1. A TDDFT study on the dichroism in the photoelectron angular distribution from a chiral transition metal compound
M. Stener
Dipartimento di Scienze Chimiche
Università degli Studi di Trieste
Via L. Giorgieri 1, TRIESTE - ITALY
Gordon Research Conference on Photoions, Photoionization & Photodetachment
January 31st - February 5th, 2010 Hotel Galvez Galveston, TX

2. (RANDOMLY ORIENTED MOLECULES) PARTIAL DIFFERENTIAL CROSS SECTION:
GAS PHASE EXPERIMENT
(RANDOMLY ORIENTED MOLECULES) PARTIAL DIFFERENTIAL CROSS SECTION:
In this work only Electric Dipole (E1) transition moments are considered:

3. CHIRAL MOLECULES AND CIRCULARLY POLARIZED LIGHT
: Cross section
D: Dichroism
: Asymmetry parameter
emission angle: between photoelectron k and light propagation
mr: +1 or -1 for left/right circular polarization
D has opposite sign for enantiomeric pairs
Dichroism D: Circular Dicroism in Angular Distribution (CDAD)

4. M. Stener G. Fronzoni and P. Decleva Chem. Phys., 361, 49 - 60 (2009).
Theoretical Method
Esplicit treatment of photoelectron continuum
Multicentric B-spline basis set
Formalism: TDDFT
Parallel implemetation
Large matrices dim(H)  20000, 1 energy point: 1h with 256 cpu
M. Stener, G. Fronzoni and P. Decleva, J. Chem. Phys., (1-11) (2005).
M. Stener G. Fronzoni and P. Decleva Chem. Phys., 361, (2009).
What is new?
First TDDFT calculation of dichroic parameter D
First application on a chiral transition metal compound
First calculation of dichroism D over autoionization resonance (only TDDFT can do it!)

5. Previous applications (Kohn-Sham) Circular Dichroism in Angular Distribution of Photoelectrons from Chiral Molecules: S(-) methyl-oxirane
Good agreement KS Theory vs. Exp.
Dichroism decays to zero within few eVs above threshold
S. Stranges, S. Turchini, M. Alagia, G. Alberti, G. Contini, P. Decleva, G. Fronzoni, M. Stener, N. Zema and T. Prosperi
J. Chem. Phys (1-6) (2005).

6. Chiral transition metal compound: D-Co(acac)3
D3 point group symmetry

7. PES D-Co(acac)3 C
B’’ K L B’ M

8. Electronic structure: D-Co(acac)34p 3d 3p
LP-
LP+

9. PES D-Co(acac)3 K K: 30e L L: 18a1 + 15a2 M M: 29e + 14a2 B’ B’: 28e
B”: 27e + 17a1
B’’

10. Electronic structure: D-Co(acac)3
30e: Co 3d – 3p antibonding

11. Electronic structure: D-Co(acac)3
18a1: Co 3d

12. Electronic structure: D-Co(acac)3
15a2: 3p

13. Electronic structure: D-Co(acac)3
29e: Co 3d – 3p bonding

14. Electronic structure: D-Co(acac)3
14a2: ligand LP-

15. Electronic structure: D-Co(acac)3
28e: ligand LP-

16. Electronic structure: D-Co(acac)3
17a1: ligand LP+

17. Electronic structure: D-Co(acac)3
27e: Co 3d + ligand LP+
bonding

18. Electronic structure: D-Co(acac)3
31e: Co 3d + ligand LP+
antibonding

19. Co(acac)3: “Giant Autoionization”E
Direct ionization
E = 0
(30e)-1 M+
Autoionization
Co(3p)-1 Co(31e)+1 M*
Excitation
“Giant” because the same principal Q.N.: Co 3p → Co 3d
GS: Cr(3p)6 (…) (30e)6 (31e)0 M

20. Dichroism: D-Co(acac)3
30e: Co 3d – 3p antibonding
18a1: Co 3d
“Giant” autoionization: Co 3p → 3d
Similar!
Different!

21. Dichroism: D-Co(acac)3
28e: ligand LP-
Small Co 3d contribution:
Very weak resonance in cross section
But … very strong ‘window’ resonance in dichroism!!!
D is very sensitive!

22. Dichroism: D-Co(acac)3
Theory (TDDFT) vs experiment:
Preliminar experiment: D. Catone (private communication)
Elettra Sinchrotron (Trieste ITALY)

23. Dichroism: D-Co(acac)3
Theory (TDDFT) vs experiment:
Preliminar experiment: D. Catone (private communication)
Elettra Sinchrotron (Trieste ITALY)
Complete disagreement!!!

24. Dichroism: D-Co(acac)3
Theory (TDDFT) vs experiment:
Preliminar experiment: D. Catone (private communication)
Elettra Sinchrotron (Trieste ITALY)

25. Dichroism: D-Co(acac)3
Theory (TDDFT) vs experiment: Preliminar experiment: D. Catone (private communication) at Elettra Sinchrotron (Trieste ITALY)
Alternative assignment of B’ and B” bands: better agreement!!!

26. Cross section near the resonance: D-Co(acac)3
Theory (TDDFT) vs experiment: Preliminar experiment: D. Catone (private communication) Elettra Sinchrotron (Trieste ITALY)

27. Conclusions Method: Parallel multicenter B-spline TDDFT continuum.
Calculation of Dichroism (D) of D-Co(acac)3.
Strong sensistivity of D parameter.
Comparison with preliminar experimental dichroism, possible revision of previous assignment.
Co 3p → 3d autoionization: Dichroism sensitive even for ligand orbitals.
Future perspectives: dichroism experiment on Co 3p → 3d autoionization.

28. Thank you for your attention!
Acknowledgments:
Trieste University:
Prof. Piero Decleva
Prof. Giovanna Fronzoni
Dott. Daniele Toffoli
Dott. Devis Di Tommaso
Elettra Sinchrotron Trieste:
Dott. Daniele Catone
Dott. Tommaso Prosperi
Dott. Stefano Turchini
Thank you for your attention!

30. Additional slides

32. Density Functional Theory for Photoionization: the Kohn-Sham approach
hKS : bound and continuum states can be extracted, and photoionization parameters calculated (, , D)
Well known limitation of the KS scheme:
It is static: the response effects to the external time dependent electromagnetic field are neglected

33. Basis set approach The main issue is proper basis set choice
B-splines: piecewise polynomials defined over an arbitrary grid
Polynomial order k
-Knot sequence {t0  t1 … tn} over [t0, tn] = [0, Rmax]

34. B-spline functions

35. One center expansion (OCE)
All functions centered on a common origin 0
Multicenter expansion (LCAO)
{ (r0) }  { 1(r1) }  …  { p(rp) }
OCE: very stable and robust, shows smooth but slow convergence with LMAX0
LCAO: converges much more quickly, but less stable, careful choice of numerical parameters. The basis becomes easily overcomplete

37. Continuum: Kohn-Sham In the basis Hc = ESc
Bound states : standard diagonalization
Continuum states: Least Squares Approach
A(E) = H – ES, N0 lowest eigenvalues ai  0
Works fine, even with N0 a few hundred
Poisson equation VC = -4
is solved in the same basis. Gives the coulomb potential VC, avoiding the need of two electron integrals.

38. Linear response : general theory
External TD perturbation, with  frequency (dipole)
Induced density by the external field
Dielectric susceptibility, not easy to calculate

39. TDDFT: general theory
TDDFT: instead of , use S of a model system of non-interacting electrons and a modified external potential: SCF
Coupled, but linear!
K(r,r’) (kernel)

40. TDDFT: Direct (not iterative) algorithm
Exploit linearity of the problem:
defines the kernel K
defines the susceptibility 
The Response Equation becomes:
To solve : represent the response equation in the B-spline basis set
M. Stener, G. Fronzoni and P. Decleva, J. Chem. Phys., (1-11) (2005).

41. Properties Dynamical polarizability: Total cross section:
Partial cross section:

42. Well known limitation of the KS scheme:
It is static: the response effects to the external time dependent electromagnetic field are neglected
The TDDFT includes such response effects:
better agreement with experiment
New effects can be modelled by theory: Autoionization

43. Cr(CO)6: Autoionization analysis
“Giant” autoionization:
Cr 3p → Cr 3d
Parallel implementation:
M. Stener G. Fronzoni and P. Decleva Chem. Phys., 361, (2009).

44. Explicit expressions for s, b and D
Angular momentum transfer formalism, N. Chandra, J. Phys. B, 20 (1987) 3405.

45. Explicit expressions for s, b and D
Angular momentum transfer formalism, N. Chandra, J. Phys. B, 20 (1987) 3405.

46. Explicit expressions for s, b and D
Angular momentum transfer formalism, N. Chandra, J. Phys. B, 20 (1987) 3405.

47. Explicit expressions for s, b and D
Angular momentum transfer formalism, N. Chandra, J. Phys. B, 20 (1987) 3405.

48. Photoionization from chiral molecules
Linearly polarized light
Chiral molecule, Circularly polarized light
Forward-Backward asymmetry in the angular distribution
Or, switching the polarization of the light at the magic angle P2(cos)=0

49. Electronic structure: D-Co(acac)3
acac ligand 4p 3p
LP-
LP+