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4 -- Part 1: .1 Introduction

- There is a lot going on in this chapter.
- It collects together a number of disparate topics, all of which have some bearing on the application of symmetry arguments to molecular properties in general and optical activity.

- The application of symmetry principles can be applied to optical activity.

- The fundamental symmetries of space inversion, time reversal and even charge conjugation have something to say about optical activity at all levels: the experiments that show up optical activity observables, the objects generating these observables and the nature of the quantum states.
- Simplistic and evaluation of matrix elements using irreducible tensor methods is a topic of great importance in magnetic optical activity.
- The application of permutation symmetry to ligand sites on molecular skeletons is one of the topics set apart from the others.

- In this book, a lot of use is made of a cartesian tensor notation, and the symmetry aspects of various phenomena discussed in terms of the transformation properties of the corresponding tensors.
- There is a need for a review of the theory of cartesian tensors.

- It is assumed that there is a knowledge of elementary vector algebra.

- The components are resolved along three mutually perpendicular directions.

- The use of the following notation makes it much simpler to perform speaches.

- Greek letters are used for free or dummy suffixes, whereas Roman letters are used for specific components.

- It is more systematic to generalize the definition of a tensor than it is for physical quantities associated with two or more directions.
- A number unrelated to any axis is used to specify a tensor of rank zero.
- The first rank is specified by three numbers, each of which is associated with a coordinate axis.
- Nine numbers are associated with two coordinate axes for the second rank.
- Natural extensions of higher rank can be specified by 27 numbers, not a square array.
- In the case of a tensor of zero rank, the values of the numbers in the array will change as the coordinate axes are changed, but they are associated with both the axes and with the tensor quantity itself, which is a physical entity that retains its identity.
- An indication of the essential character of a particular tensor will be provided by a study of the relationships between components in one coordinate system and those in another.

- The contraction is the equivalent of the scalar product.

- The second-rank antisymmetric tensor has zero diagonal elements.
- The definition may be extended to tensors of higher rank, the symmetry or antisymmetry being defined with respect to a particular pair of suffixes.

- The construction of irreducible tensorial sets will be encountered later.

- The orientation of a rotating axis system is determined by the direction cosines.

- There are important relations between direction cosines.

- The sum of the squares of the direction cosines relating a particular axis in one coordinate system to the three axes in the other system are unity.

- Under a rotation of the axes, the components of a vector transform.

- This result shows the economy of the dummy suffix notation since the single equation represents nine equations, each with nine terms on the right hand side.

- Second-rank tensors should not be confused with direction cosines.

- This is the reason why we said that the two are considered to be rank zero and rank one, respectively.
- According to this definition, we can't describe a tensor without reference to a coordinate system, but we can distinguish it from other descriptions.
- There is no meaning to asking if a set of numbers is a tensor.
- When we are given a rule for obtaining the corresponding set of numbers in any other coordinate system, we can compare the rule with the one we were given before.

- It is necessary to distinguish between polar and axial tensors in order to generalize the transformation law.
- Section 1.9.2 shows that an axis of motion does not sign under space inversion.
- If we invert the coordinate axes, we will see that the components of a polar vector will change sign and the components of an an axis will not.

- The transformation laws are used to determine the effect of an inverse of the coordinate axes on a polar and an axon.

- A more fundamental description of the corresponding physical entity can be provided by an antisymmetric polar tensor of higher rank.

- All proper transformations between molecule-fixed and space-fixed axes are possible in a collection of freely rotating molecules.
- In the next section, we will see that the isotropic averages of components are always expressed in terms of isotropic tensors.

- The evaluation of isotropic averages of tensor components is a problem encountered frequently in the theory of light scattering.
- Since we want to relate the observable to the intrinsic properties of the molecule, we must transform a set of axes fixed to the molecule's frame.
- The expressions must be averaged over all orientations if the molecule is tumbling freely.

- A simple trigonometric analysis can be used to get the first few averages.

- It is important to note the average of certain trigonometric func tions over a sphere.

- Consider the isotropic average of a product.

- Consider the isotropic average of a product.

- The isotropic average is a product of four direction cosines.

- The isotropic averages of products of pairs of identical direction cosines can be obtained from the orthonormality relations.

- The isotropic average is zero.

- A product of five direction cosines is the last isotropic average required in this book.

- Any symmetry elements present in a molecule are associated with the principal axes.

- A rotation axis is always a principal axis and a reflection plane always has two of the principal axes.

- Consider a molecule with an axis of symmetry.

- The polarizability written in the form (4.
- 2.58) facilitates the development of theories of optical activity in the form of bonds and groups.

- The classification of quantum states and operators with respect to space inversion and time reversal is a cornerstone of atomic and molecular physics.
- Some aspects that are relevant to light scattering are reviewed here.

- If the spin of the particle is half odd-integral, it will be +2 or +-1.
- The numbers are said to be pseudoscalar.
- The intrinsic parity of the photon is defined from theoretical considerations to be negative, whereas the intrinsic parities of electrons and nucleons are taken by convention to be positive.
- Particle wavefunctions that are not eigenfunctions of the parity operator can be assigned to intrinsic parities.

- All observables can be classified as odd if their operators don't change sign under the coordinates.

- The hydrogen atom is an example of a mixed parity system.

- Section 1.9.2 states that the natural optical rotatory is a pseu doscalar and has odd parity.
- If one inverts the entire experiment (light beam plus active medium) the resulting experiment is also realized in nature.
- There are quantum states of mixed parity.

- The adiabatic potential energy function would have the form shown on the left if the configuration were stable.
- If a potential hill is raised gradually in the middle, the two pyramidal configurations become the most stable and the energy levels approach each other in pairs.
- The pairs of energy levels are not the same for an infinitely high potential hill.
- The wavefunctions are not destroyed by the rise of the central potential hill.
- The system in the lowest state of oscillation and the system in the left and right wells are not true stationary states.

- A molecule can have two equivalent configurations.
- There is complete certainty about whether the molecule is in the left or right well in the two mixed parity states of R.

- The general time dependent wavefunction of a degenerate two-state system is one of the specializations of the wavefunctions.

- If the barrier is infinite, the tunnelling splitting is zero.

- Since the rotation doesn't affect the wavefunctions, their parities can be determined from their reflection across the plane of the nucleus.

- A potential energy diagram with a very high barrier separating the left and right wells can be drawn for any resolvable chiral molecule: the horizontal axis might represent the position.
- If the tunnelling splitting is finite and the state is prepared, it will be a superposition of two states of different energy.
- The lifetime of the resolved enantiomer is related to the time scale of the optical activity measurement.
- Unless the duration of the experiment is infinite, the resolution of the definite parity states of an enantiomer of tartaric acid is impossible.

- It is possible to classify time-even and time-odd operators and the corresponding time-even and time-odd observables, depending on whether they do not or do change sign under time reversal.

- For convenience, we recall here a few definitions.

- 2 are both positive.

- It can be shown a lot.

- 2 is unaffected by a phase change.

- The earlier demonstration showed that a state and its conjugate are linearly independent, leading to an even-fold degeneracy.
- The expectation values with time-even and time-odd operators disappear when states are constructed.

- This is similar to the development below.
- Statements about matrix elements can now be made.

- The proof is similar to (4.3.33).

- A time-odd operator has opposite expectation values.
- The proof is similar.

- If there are even or odd number of electrons, 2 is equal to 1.

- The existence of permanent electric and magnetic dipole moments in atoms is an important example of the generalized selec tion rule.

- Consider a molecule with two parts.
- If it has an odd number of electrons.

- Judd and Runciman gave us a choice of phase factors and our derivation is based on them.

- The result assumes that the functions behave the same.

- Section 1.9.2 states that the optical rotation observable is a time-even pseudoscalar.
- The direction of propagation of the light beam is immaterial, so this classification seems reasonable for natural optical rotation.
- When we apply it to magnetic optical rotation, the direction of the light beam relative to the magnetic field is crucial.

- We leave the observer and his probe light beam alone and apply space inversion and time reversal to just the sample and any applied fields.

- The observer will measure an equal and opposite optical rotation when the isotropic collection of chiral molecule is replaced by a collection of enantiomeric molecule.
- It is easy to deduce that the observable is a pseudoscalar because it is invariant with respect to any proper rotation in the sample.

- The optical rotation is not changed under time reversal.

- A collection of achiral molecule in a magnetic field.
- The same magnetic optical rotation will be observed under space inversion.
- If the field is not present, the collection of molecules can be regarded as unchanged, even though individual paramagnetic molecules will change to their Kramers conjugates.

- To apply quantum mechanical arguments to the symmetry classifica tion of natural and magnetic optical rotation observables, it is necessary to specify corresponding operators with well defined behaviour.
- The introduction of effective polarizability and optical activity operators has made a good start.

- The expectation values are imaginary since the operators are anti-Hermitian.
- Natural optical rotation, being a pseudoscalar observable, is generated by a time-even odd-parity operator, while magnetic optical rotation is generated by a time- odd axis.

- Section 4.3.1 states that since the natural optical rotation observable has odd parity, a resolved chiral molecule must exist in a state of mixed parity.

- The odd-parity observables of natural optical rotation and a perma nent space fixed electric dipole moment require mixed parity quantum states.
- Time reversal invariance provides a fundamental quantum mechanical distinction between the two odd-parity observables.
- The vanishing of a permanent electric dipole moment in a stationary state is well known in elementary particle and atomic physics.

- The discussion of parity and reversality requirements has been limited to atoms.
- When the space-fixed axes are transformed, we must determine the behavior of the molecule-fixed axes.
- The directions of the space-fixed axes are reversed as a result of the operation.
- The operation of the space fixed axes must be accompanied by a reflection in a plane passing through the symmetry axis of the molecule.
- This important point is elaborated in two books.

- The arguments can be extended to show that this is also prohibited by reversality.

- Despite the fact that many asymmetric tops and certain linear tops have molecule-fixed permanent electric dipole moments, they usually do not show first-order Stark effects.

- Section 4.3.1 shows how the mixed parity states of a resolved chiral molecule can be pictured in terms of a double well potential.

- The stationary states do not correspond to a motion around one of the two equilibrium configurations if a molecule admits two different nuclear configurations.
- Each stationary state is composed of left- and right-handed configurations in equal shares.
- The existence of optical isomers may be in doubt because the right-handed or left-handed configuration of a molecule is not a quantum state.

- A system with sharp energy is inactive.

- The energy function (Hamiltonian) is always invariant with respect to space inversion, since each molecule consists of point charges interacting via Coulomb's law.
- There was no optically active molecule, which was contrary to experience.

- The translated quotations are from a review by Pfeifer.

- The lifetime of a prepared enantiomer is virtually infinite according to the type of arguments given in Sec tion 4.3.1.
- The Hamiltonian is brought up to date by injecting a small parity-violating term into it, which results in the two enantiomeric states becoming the true stationary states.

- It is appropriate to follow a treatment given by Bohm.

- The wavefunction changes with time.

- 2 is similar to two weakly coupled classical harmonic oscillators.
- If only one of the pendulums is made to swing, the energy is transferred back and forth between the two pendulums.
- If the two pendulums are made to swing at the same time, there are two states of stationary oscillation that correspond to the in-phase and out-of-phase local oscillations.
- The local coordinates are not 'diagonal' in the sense that they couple with each other, whereas the normal coordinates of the vibrating system are.

- If a two-state system is prepared in a nonstationary state, it might appear to be influenced by a time-dependence on the internal Hamiltonian of the system.

- This identification allows for the recovery of (4.3.10).

- The convenience of tunnelling through the barrier in the double well potential is emphasized.

- The time reversal arguments in Section 4.4.3 show that R is not real.

- R is zero.

- The Hamiltonian of the chiral molecule that lifts the exact degeneracy of the mirror image enantiomers is described in Section 1.9.6.
- The weak neutral current interaction creates parity between electrons and nucleons.

- The electron-electron interaction is usually neglected.

- The expectation values are zero.
- It is necessary to invoke a magnetic perturbation of the wavefunction that involves spin.
- The method for calculating the tiny parity-violating energy differences between enantiomers is tractable.

- The optical activity is asymmetrical.

- The true stationary states are R.

- The ultimate answer to the 'paradox' of the stability of optical enantiomers lies in the weak interactions.
- The situation is more complicated because of the influence of the environment.

- The measurement of the parity-violating energy differences between enantiomers and the detection of manifestations of parity violation in chiral molecule are major challenges for molecular physics.
- There is a lot of discussion of possible experimental strategies that exploit different aspects of the quantum mechanics of the two-state system perturbed by parity violation.

- If a state cannot be classified according to an irreducible representation of the symmetry group of the Hamiltonian, it has broken symmetry.
- The phenomenon of natural optical activity arises from the fact that a resolved chiral molecule displays a lower symmetry than its associated Hamiltonian.
- If the small parity-violating term in the Hamiltonian is neglected, the symmetry operation that the Hamiltonian possesses but the chiral molecule lacks is parity, and it is the parity operation that interconverts the two enantiomeric parity-broken states.
- In the context of nuclear physics, broken symmetry states are often called deformed states.

- A symmetry violation may be seen as a symmetry breaking with respect to some new and previously unsuspected deeper symmetry operation of the Hamiltonian.

- The double well model states that there is no correlation between parity violation and the stabilization of chiral molecules.

- The natural optical activity is observable only if the observation time is short and the interconversion time between enantiomers is proportional to the inverse of tunnelling splitting.
- An important criterion for distinguishing between natural optical activity generated through parity breaking and that generated through parity violation is led by these considerations.
- The tiny natural optical rotation shown by a free atomic vapour is constant since it is due to parity violation.

- The degree of chirality of individual structures in the form of some fundamental time-even pseudoscalar quantity is analogous to, say, the degree of chirality of individual structures in the form of some fundamental time-even pseudoscalar quantity.
- Under close quantum mechanical scrutiny, any pseudoscalar quantity will average to zero on an appropriate timescale because the degree of chirality is not in stationary states of the Hamiltonian.

- In ferromagnetism, symmetry breaking is associated with phase transi tions in which large numbers of particles cooperate to produce sudden transitions between symmetric and asymmetric states.
- Under spatial rotation, the Hamiltonian of an iron crystal is invariant.
- The ground state of a magnetized sample, in which all the magnetic dipole moments are aligned in the same direction, distinguishes a specific direction in space, the direction of magnetization.

- The rotation and time reversal symmetries become manifest when the temperature is raised above the Curie point.
- In the ferromagnetic phase, the sense of magnetization with respect to space-fixed axes is arbitrary.
- The full symmetry of the Hamiltonian can be recovered at sufficiently high temperature.
- Molecules do not support sharp phase transitions between symmetric and asymmetric states.
- There has been a lot of discussion about the relationship between the microscopic and the macroscopic aspects of the broken-parity states of chiral systems.

- In the case of ferromagnetism, this expression is derived from spontaneously magnetization.
- The order of ferromagnetism is the magnetization.
- A phase transition from an achiral state to a chiral state would be characterized by an order parameter.

- Heisenberg said that particles are more similar to Molecules than to atoms.

- The term mixes the definite parity states of a molecule.

- 2 are intermediates between the worlds of matter and antimatter.
- Mesons have zero numbers.

- The nonvanishing of components of property or transition tensors in a molecule with a given spatial symmetry and in a given quantum state can be established with the help of time reversal arguments.
- Section 4.3.2 shows an example of permanent electric and magnetic dipole moments.

- Section 1.9.2 shows that it is possible to classify physical quantities under space and time reversal.
- If the time reversal of the two measurable quantities is known, the property can be classified immediately.
- The characteristics listed in Table 4.1 are deduced by applying these considerations to the general expressions.

- Density, electric polarizability, and the applied uniform electric field are related to a system's physical property.

- Neumann's principle states that any component representing a physical property of a system must transform into a completely symmetric representation of the system's symmetry group.
- Neumann's principle in terms of asymmetry rather than symmetry was provided by Curie.
- There is no asymmetry in a property that is not in the system.
- The principle of Neumann's principle has been discussed by several people.

- The principle of time reversal symmetry is not applicable to phenomena where the system's entropy is changing.
- This approach is most appropriate when considering the magnetic properties of crystals.

- Since we are interested in the quantum mechanical properties of individ ual atoms and molecules in this book, we incorporate time reversal into our symmetry arguments using an alternative approach based on the generalized symmetry selection rule.
- The time reversal characteristics of a physical property are taken into account by specifying a corresponding time-even or time-odd operator, and the molecule's odd or even number of electrons are taken into account by using a single or double point group.
- The diagonal matrix elements give the same property in different quantum states as the off-diagonal matrix elements.
- An atom or molecule with a magnetic moment would not have time reversal symmetry if it were not for a magnetic field that lifts the degeneracy.

- Time reversal symmetry is said to be responsible for the symmetry of matter.
- We show how time reversal arguments in a quantum mechanical context can be used to glean more detailed information about the properties of the molecule than is given by the classical method.
- Powerful statements concerning the permutation symmetry emerge in the case of the polarizability.
- Other useful results are obtained despite the fact that analogous statements are not possible.

- The corresponding time-reversed states are 2.

- It is necessary to specify a scattering operator with well defined behavior under time reversal in order to apply light scattering.

- The result may be valid for all Raman processes despite the approximations used in the derivation.

- The initial and final states can either be even or odd with respect to time reversal.

- It is not possible to construct states that are even or odd with respect to time reversal since a single application of the time reversal operator always creates a state that is close to the original one.
- When the initial and final states are components of a twofold Kramers level, we consider the most common situation.
- When the zeroth order Born-Oppenheimer approximation is invoked, we must take the purely electronic part of the transition polarizability that results, and so the conclusions only apply to the scattering and resonance of the atoms.

- Section 2.8.1 shows that antisymmetric Rayleigh scattering is only possible from systems that are in a state of decay.
- The degeneracy must be such as to support transitions that generate a real antisymmetric tensor for even electron systems.
- There are many possibilities for the case of Rayleigh scattering from atoms.

- Since it is diagonal, symmetric, the complex transition polarizability is real.

- The conclusions were reached by considering a com plex atomic wavefunction which is neither odd nor even under time reversal.
- Since atoms are spherically symmetric, the transition polarizability will always be diagonal with respect to its spatial components.

- The general results for the transition polar izability are developed in more detail in Chapter 8.

- Since the real and imaginary parts of the optical activity are not defined by permutation symmetry, relationships similar to (4.4.2) can be written.

- In discussing natural and magnetic optical rotation from systems in degenerate states, it must be remembered that only diagonal transitions can contribute because the phases of the initial and final states must be the same.

- It is included in discussions of Jones birefringence and gyrotropic birefringence.

- The odd electron chiral molecule could show new properties.

- Neumann's principle can be used to reduce a given property to its simplest form in a particular point group.
- This involves the specification of which components are zero and any relationships between them.

- Neumann's principle is that if the coordinate transformation corresponds to one of the symmetry operations of the molecule's point group, the corresponding property tensor components are invariant.
- Since free space is isotropic, a property can only be determined by the relative orientation of the molecule and the coordinate axes.

- One conclusion we can draw is that the polar tensors of odd rank and even rank disappear for point groups containing the inversion operation.

- A simple example is a molecule with a threefold rotation axis.

- By applying the appropriate set of symmetry matrices to (4.4.16), it is possible to achieve the maximum simplification of a polar or axial tensor of any rank for a molecule belonging to a particular point group.
- Since there is usually a smaller set of generating operations from which the complete set of symmetry operations can be obtained, it is not necessary to apply a symmetry matrix for every operation of a point group.

- It is only necessary to take the set of generating matrices in order to get the maximum simplification of a tensor.

- The forms of polar and axial tensors up to the fourth rank in the important molecular point groups are displayed in Tables 4.2, which were derived using the methods outlined above.
- Each column displays the components to which the component at the top of the column reduces in the various point groups; so each row is a list of equalities between pairs of components.

- Natural optical rotation in isotropic samples, and in oriented samples for light propagating along the principal molecular symmetry axis, is supported only by chiral molecule.
- Section 1.9.1 states that natural optical rotation is possible in some oriented achiral molecules.

- Further simplification may be brought about by additional physical considerations.

- From a consideration of the symmetry operations of the molecule's point group in effect, the tensor components are determined.
- This classification should be extended to all the irreducible representations of the point groups.
- The task has been carried out by two authors, one of which considered just the components of a second-rank polar tensor, and we refer to them for the results.
- It is possible to get the information for certain components simply by consulting standard point group character tables.

- The symmetry operations of the sphere include improper as well as proper rotation.

- The laws of physics are independent of the choice of a coordinate system, so the two sides of an equation must transform in the same way.
- To cast both sides of the equations in the form of tensorial sets is convenient.
- Resolving these sets into irreducible subsets pushes the process of simplification to its limit because one disentangles the physical equations into a maximum number of separate, independent equations.

- Table 4.3 contains the results for tensors up to rank six.

- A simple but important application of the decomposition is to the derivation of angular momentum selection rules.
- The restrictions imposed by time reversal are discussed in Section 4.4.3.

- Young tableaux is the standard general method for reducing an arbitrary cartesian tensor.
- The irreducible third-rank cartesian tensors are written out explicitly.
- We simply quote the results of the work done by Andrews and Thirunamachandran.

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