Derived polarizability tensor

Mn(CO)5Br bears a close structural relationship to the hexacarbonyls so that the values of the ratios of its j>(CO)-derived polarizability tensors are of interest. For the axial carbonyl the ratio ai/a t is either —0.99 or —0.20 (the two roots of a quadratic equation obtained with neglect of L-matrix effects) compared with a value of ca. —0.23 for the hexacarbonyls08. The derived polarizability tensor for each... 

In general, the derived polarizability tensor will have a biaxial form,... 

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... 

Of particular interest in connection with the derivation of the stress-optical law is, that an analogous equation exists for the average polarizability tensor of the single chain 62). This expression is derived from the work of Kuhn and Grun (64) and reads ... 

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)... 

An alternative and simpler approach to deriving the result in equation (4.12) is to express the polarizability tensor as a general expansion in the two orthogonal unit vectors, u and p, embedded on the principal axes shown in Figure 4.4. Evidently, using Einstein notation, the polarizability can be written as... 

Fir systems of the complexity envisaged in the FrBhlich model, biopolymers and/or portions of the membrane, the magnitude of the terms in Eqs. (28) and (29) is difficult to ascertain, and nonvanishing higher-order terms are not unexpected. Even if the set of vibrations is mechanically uncoupled (161), i.e. terms of the form [32V(Q)/3Qk3Qj]= 0, kf], non-linear terms in p make the related photon-induced transitions non-vanishing. The Raman intensities, Eq. (25), can be expressed in a similar fashion in terms of the derivatives of the components of the polarizability tensor, att, t=x,y,z with respect to the vibrational coordinates ... 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

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. 

The traditional approach to evaluating RR intensities involves a summation over all unperturbed eigenstates of the resonant electronic state. This is a direct consequence of the quantum-mechanical derivation of the polarizability tensor components employing second order perturbation theory as given by the Kramers-Heisenberg-Dirac (KHD) relation for the transition polarizability tensor ... 

The occurrence of Raman scattering is connected to the change in polarizability during the transition of the molecule from one vibrational state to the other. Circular polarization ROA arises from interference of the electric dipole electric dipole polarizability tensor with the electric dipole - magnetic dipole and the electric dipole electric quadrupole optical activity tensors. Due to limited space, no rigorous derivation of the theory will be given here, but only the most important results shall be shown. 

The general SOS formula for the n-th order polarizability tensor component derived from time-dependent perturbation theory, Wj, Wj,..., w ) with... 

Note that a distinction is made between electrostatic and polarization energies. Thus the electrostatic term, Ue e, here refers to an interaction between monomer charge distributions as if they were infinitely separated (i.e., t/°le). A perturbative method is used to obtain polarization as a separate entity. The electrostatic and polarization contributions are expressed in terms of multipole expansions of the classical coulomb and induction energies. Electrostatic interactions are computed using a distributed multipole expansion up to and including octupoles at atom centers and bond midpoints. The polarization term is calculated from analytic dipole polarizability tensors for each localized molecular orbital (LMO) in the valence shell centered at the LMO charge centroid. These terms are derived from quantum calculations on the... 

Allowing X to be a vertor would define all nine components of the second derivative tensor (d E/dkjdkj).] For the polarizability tensor where X6 = r,-... 

All of the EFP interaction terms are pairwise additive, except the induction (polarization) energy, which is modeled with asymmetric anisotropic polarizability tensors located at the centroids of the localized molecular orbitals. The analytic energy gradients (both forces and torques) for anisotropic polarizability tensors are derived via a direct differentiation approach in the form of matrix equations. 

JUU.C KXxy — Xyx, xxz — Xix> Xyz — zy> there are only six independent tensor elements. The intensity of the Raman scattering light on the surface is directly related to the derivative of the polarizability tensor, with respect to the given normal coordinates. In order to determine these tensor elements, it is necessary to consider the system, including the molecule and surface atoms, and to define the Cartesian coordinate axis, z, along the surface normal and others along the surface parallel. 

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