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Department of Chemistry
Chapter 3
Theories of optical activity and circular dichroism
Department of Chemistry
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
Department of Chemistry
)()( 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
Department of Chemistry
)()( 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
Department of Chemistry
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
Department of Chemistry
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
Department of Chemistry
Field view of optical activity
η�φ
VORD(in phase)
VCD(π/2 phase shift)
Circular Birefringence Circular Dichroism
Department of Chemistry
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
Department of Chemistry
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.
Department of Chemistry
[ ] 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
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( )( )( )( ) ( )( )[ ] ( )( ))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
=⋅⋅∫
Department of Chemistry
( ) ( )( )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
Department of Chemistry
...'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…
Department of Chemistry
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
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Q.M. Treatment of Circular Dichroism
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
Department of Chemistry
−
−==⇒
+ 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
Department of Chemistry
−=
+
−
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
Department of Chemistry
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
Department of Chemistry
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
Department of Chemistry
chromophore 1 chromophore 2
=
2
1
εβ
βεH
coupled
+ϕ
−ϕ01
10
00
m+ m− µ+
µ−
Exciton Model of Circular Dichroism
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
Department of Chemistry
( )][][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
Exciton Model of Circular Dichroism
Department of Chemistry
)()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
Exciton Model of Circular Dichroism
Department of Chemistry
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
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
Department of Chemistry
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!
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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
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37−β
Validity of TDC for VCD
3,7-dioxo-5beta-cholanic acid
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.
3,12-dioxo-5beta-cholanic acid
predicted TDC too smallpredicted couplet not observed
H
H
H
H
O
OH
O
OH
H
H
H
O
OH
OO
Department of Chemistry
( ) ( )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
Department of Chemistry
Coupled transition dipoles LH2Bacteria 77K
B850
B800
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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
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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
Department of Chemistry
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
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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
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Population time=0 fs
B850
B800
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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∗
Department of Chemistry
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∗
Department of Chemistry
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∗
Department of Chemistry
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‘
Department of Chemistry
(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
Department of Chemistry
Q.M. Treatment of Circular Dichroism
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:
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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!
Department of Chemistry
VCD Theory - Summary
Click!
# opt freq=vcd b3lyp/6-31g(d,p) geom=connectivity
Use polarizable basis functions (**)