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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, 34127 TRIESTE - ITALY
Gordon Research Conference on Photoions, Photoionization &
Photodetachment
January 31st - February 5th, 2010 Hotel Galvez Galveston, TX
GAS PHASE EXPERIMENT
(RANDOMLY ORIENTED MOLECULES) PARTIAL
DIFFERENTIAL CROSS SECTION:
e-
d I
 , 
d
k
h
IN-1
M

M+I
In this work only Electric Dipole (E1) transition moments are considered:
d I
 ,   4 2 I(k) T 0N
d
2
 4  
2
()
k
t I
2
CHIRAL MOLECULES AND CIRCULARLY
POLARIZED LIGHT
d I
 I ( )  1

,  
1   I ( ) P2 cos   mr DI  P1 cos 

d
4  2

: Cross section
: Asymmetry parameter
D: Dichroism
•  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)
Theoretical Method
1. Esplicit treatment of photoelectron continuum
2. Multicentric B-spline basis set
3. Formalism: TDDFT
4. Parallel implemetation
5.
M. Stener, G. Fronzoni and P. Decleva, J.
Chem. Phys., 122 234301(1-11) (2005).
M. Stener G. Fronzoni and P. Decleva
Chem. Phys., 361, 49 - 60 (2009).
Large matrices dim(H)  20000, 1 energy point: 1h with 256 cpu
What is new?
1. First TDDFT calculation of dichroic parameter D
2. First application on a chiral transition metal compound
3. First calculation of dichroism D over autoionization resonance
(only TDDFT can do it!)
Previous applications (Kohn-Sham) Circular Dichroism in
Angular Distribution of Photoelectrons from Chiral
Molecules: S(-) methyl-oxirane
O
H
CH3
0.05
14a (II)
11a (II)
0.00
0.1
-0.05
D
H
H
0.2
0.0
-0.10
-0.1
2. Dichroism decays to zero
within few eVs above
threshold
12a (II)
15a (II)
0.10
0.1
D
0.05
0.00
0.0
-0.05
-0.10
-0.1
-0.15
13a (II)
16a (II)
0.05
0.05
0.00
D
1. Good agreement KS Theory
vs. Exp.
-0.15
0.00
S. Stranges, S. Turchini, M. Alagia, G. Alberti, G.
Contini, P. Decleva, G. Fronzoni, M. Stener, N.
Zema and T. Prosperi
J. Chem. Phys. 122 244303 (1-6) (2005).
-0.05
-0.05
-0.10
-0.10
0
10
20
30
-0.15
0
40
10
Photoelectron Energy (eV)
20
30
40
Chiral transition metal compound: D-Co(acac)3
D3 point group symmetry
PES D-Co(acac)3
C
B’’
K
L B’
M
Electronic structure: D-Co(acac)3
KS Eigenvalues (eV)
acac
-8
Co(acac)3
Co
4
3d
-10
3
LP-
-12
(acac)3
LP+
PES D-Co(acac)3
KS Eigenvalues (eV)
-8
-9
-10
-11
K
K: 30e
L
L: 18a1 + 15a2
M
M: 29e + 14a2
B’
B’: 28e
-12
B’’
B”: 27e + 17a1
Electronic structure: D-Co(acac)3
KS Eigenvalues (eV)
-8
-9
-10
-11
-12
30e: Co 3d – 3 antibonding
Electronic structure: D-Co(acac)3
KS Eigenvalues (eV)
-8
-9
-10
-11
-12
18a1: Co 3d
Electronic structure: D-Co(acac)3
KS Eigenvalues (eV)
-8
-9
-10
-11
-12
15a2: 3
Electronic structure: D-Co(acac)3
-8
KS Eigenvalues (eV)
29e: Co 3d – 3 bonding
-9
-10
-11
-12
Electronic structure: D-Co(acac)3
-8
KS Eigenvalues (eV)
14a2: ligand LP-
-9
-10
-11
-12
Electronic structure: D-Co(acac)3
-8
KS Eigenvalues (eV)
28e: ligand LP-
-9
-10
-11
-12
Electronic structure: D-Co(acac)3
-8
KS Eigenvalues (eV)
17a1: ligand LP+
-9
-10
-11
-12
Electronic structure: D-Co(acac)3
-8
KS Eigenvalues (eV)
27e: Co 3d + ligand LP+
-9
-10
-11
-12
bonding
Electronic structure: D-Co(acac)3
-8
KS Eigenvalues (eV)
31e: Co 3d + ligand LP+
-9
-10
-11
-12
antibonding
Co(acac)3: “Giant Autoionization”
E
E=0
(30e)-1 M+
Direct ionization
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
Dichroism: D-Co(acac)3
30e: Co 3d – 3 antibonding
18a1: Co 3d
20
25
Cross Section (Mb)
20
D-Co(acac)3 18a1
KS
TDDFT
“Giant”
autoionization
: Co 3p → 3d
15
10
5
Cross Section (Mb)
D-Co(acac)3 30e
15
10
5
0
0
Asymmetry Parameter ()
Similar!
1
1
0
-1
-1
0.1
0.1
Dichroism (D)
Asymmetry Parameter ()
0
Dichroism (D)
KS
TDDFT
0.0
0.0
-0.1
-0.1
-0.2
-0.2
0
20
40
60
80
100
Different!
Photon Energy (eV)
0
20
40
60
80
100
Photon Energy (eV)
Fig. 1
Fig. 2
Dichroism: D-Co(acac)3
25
D-Co(acac)3 28e
Cross Section (Mb)
20
28e: ligand LP-
KS
TDDFT
15
10
5
Asymmetry Parameter ()
0
Small Co 3d contribution:
1. Very weak resonance in cross
section
2. But … very strong ‘window’
resonance in dichroism!!!
3. D is very sensitive!
1
0
Dichroism (D)
-1
0.1
0.0
-0.1
-0.2
0
20
40
60
80
100
Photon Energy (eV)
Fig. 6
Dichroism: D-Co(acac)3
Theory (TDDFT) vs experiment:
Preliminar experiment: D. Catone (private communication)
Elettra Sinchrotron (Trieste ITALY)
Dichroism: D-Co(acac)3
Theory (TDDFT) vs experiment:
Preliminar experiment: D. Catone (private communication)
Elettra Sinchrotron (Trieste ITALY)
Complete
disagreement!!!
Dichroism: D-Co(acac)3
Theory (TDDFT) vs experiment:
Preliminar experiment: D. Catone (private communication)
Elettra Sinchrotron (Trieste ITALY)
Dichroism: D-Co(acac)3
Theory (TDDFT) vs experiment: Preliminar experiment: D. Catone
(private communication) at Elettra Sinchrotron (Trieste ITALY)
0.2
Dichroism (D)
D-Co(acac)3 Band B' (17a1)
0.1
Alternative assignment of B’ and
B” bands: better agreement!!!
0.0
-0.1
-0.2
Dichroism (D)
D-Co(acac)3 Band B" (27e + 28e)
0.1
0.0
-0.1
-0.2
0
20
40
60
Photon Energy (eV)
80
100
Cross section near the resonance: D-Co(acac)3
Theory (TDDFT) vs experiment: Preliminar experiment: D. Catone
(private communication) Elettra Sinchrotron (Trieste ITALY)
D-Co(acac)3
Cross Section (Mb)
10
K
L
M
5
0
58
60
62
64
66
68
70
72
Photon Energy (eV)
D-Co(acac)3
Cross Section (Mb)
10
B1
B2
C
5
0
58
60
62
64
66
68
Photon Energy (eV)
70
72
74
74
Conclusions
1. Method: Parallel multicenter B-spline TDDFT continuum.
2. Calculation of Dichroism (D) of D-Co(acac)3.
3. Strong sensistivity of D parameter.
4. Comparison with preliminar experimental dichroism, possible
revision of previous assignment.
5. Co 3p → 3d autoionization: Dichroism sensitive even for ligand
orbitals.
6. Future perspectives: dichroism experiment on Co 3p → 3d
autoionization.
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!
Additional slides
O
H
H
H
CH3
Density Functional Theory for
Photoionization: the Kohn-Sham approach
hKS  
VC (1) 

1
D  Vnucl ( r )  VC ( r )  V XC ( r )
2
 (2)
r12
dr2
VXC (1)  VXC [  (1)]
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
Basis set approach
 E      CE
The main issue is proper basis set choice
 ilm
1
 Bi ( r )Ylm ( ,  )
r
B-splines: piecewise polynomials defined over an arbitrary grid
-Polynomial order k
-Knot sequence {t0  t1 … tn} over [t0, tn] = [0, Rmax]
B-spline functions
One center expansion (OCE)
{ (r0) }
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
One Center Expansion: {}
Multicenter expansion: {p}
In the basis
Hc = ESc
Bound states : standard diagonalization
Continuum states: Least Squares Approach
min || ( H  E ) 
2
|| R 

A Ac  ac
A(E) = H – ES,
N0 lowest eigenvalues ai  0
Works fine, even with N0 a few hundred
Poisson equation
DVC = -4
is solved in the same basis. Gives the coulomb potential VC,
avoiding the need of two electron integrals.
Linear response : general theory

EXT

(r ,  )

nr ,  
External TD perturbation, with 
frequency (dipole)
Induced density by the external field

  
EXT 
nr ,     dr  r , r ,   (r ,  )
Dielectric susceptibility, not easy to calculate
TDDFT: general theory
TDDFT: instead of , use S of a model system of noninteracting electrons and a modified external potential:
SCF


 
SCF 
nr ,     dr  S r , r ,   (r ,  )
Coupled, but linear!


LDA
 nr ,   dVXC nr  
SCF 
EXT 
 (r ,  )   (r ,  )   dr    
nr ,  

r  r
dnr 
K(r,r’) (kernel)
Exploit linearity of the problem:
V  Kn
n   SVSCF
defines the kernel K
defines the susceptibility 
VSCF  Vext  V
The Response Equation becomes:
( K S  1)VSCF  Vext
To solve : represent the response equation in the B-spline basis set
M. Stener, G. Fronzoni and P. Decleva, J. Chem. Phys., 122 234301(1-11) (2005).
Dynamical
polarizability:
Total cross section:
Partial cross section:


      dr znr ,  
4
   
Im  
c
4 2
 i   
ni  1  n j  i  SCF r ,   j
3c
j
2
    j   i 
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
Cr(CO)6: Autoionization analysis
“Giant” autoionization:
Cr 3p → Cr 3d
Parallel implementation:
M. Stener G. Fronzoni and P. Decleva
Chem. Phys., 361, 49 - 60 (2009).
Explicit expressions for ,  and D
Angular momentum transfer formalism, N. Chandra, J. Phys. B, 20 (1987) 3405.
Explicit expressions for ,  and D
Angular momentum transfer formalism, N. Chandra, J. Phys. B, 20 (1987) 3405.
Explicit expressions for ,  and D
Angular momentum transfer formalism, N. Chandra, J. Phys. B, 20 (1987) 3405.
Explicit expressions for ,  and D
Angular momentum transfer formalism, N. Chandra, J. Phys. B, 20 (1987) 3405.
Photoionization from chiral molecules
Linearly polarized light
d  0

[1   P2 (cos )]
d 4
Chiral molecule, Circularly polarized light
d  0
1

[1  mr D cos  P2 (cos )]
d 4
2
Forward-Backward asymmetry in the angular distribution
0
 ( )   (   ) 
2 D cos
4
Or, switching the polarization of the light at the magic angle P2(cos)=0
I (  )  I ( )
D

I (  )  I ( )
3
Electronic structure: D-Co(acac)3
KS Eigenvalues (eV)
acac ligand
-8
4
-10
-12
3
LPLP+