##### 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

wavefunctions.)

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

ψ

(0)

± =

1√

2

(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√

2

(

ψ

(0)

+ 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

(0)

+ , ψ

(0)

− ) 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

(0)

+ , ψ

(0)

− ). 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]

1

2 , (4.3.61)

ψ

(0)

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

ψ

(0)

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

with

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

(0)

− 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

(0)

− 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,

387–8

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,

275–6

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

reversal

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,

347

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,

204–7

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,

219–22

vibronic development of, 388–93; antisymmetric,

401–2

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,

192–3

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,

304–10

vibronic coupling, 120–2, 305–7, 388–93

and antisymmetric scattering, 401–2

viruses, natural Raman optical activity,

383–4

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,

409–16

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

4.3 Inversion symmetry in quantum mechanics 209

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

wavefunctions.)

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

ψ

(0)

± =

1√

2

(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√

2

(

ψ

(0)

+ 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

(0)

+ , ψ

(0)

− ) 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

(0)

+ , ψ

(0)

− ). 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]

1

2 , (4.3.61)

ψ

(0)

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

ψ

(0)

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

with

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

(0)

− 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

(0)

− 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,

387–8

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,

275–6

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

reversal

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,

347

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,

204–7

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,

219–22

vibronic development of, 388–93; antisymmetric,

401–2

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,

192–3

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,

304–10

vibronic coupling, 120–2, 305–7, 388–93

and antisymmetric scattering, 401–2

viruses, natural Raman optical activity,

383–4

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,

409–16

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