Chapter 3 - UZHffffffff-cd8c-a533-0000-00007e0e1f31/... · Department of Chemistry Driven...

Department of Chemistry

Chapter 3

Theories of optical activity and circular dichroism


Optical Activity- New polarizationcomponent

L

R

L

R

Circular Birefringence Circular Dichroism

∆∆

∆−

∆

=

2cosh

2sinh

2sinh

2cosh

αα

αα

i

iM CD

∆∆−

∆∆

=

2cos

2sin

2sin

2cos

φφ

φφ

ORDM

=

0

x

in

EEr

=

y

x

in iE

EEr

≈

ηx

out

EEr


)()( tBtE ⇒&

e- motion in direction of E-field causes circular current

)()( tEtB ⇒&

B-field induced circular e- motion causes charge flow along axis

Macroscopic View

componenton polarizati new ⇒⊥ EBrr

Time-varying current (accelerated charge) give rise to radiation


)()( tBtE ⇒&

)()( tEtB ⇒&

Additional polarization terms

in Maxwell equations:

Macroscopic View

t

DjH

t

BE

BD

f

f

∂

∂+=×∇

∂

∂−=×∇

=⋅∇=⋅∇r

rrr

r

rr0ρ

)(,1

)()(,

0

0

tEbMMBH

tBbtEaPPED

&rrrrr

&rrrrrr

=−=

−=+=

µ

ε


Driven oscillator

Amplitude

-π

0

-π/2

phase

� On resonance the oscillation is phase-shifted by π/2 with respect to the drivingfield

driving frequency


Ein

Esig

� half of the π phase shift of is (quantum) mechanical� the second half of the phase shift is due to integration over a slab of (randomlyoriented) oscillators (Maxwell) → „phase matching“

Field view of absorption


Field view of optical activity

η�φ

VORD(in phase)

VCD(π/2 phase shift)

Circular Birefringence Circular Dichroism


Circular Dichroism Origins

λπ

rirkie

rki 211 +=⋅+≈⋅ rrrr

ratio r/λ

UV: 10-2

IR: 10-3-10-4

molecule

E(r)

rr

rkiRkiRkieee

rrrrrr⋅⋅⋅ = 0

0Rr

)(

0)( RktieEtErr

⋅−−= ω

Rr

el. dipole approximation

rr

λ≈50‘000 A

change of phase on molecular scale


m

AepAAp

m

eV

m

pV

m

eApH

H

2)ˆˆ(

22

ˆ

2

)ˆ( 2222

0

+⋅+⋅−+=+−

=rr

321

)ˆ(

1

:PotentialVector

Akc

iAB

Ac

iA

cE t

rrr

rrr

×−

=×∇=

−=∇−∂−=

ω

ωφ

Q.M. Treatment of Circular Dichroism

In free space

( )prm

em ˆˆˆ

:moment Magnetic

×=

( )rkeApAp

A

ti rr

321rr

r

⋅−⋅≈⋅ 1ˆˆ

0

0

ωε

reˆˆ

:moment Dipole

=µ

Potential at origin (center of molecule)

Idea: instead of considering the r-dependenceof the field, we let it interact at the origin withdifferent multipole moments of the molecule

( ) ( ) ( )( ) ( )( )cbdadbcadcbarrrrrrrrrrrr

⋅⋅−⋅⋅=×⋅×

After some vector gymnastics using things like

we obtain…

.)](exp[

: waveticmonochroma

0 ccrktiAA +⋅+−=rrrr

ωε( ) ( ) ( ) ( )

( )( ) ( )( )kpArApkr

Akprmc

eiApr

m

eBm

ˆˆˆˆ

ˆˆˆˆ

00

000

⋅⋅−⋅⋅=

×⋅×−=×∇⋅×=⋅

rrrr

rrrrr ω

For details see: Craig, D. P.; Thirunamachandran, T., Molecular Quantum Electrodynamics An Introduction to Radiation Molecule Interactions. Dover Publications, Inc. 1998 ed.; Academic Press: London, 1984.


[ ] int. quadrupole)ˆ(ˆˆ

/ˆˆ

'

'

0

0

0

+×⋅+⋅+=

∂∂−⋅−⋅−=

+=

444 3444 21

rr

rr

V

jiij

AkmAc

iH

rEQBmEH

VHH

µω

µ

( )( )2222

2

ˆ)]ˆ(ˆˆRe[2ˆ

)ˆ(ˆˆ)ˆ(ˆˆ'

bmaAAkambbaAbaA

AkambAabAkbmaAbabVa

m

rr

321321

rr

rrrr

rr

+×⋅⋅+=

×⋅+⋅×⋅+⋅=

µ

µµ

µµ

weak

Multipolar Coupling Hamiltonian

No contribution in isotropic sample

Transition Probablility


( )( )( )( ) ( )( )[ ] ( )( ))ˆ()ˆ()ˆ(Re2

)ˆ(ˆˆ)ˆ(ˆˆ'

***

**2

AkmAkmAkmAAA

AkambAabAkbmaAbaaVb

baabbaabbaab

rrrrrrrrrrrr

rrrr

×⋅×⋅+×⋅⋅+⋅⋅=

×⋅+⋅×⋅+⋅=

µµµ

µµ

Transition Probability

abbam µrr

<<

( )( ) )]ˆ(][[3

1)ˆ(*

AkAmAkmA baab

ondistributiisotropic

baab

rrrrrrrr×⋅⋅=×⋅⋅∫ µµ

( )( ) 22*

3

1ab

ondistributiisotropic

baab AAA µµµrrrrrr

=⋅⋅∫


( ) ( )( )2

2112

2

2121

2*

21

)]ˆ()ˆ([

)ˆ(

:light polarizedcircular

AikkAi

ikiAAkA

i

rrrrrr

rrrrrrrr

rrr

±=×⋅−×⋅±=

−×+=×⋅⇒

±=

εεεε

εεεε

εεε

)]Im[())]ˆ()(Re[(2

*mAAkAmrrrrrrr

⋅=×⋅⋅⇒ µµ

⋅±=∝→ ]ˆˆIm[2ˆ

3

1'

222

321321

r

rrm

ba ambbabaAbVaP

µ

µµ

]ˆˆIm[ strengthRotary 321321

rrm

ab ambbaR ⋅=

µ

µ


...'ˆ

2

1'ˆ1

'

153

+⋅

+⋅

+=−

∑r

rrQ

r

rr

rrr

ji

ij r

r

r

r

rrr

...ˆ

'

)'()'(

3+

⋅+=

−= ∫

r

r

r

q

rr

rr r

r

rrr

rr µρ

φ

...ˆ

...)'('ˆ1

...)'('ˆ1

0'

)'(1)'(

33

3

+×

=+××−=

+⋅⋅+=−

=

∫

∫∫

r

rmrjrr

rc

rjrrrcrr

rj

crA

r

rrrr

r

rrrrrr

rrrr

)(2

)'('

)'('

prm

erjrm

rerr

rrrrrr

rrrr

×=×=

==

∫

∫ ρµpoint charge

')'()'''3( 2rdrrrrQ ijjiij

rrρδ∫ −=

Multipole Expansion – just showing off…


Krr

Krr

+∂

∂−⋅−Φ=

+∂∂

Φ∂+Φ∇⋅+Φ=Φ

∑

∑

ji i

j

ji

ji ji

ji

r

ErrEr

rrrrrr

,

,

2

)0(2

1)0()0(

)0(2

1)0()0()(

)'(' rrrrr

∫= ρµ

j

jr

EE∂

Φ∂−=Φ−∇= ,

r

')'()'( rdrrWrrr

ρ∫Φ=

Jacskon, chapter 4.2

Krr

+∂

∂−⋅−Φ= ∑

ji i

j

ijr

EQEqW

,

)0(6

1)0()0( µ

')'()'''3( 2rdrrrrQ ijjiij

rrρδ∫ −=

∑∂

∂=⋅∇=

j jr

ErEr

22

6

1

6

10

rsubtract:

Note: Nabla E=0 because the external field is not influenced by the charge distribution

Energy of charge distribution in external potential - idem



jiij rEQBmEH

VHH

∂∂−⋅−⋅−=

+=

/ˆˆ

'

0

0rr

µ

Interaction with field

Mol. Hamiltonian

No contribution in isotropic sample

0

1

Dipole Strength D Rotatory Strength R

Transition Probability (+ for left circular, - for right circular light):

]ˆˆIm[2ˆ' 0101

2

01

2

0110 4342143421rrm

mVP ψψψµψψµψψψ

µ

⋅±∝∝→

rkierki rrrr

⋅+≈⋅ 1

new terms

4R

D

ω12


−

−==⇒

+ radiation magneticfor )1(

radiation electricfor )1(

onconservatiParity

1l

l

flfi ππππ

changeparity dipole magnetic

changeparity nodipole electric1

forbidden0

=

=

l

l

changeparity nohexapole electric3l

changeparity quadrupole magnetic

changeparity noquadrupole electric2

=

=l

)()( ˆ

ˆ

prprP

rrPrrrr

rr

×=×

−=

Parity Conservation


−=

+

−

01

10

cossin

sincos

νν

νν

ϕ

ϕ

T

E

E

E

E

−

−=

−

+

νν

νν

β

β

νν

νν

cossin

sincos

cossin

sincos

0

0

2

1

−=

+

−

2

1

cossin

sincos

µ

µ

νν

νν

µ

µr

r

r

r

Exciton Model of Circular Dichroism

coupled

+ϕ

−ϕ01

10

00

µ2 µ1µ+

µ−

0

E1

E2

E-

E+

10ˆ0

01ˆ0

ˆ0

2

1

µµ

µµ

ϕµµ

=

=

= ±±

r

r

r

=

2

1

εβ

βεH


Origin-dependence of magnetictransition dipole moment

rr

pr

rr

pr

0≠× prrr

0=× prrr

jrr

'rr

jprm

em j

ˆˆ02

×=r

jpm

ermjprr

m

ejjj

ˆ02

ˆ)ˆ(02

×+=×+rrr

Monomer in its own reference system Monomer in shifted reference system


Origin-dependence of magnetictransition dipole moment

jrr

'rr

( )

( )j

j

j

iEEjre

i

jrHjHrie

jrHie

jpm

e

µω r

h

hh

2ˆ0

2

ˆ0ˆ02

]ˆ,[02

ˆ02

0 =−=

−=−=

]ˆ,[ˆ rHim

ph

−=

jj

j

j

jjj

ri

m

jpm

ermjprr

m

e

µω rrr

rrr

×+=

×+=×+

2

ˆ02

ˆ)ˆ(02


chromophore 1 chromophore 2

=

2

1

εβ

βεH

coupled

+ϕ

−ϕ01

10

00

m+ m− µ+

µ−


12rr

222

22

µω rrr

×+ ri

m1µr

2µr

Magnetic transition dipolemoment of 2 in coordinatesystem centered r2 away

]Im[]0ˆˆ0Im[ ±±±±± ⋅=⋅= mmRab

rrµϕϕµ

2rr

1rr

111

12

µω rrr

×+ ri

m


( )][][cossin2

)(cossin

)(sin)(cos

22121121

1221

22

2

11

2

µµωµµωνν

µµνν

µνµν

rrrrrr

rrrr

rrrr

×⋅+×⋅−

⋅+⋅−

⋅+⋅=

rri

mm

mm

( )

×+−

×+⋅−=

⋅ −−

νµω

νµω

νµνµ

µ

sin2

cos2

sincos 222

2111

121

rrrrrrrr

rr

ri

mri

m

m

( )444444 3444444 21rrrrrr

rrrr

rrrr

rrr][

21222111

1221

22

2

11

2

2112

][][2sin4

)(2sin2

1

)(sin)(cos

µµϖ

µµωµµων

µµν

µνµν

×⋅=

×⋅+×⋅−−

⋅+⋅−

⋅+⋅=

r

rri

mm

mm



)()2(sin4

))(2(sin2

1

)(cos)(sin

2112

1221

22

2

11

2

µµνω

µµν

µνµν

rrr

rrrr

rrm

rr

×⋅±

⋅+⋅±

⋅⋅±=±

r

mmi

mmiRab

Survives for coupledchomophores withoutintrinsic rotary power

]Im[]0ˆˆ0Im[ ±±±±± ⋅=⋅= mmRab

rrµϕϕµ

Electric dipole of onechromophore and magneticdipole of the other

Intrinsic chirality of theindvidual chromophores

note: the magnitue of the magnetic dipole moments depends on thechoice of origin. The rotary strengt R will of course not!Here we assume: m is purely imaginary, µ real



Transition Dipole Coupling

( )( )

+⋅⋅

−⋅

+⋅

−⋅

+=

−

⋅= ∫∫

...34

1

''''''

)''()'(

4

1

5333

0

0

4444 34444 21

r

rrrr

r

rr

444 3444 21

r

rr

r

rv

r

rr

rr

jk

jkkjkj

jk

kj

jk

jkkj

jk

jkjk

jk

jk

jk

jk

r

rr

rr

rq

r

rq

r

qq

dVdVrr

rrV

µµµµµµ

πε

ρρ

πε

jrr

'rr

krr

''rr

jkrr

)'(rk

rρ

)''(rj

rρ

Coulomb Interaction

')'( Vdrqk ∫=r

ρ ')')('( Vdrrr kk ∫ −=rrrr

ρµ

Dipole-dipole Interaction


Transition Dipole Coupling

⋅⋅−

⋅=

53

0

))((3

4

1

jk

jkjjkk

jk

jk

jkr

rr

r

rrrrrrµµµµ

πεβ

O

O

N

NC

Cr

12

1µr

2µr

=

2

1

εβ

βεH

T

E

E

E

E

−

−=

−

+

νν

νν

β

β

νν

νν

cossin

sincos

cossin

sincos

0

0

2

1

)()2(sin4

2112 µµνω rrr

×⋅±=±rR

All quantities defined by transitiondipoles and distances betweenthem!


36−β

Validity of TDC for VCD

3,6-dioxo-5beta-cholanic acid methyl ester

712−β

Narayanan, U.; Keiderling, T. A., Coupled oscillator interpretation of the vibrational circular dichroism of several

dicarbonyl-containing steroids. J. Am. Chem. Soc. 1983, 105 (21), 6406-6411.

7,12-dioxo-5beta-cholanic acid

O

OH

H

H

H

O

OMe

H

H

H

H

O

OH

O

O


37−β

Validity of TDC for VCD


312−β

Narayanan, U.; Keiderling, T. A., Coupled oscillator interpretation of the vibrational circular dichroism of several

dicarbonyl-containing steroids. J. Am. Chem. Soc. 1983, 105 (21), 6406-6411.


predicted TDC too smallpredicted couplet not observed

H

H

H

H

O

OH

O

OH

H

H

H

O

OH

OO


( ) ( )T

ik

nn

n

ik

n E

E

E

E

α

β

β

α

=

L

MOM

L

O

1

111

~00

00

00~

Transition Dipole Couplingmultiple dipoles

O

O

N

NC

Cr

12

1µr

2µr

×⋅−⋅⋅

⋅=⋅= ∑∑∑

===

n

j

jjkj

n

j

jkj

n

s

skskkk rimmR111 2

Im]~~Im[ µαω

αµαµrrrr

×⋅−−= ∑ ∑

−

= +=

1

1 1

, )()(2

n

j

n

js

sjsjkskjcouplingk rrR µµααω rrrr

Exciton basis Site basis

0

real

=j

j

mr

rµ


Coupled transition dipoles LH2Bacteria 77K

B850

B800


Figure 4 A: Modeled (solid) and measured (dashed) absorption (top) and CD (bottom) spectra of a LH1 ring with 16-fold symmetry that contains both the BChl (QY, QX) and Soret (BY and BX) transitions and the carotenoid transitions (S0 → S2). The measured spectra from Rba. sphaeroides WT LH1 contain a mixture of carotenoids (spheroidene/spheroidenone). The absorption spectra are normalized on the QY band, and the CD spectra are scaled on the normalized absorption. For more details on measured spectra, see companion paper.44 B: Modeled absorption (top) and CD (bottom) spectra of a LH1 ring with 15 building blocks (solid), simulating a LH1 ring that is missing one BChl pair and one carotenoid and with 16 building blocks (dashed), simulating a closed LH1 ring. The insert shows a two-dimensional picture of the positions of the pigments in an open ring structure (in black balls) and in a closed ring structure (in white circles). The outer ring shows the centers of the BChls, and the inner ring, the centers of the carotenoids. The parameters used for the modeling for both A and B are listed in Tables 1 and 3.

Published in: Sofia Georgakopoulou; Rienk van Grondelle; Gert van der Zwan; J. Phys. Chem. B 2006, 110, 3344-3353.DOI: 10.1021/jp051794cCopyright © 2006 American Chemical Society


LH2 of plants

From the following article:Crystal structure of spinach major light-harvesting complex at 2.72 Å resolution

Zhenfeng Liu, Hanchi Yan, Kebin Wang, Tingyun Kuang, Jiping Zhang, Lulu Gui, Xiaomin An and Wenrui ChangNature 428, 287-292(18 March 2004)doi:10.1038/nature02373


Figure 1 Directions of the transition dipole moments of the Chls and carotenoids as assigned in the modeling program.

Published in: Sofia Georgakopoulou; Gert van der Zwan; Roberto Bassi; Rienk van Grondelle; Herbert van Amerongen; Roberta Croce;Biochemistry 2007, 46, 4745-4754.DOI: 10.1021/bi062031yCopyright © 2007 American Chemical Society

Coupled transition dipoles LH2Plants


Figure 2 WT LHCII monomer measured (solid line) and modeled (dashed line) absorption (top) and CD (bottom) spectra. CD spectra are scaled to the normalized absorption spectra. Also shown are the intrinsic CD signal of Chls (dash-dotted line) and the modeled spectrum with the intrinsic CD added to it (dotted line). The insert shows in more detail the region around 670 nm.

Published in: Sofia Georgakopoulou; Gert van der Zwan; Roberto Bassi; Rienk van Grondelle; Herbert van Amerongen; Roberta Croce;Biochemistry 2007, 46, 4745-4754.DOI: 10.1021/bi062031yCopyright © 2007 American Chemical Society

Coupled transition dipoles LH2Plants


Population time=0 fs

B850

B800


Source: http://cnx.org/content/m38277/latest/Original caption: Figure 4: CD spectra of samples with representative conformaitons. Adapted by permission from

N. Greenfield, Nat. Proto., 2006, 1, 6.

N

O

H

CH

R96°

xµππ∗

mnπ∗

UV transition dipoles of a peptidebond

ππ∗ nπ∗


εθ ∆≈ 3298]/ degree][[ 2dmolcm

Perfect α-helix:

residueper //10

residueper / degrees1039][ 23

222

cmmolel

dmolcmnm

≈∆→

⋅=

ε

θ

Absorption ECD

1 mOD signalfor OD 1 solution32

*

4

*

1010)230(

10)195(

−≈

≈

−

−

nm

nm

n π

ππ

ε

ε

32 1010 −− −=∆

ε

ε

Circular Dichroism – secondarystructure

ππ∗ nπ∗


Coupled Oscillator Modelpeptides and proteins

Electronic CD

1 mOD signalfor OD 1 solution

32 1010 −− −=∆

ε

ε

)()2(sin4

))(2(sin2

1

)(cos)(sin

2112

1221

22

2

11

2

µµνω

µµν

µνµν

rrr

rrrr

rrm

rr

×⋅±

⋅+⋅±

⋅⋅±=±

r

mmi

mmiRab

coupled electricdipoles

Electric dipole of onechromophore and magneticdipole of the other

NH

O

NH

O

mnπ∗

µππ∗


Coupled Oscillator Modelpeptides and proteins

)( 2112

12

µµβ

ωrrr

×⋅−

±∝ REE

VCD

VCD 0.05 mOD signalfor OD 1 solution

54 1010 −− −=∆

ε

ε O

O

N

NC

CR

12

1µr

2µr

K.-K. Lee, K.-I. Oh, H. Lee, C. Joo, H. Han and M. Cho, ChemPhysChem 2007, 8, 2218-2226.

Isotope labelling of one carbonyl increases energy difference between localC=O stretch trasitions (at constant β)and strongly reduces the VCD: ‚decouping‘


(LKKL)n

(LK)n

(K)n

α-helix

(β-sheet)

coil

T. A. Keiderling, J. Kubelka and J. Hilario in Vibrational Circular Dichroism of Biopolymers. Summary of Methods and Applications, Vol. Eds.: M. Brainman and V. Gregoriou), Marcel Dekker, New York, 2005, p. 253.

K=Lysine

L=LeucineN

O

N

O

N

VCD and secondary structure



nuclJelecJJ J

BORRM

HH

∂

∂

∂

∂−= ∑

αα ,,

0

h

Kinetic energy operator acting on nuclei and electrons (non-BornOppenheimer)

α

α

,

,

JJ

nuclJ

RMi

R&

h=

∂

∂

),,(),,()( RRrERRrRR

iRHCA

el

CA

elel

&rr&rr&rrhr

ψψ =

∂

∂−

−

∂∂+≈Ψ ∑

≠gs

BO

s

gs

BO

g

BO

sBO

gvib

CA

vibel RrEE

RirRRRr

&rr

hr&rr

)(/

)()(),,(, ψψψ

ψφ

Substitute operator by variable:

New separation of coordinates:

First order perturbation of wavefunction:


Q.M. Treatment Circular Dichroism

00,

0,

~~

==

≠=

∂

∂

∂

∂=

−

∂

∂= ∑

H

g

RJ

g

es

BO

s

gs

BO

g

BO

s

RJ

gJ

HR

EE

m

RI

rr

r

βα

β

α

αβ

ψψ

ψψψψ

Stephens 1985, 1988

gψ~ - Born Oppenheimer wavefunction in external H-field

Sum over all excited states s

Magnetic moment origin dependent

Implementation into Gaussian 98 – Chose polarizable basis set!


VCD Theory - Summary

Click!

# opt freq=vcd b3lyp/6-31g(d,p) geom=connectivity

Use polarizable basis functions (**)

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