Tom Ziegler Department of Chemistry University of Calgary,Alberta, Canada T2N 1N4 Magnetically Perturbed Time Dependent Density Functional Theory. Applications

Tom Ziegler Department of Chemistry University of Calgary,Alberta, Canada T2N 1N4

Magnetically Perturbed Time Dependent Density Functional Theory.Applications and Implementations

Tuesday November 11 11:30 am - 12:10 pm

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

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 :

T. Ziegler,M.Seth,M.Krykunov,J.AutschbachA Revised Electronic Hessian for Approximate Time-Dependent Density Functional TheorySUBMITTED, J.C.P.

T. Ziegler,M.Seth,M.Krykunov,J.AutschbachA Revised Electronic Hessian for Approximate Time-Dependent Density Functional TheorySUBMITTED, J.C.P.



€


∂Pjb

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟0

€

Bia,bj =∂F ia

∂Pbj

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟0

Corredted Definition of A and B Matrices :Corredted Definition of A and B Matrices :



€

ΩF (λ ) =Wλ2F (λ )

€

Ω=−S−1/2(A + B)S−1/2

€

S−1/ 2 = (A − B)1/ 2

Where :Where :

€

+ f1

2[Jaa,aa − Kaa,aa + Jii,ii − K ii,ii − 2Jaa,ii + 2Kaa,ii]

€

+ f1

2[Jaa,aa − Kaa,aa + Jii,ii − K ii,ii − 2Jaa,ii + 2Kaa,ii]



€


∂Pjb

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟0

€

Bia,bj =∂F ia

∂Pbj

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟0

Corredted Definition of A and B Matrices :Corredted Definition of A and B Matrices :



€

ΩF (λ ) =Wλ2F (λ )

€

Ω=−S−1/2(A + B)S−1/2

€

S−1/ 2 = (A − B)1/ 2

Where :Where :

Spin-flip transitions using non-collinear functionalsLiu (2004),Ziegler+Wang (2005),Vahtras (2007)


€

Wλ = ΔE o,λTransition Energy :Transition Energy :

Basic Equation :Basic Equation :M.E.CasidaM.E.Casida


€

ΩF (λ ) = Wλ2F (λ )

Electric Transition Dipole Moment :Electric Transition Dipole Moment :

€

Aα ˆ M Jλ =1

WJ

μ iaFia(Jλ ) (εa − εi)

ia

∑

€

μ ia = − ir r a

€

Jλ ˆ L Aα = WJ liaFia( Jλ ) 1

(ε a −ε i )ia

∑

Magnetic Transition Dipole Moment :Magnetic Transition Dipole Moment :

€

l jb = −iμB jr r ×

r ∇ b

A

C

B

A

B

C

Absorption Spectra and TD-DFTAbsorption Spectra and TD-DFT

Transition Energy :Transition Energy :

€

fλ =2

3μ iaFia

(Jλ ) (εa − εi)ia

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥⋅ μ jbF jb

(Jλ ) (εb − ε j )jb

∑∫ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

Wλ = ΔE0,λ

hv

Cl

C

C

C

C

C

Si Zr

C

C

C

C

Cl

C

Inorganic SpectroscopyInorganic Spectroscopy

N

MN N

N

H

Why MCD and MOR ? Why MCD and MOR ?

Magnetic Circular Dichroism (MCD) SpectroscopyMagnetic Circular Dichroism (MCD) Spectroscopy

More information about each excited stateMore information about each excited state

In absorption spectroscopy onlypositive (often overlapping) bandsIn absorption spectroscopy onlypositive (often overlapping) bands

Why MCD ? Why MCD ?


In MCD bands of different shapesMore information about each excited stateIn MCD bands of different shapesMore information about each excited state


€ €

Origin of MCD ? Origin of MCD ?

€

AJ = γoω(NAαg − N Jλ j )

Nαλgj∑ Aαg ˆ M Jλ j

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟× ρ Aαg.Jλ j (ω)

€



2 ⎛ ⎝ ⎜ ⎞


Electric dipole operator:Electric dipole operator:

€

ˆ M = ˆ m ii∑ = − (xii

∑ r e xi

+ yi

r e yi

+ yi

r e yi

)

€

ˆ M = ˆ m ii∑ = − (xii

∑ r e xi

+ yi

r e yi

+ yi

r e yi

)

Absorbance in dipole approximation.Absorbance in dipole approximation.

Aα

Jλ


€ €


€



2 ⎛ ⎝ ⎜ ⎞


€



2 ⎛ ⎝ ⎜ ⎞


Absorbance in dipole approximation.Absorbance in dipole approximation.

Aα

Jλ

€

ρAαg.Jλj (ω) ≈ fJ (ω −ωJλ ) =1

πWJ

e−

ωJλ −ω

WJ

⎛

⎝ ⎜

⎞

⎠ ⎟

2


Electric dipole operatorFor circular polarizedLight:

Electric dipole operatorFor circular polarizedLight:

€

ˆ M ± = ˆ m ±,ii∑

€

ˆ M ± = ˆ m ±,ii∑€

ΔAJ

ω= γo

(NAαg − N Jλj )

Nαλgj∑ Aαg ˆ M − Jλ j

2− Aαg ˆ M + Jλ j

2 ⎛ ⎝ ⎜ ⎞


Difference in absorbance of left and right circular polarized lightDifference in absorbance of left and right circular polarized light

Circular Polarized LightCircular Polarized Light

€

ˆ m − =1

2(x

r e x − iy

r e y )

€

ˆ m + =1

2(x

r e x + iy

r e y )



€

ΔAJ'

ω=

A−,J'

ω−

A+,J'

ω

γo

(NAαg − N Jλj )

Nαλgj∑ Aαg ˆ M − Jλ j

2− Aαg ˆ M + Jλ j

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟.ρ Aαg.Jλ j (ω)

⎛

⎝ ⎜

⎞

⎠ ⎟

'

The difference in absorption of left and right circularly polarized light in the presence of a magnetic field as a function of photon energy

The difference in absorption of left and right circularly polarized light in the presence of a magnetic field as a function of photon energy


€

ρAαg.Jλj (ω) ≈ fJ (ω −ωJλ, ) =

1

π WJ

e−

ω Jλ, −ω

WJ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2


€

ΔAJ'

hω=

γo

3 γ

3

∑ (NAα − N Jλ )

Nαλ∑ Aα ˆ M −

γ Jλ2

− Aα ˆ M +γ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟ ∂

∂Bγ

fJ (ω −ωJλ )( )oB

+γo

3 γ

3


Nαλ∑ ∂

∂Bγ

Aα ˆ M −γ Jλ

2− Aα ˆ M +

γ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

fJ (ω −ωJ )B

+γo

3

∂

∂B γ

3

∑ (NAα − N J )

N

⎛

⎝ ⎜

⎞

⎠ ⎟o

α∑ Aα ˆ M −

γ Jλ2


2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟ fJ (ω −ωJ )B

€

ΔAJ'

hω=

γo

3 γ

3


Nαλ∑ Aα ˆ M −

γ Jλ2


2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟ ∂

∂Bγ

fJ (ω −ωJλ )( )oB

+γo

3 γ

3


Nαλ∑ ∂

∂Bγ

Aα ˆ M −γ Jλ

2− Aα ˆ M +

γ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

fJ (ω −ωJ )B

+γo

3

∂

∂B γ

3

∑ (NAα − N J )

N

⎛

⎝ ⎜

⎞

⎠ ⎟o

α∑ Aα ˆ M −

γ Jλ2


2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟ fJ (ω −ωJ )B

€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )

∂ωB +γ0B J fJ (ω −ωJ )B +γoC J fJ (ω −ωJ )B

€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )

∂ωB +γ0B J fJ (ω −ωJ )B +γoC J fJ (ω −ωJ )B


€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )

∂ωB +γ0B J fJ (ω −ωJ )B

+γoC J fJ (ω −ωJ )B

€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )




The MCD disprsion The MCD disprsion

P.J.Stephens. Ph.D. Thesis 1964

A

B

C(T)

A

€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )



€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )






Positive A-term

Absorption band Negative A-term

Degenerate ground- or (and) excited state

€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )



€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )






Absorption band

All cases

Negative B-term

Positive B-term

Negative B-term

Positive B-term

€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )



€

ΔAJ'

ω= −γoA J

∂fJ (ω −ωJ )






Absorption band

Space and(or) spin-degenerate ground state

Negative C-term

Positive C-term

Origin of B-TermOrigin of B-Term

The B term The B term

O

X+iaY

Y-iaX

M- M+

B>0

ΔA-A+

A-

B>0

O

X

Y

M- M+

B=0

-A+

A-ΔA

B=0

€

ΔA '

ω= γoB -A J

∂fJ (ω −ωJ )B

∂ω+(B J +

C J

kT) fJ (ω −ωJ )B

⎡

⎣ ⎢ ⎤

⎦ ⎥J∑

€

ΔA '

ω= γoB -A J

∂fJ (ω −ωJ )B

∂ω+(B J +

C J

kT) fJ (ω −ωJ )B

⎡

⎣ ⎢ ⎤

⎦ ⎥J∑

€

B J =1

3 γ

3

∑λ

∑ ∂

∂Bγ

Aα ˆ M −γ Jλ

2− Aα ˆ M +

γ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

Expression for the B-TermExpression for the B-Term

The B term The B term

€

B J =1

3 γ

3

∑λ

∑ ∂

∂Bγ

Aα ˆ M −γ Jλ

2− Aα ˆ M +

γ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

Or by using the identity Or by using the identity

€

t

∑ A ˆ M t J2

− A ˆ M t J2

= i ε rst

r,s ,t

∑ A ˆ M r J J ˆ M s A = i ε rst

r ,s,t

∑ α rs (ωL )

€

Here ε rst is the three - dimensional Levi - Civita symbol

€

Here ε rst is the three - dimensional Levi - Civita symbol

We thus haveWe thus have

€

BJ =i

3 s,t ,u

3

∑ εstu

∂α st (ω)

∂Bu

⎛

⎝ ⎜

⎞

⎠ ⎟ω=ωL( )


The Calculation of the B-termThe Calculation of the B-term

The B term : practical calculations The B term : practical calculations

We have:We have:

TD-DFT calculationsTD-DFT calculations

€

€

B J =i

3ε stu

∂α st (ω)

∂Bu

⎛

⎝ ⎜

⎞

⎠ ⎟

s,t ,u

∑ω=ωL

Where:Where:

€

α st (ω)ω=ωL= m s[XL (ω) + YL (ω)][m t (XL (ω) + YL (ω)]

€

ωC 0

0 −C

⎛

⎝ ⎜

⎞

⎠ ⎟X

Y

⎛

⎝ ⎜

⎞

⎠ ⎟=

A B

B* A*

⎛

⎝ ⎜

⎞

⎠ ⎟X

Y

⎛

⎝ ⎜

⎞

⎠ ⎟

Early work:

J.Michl, J.Am.Chem.Soc. 100,6801 (1978)

Early work:

J.Michl, J.Am.Chem.Soc. 100,6801 (1978)

€

m s = i(0) m s a(0)


We have:We have:

TD-DFT calculationsTD-DFT calculations

Solve:

€

€

BJ = −i

3εstu

∂α st (ω)

∂Bu

⎛

⎝ ⎜

⎞

⎠ ⎟

s,t ,u

∑ω=ωL

Where:Where:

€

ω(0) C 0

0 −C

⎛

⎝ ⎜

⎞

⎠ ⎟X (0)

Y (0)

⎛

⎝ ⎜

⎞

⎠ ⎟=

A(0) B(0)

B(0) A(0)

⎛

⎝ ⎜

⎞

⎠ ⎟X (0)

Y (0)

⎛

⎝ ⎜

⎞

⎠ ⎟

€

BAJ = −2i

3εstu[

stu

∑ m s(1)u(XJ(0) −YJ

(0))m t(0)(XJ(0) + YJ

(0))

+m s(0)(XJ(1)u + YJ

(1)u)m t(0)(XJ(0) + YJ

(0))]



By differentiation ofBy differentiation of

€

ω(0) C 0

0 −C

⎛

⎝ ⎜

⎞

⎠ ⎟X (0)

Y (0)

⎛

⎝ ⎜

⎞

⎠ ⎟=

A(0) B(0)

B(0) A(0)

⎛

⎝ ⎜

⎞

⎠ ⎟X (0)

Y (0)

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Implementation - X(1), Y(1)( )

€

The equation that we use for evaluating (X(1), Y(1) ) is

€

A(0) B(0)

B(0)* A(0)*

⎛

⎝ ⎜

⎞

⎠ ⎟−ωI

(0) −I 0

0 I

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟X I

(1)

YI(1)

⎛

⎝ ⎜

⎞

⎠ ⎟=

ωI(1) −I 0

0 I

⎛

⎝ ⎜

⎞

⎠ ⎟−

A(1) B(1)

B(1)* A(1)*

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟X I

(0)

YI(0)

⎛

⎝ ⎜

⎞

⎠ ⎟


€

BAJ = −2i

3εstu[

stu

∑ m s(1)u(XJ(0) −YJ

(0))m t(0)(XJ(0) + YJ

(0))

+m s(0)(XJ(1)u + YJ

(1)u)m t(0)(XJ(0) + YJ

(0))]


€

Evaluation of - X(1),Y(1)( )

€

Evaluation of - X(1),Y(1)( )

€

Introducing the unitary transformation U =1 1

1 -1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

UA(0) B(0)

B(0)* A(0)*

⎛

⎝ ⎜

⎞

⎠ ⎟−ωI

(0) −I 0

0 I

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟U

+UX I

(1)

YI(1)

⎛

⎝ ⎜

⎞

⎠ ⎟=U ωI

(1) −I 0

0 I

⎛

⎝ ⎜

⎞

⎠ ⎟−

A(1) B(1)

B(1)* A(1)*

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟U

+UX I

(0)

YI(0)

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ω I(0)I −Ω[ ]Z I

(1) = −ωI(0)S1/2(A(1) + B(1) )S−1/2FI

(0)

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

(0)

Here:Here:

€

Z I(1) = ωI

(0) S1/2(X I(1) +YI

(1) )

AffordsAffords




€

ω I(0)I −Ω[ ]Z I

(1) = −ωI(0)S1/2(A(1) + B(1) )S−1/2FI

(0)

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

(0)

€

An Expression for Kai,bj(1)

€

An Expression for Kai,bj(1)

€

We need φp(1). A well known expresson exists that is particularly simple because we have an

imaginary perturbation

€ €

€

φp(1) = Uqp

(1)φq(0) Uqp

(1) =H pq

(1)

εq(0) −ε p

(0)q≠ p

∑

€

φp(1) = Uqp

(1)φq(0) Uqp

(1) =H pq

(1)

εq(0) −ε p

(0)q≠ p

∑

€

Where H(1) is the Hamiltonian describing the perurbation

Thus

€

Kai,bj(1) = U pa

(1)*K pi,bj(0)

p≠a

∑ + U pi(1)*Kap,bj

(0) + U pb(1)*Kai,pj

(0)

p≠b

∑p≠i

∑ + U pb(1)*Kap,bp

(0)

p≠ j

∑

€

Kai,bj(1) = U pa

(1)*K pi,bj(0)

p≠a

∑ + U pi(1)*Kap,bj

(0) + U pb(1)*Kai,pj

(0)

p≠b

∑p≠i

∑ + U pb(1)*Kap,bp

(0)

p≠ j

∑


Seth+Ziegler JCP,2008,in pressSeth+Ziegler JCP,2008,in pressM.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

The B term : Direct method The B term : Direct method

We must solve We must solve

€

AX = b€

ω I(0)I −Ω[ ]Z I

(1) = −ωI(0)S1/2(A(1) + B(1) )S−1/2FI

(0)

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

(0)

€

Our equation has the form

€

Our equation has the form

€

With A a known matrix, b a known vector and X the unknown vector to be determined. This

equation can be solved easily if we have A−1. There are two problems however

€

With A a known matrix, b a known vector and X the unknown vector to be determined. This

equation can be solved easily if we have A−1. There are two problems however

€

(a) The matrix A =ωI(0)I −Ω. This matrix has no inverse because ωI

(0)I is an eigenvalue of Ω

€

(b) The matrix A is extremely large and we don' t want to try and invert it directly.

€

To avoid this problem we :

€

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 the ith iteration)

€

(ii) Project out from the Krylov supspaces any contribution from FI(0)

The Calculation of the B-term by Direct MethodThe Calculation of the B-term by Direct Method

Seth+Ziegler JCP,2008Seth+Ziegler JCP,2008

The B term : Direct Method The B term : Direct Method

We must solve We must solve

€

Ax = b

€

(i) Can be used in conjunction with an unperturbed

TDDFT calculation that yields only a few solutions F(0).

€

(ii)Degree of convergence is known

ProsPros

ConsCons

€

(i) The iterative procedure is often slowly convergent.

We are attempting to improve convergence by adding

the unperturbed TDDFT solutions FJ(0), J ≠ I to Krylov subspace

€

BAJ = −2i

3εαβγ

αβγ

∑ M β S−1/2

ωJ(0)

(Z J(1)α )M λ (XJ

(0) +YJ(0) )

€

ω I(0)I −Ω[ ]Z I

(1) = −ωI(0)S1/2(A(1) + B(1) )S−1/2FI

(0)

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

(0)

The Calculation of the B-term by Direct MethodThe Calculation of the B-term by Direct Method

Seth+Ziegler JCP,2008,in pressSeth+Ziegler JCP,2008,in pressM.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

The B term : Sum Over State The B term : Sum Over State

€

ω I(0)I −Ω[ ]Z I

(1) = −ωI(0)S1/2(A(1) + B(1) )S−1/2FI

(0)

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

(0)

€

Z (1) by Sum - Over - State

€

Z (1) by Sum - Over - State

€

Z I(1) = CJIFJ

(0)

J≠I

∑

€

Z I(1) = CJIFJ

(0)

J≠I

∑

€

Substitute into first order equation and multiply by FJ(0) from left affords

€

FJ(0)+ ωI

(0)I −Ω[ ]( (CJIFJ(0)

J≠I

∑ ) = −ωI(0)FJ

(0)+S1/2 (A(1) + B(1) )S−1/2FI(0)

−ωI(0)FJ

(0)+S−1/2(A(1) − B(1) )S1/2FI(0)

OrOr

€

CJI =−ωI

(0)(FJ(0)+S1/2(A(1) + B(1) )S−1/2FI

(0) + FJ(0)+S−1/2(A(1) − B(1) )S1/2FI

(0)

ωI(0) −ωJ

(0)

Writing Z(1) in terms of the complete set F(0) affordsWriting Z(1) in terms of the complete set F(0) affords

The Calculation of the B-term by Sum-over-State MethodThe Calculation of the B-term by Sum-over-State Method

Seth+Ziegler JCP,2008,134108Seth+Ziegler JCP,2008,134108M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.

Phys. J. Chem. Phys. 128, 144105 (2008)

The B term : Sum Over State The B term : Sum Over State

€

Z (1) by Sum - Over - State : Z I(1) = CJI

J≠I

∑ FJ(0)

€

CJI =−ωI

(0)(FJ(0)+S1/2(A(1) + B(1) )S−1/2FI

(0)

ωI(0) −ωJ

(0)+

+FJ(0)+S−1/2(A(1) − B(1) )S1/2FI

(0)

ωI(0) −ωJ

(0)

€

BAJ = −2i

3εαβγ

αβγ

∑ M β S−1/2

ωJ(0)

(Z J(1)α )M λ (XJ

(0) +YJ(0) )

€

BAJ = −2i

3εαβγ

αβγ

∑ M β S−1/2

ωJ(0)

(Z J(1)α )M λ (XJ

(0) +YJ(0) )

ProsPros

€

Interpretation easy in terms of contributions

from different excited statesConsCons

€

May need to calculate many FJ(0) in unperturbed

TDDFT and convergence of summation is unknown

The Calculation of the B-term by Sum-over-State MethodThe Calculation of the B-term by Sum-over-State Method

Seth+Ziegler JCP,2008,134108Seth+Ziegler JCP,2008,134108

Other B-term implementations Other B-term implementations

7S.Coriani, P.Jørgensen, T.Helgaker J.Chem.Phys. 113,3561,2000

HF+CI

CCSD(T)

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)

T.Kjœrgaard, B.Jansik, P.Jørgensen,S.Coriani, J.Michl, J.Phys.Chem. A 111,11278 (2007))

DFT

H.Solheim; L.Frediani; K.Rudd; S.Coriani Theor.Chem.Acc 119,231,2007

DFT-SOS

M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107

M.Seth,T.Ziegler,J.Autschbach J.Chem.Theory.Comp.3,434,2007

J.Michl J.Am.Chem.Soc. 100, 6801, 1978


Convergence of SOS-method for Ethylene Convergence of SOS-method for Ethylene

Comparison of Sum-over-State and Direct Method for B-termsComparison of Sum-over-State and Direct Method for B-terms

€

π → π *

€

π → 3s

€

π → 3s

€

π → π *

Seth+Ziegler JCP,2008Seth+Ziegler JCP,2008

Comparison of Direct Method for B-terms with ExperimentComparison of Direct Method for B-terms with Experiment

Exp: J.W.Waluk, J.Michl Inorg.Chem. 21,556,1982) Exp: H.-P.Klein, R.T. Oakley, J.Michl Inorg.Chem. 25,3194 (1986)

S4N3+ S4N2

Comparison of Direct Method for B-terms with ExperimentComparison 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,2008Seth+Ziegler JCP,2008

Comparison of Direct Method for B-terms with Experiment and other MethodsComparison 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)


TD-DFT calculations of B-term. TD-DFT calculations of B-term.

Furan

Thiophene

Selenophen

Tellurophen

S

Se

Te

OW. Hieringer, S. J. A. van Gisbergen, and E. J. BaerendsJ. Phys. Chem. A 2002, 106, 10380

X

1a2

1b1

2b1

11A1 --> 21A1

11A1 --> 11B2

X

X

X 1b2 11A1 --> 11B1


TD-DFT calculations of B-term. Sum-over-state formulationTD-DFT calculations of B-term. Sum-over-state formulation

Norden, B.; Hansson, R.; Pedersen, P. B.; Thulstrup, E. W. Chem.Phys. 1978, 33, 355.


Norden, B.; Hansson, R.; Pedersen, P. B.; Thulstrup, E. W. Chem.Phys. 1978, 33, 355.


Furan

O

6.0 6.2

0.20

-5.05 0.0 3.37

.13

€

1a2 → 3b1

€

2b1 → 3b1



Thiophene

S

5.5 5.7

450 6-477

.04

5.9

.13

€

1a2 → 3b1

€

2b1 → 3b1



Selonophene

Se

5.1 5.3

0.22

59.1 -3 -101

.07

5.5

€

1a2 → 3b1

€

2b1 → 3b1



TellurophenX

1a2

1b1

2b1

11A1 --> 21A1

11A1 --> 11B2

X

X

X 1b2 11A1 --> 11B1

Te

4.4 4.8

0.64

-5.1 -28.012.8

5.24.4 4.8

0.64

-5.1 -28.012.8

5.2


A-term of MCDA-term of MCD

€

ΔAJ'

hω=

γo

1

3γ

∑ (NA − N Jλ )

NJλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

∂

∂Bγ

f (ω −ωJλ )( )0B

+γo

1

3γ

∑ (NA − N Jλ )

N

∂

∂BγJλ

∑ A ˆ M γ− Jλ

2− A ˆ M γ

+ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

⋅ f (ω −ωJλ )( )B

+γo

1

3γ

∑ ∂

∂Bγ

(NA − N Jλ )

N

⎛

⎝ ⎜

⎞

⎠ ⎟0

Jλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅ f (ω −ωJλ )( )B

€

ΔAJ'

hω=

γo

1

3γ

∑ (NA − N Jλ )

NJλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

∂

∂Bγ

f (ω −ωJλ )( )0B

+γo

1

3γ

∑ (NA − N Jλ )

N

∂

∂BγJλ

∑ A ˆ M γ− Jλ

2− A ˆ M γ

+ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

⋅ f (ω −ωJλ )( )B

+γo

1

3γ

∑ ∂

∂Bγ

(NA − N Jλ )

N

⎛

⎝ ⎜

⎞

⎠ ⎟0

Jλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅ f (ω −ωJλ )( )B

€

ΔAJ'

ω= −γoA J

∂f (ω −ωJλ )

∂ω+γ0B J f (ω −ωJλ )B +γoC J f (ω −ωJλ )B

€

ΔAJ'

ω= −γoA J

∂f (ω −ωJλ )

∂ω+γ0B J f (ω −ωJλ )B +γoC J f (ω −ωJλ )B

Origin of A-term Origin of A-term

M.Seth,T.Ziegler,J.Chem.Phys. 2004,120,10943


The A-term of Magnetic Circular Dichroism (MCD) SpectroscopyThe A-term of Magnetic Circular Dichroism (MCD) Spectroscopy

The A term The A term

€

γo -A J

∂f (ω −ωJ )

∂ω

⎡ ⎣ ⎢

⎤ ⎦ ⎥J

∑ B

= γo

1

3γ

∑ (NA − N Jλ )

NJλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

∂

∂Bγ

f (ω −ωJλ )( )0B

= γo

1

3γ

∑ (NA − N Jλ )

NJλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

∂ωJλ

∂Bγ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟0

∂f (ω −ωJ )

∂ωB

ThusThus

€

A J = −λ

∑ 1

3γ

∑ (NA − N Jλ )

NA ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟0

⋅∂ωJλ

∂Bγ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟





€

A J = −λ

∑ 1

3γ

∑ (NA − N Jλ )

NA ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟0

⋅∂ωJλ

∂Bγ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

We haveWe have

ThusThus


HereHere

€

ωλ2 = FJλ

T ΩFJλ


€

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∑

-A+

A-ΔA

B=0

RCP LCP

1S

1P

B=0

-A+

A-

ΔA-

B>0

RCP LCP

O

-1

1

0

B>0

€

A ˆ M Jλ =r

∑ M rS−1/ 2FLλ(0)v

e r

€

A ˆ M Jλ =r

∑ M rS−1/ 2FLλ(0)v

e r


€

ˆ L = ˆ l ij=1

n

∑ = −ir r i

j=1

n

∑ ×r

∇ i

Other A-term implementations Other A-term implementations

7S.Coriani, P.Jørgensen, T.Helgaker J.Chem.Phys. 113,3561,2000

HF+CI

CCSD(T)

Y.Honda, M.Hada, M.Ehara, H.Nakatsuiji,J.Downing,J.Michl , Chem.Phys.Lett 355,219, , 2002


DFT

H.Solheim; ; K.Rudd; S.Coriani ,P.Norman J.Chem.Phys. 128,094193,2008

J.Michl J.Am.Chem.Soc. 100, 6801, 1978

Y.Honda, M.Hada, M.Ehara, H.Nakatsuiji,J.Michl , J.Chem.Phys. 123,164113 (2005)


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

Se42+ Te4

2+

Fe(CN)64-

Ni(CN)42- C6Cl6

C6H3Br3

Oh

D4hD4h

D6h

D3h

Exp: 0.72Calc: 0.63

Exp: 0.60Calc: 0.55Exp: 0.40 Calc: 0.48

Exp:-0.66Calc:-0.72Exp:-0.50Calc:-0.80


€

AD

∝∂ω∂B

Positive A-term

Negative A-termNegative

B-termPositive B-term

Different MCD-terms

Absorption band

3t2

2e

t1

2t2

Metal

Ligand

3t2

2e

t1

2t2

Metal

Ligand


MCD-terms for Oxyanions MCD-terms for Oxyanions


Exp.Theor

MCD-terms for Thioanions MCD-terms for Thioanions

M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. Submitted

N

MN N

N

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.

MCD spectra of Porphyrins containing Mg,Ni and ZnMCD spectra of Porphyrins containing Mg,Ni and Zn

21Eu 31Eu

5 10-2

2e1.g

2a2.u

1a2.u

1a1.u

1b2.u1e1.g

ZnP

1b1.g

Orbital level diagram for ZnPOrbital level diagram for ZnP

2a2u

1a1u

1b2u

E.J. Baerends , G. Ricciardi , A. Rosa , S.J.A van GisbergenJ.Phys.Chem. A2001,105,3311

E.J. Baerends , G. Ricciardi , A. Rosa , S.J.A van GisbergenCoord.Chem.Rev. 2002,230,5

2eg2

2eg1

21Eu 31Eu

5 10-2

Exc. Energ. (eV) Complex Symmetry

exp. calc. Composition % h

f Assig n.

2a 2u -> 2e g 52.10 1E u

2.03 c, 2.21 d,

2.23 e , 2.18 f

2.28 1a 1u -> 2e g 46.63

0.001 Q

1b 2u -> 2e g 68.44

1a 1u -> 2e 1g 17.54 2E u 2.95 c, 3.09 d,

3.18 e

, 3.13 f 3.25

2a 2u -> 2e g 10.05 0.496

1b 2u -> 2e g 29.88 2a 2u -> 2e g 29.31 1a 1u -> 2e g 27.13

ZnP

3E u

3.32

1a 2u -> 2e g 10.30

0.943

Bg

Experimental Spectrum for ZnPExperimental Spectrum for ZnP



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


2Eu

3Eu

C2Φ(2a2u → 2eg)-C1Φ(1a1u --> 2eg)Conjugated Gouterman State

Φ(1b2u → 2eg)

1A1g Ground State

2e1.g

2a2.u

1a2.u

1a1.u

1b2.u1e1.g

ZnP

1b1.g

21Eu 31Eu

5 10-2

21Eu 31Eu

5 10-2

Experimental Spectrum for ZnPExperimental Spectrum for ZnP

L.Edwards,D.H.Dolphin,M.Goutermn J.Mol.Spectrosc 35(1970)90



2Eu

3Eu

C2Φ(2a2u → 2eg)+C1Φ(1a1u --> 2eg)Conjugated Gouterman State

Φ(1b2u → 2eg)

1A1g Ground 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


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


0.50.0

D4h

€

B(nB1) = Im−2

3

nB1ˆ L z nB2 A1


ΔWn

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

€

B(nB2 ) = Im2

3

nB1ˆ L z nB2 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 M y A1g

W (3E1uy ) −W (2E1uy )

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

€

B (3Eu ) =

Im−4

3 p≠n

∑2Eux


ˆ M y A1g

W (2E1uy ) −W (3E1uy )

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

= −B (2Eu )



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



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)


0.5

0.0

NiTPP

N

NiN N

N

€

A (1Eu )

€

B (2Eu )

€

B (3Eu )



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)


0.5

0.0

ZnTPP

x10

N

ZnN N

N

€

A (1Eu )

€

B (2Eu )

€

B (3Eu )

€

A (1Eu )



Inorg. Chem. 2007,46, 9111-9125.

Tetraazaporphyrins and PhthalocyaninesTetraazaporphyrins and Phthalocyanines

N

M

m

β

N N

N


N

M

m

β

N N

N

N

M

N N

N N

NN N

MTAPtetraazaporphyrin


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



Inorg. Chem. 2008,46, 9111-9125.


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

Limitations of Traditional TD-DFTLimitations of Traditional TD-DFT

ix iy

What do we do with adegenerate ground statethat can not be representedby single Slater determinant ?

What do we do with adegenerate ground statethat can not be representedby single Slater determinant ?

Degenerate Ground StateDegenerate Ground State

a

ix iy

a

ix iy

What are thefundamentalequations ?


How do we calculateexcitationenergies


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

• Degenerate ground states are generally treated within DFT by fractional occupations of the degenerate orbital. This gives a ground state of indeterminent symmetry.

• Degenerate ground states are generally treated within DFT by fractional occupations of the degenerate orbital. This gives a ground state of indeterminent symmetry.

ChallengesChallenges

• 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

• 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

Transformed Reference with an Intermediate ConfigurationKohn Sham (TRICKS) TDDFT

Solution:Solution:


Idea:Idea:Avoid problems with a degenerate ground state by taking an excited state that is nondegenerate as the (Transformed) Reference Intermediate Configuration.

Example 1:d1 transition metal complexes of Oh symmetry,d-d transition

Example 1:d1 transition metal complexes of Oh symmetry,d-d transition

Application of the TRIC methodApplication of the TRIC method

€

TiF63−

Results 1:d1 transition metal complexes of Oh symmetry,d-d transition.

Results 1:d1 transition metal complexes of Oh symmetry,d-d transition.


€

TiF63−

Example 2:d1 transition metal complexes of Td symmetry,d-d transition



€

VCl4

Result 2:d1 transition metal complexes of Td symmetry,d-d transition



€

VCl4

Example 3:d1 transition metal complexes of Td symmetry,charge transfer



€

VCl4

Result 3:d1 transition metal complexes of Td symmetry,charge transfer



€

VCl4

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


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


|A>

|J>

|K>

JA <A|LAJ|J>

<J|r|A>

€

ε(1)

hω

<K|LAJ|J>

KJ

|A>

|J>

€

ε(2)

hω

|K>

<J|r|A><A|r|K>|A>

|J>

€

ε(3)

hω

<J|r|A>

|K><K|LAJ|A> KJ

Spin-degenerate Ground State MCD via Spin-orbit Coupling

M.L.Kirk Curr.Op.Chem.Bio 2003,220

Application to Plastocyanin

85 §M.E. I. Solomon, R.K. Szilagyi, S. D. George and L. Basumallick, Chem. Rev, 104, 419, 2004.

Application to Plastocyanin

<K|LAJ|J>

KJ

|A>

|J>

€

ε(2)

hω

|K>

<J|r|A><A|r|K>

Application to Sulfite Oxidase Application to Sulfite Oxidase

87 §M.E. Helton, A. Pacheco, J. McMaster, J.H. Enemark and M. Kirk, J. Inorg. Biochem., 80, 227, 2000.

Application to Sulfite Oxidase

L1: -SCH3. L2: -OH. L3: -S(CH2)2S-.

|A>

|J>

<J|r|A>

|K><K|LAJ|A> KJ

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

€


€

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,


€

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


TD-DFT formulation with damping TD-DFT formulation with damping

€


€


€

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

With:

€

(X(ω)+Y (ω)) = 2S−1/2[(ω + iγ )2 −Ω]S−1/2V (ω)L.Jensen; J.Autchbach; G.C.Schatz J.Chem.Phys.2005,122,224115



€

Vresdm (ω) =Vres

R,dm (ω)+ iVresI ,dm

HereHere

€

VresR,dm (ω) =Vsos

R,dm (ω) ≡J

∑ 2ω2 (ωJ2 −ω2 )B J

(ωJ2 −ω2 )2 + 4ω2γ 2

€

VresI ,dm (ω) =Vsos

I ,dm (ω) ≡J

∑ 4ω3γB J

(ωJ2 −ω2 )2 + 4ω2γ 2andand

M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted



€

VresR,dm (ω) =Vsos

R,dm (ω) ≡ γ0

J

∑ 2ω2(ωJ2 −ω2 )B J

(ωJ2 −ω2 )2 + 4ω2γ 2

oror

€

VsosR,dm (ω) =Vsos

udm (ω) fd (ω)

€

VsosR,dm (ω) =Vsos

udm (ω) fd (ω)

€

fd (ω) =

2(ωJ2 −ω2 )2

(ωJ2 −ω2 )2 + 4ω2γ 2

HereHere

€

Vsosudm (ω)

€

VsosR,dm (ω)




€

VresI ,dm (ω) =Vsos

I ,dm (ω) ≡J

∑ 4ω3γB J

(ωJ2 −ω2 )2 + 4ω2γ 2

€

VresI ,dm (ω) /ω = γ0 ωB J

J

∑ fJ,B (ω) = Δε MCD (ω)oror

€

fJ,B (ω) =4ω γ

(ωJ2 −ω2 )2 + 4ω2γ 2

€

fJ,B (ω)




€


J

n

∑ fJ,B (ω) = Δε MCD (ω)

€

We can obtain B J (j =1,n) from a least square fit of

€

D =i=1

m

∑ VresI ,dm (ωi ) /ωi −γ0 ωiB J

J

n

∑ fJ,B (ωi ) ⎛

⎝ ⎜

⎞

⎠ ⎟

2

For m>n

MCD spectra of Porphyrins containing Mg,Ni and ZnMCD spectra of Porphyrins containing Mg,Ni and Zn

N

MN N

N

N

MN N

N

MTPP

tetraphenylporphyrin

MOEP

tetraphenylporphyrin

N

M

N N

N N

NN N

MTAP

N

M

N N

N N

NN N

tetraazaporphyrin MPcphthalocyanine

porphyrin

MP

N

M

m

β

N N

N

Example 2:Double ExcitationsExample 2:Double Excitations


Seth,M., Ziegler,T. , J. Chem. Phys., 2005,123, 144105,

Seth,M. ; Ziegler,T. J. Chem. Phys. 2006, 124, 144105

Groundstate

Excited state

TRIC state


€

where A, B C are defined by

€

where A, B C are defined by

€

Aaiσ ,bjτ =δστ δabδij

εbτ −ε jτ

nbτ − n jτ

− Kaiσ ,bjτ

€

Kaiσ , jbτ = dr dr 'ϕ aσ* (r)∫∫ ϕ iσ (r)

1

r − r'ϕ bτ (r' )ϕ jτ

* (r' )

+ dr dr 'ϕ aσ* (r)∫∫ fXC (r,r',ω)ϕ bτ (r' )ϕ jτ

* (r' )

€

Baiσ ,bjτ = Kaiσ , jbτ

€

Caiσ ,bjτ =δστ δabδij

1

nbτ − n jτ


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





Transformed Reference with an Intermediate ConfigurationKohn Sham (TRICKS) TDDFT

Solution:Solution:


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




€

VCl4




€

VCl4




€

VCl4




€

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.


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

∑

€


3

hω2B J

WJ2 − hω2

J

∑

aM. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,

A.Banerjee,J.Autschbach,T.Ziegler Int.J.Quant.Chem.2006,101,572


€

€


3

hω2B J

ωJ2 −ω2

J

∑

€


3

hω2B J

ωJ2 −ω2

J

∑€

VRe s (ω)€

VRe s (ω) =ωμ ocN

12Im[

∂

∂Bx

{α yz (ω) −α zy(ω}]ω≠ωJ

+ωμ ocN

12Im[

∂

∂By

{α zx (ω) −α xz (ω}]ω≠ωJ

+ωμ ocN

12[Im

∂

∂Bz

{α xy(ω) −α yz (ω)}]ω≠ωJ

ωμ ocN

12Imε stu

∂

∂Bu

(α st (ω)) Bu =0( )

€

VRe s (ω) =ωμ ocN

12Im[

∂

∂Bx

{α yz (ω) −α zy(ω}]ω≠ωJ

+ωμ ocN

12Im[

∂

∂By

{α zx (ω) −α xz (ω}]ω≠ωJ

+ωμ ocN

12[Im

∂

∂Bz

{α xy(ω) −α yz (ω)}]ω≠ωJ

ωμ ocN

12Imε stu

∂

∂Bu

(α st (ω)) Bu =0( )

€

∂αst (ω)

∂Bu

⎡

⎣ ⎢

⎤

⎦ ⎥ω≠ωJ

=

−Im [∂

∂Bu

Δρ (t )(ω,r r )]ω≠ωJ∫ xsd

r r

€

∂αst (ω)

∂Bu

⎡

⎣ ⎢

⎤

⎦ ⎥ω≠ωJ

=

−Im [∂

∂Bu

Δρ (t )(ω,r r )]ω≠ωJ∫ xsd

r r

€

B J

€

∂ααβ (ω)

∂Bγ

⎡

⎣ ⎢

⎤

⎦ ⎥

ω=ωJ

=

−Im [∂

∂Bλ

Δρ (β )(ω,r r )]ω=ωJ∫ xα d

r r

€

∂ααβ (ω)

∂Bγ

⎡

⎣ ⎢

⎤

⎦ ⎥

ω=ωJ

=

−Im [∂

∂Bλ

Δρ (β )(ω,r r )]ω=ωJ∫ xα d

r r

€

B J =

−i

3ε rst

∂α st (ω)

∂Br

⎛

⎝ ⎜

⎞

⎠ ⎟

r,s ,t

∑ω=ωL

M. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,


€


J

n

∑ fJ,B (ω) = Δε MCD (ω)

€

We can obtain B J (j =1,n) from a least square fit of

€

D =i=1

m

∑ VresI ,dm (ωi ) /ωi −γ0 ωiB J

J

n

∑ fJ,B (ωi ) ⎛

⎝ ⎜

⎞

⎠ ⎟

2

For m>n



€

Δε

Furan Thiophene

Selenophen Tellurophene


Documents

Tom Ziegler Department of Chemistry University of Calgary,Alberta, Canada T2N 1N4 Magnetically Perturbed Time Dependent Density Functional Theory. Applications