Density Functional Implementation of the Computation of Chiroptical Molecular Properties With Applications to the Computation of CD Spectra Jochen Autschbach & Tom Ziegler, University of Calgary, Dept. of Chemistry University Drive 2500, Calgary, Canada, T2N-1N4 1

Motivation 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- principles quantum chemical methods will be an important tool to asssist chemical and biochemical research and enhance our under- standing of optical activity 2

Quantifying Optical Activity Methodology 3 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 with a chiral molecule perturbed electric & magnetic moments  is the opticalrotationparameter or CH 3 O

Sum-Over-States formalism yields Methodology 4 Excitation Frequencies   Rotatory Strengths R  frequency dependent optical rotation para- meter  ORD spectra Related to the CD spectrum electric transition dipole magnetic transition dipole

Direct computation of  and R with TDDFT Methodology 5 Frequency dependent electron density change (after FT)  = molecular orbitals, occupation # 0 or 1 Fourier-transformed density matrix due to the perturbation ( E(t) or B(t) )

Direct computation of  and R with TDDFT Methodology 6 RPA-type equation system for P, i  occ, a  virt X = vector containing all ( ai ) elements, etc… 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

Direct computation of  and R with TDDFT Methodology 7 Definitions: The F ’s are the eigenvectors of ,   its eigenvalues (  = excitation frequencies) Skipping a few lines of straightforward algebra,we obtain

Direct computation of  and R with TDDFT Methodology 8 Comparison with the Sum-Over-States Formula yields for R  Therefore consistent with definition of oscillator strength in TDDFT, obtained as

Implementation into ADF Excitation energies and oscillator strengths al- ready available in the Amsterdam Density Functional Code (ADF, see Only M ai matrix elements additionally needed for Rotatory Strengths ( , D, S, F already available) Computation of M ai by numerical integration Abelian chiral symmetry groups currently sup- ported for computation of CD spectra ( C 1, C 2, D 2 ) Implementation for  in progress (follows the available implementation for frequency dependent polarizabilities 9

Implementation 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 R  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 10

(R)-Methyloxirane Applications 11 Excit.ADF GGA a) ADF SAOP b) Other Ref [1] Other Ref [2] Expt. Ref [2] 1 E/eV f R /10 40 cgs /eV ff  R /10 40 cg s [1] TD LDA: Yabana & Bertsch, PRA 60 (1999), 1271 [2] MR-CI: Carnell et al., CPL 180 (1991), 477 a) BP86 triple-zeta + diff. Slater basis b) SAOP potential

(S,S)-Dimethyloxirane Applications 12 ADF CD Spectra simulation *) *) Assumed linewidth proportional to  E (approx eV), Gaussians centered at excitation energies reproducing R, ADF Basis “Vdiff” (triple-  + pol. + diff) Exp. spectrum / MR-CI simulation [1] R calc = 7.6 R exp. = 9.5 calc. predicts large neg. R for this excitation low lying Rydberg excitations, sensitive to basis set size / functional good agreement with exp. and MR-CI study for R of the 1 st excitation  E for GGA ~ 1eV too small, but well reproduced with SAOP potential [1] Carnell et al., CPL 179 (1994), 385

Cyclohexanone Derivatives Applications 13 H ?  CH 3 a)  E calc /eV R calc GGA b) R Other Ref [1] R Other Ref [2] R Expt. Ref [1] c) none 3.94 (4.3) b) 0000 H7H (4.3) (small) H9H (4.3) d) H 7 H (4.3) H 7 H 13 H (4.3) [1] CNDO: Pao & Santry, JACS 88 (1966), [2] Extended Hückel: Hoffmann & Gould, JACS 92 (1970), 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 C=O ~290 nm (4.4 eV)  * transition

HexaheliceneApplications 14 ADF CD Spectra simulation *) [1] TDDFT/Expt. Furche et al., JACS 122 (2000), 1717 Exp. / theor. study [1]  R exp = 331  R theo = 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

Chloro-methyl-aziridinesApplications 15 SAOP yields com- parable  E than GGA Exp. spectra quali- tatively well repro- duced, for 1a,1b magnitudes for  also comparable to experiment (+)Band at ~260 nm for 2 much stronger in the simulations (low experimental resolution ?) Blue shift for 1b is not 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 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 3 rd 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 progress 16