5.1 Introduction

• There is only one alternative to being right or wrong.
• It is possible that a model is right, but irrelevant.

• Natural optical activity in the electronic spectrum can be seen in the absence of an external influence such as a static magnetic field.

• The HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax HairMax We only talk about the most important situations in the field, such as complete isotropy, as in a liquid or solution, and isotropy in the plane.
• Light propagation in the direction of a static field applied to an isotropic medium is specified in the language of crystal optics.

• The refringent scattering approach was used to derive expressions for natural optical rotation and circular dichroism.

• The quadrupole contributions average zero.

• It is easy to see that the combinations of components specified in (5.2.1) are independent of the choice of origin.
• The difference in one contribution on moving the origin is canceled by the difference in the other.
• The analysis of optical rotation or circular dichroism data on oriented systems can be quite wrong if only the electric dipole-magnetic dipole contribution is considered.

• The refringent scattering approach provides the most complete description of optical rotation and circular dichroism, but it is less well known than the description in terms of circular differential refraction.
• The basic equations are derived using a more conventional approach.
• Buckingham and Dunn gave a derivation.

• The equivalent expression derived in the theory of crystal optics can be used to account for optical activity.

• There are more developments of the circular differential refraction ap proach to natural optical rotation and circular dichroism in the books of Theron and Cloete and Kaminsky.

• It should be translated into experimental units for applications.

• The ratio of the circular dichroism to the absorp tion is given by the dissymmetry factor.

• The point groups are referred to as the chiral point groups.

• The sum is over all the states, not just the electronic ones.

• Both isotropic and oriented samples tend to have zero optical rotation at very low and high frequencies.
• This behavior follows 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 The high frequency behavior is related to the sum rules.

• There is another version of the Condon sum rule that follows the relations outlined in Section 2.6.4.

• Since optical rotation and circular dichroism are determined by the dispersive and absorptive parts of the optical activity, we can write Kramers-Kronig relations directly for them.

• Moscowitz developed the application of Kramers-Kronig relations to optical rotation and dichroism.

• The stimulation by the light wave of electric quadrupole moments within the molecule which mutually interfere is the essential feature of any source of natural optical activity.
• Knowledge of the ground and excited state wavefunctions is required for quantum chemical computations of natural optical activity observables.
• Accurate determination of the wavefunctions is still a difficult problem, and we refer to Koslowski, Sreerama and Woody (2000) for an account of such calculations.

• All groups within a molecule are inherently achiral and there is no electron exchange between them.
• Any optical activity is assumed to arise from the changes in the electronic states of the group.
• The models provide rules which relate the signs and magnitudes of rotatory dispersion and circular dichroism bands to stereochemical and structural features.
• A model that provides the correct sign for the specific rotation and a magnitude comparable in accuracy is useful.

• There are two types of model.

• A molecule is a system of units that are fixed relative to one another.
• Each of these units have the property of assuming an idiosyncrasy under the action of an electric field.
• When a beam of light is incident upon a molecule, it causes the components to become polarized.
• The field of force produced by each of these units acts upon each of the other units.
• The combined influence of the applied external field and the fields created by all the other units of the molecule determines the polarization of each unit.
• The state of one unit of a molecule can be influenced by the state of other units of the same molecule.

• The static and dynamic models can make similar contributions in the same molecule, and there can be higher order terms involving simultaneous static and dynamic perturbations.

• Cotton effects at absorbing frequencies in any chiral molecule can be applied to optical rotation at transparent frequencies, but they have been most successful in situations where Cotton effects are induced in electronic transitions of a single achiral chromophore.
• In such a situation, the perturbing fields at the chromophore often originate in one of the several other groups in the molecule, and so the problem can be reduced to considerations of a simple two-group structure comprising a chromophore and a.

• Section 4.4.4 outlines general methods for this.
• The irreducible representations spanned by components of the electric dipole, magnetic dipole and electric quadrupole moment operators can be read directly from character tables.

• The electric and magnetic dipoles can be used.

• The electric and magnetic dipoles can be parallel.

• The reason given in Section 4.5.7 for not using the algebra of chirality functions is that it is restricted in its present form to ligands, and most of the results developed below specify anisotropic group properties.

• For a detailed account of the degenerate coupled oscillator model and its application to some typical chiral molecules, see also Rodger andNorden.

• The optical activity generated by two groups 1 and 2 is considered to be a chiral structural unit.
• The optical activity is assumed to arise from static fields from other groups.

• The optical activity of the two-group structure is determined by the chromophore transition between appropriate initial and final electronic states of 1 perturbed by the static fields from 2.

• The interactions of individual groups with the chromophore are summed in molecule containing more than two groups.
• Since the free atoms are uncharged and nondipolar, any associated fields in the molecule originate in effects such as incomplete shielding of nuclear charges at short distances.
• The effects are usually small, and the optical activity of such molecule is likely determined by dynamiccoupling, as discussed in the next section.

• The molecule is divided into a set of atoms or groups such as a carbon atom and its four substituent groups.
• The sums of the moments in individual groups referred to as local group origins, together with additional contributions from the origin- dependent moments, are referred to as the oscillating multipole moments.
• Both the direct influence of the radiation field on individual groups and the secondary fields arise from moments generated in other groups.

• The optical activity tensors can be obtained via the electric dipole moment, but the calculation is more complicated.

• The optical activity generated by two neutral groups is considered first.
• The differentials are performed when the optical activity tensors are calculated.
• The local origin on 1 is the origin of the group multipole moments.
• All observables obtained are independent of this choice of origin.

• Section 2.4.5 contains three different expressions for the electric and magnetic fields arising from multipole moments, depending on the distance and wavelength.

• There is no contribution to the electric field from the magnetic dipole moment in this approximation.
• The separation of charges within a group is not the same as the separation of charges within a group that is not in a group.

• The result in terms of group polarizabilities was achieved by introducing the dynamic dipole-dipolecoupling as a perturbation of the electronic wavefunctions of the two groups.

• The choice of local origins within the two groups affects the term Kirkwood.

• When specified components are used in expressions for observables, the results are origin invariant.

• There are equal and opposite terms that cancel and identical terms that reinforce when it comes to the Kirkwood contributions to the optical rotation mechanisms.

• The term can be given a tractable form if the two groups have threefold or higher rotation axes.

• The form of the second part depends on the symmetry of the groups.

• If one of the two groups is isotropically polarizable, the Kirkwood term does not contribute to optical rotation in an isotropic sample; likewise if the symmetry axes of the two groups lie in the same plane, both situations correspond.
• The dynamic results can be extended to more than one group in a molecule.
• If a molecule contains three groups, at least one of them must be anisotropically polarizable.

• In the Born-Boys model, the light wave is transmitted from the first group to the other three groups.
• The sum of the moments at each group by a wave that has suffered sequential scattering from each of the other three groups is the induced magnetic dipole moment of the complete system.

• The Born- Boys contribution to optical rotation for light propagating in an oriented sample can now be written down immediately.

• If each pair of atoms is considered to be an anisotropically polarizable group, all the other terms correspond to dynamiccoupling between pairs of polarizable groups with their symmetry axes lying in.
• In order for optical rotation to be generated in an isotropic sample, dynamiccoupling must extend over all four atoms.

• The Born-Boys model would not provide the lowest order contribution to optical rotation because the C-X bonds are anisotropically polarizable groups.
• A bond and a pair of atoms can be combined to generate optical rotation.
• For details of how complicated general formulae can be used in numerical calculations, we refer to Applequist.

• The lowest order mechanism that can generate optical rotation is the one with the least number of deflections.
• The picture is oversimplified because the waves scattered from a large number of molecules must first be combined into a net plane wave moving in the forward direction before interfering with the unscattered wave.
• The basis of a treatment of optical activity is based on pictures.

• The perturbation of the group 1 chromophore is simply described by the field from group 2.

• When the Kirkwood contribution is not symmetry forbidden, we have included terms of higher order.

• There is a similarity between the dispersion contribution to intermolecular forces and the dynamiccoupling mechanism.
• This contribution is considered to be a dispersion mechanism at visible and ultraviolet frequencies, but a dynamiccoupling mechanism at infrared frequencies.

• We have used wave functions on two or more separate groups of a structure.

• The interaction Hamiltonian is equivalent to the interaction energy between two charge distributions.

• The symmetric state has more energy.

• The terms relating to the strengths of the two groups have been dropped.

• Absorptive lineshape functions can be introduced.

• There are two opposite signs.

• The opposite absolute signs can be obtained when the two factors have the same absolute signs.

• The exciton treatment falls within the model since the exci ton splitting is caused by an interaction between the electric dipole moments of the states excited by the light wave.
• In the limiting case of frequencies larger than the linewidth, the exciton treatment is appropriate.

• The geometry of a two-group structure is being defined.

• The optical rotation of the structures can be deduced.

• When viewed towards the light source, 12 leads to a positive angle of rotation.

• The optical rotation of the structures can be deduced.

• The isotropic result is recovered if this is averaged over all orientations.

• The results are obtained when the model is applied.

• There are two factors that have opposite signs.
• The chirality factor is zero when the two transition moments are parallel and a maximum when they are perpendicular.

• Since the electronic absorption band is weak, there is not much light available for measurement, but the associated Cotton effects can be large.
• There is a large body of experimental data from which symmetry rules have been deduced, and this allows the relative importance of the static and dynamiccoupling mechanisms to be assessed.

• It is a singlet.
• A singlet and a triplet are generated.

• The electronic transitions of the carbonyl chromophore.
• The mid point of the CO bond is the origin of the coordinate system.

• For example, King, 1964.

• Section 5.5.3 shows that the rotational strength associated with a vibronic transition can require a different symmetry rule, and that there is a different sign from the ground electronic state to the ground state of the vibronic transition.

• This requires a different environment.

• The perturber makes contributions to the strength.
• The charge could come from an ionic atom or group, or it could come from incomplete shielding of the nuclear charges in a neutral atom or group.

• It is difficult to assess the relative importance of other excited states.

• If the perturber is neutral and spherical with no electric multipole moments, only dynamiccoupling contributes to the strength.

• If the perturber is dipolar, static and dynamiccoupling contribute to the strength.
• A perturber could be a bond, a group of electrons, or a nitrogen atom.

• One part of this more general expression is the contribution from a neutral spherical perturber.
• Information about the orientation of the dipolar anisotropic perturber is required by all the other terms.

• The signs of the transition moments and the polarizability anisotropies are some of the factors that determine the absolute signs of the various terms.
• For a detailed discussion of the absolute signs within this model, we refer to H"ohn and Weigang and Buckingham and Stiles.

• There are many contributions to the carbonyl chromophore that give conflicting symmetry rules.
• Selecting the term which will dominate in a particular molecule and predicting the symmetry rule is hazardous and even more so if you want to deduce the absolute configuration from the dominant term.
• If vibronic structure components of opposite sign are present, these problems are compounded.
• It is possible that a neutral isotropically polarizable perturber is the dominant term in most molecules.
• When the perturber is a fluorine atom, there is an antioctant rule.
• The usual perturbers (alkyl groups, hydrogen and halogen atoms, etc.)
• are polarizable.
• 2 of the perturber could cause the antioctant behavior.
• Further details of the rule can be found in two books.

• The listings in Tables 4.2 give the allowed components of polar and axial property in the important point groups, which correspond to the rotational strength expressions deduced above.

• In transition metal complexes, the central metal ion can be stimulated by a chiral arrangement of ligands.

• The first clear example of electric dipole-electric quadrupole optical activity was provided by the 3 axis of the ion of uniaxial crystals.
• The detailed electronic mechanisms are complicated so we will confine the discussion to generalities.

• Most of the observed intensity comes from the vibronic electric dipole allowed transitions.
• The structure of the circular dichroism bands can be generated by vibronic transitions.

• The strength is determined by the allowed magnetic and electric quadrupole transition moments and the small electric dipole transition moment, which are caused by the chiral environment.

• The absorption band is associated with strong magnetic dipole-weak electric dipole circular dichroism, and the band is associated with strong electric quadrupole-weak electric diole circular dichroism.
• The transition matrix elements can be developed explicitly using the irreducible tensor methods outlined in Section 4.4.6.

• The circular dichroisms should be of opposite sign.

• The dichroism is increased by an order of magnitude.

• Adapted from McCaffery and Mason.

• The Rh complex has larger rotational strength.

• The first symmetry allowed high order contributions to the strength.
• Mason and Richardson reviewed the theory of electronic optical activity in transition metal complexes.

• The model for discussing the generation of natural optical activity within molecules with a finite helical structure is provided by the severely overcrowded hydrocarbon hexahelicene.

• We will calculate the optical rotation at transparent wavelength by considering the dynamic correlation between all 15 pairs of benzenoid rings.
• The theory assumes that the electronic transitions are in the groups.
• A good answer is obtained for the specific rotation, and the correct absolute configuration is deduced.
• It is possible that electron delocalization could be incorporated into the approach by summing contributions from appropriately weighted bond structures.

• Fitts and Kirkwood first came up with the equation in 1955.

• The helix optical activity, at least in the form of circular dichroism, is generally found to have opposite signs for light propagating parallel to the helix axis.

• The three components have not yet been isolated.

• The discussion of the generation of natural electronic optical activity within chiral moleculues has been focused on allowing contributions to the rotational strength.
• This depends on the electronic chirality when the nuclei are at their equilibrium positions in the ground electronic state, and reflects the molecular chirality which might be correlated with the sign and magnitude of the rotational strength by a symmetry rule.

• There is an additional contribution to the strength because the elec tronic chirality changes as the nucleus undergoes motion.
• The magnitude of these vibronic contributions is less direct.

• We will look at this topic by considering the ground and excited electronic states in the quantum mechanical expression for the isotropic rotational strength.

• The terms for overtone and combination transitions are no longer used.
• A series of single quantum vibronic bands separated by their respective fundamental frequencies form the circular dichroism spectrum.

• When delocalized over a completely asymmetric structure, all transition moments are fully allowed, and all normal modes are completely symmetric.

• The potential energy surfaces of the ground and excited electronic states are different.
• The progressions and combination vibronic bands can now be formed with the first members.
• The second term of the general vibronic rotational strength can now contribute.
• The third term can contribute to the strength of the vibronic transition to a single symmetric mode in a state of excitation associated with an odd number of quanta.
• The combination of an odd quanta non totally symmetric mode with both even and odd quanta totally symmetric modes is often observed as a single quantum of a non totally symmetric mode with a totally symmetric progression.

• The first term shows that the sum of the individual vibronic rotational strengths is equal to the rotational strength for the 0-0 transition.
• The disappearance of the second term on performing the summation indicates that it makes no net contribution to the integrated rotational strength, and that its contribution to a particular vibronic rotational strength could equally well be positive or negative, regardless of the sign of the zero-order rotational strength.

• The carbonyl chromophore is illustrated with some general remarks.

• We consider the application of the third term to describe the generation of vibronic circular dichroism.

• vibronic mixing of the ground with excited electronic states can often be overlooked compared to mixing excited state with other excited states.

• Section 5.4.1 has symmetry rules that were derived for the generation of optical activity in the carbonyl chromophore through the use of a chiral environment.

• Adapted from a movie.

• The appropriate vibronic mixing is allowed at the lowest.
• The equation predicts an octant rule.

• The hypersensitivity of the vibronic structure of a circular dichroism band to the solvent medium is the reason why the detailed application of such expressions is not attempted here.
• A negative 'allowed' progression of the 1200 cm-1 carbonyl stretching mode in the excited electronic state is complemented by a positive 'forbidden' band system based on the same 1200 cm-1 totally symmetric progression.

• One explanation is that the excited electronic state carbonyl stretching mode enhances progressions based on corresponding bending modes.

• I have been able to illuminate a magnetic line and magnetize a ray of light.

• The visible and near ultraviolet optical rotation and circular dichroism that all molecule show in a static magnetic field is the focus of this chapter.
• This chapter deals with a liquid or solution sample in a static magnetic field, which constitutes a uniaxial medium for light propagating along the field direction.

• The form of magnetic optical rotation and circular dichroism developed below is based on an article by Buckingham and Stephen.
• Although the correct quantum mechanical description had been given by Serber, it was the Buckingham- Stephens work that started a new era in magnetic optical activity.

• The difference is that magnetochiral phenomena are supported only by the same molecule.

• The refringent scattering approach was used to derive expressions for magnetic optical rotation and circular dichroism.

• The nonzero spatial averages are generated by the component of the magnetic field in the direction of the light beam.

• The above equations can only be valid if the removal of the degeneracy by the magnetic field is not resolved.

• There is a more detailed discussion of the delicate problem of lineshapes in magnetic optical rotation and circular dichroism.

• The reduced electric dipole moment matrix elements cancel out, leaving simple factors that can be compared with ratios from measured circular dichroism.

• Section 1.3 states that the Zeeman and Faraday effects are related.
• Only when the ground or excited state is a member of a degenerate set can the matrix elements of the magnetic dipole moment operator be diagonal.
• When the ground state is in a set, the term is nonzero.
• The term ground state orbital degeneracy is complicated by Jahn-Teller effects.
• Terminating a molecule with an odd number of electrons is important.
• The term involves only the off-diagonal matrix elements of the magnetic dipole moment operator.

• Adapted from Buckingham and Stephens.

• They generate a circular motion of charge when they are out of phase.
• The magnetic field results in an incomplete cancellation that results in optical rotation and circular dichroism lineshapes.

• The transition resulted in incomplete cancellation of the magnetic optical rotation and circular dichroism lineshapes.

• The term comes from the calculation.

• Adapted from Buckingham and Stephens.

• Two electric dipole transition moments and one magnetic dipole transition moment are involved.
• The two different states being connected by a magnetic dipole interaction are just one state in common.
• In the case of metal-free porphyrins, the states that are coupled by the magnetic field can be correlated with components of the other states in equivalent molecule of higher symmetry.

• The magnetic optical rotation and circular dichroism curves are associated with diamagnetic samples.
• Since paramagnetic samples require a ground state magnetic moment, and generate the magnetic optical rotation and circular dichroism curves associated historically with paramagnetic samples, term can only exist in paramagnetic samples.

• A simple treatment involving one electron promotion from the highest filled to the lowest empty molecular orbitals provides a description for our purposes.

• The Soret band is an order of magnitude stronger.

• The visible and near ultraviolet spectrum of metal porphyrins is generated by the absorption of the one electron transitions.

• The occurrence is at about 540 nm.

• The reduced magnetic moment matrix element will not be evaluated.

• For more detailed theoretical discussions of the effects of Faraday on porphyrins, we refer to McHugh, Gouterman and Weiss.

• There is no showing of spin-orbit splitting.

• Magnetic moments disappear from the ground state.
• The spin quantum numbers are dropped from the states and we look at the contribution to the ground state magnetic moment.

• The term is usually straightforward since the chromophore symmetries must be high to support essential degeneracies.

• Since all groups within a molecule show intrinsic magnetic optical activity, the influence of static and dynamic perturbations from other groups is expected to be less significant than in natural optical activity.
• The magnetic optical activity associated with electronic transitions for which all components are forbidden is a notable exception.
• All components of the transition are forbidden by the electric dipole and are being gained through structural and vibrational changes.

• The term is the second order in any perturbation that can cause electric dipole strength.

• The natural optical activity expressions are barely tractable.
• Dynamiccoupling between a perturbed group and an unperturbed chromophore can provide a contribution to the magnetic optical activity of the perturbed group.

• The term is erated entirely by resonances.

• Since it is difficult to predict the absolute signs, the signs observed in formaldehyde are used as a basis for the analysis of the magnetic circular dichroism spectrum of other carbonyl systems.
• There are two modes on the magnetic circular dichroism spectrum.

• The rigidity of the adamantanone skeleton causes 1 bending.
• The weaker negative band is generated by 1 vibration.

• The analysis becomes very complicated when these structural perturbations are compared with the vibronic perturbations.

• This is a good place to start developing quantum mechanical expressions for dichroism and magnetochiral birefringence.
• The resulting expressions show how the interplay of magnetism and chirality creates subtle phenomena.
• A quantum statistical average is used to replace a classical Boltzmann average in order to rewrite the classical expression for magnetochiral birefringence.

• The treatment of the Faraday effect is given in Section 6.2.1.

• The application of symmetry arguments shows that the magnetochiral effect can only be supported by certain types of molecule.

• The same conclusion can be shown to be followed by a consideration of the other specified components.

• Population differences between components of the split ground level are what start the term.

• Since the ground or excited state is a member of a degenerate set, only the matrix elements of the magnetic dipole moment operator can be diagonal.
• A threefold or higher rotation axis is needed for excited state orbital degeneracy to be possible.
• The term involves only the off-diagonal matrix elements of the magnetic dipole moment operator.
• The number of electrons in a molecule will be important.

• There are no model examples of how the effects are generated, and few experimental observations of magnetochiral birefringence and dichroism have been reported to date.
• The generation of natural optical activity and considerations of magnetic structure will need to be considered together.

• The newer topic of optical activity is related to transitions between the levels of chiral molecule.
• There are two different methods for obtaining a vibrational spectrum.
• We will be concerned with all the manifestations of optical activity in the spectrum.
• The form of optical rotation and circular dichroism is described in Section 1.5.

• The variation of these tensors with the normal coordinates of vibration is what brings about the transitions.

• 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.