Antisymmetric scattering

The occurrence of antisymmetric scattering contributions is invariably associated with processes in which the symmetry of particular electronic states becomes important. In the non-resonance electronic (or the vibro-electronic) Raman effect (ERE or VERE), the initial and final states are different electronic (or vibronic) states. Thus, if these states are of appropriate symmetry, an antisymmetric contribution may be present in the scattering 46). The same is true for the special case of the VERE represented by vibrational Raman scattering in systems possessing degenerate electronic ground states (see Section 2.11). In the resonance ERE and resonance VERE as well as the vibrational resonance-Raman effect, it is the symmetry of the intermediate state which becomes significant. 

As a further example, the V coefficient relevant to antisymmetric scattering in cubic symmetries is 48)... 

Returning now to Eqs.(28), it can be seen that, because of the antisymmetry relations, the resonant and non-resonant parts of each tensor element tend to cancel in the off-resonance limit Eq Ej > El > hco. In addition, the contributions from le> and s> states to each term in Eqs.(28) are of comparable magnitude off resonance. These contributions tend to cancel because of the difference in sign of the HT perturbation energy denominators, viz. (Ee - E )" = - (Es - Ee) . The net result is the disappearance of antisymmetric scattering in the off-resonance region. 

The symmetry properties of resonance Raman lines can be predicted on the basis of Eq. 19.15. For totally symmetric vibrations, a and =0. Then Eq. 19.15 gives 0 sp <. Nontotally symmetric vibrations (q=0) are classified into two types those which have symmetric scattering tensors, and those which have antisymmetric scattering tensors. If the tensor is symmetric, = 0 and y, 9 0. Then Eq. 19.15 gives Py, = (depolarized). If the tensor is... 

S,R,Q,P,0 branches of the rotational Raman spectrum. The treatment outlined here is a little more general than usual since we have retained the antisymmetric operator contribution (2,.39b), which generates the less familiar selection rules for antisymmetric scattering, namely at = I with AX = O forbidden if K = 0 [18]. 

Depolarization ratio anomalies are the hallmark of antisymmetric scattering. It can be seen from the general expression (2.27) for the depolarization ratio that, for pure antisymmetric scattering arising from the invariant C alone, the depolarization ratio is infinite. This compares with zero for pure isotropic scattering and % for pure anisotropic scattering. Thus a depolarization ratio larger than expected indicates antisymmetric contributions. 

Resonance Raman and antisymmetric scattering are involved in a novel technique involving spin-flip Raman transitions in paramagnetic molecules that can function as Raman electron paramagnetic resonance. Figure 3.2a shows a conventional vibrational Stokes resonance Raman process, while 3.2b and 3.2c show the polarization characteristics of the two distinct spin-flip Raman processes for scattering at 90°... 

The external reflection of infrared radiation can be used to characterize the thickness and orientation of adsorbates on metal surfaces. Buontempo and Rice [153-155] have recently extended this technique to molecules at dielectric surfaces, including Langmuir monolayers at the air-water interface. Analysis of the dichroic ratio, the ratio of reflectivity parallel to the plane of incidence (p-polarization) to that perpendicular to it (.r-polarization) allows evaluation of the molecular orientation in terms of a tilt angle and rotation around the backbone [153]. An example of the p-polarized reflection spectrum for stearyl alcohol is shown in Fig. IV-13. Unfortunately, quantitative analysis of the experimental measurements of the antisymmetric CH2 stretch for heneicosanol [153,155] stearly alcohol [154] and tetracosanoic [156] monolayers is made difflcult by the scatter in the IR peak heights. 

As the electrons are indistinguishable in the antisymmetrized wave function, the one-electron scattering can be obtained by integration over all coordinates but those of they th electron. Summation over all equivalent electrons then leads to... 

The atom-centered models do not account explicitly for the two-center density terms in Eq. (3.7). This is less of a limitation than might be expected, because the density in the bonds projects quite efficiently in the atomic functions, provided they are sufficiently diffuse. While the two-center density can readily be included in the calculation of a molecular scattering factor based on a theoretical density, simultaneous least-squares adjustment of one- and two-center population parameters leads to large correlations (Jones et al. 1972). It is, in principle, possible to reduce such correlations by introducing quantum-mechanical constraints, such as the requirement that the electron density corresponds to an antisymmetrized wave function (Massa and Clinton 1972, Frishberg and Massa 1981, Massa et al. 1985). No practical method for this purpose has been developed at this time. 

The amplitude fu is antisymmetric with respect to interchange of the nuclei, which is a direct reflection of the symmetry property of the corresponding electronic wave function. This implies that the cross section need not be symmetric about 0CM = 9O°. We can define a scattering amplitude fd(6) for direct scattering... 

FIGURE 5. Schematic diagram of low resolution (0.8 nm) Raman scattering spectrum of O3 excited at 266 nm. The spectrum consists of overtones and combination bands in v-] (antisymmetric stretch) and v3 (antisymmetric stretch up to v" = 7 and V3 = 6 No bands with >2 (bending) are evident, suggesting that the bond angle remains the same during the transition. Reproduced from reference (80) with permission from the American Chemical Society. 

These equations use Cartesian tensor notation in which a repeated Greek suffix denotes summation over the three components, and where ay7 is the third-rank antisymmetric unit tensor. For a molecule composed entirely of idealized axially symmetric bonds, for which [3 (G )2 = /3(A)2 and aG1 = 0 [13, 15], a simple bond polarizability theory shows that ROA is generated exclusively by anisotropic scattering, and the CID expressions then reduce to [13]... 

Ordinary Raman scattering is determined by derivatives of the electric dipole-electric dipole tensor ae, and ROA by derivatives of cross-products of this tensor with the imaginary part G,e of the electric dipole-magnetic dipole tensor (the optical activity tensor) and the tensor Ae which results from the double contraction of the third rank electric dipole-electric quadrupole tensor Ae with the third rank antisymmetric unit tensor s of Levi-Civita. The electronic property tensors have the form ... 

In view of Equation (2.133), we have specified in Equations (2.139)-(2.143) the isotropic and anisotropic part only for af, . As its antisymmetric part vanishes outside resonance, antisymmetric invariants do not occur in ordinary Raman and ROA scattering. The expressions for the tensors Vnfj have been kept general and the symmetric nature of ae has not been used to simplify them. We further note that the tensor Af t does not give rise to an isotropic invariant as it is traceless because Ae is symmetric in the second and third indices. 

Both formal analysis and computational developments associated with DFT can be carried over intact to nDFT. For example, the exact two-particle ground-state density, no(x), can be determined through a constrained search [34] for that many-particle, properly symmetrized or antisymmetrized wave function, with symmetry imposed with respect to ordinary particles, which yields n0 and also minimizes the many-particle energy, T + Vpp, where Vpp denotes the interparticle interaction in two-particle space. Essentially any method developed within a single-particle application of DFT for the study of electronic structure can, with appropriate technical modifications, be extended to two-, or rc-particle states. The use of multiple-scattering theory to calculate fully correlated two-particle densities in solids will be given in a future publication. 

Consider first the nature of the d,(y) that enter Eq. (3.79), prior to averaging over scattering angles. We denote this as dq(ij k), where k is the scattering n. Since 3) is symmetric and E2) is antisymmetric, and adopting the... 

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