235nd ACS National Meeting April 5-10, 2008 New Orleans, Louisiana COMPUTATIONAL SPECTROSCOPY ike Seth, Mykhaylo Krykunov, Tom Ziegler epartment of Chemistry niversity of Calgary,Alberta, Canada T2N 1N4 ndent density functional theory as a practical tool in the st d CD spectra of transition metal complexes. Implementations cations. New Orleans National Meeting Organizers: H. B. Schlegel and Krishnan Raghavachari Preciding:H. B. Schlegel Wednesday April 9 10:20 -11:00 am Morial Convention Center -- Rm. 342,
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235nd ACS National Meeting April 5-10, 2008 New Orleans, Louisiana COMPUTATIONAL SPECTROSCOPY Mike Seth, Mykhaylo Krykunov, Tom Ziegler Department of Chemistry.
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235nd ACS National Meeting April 5-10, 2008 New Orleans, Louisiana
COMPUTATIONAL SPECTROSCOPY
Mike Seth, Mykhaylo Krykunov, Tom Ziegler Department of Chemistry University of Calgary,Alberta, Canada T2N 1N4
Time-dependent density functional theory as a practical tool in the study of MCD and CD spectra of transition metal complexes. Implementations and applications.
New Orleans National Meeting
Organizers: H. B. Schlegel and Krishnan RaghavachariPreciding:H. B. SchlegelWednesday April 9 10:20 -11:00 am Morial Convention Center -- Rm. 342,
ADF• Solves Kohn-Sham equations• Properties
– NMR, EFG, EPR, Raman, IR, UV/Vis, NLO, CD, MCD…– 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
hv
Cl
C
C
C
C
C
Si Zr
C
C
C
C
Cl
C
Inorganic SpectroscopyInorganic Spectroscopy
Basic Time Dependent Density Functionl TheoryBasic Time Dependent Density Functionl Theory
Basic Equation :Basic Equation :
€
Aia, jb = (εa − εi)0δ ijδab +∂F ia
∂Pjb
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟0
€
Bia,bj =∂F ia
∂Pbj
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟0
Definition of A and B Matrices :Definition of A and B Matrices :
M.E.CasidaM.E.Casida
Gross,E.K.; Kohn W.Gross,E.K.; Kohn W.
€
ΩF (λ ) =Wλ2F (λ )
€
Ω=−S−1/2(A + B)S−1/2
€
S−1/ 2 = (A − B)1/ 2
Where :Where :
Basic Time Dependent Density Functionl TheoryBasic Time Dependent Density Functionl Theory
Consider a planar polarized light traveling a distance l through a media of randomly oriented molecules along the direction ofa constant magnetic field with strength B.
Consider a planar polarized light traveling a distance l through a media of randomly oriented molecules along the direction ofa constant magnetic field with strength B.
€
€
α =V (ω)Bl
Here V( ) is called the Verdet constantHere V( ) is called the Verdet constant
BE E
α
l
For such a system the plane of polarization will rotate by an angle given byFor such a system the plane of polarization will rotate by an angle given by
€
Vsos (ω) ≡ −μ 0cN
3
hω2B J
WJ2 − hω2
J
∑
€
Vsos (ω) ≡ −μ 0cN
3
hω2B J
WJ2 − hω2
J
∑
aM. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,
M. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,
MOR and MCDMOR and MCD
€
Vsos (ω) = −γω2B J
ωJ2 −(ω)2
J
∑
€
Vsos (ω) = −γω2B J
ωJ2 −(ω)2
J
∑The expressionThe expression
Allows us to calculate the MOR parameter V( ) from the MCDparameters BJ after summing over all statesAllows us to calculate the MOR parameter V( ) from the MCDparameters BJ after summing over all states
aM. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,
MOR and MCDMOR and MCD
€
V (ω) = −γω2B J
ωJ2 −(ω)2
J
∑
€
V (ω) = −γω2B J
ωJ2 −(ω)2
J
∑The expressionThe expression
Vres(
€
Vdamp(ω)€
The expression for V(ω) diverges for ω = ωJ
€
The expression for V(ω) diverges for ω = ωJ
We need a TD-DFT formulation in which damping includedWe need a TD-DFT formulation in which damping included
MOR and MCD`MOR and MCD`
TD-DFT formulation without damping TD-DFT formulation without damping
€
We solve the equation
€
We solve the equation
€
ˆ h ks (v r )+V ext (
v r , t) − i
∂
∂t
⎡ ⎣ ⎢
⎤ ⎦ ⎥φk (r)× exp[−iε kt] = 0
To obtain the solutionTo obtain the solution
€
φk' (
v r , t) = C j (t)φ j
j≠i
∑ (v r )× exp[−iε jt]
€
Δρ ( y)(ω,r r ) = [X(ω)ai +Y (ω)ai ]φaφi
a
vir
∑i
occ
∑
From which we obtain density change in frequency domainFrom which we obtain density change in frequency domain
With:With:
€
(X(ω)+Y (ω)) = 2S−1/2[ω2 −Ω]−1S−1/2V (ω)
MOR and MCD`MOR and MCD`
TD-DFT formulation with damping TD-DFT formulation with damping
€
We solve the equation
€
We solve the equation
€
ˆ h ks (v r )+V ext (
v r , t) − i
∂
∂t
⎡ ⎣ ⎢
⎤ ⎦ ⎥φk (r)× exp[−iε kt]exp[−Γt] = 0
To obtain finite lifetime solutionsTo obtain finite lifetime solutions
€
φk' (
v r , t) = C j (t)φ j
j≠i
∑ (v r )× exp[−iε jt]exp[−λ t]
€
Δρ ( y)(ω,r r ) = [X(ω)ai +Y (ω)ai ]φaφi
a
vir
∑i
occ
∑
From which we obtain density change in frequency domainFrom which we obtain density change in frequency domain
E.J. Baerends , G. Ricciardi , A. Rosa , S.J.A van GisbergenCoord.Chem.Rev. 2002,230,5
1Eu C1Φ(2a2u → 2eg)+C2Φ(1a1u --> 2eg)Gouterman State
2Eu
3Eu
C2Φ(2a2u → 2eg)+C1Φ(1a1u --> 2eg)Conjugated Gouterman State
Φ(1b2u → 2eg)
1A1g Ground State
1Eu C1Φ(2a2u → 2eg)+C2Φ(1a1u --> 2eg)Gouterman State
2Eu
3Eu
C2Φ(2a2u → 2eg)+C1Φ(1a1u --> 2eg)Conjugated Gouterman State
Φ(1b2u → 2eg)
1A1g Ground State
€
D(1Eu ) = C1 2a2u y 2egy +C2 1a1u y 2egx[ ]2
1
22.92 −
1
23.25
⎡ ⎣ ⎢
⎤ ⎦ ⎥
2
= 2.27x10−2
€
D(3Eu ) = C1 2a2u y 2egy +C2 1a1u y 2egx[ ]2
1
22.92 +
1
23.25
⎡ ⎣ ⎢
⎤ ⎦ ⎥
2
= 9.51
Simulated Spectrum for ZnP with A-term onlySimulated Spectrum for ZnP with A-term only
ZnP Exp
A-only
Comp l ex Sy m m e try h
A
h
A/D
1E u 0.05 5.49
2E u - 3.37 - 1.62
ZnP
3E u - 0.57 - 0.15
1Eu
2Eu+3Eu
Q
Q
S
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2007,46, 9111-9125. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2007,46, 9111-9125.
Influence of ring distortion on MCD spectrum of ZnPInfluence of ring distortion on MCD spectrum of ZnP
€
B(nB1) = Im−2
3
nB1ˆ L z nB2 A1
ˆ M x nB1 nB2ˆ M y A1
ΔWn
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
€
B(nB2 ) = Im2
3
nB1ˆ L z nB2 A1
ˆ M x nB1 nB2ˆ M y A1
ΔWn
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
€
B (nB1)
€
B (nB2)
€
B (nB1) + B (nB2)
€
A (nEu)
N
N
N
N
M
ω
nB1
nB2
C2vD4h
nEu
N
N
N
N
M
Influence of ring distortion on MCD spectrum of ZnPInfluence of ring distortion on MCD spectrum of ZnP
2.00 2.50 3.00 3.50E(eV)
ZnPx10
Normalized Intensity-0.5
0.50.0
Dist C2V
2.00 2.50 3.00 3.50E(eV)
ZnPx10
Normalized Intensity-0.5
0.50.0
D4h
€
B(nB1) = Im−2
3
nB1ˆ L z nB2 A1
ˆ M x nB1 nB2ˆ M y A1
ΔWn
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
€
B(nB2 ) = Im2
3
nB1ˆ L z nB2 A1
ˆ M x nB1 nB2ˆ M y A1
ΔWn
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
N
N
N
N
M
ω
nB1
nB2
C2vD4h
nEu
N
N
N
N
M
Simulated Spectrum for ZnP with B-term onlySimulated Spectrum for ZnP with B-term only
Exp.
€
B(nEu ) = Im−4
3 p≠n
∑nEux
ˆ L z pEuy A1gˆ M x nEux pEuy
ˆ M y A1g
W ( pE1uy )− W (nE1uy )
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
1Eu 2Eu3Eu
B-terms
Simulated Spectrum forZnP with A+B-term onlySimulated Spectrum forZnP with A+B-term only
E (eV) E (eV)
x 100
ZnP
-0.50
0.00
0.50
1.00
Normalized Intensity
2.00 2.50 3.00 3.50
Exp.
€
B (3Eu ) =
Im−4
3 p≠n
∑3Eux
ˆ L z 2Euy A1gˆ M x 3Eux 2Euy
ˆ M y A1g
W (2E1uy ) −W (3E1uy )
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
€
B (2Eu ) =
Im−4
3 p≠n
∑2Eux
ˆ L z 3Euy A1gˆ M x 2Eux 3Euy
ˆ M y A1g
W (3E1uy ) −W (2E1uy )
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
€
B (3Eu ) =
Im−4
3 p≠n
∑2Eux
ˆ L z 3Euy A1gˆ M x 3Eux 2Euy
ˆ M y A1g
W (2E1uy ) −W (3E1uy )
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
= −B (2Eu )
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
Simulated Spectrum for MgP and NiP with A+B-termSimulated Spectrum for MgP and NiP with A+B-term
2eg 2eg 2eg
2a2u2a2u2a2u
1a1u
1a2u 1a2u 1a2u
1a1u1a1u
1b2u1b2u1b2u
dxy
1eg 1eg
1egdxz, dyz
1eu
dxz, dyz
dx2-y2
dz2
MgP NiP ZnP-7.00
-9.50
-12.00
E(eV)
1b1g
2eg 2eg 2eg
2a2u2a2u2a2u
1a1u
1a2u 1a2u 1a2u
1a1u1a1u
1b2u1b2u1b2u
dxy
1eg 1eg
1egdxz, dyz
1eu
dxz, dyz
dx2-y2
dz2
MgP NiP ZnP-7.00
-9.50
-12.00
E(eV)
1b1g
2.0 2.5 3.0 3.5
E(eV)
(a) MgP
0.0
0.5
0.5
x100
2.0 2.5 3.0 3.5
E(eV)
(a) MgP
0.0
0.5
0.5
x100
2.0 2.5 3.0 3.5
E(eV)
0.0
0.5
0.5
(b) NiP
x100
2.0 2.5 3.0 3.5
E(eV)
0.0
0.5
0.5
(b) NiP
x100
1Eu
2Eu3Eu
Substituted Porphyrins Substituted Porphyrins
N
M
m
β
N N
N
N
MN N
N
MTPP
N
MN N
N
MOEPtetraphenylporphyrin octaethylporphyrin
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
Excited States for Substituted Porphyrins Excited States for Substituted Porphyrins
2.00 2.50 3.00 3.50E(eV)
Normalized Intensity-0.5
0.5
0.0
NiTPP
N
NiN N
N
€
A (1Eu )
€
B (2Eu )
€
B (3Eu )
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
Excited States for Substituted Porphyrins Excited States for Substituted Porphyrins
2.00 2.50 3.00 3.50E(eV)
Normalized Intensity-0.5
0.5
0.0
ZnTPP
x10
N
ZnN N
N
€
A (1Eu )
€
B (2Eu )
€
B (3Eu )
€
A (1Eu )
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
Tetraazaporphyrins and PhthalocyaninesTetraazaporphyrins and Phthalocyanines
N
M
m
β
N N
N
Tetraazaporphyrins and PhthalocyaninesTetraazaporphyrins and Phthalocyanines
N
M
m
β
N N
N
N
M
N N
N N
NN N
MTAPtetraazaporphyrin
Tetraazaporphyrins and PhthalocyaninesTetraazaporphyrins and Phthalocyanines
N
M
m
β
N N
N
N
M
N N
N N
NN N
MTAPtetraazaporphyrin
N
M
N N
N N
NN N
MPcphthalocyanine
Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem.
Inorg. Chem. 2007,46, 9111-9125.
Magnetic Circular Dichroism (MCD) SpectroscopyMagnetic Circular Dichroism (MCD) Spectroscopy
The C term The C term
€
A−'
hω−
A+'
hω=
ΔA'
hω= γβB -A J
∂ρA .J (hω)
∂(hω)+ (B J +
C J
kT)ρA .J (hω)
⎡
⎣ ⎢ ⎤
⎦ ⎥J∑
€
A−'
hω−
A+'
hω=
ΔA'
hω= γβB -A J
∂ρA .J (hω)
∂(hω)+ (B J +
C J
kT)ρA .J (hω)
⎡
⎣ ⎢ ⎤
⎦ ⎥J∑
1P
1S
M- M+
B=0
-A+
A-ΔA
B=0
1P+
1S
M- M+
B>0
1P-
ΔA-A+
A-
B>0
If
€
NP+− NP+
N tot
≈EP+
− EP+
3kT
€
EP+− EP+
<< kT
€
C = −i
3 AAα '
αa 'λ
∑ ˆ L Aα ⋅ Aα ˆ M Jλ × Jλ ˆ M Aα ' ⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
C = −i
3 AAα '
αa 'λ
∑ ˆ L Aα ⋅ Aα ˆ M Jλ × Jλ ˆ M Aα ' ⎛ ⎝ ⎜
⎞ ⎠ ⎟
Electron configuration t1u6t2u
6t1u6t2g
5
Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412
Application: Fe(CN)63-Application: Fe(CN)63-
Electron configuration t1u6t2u
6t1u6t2g
5
Excitations are ligand-metal charge transfer. C term of a transition to a T1u state is positive andto 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- [Fe(CN)5SCN]3-
MnPc
Exp. Calc.
0.58 0.84
-0.60 -0.84
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
Exp. Calc.
0.03 0.90
0.23 0.90
Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412
Limitations of Traditional TD-DFTLimitations of Traditional TD-DFT
ix iy
What do we do with adegenerate ground statethat can not be representedby a single Slater determinant ?
What do we do with adegenerate ground statethat can not be representedby a single Slater determinant ?
Degenerate Ground StateDegenerate Ground State
a
ix iy
a
ix iy
What are thefundamentalequations ?
What are thefundamentalequations ?
How do we calculateexcitationenergies
How do we calculateexcitationenergies
Transformed Reference with an Intermediate ConfigurationKohn Sham (TRICKS) TDDFT
Solution:Solution:
TRICKS of the Trade: Calculating the Excitation Energies of Molecules with Degenerate Ground States using TD-DFTTRICKS of the Trade: Calculating the Excitation Energies of Molecules with Degenerate Ground States using TD-DFT
Idea: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
Example 2:d1 transition metal complexes of Td symmetry,d-d transition
Example 2:d1 transition metal complexes of Td symmetry,d-d transition
Application of the TRIC methodApplication of the TRIC method
€
VCl4
Result 2:d1 transition metal complexes of Td symmetry,d-d transition
Result 2:d1 transition metal complexes of Td symmetry,d-d transition
Application of the TRIC methodApplication of the TRIC method
€
VCl4
Example 3:d1 transition metal complexes of Td symmetry,charge transfer
Example 3:d1 transition metal complexes of Td symmetry,charge transfer
Application of the TRIC methodApplication of the TRIC method
€
VCl4
Result 3:d1 transition metal complexes of Td symmetry,charge transfer
Result 3:d1 transition metal complexes of Td symmetry,charge transfer
Application of the TRIC methodApplication of the TRIC method
€
VCl4
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.
TD-DFT/MCD
Dr. Mike SethDr.Jochen Autschbach
Alejandro Gonzalez Peralta
Dr. Mykhaylo Krykunov
Fan Wang
Hristina Zhekova
PRF
Mitsui
MOR and MCD`MOR and MCD`
TD-DFT formulation without damping TD-DFT formulation without damping
€
We solve the equation
€
We solve the equation
€
ˆ h ks (v r )+V ext (
v r , t) − i
∂
∂t
⎡ ⎣ ⎢
⎤ ⎦ ⎥φk (r)× exp[−iε kt] = 0
To obtain the solutionTo obtain the solution
€
φk' (
v r , t) = C j (t)φ j
j≠i
∑ (v r )× exp[−iε jt]
€
Δρ ( y)(ω,r r ) = [X(ω)ai +Y (ω)ai ]φaφi
a
vir
∑i
occ
∑
From which we obtain density change in frequency domainFrom which we obtain density change in frequency domain
With:With:
€
(X(ω)+Y (ω)) = 2S−1/2[ω2 −Ω]−1S−1/2V (ω)
MOR and MCDMOR and MCD
€
Vsos (ω) = −γhω2B J
WJ2 −(hω)2
J
∑
€
Vsos (ω) = −γhω2B J
WJ2 −(hω)2
J
∑The expressionThe expression
Allows us to calculate the MOR parameter V( ) from the MCDparameters BJ after summing over all statesAllows us to calculate the MOR parameter V( ) from the MCDparameters BJ after summing over all states
aM. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,
MOR and MCDMOR and MCD
€
V (ω) = −γhω 2B J
WJ2 − (hω)2
J
∑
€
V (ω) = −γhω 2B J
WJ2 − (hω)2
J
∑The expressionThe expression
Vres(
€
Vdamp(ω)
€
The expression for V(ω) diverges for hω = WJ
€
The expression for V(ω) diverges for hω = WJ
We need a TD-DFT formulation in which damping includedWe need a TD-DFT formulation in which damping included
MOR and MCD`MOR and MCD`
TD-DFT formulation with damping TD-DFT formulation with damping
€
We solve the equation
€
We solve the equation
€
ˆ h ks (v r )+V ext (
v r , t) − i
∂
∂t
⎡ ⎣ ⎢
⎤ ⎦ ⎥φk (r)× exp[−iε kt]exp[−Γt] = 0
To obtain finite lifetime solutionsTo obtain finite lifetime solutions
€
φk' (
v r , t) = C j (t)φ j
j≠i
∑ (v r )× exp[−iε jt]exp[−λ t]
€
Δρ ( y)(ω,r r ) = [X(ω)ai +Y (ω)ai ]φaφi
a
vir
∑i
occ
∑
From which we obtain density change in frequency domainFrom which we obtain density change in frequency domain