the multifaceted problem of double and higherexcitations...

Mark E. CasidaLaboratoire de Chimie Théorique (LCT)Département de Chimie Moléculaire (DCM, UMR 5250)Institut de Chimie Moléculaire de Grenoble (ICMG, FR 2607)Université Joseph Fourier (Grenoble I)F38041 GrenobleFranceemail: Mark.Casida@UJFGrenoble.FRhttp://dcm.ujf-grenoble.fr/PERSONNEL/CT/casida/

The Multifaceted Problem of Double and HigherExcitations in TDDFT

TDDFT workshopBenasque, Spain

50 min.Weds. 13 January 2010

mailto:[email protected]

TDDFT Benasque 13 January 2010 2

TDDFT is the Main SingleDeterminantal Theory for Excited States

Web of Science 9 October 2009 key words:

TDDFT or timedependent densityfunctional theory


RECENT WORKS ON TDDFT

Book: TimeDependent Density Functional Theory, Edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross, Lecture Notes in Physics Vol. 706 (Springer: Berlin, 2006).

Special Issue:“TimeDependent DensityFunctionalTheory for Molecules and Solids”,Guest edited by M.E. Casida, H. Chermette, and D. Jacquemin, J. Mol. Struct. (Theochem), vol. 914 (2009).

Special Issue:“TimeDependent DensityFunctionalTheory”, Guest edited by M.A.L. Marques and A. Rubio, Phys. Chem. Chem. Phys. 11, issue 22, pp. 44214688 (2009).


John Pople's Diagram

size of system

soph

istic

atio

n o

f m

ode

l

No Computer Land

Moving boundary :computer advances (50%)theory advances (50%)

Our emphasis :advancing theory for● Spectroscopy● Photochemistry● Conductivity non excluded

We keep running into the need for an explicit description of double and higherelectron excitations!


http://dcm.ujfgrenoble.fr/PERSONNEL/CT/casida/


I. The Origin of the ProblemI. The Origin of the ProblemII. SpinContamination: The Case of [Fe(HII. SpinContamination: The Case of [Fe(H

22O)O)

66]]2+2+

III. Photochemical Funnels: SpinFlip and OxiraneIII. Photochemical Funnels: SpinFlip and OxiraneIV. Doing It Right: Polarization Propagator CorrectionsIV. Doing It Right: Polarization Propagator CorrectionsV. SummaryV. Summary


TIMEDEPENDENT KOHNSHAM EQUATION

[−1

2∇

2v

extr t ∫

r ' t

∣r−r '∣d r 'v

xcr t ]i

r t =i ∂∂ t

ir t

r t =∑in

i∣

ir t ∣2where

and vxcr t =

Axc[]

r t

(1)

(2)

(3)

[E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)]


ElectricField Induced Electronic Polarization

●Classical model of a photon

● Induced dipole moment

t =−e 0∣r∣

0t

0t ∣r∣

0

t = cos0t

v r t =et ⋅r

H HH H

O

H HH Hℏ

0

photon

t


THE DYNAMIC POLARIZABILITY

it =

i∑ j

i , j

jcos t⋯

r

i, r

j

=∑I≠0

2I

0∣r

i∣

I

I∣r

j∣

0

I

2−

2

=∑I≠0

fI

I

2−

2

fI=

2

3

I∣

0∣x∣

I∣

2∣

0∣y∣

I∣

2∣

0∣z∣

I∣

2

Sumoverstates (SOS) theorem

fI

ωI

How to make computationally convenient?


COMPUTATIONALLY CONVENIENT FORMULATION

[ AI BI

BI AI ] X I

Y I=I [1 0

0 −1 ] X I

Y I

Mark E. Casida in Recent Advances in Density Functional Methods, Part I, edited by D.P. Chong (Singapore, World Scientific, 1995), p. 155."Timedependent densityfunctional response theory for molecules''

Aij ,kl = ,i ,k j , l j− j K ij , kl

K ij , kl=∫∗i r j r f Hxc ,r ,r ' ;k r ' ∗l r ' d r d r '

Bij , kl =K ij , lk

“RPA” equation

(1)

where (2)

(3)

Coupling matrix

(4)


All is exact up to this point!

(Even fractional occupation numbers have been includedin my original formulation of LRTDDFT.)


TDDFT ADIABATIC APPROXIMATION (AA)

tr =r t

This defines “conventional TDDFT.”

vxcr t =

Axc[]

r t v

xcr t =

Exc[]

tr

Assume xcpotential responds instantaneously and without memory toany temporal change of the charge density.


CONSEQUENCE OF ADIABATIC APPROXIMATION

[ A BB A ] X I

Y I=I [1 0

0 −1 ] X I

Y I

LRTDDFT matrix is now independent of frequency.

Nocc

Nvirt

excitation solutions and Nocc

Nvirt

deexcitation solutions.

Conclude: AATDDFT only gives singleelectron excitations (albeit dressed to include some correlation)

Need frequency dependence to include double excitations!


IS THAT REALLY SO?

YesExplicit poles of (

s;

1,

2,

3) are still at oneelectron excitations:

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

NoRealtime sequential absorption of two photons can lead to twoelectron excitations. C.M. Isborn and X. Li, J. Chem. Phys. 129, 204107 (2007).


TAMMDANCOFF APPROXIMATION (TDA)

B=0 defines the TammDancoff approximation

A X I=IX I

[TDDFT reference : S. Hirato and M. HeadGordon, Chem. Phys. Lett. 314, 291 (1999)]

Loose ThomasReicheKuhn (TRK) sum rule,

∑If I=N ,

but gain elsewhere.

(1)

(2)



22O)O)

66]]2+2+



MANY INTERESTING COMPOUNDS HAVE CLOSEDSHELL GROUND STATES

Expt Coord Chem Rev. 159, 325 (1997)Extended Hückel J. Phys. Chem. 99,12141 (1995).TDB3LYP/LANL2DZ in CH3CN (100 states) J. Phys. Chem. A 111, 1487 (2007)TDB3LYP/LAN2DZ (60 states) Present work

C. Muhavini WAWIRE

t2g

eg


For certain choices of ligand, ∆EHL

is

small enough to allow thermal excitation.

Typical complex

h

[Fe(ptz)6](BF

4)

2 @ 10 K (left) & 296 K (right)

MANY INTERESTING STABLE COMPOUNDS HAVE OPENSHELL GROUND STATES


MODEL (“TEXT BOOK”) COMPOUND(OF BIOLOGICAL IMPORTANCE)

[Fe(H2O)

6]2+

A. Fouqueau, S. Mer, MEC,

L.M. Lawson Daku, A. Hauser, T. Mineva,

and F. Neese, J. Chem. Phys. 120, 9473 (2004).

t2g

eg single color,

excited state

t2g

eg multicolor,

ground state


EXPERIMENTAL SPECTRUM [Fe(H2O)

6]2+

I. Fontana, A. Lauria, G. Spinolo, Phys. Stat. Sol. B 244, 4669 (2007).

MLCT

MC


DODS FOR EXCITATIONS IN RADICALS

S. Hirata and M. HeadGordon, Chem. Phys. Lett. 302, 375 (1999).

CN radical

0 1 2 3 4 5 6 7 eV

Expt :

TDLDA :

XCIS :

ROCIS :

UCIS :

V 2 V 2


UDFT+UTDDFT PES FOR [Fe(H2O)

6]2+ BREATHING COORDINATE

t2g

eg

LC

t2g

eg

LCt2g

eg

LC

t2g

eg

LC

t2g

eg

LC

E. F

leur

inor

, S. B

rune

au,

L. J

oube

rt D

orio

l, M

EC

, unp

ublis

hed


SODS ANALYSIS OF 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

−av

∣ii a ⟩a i

∣ia v ⟩

−v i

∣iv v ⟩avv i

∣i av ⟩

v i a

v


SPIN OPERATORS

S2=∑ P nS z S z

−1 S

z=

1

2 n −n

where

n=∑ rr

P =∑ 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.


RADICAL EXCITED STATES |S,MS)

Doublets

Quadruplet

∣D2 ⟩=1

6∣i v a ⟩∣i v a ⟩−2∣iv a ⟩

∣Q ⟩= 1

3∣i v a ⟩∣i v a ⟩∣iv a ⟩

"Extended Singles"(a type of doubles)

∣D1 ⟩=1

2∣i v a ⟩−∣i v a ⟩

∣ii a ⟩ ∣iv v ⟩


TDDFT, TDHF, AND CIS GIVE

Singlet Coupling

∣TC ⟩= 1

2∣i v a ⟩∣i v a ⟩

∣D1 ⟩=1

2∣i v a ⟩−∣i v a ⟩

Triplet Coupling

Doublets

Neither a doublet nor a quadruplet!

MISSING: The quadruplet and one of the doublets!

∣ii a ⟩

∣iv v ⟩


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

!


NEED A WAY TO CALCULATE EXCITEDSTATE SPINCONTAMINATION

Ground state DFT spincontamination calculated using Löwdin's UHF formula,

Excitedstate TDDFT spincontamination could be calculated using UTDHFformula ... if there were one!

First derived: MEC, A. Ipatov, and F. Cordova, in TimeDependent DensityFunctional Theory, edited by M.A.L. Marques, C. Ullrich, F. Nogueira, A. Rubio, and E.K.U. Gross, Lecture Notes in Physics (Springer: Berlin, 2006), pp. 243257; A. Ipatov, F. Cordova, L. Joubert Doriol, and MEC, J. Mol. Struct. (Theochem) 914, 60 (2009); MEC, J. Mol. Struct. (Theochem) 914, 3 (2009).

⟨∣S2∣⟩=nS z S z−1−∑ ∣ j ,i∣

2


PROBLEM INVOLVES 2ELECTRON REDUCEDDIFFERENCE DENSITY MATRIX (2RDDM)

[ A BB A ] X I

Y I=I [1 0

0 −1 ] X I

Y I

Rowe 1RDDM:

A X I=IX I

Maurice and HeadGordon 1RDDM:

P I=X I X I−X I X I

X I X I

P I=X I X I

Y IY

−X I

X IY IY I

X I X I−Y I

Y I

Casida and Joubert Doriol 1RDDM:

F I=IF I

P I=F IF I−F IF I

F I F I

=A−B1/2AB A−B1/2

F I=A−B−1/2 X IY I


A. Ipatov, F. Cordova, L. Joubert Doriol, and MEC, J. Mol. Struct. (Theochem) 914, 60 (2009)


S=2⇒⟨ S2⟩=6

⟨ S2⟩=7⇒ Spin contaminated

SEPARATING ARTIFACTS FROM PHYSICAL STATES(TDA CALCULATION)


NO SPIN CONTAMINATION FOR MC TRANSTIONS

t2g

eg

LC

t2g

eg

LC

t2g

eg

LC

t2g

eg

LC

t2g

eg

LC


SPIN CONTAMINATION FOR MLCT TRANSITIONS

t2g

eg

LC

t2g

eg

LC

t2g

eg

LC

t2g

eg

LC

t2g

eg

LC

t2g

eg

LC

t2g

eg

LC


Sébastien BRUNEAU and Loïc JOUBERT DORIOLMasters' projects 2009

[Fe(H2O)

6]2+

Expt: I. Fontana, A. Lauria, G. Spinolo, Phys. Stat. Sol. B 244, 4669 (2007).

TDLDA/PBE/TZVP(2)d>d

no spin contaminationCT

spin contamination(but not always)



22O)O)

66]]2+2+



OXIRANE PHOTOCHEMISTRYT. Ibuki, M. Inasaki et Y. Takesaki, J. Chem. Phys. 59, 2076 (1973).


Transition State


MODERN PICTURE BASED UPON POTENTIAL ENERGY SURFACES Competing processes! React fast or loose your chance!

Image source: J. Michl and V. BonacicKoutecky, Electronic Aspects of Organic Photochemistry(Wiley: New York, 1990), p. 71. Embellishments: E. Tapvicza.

REACTANTSPRODUITS PRODUITS


CONICAL INTERSECTIONS AS A DIMENSIONALITY PROBLEM

A conical intersection (CX) is the analogue in N dimensions of avoidedcrossings in diatomics.

Ever since the 1990's we believe that many (most?) photochemical reactions proceed through CX rather than intersurface jumps atavoided crossings.


DIMENSIONALITY OF CONICAL INTERSECTIONS

A molecule with N internal degrees of freedom has an Ndimensional potential energy (hyper)surface (PES) in an (N+1)dimensional space.

In the absence of coupling 2 PESs cross in an (N1)dimensional intersection hyperline.

A conical intersection is (N2)dimensional hyperpoint.

EI=E

IQ

1,Q

2, ... ,Q

N

EIQ

1,Q

2,... ,Q

N=E

JQ

1,Q

2,... ,Q

N

0=HI , JQ

1,Q

2,... ,Q

N


(TD)DFT IS ALREADY PART OF THE PHOTOCHEMICAL MODELER'S TOOLBOX

There is presently a beautiful symbiosis between the use of (TD)DFTas a rapid, albeit only semiquantitative method, and more rigorousCASSCFbased methods for CXs.

● E.W.G. Diau, C. Kotting, A.H. Zewail, ChemPhysChem 2, 273 (2001). “Femtochemistry of Norrish type1 reactions: I. Experimental and theoretical studies of acetone and related ketones on the S

1 surface”

● E.W.G. Diau, C. Kotting, A.H. Zewail, ChemPhysChem 2, 294 (2001). “Femtochemistry of Norrish typeI reactions: II. The anomalous predissociation dynamics of cyclobutanone on the S

1

surface”● E.W.G. Diau, C. Kotting, T.I. Solling, et al., ChemPhysChem 3, 57 (2002). “Femtochemistry of Norrish typeI reactions: III. Highly excited ketones Theoretical”● T.I. Solling, E.W.G. Diau, C. Kotting, et al., ChemPhysChem 3, 79 (2002). “Femtochemistry of Norrish typeI reactions: IV. Highly excited ketones Experimental”


OXIRANE PHOTOCHEMISTRY: CO RING OPENING

1 2 3 4

5 6 7

E. Tapavicza, I. Tavernelli, U. Röthlisberger, C. Filippi, and MEC, J. Chem. Phys. 129, 124108 (2008). “Mixed TDDFT TDA/Classical Photodynamics Study of Oxirane Photochemistry”


GomerNoyes Mechanism[E. Gomer et W.A. Noyes, Jr., J. Am. Chem. Soc. 72, 101 (1950)]


Funnel Region: Involves both ground and excited states Typically involves bond breaking


WAVE FUNCTION THEORY

H2

H. + H.

H+ + H:

RHF

1g u

2

1g g

2

1u gu

3u gu

1g [ 1

2 sA sBsBsA ]

1g [ 1

2 sA

2−sB2 ]



H2

H. + H.

H+ + H:

RHF

1g u

2

1g g

2

1u gu

3u gu

1g [ 1

2 g

2−u2 ]

1g [ 1

2 g

2u2 ]



H2

H. + H.

H+ + H:

UHF

1g u

2

1g g

2

3u gu

1sAs B

Can “cheat” for ground state by breaking symmetry but not for excited states!

1g [ 1

2 g

2u2 ]

1u gu


DENSITYFUNCTIONAL THEORY

Exact theoryGround state singlet belonging to the totally symmetric irrep

v xc =v xc

⇒ = ⇒ ⇒ No symmetry breaking expected!

Assumes noninteracting vrepresentability (NVR)

NVR: There is a fictitious KohnSham system of noninteracting electronswith integer occupation number whose ground state gives the density of the interacting system.

Traditional workaroundis the ensemble formulationwith fractional occupationnumbers.


DENSITYFUNCTIONAL THEORY

Practical theory

⇒ Symmetry breaking lowers the energy !

Frequently encounter effective violation of noninteracting vrepresentability (But CoulsonFisher point is further out than for HF)

Trivial example: In H2, triplet falls below RKS singlet ground state.

Convergence problems due to enforcementof Aufbau principle!

Ground state singlet belonging to the totally symmetric irrep



Important MOs for the classic WoodwardHoffman model

Important MOs for the UVspectrum

O

CH2H

2C

oxirane,ethylene oxide,epoxyethane,dimethylene oxide



40 60 80 100120140160180200.00

5.00

10.00

15.00

20.00

25.00

1 & 2 ^1A_1

CAS(4,4) 1^1A_1SA-CAS(4,4) 1^1A_1CAS(4,4) 2^1A_1SA-CAS(4,4) 2^1A_1HF (sigma)^2HF (sigma*)^2

Angle

En

erg

y (

eV

)

Cusp!

Similar situation for rotation around double bond in CH2=CH

2 .

2

2

(2

(2

Oxirane CC ring opening O

CH2H

2C

Warning: CASactive space is toosmall!


SPINFLIP THEORY

ia

ia

ia

ia

i i

a a

a i

i a∣ii∣

∣i a∣

∣i a∣∣i a∣

∣a a∣ ia

∆MS=1 excitations from the lowest triplet allow a simple description

where ground, 1electron singlet, and 2electron singlet excited statescan mix.

Ana Krylov's group has applied this concept at many levels of electronic structure theory.


NONCOLLINEAR SPINFLIP TDDFT[F. Wang and T. Ziegler, J. Chem. Phys. 121, 12191 (2004).]

E xc[ , ]E xc[ ,− ]

where± r =

12r ±s r

and the density and spin density are defined in a rotationally invariant way

r = , r , r

s2r = , r − ,r

22 ,

2r ,

2r

All the usual definitions are recovered in the collinear limit!

(1)

(2)

(3)

(4)


[f xc , f xc

, f xc , f xc

,

f xc , f xc

, f xc , f xc

,

f xc , f xc

, f xc , f xc

,

f xc , f xc

, f xc , f xc

, ]=[f xc , f xc

, 0 0

f xc , f xc

, 0 0

0 0v xc −v xc

−0

0 0 0vxc −vxc

−

]NONCOLLINEAR SPINFLIP TDpureDFT

In the collinear limit,


DISSOCIATION OF H2

H+ + H

H. + H.


OXIRANE C2v

RING OPENING AVOIDED CROSSING

B. Natarajan, M. Huix-Rotllant, A. Ipatov, C. M. Wawire, T. Deutsch, and M. E. Casida,manuscript in preparation: Behaviour of Conical Intersections within Noncollinear Spin-Flip Time-Dependent Density-Functional Theory: Oxirane as Test Case

2

2

(2

(2


No more effective violation of noninteracting vrepresentability.

Convergence is much easier!


OXIRANE PHOTOCHEMISTRY: CO RING OPENING(Favored by alkyl substitution)

1 2 3 4

5 6 7

E. Tapavicza, I. Tavernelli, U. Röthlisberger, C. Filippi, and MEC, J. Chem. Phys. 129, 124108 (2008). “Mixed TDDFT TDA/Classical Photodynamics Study of Oxirane Photochemistry”


CASSCF CONICAL INTERSECTION GEOMETRYAND BRANCHING COORDINATES

Derivative coupling (DC, NAC) vector

Unscaled gradient difference (UGD, GD)vector

hm ,n

qI =Cm ∂H

∂ qI

Cn

gm ,n

qI =Cm ∂H

∂ qI

Cm−C

n ∂H

∂ qI

Cn


CASSCF CONICAL INTERSECTION

S1/S0 I=E I−E0


DIMENSIONALITY OF CONICAL INTERSECTIONS

A molecule with N internal degrees of freedom has an Ndimensional potential energy surface (PES) in an (N+1)dimensional space hypersurface.

In the absence of coupling 2 PESs cross in an (N1)dimensional intersection hyperline.

A conical intersection is (N2)dimensional hyperpoint.

EI=E

IQ

1,Q

2, ... ,Q

N

EIQ

1,Q

2,... ,Q

N=E

JQ

1,Q

2,... ,Q

N

0=HI , JQ

1,Q

2,... ,Q

N

Brillouin's theoremmeans no CX in CIS


CASSCF

CIS SEAM

S1/S0 I=E I−E0

CIS

CASSCF

CIS


H2O

ConicalIntersection

CASSCF

CIS TDB3LYP

B.G. Levine, C. Ko, J. Quenneville, and T.J. Martinez, Mol. Phys. 104, 1039 (2006).

CONICAL INTERSECTIONS DO NOT EXIST IN AA TDDFT


CASSCF

SURPRISING TDDFT

S1/S0 I=E I−E0

CASSCF

TDA TDPBE TDA TDPBE


EFFECTIVE VIOLATION OF NONINTERACTING vREPRESENTABILITY

Grossly oversimplified!


SFTDDFT

TDPBE TDACASSCF CIS

SFTDLDA TDA

reference islowest triplet

Easier convergence Coupling => true CX But the CX is rather

approximate


PROBLEMS WITH SFTDDFT

The initial state wave function can only be rigorously eliminated from TDDFT when the reference is the ground state. Tripletreferenced SFTDDFT and singletreferenced SPTDDFT treat

relaxation differently.

Mixed spin symmetry states occur. In UKS, the ith spin and the ith

spin orbitals may sometimes be very different.


O. Vahtras and Z. Rinkevicius, J. Chem. Phys. 126, 114101 (2007).“General excitations in timedependent density functional theory”

Even Better SFTDDFT ?

SFTDDFT ROKS Explicit spin coupling

But explicit 2 and higherelectron excitations still needed ...


Bhaarathi NATARAJAN

M. HuixRotllant, B. Natarajan, A. Ipatov, C.M. Wawire, M.E. Casida, and T. Deutsch, “Assessment of Noncollinear SpinFlip TammDancoff Approximation TimeDependent DensityFunctional Theory forthe Photochemical RingOpening of Oxirane”, in preparation.



22O)O)

66]]2+2+



2electron and higher excitations needed for Polyene spectra Spectra of molecules with openshell ground states Also implicated in conical intersections

Exact TDDFT quantity

Frequencydependent localizer

MBPT polarization propagator

f Hxc =s [ s−1 −−1 ]

WHAT IS THE ANALYTIC STRUCTURE OF THE EXACT xcKERNEL?


BETHESALPETER EQUATION

L=LsLsHxcL

BSE is an equation for a 4spatial, 4time coordinate quantity

Resembles TDDFT equation which is for a 2spatial, 2time quantity

Previous work by Reining, Rubio, Onida, ... based on BSE

= s s f Hxc

Polarization propagator is a 4spatial, 2time coordinate quantity

1,2,3,4 ; t−t ' =L1t ,2t ;3t ' ,4t '

(1)

(2)

(3)

= s s s−1−−1 (4)


LINEAR RESPONSE THEORY (MBPT)

Response of the density matrix to a possibly nonlocal perturbation

1,1 ' ;=∫1,1 ' ;2,2 ' ;wappl 2,2 ' ;d2 d2 '

Polarization propagator

1,1 ' ;2,2 ' ;=∑I

⟨0∣ 1 ' 1∣I ⟩ ⟨ I∣

2 2 ' ∣0⟩−I

Poles give excitation energies

−∑I

⟨0∣ 2 2 ' ∣I ⟩ ⟨ I∣

1 ' 1∣0 ⟩I

=wappl

(1)

(2)

(3)

I=E I−E0 (4)

Residues give absorption intensities.

⇒ =

wappl


LINEAR RESPONSE TDDFT (LRTDDFT)

1 ;=∫1 ;2; vappl 2 ;d2

Since everything is local,

where1 ;2 ;=1,1 ;2,2;

Harriman's collapse operator,

=

(1)

(2)

(3) x 1,2= x 1,1

vx 1=v x 11−2

⇒

(4)

(5)

Expansion operator,

J.E. Harriman, PRA 27, 632 (1983); PRA 34, 29 (1986).


LINEAR RESPONSE TDDFT (LRTDDFT)

= vappl

veff = vappl v Hxc

Density of interacting and noninteracting systems must be the same :

Here

(1)

(2)

(4)

=ss f Hxc

(5)s−1=

−1f Hxc

(6)

=

vappl

=s veff

⇒

⇒ s =

veff

(3)

veff

= vappl

v Hxc

Hxckernel


[s ] f Hxc [

]=[ s ] [s−1−−1 ] [ ]

f Hxc =s [ s−1 −−1 ]

=[]−1[ ]The “Localizer”

LOCALIZATION IN SPACE => FREQUENCY DEPENDENCE

We have used this to recover Görling's TDEXX.


f Hxc =s [ s−1 −−1 ]

f Hxc =s [ s−1 −−1 ]s

STRUCTURE PRESERVING APPROXIMATION

Nanoquantalikeapproximation

Essential for 2electron excitations!Previous workMEC, J. Chem. Phys. 122, 054111 (2005)D. Romaniello et al., J. Chem. Phys. 130, 044108 (2009)D. Sangalli and G. Onida, in preparation


GONZESCHEFFLER RELATION[PRL 82, 4418 (1999)]

ai∣f Hxc ai∣ia=ai∣s aiK ais ai∣ia=K ai , iaai

K = s−1−

−1

“Coupling Matrix”

GS relation

ai=a−i

HartreeFock case

K ai ,ia=ai∣∣ia−aa∣∣ii

Recover CIS (TDHF TDA) from exact TDDFT!


E

D

S

a

b

S D

a b

f

f a

f b

DRESSEDTDDFTN.T. Maitra, F. Zhang, R.J. Cave, and K. Burke,

J. Chem Phys. 120, 5932 (2004)


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

!


DRESSEDTDDFTN.T. Maitra, F. Zhang, R.J. Cave, and K. Burke, J. Chem Phys. 120, 5932 (2004)

Twoorbital model

=aiia∣f Hxc ∣ia

Assume AA gives

S

Have theory which separates S and the

D correction

NA is the D correction

f Hxc =f HxcAAf xc

NA

Beautiful but ad hoc model

Can we make this into a first principles theory?


s−1−

−1GET FROM SOPPA OR ADC(2)

− sr , qp≈ p q∣T P−1

T ∣r s

Resummation incompatible with dressed interaction (W)

400 terms resummedorderconsistentfirstprinciples theory

Approximation:


ORDERCONSISTENT SEPARATION INTO SINGLES AND DOUBLES

s−1−

012 ,−1=[ 1,1

12 − 1,112

−1,112

1,112 ]

AA

[ 1,212,2

0 ,−1 2,11 0

0 1,212,20 ,−1

2,11 ]

NA

≈P−1

Inverse:

=T∣1−H −1∣T


THEORY SEPARATING S AND

D CORRECTION

(PRESENT WORK)

K =1,1−P =K SKD

P−1= 1,1

−1 1,1

−1K 1,1

−1

PseudoBSE

Natural separation

≈aiFor , make GonzeSchefflerlike approximation

Twoorbital model

=aiia∣s K Ds∣ai

=aiK ia , aiD

(1)

(2)

(3)

(4)


WHAT IS NEW?

Orderconsistent formulae Firstprinciples estimation of

D

Proper accounting for frequencydependent localization


See poster byMiquel HUIXROTLLANT

M. HuixRotllant and M. E. Casida, “Formal foundations of dressed timedependent density functional theory for manyelectron excitations”, in preparation.



22O)O)

66]]2+2+



f Hxc =s [ s−1 −−1 ]

photochemical funnels

MBPT

spincontamination

Need for explicit 2 and higherelectron excitations:

How to get explicit 2 and higherelectron excitations:

SFTDDFTf xc ,

=v xc −vxc

−

Need for proper treatment of fractional occupation numbers!


Thanks to the organizers.

Many thanks for your attention!


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