpolarization effects are absent, the induced dipoles respond linearly relative to the electric field. In this case, the induced dipole p on an atom is the product of the total electric field E and the atomic polarizability tensor a. 

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... 

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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