The changing of bodies into light and light into bodies is very comformable to the course of Nature.
The theory of optical activity developed in this book is based on a semi-classical description of the interaction of light with molecule, which is a quantum mechanical object.
The most complete account to date of the radiation field and its interactions with molecule is not used in this book since the required results can be obtained more directly with semiclassical.
The chapter reviews the classical aspects of mechanical perturbation theory required for the semiclassical description of the scattering of light by Molecules.
The methods are based on theories developed in the 1920s and 1930s when quantum mechanics was applied to the interaction of light and atoms.
This book uses a cartesian tensor notation, which is elaborated in Chapter 4: this is essential if the delicatecouplings between electromagnetic field components and components of the molecule responsible for optical activity are to be manipulated succinctly.
The recently published comprehensive treatise on the theory of the Raman effect by Long should be consulted for further details.
The fields depend on only one space coordinate in the special case of electromagnetic waves.
It is normal to the direction of propagation.
The waves have no field components in the direction of propagation.
The choice of a minus sign is universal in quantum mechanics and is an advantage in a work on molecular optics.
A tilde is used throughout the book to indicate a complex quantity.
The two equations are uncoupled because of the arbitrariness of the potentials.
Recently, it was pointed out that this is a case of mistaken paternity.
This condition and the associated gauge should be attributed to the physicist L. Lorenz.
The polarization is circular after 2 out of phase.
The ellipticity and azimuth are in line with the convention for a positive ellipticity and angle of optical rotation used in Chapter 1.
The quantity is no longer ellipticity if the wave is only partially polarized.
The set of four intensity measurements are required to determine the state of a light beam.
Two optical elements are required, one for an analyzer for which the beam is linearly polarized along the transmission axis, and the other for a retarder, which alters the phase relationship between the beam components.
There is excess intensity transmitted by a device which accepts both right- and left-circularly polarized light.
The electric field of the light is always completely polarized, with the tip of the electric field at each point in space 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- Quasi-monochromatic light has an extradimension in its range of possible polarizations, because the component waves can have different phases.
The net electric field of quasi-monochromatic light can have the properties of a completely monochromatic wave and the light is said to be completely polarized.
The opposite extreme is unpolarized or natural light, where the tip of the net electric field is not always straight.
This is the case of scattered light.
Time averages of real functions of the fields are measured in measured intensities.
The time averaged products of the electric field are used to define the parameters of the quasi-monochromatic beam.
A beam can be divided into two parts, one of which is unpolarized and the other of which is partially polarized.
The partial polarization can be specified with the polarization tensor.
The basis of the calculation in Chapter 3 of birefringent polarization changes comes from interference between the transmitted and the forward-scattered waves.
A four-dimensional real space has a zero.
The mathematical form of a first-rank spinor can be found in the Jones vector.
The density matrix description of the state of a system in quantum mechanics is similar to the wave function description.
A pure state can only be specified by a quantum mechanical wave function.
A mixed quantum mechanical state is an incoherent superposition of pure states and must be specified by a density matrix.
A transition between two quantum states of an atom or molecule is what causes light to be generated.
If the quantum states of the emitter are precisely defined before and after the transition, complete polarization results.
Fano discussed the question in detail.
The structures of charge and current distributions are being investigated.
Magnetic multipole moments and electric multipole moments are used to develop charge distributions.
The electric dipole moment is independent of the choice of the origin if the net charge is zero.
If the net charge and the dipole moment are both zero, the electric quadrupole moment is independent.
The traceless definition is preferred here since it automatically emerges as the source of a well-defined part of the scalar potential generated by a static charge distribution.
It vanishes for a spherical charge distribution is a related reason for preferring the traceless definition.
This number may be reduced by symmetry.
It is possible to define complex multipole moments as spherical harmonics.
The real form of the molecule is better for our purposes because it has a natural cartesian frame instead of a natural polar frame.
Magnetic monopoles have not been observed.
The symmetry with respect to exchange of the suffixes leads to an uncharacteristic form for the magnetic fields that are caused by a quadrupole moment.
Magnetic quadrupole moments do not invoke this point.
The electric field is generated by a charge distribution.
The physical interpretation is as follows.
The first term disappears if the collection of charges is neutral.
A repeated suffix reduces a tensor to zero.
The static magnetic field is generated by a system of charges.
We are interested in a constant current which can be generated through a circulatory character and a static magnetic field.
During the motion of the charges, this change occurs.
Since the time average of the linear velocity of a particle constrained to move within a small volume is zero, the first term is over.
The magnetic field is derived from the magnetic dipole contribution.
Specific contributions from time-varying electric and magnetic multipole moments make up the radiation field generated by a system of time-varying charges and currents.
The charge and current densities are assumed to vary with time.
We don't include tildes over complex quantities in this section.
It is necessary to relate the current to the moments of the charge distribution in order to develop the dynamic vector potential.
The equation of continuity is the starting point.
The system is neutral for this treatment to be consistent.
Two important limits are now considered.
The expressions used in Buckingham and Raab are the same as those used in Landau and Lifshitz.
They can be used to check that each term has the correct behavior under space inversion and time reversal.
We now consider the energy of a system of charges and currents bathed in both static and dynamic electric and magnetic fields and develop expressions which, in operator form, constitute convenient Hamiltonians for subsequent quantummechanical calculations.
The Hamiltonian can be used to get explicit multipole terms for the interaction energy between a system of charges and currents and static electric and magnetic fields.
The vector potential is held for by 0
The magnetic potential energy was derived from a uniform magnetic field.
If the field is not uniform and higher multipole interaction terms are required, a general expansion about the origin must be used.
In multipole form, the interaction energy between a system of charges and currents and static electric and magnetic fields is obtained.
The development of the interaction energy in multipole form is more difficult when the fields are dynamic.
The interaction energy of charge dis tributions 1 and 2 is now considered.
The same interaction energy is given when the roles of 1 and 2 are changed.
Magnetic monopoles don't exist so Magnetic analogues of the lower order terms don't arise.
The Hamiltonian is important for the case of charges and currents in electric and magnetic fields.
The most widely used method is to invoke the quantum mechanical commutation relations between the coordinates and the Hamiltonian of the charges and currents.
The dynamic interaction Hamiltonian is effectively in mul tipole form, but it is not as clean as the static multipole interaction Hamiltonians.
The fundamental interaction Hamiltonian is simply equal to the multipole Hamiltonian thanks to a simple method given here.
The two Hamiltonians should give the same results if applied consistently.
Dirac gave an early example of this equivalence by giving a derivation of the Kramers-Heisenberg dispersion formula for the scattering of a photon by an atom or molecule.
The interaction describes simultaneous photon absorption and emission.
This feature is only found in a quantized radiation field.
2 terms do not make a contribution.
The incident light wave has no components in the 2 term.
The transformation to a multipole form is more delicate if the Hamiltonian contains spin-orbit interac tion.
The matter has been discussed in detail.
In this section, perturbation theory is used to derive quantum mechanical expressions for the molecular property tensors that describe the response of a molecule to a particular electric or magnetic field component.
The property tensors appear later in the expressions for the observables, such as the angle of optical rotation.
The electric and magnetic multipole moments appearing in the expressions for the interaction of energy of a system of charges and currents with external electric and magnetic fields can be permanent attributes of the system.
The static molec ular property is a quantum mechanical form.
We refer to standard works for the development of these approximate expressions.
Similar expressions can be found for the other static property tensors, but they are not reproduced here since only the dynamic versions are required in what follows, and these are derived below.
Buckingham gave a full account of the static electric molecular property.
A radiation field can cause multipole moments in a molecule.
The moments are related to the electric and magnetic field components of the radiation field.
We use the second procedure of taking expectation values of the multipole moment operators using wave functions perturbed by the radiation field to identify the dynamic molecular property tensors.
The final results should be the same regardless of which Hamiltonians are used.
Since it involves less work, we use the multipole Hamiltonian.
The products of the same multipole transition moments are related to the other property tensors.
The property tensors involving products of different multipole transition moments are not analogous to symmetric and antisymmetric parts.
The real multipole moments are presented in a complex form.
The tildes over complex quantities were omitted in Section 2.4.5 in the interests of economy, which facilitates the application of expressions such as (2.4.43).
The contribution of a number of these dynamic molecular property tensors to particular light scattering phenomena are discussed in detail in subsequent chapters.
According to the uncertainty principle, the electronic energy levels of the molecule have an infinite lifetime.
There is a chance of absorption of radiation by the molecule if the polarizabilities are near resonance.
Incorporating the finite energy width of the excited states of the molecule allows for a finite lifetime.
The emission of radiation by molecule in excited states is caused by the finite lifetime.
The lifetimes of excited states and the width of energy levels are discussed in further quantum-mechanical terms by Davydov (1976).
The force on the system of charges.
The real parts of is what it is.
The response functions are a class of functions that include the property tensors.
Some of the general properties of such functions are independent of the theoretical model used in the book.
For a detailed derivation, we refer to works such as Lifshitz and Pitaevski.
The response function's dispersive and absorptive parts are connected by the Kramers-Kronig relations.
If the dispersive part is constant, the absorptive part is zero.
There can be no absorption of energy from a static applied field.
The derivation of sum rules is an important application of Kramers-Kronig relations.
This is an alternative statement to the sum rule.
The real and imagi nary parts of a complex response tensor are referred to in other treatments.
We have refrained from using this terminology, instead referring to the dispersive and Absorptive parts.
The real and imaginary parts of a response tensor are applied to the real and imaginary parts of a complex field.
The sum rule for the antisymmetric polarizability is similar to the sum rule for the symmetric polarizability.
Some of the operators specified in the transition moment products do not commute, so care is needed in the extension of the sum rules to other property tensors.
It is difficult to evaluate the sum over all excited states.
It can be avoided by using a static approximation.
The results allow the polarizability and optical activity tensors to be obtained from calculations of the molecular orbitals perturbed by a static electric field and a static magnetic field.
They are useful for calculations of optical rotation and activity.
Similar expressions are obtained for internal and external perturbations of the same type.
The dependence of perturbed dynamic molecular property ten sors in the region of an isolated absorption band is easily deduced.
When the peturbation is much smaller than the linewidth, these are valid.
The results apply when the frequencies are much smaller than the width of the absorption band and in a magnetic field.
The generation of characteristic rotatory dispersion and circular dichroism lineshapes through large exciton splittings is an important example of the latter situation.
The exposition has come up with the multipole moments that are the same Frequency and phase as the inducing light wave.
Radiation from those moments is responsible for scattering.
The scattered light waves have different frequencies than the incident light waves, and are usually unrelated in phase to the incident light wave.
Similar expressions, but without the decomposition into real and imaginary products of transition moments, have been derived by Placzek.
The real and imaginary products are clearly displayed here.
Time reversal has the effect of replacing the time-independent part of a wavefunction with a complex conjugate.
It's pure real.
The scattering operator is exact, but we will find more useful an approximate operator which breaks down into parts with better defined Hermiticity and time reversal characteristics.
There is no static antisymmetric polarizability.
Even though there is no well defined permutation symmetry, the superscripts's' and 'a' are retained to conform with the corresponding parts of the effective polarizability operator.
Linear response theory can be used to derive effective polarizability and optical activity operators.
Nonsingular versions of these operators are valid for all Raman processes because they do not rely on the average energy approximation.
For more details, we refer to Hecht and Barron.
The electrons follow the nuclear motions.
An electron does not make transitions from one state to another in an adiabatic motion.
The explanation for the adiabatic approximation lies in the slow nuclear motion compared with the electronic motion because of the large disparity between the nuclear and the electronic mass.
The nuclear motion has a number of con tributions which can be separated to a good approximation.
The elimination of translational motion is accomplished by using a set of axes.
We will often use simplified notations that are clear.
Section 2.8.4 outlines the extension to orbitally-degenerate states in the simplified context of the 'crude' adiabatic approximation.
The adiabatic approximation can be used to make the transition tensors simpler.
The adiabatic wavefunction and energy are used first to separate the parts of the general Raman transition polarizabilities.
For incident radiation at microwave frequencies, these terms are only significant.
If the lifetimes of excited states are taken into account, the approximation should be good at absorbing frequencies.
The rotational states have been dropped since we are only concerned with the scattering of strontium from fluids.
The molecule remaining in the ground electronic state is known as the ionic part of the vibrational transition polarizability.
This term can be ignored if the exciting light is in the infrared region or below.
The real part of a quantity minus its complex conjugate, which is pure imaginary, disappears in this approximation.
The imaginary part of a quantity plus its complex conjugate is real and the symmetric part is gone.
There is no separation of the optical activity tensors into symmetric and antisymmetric parts in Placzek's approximation.
The adiabatic wavefunction and energy were arrived at through the neglect of electronic and nuclear motions.
The 'vibronic'coupling can give rise to some of the structures of electronic absorption and circular dichroism bands.
In reviews such as Longuet-Higgins, Englman, Ozkan and Goodman, and Ballhausen, vibroniccoupling can be considered at various levels of sophistication.
The Herzberg-Teller method provides a simple framework on which to hang symmetry arguments, which is our main concern, even though the numerical aspects of the crude adiabatic approximation are not even qualitatively correct.
The second and higher term in the expansion of the electronic Hamiltonian in the normal coordinates is the perturbation that mixes the electronic states.
The Herzberg-Teller approach must be reform if the nuclear motion mixes electronic states.
0 is not an equilibrium configuration, but must have enough symmetry for the degeneracy to be nonaccidental.
2 0 ignores mixing with all other electronic states.