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.