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VIBRATIONAL RAMAN OPTICAL ACTIVITY SPECTRA
OF CHIRAL MOLECULES
SANDRA LUBER AND MARKUS REIHER
Laboratorium fur Physikalische Chemie, ETH Zurich, Wolfgang-Pauli-Str. 10, 8093 Zurich, Switzerland, email:
{sandra.luber,markus.reiher}@phys.chem.ethz.ch
Vibrational Raman Optical Activity
• Vibrational Raman Optical Activity (ROA):probes the chirality of the molecule.
• Measured quantity: IR−IL, where IR and IL are the scattering intensitiescorresponding to incident right- and left-circularly polarized light.
• A scattering angle of 180◦ (backward scattering) is often employed.
• No generally applicable rules about the relationship between molecularstructure and ROA intensity differences are available⇒ quantum chemical calculations needed for interpretation of spectra.
• Till recently, the largest molecule for which ROA calculations with den-sity functional theory (DFT) were carried out was all-(S)-decaalanine(103 atoms)a by employing the mode-tracking protocolb. With our imple-mentation, ROA spectra of molecules with some hundred atoms can rou-tinely be calculatedc.
aC. Herrmann, K. Ruud, M. Reiher, ChemPhysChem 2006, 7, 2189.bM. Reiher, J. Neugebauer, J. Chem. Phys. 2003, 118, 1634.cS. Luber, M. Reiher, Chem. Phys. 2008, 346, 212; C. R. Jacob, S. Luber, M. Reiher, submitted.
ROA theory
• Semiclassical description: Molecule is treated quantum mechani-cally, the radiation classically.
• Time-dependent perturbation theory is applied for the calculationof the molecule–light interaction.
• The induced electric-dipole, electric-quadrupole and magnetic-dipole moments are needed⇒ three polarizability tensors in the far-from-resonanceapproximation:
– electric-dipole–electric-dipole tensor α,
– electric-dipole–magnetic-dipole tensor G′,
– electric-dipole–electric-quadrupole tensor A.
• Taking the perturbed time-dependent wavefunction to be real andomitting imaginary damping terms, α, G′, and A are given as(Hartree atomic units are employed throughout this work)
ααβ =∑
j 6=m,n
[
〈m|µα|j〉〈j|µβ|n〉
ωjn − ωL
+〈m|µβ|j〉〈j|µα|n〉
ωjm + ωL
]
,
G′αβ =
∑
j 6=m,n
[
〈m|µα|j〉〈j|mβ|n〉
ωjn − ωL
+〈m|mβ |j〉〈j|µα|n〉
ωjm + ωL
]
,
Aα,βγ =∑
j 6=m,n
[
〈m|µα|j〉〈j|θβγ|n〉
ωjn − ωL
+〈m|θβγ |j〉〈j|µα|n〉
ωjm + ωL
]
;
µα: α component of the electric-dipole moment operator;mα: α component of the magnetic-dipole moment operator;
θβγ : βγ component of the electric-quadrupole moment operator;|n〉, |j〉, |m〉: wavefunctions of initial, intermediate and final states;ωjn, ωjm: angular transition frequencies between states |j〉 andeither |n〉 or |m〉, respectively;ωL: angular frequency of the incident light.
• Within the Placzek polarizability theory, we obtain for the polariz-ability tensor elements and normal coordinates Qk
ααβ(Q) = (ααβ)0 +∑
k
(
∂ααβ
∂Qk
)
0
Qk.
• Averaging over all possible molecular orientations yields theROA invariants αG′, β(G′)2 and β(A)2.
• In the case of backward scattering, the ROA intensity difference isgiven as
(IR − IL)(180◦) ∝ 96[β(G′)2 +1
3β(A)2]
1
c,
where β(G′)2 and β(A)2 are written as
β(G′)2 =1
2(3ααβG′
αβ − αααG′ββ),
β(A)2 =1
2ωLααβǫαγδAγ,δβ .
ǫαγδ is the αγδ component of the third-rank antisymmetric (Levi–Civita) unit tensor and c the velocity of light.
Quantum chemical methods
• Structure optimizations, gradients and property tensors calculated withTURBOMOLE:
– DFT(BP86/RI)
– Ahlrichs’ TZVP (metal complexes) and TZVPP (L-tryptophan and 1,6-anhydro-β-D-glucopyranose) basis sets
• Interfaces:
(1) SNF — a program for the quantum chemical calculation of vibra-tional spectraa
(2) AKIRA — a program for the selective calculation of normal modesb
- Efficiency through numerical differentiation of analytical gradientsand the molecular property tensors along normal modes
- Massive-parallel calculation
aJ. Neugebauer, M. Reiher, C. Kind, B. A. Hess, J. Comput. Chem. 2002, 23, 895;www.reiher.ethz.ch/software/snf.
bM. Reiher, J. Neugebauer, J. Chem. Phys. 2003, 118, 1634; www.reiher.ethz.ch/software/akira.
The first ROA spectra of chiral metalcomplexes
• No experimental or calculated spectra were available for metal com-plexes.
• We calculated the first ROA spectra of chiral metal complexesa.
• Example: Λ-tris(acetylacetonato)cobalt(III)
O
O
O
Co
O
O
O
Calculated backscattering ROA spectrum (BP86/RI/TZVPP) of Λ-
tris(acetylacetonato)cobalt(III). The upper part of the spectrum shows the
plot without the A tensor contribution, the lower panel provides the full
reference spectrum.
• Example: dichloro(sparteine)zinc(II)
Cl
N
Zn
N
Cl
Calculated backscattering ROA spectrum (BP86/RI/TZVPP) of dichloro-
(sparteine)zinc(II). The upper part of the spectrum shows the plot without
the A tensor contribution, the lower panel provides the full reference spectrum.
• Deviations due to neglect of the A tensor contribution are small in thewavenumber region below 2000 cm−1, but may be larger at higher wavenumbers.
• The changes due to the omitted A tensor contribution are smaller thanthe ones observed for organic moleculesb.
aS. Luber, M. Reiher, Chem. Phys. 2008, 346, 212.bS. Luber, C. Herrmann, M. Reiher, J. Phys. Chem. B 2008, 112, 2218.
Intensity-tracking for Raman and ROAspectroscopy
• Calculations of Raman and especially ROA spectra for large molecules arecomputationally expensive.
• Intensity-trackinga: modes with high intensity are selectively calculated⇒ reduction of computational effort.
• Complement to the mode-tracking protocol for the selective calculationof normal modes via subspace iteration schemesb.
• The accuracy of the normal modes and the corresponding intensities canbe controlled by suitable convergence criteria.
• The starting guess vector for the intensity-tracking calculation is a hypo-thetical vibration which carries the maximum Raman/ROA intensityc.
The hypothetical modes of L-tryptophan with maximum Raman (left) and
[backscattering] ROA (right) intensity.
0 1000 2000 3000
wavenumber / (1/cm)
rela
tiv
e in
ten
sity
Iteration 10: 51 basis vectors
Iteration 12: 59 basis vectors
Iteration 2: 2 basis vectors
conventional full calculation
Iteration 1: 1 basis vector
Iteration 7: 20 basis vectors
Iteration 9: 45 basis vectors
converged
Iteration 14: 63 basis vectors intensity−tracking
Approximate ROA spectra of L-tryptophan obtained with an ROA intensity-
tracking (bottom and middle) and a conventional full calculation (top).
aK. Kiewisch, J. Neugebauer, M. Reiher, J. Chem. Phys. 2008, 129, 204103; S. Luber, J. Neuge-bauer, M. Reiher, J. Chem. Phys. 2009, 130, 064105.
bM. Reiher, J. Neugebauer, J. Chem. Phys. 2003, 118, 1634; M. Reiher, J. Neugebauer,Phys. Chem. Chem. Phys. 2004, 6, 4621.
cS. Luber, M. Reiher, ChemPhysChem 2009, DOI: 10.1002/cphc.200900255.
Acknowledgements
This work has been supported by the Swiss National Science Foundation(project 200020-113479).
Solvent effects in sugar ROA spectra
• Our example: 1,6-anhydro-β-D-glucopyranose (AGP)a
−0.002
−0.0015
−0.001
−0.0005
0
0.0005
0.001
0.0015
0.002
600 800 1000 1200 1400 1600
(I R
− I L
) / (
Å4 a
.m.u
.−1 )
wavenumber / cm−1
24
16
17
51
93
8
10
7 6
1120
21
15
18
19
12
13
14
g−g+t
The optimized (TZVPP/RI/BP86) structure of the g−g+t conformer of AGP in the
chair conformation (right-hand side) and the corresponding backscattering ROA
spectrum (TZVPP/RI/BP86; left-hand side); the line spectrum is scaled by a factor
of 0.05.
• Solvent effects are included via the COSMO continuum model and explicitsolvation with water molecules:
−0.0015
−0.001
−0.0005
0
0.0005
0.001
0.0015
0.002
600 800 1000 1200 1400 1600
(I R
− I L
) / (
Å4 a
.m.u
.−1 )
wavenumber / cm−1
−0.0015
−0.001
−0.0005
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
600 800 1000 1200 1400 1600
(I R
− I L
) / (
Å4 a
.m.u
.−1 )
wavenumber / cm−1
cosmo+expl. solvation
cosmo
Calculated backscattering ROA spectra (TZVPP/RI/BP86) of the g−g+t con-
former of AGP in the chair conformation obtained by employing the continuum
model COSMO (left-hand side) and by explicit solvation with water molecules
and COSMO (middle); the line spectra are scaled by 0.05. The optimized
(TZVPP/RI/BP86) structure of the explicitly solvated g−g+t conformer is shown
on the right-hand side.
• The final spectra are constructed by taking all possible chair conforma-tions into account (no weighting of the conformers is included since noimprovement is found when weighting the spectra according to the pop-ulations obtained from electronic energy or Gibbs enthalpy differences):
0
2
4
6
8
10
12
14
16
600 700 800 900 1000 1100 1200
Ram
an a
ctiv
ity /
(Å4 a
.m.u
.−1 )
wavenumber / cm−1
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
600 700 800 900 1000 1100 1200
(I R
− I L
) / (
Å4 a
.m.u
.−1 )
wavenumber / cm−1
600 700 800 900 1000 1100 1200
Ram
an a
ctiv
ity
wavenumber / cm−1
0
600 700 800 900 1000 1100 1200
(I R
− I L
)
wavenumber / cm−1
Raman (top) and ROA (bottom) spectra of the backscattering direction
(TZVPP/RI/BP86) obtained by overlapping the spectra of all explicitly solvated
chair conformers (left-hand side) and the experimental spectra (right-hand side;
reproduced from L. D. Barron et al. Carbohydr. Res. 1991, 210, 39-49; the line
spectra are scaled by 0.1).
aS. Luber, M. Reiher, J. Phys. Chem. A 2009, in press.
Conclusion
(1) We present a semi-numerical density-fitting-based implementation forthe calculation of ROA spectra. With this set-up, it is possible to obtainforce fields and ROA spectra of large molecules with large basis setsa.
(2) The first ROA spectra of chiral metal complexes were calculateda.
(3) The determination of Raman/ROA intensity-carrying modes, which are hy-pothetical modes with maximum Raman/ROA intensity, has been devel-opedb.
(4) The first intensity-tracking calculation, in which selectively normal modeswith high ROA intensity have been determined, has been performed lead-ing to an additional saving in computational timeb.
(5) For the ROA spectra calculation of the sugar molecule 1,6-anhydro-β-D-glucopyranose, the inclusion of explicit solvation with water moleculeshas been found to be important in order to get good agreement with ex-perimental datac.
aS. Luber, M. Reiher, Chem. Phys. 2008, 346, 212.bS. Luber, M. Reiher, ChemPhysChem 2009, DOI: 10.1002/cphc.200900255.cS. Luber, M. Reiher, J. Phys. Chem. A 2009, in press.
Determination of the Ramanintensity-carrying modes
• The Raman intensity for the normal coordinate Qs is given as
Is = 123∑
l=1
(
∂αll
∂Qs
)2
0
+3
2
3∑
l, k = 1
l 6= k
(
∂αll
∂Qs
)
0
(
∂αkk
∂Qs
)
0
+21
2
3∑
l, k = 1
l 6= k
(
∂αlk
∂Qs
)2
0
,
where the αlk denote the components of the polarizability tensor and thederivatives are taken at the molecule’s equilibrium structure.
• Writing(
∂αlk
∂Qs
)
0=∑3M
i=1
(
∂αlk
∂Rmw
i
)
0
(
∂Rmw
i
∂Qs
)
0=∑3M
i=1 Umwlk,i Lsi = Umw
lk · Ls ,
the condition that the intensity should be stationary with respect to the
components of the guess vector L(0)s , i.e., ∂Is/∂L
(0)sj = 0 for every j, leads
to the eigenvalue equation
MLs = asLs
with Mij = 8∑3
l=1 Umwll,i Umw
ll,j +∑3
l.k = 1
l 6= k
Umwll,i Umw
kk,j +7∑3
l, k = 1
l 6= k
Umwlk,i U
mwlk,j and
the eigenvalue ak determines the Raman intensity of the vibration alongthe hypothetical mode Qs (analogously for ROA intensities)a.
• Six hypothetical modes with high Raman intensity are found, which arethe so-called intensity-carrying modes.
aS. Luber, M. Reiher, ChemPhysChem 2009, DOI: 10.1002/cphc.200900255.