Mark E. CasidaInstitut de Chimie Moléculaire de Grenoble (ICMG)Laboratoire d'Études Dynamiques et Structurales de la Sélectivité (LÉDSS)Équipe de Chimie Théorique (LÉDSS­ÉCT) Université Joseph Fourier (Grenoble I)F­38041 GrenobleFrancee­mail: Mark.Casida@UJF­Grenoble.FR

PROPAGATOR CORRECTIONS TO LINEAR RESPONSETIME­DEPENDENT

DENSITY­FUNCTIONAL THEORY (TDDFT)

1

11th Nanoquanta Workshop Houffelize, Belgium 20 September 2006 30 min

What chemists should not do ...

With permission from Sidney Harriswww.sciencecartoonsplus.com

2

Difficultés de Communication (Language Problems)

Chemist = chemical physicistPhysicist = solid state physicist

DFT≠abinitio

For the purposes of this talk :

3

I. Ionization Spectra II. TDDFTIII. Green Functions from a Chemist's ViewpointIV. Polarization Propagator CorrectionsV. Conclusion

4

Electron Momentum Spectroscopy (EMS)

e­ e­

e­Energy of the ionized electron

ℏ=ℏ

2m k1

2ℏ

2m k2

2−ℏ

2m k0

2

k=k1k2−k0

Momentum of theionized electron

k1k2

k0

HCCH

5

ℏ=ℏ

2m k1

2ℏ

2m k2

2−ℏ

2m k0

2

k=k1k2−k0

P. Duffy, S.A.C. Clark, C.E. Brion, et al. Chem. Phys. 165, 183 (1992)

Experimental Data

6

Ionization spectra are traditionally modeled using Green functionmethods. We would like to be able to be able to obtain the samething from DFT.

Patrick Duffy, Delano Chong, Mark E. Casida, and Dennis R. Salahub, Phys. Rev. A 50, 4707 (1994). "Assessment of Kohn­­Sham Density­Functional Orbitals as Approximate Dyson Orbitals for the Calculation of Electron­Momentum­Spectroscopy Scattering Cross Sections''

S. Hamel, P. Duffy, M.E. Casida, and D.R. Salahub, J. Electr. Spectr. and Related Phenomena 123, 345 (2002). "Kohn­Sham Orbitals and Orbital Energies: Fictitious Constructs but Good Approximations All the Same"

7

Ionization Spectra as Excitation Spectra

HCCH

HCCH+

⋯2g22u

23g21u

4

1u−1

2u−1

3g−1

2g−1 ?

2g−1 ?

∆SCF works forprincipal ionizations

Shake­ups mightbe best obtainedas excitations

8

What is a shake­up ionization?

1u

1g

3g

2u

2g

1u

1g

3g

2u

2g

Mixing of 1h and 2h1p states

3u 3u

9

I. Ionization Spectra as Excitation SpectraII. TDDFTIII. Green Functions from a Chemist's ViewpointIV. Polarization Propagator CorrectionsV. Conclusion

10

Pour

For a system, intially in its ground state, exposed to time­dependentperturbation :

1st Theorem: vext

(rt) is determined by ρ(rt) up to an additive function oftime

Corollary: rt N , vext r t C t H t C t t e−i∫t0

tC t ' dt '

2nd Theorem: The time­dependent density is a stationary point of theaction

A[]=∫t 0

tt ' ∣i ∂

∂ t '− H t ' ∣t ' dt '

[Actually, the time­dependent density should rather be generated by the Keldysh action. See R. van Leeuwen, Int. J. Mod. Phys. B 15, 1969 (2001).]

TIME­DEPENDENT DENSITY­FUNCTIONAL THEORY (TDDFT)[according to E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984)]

11

[−12∇2vext r t ∫ r ' t

∣r−r '∣d r 'vxc r t ]i r t =i ∂

∂ ti r t

r t =∑if i∣i r t ∣2

vxc r t = Axc []

r t

THE TIME­DEPENDENT KOHN­SHAM EQUATION

where

and

12

Classical model of a photon

Induced dipole moment

t =−e0∣r∣0t 0t ∣r∣0

t = cos0 t

v r t =et ⋅r

O

H H

O

H H

O

H H

O

H Hℏ0

photont

EXCITED­STATES ARE OBTAINED FROM LINEAR RESPONSE THEORY

13

i t =i∑ ji , j j cos t⋯

r i , r j=∑I≠0

2I0∣r i∣II∣r j∣0

I2−2

=∑I≠0

f I

I2−2

f I=23I ∣0∣x∣I∣

2∣0∣y∣I∣2∣0∣z∣I∣

2

Sum­over­states theorem (SOS)f

I

ωI

DYNAMIC POLARIZABILITY

14

xz =−2xA−B1/2 [2 1−A−B1/2ABA−B1/2]A−B1/2z

where

Aij kl= ,i , k j , l j− jK ij , kl

Bij , kl=K ij , lk

MATRIX EQUATIONS[Mark E. Casida in Recent Advances in Density Functional Methods, Part I, edited

by D.P. Chong (Singapore, World Scientific, 1995), p. 155]

15

The poles of the dynamic polarizability are the solutions of the pseudo­eigenvalue problem,

I F I=I2 F I

=A−B1/2ABA−B1/2

PSEUDO­EIGENVALUE PROBLEM

where

In principle these equations take all spectroscopically allowed transitionsinto account when the functional is exact. This is not a problem becauseit is well­established that the number of solutions of such an equationcan exceed the dimension of the matrix Ω.

These equations are variously known as the RPA equations (which is really confusing) the "Casida equations"

16

THE TDDFT ADIABATIC APPROXIMATION

vxc r t = Axc []

r t

vxc r t =E xc [t ]

t

where t r =r t

Supposing that the reaction of the self­consistant field to variationsin the charge density is instantaneous and without memory givesthe adiabatic approximation,

In general

17

PROBLEM

Theorem: The adiabatic approximation limits TDDFRT to 1e excitations.

Reasoning: Counting argument.

f xc = f xc

independent of frequency means that the eigenvalue problem hasexactly the same number of solutions as the number of single excitations.

Alternatively: The "Casida equation" includes TDHF. Make the Tamm­Dancoff approximation, B=0. The "Casida equation" thenreduces to CIS.

Note however that adiabatic TDDFT 1e excitations include somecorrelation effects (they are "dressed"). 18

PROBLEM2

Theorem: The adiabatic approximation limits the poles of the nonlinearresponse to 1e excitations.

Proof: Complicated, but basically it is related to the idempotency of theKS density matrix. Tretiak et Chernyak have shown that singularities of the adiabatic TDDFT 2nd hyperpolarizability occur only at double excitations which are 1e excitations*.

* S. Tretiak and V. Chernyak, J. Chem. Phys. 119, 8809 (2003).

=13 !

[ I II ⋯VIII ]

For example,

I =∑ , ,

−

−

−SSS

[−3 ] [−2 ] [− ]

19

EXCITATIONS IN RADICALS

i

v

a

i

v

a

i

v

a

i

v

a

i

v

a

i

v

a

∣ii v ⟩

∣ai v ⟩

a i

−a v

∣ii a ⟩ a i

∣i a v ⟩

−v i

∣iv v ⟩a v v i

∣i av ⟩

v i a

v

20

SPIN OPERATORS

S2=∑ P nS z S z−1

S z=12

n−n

wheren=∑ r

rP =∑ r

s s r

Single determinants are eigenfunctions of Sz but not necessarily of S2

Eigenfunctions of S2 are linear combinations of determinants with different distributions of the same number of up and down spins.

21

RADICAL EXCITED STATES |S,MS)

Doublets

Quadruplet

∣D2 ⟩= 1

6∣i v a ⟩∣i va ⟩−2∣iv a ⟩

∣Q ⟩= 1

3∣i v a ⟩∣i va ⟩∣iv a ⟩

"Extended Singles"(a type of doubles)

∣D1 ⟩= 1

2∣i v a ⟩−∣i va ⟩

∣ii a ⟩ ∣iv v ⟩

22

TDDFT, TDHF, AND CIS GIVE

Singlet Coupling

∣TC ⟩= 1

2∣i v a ⟩∣i va ⟩

∣D1 ⟩= 1

2∣i v a ⟩−∣i va ⟩

Triplet Coupling

Doublets

Neither a doublet nor a quadruplet!

MISSING: The quadruplet and one of the doublets!

∣ii a ⟩∣iv v ⟩

23

CONSEQUENCES FOR OPEN­SHELL MOLECULES

In the adiabatic approximation, Only transitions which conserve S2 have correct symmetry There are too few transitions conserving S2

intensity=1

ω(S)

ω(Ψ) ω(Ψ')

intensity =sin2 θ

intensity =cos2 θ

24

ALSO A PROBLEM FOR THE SPECTRUM OF CLOSED­SHELL MOLECULES

,ia

=S sinD cos '=S cos−Dsin

S

ia D

An example is found in the spectrum of butadiene, the smallest polyacetylene oligamer.

intensity=1

ω(S)

ω(Ψ) ω(Ψ')

intensity =sin2 θ

intensity =cos2 θ

25

I. Ionization Spectra as Excitation SpectraII. TDDFTIII. Green Functions from a Chemist's ViewpointIV. Polarization Propagator CorrectionsV. Conclusion

26

PROPAGATOR CORRECTIONS TO TDDFT

Carry out many­body theory with the KS system as reference system. Some people seek to calculate the exact v

xc and f

xc this way

[optimized effective potential (OEP) approach]. We seek many­body corrections to adiabatic TDDFT. There is

necessarily an interface problem between TDDFT and propagator (Green's function) theory.

Basic Idea :

Choices of Propagator Formalism :

Functional derivative (Schwinger­type) Diagramatic (Feynman­Dyson) Equations­of­Motion (EOM) Superoperator

27

ALWAYS INCLUDE EXCHANGE, ALWAYS WORK WITH 4­POINT QUANTITIES

Hugenholtz Brandow Feynman

+

28

SCHWINGER'S FUNCTIONAL DERIVATIVE FORMALISM

See e.g. G.Y. Csanak, H.S. Taylor, and R. Yaris, Advances in Atomic and Molecular Physics, vol. 17, (Academic Press: New York, 1971), pp. 287­361.“Green's function technique in atomic and molecular physics”

G 11 ' =G0 11 ' ∫G01222 ' G 2 ' 1 ' d2 d2 '

12=−i∫G−132G214 ;34v 41d4

R 121 ' 2 ' =G 12 ' G 21 ' ∫G 13G 3 ' 1 33 ' G 4 ' 4

R 4 ' 242 ' d3 d3 ' d4 d4 '

R 121 ' 2 ' =[ G 11 ' [U ]

U 2 ' 2 ]U=0

R 121 ' 2=i 11 ' U 2

generalized response function

Bethe­Salpeter equation

Dyson equation

self­energy

2­electron Green functionG212 ;1 ' 2 ' =G 11 ' ' G 22 ' −G 12 ' G 21 ' ∫G 13G 41 ' 34

G 34 ' R 3 ' 24 ' 2 ' d3 d3' d4 d4 '

29

W. von Niessen, J. Schirmer, and L.S. Cederbaum, Computer Physics Reports1, 57 (1984). “Computational methods for the one­particle Green's function”

30

ALGEBRAIC DIAGRAMMATIC CONSTRUCTION (ADC)

pq =U p 1−K−C −1U q

U p=U p1U p

2⋯

C=C1C2⋯K is a diagonal matrix of orbital energy differences

Expand in electron interaction

Obtain Up and C by analyzing correspondence with finite order expressions

pq ;3=U p1 1−K −1U q

1U p2 1−K −1U q

1

U p1 1−K −1 Uq

2U p1 1−K −1 C11−K −1 U q

1O 4

This means that the self­energy can go beyond the quasi­particle regimeto describe shake­up ionizations.

(See von Niessen et al.)

31

HEDIN'S EQUATIONS

L. Hedin, Phys. Rev. 139, A796 (1965). “New method for calculating the one­particle Green's function with application to the electron gas problem”

W 12=v 12∫ v 13P 34W 42d3 d4

G 12=G012∫G01334G 42d3 d4

12=i∫G 13 324W 41d3 d4

P 12=−i∫G 13G 41 342d3 d4

123=1213∫ 12G 45

G 46G 75 673d4 d5 d6 d7vertex function

self­energy

polarization

screened potential

Dyson equation

32

G0W0

G 12=G012∫G01334G 42d3 d4

12=i G012W 021

P012=−i G012G021

0123=1213vertex function

self­energy

polarization

screened potential

Dyson equation

33

W 012=v 12−i∫ v 13G034G043W 042d3 d4

Could it possibly have any value for molecules?

M.E. Casida and D.P. Chong, Phys. Rev. A 44, 4837 (1989); Erratum, ibid, 44, 6151 (1991) "Physical interpretation and assessment of of the Coulomb­hole and screened­exchange approximation for molecules"

M.E. Casida and D.P. Chong, Phys. Rev. A 44, 5773 (1991). "Simplified Green­function approximations: Further assessment of a polarization model for second­order calculation of outer­valence ionization potentials in molecules" GW2 = GF2 + self­polarization correction

G. Onida, L. Reining, R.W. Godby, R. Del Sole, and W. Andreoni, Phys. Rev. Lett. 75, 818 (1995). “Ab initio calculations of the quasiparticle and absorption spectra of clusters: the sodium tetramer”

C.­H. Hu, D.P. Chong, and M.E. Casida, J. Electron Spectr. 85, 39 (1997). “The parameterized second­order Green function times screened interaction (pGW2) approximation for outer valence ionization potentials”

Y. Shigeta, A.M. Ferreira, V.G. Zakrzewski, and J.V. Ortiz, Int. J. Quant. Chem. 85, 411 (2001). “Electron propagator calculations with Kohn­Sham reference states” GW2 works surprisingly well but need to use quasi energies in denominator

P.H. Hahn, W.G. Schmidt, and F. Bechstedt, Phys. Rev. B 72, 245425 (2005). “Molecular electronic excitations from a solid­state approach” 34

GW2

212=i G012W 0221

self­energy

screened potential

35

W 0212=v 12−i∫ v 13G034G043v 42d3 d4

+Σ =

10 11 12 13 14 15 16 17 18 19 20 21 22

10

11

12

13

14

15

16

17

18

19

20

21

22

Expt

GW

2IP in eV from Shigeta et al.

KS orbitals Pseudo HF orbital energies calculated from KS orbitals

36

I. Ionization Spectra as Excitation SpectraII. TDDFTIII. Green Functions from a Chemist's ViewpointIV. Polarization Propagator CorrectionsV. Conclusion

37

CHOICES OF PROPAGATOR FORMALISM

Functional derivative (Schwinger­type) Diagramatic (Feynman­Dyson) Equations­of­Motion (EOM) Superoperator

Choose this because it is contains but is more versatile than the diagrammatic formalism (question of taste).

38

PROPAGATORS

⟨ ⟨ At ; B ⟩ ⟩=±it ⟨0∣AH t B∣0 ⟩i∓t ⟨0∣B AH t ∣0 ⟩

The upper sign is the usual (i.e. causal) propagator.The lower sign is the response function (retarded propagator).

AH t =eiHt Ae−iHt

⟨ ⟨ A; B ⟩ ⟩=∫−∞

∞ei t ⟨ ⟨ At ; B ⟩ ⟩

=∓∑I

⟨0∣A∣I ⟩ ⟨I∣B∣0 ⟩−Ii

±∑I

⟨0∣B∣I ⟩ ⟨I∣A∣0 ⟩I∓i

39

SUPEROPERATOR FORMALISM : LIOUVILLIAN

H A=[H , A] 1 A=ALiouvillian identity

AH t =eit H A

Specializing to the causal propagator allows us to write this as a commutator(in the limit ),

⟨ ⟨ A; B ⟩ ⟩= A∣ 1− H −1∣B

0

Note the introduction of the superoperator metric,

A∣B = A∣B 0= ⟨0∣[ A , B ]∣0 ⟩ (Number conserving

operators only)40

SUPEROPERATOR ALGEBRA : BASIS SET

T =T 1 , T 2

, T 3 ,

1e excitation and de­excitation operators,T 1=E1,

E1

2e excitation and de­excitation operators,

T 2=E2,

E2

E1=a i , b j , excitation

E1=i a , j b , de­excitation

E2=a b ji , c d lk ,

E2=i j ba , k l dc ,excitation

de­excitation41

OPEN­SHELL PROBLEMS

MCSCF calculation gives Ψ0 , Ψ

1 , Ψ

2 , etc.

Accuracy considerations may require inclusion of state transfer operators,

∣I ⟩ ⟨0∣ ∣0 ⟩ ⟨I∣

D.L. Yeager and P. Jorgensen, Chem. Phys. Lett. 65, 77 (1979).“A multiconfigurational time­dependent Hartree­Fock approach”

42

in the superoperator basis sets.

SUPEROPERATOR ALGEBRA : EXCITATION ENERGIES

r i , r j=r i∣ 1− H −1∣r j

=r i∣T T ∣ 1− H∣T −1 T ∣r j =∑I r i∣T U I −I

−1 U I S1/2 S−1/2 T ∣r j

Sum­over­states theorem :=∑I

f I

I2−2

Tells us that the excitation energies are at the poles of the dynamic polarizability

T ∣ H∣T U I=I T ∣T U I

H U I=I S U I

So it suffices to solve

With the normalisationU I S U J=I , J 43

SUPEROPERATOR ALGEBRA : LÖWDIN (FESHBACH) PARTITIONNING

[ T 1∣ H∣T 1

T 1∣ H∣T 2

T 2

∣ H∣T 1 T 2

∣ H∣T 2 ] U I

1

U I2 =I [ T 1

∣T 1 T 1

∣T 2

T 2 ∣T 1

T 2 ∣T 2

] U I1

U I2

Can be rewritten as

T 1∣ H∣T 1

K I U I1=I T 1

∣T 1 U I

1

K I =[ T 1∣ H∣T 2

−I T 1∣T 2

]×[I T 2

∣T 2 −T 2

∣ H∣T 2 ]

−1

×[ T 2 ∣ H∣T 1

−I T 2 ∣T 1

]

where

Bethe­Salpeterequation

44

PROPAGATOR CORRECTIONS TO TDDFT : BASIC IDEA

Kohn­Sham reference state 0=s⋯

A∣B = A∣B s[ A∣B − A∣B s ]= A∣B ss A∣B

T 1∣hs∣T 1

sK Hxc I U I1=I T 1

∣T 1 s U I

1

Then the exact “Casida equation” is

The TDDFT adiabatic approximation is

K Hxc I =T 1∣ H−hs∣T 1

ss T 1∣ H∣T 1

−Is T 1∣T 1

K I

K Hxc 0=T 1∣ H−hs∣T 1

ss T 1∣ H∣T 1

K 0

45

WHY “CASIDA'S EQUATION” IS NOT CASIDA'S EQUATION

same excitation energies same transition densities different transition density matrices

So what is wrong?

46

4­POINT VERSUS 2­POINT “BETHE­SALPETER” EQUATIONS

R 121 ' 2 ' =G 12 ' G 21 ' ∫G 13G 3 ' 1 33 ' G 4 ' 4

R 4 ' 242 ' d3 d3 ' d4 d4 '

or411 ' ;22 ' =s

411 ' ;22 ' ∫s411 ' ;33 ' K 33 ' ; 44 ' 444 ' ;22 ' d3 d3 ' d4 d4 '

Also21 ' ;2=s

21 ;2∫s41 ;3 f xc 3424 ;2d3 d4

21 ;2=411 ;22

So

f xc 12=∫s2−11 ;1 ' s

41 ' 1 ' ;33 ' K 33 ' ; 44 ' 444 ' ; 2 ' 2 ' 2−12 ' ; 2d1' d2 ' d3 d3 ' d4 d4 '

However Casida's equation is a 4­point equation for the noninteractingsystem rather than a 2­point equation, so we will continue with the provisionalidentification of “Casida's equation” and Casida's equation.

47

PROPAGATOR CORRECTIONS TO TDDFT : ALMOST EXACT EXCHANGE

T 1∣hs∣T 1

sK Hx U I1=I T 1

∣T 1 s U I

1

In this approximation,

K Hx=T 1∣ H−hs∣T 1

s

When correlation effects are neglected, the principal difference between theKS and HF orbitals is a unitary transformation among occupied orbitals.

48

PROPAGATOR CORRECTIONS TO TDDFT : ALMOST EXACT EXCHANGE

[i , ja , b a−i K ai , bjHx K ai , jb

Hx

K ia , bjHx i , ja , b a−i K ia , jb

Hx ] X I

Y I=I [i , ja , b 0

0 −i , ja , b] X I

Y I

K ai , bjHx =K ia , jb

Hx =i , j ⟨a∣hHF−hKS∣b ⟩−a , b ⟨ j∣hHF−hKS∣i ⟩ai∣∣ jb −ab∣∣ ji K ia , bj

Hx =K ai , jbHx =ib∣∣ ja − jb∣∣ia

Which is a generalization of the result,

K ai , aiHx = ⟨a∣hHF−hKS∣a ⟩− ⟨ i∣hHF−hKS∣i ⟩−aa∣∣ii

Given in X. Gonze and M. Scheffler, Phys. Rev. Lett. 82, 4416 (1999).

T 1∣hs∣T 1

sK Hx U I1=I T 1

∣T 1 s U I

1

49

PROPAGATOR CORRECTIONS TO TDDFT : APPROXIMATIONS

K Hxc =T 1∣ H−hs∣T 1

ss T 1∣ H∣T 1

−s T 1∣T 1

K

Exact formulae

Simplifying approximations

0HF1)

2) HF=KS

T 1∣hs∣T 1

sK Hxc I U I1=I T 1

∣T 1 s U I

1

(In terms of perturbation theory, we really should be more careful to keep track of and be consistant in the order of perturbation.)

50

PROPAGATOR CORRECTIONS TO TDDFT : CONSEQUENCES

T 1∣hs∣T 1

sK Hxc I U I1=I T 1

∣T 1 s U I

1

K Hxc =T 1∣ H−hs∣T 1

sadiabatic TDDFTx

K propagator correction

K =T 1∣ H∣T 2

s [ T 2 ∣T 2

s−T 2 ∣ H∣T 2

s ]−1

T 2 ∣ H∣T 1

s

Note that the propagator correction contains the correlation part of the adiabatic coupling matrix, K(0).

51

PROPAGATOR CORRECTIONS TO TDDFT : PHILOSOPHY

The propagator correction should only include thoseexcitations missing in adiabatic TDDFT and trulynecessary for a correct physical description.

PROPAGATOR CORRECTIONS TO TDDFT : DRESSED TDDFT

[MZCB04] N.T. Maitra, F. Zhang, F.J. Cave, and K. Burke, J. Chem. Phys. 120, 5932 (2004). “Double excitations within time­dependent density functional theory linear response” [CZMB04] R.J. Cave, F. Zhang, N.T. Maitra, and K. Burke, Chem. Phys. Lett. 389, 39 (2004). “A dressed TDDFT treatment of the 21Ag states of butadiene and hexatriene”

The present treatment is essentially equivalent to “dressed TDDFT” for the closed­shell case :

2 [q∣ f xc ∣q ]=2 [q∣ f xcA q∣q ] ∣H qD∣

2

−H DD−H 00

Modifications are needed for the open­shell case.

!

53

PROPAGATOR CORRECTIONS TO TDDFT : TDA

Tamm­Dancoff approximation (TDA)

T E=E1 , E2

, E3 ,

E1=a i , b j ,

Excitation operators only!

E2=a b ji , c d lk ,

The resultant formulae are closely related to configuration interaction (CI).

54

EXTENDED SINGLES TDDFT TDA

∣ii a ⟩∣iv v ⟩∣ai v ⟩∣i a v ⟩∣iv a ⟩ [a

−vva∣va Hxc

, av∣iv Hxc

, va∣ia Hxc

, va∣ia Hxc

,I

iv∣va Hxc

,v−i

iv∣iv Hxc

, iv∣ia Hxc

, iv∣ia Hxc

,J

ia∣va Hxc

, ia∣iv Hxc

,a−i

ia∣ia Hxc

, ia∣ia Hxc

,K

ia∣va Hxc

, ia∣iv Hxc

, ia∣ia Hxc

,a−i

ia∣ia Hxc

,L

I J K L H]

va i v ia i a i vv a

K pq , sr Hxc = pq∣rs Hxc

,= pq∣rs −

, ps∣rq

Mullikan ("charge cloud") notation ) :

Reduces to CIS when :

K pq , sr Hxc = pq∣rs Hxc

,= pq∣rs pq∣ f xc

,∣rs

55

U=[1 0 0 0 00 1 0 0 0

0 0 1

21

61

3

0 0 1

2−

1

6−

1

3

0 0 0 −2

61

3

]=

∣ii a ⟩∣iv v ⟩∣ai v ⟩∣i a v ⟩∣iv a ⟩

[ E G C C IG F D D JC D A B KC D B A LI J K L H

]EXTENDED SINGLES (XS)

UU will be blocked

56

ENFORCING THE BLOCK FORM

I= C−CJ= D−D

KL= A−AB−B

H=A A

2−

BB

2−

K−L2

K−L=?

Symmetry requires

UU=

∣ii a ⟩∣ai v ⟩∣D1⟩∣D2 ⟩∣Q ⟩ [

E G C C2

C− C−2 I6

C− CI3

G F D D2

D− D−2 J6

D− DJ3

C C

2D D

2A A

2

BB

2A− AB−B−2KL

23A− AB−BKL

6C C−2 I

6D D−2 J

6A− AB−B−2KL

23A A−B−B−2K−L −2K−L4 H

6A A−B−BK−L−2K−L−2 H

32C− CI

3D− DJ

3A− AB−BKL

6A A−B−BK−L−2 K−L−2 H

32A A−B−BK−L K−LH

3

]57

CLOSED­SHELL DRESSED TDDFT WOULD GIVE

∣ii a ⟩∣iv v ⟩∣ai v ⟩∣i a v ⟩∣iv a ⟩ [a

−vva∣va Hxc

, av∣iv Hxc

, va∣ia Hxc

, va∣iv Hxc

, iv∣aa iv∣va Hxc

,v−i

iv∣iv Hxc

, iv∣ia Hxc

, iv∣ia Hxc

,−ii∣va

ia∣va Hxc

, ia∣iv Hxc

,a−i

ia∣ia Hxc

, ia∣ia Hxc

, av∣av ia∣va Hxc

, ia∣iv Hxc

, ia∣ia Hxc

,a−i

ia∣ia Hxc

, av∣av iv∣aa −ii∣va −iv∣iv av∣av HFa

−HFi−ii∣aa

]va i v ia i a i vv a

I= C−C=va∣iv Hxc

,−va∣ia Hxc

,≠iv∣aa

J= D−D=iv∣ia Hxc

,−ia∣ia Hxc

,≠−va∣ii

This violates spin­symmetry!

We need symmetry to get symmetry­pure excitations.

58

TDDFT TDA WITH XS EXCITATIONS

∣ii a ⟩∣iv v ⟩∣ai v ⟩∣i a v ⟩∣iv a ⟩ [ a

−vva∣va Hxc

, av∣iv Hxc

, va∣ia Hxc

, va∣iv Hxc

, va∣iv Hxc

,−va∣ia Hxc

,

iv∣va Hxc

,v−i

iv∣iv Hxc

, iv∣ia Hxc

, iv∣ia Hxc

, iv∣ia Hxc

,−ia∣ia Hxc

,

ia∣va Hxc

, ia∣iv Hxc

,a−i

ia∣ia Hxc

, ia∣ia Hxc

,K

ia∣va Hxc

, ia∣iv Hxc

, ia∣ia Hxc

,a−i

ia∣ia Hxc

,L

va∣iv Hxc

,−va∣ia Hxc

, iv∣ia Hxc

,−ia∣ia Hxc

,K L H

]va i v ia i a i vv a

The I and J blocks are completely determined by symmetry.A knowledge of H determines K and L.

K= A−B−H=a−i

ia∣ia Hxc

,−ia∣ia Hxc

,−H

L=B−AH=i−a

ia∣ia Hxc

,−ia∣ia Hxc

,H

59

CHOICE OF H

* Y. Shao, M. Head­Gordon, and A.I. Krylov, J. Chem. Phys. 118, 4807 (2003). “The spin­flip approach within time­dependent density functional theory: Theory and applications to diradicals”

H=HFa−HFi

−ii∣aa 1) “dressed TDDFT” (recommended first­principles propagator correction)

2) translation rule (spin­flip approach*)

H=a−i

i a∣ f Hxc∣ia ,

3) “physical intuition”

H=vai via∣ia Hxc

,−ia∣ia Hxc

,

Closed­shell : N.T. Maitra, F. Zhang, R.J. Cave, and K. Burke, J. Chem. Phys. 120, 5932 (2004). “Double excitations within time­dependent density functional linear response”Open­shell : present work

60

CONCLUSION

Adiabatic TDDFT is limited to 1e excitations. Propagator corrections allow explicit incorporation of 2e excitations. The Gonze­Scheffler result has been generalized. “Closed­shell dressed TDDFT” has been generalized. The basic ideas needed to extend “dressed TDDFT” to the open­shell

case have been presented.

AP A P=P AATDDFT PP APC P

[ A BB A ] XY = [1 0

0 −1 ] XY A X= X

P = symmetry projector

TDA

ATDDFT

61

Jingang GuanSerguei TretiakXavier Gonze

Careful reading and comments :Kieron BurkeNeepa MaitraAnna Krylov

HELPFUL DISCUSSIONS

MANUSCRIPTM.E. Casida, J. Chem. Phys. 122, 044110 (2005). “Propagator Corrections to Adiabatic Time­Dependent Density­Functional Theory Linear Response Theory”

FINANCIAL SUPPORT

COST Chemistry Working Group Action D6ECOS­Nord project M02P03GdR­COMESGdR­DFT

Lucia ReiningAngelo RubioAndrei Ipatov

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