Superoperator TDDFT Density

Mat,rix Approach to Spontaneous

Density Fluctuations and N onlinear

Shaul MukamelDepartment oí Chemistry U niversity oí California

Irvine, CA 92697-2025

Response

1

. Superoperator (Liouville space

dependent density functional theory (TDDFT).

. Develop a real-space analysis of nonlinear response

functions in terms of quasiparticles, collective elec-

tronic oscillators (CEO).

. Unified discription of spontaneous density fluctua-

tions and nonlinear response functions.

Expand charge ftuctuations and intermolecular

forces in CEO modes using generalized response

functions (G RF).

.

. Resolve the causality paradox oí TDDFT in Liou-

vine space.

. Connections to non-equilibrium (Keldysh) Green

functions, GW approach and open systems.

Goals

formulation of time

1

PRINCIPLES OF

Nonlinear Oplical Spectroscopy

Shaul Mukamel

O., fo (I.p

fA fEILgt1c k

Q )'JIJ>¡V. y"

(I~f1f1)

. Kohn-Sham

HKS(n)(r, t)

\72 .

~ ~ Kinetic Energy,2m

j dr,n(r', t)Ir - r'l

Uxc( n( r, t)) -7 Exchange Correlation Potential

. Second quantization: Field operators 1jJ(r):

'lj;(r) -

with wavefunctions cPa(r).a

. Anticommutation relations (Fermions):

[Ca, c¡]+ = 8ab, [cl, c¡]+ = [Ca, Cb]+

['ljJt(r), 'ljJ(r')] + 8(r r'), ['ljJt(r), 'ljJt(r')]+=['ljJ(r), 'ljJ(r')]+=O

(KS) Hamiltonian ..

Jdr' n(r', t)Ir - r'l-- + Uo(r)

2mUxc(n(r, t)) + Uex(r,t)

+-

Uo (r) ---+ Nuclear Potential

Uex(r, t) -+ Ext. Pote--+ Hartree Pot.,

a particle in state

o

4

. Red uced single-electron density matrix:

< >

= p(r, r, t). n(r, t)

Diagonal

density.

. p(r, r'), r =f r'

Off-diagonal elements represent the elec-

tronic coherences.

. p9(rI, r2) = (gl7,bt(r)7,b(r') Ig)

Densi ty Matrix o btained

statinary Kohn-Sham equations

p ( r, r', t) = (~t ( r, t) ~ ( r', t))

Trace Over the Ground State.

elements represent the charge

Ground State

by solving the

[fhs,fJ9]o-

5

. Change in density matrix induced by a time

dependent external field

8p(rl, r2, t) == p(rl, r2, t) - pg(rl, r2)

. Time dependent 81111Ji8ld KS equations

of motion for p

[HKS(n), p].82 8t (j p =

. TDDFT gives the ordinary (causal) re-

sponse of charge density to the external po-

tential Uex.

A A

UeXP + pU

6

~

Coupled Electronic Oscillator

( CEO) Representation of the

. TDDFT density

to a many-electron wavefunction given by a

single Slater determinant at all times.

. U sing the ground state Kohn-Sham orbital

basis it can be divided into four blocks.

Inter band

'"

~-

density matrix

matrix, p(t), corresponds

~

8p(t) =

( e- h ) Intraband (e-e,e-h)

~~

oT(~) -

o

~(t) T ( ~) ( t )8p(t) +-

7

A

.~

. Idempotent property:

T() = ~(j - 2f;9) {j - Vi - 4}=?

The e - e and h - h part

T(~) = (1 - 2,8g)[~~

A. 8 P can be expressed solely in terms of ~.

The elements of ~ (but not of 8jJ) constitute

independent coordinates for the electronic

structure and can be viewed as classical os-

cillators.

~ = [[6 p, pg], pg]

(¡JJ + 6p)2 = (pg + ¿jp)

(T)A

~

~~~~ + oo.]

is uniquely

8

. Eigenmodes of the linearized TD D FT equa-

tion for the density matrix

Linearized Liouville spaceA A A KS - A .A KS Ag

L~a = [Ha (n), ~a] + [Hl (~a)' p ]

Ground State Charge Densityn-+

. fIKS and f¡KS. o 1obtained by linearizing the KS Hamiltonian

in 8n.H!!S(ñ)(rl,r2) = 8(rl - r2)H{!S(n) (rl)

KS .

)H1 (~a)(rI, r2

A A A

L~a = r2a~a

operator( L):

are diagonal (local) matrices

8(rl--

.+ ~a(r3, r3)!xc(rz, r3)

f xc(rl, r2)-

9

. Each mode, represented by a matrix (~a)'

describes an electronic transition between

the ground state and an electronically ex.

cited state (0:).

. ~a come in pairs labled by positive and neg-

ative values of a. n-a =

Each.

palr.electronic oscillator (CEO).

1,;/ ~ CY.Qa =

. Scalar product of two interband matrices:

(~I1}) =::::}

Adopt the following normalization:"t "

(~al~J3) = 8a(3,

usual: N orm is zeroUn,

" - "t-Den ~-a-~ .

of modes represents a collective

r¿J2 (~a - ~~a)+ ~~Q) Pr-

a

= Tr{¡J9[tt, r]]}(~I7})(7}t I~t) *

( ~ I ~t) o--

(~QI~Jj) o-

10

'"

~ can be expanded in CEO modes,.

Using our scalar product

Za( t) =

Density matrix (8p =

CEO amplitudes (za):

.

Jp(rI, r2, t)= LJ.la(rl, r2)Za( t) + 2a

1+ 3 L I-LO:(J,/(rl, r211)zo:(t)z(J(t)z,/(t)...

aj3,

AII coefficients are expressed in terms of

~

~(t)'"

Za(t)~aLa=:f:l,:i:2,..

-

.

.

(!I(t))

T ( t) )'"

expanded in~

1L J-la,/3( rl, r2)Za( t )Zj3( t)

a,¡J

'"

~Q

...

~Q

(2pg

A

{la

A

{la/3A

{la/3,

-

- 1) (~a~(3 + ~(3~Q)-

. -~a (~{3~1 + ~1~{3)

11

. Equations of motion for Za are obtained by

projecting the TDDFT equation anta the

interband space (¿ = [[6,8, ,89], ,89]) and using

the normalization of the eigenmodes.

.dza( t)r¿ dt

K-a(t)

K-a{3(t)

K -a{3, (t)

v --af3,

- A KS . A KSV(~a)(rl)~J1 = Hl (~a)(rl)~J1 + H2 (~a~J1)(rl)pg

= r2aZa(t) + K-a(t) + K -a{3 ( t ) Z {3 ( t)

-a{3-y1i (t) zj3( t) z-y( t) ZIi( t) + . . .

drlU(rl, t)JL-o:(rl)

drlU(rl, t)JL-a¡3(rl)

-

-

Jdrl U (rl, t) ~-Q;{3,( rl)-

~Tr {¡1a(3V(~'Y) + ¡1(3'YV(~a) + M'Ya V(~¡3)}

12

is obtained by expanding KS Hamilto-nKS2.

nlan

.to second arder in bn.

H!jS(~a, ~J1)(rl)=

. Each electronic excitation can be viewed as

a classical oscillator with variables Za and

z~, and frequency OQ8

. The system

tronic anharmonic oscillators.

-

fxc( r2, r3) ) ~a(r3, r3)

JJ~2~3~C(rl,r2,r3)~(r2,r2)~(r3,r3)

9xc(rl,r2,r3) =ñ

elec-

13

. Classical Hamiltonian (to third arder):

H(z(t)) -

+ Uex(t)6p(z)

Poisson brackets:

A A At At{~a' ~JJ} = {~a' ~~} = O,

{Za, Z~} = iba/3,

{Za, Z,aZ,}

classical Hamilton equations give the

TDDFT equations of motion for the oscil-

lator amplitude

1+~

3¿ Va¡3,Za(t)Z¡3(t)z,(t)¿ ~aZa(t)Z-a(t)

a>O a/3,

At A

{~a,~J3} i6a{3

{Za, ZJ1} = -{ZJ1, Za}

ZjJ{Za, Z"(}{Za, Z¡1}Z¡ +-

Za.

14

CEO Representati

. Expand Za = Z(l) + Z(2) +

To linear arder:

<5p(r, r, t)

Zil)(t) = r¿sa:

Ga(t - r)

+1

-1

So:-

-

on of N onlinear

Response Functions

in powers of Uex. .

Zil) (t)/-la(r)La=:f:l,::t2,..

Ón(r, t) -

1: dr J r)drU(r, r)J-l-a(r)Ga(t -

r) exp { -iDa(t - T)}=.8(t -

For Positive Modes (a > O)

For Negative Modes (a < O)

15

. The linear density response function, X(1):

Ón(r, t) = 1: dr J

X(l) (rt, r't')

. In frequency domain:

X(l) (rlWI, r2W2) = 8(Wl + W2) L

. Higher arder response

computed similarly.

drIU(r/, r)x(l)(rt, r/T)

t')L isal-la(r)l-l-a(r')Ga(t --a

SaMa (rI) M-a (r2)'-VI - Da + it

a

functions can be

16

. The second arder response function:

X(2) (rltI, r2t2, r3t3)

LSaS/3J-L-a/3(rl) J-la(r2) J-L-¡3( r3)Ga( tI

a,/3

LSaS/3J-La/3(rl)J-L-a( r2) J-L-/3( r3)Ga( tI

a,¡3

-2

+i

drGa(tl - r)Gj3(r - t2)G,(r - t3)x

. In frequency domain:

X(2) (rlúJI, r2úJ2, r3, úJ3)

-

1¿a.{3

+-2

1+-

2¿a:{3

La.j3

-2

- t2)G/3(t2 - t3)

- t2)G (3( tI - t3)

-0.,/3,"'(

sQS/3¡.tQ(rl)¡.t-o:j3(r2)¡.t~/3(r3)Ó(Wl + w2 + w3)(W2 + W3 + Da - it)(w3 + DjJ - iE)

SaSj3J.La(rl)J.L-cxj3(r3)J-l-/3(r2)6(Wl + W2 + W3)

(W2 +W3 + Da - i)(W2 + rlj3 - iE)

sQs/3J1Q/3(rl)J1-Q:(r2)J1~/3(r3)¿j(f.J.Jl + '-V2 + '-V3)(W2 + 0('( - i)(f.J.J3 + 0/3 - i)

17

. Third arder response function:

- iP-a/3(r)p-/Jy(rl)Pa(r2)p-"((r3)Ga(t - tl)G/3(tl - t2)

G"((t2 - t3) +2 LP-a/3(r)V~/3"(óPa(rl)p-"((r2)p-Ó(r3)sÓ

,5

(tdrGa(t - tl)G/3(tl - T)G"((r - t2)GÓ(r - t3)Jt3iP-a/3"((r)Pa(rl)p-/3(r2)p-"((r3)Ga(t - tl)G/3(tl - t2)G"((tl

x

x

drGa(t - r)G/3( r - tl)G1( 1" ~ t2)G(j(t2 - t3)

x Ga(t - 7JG(3( 7 - tl)G"{( 7 - 7')GJ( 7' ~ t2)Gr¡( 7' - t3)

+

x

+ 2ipCt{3(r)p-{3,./rl)p-Ct(r2)p-~,/r3)GCt(t - tl)G{3( 'T - t2)G"((t2 - t3)

x

6¿SaSj3S'"'(-a. /31

t3)

{t {r drdr'Jt3 Jt3

17

Liouville Space

Generalized

. Hilbert space N x N operators such as p(t),

are viewed as vectors oí length N2 in Liou-

ville space.

. Operators of N2 x N2 dimension in Liouville.space are superoperators.

. With each Hilbert space operator A we can

associate two superoperators AL (left) and

( right ) :AR

ALX = AX

Hilbert Space (Ordinaryx

~

Superoperators and

ARX=XAand

Operator

18

. Symmetric and antisymmetric combina-

tions:

1-(AL2

-

. The commutator and anticommutator op-

erations in Hilbert space can be imple~

mented with a single multiplication by a "-"

and "+" superoperators, respectively.

. Superoperator algebra:

For two operators A and B:

(AB)L = ALBL,

1(AB)+X=2(ALBLX

(AB) _x = (ALBL-

=> (AB)+

(AB) -

(AL+AR) and A- -

(AB)R == ARBR

1== (A+B+ + -A_B_)X

4~ + ARBRX) == (A+B+ + ~A_B-

4. ARBR)X = (A+B- + A_B+)X

1= A+B+ + - A_B-

4= A+B- + A_B+A+B- + A_B+

19

In Hilbert Space:.

sjn) (t)

I'l/Jjm) (t))-+ Wavefunction to m'th order in E

. In Liouville Space:

To n' th arder in E:

p(n)(t)

=} S(t)

Expectation val ue of A

L Pj(1/Jj(t) I

J

S(t) -

L Pjsjn)(t)),n

S(t) -

n

L(~Jm)m=O

(t) IAI~)n-m) (t))-

Tr {Ap(t)}S(t) -

n

LPjLj m=O

¿Tr{

-

n

58

. Backward propagation of the bra:

Ud (t, to)

uJ (t, to)

-T Hilbert Space Anti- Time Ordering

Operator

U(t, to)

ut(t, to)

(~j(to)I[!J(t, lo)

(~j(to)I[!J(t, lo)

~dTHo(

dT Hint

(~J(t)1

(~j(t)1

-

}~1t. {t

~ lto

-Texp )T-

}-

(T)T

Forward

Backward Propagation

60

Dynamics in Liouville Space

. Liouville equation:

8 i8tP(t) = ñ (HT(t), p(t)]

e evolution of density matrix:

p(t) = Q(t, to)p(to)

g(t, lo) = T exp { - ~ it t:,(r)dr }

. Tim

Q(t) ~Liouville Space Time Evolution

¡:, ~ Liouville Operator for H T

. U nlike Hilbert space, Liouville space expec-

tation values only require propagating the

density matrix forward in time.

~~~~

61

Linear response.

Hilbert Space

X(I)(t2' tI)

Liouville Space

. X(I)(t2' tI) = Tr [ALQ(t2, tl)ALPeq] + C.C.

.

.

.~ L Pj(1jJjIUt(t2' tI)

JAU(t2, tl)AI~j) + c. c.-

63

. Third arder response:( .) 3 4

X{31(t4, t3, t2, tI) = ~ ~~ PjX~3)(t4' t3, t2, tI)+C.C.

Hilbert Space

Xi3) = (~jIUt(t2' tl)AUt(t3, t2)AUt(t4' t3)AU(t4, tl)AI~j)

X~3) = (~jIU(t2, tl)AUt(t3, tl)AUt(t4, t3)AU(t4, t2)AI~j)

X~3) = (~jIU(t3, tl)AUt(t2, tl)AUt(t4' t2)AU(t4, t3)AI~j)

(3) IX4 = ('lf¡jIUt(t4' tl)AU(t4, t3)AU(t3, t2)AU(t2, tl)AI'lf¡Ú

Liouville Space

XP) = Tr[ALQ(t4, t3)ARQ(t3, t2)AR9(t2, tl)ALPeq]

X~3) = Tr[ALQ(t4, t3)ARQ(t3, t2)ALQ(t2, tl)ARPeq]

X~3) = Tr[ALQ(t4, t3)ALQ(t3, t2)ARQ(t2, tl)ARPeq]

Xi3) = Tr[ALQ(t4, t3)ALQ(t3, t2)ALQ(t2, tl)ALPeq]

64

. Liouville space time ordering operator T

T

v== +/- or L/ R

. Expectation (A+(t)) ---+ Tr{A+(t)Peq} Causal-

ity in Liouvile Space:

. (B_(t')A+(t)) = Tr{B_(t')A+(t)Peq}

1= 2 Tr{[B(t'), (A(t)Peq + PeqA(t))]}

= O (Trace of a cornmutator)

. Expectation value of the product A+B_:

(7 A+(t)B_(t'))

(7 A+(t)B_(t')) is

.

.

AvCTl)B¡t( 1"2) T2 < TI

B¡.¿( 72)Av( 71) TI < T2

~[Av( 71) B/-l ( 71) + B/-l( Tl)Av( TI)] 72 -1"1

tI < t

tI> t

(A+( t)B~( tI))

(B~(tl)A+(t)) = O

-

causal, vanishes

20

for t' > t.

. The p'th arder response function

(p)( ) - (- ")p "x. Xp+ 1, ..., Xl - r¿ ~

x(p) describes the response generated at

point rp+ltp+l to p fields interacting with

.

the system at points rl tl...r p tp and n is the

charge-density operator.

. The step functions 8(t) keep track of time

ordering and guarantee causality tl...tp <

tp+l. pFrm is a sum ayer the' p! permutations

of Xl"'Xp.

r, t):(: -x

o L B( tp+l - tp) . . . B( t2 - tI)

perm

[[n(Xp+l), n(xp)] ,n( Xp-l)]... f¿(Xl)])([ .

21

Superoperator

(p)( )X Xp+ 1 . . . Xp

.

-

. p'th arder generalized response functions

(GRF):

X1/p+l...1/1 (Xp+l

Superoperator Index (+ or -)l/p

p' N umber

Vp+b Vp, .., VI

of response:representaion

= (-i)P (Tñ+(xp+l)ñ- (Xp)...ñ-(Xl))

1

2(nL"'~ '" '"n == nL - nR+ nR),

( -i)pl (7 ñVp+l (Xp+l) . . . ñVl (Xl)).

indices in the setof "-"

22

. Ordinary (causal)

correspond to GRF with one "+" and p "-".f

r "'superoperator indices (x+ ~ ~ ... ~) .

. Other functions (with more than one "+"

index) represent the correlations of sponta-

neous fluctuations or their response to the

external potential.

.GRF Sn.A..

r "'(x - - -. . .

(x(p))response functions

indicesall " "-with vanish

O).-

23

+ 1), p'th arder generalized re-

sponse functions with n = 0,1, ..., p "minus"

. There are (p

indices.

Linear GRFs (p=l):

X++(x,x/) = (Tñ+(x)f¿+(x/)) (Density Fluctuations)

X+-(x, x') = -i(Tñ+(x)ñ-(x/)) (Density Response)

.

. x+- is causal: For t' > t

X+-(X, X')

=> X+-(X, X') = -i()(t - t') (ñ+(x)ñ+(x'))

. x++ is non-causal:

x++(X, X') = 8(t - t') (ñ+(x)ñ+(x/))

+ 8(t' - t)(ñ+(x/)ñ+(x))

Tr{n-(x')n+(x)p}-

Tr{[n(x'), n(x)p + pn(x)]} == o

24

. Linear G RF x++ and x+- are not indepen-

dent. They are related by the fluctuation-

dissipation (FD)

x+-(rI, r2, t)

x++(rl, r2, t)

a++(rl, r2, LV)

. Classical, high temperature, limit of FD:

* x+-(r¡, tI, r2, tI + t) =

relation .

r¿100 +- -il.AJt

a (r1, r2, ~)e d~1r -00

ñ j oo . . - a ++(r r W)e-u.lJtdw2 1, 2,

1r -00

-

-

(¡3~) a+-(rl,r2,úJ)coth-

)({31iwcoth f",.J

()(t) d- KBT dtx++(r1, t1, r2, t1 + t)

25

. p'th arder GRF is a combination oí 2p+l Li-

ouville space pathways(LSP) with different

number of "left" and "right" operators.

Four LSP contribute to the linear G RF:.XLL(Xl, X2) = (TnL(Xl)nL(X2))

XRR(Xl, X2) = (7 nR(Xl)nR(X2))

. XLR(Xl, X2) = (TnL(Xl)nR(X2))

XRL(Xl, X2) = (TnR(Xl)nL(X2))

X++(Xl, X2)

X+-(Xl, X2)

-

26

. AII LSP are non-causal and symmetric with

respect to the change of their superoperator

and space-time arguments.

. Ordinary response

nation of 2p+l LSP. Other combinations rep-

resent spontaneous fluctuations and their

responses to external field.

(x(p)) is a specific combi-

27

Dynamical Approach toIntermolecular Forces U sing

Generalized

. Two well separated coupled molecules ( a

b):and

N o Charge Over lap, No Exchange=>

H = Ha + Hb + AHab,

. Electrostatic interaction:

Hab

Charge Density of Molecule a

Charge Fluctuation of Molecule

na

Óna

Functions

0>'\>1

¡¡drdr'na(r )nb(r')J(lr r'/)- -

e2J(lr r'l) =

na ( r )

Ir - r'l

ña+ Óna-

a

28

. Hellmann-Feynmann Theorem:

. Interaction

w=

w(O)

W(I) --

+

W(II)

~

8HA8>"

\Ir»)(WA-

energy:

11d>"(Hab) A- E(A = O) :=

r')ña (r )nb(r')

r') [na(r) (Ónb(r'));\J(r ~dA

,

11 dA J J drdr' J(r - r') (óna(r )8nb(r')) A

29

. McLachlan's (196:

Waals interaction:

For T = O

WML = 2 roo~JL'illf- 2ñ J-oo ]

(};b(r'l, r'2, w)coth.

(}:a(rl, r2, úJ) --* Linear Polarizability of

For T > o

WML-

J(rln

21rn//3ñúJn -

963) express ion for van der

f(t

(drlJe

¡J~)

j dr' 2Üa (r¡, r2, (.¡))

r'r)J(r2 - r'2) I

dw dr1

J(rl

Molecule a

aa(rln, r2n, iwn)ab(r'rn, r/2n, iwn)

r' 2n)ln)J(r2n -

Matsubara Frequencies

30

. In superoperator notation:

(c5na(r )c5nb(r')) A

. In the interaction picture, ground state

density matrix of interacting system (Peq) is

generated from the non-interacting density

matrix (Po = p( -00 )) by Adiabatic switching

of interactions.

Peq

Expectation value of an operator A(x):.(A+(x)) = Tr {A+(x)p}

(8n~(r )8nt(r')) A

. f a- ~ -00 }{ drV-(T)Texp Po-

= (r A+(x) exp {~ ¡~ drV-(r)} )0

A+(x) = Qo(t)A+(x)Qo(t)

Qo(t) = B(t) exp { - h'Hüt}

31

. In superoperator notation:

(8na(r )8nb(r')) A

. In the interaction picture, ground state

density matrix of interacting system (Peq) is

generated from the non-interacting density

matrix (Po = p( -00 )) by Adiabatic switching

of interactions.

Peq == T exp

. Expectation value of an operator A(x):

(A+(x)) = Tr {A+(x)p}

(8n~ (r )8nt (r/)) A

}r¿

{-

drV-Cr) Po1i

O ¡ f

~ -00 }){\ drV-(T)}i+(x)T exp-o

A+(x) = Qo(t)A+(x)Qo(t)

Qo(t) = O(t) exp { - ~1tot}

31

. Superoperator Rules: For ordinary opera-

tors A and B

Hab thus contains n; and nt:.Hab(r, r', t) = [ña(t, t)ñb(r', t)]-

Response of the coupled system is recast in

terms of GRFs of individual molecules:

R(n)(t, tI, ..., tn) =

1A+ B+ + -A- B-

4

(AB) +

(AB)-

-

A+B- +A-B+

= nd"(r, t}n¡;(r', t) + n;;(r, t)nt(r', t)

n-m mn ~~

'""""X+.. . + - . . . -(t t t )L ,.¡ a , 1, ..., n

m=Om n-m

~~+...+ (t t t )Xb ' 1,.. ., . nx

33

. The expectation value (c5na(r)) A can be ex-

panded perturbatively in A and expressed

in terms of the n'th arder joint response

function:

R(n)(t, tI, ..., tn)

< >0,--+.. .-

liñ, Hab

. The ground state density matrix :

., in) = (-i)n(T8ñ+(t)Hab(tl)' ..., Hab(tn))O

-+ Trace ayer ground state (A = O)

In interaction picture

-

fJ9 = p~pg

32

. Interaction energy (W) depends on all

GRFs of individual molecules. Tú sixth or-

der in charge fiuctuations

w(n) ~ Contribution

Charge Fluctuation

W(l) o-

W(2)= -1

2ñ

Xd-(XI, X2)J(rl - r'1)J(r2 - r'2)

¡ tI =~L

6ñ2 p -00

J(rl ~ r'1)J(r2 - r'2)J(r3

X~--(Xl' X2, x3)ñb(r'2)ñb(r'3)

W(3)

x

x

6

L w(n)w -n=l

From

~ ¡t~dt2 J J J J drldr' ldr2dr' 2fib(r'l)fib(r' 2)

dt{t!t3 J J J J J J drl dr' 1 dr2dr;dr3dr~1

r'3)nb(r\)~

34

. McLachlan expression:

(tI~ Loo dt2

J(rl - r'¡)J(r2 - r'2)Xd+(Xl, X2)Xt-(X'I, X'2)

6~2 ~ 1: dt{: dt3j j j j j j drl dr'¡ dr2dr' 2dr3dr~

J(rl - r'¡)J(r2 - r'2)J(r3 ~ r'3)

[na(r2)nb(r'3)X~-(Xl' X3)Xt-(X'r, X'2)

+ nb(r'r)na(r3)X~-(Xl, X2)Xt-(X'2, X'3)]

W(4) -

x

+

x

x

. First term of W(4) reproduces McLachlan's

result for van der Waals interaction. This

can be expressed in terms of x+- alone since

x++ and x+- are related through FD theo-

reme

J J J J drl dr\ dr2dr' 21

2ñ

35

. Fifth and Sixth arder terms:

W(5) = 6~~ ¡t~p j-C(

W(5) = 6~~p ~¡~t1~t3J J J J J ftrl dr'l dr2dr' 2dr3drí

x J(rl - r'¡)J(r2 - r'2)J(r3 - r'3) { xt-- (x\, X'2, X'3)

x [ña(r1)X~+(X2, X3) + ña(r2)X~+(XI, X3)

+ ña(r3)X;+(XI, X2) + ña(rl)X~-(XI, X3)]

++- ( ' , , )[- ( ) +- ( )+ Xb x 1, X 2, X 3 na r2 Xa Xl, X3

J(rl - r'r)J(r2x

x

+

ñb(r'3)xt-(X't, X'2)] }+

w( 6) = 6~j)p : ¡~ti~t3J J J J J frl dr\ dr2dr' 2(

X J(rl - r'I)J(r2 - r'2)J(r3 - r'3)

[ +++( ) +-- ( ' , ' )X Xa Xl, X2, X3 Xb X 1, X 2, X 3

++- ( ) ++- ( ' , ' )]+ Xa XI,X2,X3 Xb XI,X2,X3

>' ~ Sum ayer Single Permutation Xn ~ x' nJ

x

x

¿p

I

r[ / /. /}drldr ldr2dr 2dr3dr3

~

36

. Complete set of ordinary causal) response

functions CX+ ) is not sufficient to calcu-

late intermolecular interaction.

AII GRFs are required

hysical Reason:

uctuations are not described by ordinary

Mathematical Reason:

(na(r )nb(r'))-

.

Correlated spontaneous

functions.

n;(r)nt(r')nd(r)nb(r')-

37

Computing Generalized Response

. TDDFT equation for density matrix can be

generalized as: p

a

. The system is coupled to two external po-

tentials, "left" UL acting on the ket and

"right" UR acting on the bra.

. In superoperator notation:

i :l(rl, r2, t) = 1íKSp(rl, r2, t) - U-(rl, r2, t)p(rl, r2, t)

HKsjJ=[HKS, jJ], U- jJ=[U+, jJ], U+jJ=[U_, jJ]

Functions by TDDFT

A A

ULp - URp[HKs(n), ,8]i 8t óíJ +-

- U+(rl, r2, t)p(rl, r2, t)

38

. When UL = U R, P is the density matrix. The

complete set of GRF can be obtained by

allowing U L 'and U R to be different.

Ordinary, for example,

function x+- represents

sponse of the system to an applied poten-

tial that couples to charge density through

a commutator.

x++ can be formally obtained as the re-

sponse to an artificial external potential,

U+, that couples to the charge density

through

linear response

the density re-

ti commutatoran an .

39

. Interband matrix

. Za is expanded in external potential Uv:

- (t) - vI + vI v2+Za - Za Za ...,

ZVl...Vp ---t p'th arder term in

Linear..8Z~1 (t)r¿ 8t .

K-a(t) =

"'-

/.La:-

A

is expanded in ~a:A

~

L taza(t)A

~(t) =

a=:t:::l,:!:2..

Vn == +

U 1/1 . .. U 1/p .

-

tion:

QaZ~l ( t )

L

A+/-la

equa

VI == + or+ K-a(t)

drlUv(rl, t)fLV a(rl)

Aila=(2pg - I)~a.

--

Z/

and

40

. Solution for z~ :

z¿ (tI)

Za (tI)

-

-

. Linear G RFs:

x++(rltl, r2t2) =

x+-(rltl, r2t2) = ~ie(tl - t2f2:ISaJJta(rl)¡J,~(}!(r2)e

. Frequency domain: ,x++(rlWl, r2W2) = i8(Wl + W2)

Pa(r2)P-a(W2 + Sta -

+ ~

X -(rlWl, r2W2) = 8(Wl + W2)x+-(rlWl, r2W2)

J dI

:t2 J

i 1: dt2

iBa {'x; dJ-oo

drlU+(rl, t2)[l-a(rl)Ga(tl - t2)

r drlU-(rl, t2)/La(rl)Ga(tl - t

.r¿

t2)dt2

t2) L [la (rl) p=fJ!(r2)e -iOQ(tl-t2)e (tI -a

e(t2 - tI) 2: [la(r2)[l-a(rI)eiD.a(tl -t2)

a

,-iOa(tl-t2)

a

[ fla(rl)fl-a( r.2)

w2 - Da + ~ELa

W2

t'

a

41

. Second order G RF:

x+-- is the ordinary X(2) obtained from the

ordinary TDDFT density matrix equation.

~

x+++(rltl, r2t2, r3t3) =

L fl-{3(r3)Ga(tl - t2)

a,{3

- 2J.La/3(rl)[L-a(r2)G/3(tl - t3) ]1

~

[tL-a¡3(r2)fJa(rl)G¡3(t2 - t3)

l-la(rl)p-j3(r2)p-"((r3) V-a,j3,"(

d'T Ga(tl-'T) G/3(r-t2) G"((r-t3)

Sum ayer Permutation r jtj ~ rktk

42

X++-(Xl, X2, X3) =

L0.,/3

SaS/3Jt-a/3(r2)Pa(rl)p-/3(r3)Ga(tl - t2)G{3(t2 - t3)

¿a,(3

-2

LL{XI,X2} aj3¡

+ 2i

1:

¿ ~ Sum ayer Permutation r jtj +-+ rktk

{Xj,Xk}

- t2)G/3(tl - t3)s /3/-la/3 (rl) íl-a( r2) tL~{3 (r3) G a (tI

Sa S1 f-la(rl)p-/3(r2)f-l-1(r3)V-a,/3,1

- t3)dT GoJt1 - T) G/3(T -

43

. Secand arder G RF

x+--(rlúJl, r2úJ2, r3, úJ3)=

1La{3

8a8j3,

( 28, V-aj3, /-La (rl) /-L- /3 (r2) /-L-, (r3)(W3 + O, - i)

f-ta( rl) f-t-a(3( r2) f-t-(3( r3) f-ta( rl) f-t-a(3( r3) f-t-(3 ( r2) )('"-'3 + Dj3 - i) + ('"-'2 + Dj3 - iE)

/Lc«3(rl)/L-c,(r2)/L-(3(r3) -

](W2 + Oa - i)(W3 + 013 - i)

-2

x

+

frequency domain):

c5(Wl + W2 + w3)

44

x+++(rlWl, r2W2, r3, W3)

+ J-tCt (r3)íl-{3(rl)íl-'Y (r2) ](W2+ W3- °'Y+ i) ("'-'2 + O{3- iE)(W3- O~/+ iE) 8(Wl + W2+ W3)

+~~a:{3

¡.ta{3( r2) fl-a(rl) fl-{3( r3)

(W2+ W3- na+ i)(W3+ n{3- i~)

¡.ta{3( r3) fl-a( r2) P-{3( rl)

(W2+ W3- na- i)(W2+ n{3- i)

-

f..la:p( rl) íl-a (r2) íl-/3( f3)

(w2+ Da- iE)(úJ3+ D/3- ie)

}8("'-"1 + "'-"2+ "'-"3)

42

x++-(rlÚ)l, r2Ú)2, r3, ú)3) =

S, V-a{3,¡.ta(r2)íl-j3(rl)¡;,~,(r3)8{úJl + úJ2 + úJ3)(úJ2 - Da + i)("'-'2 + "'-'3 - O{3 + iE)(úJ3 + D, - iE)+

Sa¡.ta( rl) [¡.t-aj3( r3),u-j3( r2) - Sj3,u-aj3( r3)¡.t-j3( r2)]

(W2 + W3 + r2a - i)(W2 + r2¡1 - i)+

Sa~a(r2) [~-at3(r3)p--t3(rl) - St3P--at3 (r3)Jl-¡3 ( fl)]+ _u

("-'2 + (.V3 - r2t3 + i)("-'2 - [la + i)

s/3J.L-/3(r3) (~Q¡J(r2),ü-Q(rl)+ ("-'3 + r2/3 - i) ("-'2 + Oa - i)

+ J.La/3(rl)jl~a(r2)) ](""2 + ""3 - na + iE) 8((.¡)1 + (.¡)2 + (.¡)3)

Expanding the Intermolecular

We have expressed intermolecular ener-

gies in terms of GRF which, in turn, are

computed using CEO múdese Combining

the two yields closed expressions for inter-

molecular energy (W) to any order in CEO

.

modes.

. To fourth arder charge fiuctuations:4

Lw(n)n 2

w=

1

-2ñLpW(2) -

L s c/ /-L-a' ( r't) /-La' ( r' 2)

ei Oalx

energy in CEO modes

drldr2dr'ldr'2

1111 rl)na(r2)Irl - r'l

46

W(3) =

ña (rl)ña (r2)ña( r3)Sc/S {3'x

x

1W( 4) ---- 2ñ

x

x

[ña(rl)Jta(r2) + ña(r2)Jta(rl)]x

L 1J/J1J drl dr2dr3dr'l dr' 2dr' 3p Irl - r'll/r2 - r'211r3 - r'31

47

Superoperators A utomatically

Resolve the TDDFT Causality

. Density as. a functional derivative of an ac-

tion:

. Is it impossible to construct this functional

. The paradox: We are in trouble !

[Gross, 1996]

6"2

8 v ( rt ) 8 v ( r' t')

. The RHS is the density response function

which must be causal i.e. vanish for t' > t,

whereas the LHS is symmetric to the inter-

change of its space and time arguments.

Paradox

.-

8A[v(r, t)]8v(rt)

n(rt) =

-A[v(rt)] 8n(rt)

8v (r' t')

( I IX rt, r t )-

48

. Liouville space action:

A[VL, ~

( iñ)ln

VR]

: (~Texp{ -~J

. Liouville space pathways can be generated:

X~P+ll1p...lIl (Xp+l, Xp...Xl) =

.( -~ )P

!i

XC Connected LSP..There is no "causality paradox" since all

LSP are non-causal.

. AII causal and non-causal functions can be

constructed as

space pathways.

-

dX[VL (X)1ÍL (X) + VR(X)ñR(X)]})-

VL=VR=O

8p+l A[VL, VR]

8VVp+l (xp+l)8vllp(xp).. ,ÓVVl (Xl)

combinations of Liouville

50

. n'th arder G RF:

Xlln+l...lIl (tn+l

. Generating functional for GRF:

S1(t)-O ¡ t

~ dñ -00

X lIn+looolll (t t )n+ 1. .. 1

< 000 >0 --+ Trace ayer Po

. Response functions (Xn) are calculated in

terms of known (trace over Po) quantities

x(n) (tn+b . . .tl) =

- - - - ,(T A+(tn+l)A-(tn)... A-(tl)V-(Tm)x

n

(TAj. tI) -l(tn+l) . . . AVl(tl)l/n + ]

}).2

~díV-( í)xo

T)A+(T))] )dr

o

Ón+lS( t)iñ óE"n+! (tn+i). ..El/! (ti) E+=E-=O-

00

Lm=O

( i ) m+l tn+l I j 'Tn+l I - d71. . . d7 m

ñ -00 -00

( -l)mm!

. . V- ( iD) o

69

tI

qL (O)-

qR (O)

The Keldysh time loop. Going along the loop,we can arrange al! the operators in the properarder.

,.-,I

~t

q\.l~)t3

t4l~)'IR

t2

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