Tuesday November 11 11:30 am - 12:10 pm
Tom Ziegler
Department of Chemistry
University of Calgary,Alberta, Canada T2N 1N4
Magnetically Perturbed Time Dependent Density Functional Theory.
Applications and Implementations
ADF
•
•
•
•
•
Solves Kohn-Sham equations
Properties
– NMR, EFG, EPR, Raman, IR, UV/Vis, NLO, CD, …
– Potential energy surfaces (transition states, geometry optimization)
Environment effects
– QM/MM, COSMO
Relativistic effects
– Scalar relativistic effects, spin-orbit coupling
– Transition and heavy metal compounds
Uses Slater functions
Inorganic Spectroscopy
hv
C
C
C
C
C
Cl
Zr
Si
C
Cl
C
C
C
C
Basic Time Dependent Density Functionl Theory
Basic Equation :
F
()
 W F
2
M.E.Casida
()
Gross,E.K.; Kohn W.
Where :
1/2 (A  B)S 1/2
  S
S1/ 2  (A  B)1/ 2
Definition of A and B Matrices :
Aia, jb
 

Fia
 (a  i ) 0  ij ab  
P 

 jb 0
Bia,bj
T. Ziegler,M.Seth,M.Krykunov,J.Autschbach
A Revised Electronic Hessian for Approximate
Time-Dependent Density Functional Theory
SUBMITTED, J.C.P.

F 
ia
 
P 

 bj 0
Basic Time Dependent Density Functionl Theory
Basic Equation :
F
()
 W F
2
M.E.Casida
()
Gross,E.K.; Kohn W.
Where :
1/2 (A  B)S 1/2
  S
S1/ 2  (A  B)1/ 2
Corredted Definition of A and B Matrices :

Aia, jb

F 
ia
 (a  i ) 0  ij ab  
P 

 jb 0
Bia,bj
1
 f [Jaa,aa  Kaa,aa  Jii,ii  Kii,ii  2Jaa,ii  2Kaa,ii ]
2

F 
ia
 
P 

 bj 0
Basic Time Dependent Density Functionl Theory
Basic Equation :
F
()
 W F
2
M.E.Casida
()
Gross,E.K.; Kohn W.
Where :
1/2 (A  B)S 1/2
  S
S1/ 2  (A  B)1/ 2
Corredted Definition of A and B Matrices :

Aia, jb

F 
ia
 (a  i ) 0  ij ab  
P 

 jb 0
Bia,bj
Spin-flip transitions using non-collinear functionals
Liu (2004),Ziegler+Wang (2005),Vahtras (2007)

F 
ia
 
P 

 bj 0
Basic Time Dependent Density Functionl Theory
Basic Equation :
F
()
W F
2

()
M.E.Casida
Gross,E.K.; Kohn W.
W  = E o,
Transition Energy :

Electric Transition Dipole Moment :
 1
ˆ
A M J 
WJ

(J)
F
(a  i )
ia ia
ia   i r a
ia
Magnetic Transition Dipole Moment :
J Lˆ A  WJ lia Fia( J )
ia
1 
( a  i )
l jb  iB j r   b
Absorption Spectra and TD-DFT
W   E 0,
Transition Energy :
AA


 
2 
(J)
(J)
f    ia Fia (a  i )   jb F jb (b   j ) 
3 ia

 
 jb


B
C
C
Inorganic Spectroscopy
hv
C
C
C
C
C
N
Cl
Si
N
M
C
Zr N
Cl
C
N
C
C
C
H
Magnetic Circular Dichroism (MCD) Spectroscopy
Why MCD and MOR ?
In absorption spectroscopy only
positive (often overlapping) bands
More information about each excited state
Magnetic Circular Dichroism (MCD) Spectroscopy
Why MCD ?
In MCD bands of different shapes
More information about each excited state
Magnetic Circular Dichroism (MCD) Spectroscopy
Origin of MCD ?
Absorbance in dipole approximation.
J
A


2 
(N Ag  N Jj ) 
ˆ
 Ag M Jj   Ag.Jj ( )
AJ   o
gj


N
Electric dipole operator:
ˆ  m
ˆ i   (xi exi  yi eyi  yi eyi )
M
i
i
Magnetic Circular Dichroism (MCD) Spectroscopy
Origin of MCD ?
Absorbance in dipole approximation.
J
A


2 
(N Ag  N Jj ) 
ˆ
 Ag M Jj   Ag.Jj ( )
AJ   o
gj


N
1
 Ag.Jj ( )  f J (   J ) 
e
 WJ
 J  2


W


J
Magnetic Circular Dichroism (MCD) Spectroscopy
Origin of MCD ?
Circular Polarized Light
mˆ  
1
(xex  iyey )
2
mˆ  
1
(xex  iyey )
2
Difference in absorbance
polarized light
 of left and right circular

(N Ag  N Jj )
ˆ Jj
 Ag M
 o

gj


N
AJ
Electric dipole operator
For circular polarized
Light:
2
ˆ  m
ˆ ,i
M

i
2 
ˆ
 Ag M Jj   Ag.Jj ( )

Magnetic Circular Dichroism (MCD) Spectroscopy
Origin of MCD ?
The difference in absorption of left
and right circularly polarized light
in the presence of a magnetic field
as a function of photon energy
AJ'


'
A,J


'
A,J


(N Ag  N Jj ) 
ˆ Jj
 Ag M
 o gj


N

'

2
2 
ˆ
 Ag M  Jj . Ag.Jj ( )


1
Ag.Jj ( )  f J (   ) 
e
W
,
J
 ,  2
J


 W

J



Magnetic Circular Dichroism (MCD) Spectroscopy
Origin of MCD ?
AJ'

o
3


3
 

o
3
o
3
 

(N A  N J )
ˆ  J
 A M


N
2
2  

ˆ
 A M J   f J (   J )o B
B
(N A  N J )  
ˆ  J
 A M

N
B 
 3
 

3  B 
(N A  N J )  
ˆ  J
A M

 

N

o 
2
2
2 

ˆ
 A M J  f J (   J )B
o
ˆ  J 2 
 f J (   J )B
 A M


f J (   J )
  oA J
B   0BJ f J (   J )B   oC J f J (   J )B


AJ'
Magnetic Circular Dichroism (MCD) Spectroscopy
The MCD disprsion
AJ'
f J (   J )
  oAJ
B   0BJ f J (   J )B


 oC J f J (   J )B
P.J.Stephens.
Ph.D. Thesis 1964
C(T)
A
B
A
Magnetic Circular Dichroism (MCD) Spectroscopy
The MCD disprsion
AJ'
f J (   J )
  oAJ
B   0BJ f J (   J )B


 oC J f J (   J )B
Degenerate ground- or (and) excited state
Absorption band
Positive
A-term
P.J.Stephens.
Ph.D. Thesis 1964
Negative
A-term
Magnetic Circular Dichroism (MCD) Spectroscopy
The MCD disprsion
AJ'
f J (   J )
  oAJ
B   0BJ f J (   J )B


 oC J f J (   J )B
P.J.Stephens.
Ph.D. Thesis 1964
All cases
Absorption band
Negative
B-term
Positive
B-term
Magnetic Circular Dichroism (MCD) Spectroscopy
The MCD disprsion
AJ'
f J (   J )
  oAJ
B   0BJ f J (   J )B


 oC J f J (   J )B
Space and(or) spin-degenerate ground state
Absorption band
Negative
Negative
B-term
C-term
P.J.Stephens.
Ph.D. Thesis 1964
Positive
Positive
B-term
C-term
Origin of B-Term


C
f (   J )B
  o B -A J J
 (BJ + J ) f J (   J )B
J 


kT

The B term
A'
B=0
Y
Y-iaX
X
X+iaY

A
-A+
A-
M-
A-
M-
M+
M+
O
O
A
-A+
B>0
B=0
3
1
 
ˆ  J
 A M
BJ   

3   B 
B>0
2
2 

ˆ
 A M J 
o
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)
Expression for the B-Term
The B term
3
1
 
ˆ  J
 A M
BJ   

3   B 
2
2 

ˆ
 A M J 
o
Or by using the identity

t
ˆtJ
AM
2
ˆtJ
 AM

2
ˆrJ JM
ˆ s A  i  ( )
 i rst A M
rst rs
L
r,s,t
r,s,t
Here  rst is the three - dimensional Levi - Civita symbol
We thus have
3
 st ( ) 
i
BJ   stu 

3 s,t,u
 Bu    L 
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)
The Calculation of the B-term
The B term : practical calculations
 st ( ) 
i
BJ   stu 

3 s,t,u  Bu  
We have:
Where:
L
st ()  ms[XL ()  YL ()][mt (XL ()  YL ()]
L

Early work:

J.Michl, J.Am.Chem.Soc. 100,6801 (1978)
m s  i(0) m s a(0)

TD-DFT calculations
C 0 X   A

   *

0 CY  B
B X 
 
* 
A Y 
The Calculation of the B-term
The B term : practical calculations
We have:
 st ( ) 
i
BJ   stu 

3 s,t,u  Bu   
L
Where:
2i
BAJ   stu [m s(1)u (X J(0)  YJ(0) )m t(0) (X J(0)  YJ(0) )
3 stu

m s(0) (X J(1)u  YJ(1)u )m t(0) (X J(0)  YJ(0) )]

TD-DFT calculations

Solve:
(0)   (0)



C
0
X
A
(0)
 
 (0)   (0)
0 CY  B
(0) 


B
X
 (0) 
(0) 
A Y 
(0)
The Calculation of the B-term
The B term : practical
calculations
By differentiation
of

BAJ  
2i
stu [m s(1)u (X J(0)  YJ(0) )m t(0) (X J(0)  YJ(0) )

3 stu
m s(0) (X J(1)u  YJ(1)u )m t(0) (X J(0)  YJ(0) )]
(0)   (0)



C
0
X
A
(0)
 
 (0)   (0)
0 CY  B
Implementation
- X ,Y
(1)
(1)
B(0) X (0) 
 (0) 
(0) 
A Y 

T he equation 
that we use for evaluating(1)(X
, Y(1) ) is
A(0)

B(0)*

 I
(1)


 I 0
 
B(0)  (0) I
  I 
(0)* 
A 
0
0 A(1)
  (1)*
I  B
0X I(1) 
 (1) 


I YI 
B(1) X I(0) 
 (0) 
(1)* 
A YI 
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)
The Calculation of the B-term
Evaluation of- X(1) , Y(1) 
1 1 
Introducing the unitary transformation= 
U

1 -1
A(0)
U 
B(0)*

B(0)  (0) I
  I 
(0)* 
A 
0
0   X I(1)   (1) I
 I 
U U  (1)  U 

I 
YI   0

I  Z
  S (A  B )S
0 A(1)
 
I  B(1)*
B(1)   X I(0) 
U U  (0) 
(1)* 
A 
YI 
Affords

(0)
I
 S
(0)
I
Here:
1/2
(1)
I
(0) 1/2
I
(1)
(1)
1/2
(0)
I
F
(A  B )S F
(1)
(1)
1/2
(0)
I
Z   S (X Y )
(1)
I
(0)
I
1/2
(1)
I
(1)
I
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)
The Calculation of the B-term
The B term : practical
calculations
(0)
(1)
(0) 1/2
(1)
(1)
1/2 (0)

I


Z



S
(A

B
)S
FI
I
I
I
1/2
(1)
(1)
1/2 (0)
 (0)
S
(A

B
)S
FI
I
(1)
An Expression for K
ai,bj
We needp(1) . A well known expresson exists that is particularly simple because we have a

imaginary perturbation
(1) (0)
 (1)

U

p
qp q
q p

(1)
H
(1)
Uqp
 (0) pq (0)
q   p

Where H(1) is the Hamiltonian
describing t he perurbation
T hus


(1)
(1)* (0)
(0)
(1)* (0)
(1)* (0)
Kai,bj
 U pa
K pi,bj  U pi(1)*Kap,bj
 U pb
Kai,pj  U pb
Kap,bp
pa
pi
pb
p j
M.Seth,T.Ziegler,
J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)
Seth+ZieglerM.Krykunov,
JCP,2008,in press
The Calculation of the B-term by Direct Method
The B term : Direct method
We must solve

(0)
I
1/2
I  Z I(1)   (0)
(A (1)  B (1) )S 1/2 FI(0)
I S
1/2
 (0)
(A (1)  B (1) )S1/2 FI(0)
I S
Our equation has the form
AX  b
Seth+Ziegler JCP,2008

WithA a known matrix,b a known vector and
X the unknown vector to be determined. T his
equation can be solved easily if we have
A1 . T here are two problems however
(0)

(a) T he matrixA  (0)
I

.
T
his
matrix
has
no
inverse
because

I
I I is an eigenvalue of
(b) The matrix A is extremely large and we don' t want to try and invert it directly.
To avoid this problem we :
(i) Solve the equations iteratively by expanding the solution in a Krylov
subspace(the space b,Ab, A2b,...Aib in theith iteration)
(0)
(ii) Project out from the Krylov supspaces any contribution from
F
I
The Calculation of the B-term by Direct Method
The B term : Direct Method
We must solve
Ax  b
1/2
2i
 S
BAJ    M
(Z J(1) )M  (X J(0) YJ(0) )
3 
 (0)
J

(0)
I
1/2
I  Z I(1)   (0)
(A (1)  B (1) )S 1/2 FI(0)
I S
1/2
  (0)
(A (1)  B (1) )S1/2 FI(0)
I S
Pros

(i) Can be 
used in conjunction
with an unperturbed
(0)
T DDFT calculation that yields only a few solutions
.F
(ii)Degree of convergence is known
Seth+Ziegler JCP,2008,in press
Cons
(i) T he iterative procedure is often slowly convergent.
We are at tempting t o improve convergence by adding
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)
the unperturbed T DDFT solutionsJ(0)F, J  I to Krylov subspace
The Calculation of the B-term by Sum-over-State Method
The B term : Sum Over State

(0)
I
1/2
I  Z I(1)   (0)
(A (1)  B (1) )S 1/2 FI(0)
I S
1/2
 (0)
(A (1)  B (1) )S1/2 FI(0)
I S
Z (1) by Sum - Over- State
Writing Z(1) in terms of the complete set F(0) affords
Z I(1) = CJI FJ(0)
JI
(0)
Substitute into first order equation and multiply by
from
F
left affords
J
(0)
(0) (0) 1/2
FJ(0)  (0)
S (A(1)  B(1) )S 1/2 FI(0)
I I  ((C JI FJ )   I FJ
JI
(0) 1/2
 (0)
S (A(1)  B(1) )S1/2 FI(0)
I FJ
Or
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.
Seth+Ziegler JCP,2008,134108
Phys. J. Chem. Phys. 128, 144105 (2008)
(0) 1/2
(1)
(1)
1/2 (0)
(0) 1/2
(1)
(1)
1/2 (0)
 (0)
(F
S
(A

B
)S
F

F
S
(A

B
)S
FI
I
J
I
J
C JI 
(0)
 (0)


I
J
The Calculation of the B-term by Sum-over-State Method
The B term : Sum Over State
1/2
2i
 S
BAJ    M
(Z J(1) )M  (X J(0) YJ(0) )
3 
 (0)
J
Z (1) by Sum - Over- State: Z I(1)  CJI FJ(0)

JI
(0) 1/2
 (0)
S (A(1)  B(1) )S 1/2 FI(0)
I (FJ
C JI 

(0)
(0)
I  J
FJ(0) S 1/2 (A(1)  B(1) )S1/2 FI(0)
(0)
 (0)


I
J
Pros

Cons

Seth+Ziegler JCP,2008,134108
Interpretation easy in terms of contributions
from different excited states
May need to calculate manyJ(0)Fin unperturbed
T DDFT and convergence of summation is unknown
Other B-term implementations
J.Michl J.Am.Chem.Soc. 100, 6801, 1978
HF+CI
E.Dalgaard Phys.Rev. A 42 42 1982
J.Olsen; P. Jørgensen J.Chem.Phys. 82 3235 (1985)
W.A.Parkinson; J.Oddershede J.Chem.Phys. 94,7251 (1991)
W.A.Parkinson; J.Oddershede) Int.J.Quantum Chem. 64,599 (1997)
CCSD(T)
7S.Coriani, P.Jørgensen, T.Helgaker J.Chem.Phys. 113,3561,2000
T.Kjœrgaard, B.Jansik, P.Jørgensen,S.Coriani, J.Michl, J.Phys.Chem. A 111,11278 (2007))
DFT
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)
H.Solheim; L.Frediani; K.Rudd; S.Coriani Theor.Chem.Acc 119,231,2007
DFT-SOS
M.Seth,T.Ziegler,J.Autschbach J.Chem.Theory.Comp.3,434,2007
M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107
Comparison of Sum-over-State and Direct Method for B-terms
Convergence of SOSmethod for Ethylene
  *


 3s
  *
 3s


Seth+Ziegler JCP,2008
Comparison of Direct Method for B-terms with Experiment
S4N3+
Exp: J.W.Waluk, J.Michl Inorg.Chem. 21,556,1982)
S4N2
Exp: H.-P.Klein, R.T. Oakley, J.Michl Inorg.Chem. 25,3194 (1986
Comparison of Direct Method for B-terms with Experiment
Exp: H.-P.Klein, R.T. Oakley, J.Michl Inorg.Chem. 25,3194 (1986)
Exp: H.-P.Klein, R.T. Oakley, J.Michl Inorg.Chem. 25,3194 (1986)
Seth+Ziegler JCP,2008
Comparison of Direct Method for B-terms with Experiment and other Methods
Exp: H.-P.Klein, R.T. Oakley, J.Michl Inorg.Chem. 25,3194 (1986)
T.Kjœrgaard, B.Jansik, P.Jørgensen,S.Coriani, J.Michl, J.Phys.Chem. A 111,11278 (2007))
TD-DFT calculations of B-term.
O
W. Hieringer, S. J. A. van Gisbergen,
and E. J. Baerends
J. Phys. Chem. A 2002, 106, 10380
Furan
1b2 11A1 --> 11B1
X
S
2b1
X
11A1 --> 21A1
11A1 --> 11B2
Thiophene
Se
X
1a2
1b1
X
Selenophen
Te
Tellurophen
Seth+Ziegler JCP,2008,134108
TD-DFT calculations of B-term. Sum-over-state formulation
Norden, B.; Hansson, R.; Pedersen, P. B.; Thulstrup, E. W.
Chem.Phys. 1978, 33, 355.
TD-DFT calculations of B-term. Sum-over-state formulation
Norden, B.; Hansson, R.; Pedersen, P. B.; Thulstrup, E. W.
Chem.Phys. 1978, 33, 355.
TD-DFT calculations of B-term. Sum-over-state formulation
O
-5.05
0.0
.13
0.20
Furan
3.37
6.0
6.2
1a2  3b1
2b1  3b1
Seth+Ziegler JCP,2008,134108


TD-DFT calculations of B-term. Sum-over-state formulation
S
450 -477
.04
6
.13
Thiophene
5.5
5.7
2b1  3b1 1a2  3b1

5.9

Seth+Ziegler JCP,2008,134108
TD-DFT calculations of B-term. Sum-over-state formulation
Se
59.1
.07
-101
-3
0.22
Selonophene
5.1
2b1  3b1

5.3
1a2  3b1

Seth+Ziegler JCP,2008,134108
5.5
TD-DFT calculations of B-term. Sum-over-state formulation
Te
1b2 11A1 --> 11B1
X
0.64
2b1
X
Tellurophen
4.4
11A1 --> 21A1
1b1
X
11A1 --> 11B2
X
-28.0
-5.1 12.8
1a2
Seth+Ziegler JCP,2008,134108
4.8
5.2
A-term of MCD
Origin of A-term
AJ'


1 (N A  N J ) 
ˆ  J
 A M
 o  J 


3
N

2
ˆ  J
 AM

1 (N A  N J )  
ˆ  J
 A M
 o  

J

3
N

B


1 
 o  
J
 3 B
2
2
 

 f (   J )0 B
 B
ˆ  J
 AM

(N A  N J )  
ˆ  J
AM

 

N

0 
2
2

   f (   J )B
o
ˆ  J
 AM

2

  f (   J )B

f (   J )
  oA J
  0BJ f (   J )B   oC J f (   J )B


AJ'
M.Seth,T.Ziegler,J.Chem.Phys. 2004,120,10943
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)
The A-term of Magnetic Circular Dichroism (MCD) Spectroscopy
The A term


 o J -A J
f (   J ) 
B



1 (N A  N J ) ˆ 
 A M J
 o 
J

3
N

1 (N A  N J ) ˆ 
 A M J
 o 
J

N
 3
2
2
2  

ˆ
 A M J 
 f (   J )0 B
 B
2    f (   )

J
J
ˆ
 A M J  
B



  B 

0
Thus

AJ  

1 (N A  N J )  ˆ 
 3 N  A M J

2
 

ˆ J    J 
 AM


0 

B
  

M.Seth,T.Ziegler,J.Chem.Phys. 2004,120,10943
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)
2
The A-term of Magnetic Circular Dichroism (MCD) Spectroscopy
The A term
AJ  

Here

1 (N A  N J )  ˆ 
 3 N  A M J

2
 

ˆ J    J 
 AM


0 
 B 

2
  F FJ
2
T
J
We have

Thus
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)
The A-term of Magnetic Circular Dichroism (MCD) Spectroscopy
'


C
A'
A' A
 ( )


 B -A J A.J
 (BJ + J )A.J ( )
J 



( )
kT

The A term
1
1P
B=0

A
B>0
0
-1
A-
A-
ARCP
-A+
LCP
RCP
1
S
B=0
n
n
j1
j1
ˆ  lˆ  ir  
L
i
i
i
O
LCP
-A+
B>0
ˆ J   M r S1/ 2F (0)e
AM
L r
r
Other A-term implementations
J.Michl J.Am.Chem.Soc. 100, 6801, 1978
HF+CI
Y.Honda, M.Hada, M.Ehara, H.Nakatsuiji,J.Downing,J.Michl , Chem.Phys.Lett 355,219, , 2002
Y.Honda, M.Hada, M.Ehara, H.Nakatsuiji,J.Michl , J.Chem.Phys. 123,164113 (2005)
CCSD(T)
7S.Coriani, P.Jørgensen, T.Helgaker J.Chem.Phys. 113,3561,2000
T.Kjœrgaard, B.Jansik, P.Jørgensen,S.Coriani, J.Michl, J.Phys.Chem. A 111,11278 (2007))
DFT
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)
H.Solheim; ; K.Rudd; S.Coriani ,P.Norman J.Chem.Phys. 128,094193,2008
M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107
M.Seth,T.Ziegler, E.J.Baerends J.Chem.Phys. 2004,120,10943
M.Seth,T.Ziegler, J.Chem.Phys. 2007,127,134108
Applications:A/D
Se4
2+
D4h
Te4
Exp:-0.66 Calc:-0.72 Exp:-0.50
Fe(CN)64-
Exp: 0.40
2+
Ni(CN)42-
Calc:-0.80
Oh
C6Cl6
Exp: 0.72
A 

D B
Calc: 0.48

D4h
M.Seth,T.Ziegler,J.Chem.Phys. 2004,120,10943
C6H3Br3
Exp: 0.60
D6h
Calc: 0.63
D3h
Calc: 0.55
Different MCD-terms
Negative
B-term
Positive
B-term
Negative
A-term
Positive
A-term
3t2
Metal
2e
t1
Ligand
Absorption band
2t2
M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)
MCD-terms for Oxyanions
MCD-terms for Thioanions
Theor
Exp.
M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. Submitted
MCD spectra of Porphyrins containing Mg,Ni and Zn
5 10-2
21Eu
N
31Eu
N
M
N
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
N
Orbital level diagram for ZnP
ZnP
2eg1
2e1.g
2eg2
5 10-2
21Eu
2a2u
2a2.u
1a1.u
31Eu
1a1u
1b1.g
1b2.u
1e1.g
1a2.u
1b2u
E.J. Baerends , G. Ricciardi , A. Rosa , S.J.A van Gisbergen
J.Phys.Chem. A2001,105,3311
E.J. Baerends , G. Ricciardi , A. Rosa , S.J.A van Gisbergen
Coord.Chem.Rev. 2002,230,5
ZnP
Experimental Spectrum for ZnP
3Eu
2Eu
C2(2a2u eg)-C1(1a1u --> 2eg)
Conjugated Gouterman State
2e1.g
(1b2u eg)
5 10-2
21Eu
1Eu
1A1g
2a2.u
1a1.u
31Eu
C1(2a2u eg)+C2(1a1u --> 2eg)
Gouterman State
1b1.g
1b2.u
1e1.g
1a2.u
Ground State
Complex Symmetry
1Eu
Exc. Energ. (eV)
exp.
calc.
c
d
2.03 , 2.21 , 2.28
e
f
2.23 , 2.18
c
2Eu
ZnP
%
h
f
Assign.
-> 2eg
-> 2eg
52.10
46.63
0.001
Q
3.25
1b2u -> 2eg
1a1u -> 2e1g
2a2u -> 2eg
68.44
17.54
10.05
0.496
3.32
1b2u
2a2u
1a1u
1a2u
29.88
29.31
27.13
10.30
d
2.95 , 3.09 ,
e
f
3.18 , 3.13
Composition
2a2u
1a1u
g
E.J. Baerends , G. Ricciardi , A. Rosa ,
S.J.A van Gisbergen
Coord.Chem.Rev. 2002,230,5
3Eu
->
->
->
->
2eg
2eg
2eg
2eg
B
0.943
Experimental Spectrum for ZnP
5 10-2
21Eu
C2(2a2u eg)+C1(1a1u --> 2eg)
Conjugated Gouterman State
3Eu
2Eu
(1b2u eg)
31Eu
C1(2a2u eg)+C2(1a1u --> 2eg)
Gouterman State
1Eu
1A1g
Ground State

D(1Eu )  C1 2a2u y 2egy  C2 1a1u y 2egx

2
1

1
2
2.92
3.25

2.27x10


2

2
2
L.Edwards,D.H.Dolphin,M.Goutermn
J.Mol.Spectrosc 35(1970)90
E.J. Baerends , G. Ricciardi , A. Rosa ,
S.J.A van Gisbergen

Coord.Chem.Rev. 2002,230,5

D(3Eu )  C1 2a2u y 2egy  C2 1a1u y 2egx
1

1
2.92
3.25
 9.51


2

2
2

2
Simulated Spectrum
for ZnP with A-term only
A-only
1Eu
2Eu+3Eu
Q
Complex
ZnP Exp
Symmetry
h
A
h
A/D
1Eu
0.05
5.49
2Eu
-3.37
-1.62
3Eu
-0.57
-0.15
ZnP
Q
S
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem.
Influence of ring distortion on MCD spectrum of ZnP
N
N
M
N
N
N

nB1
D4h
C2v
N
 nB Lˆ nB A Mˆ nB nB Mˆ A 
2
1 z
2
1
x
1
2
y 1 
B(nB2 )  Im

3
W


n


nB2
nEu
N
M
N

 nB Lˆ nB A Mˆ nB nB Mˆ A
2
1 z
2
1
x
1
2
y 1
B(nB1)  Im
3
Wn


B(nB2 )





B(nB1)  B(nB2 )



A (nE u )
B(nB1 )
Influence of ring distortion on MCD spectrum of ZnP
N
N
M
N
N
N
N
M
N

N
 nB Lˆ nB A Mˆ nB nB Mˆ A 
2
1 z
2
1
x
1
2
y 1 
B(nB2 )  Im

3
W


n


nB2
nEu
nB1
D4h
C2v
0.5


D4h
ZnP
 nB Lˆ nB A Mˆ nB nB Mˆ A
2
1 z
2
1
x
1
2
y 1
B(nB1)  Im
3
Wn


ZnP
0.5
x10
0.0
-0.5
-0.5
2.00
x10
N
o
rm
al
iz
ed
In
te
n
si
ty
N
o
rm
al
iz
ed
In
te
n
si
ty
0.0
Dist C2V
2.50
E(eV)
3.00
3.50
2.00
2.50
E(eV)
3.00
3.50





Simulated Spectrum for ZnP with B-term only
B-terms
Exp.
3Eu
1Eu

4
B(nEu )  Im 
3

 pn
2Eu
nEux Lˆz pEuy A1g Mˆ x nEux pEuy Mˆ y A1g 


W ( pE1uy ) W (nE1uy )


1.00
Simulated Spectrum for
ZnP with A+B-term only
Normalized Intensity
ZnP
0.50
0.00
x 100
-0.50
2.00
2.50
3.00
E (eV)
E (eV)
3.50

Exp.

B(2Eu ) 

4
Im  
3

 pn
2Eux Lˆ z 3Euy A1g Mˆ x 2Eux 3Euy Mˆ y A1g 


W (3E1uy ) W (2E1uy )


B(3Eu ) 

4
Im  
3

 pn
3Eux Lˆ z 2Euy A1g Mˆ x 3Eux 2Euy Mˆ y A1g 


W (2E1uy ) W (3E1uy )


B(3Eu ) 

4
Im  
3

 pn
 B(2Eu )
2Eux Lˆ z 3Euy A1g Mˆ x 3Eux 2Euy Mˆ y A1g 


W (2E1uy ) W (3E1uy )


Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46,
 9111-9125.
Simulated Spectrum for MgP and NiP with A+B-term
E(eV)
MgP
NiP
2eg
-7.00
ZnP
2eg
(a) MgP
2eg
0.5
2Eu
dx2-y2
3Eu
0.0
x100
1Eu
N
or
m
al
iz
ed
In
te
n
si
ti
es
0.5
2a2u
1a1u
-9.50
1a1u
2a2u
dz2
2a2u
1a1u
1a2u
dxy
dxz, dyz
1eg
3.5
(b) NiP
1b1g
1b2u
1eg
1a2u
0.5
0.0
x100
0.5
N
o
rm
a
li
ze
d
In
te
n
si
ti
es
1a2u
1b2u
3.0
E(eV)
dxz, dyz
1b2u
1eg
2.5
2.0
1eu
-12.00
2.0
2.5
E(eV)
3.0
3.5
Substituted Porphyrins

m
N
N
N
M
N
N
N
M
N
N
M
N
N
MTPP
MOEP
N
tetraphenylporphyrin
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
N
octaethylporphyrin
Excited States for Substituted Porphyrins
NiTPP
0.5

B(3Eu )

N
or
m
al
iz
ed
In
te
ns
it
y
0.0
A(1Eu )
N
N
-0.5
Ni
N
N
B(2Eu )
2.00
2.50
3.00
E(eV)

Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
3.50
Excited States for Substituted Porphyrins
ZnTPP
B(3Eu )
0.5
A(1Eu )

0.0

N
or
m
al
iz
ed
In
te
ns
it
y
A(1Eu )
x10
N
N
-0.5
Zn
N
N
B(2Eu )

2.00
2.50

E(eV)
3.00
3.50
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
Tetraazaporphyrins and Phthalocyanines

m
N
N
M
N
N
Tetraazaporphyrins and Phthalocyanines

m
N
N
N
N
N
M
N
N
M
N
N
N
N
N
MTAP
tetraazaporphyrin
Tetraazaporphyrins and Phthalocyanines

N
m
N
M
N
N
N
N
N
N
M
N
N
M
N
N
N
N
N
N
N
N
N
N
MTAP
tetraazaporphyrin
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2008,46, 9111-9125.
MPc
phthalocyanine
Magnetic Circular Dichroism (MCD) Spectroscopy
'


C
A'
A' A
 ( )


 B -A J A.J
 (BJ + J )A.J ( )
J 



( )
kT

The C term
B=0

A
-A+
B>0
B=0
1S
1
S
A-
M1
P
M+

EP  EP  kT
A-
M- M +
1
1
If
B>0
P+
A
-A+
P-
N P  N P
N tot

EP  EP
3kT


i
ˆ
ˆ
ˆ
C 
A ' L A   A M J  J M A ' 



3 A a'

Electron configuration t1u6t2u6t1u6t2g5
Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412
Limitations of Traditional TD-DFT
What are the
fundamental
equations ?
a
a
ix
Degenerate Ground State
iy
ix
iy
ix
How do we
calculate
excitation
energies
iy
What do we do with a
degenerate ground state
that can not be represented
by single Slater determinant ?
TRICKS of the Trade: Calculating the Excitation Energies
of Molecules with Degenerate Ground States using TD-DFT
Challenges
• Degenerate ground states are generally treated within DFT by
fractional occupations of the degenerate orbital. This gives a ground
state of indeterminent symmetry.
• A degenerate ground state can be made non-degenerate by breaking
utilizing a lower symmetry point group. The amount of symmetry
breaking in this case can be large and symmetry assignments complicated
TRICKS of the Trade: Calculating the Excitation Energies
of Molecules with Degenerate Ground States using TD-DFT
Solution:
Transformed Reference with an Intermediate Configuration
Kohn Sham (TRICKS) TDDFT
Idea:
Avoid problems with a degenerate ground state by taking an excited
state that is nondegenerate as the (Transformed) Reference Intermediate
Configuration.
Application of the TRIC method
Example 1:
d1 transition metal
complexes of Oh symmetry,
d-d transition
TiF63

Application of the TRIC method
Results 1:
d1 transition metal
complexes of Oh symmetry,
d-d transition.
TiF63

Application of the TRIC method
Example 2:
d1 transition metal
complexes of Td symmetry,
d-d transition
VCl 4

Application of the TRIC method
Result 2:
d1 transition metal
complexes of Td symmetry,
d-d transition
VCl 4

Application of the TRIC method
Example 3:
d1 transition metal
complexes of Td symmetry,
charge transfer
VCl 4

Application of the TRIC method
Result 3:
d1 transition metal
complexes of Td symmetry,
charge transfer
VCl 4

Application: Fe(CN)63Electron configuration t1u6t2u6t1u6t2g5
Excitations are ligand-metal charge transfer.
C term of a transition to a T1u state is positive and
to a T2u state is negative.
Transition Exp.
Calc.
1
1.21/0.61
0.86
2
-0.68
-0.86
3
0.56
0.86
Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412
More Applications
RuCl63-
Exp
Calc
7.5
7.3
6.9
7.3
-6.9
-7.3
6.3
7.3
-3.1
-7.3
2.2
7.3
[Fe(CN)5SCN]3-
MnPc
Exp.
0.58
-0.60
Calc.
0.84
-0.84
Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412
Exp.
0.03
Calc.
0.90
0.23
0.90
Spin-degenerate Ground State MCD via Spin-orbit Coupling
<K|LAJ|J>
|K>
KJ
|J>
JA
|K>
|J>
|J>
<A|LAJ|J>
|A>
|A>
<J|r|A>
 (1)

|K>
KJ |A>
<J|r|A><A|r|K>
 ( 2)

<K|LAJ|A>
<J|r|A>
 ( 3)

2003,220
 M.L.Kirk Curr.Op.Chem.Bio

Application to Plastocyanin
Application to Plastocyanin
<K|LAJ|J>
KJ
|K>
|J>
|A>
§M.E.
<J|r|A><A|r|K>
I. Solomon, R.K. Szilagyi, S. D. George and L. Basumallick, Chem. Rev, 104, 419, 2004.
( 2)
85
Application to Sulfite Oxidase
Application to Sulfite Oxidase
|J>
L1: -SCH3. L2: -OH. L3: -S(CH2)2S-.
|K>
KJ |A>
<K|LAJ|A>
<J|r|A>
§M.E.
2000.
Helton, A. Pacheco, J. McMaster, J.H. Enemark and M. Kirk, J. Inorg. Biochem., 80, 227,
87
TD-DFT/MCD
Fan Wang
Dr. Mykhaylo Krykunov
Dr.Jochen Autschbach
Alejandro
Gonzalez Peralta
Dr. Mike Seth
Hristina Zhekova
PRF
Mitsui
MOR and MCD`
TD-DFT formulation without damping
We solve the equation
ˆ ks
 
ext
h (r )V (r ,t)  i  k (r)  exp[i k t]  0


t 
To obtain the solution
k' (r ,t)  C j (t) j (r ) exp[i j t]
ji
From which we obtain density change in frequency domain
occ vir
 ( y) (,r )  [X( )ai Y ( )ai ] ai

i a
With:
(X()Y ())  2S
1/2
[  ] S
2
1
V()
1/2
MOR and MCD
The expression
 2BJ
Vsos ( )    2
2
W

(

)
J
J
Allows us to calculate the MOR parameter V() from the MCD
parameters BJ after summing over all states

aM. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,
MOR and MCD
The expression
 2BJ
V ()   2
2
W

(
)
J
J
The expression for V(  ) diverges for
Vres(

 = WJ
Vdamp ()

We need a TD-DFT formulation in which damping included
MOR and MCD`
TD-DFT formulation with damping
We solve the equation
ˆ ks
 
ext
h (r )V (r ,t)  i  k (r) exp[i k t]exp[t]  0


t 
To obtain finite
lifetime solutions
k' (r ,t)  C j (t) j (r ) exp[i j t]exp[t]
ji
From which we obtain density change in frequency domain
occ vir

 ( y) (,r )  [X( )ai Y ()ai ] ai
With:
i
a
(X()Y ())  2S
1/2
[(  i )  ]S
2
L.Jensen; J.Autchbach; G.C.Schatz J.Chem.Phys.2005,122,224115
V()
1/2
MOR and MCD`
TD-DFT formulation with damping
V ()  V
dm
res
Here
R,dm
res
R,dm
res
V
() iV
I,dm
res
( )  V
R,dm
sos
( )  
J
and
I,dm
res
V
( )  V
I,dm
sos
( )  
J
2 (   )BJ
2
2 2
( J   )  4 
2
2
J
2 2
2
4 BJ
2
2 2
2 2
( J   )  4 
3
M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted
MOR and MCD`
TD-DFT formulation with damping
R,dm
res
V
or
( )  V
R,dm
sos
R,dm
sos
V
( )   0 
( )  V
udm
sos
Here
J
2 (   )BJ
2
2 2
( J   )  4 
2
2
J
2 2
2
() fd ()
udm
Vsos
()
R,dm
sos
V
fd ( ) 
2(   )
2
2 2
( J   )  4 
2
J
2 2
2 2

M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted
( )
MOR and MCD
TD-DFT formulation with damping
I,dm
res
V
( )  V
I,dm
sos
( )  
J
or
I,dm
res
V
4 BJ
( 2J   2 )2  4 2 2
3
( )/    0 BJ f J,B ( )   MCD ( )
J
f J,B ( )

4 
f J,B ( )  2
2 2
2 2
( J   )  4 

M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted
MOR and MCD
n
VresI,dm ( )/    0 BJ f J,B ( )   MCD ( )
J
We can obtain BJ (j = 1, n) from a least square fit of


I,dm
D   Vres (i ) / i   0 iBJ f J,B (i ) 
i1

J
m
n
For m>n
M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted
2
MCD spectra of Porphyrins containing Mg,Ni and Zn

m
N
N
M
N
N
N
N
M
N
N
MP
MOEP
MTPP
tetraphenylporphyrin
porphyrin
N
N
N
N
M
N
tetraphenylporphyrin
N
N
N
N
MTAP
tetraazaporphyrin
N
N
N
N
M
N
N
N
M
N
N
N
N
MPc
phthalocyanine
Application of the TRIC method
Example 2:
Double Excitations
Excited
state
TRIC state
Ground
state
Seth,M. ; Ziegler,T. J. Chem. Phys. 2006, 124, 144105
Seth,M., Ziegler,T. , J. Chem. Phys., 2005,123, 144105,

The Calculation of the B-term
The B term : practical calculations
where A, B C are defined by
Aai ,bj   abij
Bai ,bj  K ai ,jb
Cai ,bj
b  j
nb  n j
 K ai ,bj
1
  abij
nb  n j
1
*
K ai , jb   dr  dr  (r) i (r)
 b (r') j (r')
r  r'
' *
*
  dr  dr  a (r) f XC (r,r',  ) b (r') j (r')
'
*
a
Limitations of Traditional TD-DFT
What are the
fundamental
equations ?
a
a
ix
Degenerate Ground State
iy
ix
iy
ix
How do we
calculate
excitation
energies
iy
What do we do with a
degenerate ground state
that can not be represented
by a single Slater determinant ?
TRICKS of the Trade: Calculating the Excitation Energies
of Molecules with Degenerate Ground States using TD-DFT
Solution:
Transformed Reference with an Intermediate Configuration
Kohn Sham (TRICKS) TDDFT
Idea:
Avoid problems with a degenerate ground state by taking an excited
state that is nondegenerate as the (Transformed) Reference Intermediate
Configuration.
A. I. Krylov , Acc. Chem. Res. 2006, 39, 83-91
Application of the TRIC method
Example 2:
d1 transition metal
complexes of Td symmetry,
d-d transition
VCl 4

Application of the TRIC method
Result 2:
d1 transition metal
complexes of Td symmetry,
d-d transition
VCl 4

Application of the TRIC method
Example 3:
d1 transition metal
complexes of Td symmetry,
charge transfer
VCl 4

Application of the TRIC method
Result 3:
d1 transition metal
complexes of Td symmetry,
charge transfer
VCl 4

Conclusion
• Method for calculating the MCD A term (and dipole strength D) within
TD-DFT is outlined. Procedure for calculating C/D more
straightforward.
• Implemented into the Amsterdam Density Functional Theory (ADF)
program
• Applications to a range of small molecules
• Further information can be found in M. Seth, T Ziegler, A Banerjee, J.
Autschbach, S.J.A. van Gisbergen E. J. Baerends, J. Chem. Phys.
120,10942, 2004 and M. Seth, T. Ziegler, J. Autschbach, J. Chem.
Phys. accepted for publication.
MOR and MCD
Consider a planar polarized light traveling a distance l through
a media of randomly oriented molecules along the direction of
a constant magnetic field with strength B.

E
l
E
B
For such a system the plane of polarization will rotate by an

angle given by
  V ()Bl
0 cN
 2BJ
Vsos ( )  

3 J WJ2   2
Here V() is called the Verdet constant

A.Banerjee,J.Autschbach,T.Ziegler
Int.J.Quant.Chem.2006,101,572

aM. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,
MOR and MCD
 st ( ) 


 Bu 

{ yz ( )   zy (}] 
12
Bx
 cN

 o Im[
{ ( )   xz (}] 
12
By zx
VRe s ( ) 

  J
 Im  [
o cN
J
J

{ xy ( )   yz ( )}] 
12
Bz
o cN

Im stu
( st ( ))B 0
12
Bu


 (t ) (, r )]  x s dr
Bu
J
o cN
Im[
[Im
J
u
VRe s ( )

 2BJ
Vsos ( )  

2
2
3




J
J

 ( ) 


 B 
  J
Im  [

0 cN
BJ


 ( ) (,r )]  x dr
B
J
BJ 
 ( ) 
i
  rst  st

3 r,s,t  Br  
M. Krykunov, A. Banerjee, T. Ziegler,J.
Autschbach J. Chem. Phys. 2005, 122, 075105,
L
MOR and MCD
n
VresI,dm ( )/    0 BJ f J,B ( )   MCD ( )
J
We can obtain BJ (j = 1, n) from a least square fit of


I,dm
D   Vres (i ) / i   0 iBJ f J,B (i ) 
i1

J
m
n
For m>n
M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107
2




MOR and MCD
Thiophene
Furan






Selenophen
Tellurophene
M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107