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5 -- Part 2: .1 Introduction

- There is no account of the theory of molecular vibrations in this place since it is covered in many texts and needs no reformulation in order to cope with optical activity.
- We refer to Wilson, Decius and Cross.

- At the time of writing the first edition of this book, the theories of the two different types of optical activity were in a state of change.
- In the intervening years, a lot of progress has been made.
- The framework for accurate Ab initio calculations of circular dichroism is a triumph of quantum chemistry.
- There are general surveys of the theory of optical activity in Polavarapu's book and in reviews by Buckingham and Nafie.

- The general aspects of natural optical rotation and circular dichroism are the same as for the electronic case.

- There are several ways in which these expressions can be developed, of which three will be considered below.
- The magnetic dipole moment operator has zero expectation values in nondegenerate electronic states, soExplicit consideration of the electronic quantum states exposes a subtle problem.
- The problem may be solved by going beyond the adiabatic approximation and considering the dependence of the electronic wavefunction on the nuclear velocities as well as the nuclear positions.

- Since the two cases are similar, the size of the observables can be smaller.

- The atoms are taken to be the ultimate particles with residual charges determined by the equilibrium electronic distribution of the molecule.

- The fixed partial charge model can be used with normal co ordinates.

- The atomic displacement coordinates encompass rotation and translations of the molecule.

- A normal coor dinate analysis of the molecule is required for the application of these fixed partial charge expressions.
- A set of fixed partial charges is required.
- For an example of a calculation, we refer to Keiderling and Stephens.
- The fixed partial charge model at this level of approximation consistently gives a smaller than actual rotational strength, sometimes of the wrong sign.
- Charge redistribution can be done during the excursions away from the equilibrium configuration.

- The choice of internal coordinates excludes rotation and translations.
- matrix elements are determined from a coordinate analysis

- The adiabatic approximation is used to calculate the intensities within the bond dipole model.

- In Sverdlov, Kovner and Krainov, there is a detailed account of the bond dipole theory.

- The magnetic dipole moment operator is time odd and therefore has zero expectation values in nondegenerate electronic states.

- The magnetic dipole moment operator's time- odd character is now embodied in the operator.

- The inclusion of the origin dependent part of each bond is a crucial step in the development of the bond dipole theory.

- The time between the local bond and the group origin is dependent on the number of atoms in the molecule.

- This expression is used together with.

- The last term deals with the product of electric and magnetic dipole moment derivatives.
- Simple examples of the first two contributions are given later.

- Any shift of the group origins along the dipole axes are invariant to the two-group and inertial dipole terms taken together.
- It is left to the reader to verify that an origin shift along the dipole axis of a group causes a change in the two-group term.

- Since it is based on internal vibrational coordinates rather than atomic cartesian displacements, it immediately creates simple geometrical expressions for model structures and allows group optical activity approximations to be made.
- There is a computational version of the bond dipole theory that can be used.

- Accurate calculations of circular dichroism have been made possible by perturbation theories.
- The electronic contributions to the vibrational transition moments are derived using the vibronic coupling formalism described in Section 2.8.4.

- The wavefunctions are real for nondegenerate electronic states in the absence of a magnetic field.

- Similar expressions have been created.
- The required derivatives can be computed using modern methods.
- Testimony to the power of this formalism is the close agreement between the observed and calculated dichroism spectrum.

- There is a small difference in the intensity of the light in the right and left direction.
- General expressions for the optical activity observables were derived in Chapter 3.

- The geometry for light scattering.

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- The molecule is the source of the scattered light because of the electric and magnetic multipole moments in it.

- For scattering from fluids it is necessary to average these expressions over all orientations of the molecule.

- The same expressions can be used if the property tensors are replaced by corresponding transition tensors.

- Basic symmetry requirements can now be deduced.
- All of the active vibrations in a chiral molecule should show optical activity.

- There are several ways in which the further development of the optical activity expressions can proceed.
- The latter is more in keeping with the approach to electronic optical activity used in Chapter 5 in that it is based on bond or group properties.

- The first part of the calculations did not reach the high levels of accuracy that are now routine for calculations of circular dichroism, but they still proved valuable.
- The frequencies of most of the fundamental normal modes of vibration are too low to be accessible in a dichroism study.
- The calculation of high quality optical activity on hydrogen atoms has been achieved.

- Below is a detailed discussion of raman scattering.
- Although the basic results in the first part of this section can be applied to the resonance situation, the subject is still at an early stage of development.

- A lot of new Raman optical activity phenomena that do not arise in transparent scattering could be useful for the study of biomolecules in the ultraviolet region.

- There are problems with the standardization of instrumental factors that prevent absolute intensities from being measured in most light scattering work.

- The circular intensity difference is derived from the intensities measured on the same arbitary scale.

- The relative sum and difference intensities may be compared directly because common factors in the numerators and denominators of these circular intensity differences have not been canceled.
- The dependence of the circular intensity difference components on the scattering angle and the extract of the tensor invariants are discussed in the books of Hecht and Barron.

- There is an experimental configuration for circular intensity difference measurements.

- Section 3.5.4 states that the degree of circularity of the scattered light gives the same information as the circular intensity difference.

- We will not give a detailed analysis of raman optical activity.

- In the case of resonance scattering at absorbing incident wavelengths, there is an interesting Stokes-antiStokes asymmetry that arises.

- The circular intensity differences between the antiStokes and antiStokes degrees will be different.

- This involves intensity differences in the scattered light that is associated with the linear polarization states at +45* to the scattering plane in the incident or scattered radiation.

- It is clear that there are a lot of different experimental strategies that can be used to study the phenomenon.

- For most practical applications in chemistry and biochemistry measurement of either the simple incident or scattered circular polarization form of Raman optical activity provides all necessary information.
- A bond polarizability model shows that backscattering is the optimum experimental geometry for most routine applications of Raman optical activity since this provides the optimum signal-to-noise ratio.
- This discovery was crucial in the extension of the measurement of optical activity to biomolecules.

- We will pause and reflect on the relationship between the fundamental scattering mechanisms responsible for conventional optical rotation and circular dichroism on the one hand, and the other on the other.

- Chapter 5 gave us physical insight into conventional optical rotation and circular dichroism.
- Birefringence phenomena are caused by interference between transmitted and forward-scattered waves.

- It can make higher-order contributions.

- This picture can be extended to a different structure.
- Any explicit calculation would sum over all permutations.

- The mechanisms just described apply to the activity of the ras.
- It will be seen soon that different mechanisms can dominate in certain modes of vibration.

- The optical activity generated by a chiral molecule consisting of two neutral achiral groups 1 and 2 is now considered in detail.

- We assume no electron exchange between the groups, and write the polarizability and optical activity tensors of the molecule as sums of the corresponding group tensors.
- The local origin on group 1 is referred to as a fixed origin within the molecule.

- 1 is the origin and 2 is the destination.
- The methods of Chapter 5 can be used to develop the terms.

- The equations can be given a tractable form if both groups have threefold or higher rotation axes.

- This conclusion can be reached by simply using the word.

- There are combinations of polarizability-polarizability products required.

- Increasing separation of the two groups leads to increased raysy optical activity.
- The corresponding Kirkwood optical rotation decreases with increasing separation.

- A simple example of a two-group structure with symmetry axes is provided by a twisted biphenyl.
- 2 are along the sixfold rotation axes of the aromatic rings and we ignore the fact that the ring substituents required to constrain the biphenyl to a chiral conformation destroy the axial symmetry of the aromatic rings.
- The circular intensity difference is at least an order of magnitude larger than the polarized one.
- The estimates only apply to gaseous samples.
- The isotropic contribution is suppressed more than the anisotropic in liquids.

- Stone was used for an extension of the calculation to a two group structure.

- In visible light, 21 l is satisfied for most molecules.

- The geometry for scattering by a two-group molecule.

- The intensity components at the detector arise from waves that are independent of each other and from the incident light wave.
- If the two groups have threefold or higher proper rotation axes, the results are valid.

- There is no phase difference between the forward-scattered waves from the two groups, so there is no Rayleigh optical activity generated by the two-group model.

- In the Kirkwood model of optical rotation, dynamiccoupling between the groups is required to generate Rayleigh optical activity in the near-forward direction.

- The two-group model has been reexamined and provided a critical assessment from the view point.

- Placzek's approximation is the starting point for the bond polarizability model and the atom dipole interaction model.

- The second and third terms describe fundamental and first overtone and combination transitions.

- The intensity is determined by the variation of the polarizability tensor with a normal coordinate of vibration and the local internal coordinates.

- The extension to Raman optical activity involves writing the optical activity tensors as sums of corresponding bond tensors, taking care to include the origin dependent parts.

- The last terms include products of intrinsic group polarizability.
- Simple examples of different contributions are given soon.

- The question of the actual choice of origins within the groups or bonds arises when applying these optical activity expressions.

- The argument begins.

- Pure symmetric transition polarizabilities are implied by the bond polariz ability theory.
- There are exotic situations which can lead to antisymmetric transition polarizabilities.

- Their product is zero.

- The local group origins along the symmetry axes are invariant to the displacements of the two terms taken together.

- Similar results can be obtained within the bond polarizability model of Raman optical activity for a molecule composed entirely of idealized achiral groups or bonds.
- There are some valuable simplifications of the general optical activity expressions.

- A measure of the breakdown of the bond polarizability model can be found by deviating from this factor of two.

- It is easy to understand why there is no Rayleigh or Raman optical activity in the forward direction because the two waves scattered independently from the two groups have covered the same optical path distance.
- Compared with 90* scattering, the optical activity intensity is four times greater in backscattering.

- The bond polarizability model of Raman optical activity is based on a decomposition of the molecule into bonds or groups.
- If a normal coordinate analysis and a set of bond dipole and bond polarizability parameters are used, the optical activity associated with every normal mode of vibration of a chiral molecule may be calculated.
- However, due to the approximations inherent in these models, such calculations do not reproduce experimental data at all.
- In this section, we show how the two models can be applied to idealized normal modes containing just one or two internal coordiantes, of some simple chiral molecular structures.

- At high and low frequencies, the R value is the same.
- The Raman approach to optical activity uses visible exciting light, so it has a natural advantage over the infrared approach.
- The experiment is more favorable if it is over 200 cm.

- Since the structure has a twofold proper rotation axis, pairs of equivalent internal coordinates associated with the two groups will always contribute with equal weight to normal modes.

- The geometry of the two-group structure is what makes these expressions pleasing.

- These derivatives are difficult to evaluate, so empirical values are often used.

- We need only evaluate the two-group terms if we assume the connecting bond is rigid and the local group origins are the points where it joins.

- To calculate the optical activity, we need to use the group polarizability tensors of the form.

- The electric dipole moment, polarizability and optical activity of the methyl group do not change in the course of the torsion vibration.
- The rest of the molecule needs to know the origin of any optical activity.

- There is only one mechanism considered here.

- The evaluation of the insturment term is simplified if the axis of the molecule is a principal axis.

- The model is based on a single-bladed propellor.

- The internal ro tation problem has been solved with two different molecule-fixed axes systems.
- The symmetry axis of the top is not related to any of the three principal axis of the molecule.
- The internal axis method takes one of the molecule-fixed axes to be parallel with the symmetry axis of the top.

- The internal torsion mode of vibration and the rotation of the whole molecule are separated.

- The first and second terms give the energy from the complete molecule rotation and the internal rotation frozen.
- The complete Hamiltonian for rotation about the torsion axis is obtained by adding to a potential energy term.

- The instantaneous orienta tions of the two groups are relative to a principal internal axis and stationary during the torsion vibration.

- The hindered single-bladed propellor has no two-group contributions to the optical activity.

- The details of the calculation will be shown.
- The calculation is similar but simpler.

- The dependence on the molecular geometry is the same as the dependence on the circular intensity difference components.
- There are differences between the two methods of measuring optical activity.
- In the far IR, there are mazy torsions, which are well beyond the range currently accessible to circular dichroism instruments.

- The same results would be obtained if they were driving the singlebladed propellor.
- The effects are only likely to be observed with the corresponding frequencies in the accessible region of the Raman spectrum.
- The above treatment would need to be extended because of the low symmetry of the group.

- A single methyl group with its three fold axis lying along a principal intertial axis is rare.
- There is an extension to a more common situation.
- Symmetric and antisymmetric combinations of the two methyl torsions are contained in a molecule containing two adjacent groups.

- The associated optical activity is easily calculated with the 2 axis.
- The calculation on the 2 axis provides good agreement with the data from the experiment.

- Since the threefold axis of the methyl group is no longer a principal axis, the extension of the theory to a completely asymmetric molecule is difficult.
- The assignments of bands to pure methyl torsions are not expected because of the mix of low-wavenumber modes in large completely asymmetric molecules.

- The recording was made in the author's laboratory.
- The intensities are not defined, but they are significant.

- Significant contributions to the Raman optical activity can be made by achiral groups with symmetry lower than axial.

- There is a possible example of the carbonyl group in a molecule.

- The term is further developed by considering an idealized model of the carbonyl deformations.

- The circular intensity differences could be calculated from a normal coordinate analysis.

- Since a detailed consideration of the relative disposition of the two groups is required, we will not develop this contribution explicitly.

- Due to the complexity of the normal modes and the likely presence of more than one conformer, modern ab initio methods are required for reliable assignments and quantitative analysis of the optical activity of the molecule.

- The intensities are not defined, but they are significant.

- For a detailed study of optical activity in perturbed modes, we refer to Nafie, Polavarapu and Diem.

- It is possible to understand qualitatively how this isotropic Raman optical activity may be generated by considering the symmetry aspects of the optical activity.
- The 716 cm-1 Raman band is made up of a significant contribution from the methylene twist and the 765 cm-1 band is made up of a pinane-type skeletal vibration.

- The theory of natural electronic optical activity in Chapter 5 has not been invoked by the models discussed so far.
- The atom, bond or group electric and magnetic moments were taken to relate to the atom, bond or group unperturbed by nonbonded interactions with the rest of the molecule.
- We discussed briefly how electronic and vibrational interactions with the rest of the molecule can contribute to the optical activity.
- The normal modes of vibration can be determined by the set of force constants.

- One possible example of this mechanism has been identified by Barnett, Drake and Mason in 1980.

- The -NH2 group and the perturbing naphthyl group are in Groups 1 and 2.

- The static polarizability of the naphthyl group can be taken as the corresponding static polarizability.

- Adding to the group or bond electric and magnetic dipole moments in each term contribution can be used to formally incorporate electroniccoupling within the bond dipole model.
- The changes in the course of a normal mode excursion would be caused by functions of group or bond internal coordinates.
- Contributions to group or bond polarizability and optical activity tensors would be added to each term.
- In Chapter 5, the machinery for writing down explicit expressions for bond moments has been given.
- We won't write out the generalized bond dipole and bond polarizability optical activity expressions because of their complexity.

- There are small contributions from the C-N stretch and N-H deformation coordinates in the amide I mode.
- The sheet leads to strong dipolar interactions between the C and O groups.
- It is manifest as a mixing of the degenerate and near degenerate excited state wavefunctions to form delocalized excited states similar to the exciton states formed from excited electronic states.
- The inclusion of dipole-dipole interactions into computations of the normal modes of vibration of peptides was pioneered by Krimm, who treated these dipolar interactions as a set of additional force constants.

- The chapter ends with a brief account of applications of Raman optical activity in biomolecular science, which is very promising.
- The study of biomolecules can be done with more accuracy with the use of raman optical activity.
- Even though the model theories and current ab initio computational methods described above are hopelessly inadequate for Raman optical activity calculations on structures the size and complexity of biomolecules, their experimental Raman optical activity spectra have nonetheless proved rich and transparent with regard to valuable information about structure and behaviour.

- The normal modes of vibration of biomolecules can be very complex with contributions from the side chains.
- Since the largest signals are often associated with the most rigid and chiral parts of the structure, raman optical activity is able to cut through the complexity of the corresponding vibrational spectrum.
- The optical activity band patterns characteristic of the backbone are caused by these.

- Carbohydrate Raman optical activity is dominated by signals from the bones, and this time it is centred on the sugar rings and the connecting links.
- The nucleic acids' optical activity is dominated by bands from the bases with respect to each other and the sugar rings.

- The first X-ray crystallography determination of the structure and behavior of proteins was made in the late 1950s by M. F. Perutz and J. C. Kendrew.

- Each fold type has its own optical activity band patterns.
- It is possible to determine structural information by comparing the optical activity spectrum of a unknown structure with a set of known structure.
- Valuable structural information is still available from experimental Raman optical activity data despite the lack of current theories for useful calculations.

- The large contributions to some of the normal modes of vibration can be seen with large and informative Raman optical activity but only weak dichroism.
- A qualitative explanation for this may be provided by the results of Section 7.4.1 where it was shown that the bond polarizability Raman optical activity can be large.

- MOLSCRIPT diagrams are represented by the -sandwich fold.
- The top pair of the emptyprotein capsid and the middle pair of the intactprotein capsid were measured in the solution.
- The recording was made in the author's laboratory.
- The intensities are not defined, but they are significant.

- This is a type of plant virus.
- It is possible to separate virus preparations into empty capsids, capsids containingRNA1 and capsids containingRNA-2.

- The middle panel shows the spectrum of the intact capsid containingRNA-2 with bands from the nucleic acid now visible.
- The bottom panel shows the results of subtracting the top and middle spectrums.
- The difference ROA spectrum looks very similar to those of synthetic and naturalRNAs and is therefore taken as coming mainly from the viralRNA: the details reflect the single-stranded A-type helical conformation of the RNA-2 packaged in the core together with its interactions with the coat proteins.

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