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
62