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

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

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.

∂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 :

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

μ iaFia(Jλ ) (εa − εi)

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

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

Inorganic SpectroscopyInorganic Spectroscopy

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

r e yi

ˆ M = ˆ m ii∑ = − (xii

∑ r e xi

r e yi

Absorbance in dipole approximation.Absorbance in dipole approximation.

€ €

2 ⎛ ⎝ ⎜ ⎞

Absorbance in dipole approximation.Absorbance in dipole approximation.

ρAαg.Jλj (ω) ≈ fJ (ω −ωJλ ) =1

ωJλ −ω

⎝ ⎜

⎠ ⎟

Electric dipole operatorFor circular polarizedLight:

ˆ M ± = ˆ m ±,ii∑

ˆ M ± = ˆ m ±,ii∑€

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

r e x − iy

r e y )

ˆ m + =1

r e x + iy

r e y )

A−,J'

(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

ρAαg.Jλj (ω) ≈ fJ (ω −ωJλ, ) =

ω Jλ, −ω

⎝ ⎜ ⎜

⎠ ⎟ ⎟

∑ (NAα − N Jλ )

Nαλ∑ Aα ˆ M −

γ Jλ2

− Aα ˆ M +γ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟ ∂

∂Bγ

fJ (ω −ωJλ )( )oB

Nαλ∑ ∂

∂Bγ

Aα ˆ M −γ Jλ

2− Aα ˆ M +

γ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

fJ (ω −ωJ )B

∂B γ

∑ (NAα − N J )

⎝ ⎜

⎠ ⎟o

α∑ Aα ˆ M −

γ Jλ2

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟ fJ (ω −ωJ )B

Nαλ∑ Aα ˆ M −

γ Jλ2

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟ ∂

∂Bγ

fJ (ω −ωJλ )( )oB

Nαλ∑ ∂

∂Bγ

Aα ˆ M −γ Jλ

2− Aα ˆ M +

γ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

fJ (ω −ωJ )B

∂B γ

∑ (NAα − N J )

⎝ ⎜

⎠ ⎟o

α∑ Aα ˆ M −

γ Jλ2

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟ fJ (ω −ωJ )B

ω= −γoA J

∂fJ (ω −ωJ )

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

ω= −γoA J

∂fJ (ω −ωJ )

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

ω= −γoA J

∂fJ (ω −ωJ )

∂ωB +γ0B J fJ (ω −ωJ )B

+γoC J fJ (ω −ωJ )B

ω= −γoA J

∂fJ (ω −ωJ )

The MCD disprsion The MCD disprsion

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

ω= −γoA J

∂fJ (ω −ωJ )

ω= −γoA J

∂fJ (ω −ωJ )

Positive A-term

Absorption band Negative A-term

Degenerate ground- or (and) excited state

ω= −γoA J

∂fJ (ω −ωJ )

ω= −γoA J

∂fJ (ω −ωJ )

Absorption band

All cases

Negative B-term

Positive B-term

Negative B-term

Positive B-term

ω= −γoA J

∂fJ (ω −ωJ )

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

ΔA-A+

ω= γoB -A J

∂fJ (ω −ωJ )B

∂ω+(B J +

kT) fJ (ω −ωJ )B

⎣ ⎢ ⎤

⎦ ⎥J∑

ω= γoB -A J

∂fJ (ω −ωJ )B

∂ω+(B J +

kT) fJ (ω −ωJ )B

⎣ ⎢ ⎤

⎦ ⎥J∑

B J =1

∑ ∂

∂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

∑ ∂

∂Bγ

Aα ˆ M −γ Jλ

2− Aα ˆ M +

γ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

Or by using the identity Or by using the identity

∑ 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

We thus haveWe thus have

3 s,t ,u

∑ εstu

∂α st (ω)

⎝ ⎜

⎠ ⎟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)

⎝ ⎜

⎠ ⎟

s,t ,u

∑ω=ωL

Where:Where:

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

0 −C

⎝ ⎜

⎠ ⎟X

⎝ ⎜

⎠ ⎟=

⎝ ⎜

⎠ ⎟X

⎝ ⎜

⎠ ⎟

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)

⎝ ⎜

⎠ ⎟

s,t ,u

∑ω=ωL

Where:Where:

ω(0) C 0

0 −C

⎝ ⎜

⎠ ⎟X (0)

⎝ ⎜

⎠ ⎟=

A(0) B(0)

B(0) A(0)

⎝ ⎜

⎠ ⎟X (0)

⎝ ⎜

⎠ ⎟

BAJ = −2i

3εstu[

∑ m s(1)u(XJ(0) −YJ

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

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

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

By differentiation ofBy differentiation of

ω(0) C 0

0 −C

⎝ ⎜

⎠ ⎟X (0)

⎝ ⎜

⎠ ⎟=

A(0) B(0)

B(0) A(0)

⎝ ⎜

⎠ ⎟X (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

⎝ ⎜

⎠ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟X I

⎝ ⎜

⎠ ⎟=

ωI(1) −I 0

⎝ ⎜

⎠ ⎟−

A(1) B(1)

B(1)* A(1)*

⎝ ⎜

⎠ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟X I

⎝ ⎜

⎠ ⎟

BAJ = −2i

3εstu[

∑ m s(1)u(XJ(0) −YJ

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

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

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

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

Introducing the unitary transformation U =1 1

⎝ ⎜

⎠ ⎟

UA(0) B(0)

B(0)* A(0)*

⎝ ⎜

⎠ ⎟−ωI

(0) −I 0

⎝ ⎜

⎠ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟U

⎝ ⎜

⎠ ⎟=U ωI

(1) −I 0

⎝ ⎜

⎠ ⎟−

A(1) B(1)

B(1)* A(1)*

⎝ ⎜

⎠ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟U

⎝ ⎜

⎠ ⎟

ω I(0)I −Ω[ ]Z I

(1) = −ωI(0)S1/2(A(1) + B(1) )S−1/2FI

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

Here:Here:

Z I(1) = ωI

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

AffordsAffords

ω I(0)I −Ω[ ]Z I

(1) = −ωI(0)S1/2(A(1) + B(1) )S−1/2FI

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

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

εq(0) −ε p

(0)q≠ p

φp(1) = Uqp

(1)φq(0) Uqp

(1) =H pq

εq(0) −ε p

(0)q≠ p

Where H(1) is the Hamiltonian describing the perurbation

Kai,bj(1) = U pa

(1)*K pi,bj(0)

∑ + U pi(1)*Kap,bj

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

∑p≠i

∑ + U pb(1)*Kap,bp

p≠ j

Kai,bj(1) = U pa

(1)*K pi,bj(0)

∑ + U pi(1)*Kap,bj

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

∑p≠i

∑ + U pb(1)*Kap,bp

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

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

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 :

(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

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

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

−ωI(0)S−1/2(A(1) − B(1) )S1/2FI

Z (1) by Sum - Over - State

Z I(1) = CJIFJ

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

FJ(0)+ ωI

(0)I −Ω[ ]( (CJIFJ(0)

∑ ) = −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)

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

ωI(0) −ωJ

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

∑ FJ(0)

CJI =−ωI

(0)(FJ(0)+S1/2(A(1) + B(1) )S−1/2FI

ωI(0) −ωJ

+FJ(0)+S−1/2(A(1) − B(1) )S1/2FI

ωI(0) −ωJ

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

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))

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 → 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.

Thiophene

Selenophen

Tellurophen

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

11A1 --> 21A1

11A1 --> 11B2

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.

6.0 6.2

-5.05 0.0 3.37

1a2 → 3b1

2b1 → 3b1

Thiophene

5.5 5.7

450 6-477

1a2 → 3b1

2b1 → 3b1

Selonophene

5.1 5.3

59.1 -3 -101

1a2 → 3b1

2b1 → 3b1

TellurophenX

11A1 --> 21A1

11A1 --> 11B2

X 1b2 11A1 --> 11B1

4.4 4.8

-5.1 -28.012.8

5.24.4 4.8

-5.1 -28.012.8

A-term of MCDA-term of MCD

∑ (NA − N Jλ )

NJλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

∂Bγ

f (ω −ωJλ )( )0B

∑ (NA − N Jλ )

∂BγJλ

∑ A ˆ M γ− Jλ

2− A ˆ M γ

+ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

⋅ f (ω −ωJλ )( )B

∑ ∂

∂Bγ

(NA − N Jλ )

⎝ ⎜

⎠ ⎟0

Jλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅ f (ω −ωJλ )( )B

∑ (NA − N Jλ )

NJλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

∂Bγ

f (ω −ωJλ )( )0B

∑ (NA − N Jλ )

∂BγJλ

∑ A ˆ M γ− Jλ

2− A ˆ M γ

+ Jλ2 ⎛

⎝ ⎜ ⎞

⎠ ⎟o

⋅ f (ω −ωJλ )( )B

∑ ∂

∂Bγ

(NA − N Jλ )

⎝ ⎜

⎠ ⎟0

Jλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅ f (ω −ωJλ )( )B

ω= −γoA J

∂f (ω −ωJλ )

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

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 )

⎡ ⎣ ⎢

⎤ ⎦ ⎥J

∑ (NA − N Jλ )

NJλ∑ A ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

∂Bγ

f (ω −ωJλ )( )0B

∑ (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

∑ (NA − N Jλ )

NA ˆ M γ

− Jλ2

− A ˆ M γ+ Jλ

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟0

⋅∂ωJλ

∂Bγ

⎝ ⎜ ⎜

⎠ ⎟ ⎟

A J = −λ

∑ (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

hω−

hω= γβB -A J

∂ρA .J (hω)

∂(hω)+ (B J +

kT)ρA .J (hω)

⎣ ⎢ ⎤

⎦ ⎥J∑

hω−

hω= γβB -A J

∂ρA .J (hω)

∂(hω)+ (B J +

kT)ρA .J (hω)

⎣ ⎢ ⎤

⎦ ⎥J∑

RCP LCP

A ˆ M Jλ =r

∑ M rS−1/ 2FLλ(0)v

A ˆ M Jλ =r

∑ M rS−1/ 2FLλ(0)v

ˆ L = ˆ l ij=1

∑ = −ir r i

∑ ×r

Other A-term implementations Other A-term implementations

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

CCSD(T)

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

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

Fe(CN)64-

Ni(CN)42- C6Cl6

C6H3Br3

D4hD4h

Exp: 0.72Calc: 0.63

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

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

∝∂ω∂B

Positive A-term

Negative A-termNegative

B-termPositive B-term

Different MCD-terms

Absorption band

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

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

1b2.u1e1.g

Orbital level diagram for ZnPOrbital level diagram for ZnP

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

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

1a 2u -> 2e g 10.30

Experimental Spectrum for ZnPExperimental Spectrum for ZnP

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

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

Φ(1b2u → 2eg)

1A1g Ground State

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

Φ(1b2u → 2eg)

1A1g Ground State

1b2.u1e1.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

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

Φ(1b2u → 2eg)

1A1g Ground State

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

22.92 −

⎡ ⎣ ⎢

⎤ ⎦ ⎥

= 2.27x10−2

D(3Eu ) = C1 2a2u y 2egy +C2 1a1u y 2egx[ ]2

22.92 +

⎡ ⎣ ⎢

⎤ ⎦ ⎥

= 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

1E u 0.05 5.49

2E u - 3.37 - 1.62

3E u - 0.57 - 0.15

2Eu+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.

Influence of ring distortion on MCD spectrum of ZnPInfluence of ring distortion on MCD spectrum of ZnP

B(nB1) = Im−2

nB1ˆ L z nB2 A1

ˆ M x nB1 nB2ˆ M y A1

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

B(nB2 ) = Im2

nB1ˆ L z nB2 A1

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

B (nB1)

B (nB2)

B (nB1) + B (nB2)

A (nEu)

C2vD4h

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

B(nB1) = Im−2

nB1ˆ L z nB2 A1

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

B(nB2 ) = Im2

nB1ˆ L z nB2 A1

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

C2vD4h

Simulated Spectrum for ZnP with B-term onlySimulated Spectrum for ZnP with B-term only

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)

Normalized Intensity

2.00 2.50 3.00 3.50

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

1a2u 1a2u 1a2u

1a1u1a1u

1b2u1b2u1b2u

1eg 1eg

1egdxz, dyz

dxz, dyz

dx2-y2

MgP NiP ZnP-7.00

-12.00

2eg 2eg 2eg

2a2u2a2u2a2u

1a2u 1a2u 1a2u

1a1u1a1u

1b2u1b2u1b2u

1eg 1eg

1egdxz, dyz

dxz, dyz

dx2-y2

MgP NiP ZnP-7.00

-12.00

2.0 2.5 3.0 3.5

(a) MgP

2.0 2.5 3.0 3.5

(a) MgP

2.0 2.5 3.0 3.5

(b) NiP

2.0 2.5 3.0 3.5

(b) NiP

2Eu3Eu

Substituted Porphyrins Substituted Porphyrins

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)

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)

A (1Eu )

B (2Eu )

B (3Eu )

A (1Eu )

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

Tetraazaporphyrins and PhthalocyaninesTetraazaporphyrins and Phthalocyanines

MTAPtetraazaporphyrin

Tetraazaporphyrins and PhthalocyaninesTetraazaporphyrins and Phthalocyanines

MTAPtetraazaporphyrin

MPcphthalocyanine

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

The C term The C term

hω−

hω= γβB -A J

∂ρA .J (hω)

∂(hω)+ (B J +

kT)ρA .J (hω)

⎣ ⎢ ⎤

⎦ ⎥J∑

hω−

hω= γβB -A J

∂ρA .J (hω)

∂(hω)+ (B J +

kT)ρA .J (hω)

⎣ ⎢ ⎤

⎦ ⎥J∑

ΔA-A+

NP+− NP+

≈EP+

− EP+

EP+− EP+

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

Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412

Limitations of Traditional TD-DFTLimitations of Traditional TD-DFT

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

Degenerate Ground StateDegenerate Ground State

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.

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

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

Application of the TRIC methodApplication of the TRIC method

TiF63−

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

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

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

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

Application: Fe(CN)63-Application: Fe(CN)63-

Electron configuration t1u6t2u

6t1u6t2g

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-

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

JA <A|LAJ|J>

<J|r|A>

<K|LAJ|J>

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

<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>

<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-.

<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

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

⎡ ⎣ ⎢

⎤ ⎦ ⎥φk (r)× exp[−iε kt] = 0

To obtain the solutionTo obtain the solution

φk' (

v r , t) = C j (t)φ j

∑ (v r )× exp[−iε jt]

Δρ ( y)(ω,r r ) = [X(ω)ai +Y (ω)ai ]φaφi

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

Vsos (ω) = −γhω2B J

WJ2 −(hω)2

∑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

V (ω) = −γhω 2B J

WJ2 − (hω)2

∑The expressionThe expression

Vdamp(ω)

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

We solve the equation

ˆ h ks (v r )+V ext (

v r , t) − i

⎡ ⎣ ⎢

⎤ ⎦ ⎥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

∑ (v r )× exp[−iε jt]exp[−λ t]

Δρ ( y)(ω,r r ) = [X(ω)ai +Y (ω)ai ]φaφi

From which we obtain density change in frequency domainFrom which we obtain density change in frequency domain

(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

∑ 2ω2(ωJ2 −ω2 )B J

(ωJ2 −ω2 )2 + 4ω2γ 2

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

∑ fJ,B (ω) = Δε MCD (ω)oror

fJ,B (ω) =4ω γ

(ωJ2 −ω2 )2 + 4ω2γ 2

fJ,B (ω)

∑ fJ,B (ω) = Δε MCD (ω)

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

D =i=1

∑ VresI ,dm (ωi ) /ωi −γ0 ωiB J

∑ fJ,B (ωi ) ⎛

⎝ ⎜

⎠ ⎟

For m>n

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

tetraphenylporphyrin

tetraazaporphyrin MPcphthalocyanine

porphyrin

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

Aaiσ ,bjτ =δστ δabδij

εbτ −ε jτ

nbτ − n jτ

− Kaiσ ,bjτ

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

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

nbτ − n jτ

Limitations of Traditional TD-DFTLimitations of Traditional TD-DFT

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

Degenerate Ground StateDegenerate Ground State

What are thefundamentalequations ?

How do we calculateexcitationenergies

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

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

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

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

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

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.

α =V (ω)Bl

Here V( ) is called the Verdet constantHere V( ) is called the Verdet constant

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

hω2B J

WJ2 − hω2

hω2B J

WJ2 − hω2

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

hω2B J

ωJ2 −ω2

hω2B J

ωJ2 −ω2

∑€

VRe s (ω)€

VRe s (ω) =ωμ ocN

{α yz (ω) −α zy(ω}]ω≠ωJ

+ωμ ocN

{α zx (ω) −α xz (ω}]ω≠ωJ

+ωμ ocN

{α xy(ω) −α yz (ω)}]ω≠ωJ

ωμ ocN

12Imε stu

(α st (ω)) Bu =0( )

VRe s (ω) =ωμ ocN

{α yz (ω) −α zy(ω}]ω≠ωJ

+ωμ ocN

{α zx (ω) −α xz (ω}]ω≠ωJ

+ωμ ocN

{α xy(ω) −α yz (ω)}]ω≠ωJ

ωμ ocN

12Imε stu

(α st (ω)) Bu =0( )

∂αst (ω)

⎣ ⎢

⎦ ⎥ω≠ωJ

−Im [∂

Δρ (t )(ω,r r )]ω≠ωJ∫ xsd

∂αst (ω)

⎣ ⎢

⎦ ⎥ω≠ωJ

−Im [∂

Δρ (t )(ω,r r )]ω≠ωJ∫ xsd

∂ααβ (ω)

∂Bγ

⎣ ⎢

⎦ ⎥

ω=ωJ

−Im [∂

∂Bλ

Δρ (β )(ω,r r )]ω=ωJ∫ xα d

∂ααβ (ω)

∂Bγ

⎣ ⎢

⎦ ⎥

ω=ωJ

−Im [∂

∂Bλ

Δρ (β )(ω,r r )]ω=ωJ∫ xα d

3ε rst

∂α st (ω)

⎝ ⎜

⎠ ⎟

r,s ,t

∑ω=ωL

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

∑ fJ,B (ω) = Δε MCD (ω)

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

D =i=1

∑ VresI ,dm (ωi ) /ωi −γ0 ωiB J

∑ fJ,B (ωi ) ⎛

⎝ ⎜

⎠ ⎟

For m>n

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