Polarizability theory

B) THE MICROSCOPIC HYPERPOLARIZABILITY IN TERMS OF THE LINEAR POLARIZABILITY THE KRAMERS-HEISENBERG EQUATION AND PLACZEK LINEAR POLARIZABILITY THEORY OF THE RAMAN EFFECT... 

Using a simple bond polarizability theory of ROA for the case of a molecule composed entirely of idealized axially symmetric bonds, the relationships j3(G )2 = j3 (A)2 and a G = 0 are found (Barron and... 

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]... 

These results apply specifically to Rayleigh, or elastic, scattering. For Raman, or inelastic, scattering the same basic CID expressions apply but with the molecular property tensors replaced by corresponding vibrational Raman transition tensors between the initial and final vibrational states nv and rn . In this way a s are replaced by (mv aap(Q) nv), where aQ/3(<3) s are effective polarizability and optical activity operators that depend parametrically on the normal vibrational coordinates Q such that, within the Placzek polarizability theory of the Raman effect [23], ROA intensity depends on products such as (daaf3 / dQ)0 dG af3 / dQ) and (daaf3 / dQ)0 eajS dAlSf / dQ)0. 

The tensors which enter theoretical expressions are transition tensors 7, for a transition between an initial state i and a final state /. The Placzek polarizability theory for vibrational Raman scattering [56], which we use here, is valid in the far from resonance limit, i and / are then vibrational states. If we assume that they differ for normal mode p, then the transition tensors can be written as... 

The separation into a vibrational and an electronic part is implied by the Placzek polarizability theory. The further analysis of vibrational motions has in the past typically been accomplished by calculating the vibrational energy distribution in valence coordinates. For the large-scale skeletal motions often important in ROA, and for relating Raman and ROA scattering cross-sections to the vibrational motions of structural parts of an entity, a different approach is needed. 

The form of Lx and V, Equation (2.144), makes it evident that the invariants If< of products of transition tensors can be written in the frame of the polarizability theory as sums over mono- and dinuclear terms ... 

Linear response polarizability theory of spectral bandshapes was applied to the numerical analysis of the chiroptical spectra obtained for DNA-acridine orange complexes [85]. After analysis of various models of conformation, it was concluded that a dimer-pairs repeating sequence model was best able to account for the observed spectral trends. In another work, the CD induced in the same band system was studied at several ionic strengths [86]. The spectra were able to be interpreted in terms of the long-axis-polarized electronic transitions of the dyes, with the induced CD being attributed to intercalated and non-intercalated dye species superimposed by degenerate vibronic exciton interactions between these. 

As mentioned above, the basic theory of the Raman effect was developed before its discovery. However, at this time numerical calculations of the intensity of Raman lines were impossible, because these require information on all eigenstates of a scattering system. Placzek (1934) introduced a semi-classical approach in the form of his polarizability theory. This provided a basis for many other theoretical and experimental studies. Physicists and chemists worldwide realized the importance of the Raman effect as a tool for qualitative and quantitative analysis and for the detennination of structure. 

As already pointed out, this description of the Raman effect is based on the polarizability theory (Placzek, 1934) which is valid in a good approximation if the exciting frequency is much higher than the frequency of the vibrational transition // , but lower than the frequency of the transition to the electronic excited state If, on the other hand, is approaching then resonances occur which considerably enhance the intensities of the Raman lines, i.e., the resonance Raman effect. This effect and its applications are described in Sec. 6.1 and also in Secs. 4.2 and 4.8. 

Two theoretical estimates of end effects in fact suggest that they are negligible for segments smaller than this. Zimm et al. (1959), using the polarizability theory of Fitts and Kirkwood (1956a), found that most of the rotation characteristic of the helix should be present with one complete... 

Huige and Hezemans179 180 have performed extensive molecular mechanics calculations using the consistent force-field method on various oligo- and polyisocyanides. The hexadecamer of ferf-butyl isocyanide was calculated to have a helical middle section and disordered ends. The dihedral angle N=C—C=N in the middle section was found to be 78.6°, and the number of repeat units per helical turn was 3.75. The latter number is in agreement with circular dichroism calculations using Tinoco s exciton theory (3.6—4.6) and De Voe s polarizability theory (3.81). The molecular mechanics calculations further predicted that the less bulky polymers 56 and 57 form helical polymers as well, whereas a disordered structure was calculated for poly(methyl isocyanide) (55). 

In the Raman case, three distinct general computational thedries have been proposed the bond polarizability theory, the atom dipole interaction theory and localized molecular orbital theories. In the first and third of these the normal modes of vibration, and hence the vibrational quantum states, must embrace a chiral nuclear framework. They are therefore analogous to the inherently chiral chromo-phore model of electronic optical activity in which the electronic states are delo-... 

The bond polarizability theory of conventional Raman intensity is well-established 46,47). The starting point is Placzek s approximation for the vibrational Raman transition polarizability at transparent frequencies48 . On expanding the effective polarizability operator aotp(Q) in the normal vibrational coordinates Qp, the transition polarizability becomes... 

The extension of the bond polarizability theory to ROA is based on the origin-dependence of G p and AaPr Thus using (2.6) the optical activity tensors of the molecule, written as sums of corresponding bond tensors, are... 

One advantage of the bond polarizability theory is that, since it is based on a decomposition of the molecule into bonds or groups that can support local internal vibrational coordinates, it can be applied to idealized normal modes containing just a few internal coordinates and so can provide conceptual models of the generation of ROA by some simple chiral structures. Indeed, as mentioned above, the bond polarizability theory actually developed out of a synthesis of the two-group model and the inertial model, both of which have been applied in detail to a number of simple chiral structures 3 5). 

Lopes PEM, Roux B, MacKereU AD. Molecular modeling and dynamics studies with explicit inclusion of electronic polarizability theory and applications. Theor Chem Aa. 2009 124(l-2) ll-28. http //dx.doi.Org/10.1007/s00214-009-0617-x. 

Recalling Eq. (4) that gives the Raman intensity expression for a molecule based on Placzek s polarizability theory, the following gives a more complete expression with regard to the instrumental and surface factors ... 

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