Density Functional Density Functional Implementation of the Implementation of the Computation of Chiroptical Computation of Chiroptical Molecular Properties Molecular Properties

With Applications to the With Applications to the Computation of CD SpectraComputation of CD Spectra

Jochen Autschbach & Tom Ziegler, University of Calgary, Dept. of Chemistry University Drive 2500, Calgary, Canada, T2N-1N4Email: jochen@cobalt78.chem.ucalgary.ca

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MotivationMotivation Almost all biochemically relevant substances

are optically active CD (circular dichroism) and ORD (optical

rotation dispersion) spectroscopy are important methods in experimental research

Interpretation of spectra can be difficult, overlapping CD bands obscure the spectra …

Prediction of chiroptical properties by first-Prediction of chiroptical properties by first-principles quantum chemical methods will be an principles quantum chemical methods will be an important tool to asssist chemical and important tool to asssist chemical and biochemical research and enhance our under-biochemical research and enhance our under-standing of optical activitystanding of optical activity

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GDFGDFGSDGFDGFD

Quantifying Optical Activity Quantifying Optical Activity MethodologyMethodology

μr mr

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electric dipole moment in a time-dependent magnetic field (B of light wave)

magnetic dipole moment in a time-dependent electric field (E of light wave)

Light-Wave interacts witha chiral molecule

r μ '=−β

c∂B∂t

; r m '=+βc

∂E∂t

perturbedelectric &magneticmoments

is theis theoptical optical rotationrotationparameterparameter

orCH3

O

GDFGDFGSDGFDGFD

Sum-Over-States formalism yields Sum-Over-States formalism yields MethodologyMethodology

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β =2c3

Rλ0ωλ0

2 −ω2λ∑

Excitation Frequencies

Rotatory Strengths R

Rλ0 =Im(r μ 0λ ⋅ r m λ0)

frequency dependentoptical rotation para-meter ORD spectra

Related tothe CDspectrum

electrictransitiondipole

magnetictransitiondipole

R0λ =const.× dE⋅ ΔεECD Band

∫

GDFGDFGSDGFDGFD

Direct computation of Direct computation of and R with TDDFT and R with TDDFT MethodologyMethodology

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Frequency dependent electron density change (after FT)

ρ'(ω) = Pia(ω)ϕiϕa*

a∑

i∑ = molecular orbitals,

occupation # 0 or 1

iaai

aiai

PYPX

==

)()( * −= aiia PP Fourier-transformed density matrixdue to the perturbation (E(t) or B(t))

r μ ' (ω) = (Xai +Yai)⋅(−er r )aia

virt∑i

occ∑r m ' (ω) = (Xai −Yai)⋅(− e

2cr r ׈ p )ai

a

virt∑i

occ∑

GDFGDFGSDGFDGFD

Direct computation of Direct computation of and R with TDDFT and R with TDDFT MethodologyMethodology

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A BB A

⎛ ⎝ ⎜ ⎞

⎠ −ω−1 00 1

⎛ ⎝ ⎜ ⎞

⎠ ⎡ ⎣ ⎢

⎤ ⎦ ⎥

XY

⎛ ⎝ ⎜ ⎞

⎠ =VW

⎛ ⎝ ⎜ ⎞

⎠

RPA-type equation system for P, iocc, a virt

X = vector containingall (ai) elements, etc…

iaai VW = matrix elements of the external perturbation,(-dependent Hamiltonian due to E(t) or B(t))

A,B are matrices. They contain of the response of the system due to the perturbation (first-order Coulomb and XC potential)

We use the ALDA Kernel (first-order VWN potential) for XC

GDFGDFGSDGFDGFD

Direct computation of Direct computation of and R with TDDFT and R with TDDFT MethodologyMethodology

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S=−(A −B)−1 ; Ω =−S1/2(A +B)S−1/2Definitions:

The F’s are the eigenvectors of ,

its eigenvalues (= excitation frequencies)

Skipping a few lines of straightforward algebra,we obtain

β =−2Im(ωDS−1/ 2[Ω −ω]−1S+1/ 2M)

Dai =(−er r )ai dipole moment matrix elements

Mai =(− e2c

r r ׈ p ) magnetic moment matrix elements

[ω2 −Ω2]−1 =− Fλ ⊗Fλ+

ωλ2 −ω2

λ∑

GDFGDFGSDGFDGFD

Direct computation of Direct computation of and R with TDDFT and R with TDDFT MethodologyMethodology

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Comparison with the Sum-Over-States Formula yields for R

R0λ =−Im(ωλDS−1/2Fλ ⋅Fλ+S+1/ 2M)

Therefore

r μ 0λ =ωλ

−1/2DS−1/2Fλ r m 0λ =ωλ

3/2MS+1/2Fλ

consistent with definition of oscillator strength in TDDFT,obtained as

f0λ =23|DS−1/ 2Fλ |2

Implementation into ADFImplementation into ADF Excitation energies and oscillator strengths al-

ready available in the Amsterdam Density Functional Code (ADF, see www.scm.com)

Only Mai matrix elements additionally needed for Rotatory Strengths (, D, S, F already available)

Computation of Mai by numerical integration Abelian chiral symmetry groups currently sup-

ported for computation of CD spectra (C1, C2, D2) Implementation for in progress (follows the

available implementation for frequency dependent polarizabilities

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Implementation into ADFImplementation into ADF

Additionally, the velocity representations for the rotatory and oscillator strengths have been implemented (matrix elements ai)

Velocity form of R is origin-independent Differences between Rμ and R typically ~ 15% for

moderate accuracy settings in the computations Computationally efficient, reasonable accuracy for

many applications Suitable Slater basis sets with diffuse functions

need to be developed for routine applications

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GDFGDFGSDGFDGFD

(R)-Methyloxirane (R)-Methyloxirane ApplicationsApplications

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Excit. ADF GGA a)

ADFSAOP b)

OtherRef [1]

OtherRef [2]

Expt.Ref [2]

1 E/eV

f6.050.011

7.110.013

6.00.012

6.40.0004

7.120.025

R/1040cgs -10.2 -13.4 -23.0 -2.66 -11.82-4 <E>/eV 6.59 7.69 6.5 7.3 7.75

f 0.047 0.061 0.044 0.0012 0.062R/1040cgs +9.75 +14.7 +23.0 +2.24 11.8

[1] TD LDA: Yabana & Bertsch, PRA 60 (1999), 1271[2] MR-CI: Carnell et al., CPL 180 (1991), 477a) BP86 triple-zeta + diff. Slater basis b) SAOP potential

CH3

H

H

HO

GDFGDFGSDGFDGFD

(S,S)-Dimethyloxirane (S,S)-Dimethyloxirane ApplicationsApplications

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H

H

H3C

CH3O

ADF CD Spectra simulation *)

*) Assumed linewidth proportional to E (approx. 0.15 eV), Gaussians centered at excitation energies reproducing R , ADF Basis “Vdiff” (triple- + pol. + diff)

Exp. spectrum / MR-CI simulation [1]

Rcalc = 7.6

Rexp. = 9.5 calc. predicts large neg.R for this excitation

low lying Rydberg excitations, sensitive to basis set size / functionalgood agreement with exp. and MR-CI study for R of the 1st excitationE for GGA ~ 1eV too small, but well reproduced with SAOP potential

[1] Carnell et al., CPL 179 (1994), 385

GDFGDFGSDGFDGFD

Cyclohexanone Derivatives Cyclohexanone Derivatives ApplicationsApplications

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H? CH3 a) Ecalc/eV Rcalc

GGA b)

R OtherRef [1]

R OtherRef [2]

R Expt.Ref [1] c)

none 3.94 (4.3) b) 0 0 0 0

H73.96 (4.3) 0.27 0.00 9.92 +(small)

H93.96 (4.3) -1.39 -2.26 -15.11 - d)

H7H133.96 (4.3) +1.46 +3.6 +5.53 +1.7

H7H13H83.99 (4.3) +4.36 +5.3 +6.36 +6.2

[1] CNDO: Pao & Santry, JACS 88 (1966), 4157. [2] Extended Hückel: Hoffmann & Gould,JACS 92 (1970), 1813. a) Numbered hydrogens substituted with methyl groups. Same geometries used than in[1],[2] b) BP86, triple-zeta Slater basis, numbers in parentheses: SAOP functional, SAOP R’s almost identical c) As quoted in [1]. Exp. values are computed from ORD spectra d) magnitude not known

O

H7

H8

H10

H9H11

H12

H13H14

C=O ~290 nm (4.4 eV) * transition

GDFGDFGSDGFDGFD

Hexahelicene Hexahelicene ApplicationsApplications

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ADF CD Spectra simulation *)

[1] TDDFT/Expt. Furche et al., JACS 122 (2000), 1717

Exp. / theor. study [1]

Rexp = 331Rtheo = 412

*) preliminary Results with ADF Basis IV (no diff.)

Shape of the spectrum equivalent to the TDDFT and exp. spectra published in [1]magnitude of R‘s smaller than exp., in particular for the short-wavelength excitations (TDDFT in [1] has too large R ‘s for the “B” band, too small for “E” band) GGA / SAOP yield qualitatively similar results

GDFGDFGSDGFDGFD

Chloro-methyl-aziridines Chloro-methyl-aziridines ApplicationsApplications

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NCl

CH3

N

CH3

Cl

CH3Cl

CH3

N

SAOP yields com-parable E thanGGAExp. spectra quali-tatively well repro-duced, for 1a,1bmagnitudes for also comparableto experiment(+)Band at ~260 nm for 2 much strongerin the simulations(low experimental resolution ?)Blue shift for 1b isnot reproduced

1a

1b

2

GGA, shifted +0.7 eV

ADF simulation *) Exp. Spectra [1]

*) BP86 functional, ADF Basis “Vdiff” Triple-z +pol. + diff. basis

[1] in heptane, Shustov et al., JACS 110 (1988), 1719.

Summary and OutlookSummary and Outlook Rotatory strengths are very sensitive to basis set size

and the chosen density functional GGA excitation energies are systematically too low.

The SAOP potential is quite accurate for small hydrocarbon molecules with large basis sets, but not so accurate for 3rd row elements. Standard GGAs yield comparable results for these elements.

Qualitative features of the experimental CD spectra are well reproduced in particular for low lying excitations.

Solvent effects can be important in order to achieve realistic simulations of CD spectra. Currently, solvent effects are neglected.

Implementation for ORD spectra in progress16