A> |J - Cobalt - University of Calgary
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Transcript A> |J - Cobalt - University of Calgary
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
S1/ 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
S1/ 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
S1/ 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 iB 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 Ag N Jj )
ˆ
Ag M Jj Ag.Jj ( )
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 Ag N Jj )
ˆ
Ag M Jj Ag.Jj ( )
AJ o
gj
N
1
Ag.Jj ( ) 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 Ag N Jj )
ˆ Jj
Ag M
o
gj
N
AJ
Electric dipole operator
For circular polarized
Light:
2
ˆ m
ˆ ,i
M
i
2
ˆ
Ag M Jj Ag.Jj ( )
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 Ag N Jj )
ˆ Jj
Ag M
o gj
N
'
2
2
ˆ
Ag M Jj . Ag.Jj ( )
1
Ag.Jj ( ) 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 CY 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 CY 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 CY 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
0X 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 needp(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
pa
pi
pb
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
A1 . 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)
JI
(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
JI
(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)
JI
(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
j1
j1
ˆ lˆ ir
L
i
i
i
O
LCP
-A+
B>0
ˆ J M r S1/ 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
pn
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
pn
2Eux Lˆ z 3Euy A1g Mˆ x 2Eux 3Euy Mˆ y A1g
W (3E1uy ) W (2E1uy )
B(3Eu )
4
Im
3
pn
3Eux Lˆ z 2Euy A1g Mˆ x 3Eux 2Euy Mˆ y A1g
W (2E1uy ) W (3E1uy )
B(3Eu )
4
Im
3
pn
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]
ji
From which we obtain density change in frequency domain
occ vir
( y) (,r ) [X( )ai Y ( )ai ] ai
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]
ji
From which we obtain density change in frequency domain
occ vir
( y) (,r ) [X( )ai Y ()ai ] ai
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 )
i1
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 abij
Bai ,bj K ai ,jb
Cai ,bj
b j
nb n j
K ai ,bj
1
abij
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 )
i1
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