Optical rotation tensor

Formally, if one has the experimental values of the dielectric tensor e, the magnetic permeability tensor /jl, and the optical rotation tensors p and p for the substrate, one can construct first the optical matrix M, then the differential propagation matrix A, and C, which, to repeat, is the x component of the wavevector of the incident wave. Once A is known, the law of propagation (wave equation) for the generalized field vector ift (the components of E and H parallel to the x and y axes) is specified by Eq. (2.15.18). Experimentally, one travels this path backwards. 

In addition to the optical rotation tensor fi, the gyration tensor is often used as the basis of computing optical rotations, since it is more straightforward to define working equations for it in the frequency domain. The relation to the OR tensor is... 

The linear pseudoscalar G is tlie isotropic part of the optical rotation tensor, and G = (G -H G, , -H GL)/3. Time-dependent perturbation may be used to obtain a sum-over-states expression for G away from resonance [2, 7] ... 

Symmetry arguments show that parity-odd, time-even molecular properties which have a non-vanishing isotropic part underlie chirality specific experiments in liquids. In linear optics it is the isotropic part of the optical rotation tensor, G, that gives rise to optical rotation and vibrational optical activity. Pseudoscalars can also arise in nonlinear optics. Similar to tlie optical rotation tensor, the odd-order susceptibilities require magnetic-dipole (electric-quadrupole) transitions to be chirally sensitive. 

This last equation was derived originally by Rosenfeld (1928). It is sometimes convenient to have an expression for the optical rotation tensor 6 itself and not just its principal values, the... 

We see from this last equation that only the symmetric part of the optical rotation tensor contributes to the optical activity. This symmetric part can be written... 

We have used (IB-12) to replace the momentum operator by the electric dipole operator in the first terms of (IIIB-20), This equation is similar to (IIIB-18). It differs in the omission of the terms involving the permanent momentum and magnetic moment of the groups and in the fact that p and m are imaginary (Equation IB-11). Entirely analogous equations can be written for the operators needed for the components of the optical rotation tensor. 

We have not been explicitly writing out the equations for the direction-dependent optical rotation, although for each average rotational strength 1 we can write down the corresponding component of the optical rotation tensor We will only give the... 

Because the mixed electric dipole-magnetic dipole polarizability involves the magnetic dipole operator, in approximate calculations G carries an origin dependence. Indeed, the individual tensor elements of G are origin dependent. The trace of G must be origin independent, since the optical rotation is an experimental observable. In non-isotropic media, contributions to the optical rotation tensor arise from the mixed electric dipole-electric quadrupole polarizability A,... 

The specific optical rotation is related to the trace of the Rosenfeld tensor (3afj through [26]... 

For oriented samples, the rotation of the plane-polarized light becomes a tensor - that is, the optical rotation becomes directionally dependent - and includes a contribution from the electric dipole-electric quadrupole polarizability tensor, which is traceless and thus vanishes for freely rotating molecules [30], The term arising from these quadrupolar interactions can be expressed as [30]... 

In brief, the steps needed to calculate the instrumental settings P and A from given values of the permittivity e, permeability fi, and optical rotation p,p tensors are as follows ... 

In optically active substances, according to equations (12a) and (17a), the non-diagonal components of the refractive index tensor differ from zero and describe the variation in optical rotation due to a static electric field, an effect studied by Tinoco. ... 

The ab initio theoretical quantity needed to predict molecular optical rotations is the electric dipole-magnetic dipole polarizability tensor, indexelec-tric dipole-magnetic dipole polarizability tensor given by the expression ... 

Abstract The modified equation-of-motion coupled cluster approach of Sekino and Bartlett is extended to computations of the mixed electric-dipole/magnetic-dipole polarizability tensor associated with optical rotation in chiral systems. The approach - referred to here as a linearized equation-of-motion coupled cluster (EOM-CCl) method - is a compromise between the standard EOM method and its linear response counterpart, which avoids the evaluation of computationally expensive terms that are quadratic in the field-perturbed wave functions, but still yields properties that are size-extensive/intensive. Benchmark computations on five representative chiral molecules, including (P)-hydrogen peroxide, (5)-methyloxirane, (5 )-2-chloropropioniuile, (/ )-epichlorohydrin, and (75,45)-norbornenone, demonstrate typically small deviations between the EOM-CCl results and those from coupled cluster linear response theory, and no variation in the signs of the predicted rotations. In addition, the EOM-CCl approach is found to reduce the overall computing time for multi-wavelength-specific rotation computations by up to 34%. 

Anomalom optical rotation Hence the dielectric tensor may be expressed as... 

The results from recent decades allow us to describe a picture of thermal motion of long macromolecules in a system of entangled macromolecules. The basic picture is, of coarse, a picture of thermal rotational movement of the interacting rigid segments connected in chains - Kuhn - Kramers chains. One can refer to this model as to a microscopic model. In the simplest case (linear macromolecules, see Sect. 3.3), the tensor of the mean orientation (e,ej) of all (independently of the position in the chain) segments can be introduced, so that the stress tensor and the relative optical permittivity tensor can be expressed through mean orientation as... 

The quantum mechanical expression for the average optical rotation of a molecule was first given by Rosenfeld (1928). In 1937 Kirkwood (1937) and Condon (1937) presented similar derivations. A quantum mechanical expression for the optical activity tensor... 

Excited states are very important in quantum chemistry. Obviously, they are the basic quantities of interest when electronic spectra are considered. Furthermore, because the excited states form a complete basis of the Hamiltonian, all second-order properties such as polarizabilities (van der Waals forces), NMR chemical shifts, ESR g-tensors, or optical rotations of chiral molecules can be calculated quite accurately by sum-over-(excited) state expressions. It should also be clear that any attempt to model photochemical reactions must be preceded by a careful examination of the electronic spectra of the reactants and products in order to deduce the electronic character of the states involved. 

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