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TitleMolecular light scattering and optical activity
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Total Pages467
Table of Contents
                            Cover Page
About the book
ISBN 0521813417
Contents (with page links)
	1 A historical review of optical activity phenomena
	2 Molecules in electric and magnetic fields
	3 Molecular scattering of polarized light
	4 Symmetry and optical activity
	5 Natural electronic optical activity
	6 Magnetic electronic optical activity
	7 Natural vibrational optical activity
	8 Antisymmetric scattering and magnetic Raman optical activity
Preface to the first edition
Preface to the second edition
1 A historical review of optical activity phenomena
	1.1 Introduction
	1.2 Natural optical rotation and circular dichroism
	1.3 Magnetic optical rotation and circular dichroism
	1.4 Light scattering from optically active molecules
	1.5 Vibrational optical activity
	1.6 X-ray optical activity
	1.7 Magnetochiral phenomena
	1.8 The Kerr and Cotton–Mouton effects
	1.9 Symmetry and optical activity
2 Molecules in electric and magnetic fields
	2.1 Introduction
	2.2 Electromagnetic waves
	2.3 Polarized light
	2.4 Electric and magnetic multipole moments
	2.5 The energy of charges and currents in electric and magnetic fields
	2.6 Molecules in electric and magnetic fields
	2.7 A molecule in a radiation field in the presence of other perturbations
	2.8 Molecular transition tensors
3 Molecular scattering of polarized light
	3.1 Introduction
	3.2 Molecular scattering of light
	3.3 Radiation by induced oscillating molecular multipole moments
	3.4 Polarization phenomena in transmitted light
	3.5 Polarization phenomena in Rayleigh and Raman scattered light
4 Symmetry and optical activity
	4.1 Introduction
	4.2 Cartesian tensors
	4.3 Inversion symmetry in quantummechanics
	4.4 The symmetry classification of molecular property tensors
	4.5 Permutation symmetry and chirality
5 Natural electronic optical activity
	5.1 Introduction
	5.2 General aspects of natural optical rotation and circular dichroism
	5.3 The generation of natural optical activity within molecules
	5.4 Illustrative examples
	5.5 Vibrational structure in circular dichroism spectra
6 Magnetic electronic optical activity
	6.1 Introduction
	6.2 General aspects of magnetic optical rotation and circular dichroism
	6.3 Illustrative examples
	6.4 Magnetochiral birefringence and dichroism
7 Natural vibrational optical activity
	7.1 Introduction
	7.2 Natural vibrational optical rotation and circular dichroism
	7.3 Natural vibrational Raman optical activity
	7.4 The bond dipole and bond polarizability models applied to simple chiral structures
	7.5 Coupling models
	7.6 Raman optical activity of biomolecules
8 Antisymmetric scattering and magnetic Raman optical activity
	8.1 Introduction
	8.2 Symmetry considerations
	8.3 A vibronic development of the vibrational Raman transition tensors
	8.4 Antisymmetric scattering
	8.5 Magnetic Rayleigh and Raman optical activity
Index (with page links)
Document Text Contents
Page 233

4.3 Inversion symmetry in quantum mechanics 209

solutions, with the notation (±) used in Section 4.3.1 to denote even- and odd-parity

We now consider how the wavefunction changes with time. Take V12 to be real
and negative so that α = π . The amplitudes of the two perturbed wavefunctions
are then

± =


(ψ1 ∓ ψ2). (4.3.57)

Theψ (0)± are the amplitudes of stationary states with time-dependent wavefunctions

ψ±(t) = ψ (0)± e−i(W±|V12|)t/h̄. (4.3.58)
The general time-dependent wavefunction for the two-state system is now given by
the sum of the two stationary state wavefunctions:

ψ(t) = 1√


+ e

−i|V12|t/h̄ + ψ (0)− ei|V12|t/h̄
e−iW t/h̄. (4.3.59)

This can be rewritten in terms of ψ1 and ψ2:

ψ(t) = [ψ1 cos(|V12|t/h̄) + iψ2 sin(|V12|t/h̄)]e−iW t/h̄. (4.3.60)
Thus at t = 0 the system is entirely in the state ψ1 and at t = πh̄/2|V12| it is

entirely in the state ψ2 which is seen to have a phase e
−iπ/2 relative to ψ1. This

oscillation of amplitude between the two states ψ1 and ψ2 is formally similar to
that between two resonant classical harmonic oscillators, such as pendulums, that
are weakly coupled. If just one of the pendulums is made to swing, the energy is
transferred back and forth between the two pendulums at a rate proportional to
the strength of the coupling force. But if the two pendulums are made to swing
simultaneously with identical energies, two possible states of stationary oscillation
are possible (stationary in the sense that each pendulum retains constant energy)
corresponding to the in-phase and out-of-phase local oscillations. The transforma-
tion from a description in terms of local pendulum coordinates to the stationary
combinations of the local coordinates is simply a transformation to the normal co-
ordinates of the vibrating system: the local coordinates are not ‘diagonal’ in the
sense that they couple with each other; whereas there is no coupling between the
normal coordinates so they oscillate independently of each other. Likewise the set
of quantum states (ψ1, ψ2) couple with each other whereas the set (ψ

+ , ψ

− ) do

not and are true stationary states.
Thus if no external perturbation is applied to a two-state system, any ‘perturba-

tion’ which couples ψ1 and ψ2 is internal and is simply an ‘artifact’ of the chosen
representation: the Hamiltonian is the same for (ψ1, ψ2) and (ψ

+ , ψ

− ). It might

be appropriate in some situations to set up the problem in terms of perturbation

Page 234

210 Symmetry and optical activity

theory, as above, if the chosen representation is ‘almost diagonal’ in the sense that
the coupling is weak, or indeed if an external perturbation is present. But for a
general two-state system (not necessarily degenerate) the exact energy eigenvalues
and eigenfunctions are, in place of (4.3.55) and (4.3.56),

W± = 12 (H11 + H22) ± 12 [(H11 − H22)2 + 4|H12|2]
2 , (4.3.61)

+ = cos φ ψ (0)1 + sin φ ψ (0)2 , (4.3.62a)

− = − sin φ ψ (0)1 + cos φ ψ (0)2 , (4.3.62b)


tan 2φ = 2|H12|/(H11 − H22). (4.3.62c)

If ψ1 and ψ2 happen to be degenerate and are interconverted by a particular sym-
metry operation of the Hamiltonian, ψ (0)+ and ψ

− transform according to one or

other of the irreducible representations of the group comprising the identity and
the operation in question. Thus if a two-state system is prepared in a nonstationary
state, it might appear (falsely) to be influenced by a time-dependent perturbation
lacking some fundamental symmetry of the internal Hamiltonian of the system.

We now identify the two enantiomeric states ψL and ψR of a chiral molecule
with ψ1 and ψ2. Since these states are interconverted by a fundamental symmetry
operation of the Hamiltonian, the inversion, they couple with each other; whereas
the stationary states ψ (0)+ and ψ

− transform according to one or other of the ir-

reducible representations of the inversion group, ψ (0)+ ≡ ψ (0)(−) having odd par-
ity and energy W+ ≡ W (−), and ψ (0)− ≡ ψ (0)(+) having even parity and energy
W− ≡ W (+). This identification enables (4.3.10) to be recovered from (4.3.59).
The Born–Oppenheimer approximation is invoked in order to envisage this cou-
pling in terms of an overlap of ψL and ψR due to tunnelling through the barrier in
the double well potential (Fig. 4.4), but it is emphasized that this is a convenience:
the coupling is independent of any model of molecular structure. It happens that,
becausewe are able to distinguish the left- and right-handed forms of a chiral object,
we can prepare a chiral molecule in a stateψL or ψR, but these are not the stationary
states (neglecting for the moment a small parity-violating term in the Hamiltonian):
having prepared ψL or ψR, if the molecule is isolated from all external influences,
it will oscillate forever between ψL and ψR in accordance with (4.3.60).

The natural optical activity observables shown by this oscillating system are
time dependent and are given by the expectation values of the effective optical
activity operators Ĝαβ and Âα,βγ, defined in (2.8.14), for the general time-dependent
wavefunction (4.3.60). Isotropic optical rotation, for example, is proportional to the

Page 466

442 Index

selection rules
angular momentum, for Raman scattering, 237,

388, 409–10
for electric dipole transitions in atoms, 240
generalized space-time, for matrix elements, 198–9;

application to molecule-fixed electric and
magnetic dipole moments, 199; to symmetric and
antisymmetric Rayleigh and Raman scattering,

spatial, for natural optical rotation, 27, 228–9,
270–1, 274; for natural Rayleigh and Raman
optical activity, 163; for magnetochiral
birefringence and dichroism, 329

Sellmeier’s equation, 5
sodium, atomic

magnetic Rayleigh optical activity, 408–11
resonance Rayleigh scattering, 394–7

specific ellipticity, 7
specific rotation, 7, 269–70

ab initio computations of, 272
of hexahelicene, 302–4

spherical harmonics
parity of, 190
phase convention for, 200, 240–1

spherical tensor operators, 238–43
spin angular momentum, 70, 200

effective, 414
spin–orbit coupling, 224, 322, 394, 410

and antisymmetric scattering, 409–10
in atomic sodium, 394–7
in iridium (IV) hexahalides, 398–400

spontaneous symmetry breaking, 215
Stark effect

in atomic hydrogen, 190
in symmetric top molecules, 207

static coupling (one electron) model, 273,

stationary states, 89, 95, 193
and optical enantiomers, 208–13
and parity violation, 213
quasi-, 95, 192

Stokes parameters, 62–7
Stokes Raman scattering, 108–9, 348–9
sum rules

Condon, 271
and Kramers–Kronig relations, 100–2
Kuhn–Thomas, 92, 100
for the rotational strength, 271–2

symmetric scattering, 155–8
symmetry matrices, 226
symmetry and optical activity, 24–52
symmetry violation, 43–50, 208–17

see also charge conjugation; parity; and time

tartaric acid, 2, 26–8, 192
tensor, 29, 171–3

alternating (Levi-Civita), 179–81
averages, 181–5
cartesian, 170–87

invariants (isotropic tensors), 156, 181, 183–5,

irreducible: cartesian, 69, 230–8; spherical
(operators), 238–42

Kronecker delta, 176, 180–1
polar (true) and axial (pseudo), 177–80, 217, 226
rank of, 172
symmetric and antisymmetric, 173, 236–7
time-even and time-odd, 217
unit, 180–1

time reversal T, 29–33, 193–201
and angular momentum quantum states, 199–201,

classification of molecular property tensors, 217
classification of operators, 197
and matrix element selection rules, 197–9
operator, 193
and permanent electric dipole moments, 204–7
violation, 47–9

torsion vibrations, 365–7, 367–73
trans-2,3-dimethyloxirane, natural Raman optical

activity, 373
transition optical activity tensors, 110, 120, 345, 360
transition polarizability tensor, 108–14

effective operators for, 112–13
ionic and electronic parts, 119–20
permutation symmetry of, 219–22
in Placzek’s approximation, 116–20
symmetric and antisymmetric parts, 110–12, 120,

vibronic development of, 388–93; antisymmetric,

tunnelling splitting, 192, 212
two-group model

of optical rotation and circular dichroism, 274–91
of Rayleigh optical activity, 351–6

uncertainty principle and resolved enantiomers,

unitary operator, 194
units, 1
unit tensors, 180–1
uranocene, electronic resonance Raman scattering and

magnetic Raman optical activity, 414–17
universal polarimetry, 28

V coefficients, 240–2
vector, 29, 171–2

polar (true) and axial (pseudo), 29, 177–80, 217
time-even and time-odd, 30, 217

vector potential, 58–60
vector product, 178–9
velocity–dipole transformation, 93–4

vibrational, 335
Verdet constant, 10
Verdet’s law, 10
vibrational optical activity

magnetic, 19–21, 407–22
natural, 17–19, 331–84

vibrational rotational strength, 332

Page 467

Index 443

in fixed partial charge model, 336
in bond dipole model, 339

vibrational structure in circular dichroism spectra,

vibronic coupling, 120–2, 305–7, 388–93
and antisymmetric scattering, 401–2

viruses, natural Raman optical activity,

wavevector, 56
wave zone, 77
weak neutral current, 45, 211–12
Wigner–Eckart theorem, 239–42

X-ray optical activity, 21–2
in Co(en)3+3 , 300
and magnetochiral dichroism, 21

Young diagram, 248
Young operator, 251
Young tableau, 249–51

Zeeman effect, 11–12, 107, 313
and the Faraday A-, B- and C-terms, 314–16
and magnetic Rayleigh and Raman optical activity,

and the magnetochiral A-, B- and C-terms, 329

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