Kohn-Sham scheme for frequency dependent linear response

0.23 MB·

by  Ryan Requist

Kohn-Sham scheme for frequency dependent linear response
Ryan Requist∗ and Oleg Pankratov
Theoretische Festk¨orperphysik,
Friedrich-Alexander-Universit¨at Erlangen-Nu¨rnberg
Staudtstraße 7-B2, 91058 Erlangen, Germany
(Dated: January 30, 2009)
We study the Kohn-Sham scheme for the calculation of the steady state linear response
λn(ω1)(r)cosωt to a harmonic perturbation λv(1)(r)cosωt that is turned on adiabatically. Al-
though in general the exact exchange-correlation potential vxc(r,t) cannot be expressed as the
9 functional derivativeof a universal functional due to the so-called causality paradox, we show that
0 for a harmonic perturbation the exchange-correlation part of the first-order Kohn-Sham potential
20 λvs(1)(r)cosωt is given by vx(1c)(r) = δKx(2c)/δn(ω1)(r). Kx(2c) is the exchange-correlation part of the
second-order quasienergy Kv(2). The Frenkel variation principle implies a stationary principle for
n Kv(2)[n(ω1)]. We also find an analogous stationary principle and KS scheme in the time dependent
a
extension of one-matrix functional theory, in which the basic variable is the one-matrix (one-body
J
reduced density matrix).
0
3
PACSnumbers: 31.15.ee,32.10.Dk,71.15.Mb

]
r
e I. INTRODUCTION AnimportantapplicationofTDDFTisthecalculation
h of dynamic polarizabilities and excitation energies.9,10,11
t
o The description of a time dependent quantum state is Thesecanbeobtainedfromthefrequencydependentlin-
t. fundamentally differentfrom the descriptionofa ground earresponsefunctionχ(ω)=χ(r,r′;ω).32Inthetimedo-
a state because there is not a minimum principle for the main, the retarded linear response function is defined as
m former. Asdensityfunctionaltheory1 (DFT) isbasedon χ(r,t,r′,t′) = δn(r,t)/δv(r′,t′). The KS system repro-
- a minimum principle, it was not obvious that it could ducesself-consistentlythelinearresponseoftheinteract-
nd be extended to time dependent situations. Yet such an ing system to a perturbation δv(r,t). Thus, χ(r,t,r′,t′)
o extension,atimedependentDFT(TDDFT),wasestab- is related to the KS response function χs(r,t,r′,t′) =
c lishedbythe Runge-Gross(RG)theorem,2 whichasserts δn(r,t)/δvs(r′,t′) by the Dyson-like equation12
[ that the time dependent density of a many-electron sys-

2 temdeterminesthetimedependentexternalpotentialup χ(1,1′)=χs(1,1′)+ d2d3χs(1,2)
Z
v toanadditivepurelytime dependent function,assuming
7 a fixed initial state Ψ(t0) = Ψt0. The theorem applies ×(vc(r2,r3)δ(t2−t3)+fxc(2,3))χ(3,1′), (2)
7 to both interacting and noninteracting systems. This is
8 of great importance for applications because it implies where i = (ri,ti), vc(ri,rj) = ri rj −1 and fxc(i,j) =
1 that the density n(r,t) of an interacting system can be δvxc(i)/δn(j) is the exchange-|cor−relat|ion kernel. If the
2. reproduced by a noninteracting system with an effective system is in its ground state at t = t0, then χ(t,t′) de-
1 potential v (r,t), provided the initial state Φ(t ) = Φ pends only on the time difference t t′ and not t and t′
8 of the nonisnteracting system is chosen to be co0mpatibtl0e individually. Thus,fromtheFourier−transformof(2)one
0
with n(r,t).3 The potential v (r,t), which is called the obtains
: s
v time dependent Kohn-Sham4 (KS) potential, is a func-
Xi tional of n(r,t), Ψt0 and Φt0. Its exchange-correlation χ(r,r′;ω)=χs(r,r′;ω)+Z d3r2d3r3 χs(r,r2;ω)
part v (r,t) can be defined from the equation
r xc (v (r ,r )+f (r ,r ;ω))χ(r ,r′;ω) (3)
a v (r,t)=v(r,t)+v (r,t)+v (r,t), (1) × c 2 3 xc 2 3 3
s H xc
iff (ω)exists. GrossandKohnstatethat“Forthemost
where v (r,t) is the time dependent Hartree potential. xc
H generalsituation,wedonotknowwhetherf exists...”12
In contrast to v (r) in static DFT, the exact v (r,t) xc
xc xc Eq. (3) implies the following formal representation:
cannotbeexpressedasthefunctionalderivativeofauni-
versal functional of n(r,t). This is a consequence of the f (r,r′;ω)=χ−1(r,r′;ω) χ−1(r,r′;ω) v (r,r′).(4)
so-called causality paradox.5,6,7,8 In TD DFT, neither a xc s − − c
general minimum principle nor even a stationary princi- Therefore, f (ω) exists whenever χ(ω) and χ (ω) are
xc s
plehasbeenfound.31Suchaprinciplemightbehelpfulin invertible. Although the RG theorem guarantees that
the search for accurate approximations to v (r,t). The χ(t,t′) is invertible (subject to the condition that the
xc
quantum mechanical action principle does not lead to a perturbation is analytic at t=t ), the frequency depen-
0
stationary principle of the form δA = 0, where A is a dent response function χ(ω) is not always invertible.13
functional of n(r,t), because its density-functional for- As χ(ω) is the Fourier transform of χ(t t′), one might
mulation contains boundary terms.8 ask why the invertibility of χ(t t′) doe−s not imply the
−
2
invertibility of χ(ω). We shall address this question in a II. STATIONARY PRINCIPLE FOR THE
later section. The problem of the invertibility of the re- QUASIENERGY
sponse functions is an instance of the v-representability
problem in density functional theories. Inthis section,we derivea stationaryprinciple for the
In this paper, we introduce a stationary principle in quasienergyofFloquetstatesobtainedbyturningonadi-
TD DFT and use it to derive the KS equations for fre- abatically a time-periodic perturbation of the external
quency dependent linear response. Our approach is to potential.
work in the time domain, and by turning on a harmonic Webeginbyreviewingthepropertiesofthewavefunc-
perturbation λv(1)(r)cosωt adiabatically, we induce a tionwhentheHamiltonianisperiodicintime,Hˆ(t+T)=
steady state linear response density λn(1)(r)cosωt. We Hˆ(t). If Hˆ(t) is an operator on a finite-dimensional
prove that the quasienergy (an analog of the Bloch Hilbertspace,theFloquettheoremassertsthatthereex-
quasimomentum for systems periodic in time), to sec- ists a complete set of solutions of the form22,23
ond order in λ, is a stationary functional of n(1)(r).
If the linear response density λn(1)(r)cosωt can be re- Ψn(t)=ξn(t)e−iǫnt, ξn(t+T)=ξn(t), (5)
produced by a KS system with an effective potential
which are called Floquet states or quasienergy eigen-
v (r,t) = v(0)(r) + λv(1)(r)cosωt, then the station-
s s s states. The quasienergy, ǫ , is defined modulo 2π/T.
ary principle implies that the exchange-correlation part n
The periodic factor ξ (t) satisfies the equation
of v(1)(r) is the functional derivative of the exchange- n
s
correlation part of the second-order quasienergy. We Hˆ(t) i∂ ξ (t)=ǫ ξ (t). (6)
t n n n
also find ananalogousstationaryprinciple andlinear re- −
(cid:0) (cid:1)
sponse KS scheme in the time dependent extension of If the system under consideration has an infinite-
one-matrix functional theory14 (1MFT). In 1MFT, the dimensionalHilbertspace,theexistenceofFloquetstates
external potential can be nonlocal in space and spin co- isnotguaranteedbytheFloquettheorem. Formanysys-
ordinates, i.e., it acts as an integral operator with the tems,itmaybethecasethatHˆ(t) i∂ hasnonontrivial
t
kernel v(rσ,r′σ′). The corresponding many-body opera- eigenfunctions so that Floquet stat−es do not exist.24
tor is Vˆ = σσ′ d3rd3r′ψˆσ†(r)v(rσ,r′σ′)ψˆσ′(r′). Signif- Now consider an N-electron system that starts in a
icantly, a thPeoremR analogousto the RG theorem has not ground state at t = and experiences an adiabat-
−∞
been found in TD 1MFT. ically ramped (AR) periodic perturbation of the form
TD DFT for the special case of time-periodic exter- λvτ(1)(r,t) = λf(t/τ)v(1)(r,t), where v(1)(r,t + T) =
nalpotentials has been studied previously.15,16 However, v(1)(r,t) and f = f(t/τ) is a ramping function with
the scope of these and later approaches,17,18 which are time scale τ. The many-body Hamiltonian is Hˆ (t) =
τ
basedonaHohenberg-Kohn-typeminimumprinciple,are Hˆ(0)+λVˆ(1)(t) with Hˆ(0) = Tˆ+Wˆ +Vˆ(0), where Tˆ is
τ
severelylimited19,20,21 because(i)theminimumprinciple thekineticenergyoperator,Wˆ istheelectron-electronin-
isgenerallyvalidonlyforperiodicpotentialsthathaveno teraction and Vˆ(0) = d3rv(0)(r)ψˆ†(r)ψˆ(r). The ramp-
Fouriercomponentoffrequencygreaterthanthefirstex-
ing function is an arbRitrary smooth function that satis-
citationenergyand(ii)onemustassumetheexistenceof
fies f( ) = 0 and f( ) = 1. An example of a suit-
Floquet states (reviewed below). The approach we pur- −∞ ∞
able ramping function is (1 + tanh(t/τ))/2. The pre-
suehereisdistinctbecauseitemploysadiabaticramping
cise form of f is inconsequential and will be left unspec-
in real time and relies on a stationary principle instead
ified in the following. The functions v(1)(r,t) form a
of a minimum principle; hence, our results are valid for τ
one-parameter family of perturbations, and the action
allfrequencies(except, ofcourse,resonancefrequencies),
of an ideal AR perturbation is realized by taking the
and we need not assume the existence of Floquet states.
limit τ at the end of the calculation. Although
The paper is organized as follows. In Section II, we Hˆ (t) is→not∞exactly periodic, it is still possible to define
τ
review the basic properties of Floquet states, which are a quasienergy if the system approaches a steady state in
the fundamental states of a system with a time-periodic the limit (t,τ) ( , ).
Hamiltonian. The Frenkel variation principle is used to Following Re→f. 2∞5, w∞e factor the wave function as
derive a stationary principle for the quasienergy of Flo-
quet states obtained from adiabatic ramping. In Section t
Ψ(t)=ξ (t)exp i dt′K (t′) , (7)
III, we repeat the standard derivation of the frequency τ τ
− Z
dependent linear response function, except that we em- (cid:0) −∞ (cid:1)
ploy an arbitraryadiabatic ramping function. Theorems where
that establish a stationary principle and linear response
KS scheme in TD DFT are proved in Section IV. Anal- ξτ Hˆτ(t) i∂t ξτ
K (t)= − . (8)
ogous theorems in TD 1MFT are proved in Section V. τ (cid:10) (cid:12) ξ ξ (cid:12) (cid:11)
(cid:12) τ τ (cid:12)
A simple illustration of the KS scheme in 1MFT is pre- (cid:10) (cid:12) (cid:11)
sentedinSectionVI. Wecommentbrieflyonthequestion In accordance with the terminol(cid:12)ogy of Ref. 25, the over-
of the existence of f (ω) in Section VII. all phase factor will be called the secular phase, and the
xc
3
factorξ =ξ (t)willbecalledthenonsecularwave func- which suggests the definition
τ τ
tion. A system will be said to evolve adiabatically into a
steady state if the following two conditions are satisfied: K [ξ] = lim 1 t+T dt′ ξτ Hˆτ(t′)−i∂t′ ξτ
1) the nonsecular wave function tends to a unique func- v (t,τ)→(∞,∞)T Zt (cid:10) (cid:12)(cid:12) ξτ ξτ (cid:12)(cid:12) (cid:11)
tion ξ =ξ(t) with period T in the limit (t,τ) ( , ),
i.e., if for all ǫ > 0 there exist t′ and τ′→suc∞h t∞hat = 1 t+T dt′ ξ Hˆ(t′)−i∂t′ ξ ,(cid:10) (cid:12)(cid:12) (cid:11) (14)
kξτ(t)−ξ(t)k < ǫ when t > t′ and τ > τ′ and 2) all of T Zt (cid:10) (cid:12)(cid:12) ξ ξ (cid:12)(cid:12) (cid:11)
the electrons remain localized in a finite region of space (cid:10) (cid:12) (cid:11)
(cid:12)
for all time. Real systems, for which the spectrum usu- where the subscript v denotes the given v(1)(r,t). We
ally has a continuum component, are not expected to define the domain of K to be the space of all steady
v
evolve into such a steady state due to the possibility of state nonsecular wave functions ξ that can be obtained
multiquantumionization.24 However,ifthe perturbation for some v(1)(r,t). It is possible to choose a larger do-
is harmonic and the driving frequency ω = 2π/T is not main, but this is not necessary for our purposes. Let ξ
v
a resonance frequency, the first-order term of the power denote the steady state nonsecular wave function corre-
series of ξτ with respect to λ evolvesadiabatically into a sponding to the given v(1)(r,t). A variation δξ will be
unique harmonic function ξ(1) (see Section III). In such called an admissible variation if ξ +δξ is in the domain
v
cases,thesystemwillbe saidtoevolveadiabaticallyinto of K . Thus, for every admissible variation δξ, there ex-
v
a first-order steady state. The periodic function ξ, if it ists a one-parameter family of variations δξ such that
τ
exists, will be called the steady state nonsecular wave ξ +δξ evolves adiabatically into ξ +δξ. Hence, (13)
τ τ v
function, while ξ(1) will be called the first-order steady implies the stationary principle
state nonsecular wave function. The quasienergy associ-
ated with ξ is, cf. (6), δK [ξ]=0 (15)
v
K = ξ Hˆ(t)−i∂t ξ , (9) foranarbitraryadmissiblevariationδξatξ =ξv. Infact,
(cid:10) (cid:12) ξ ξ (cid:12) (cid:11) this result follows from a direct calculation of the first
(cid:12) (cid:12)
variation, assuming only the existence of ξ . We have
(cid:10) (cid:12) (cid:11) v
where Hˆ(t) is Hˆ (t) without t(cid:12)he ramping function f. carried out the above derivation based on the Frenkel
τ
variationprinciple because itwillproveusefulwhen Flo-
If the system evolves adiabatically into a steady
quet states do not exist, a case to which we now turn.
state, the Frenkel variation principle implies that the
quasienergy is a stationary functional of ξ.25 For an ar- If the system does not evolve adiabatically into an ex-
bitrary time dependent Hamiltonian Hˆ′(t), the Frenkel act steady state, the limit in (14) does not exist in gen-
eral. Nevertheless,foranARnonresonantharmonicper-
variation principle states that Ψ = Ψ(t) is the solution
turbation λv(1)(r)f(t/τ)cosωt, the system evolves adi-
of the Schr¨odinger equation with the initial condition
abatically into a first-order steady state described by
Ψ(t )=Ψ if
0 t0 ξ(0)+λξ(1), where ξ(0) is the unperturbed ground state.
0 0
δΨ Hˆ′(t) i∂ Ψ =0, (10) We nowshowthatthe second-orderquasienergyis asta-
t
− tionary functional of ξ(1).
(cid:10) (cid:12) (cid:12) (cid:11)
where δΨ=δΨ(t) is(cid:12)an arbitrar(cid:12)y variation that satisfies Consider the trial function
δΨ(t ) = 0. Setting Hˆ′(t) = Hˆ (t) and adding to (10)
0 τ ξ(r,t)=ξ(0)(r)+λξ(1)(r,t), (16)
its complex conjugate, we obtain 0
δ Ψ Hˆτ(t) i∂t Ψ +i∂t Ψ δΨ =0. (11) where r =(r1,...,rN) and ξ(1)(r,t) is an arbitrary har-
(cid:10) (cid:12) − (cid:12) (cid:11) (cid:10) (cid:12) (cid:11) monicfunctionwithfrequencyωsubjecttotheconstraint
Substituting ((cid:12)7) gives (cid:12) (cid:12) ξ(0) ξ(1) =0. Expanding(12)tosecondorderinλand
0
r(cid:10)epea(cid:12)ting(cid:11)the steps leading to (15), one obtains the sta-
∂ ξτ δξτ tiona(cid:12)ry principle25
δK (t)+i =0. (12)
τ ∂t(cid:10)ξ(cid:12)ξ (cid:11)
τ(cid:12) τ
δK(2)[ξ(1)]=0, (17)
(cid:10) (cid:12) (cid:11) v
(cid:12)
As we are interested in the response to an AR periodic
perturbation, we now consider a one-parameter family where
of variations δξ (with parameter τ) such that ξ +δξ
τ τ τ 1 t+T
estvaotleveξs+adδiξa.baTthiceanll,yufproonmtathkeinggrothuendfoslltoawteintgoliamsitteaanddy Kv(2)[ξ(1)]= T Zt dt′(cid:16)(cid:10)ξ(1)(cid:12)Hˆ(0)−E0(0)−i∂t′(cid:12)ξ(1)(cid:11)
time average,33 the secondtermof(12) vanishes,andwe + ξ(0) Vˆ(1)(t′) ξ(1)(cid:12)+c.c. . (cid:12) (18)
obtain
(cid:10) (cid:12) (cid:12) (cid:11) (cid:17)
(cid:12) (cid:12)
1 t+T Eq. (17) applies for an arbitrary admissible variation
(t,τ)→lim(∞,∞)T Zt dt′ δKτ(t′)=0, (13) δξ(1) at ξ(1) = ξv(1). The trial wave function in (16),
4
which is specified only up to first order in λ, is sufficient where Φ and E are the eigenstates and eigenenergies
k k
to obtain the quasienergy through third order,34 of Hˆ(0). The initial condition is a ( )= δ . To first
k k0
−∞
order (for k =0),
K [ξ]=E(0)+λ2K(2)[ξ(1)]+ (λ4). (19) 6
v 0 v O a(1)(t)= iV(1) t dt′f(t′/τ)cos(ωt′)eiωk0t′, (24)
This is analogous to a well-known fact concerning the k − k0 Z
−∞
variational estimate of a ground state energy: the error
intheenergyissecondorderintheerrorofthetrialwave whereVk(01) = Φk Vˆ(1) Φ0 andωk0 =Ek−E0. Treating
function. Thefirst-ordertermofthetrialfunctionin(16) first the eiωt c(cid:10)omp(cid:12)onen(cid:12)t of(cid:11)cosωt, we obtain
(cid:12) (cid:12)
can be expressed as
a(1)(t) = iV(1) t dt′f(t′/τ)ei(ω+ωk0)t′
ξ(1)(r,t)=ξ(1)(r)eiωt+ξ(1)(r)e−iωt, (20) k+ −2 k0 Z−∞
+ −
which leads to = iV(1) f(t′/τ) 1 ei(ω+ωk0)t′ t
−2 k0 (cid:20) i(ω+ω ) (cid:21)
k0 −∞
Kv(2)[ξ(1)] = Kv(2+)[ξ+(1)]+Kv(2−)[ξ−(1)]; + iV(1) t dt′df(t′/τ) 1 ei(ω+ωk0)t′
K(2)[ξ(1)] = ξ(1) Hˆ(0) E(0) ω ξ(1) 2 k0 Z−∞ dt′ i(ω+ωk0)
v± ± ± − 0 ± ±
+ (cid:10)ξ(0)(cid:12)(cid:12)Vˆ(1) ξ(1) +c.c.,(cid:12)(cid:12) (cid:11) (21) from integration by parts. The last term vanishes in the
0 ±
limit (t,τ) ( , ) if ω = ω , which can be shown
(cid:10) (cid:12) (cid:12) (cid:11) → ∞ ∞ 6 − k0
where Vˆ(1) is the many-bo(cid:12)dy op(cid:12)erator corresponding to as follows. Let
v(1)(r). ∞ df(t/τ)
I(τ) = dt ei(ω+ωk0)t
In the above derivation, we made no assumptions Z dt
−∞
about the higher order terms of the power series of ξτ ∞ df(s)
with respect to λ. The series truncated at order N, = ds ei(ω+ωk0)τs, (25)
Z ds
−∞
ξτ ≈ξ0(0)+λξτ(1)+···+λNξτ(N), (22) where s = t/τ. According to the Riemann-Lebesgue
lemma, I(τ) vanishes in the limit τ due to the
may also approach a unique periodic function. In most → ∞
rapidlyoscillatingphasefactor. Thelemmarequiresonly
physicalsystems,therewillexistanintegerN′ suchthat ∞
that the condition ds df(s)/ds < is satisfied.
the truncated series will cease to approach a periodic −∞ | | ∞
This determines theRdegree to which the ramping func-
function if N > N′, owing to multiquantum resonances
tion is arbitrary: the steady state linear response will
Nω = E E . The analysis of this section can be
k − 0 be independent of the precise form of the ramping func-
extended up to the highest order for which there are no
tion provided only that the later satisfies f( ) = 0,
multiquantum resonances. −∞
f( ) = 1, and the condition above. This conclusion is
∞
independentofthedetailsofthesystem. Thesameanaly-
sisappliesforthee−iωt componentofcosωtifω =ω .35
III. PERTURBATION THEORY FOR 6 k0
Combining the results, we find that the a(1)(t) approach
ADIABATICALLY RAMPED HARMONIC k
PERTURBATIONS the functions36
V(1) 1 1
Inordertoshowthatthe nonsecularwavefunctionξτ, a˜(k1)(t) = eiωk0t k20 (cid:20)cos(ωt)(cid:18)ω ω − ω+ω (cid:19)
k0 k0
to first order in λ, evolves adiabatically into a unique −
1 1
harmonic function, we repeat the standard calculation isin(ωt) + . (26)
of frequency dependent linear response. We employ an − (cid:18)ω ωk0 ω+ωk0(cid:19)(cid:21)
−
arbitrary adiabatic ramping function f =f(t/τ) instead
Thus,thefirst-ordertermofthenonsecularwavefunction
of the usual ramping function eηt with η 0. Without
approaches the harmonic function
→
loss of generality, we take the perturbation to be of the
form λvτ(1)(r,t) = λv(1)(r)f(t/τ)cosωt with ω ≥ 0. The ξ(1)(t)= a˜(k1)(t)e−iωk0tΦk (27)
calculationwillclarifythesenseinwhichthesteadystate kX6=0
linear response is independent of the precise form of the
The corresponding linear response density is
ramping function in the adiabatic limit τ .
→∞
We consider an N-electron system with the Hamilto- n(1)(r,t)= 2Re a˜(1)(t)e−iωk0t Φ nˆ(r) Φ
nian Hˆτ(t) = Hˆ(0) +λVˆτ(1)(t). The system is assumed kX6=0 (cid:16) k (cid:17)(cid:10) 0(cid:12) (cid:12) k(cid:11)
to start in the ground state at t = . We expand the (cid:12) (cid:12)
wave function as −∞ =cos(ωt) V(1) 1 1
k0 (cid:18)ω ω − ω+ω (cid:19)
kX6=0 − k0 k0
Ψ(t)= a (t)e−iEktΦ , (23)
k k Φ nˆ(r) Φ , (28)
Xk × 0 k
(cid:10) (cid:12) (cid:12) (cid:11)
(cid:12) (cid:12)
5
where nˆ(r) = ψˆ†(r)ψˆ(r) and we have used the fact that was shown that K(2)[ξ(1)] is stationary for an arbitrary
v
the Φk can be chosen real in the present case. The admissible variation δξ(1) at ξ(1) = ξv(1). As ξ(1) is
frequency dependent linear response function is readily (1)
a functional of n on the LR VR space, there ex-
ω
identified as
ists an admissible δξ(1) corresponding to every admis-
χ(r,r′;ω)=kX6=0(cid:18)ω−1ωk0 − ω+1ωk0(cid:19) sδoiKrbdlv(ie2n)aδ=tne(ω1(i)nδ.KtegvT(2rh)a/etδrioξenf(o1s)r,)e(a,δnξKd(1v(t)2h/)[eδnnfi(ω(ω1r1)s)]t)δifsna(ωcs1tt)oa,rtsiiousnpzaperrryeos.bsiencgaucose-
Φ0 nˆ(r) Φk Φk nˆ(r′) Φ0 . (29) This stationary principle can be used to derive an ex-
×
(cid:10) (cid:12) (cid:12) (cid:11)(cid:10) (cid:12) (cid:12) (cid:11) pression for the exchange-correlation potential of a lin-
(cid:12) (cid:12) (cid:12) (cid:12)
ear response KS system. The linear response KS sys-
IV. KOHN-SHAM SCHEME IN TIME tem is a noninteracting system that experiences the po-
DEPENDENT DENSITY FUNCTIONAL THEORY tential v (r,t) = v(0)(r)+λv(1)(r)f(t/τ)cosωt and re-
s s s
produces the first-order steady state density n(0)(r) +
We now turn to the main results of the paper. In this λn(1)(r)cosωt of the interacting system. Such a system
ω
section, we prove two theorems that establish a station- will exist if the following two v-representability condi-
ary principle and KS scheme in TD DFT for the special tions are satisfied.
case of harmonic perturbations. Condition (1a). The ground state density of the in-
In the following,we assume that the frequency depen- teracting system is noninteracting v-representable (VR-
dent density response function χ(ω) is invertible.37 This N).Thismeansthatthereexistsasystemofnoninteract-
is not alwaystrue. For example, χ(ω) will not be invert- ingelectronswithapotentialv(0)(r)suchthattheground
s
ible if it has any null eigenvalues apart from the trivial
state is nondegenerate and reproduces the ground state
one corresponding to an arbitrary constant shift of the
density of the interacting system.
external potential. Mearns and Kohn13 have given an
Condition (1b). The frequency dependent response
explicit example in which χ(ω) has nontrivialnull eigen-
function χ (ω) ofthe noninteractingsystem incondition
s
values; however, these occur only for particular isolated
(1a) is invertible on the LR VR space of the interacting
frequencies ω . The theorems below can be extended to
p system.
such cases by requiring that vω(1p)(r) and n(ω1p)(r) are or- Theorem 2. — If an interactingsystem satisfies con-
thogonal to all null “directions.” We also assume that ditions(1a)and(1b),thenthe exchange-correlationpart
ω ≥0 is not a resonance frequency, i.e., ω 6=(Ek−E0). of vs(1)(r) is given by
Consider an electron system that starts in a nonde-
generate ground state at t = −∞ and experiences an v(1)(r)= δKx(2c) , (30)
AR harmonic perturbation λvω(1)(r)f(t/τ)cosωt. The xc δn(ω1)(r)
system evolves adiabatically from the ground state with
density n(0)(r) into a first-order steady state with den- where Kx(2c)[n(ω1)] is the exchange-correlation part of the
sityn(0)(r)+λn(1)(r)cosωt. Since, byassumption,χ(ω) second-order quasienergy.
ω
Proof. — In analogy with Ref. 2, the exchange-
is invertible, the linear response density n(1)(r) deter-
ω correlation part of the second-order quasienergy is de-
mines the perturbation vω(1)(r) up to an arbitrary con- fined as
stant. The perturbation, in turn, determines the first-
1 t+T
order steady state nonsecularwavefunction ξ(1), as seen K(2)[n(1)]= dt′ ξ(1) Wˆ ξ(1) +S(2)[n(1)]
in (27). Therefore, ξ(1) is a functional of n(ω1)(r) on the xc ω T Zt (cid:10) (cid:12) (cid:12) (cid:11) W ω
space of linear response densities that can be obtained 1 (cid:12) (cid:12) n(1)(r)n(1)(r′)
forsomevω(1)(r). Thisspacewillbereferredtoasthelin- −S0(2)[n(ω1)]− 2Z d3rd3r′ ω r ωr′ ,
earresponsev-representable(LRVR)space. Letn(1)(r) | − | (31)
ω,v
bethelinearresponsedensitycorrespondingtothegiven
where
v(1)(r). We may now state the first theorem.
ω 1 t+T
Theorem 1. — The second-order quasienergy Kv(2) SW(2)[n(ω1)]= T Z dt′ ξ(1) Tˆ+Vˆ(0)−E0(0)−i∂t′ ξ(1)
is a functional of n(1)(r), and it satisfies the stationary t (cid:10) (cid:12) (cid:12) (cid:11)
ω (cid:12) (cid:12)
conditionδK(2) =0foranarbitraryadmissiblevariation and
v
δnP(ω1r)(oro)f.at n—(ω1)T(rh)e=senc(ωo1,n)vd(r-o).rder quasienergy is a func- S0(2)[n(ω1)]= T1 Z t+Tdt′ ξs(1) Tˆ+Vˆ(0)−E0(0)−i∂t′ ξs(1) ,
t (cid:10) (cid:12) (cid:12) (cid:11)
tional of n(ω1) = n(ω1)(r) by composition of Kv(2)[ξ(1)] and ξ(1) is the first-order st(cid:12)eady state nonsecular(cid:12)wave
and ξ(1)[n(ω1)]. Therefore, the domain of Kv(2)[n(ω1)] is functison of the KS system. The last term of (31) sub-
the LR VR space. A variation δn(ω1)(r) is admissi- tracts the Hartree contribution, K(2). The above func-
H
ble if n(1)(r) + δn(1)(r) is LR VR. In Section II, it tionalsareuniversalinthesensethattheydonotdepend
ω,v ω
6
onv(1)(r). However,theydodependonthegroundstate Before proceeding, it will be helpful to review some
ω
density and the value of ω, though this dependence will basic results from static 1MFT. The defining feature of
not be indicated explicitly. With these definitions, one 1MFT is that it has the capacity to treat systems in
finds K(2) = S(2) + d3r v(1)(r)n(1)(r)+K(2) +K(2). which the external potential is nonlocal with respect to
v 0 ω ω H xc
Hence, the stationaryRcondition of theorem 1 gives the space and spin coordinates. Accordingly, the neces-
sary basic variable is the one-matrix (one-body reduced
δKv(2) = δS0(2) +v(1)(r)+v(1)(r)+v(1)(r) density matrix), which is defined as
δn(1)(r) δn(1)(r) ω H xc
ω ω
γ(x,x′)=N dx ...dx ρ(x,x ,...x ;x′,x ,...x ),
=0, (32) 2 N 2 N 2 N
Z
(37)
where vH(1)(r) = δKH(2)/δn(ω1)(r). The KS system also where x = (r,σ), dx = d3r, and ρˆ =
evolvesintoafirst-ordersteadystate,andthe stationary σ
condition for its second-orderquasienergyKv(2) =S0(2)+ Pensnewmnb|lΨe nwiehiΨghnt|siswtnhesRuNch-eltehcattroPn dneRwnnsity=m1a.triBxyweitxh-
d3rvs(1)(r)n(ω1)(r) will be identical to (32) if tendingtheHohenberg-KohntheoPrem,1Gilbertproved14
R that 1) the one-matrix uniquely determines the ground
v(1)(r)=v(1)(r)+v(1)(r)+v(1)(r). (33) state wave function and 2) there is a universal energy
s ω H xc
functional E [γ] that attains its minimum at the ground
Since χ (ω) is invertible, v(1)(r) is uniquely defined. v
s s state one-matrix. There is also a KS scheme in 1MFT.
Thus, vx(1c)(r) is given by (30). This completes the proof. From the stationary condition for the energy functional,
The steady state linear response density of the in- Gilbert derived the equation
teracting system can be computed from the expression
n(1)(r,t) = N φ∗(0)(r)φ(1)(r,t) + c.c., where φ(0)(r) 1 2φ (x)+ dx′v (x,x′)φ (x′)=ǫ φ (x), (38)
are the grouPndi=s1tatie KS oirbitals and φ(1)(r,t) arie ob- − 2∇ i Z s i i i
i
tainedfromfirst-orderperturbationtheoryforthesingle- where v (x,x′)=v(x,x′)+δW/δγ(x′,x) and W =W[γ]
s
particle Schr¨odinger equation is the universal electron-electron interaction functional.
This equation can be interpreted as the single-particle
1
i∂tφi(r,t) = − 2∇2+vs(0)(r) φi(r,t) Schr¨odingerequation for the orbitals of a noninteracting
(cid:0) (cid:1) system (the 1MFT KS system). The potential v (x,x′)
+ λv(1)(r)f(t/τ)cos(ωt)φ (r,t) (34) s
s i is a functional of the one-matrix. The ground state one-
with the initial condition φ (r, ) = φ(0)(r). We re- matrixoftheinteractingsystemcanbeobtainedbysolv-
i −∞ i ing self-consistently (38) together with
mark that theorem 2 can be extended to cases where
the ground state density is not VR-N but rather EVR-
γ(x,x′)= f φ (x)φ∗(x′), (39)
N(noninteractingensemblev-representable)byfollowing i i i
an approach analogous to the one taken in the next sec- Xi
tion. where f are occupation numbers that satisfy f =
i i i
Theorem2impliesthattheexchange-correlationkernel N and 0 fi 1. Generally, fractionalPoccupa-
can be calculated as tion numbe≤rs are ≤required to reproduce the one-matrix
δ2K(2) of the interacting system, not only the values 0 and
f (r,r′;ω)= xc . (35) 1 as in DFT. This scheme was originally described as
xc δn(ω1)(r)δn(ω1)(r′) paradoxical14,26,27 because the stationary condition im-
Thestationaryprincipleforthesecond-orderquasienergy plies that essentially all of the ǫi collapse to a single
doesnotentailacausalityparadoxbecauseitsbasicvari- level. Therefore, it appeared that the single-particle
able n(1)(r) is time independent. It is straightforwardto Schr¨odinger equation would not define unique orbitals.
ω
However, when the occupation numbers are shifted
derive the following formal expression:
slightlyfromtheirgroundstatevalues,the KSequations
K(2) = 1 drdr′n(1)(r)χ−1(r,r′;ω)n(1)(r′) have a self-consistent solution for a one-matrix that is
v −2Z ω ω close to the ground state one-matrix and for which the
degeneracyis lifted.28 Thus, the correctgroundstate or-
+ drn(1)(r)v(1)(r). (36)
Z ω ω bitals can be obtained in the limit that the occupation
numbersapproachtheirgroundstatevalues. Theground
state orbitals, which are called natural orbitals, are the
V. KOHN-SHAM SCHEME IN TIME eigenfunctions of the ground state one-matrix, and the
DEPENDENT ONE-MATRIX FUNCTIONAL correspondingeigenvaluesaretheoccupationnumbers.29
THEORY As the occupation numbers are fractional, it is useful to
interpret the KS system as adopting an ensemble state.
In this section, we generalize the theorems of the pre- In the time dependent versionof 1MFT, a generalKS
vious section to TD 1MFT. scheme has not been found. Such a scheme should have
7
the capacity to treat systems in which the time depen- nonsecular wave function that would be obtained if the
dent external potential is nonlocal with respect to the system were to start in the stationary state Φ at t =
n
space and spin coordinates. In TD DFT, the existence . The ensemble-weighted second-order quasienergy,
−∞
of the KS scheme is implied by the RG theorem. But for fixed w , is a functional of γ(1) because each K(2)
n ω n
in TD 1MFT, it is not known whether there is a theo- is a functional of ξ(1) and each ξ(1) is a functional of
n n
rem as general as the RG theorem, i.e., for as general a γ(1). Each K(2) can be shown to be a functional of ξ(1)
class of time dependence. The Bogoliubov-Born-Green- ω n n
by repeating the arguments leading to (21) for a sys-
Kirkwood-Yvonhierarchyprovidesanequationofmotion
tem initially in the state Φ , assuming, as we have, that
for the one-matrix. However, this equation contains the n
Φ is not in resonance with any of the other stationary
two-matrix (two-body reduced density matrix), and it is n
states. Eachξ(1)isafunctionalofγ(1)becauseγ(1)deter-
not known whether the two-matrix is a universal func- n ω ω
tional of the one-matrix when the external potential is mines vω(1) (upto aconstant),which,inturn,determines
nonlocal. As we are interested in frequency dependent ξ(1). The first variation of K(2)[γ(1)] can be expressed
n v ω
linearresponse,weshallnarrowourattentiontothespe- as δK(2) = w (δK(2)/δξ(1))(δξ(1)/δγ(1))δγ(1), where
v n n n n n ω ω
cial case of AR harmonic perturbations. In this case, the coordinPate integrations have been suppressed. A
there is a KS scheme in TD 1MFT. Theorems 3 and 4 variation δγ(1) is admissible if γ(1) + δγ(1) is in the
ω ω,v ω
below are generalizations of theorems 1 and 2.
LR VR space of the ensemble. The first variation van-
We shall need to refer to a different linear re-
ishes for an arbitrary admissible variation δγ(1) because
sponse function. The one-matrix response function in ω
the time domain is defined as χ(x ,x′,t ;x ,x′,t ) = δKn(2)/δξn(1) = 0 for all n, which follows from a straight-
1 1 1 2 2 2
δγ(x ,x′,t )/δv(x ,x′,t ). The frequency dependent forward extension of the arguments in Sec. II.
1 1 1 2 2 2
one-matrix response function will be denoted χ(ω) = Nowconsideranelectronsystemthatstartsinanonde-
χ(x ,x′;x ,x′;ω). Similarly, the one-matrix response generategroundstateatt= andexperiencesanAR
1 1 2 2 −∞
function of the KS system will be denoted χs(ω). In or- local or nonlocal perturbation λvω(1)(x,x′)f(t/τ)cosωt.
der to emphasize the analogy between this section and Ifthe followingtwoconditionsaresatisfied,thenthereis
the previoussection,someofthe notationswillbe dupli- a linear response KS system in TD 1MFT.
cated. Condition (2a). The ground state one-matrix is
As the KS system can be interpreted as adopting an noninteracting ensemble v-representable (EVR-N). This
ensemblestate,ourfirststepwillbetoproveastationary means that there exists a system of noninteracting elec-
principle for an ensemble-weighted quasienergy. tronswithapotentialv(0)(ij)suchthatthegroundstate,
s
Consider an electron system that experiences an AR whichmaybeanensemblestateρˆ(0),reproducestheone-
local or nonlocal perturbation λvω(1)(x,x′)f(t/τ)cosωt. matrix of the interacting system.s
Suppose that the systemstartsat t= in the ensem- Condition (2b). The frequency dependent one-
ble state ρˆ(0) = nwn|ΦnihΦn|, wher−e∞Φn are orthogo- matrixresponsefunctionχs(ω)ofthenoninteractingsys-
nal stationary stPates, no two of which are in resonance, temincondition(2a)isinvertibleonthe spaceofallγ(1)
i.e., E E = ω for all m and n. Let γ(1)(x,x′) de- ω
m n ω,v that(i)areLRVRintheinteractingsystemand(ii)have
− 6
note the steady state linear response one-matrix corre-
no diagonal and degenerate components.
sponding to the given vω(1)(x,x′). Also, let Kn(2) denote The diagonal components of are simply the linear
the second-order quasienergy that would be obtained if response occupation numbers f(1) = γ(1)(ii), while
the system were to start in the pure stationary state i ω
the degenerate components are γ(1)(jk) + γ(1)(kj) and
Φ instead of the mixed state ρˆ(0). It is convenient to ω ω
n (1) (1) (0) (0)
introduce a notation in which Hermitian functions of −iγω (jk)+iγω (kj), where φj and φk are any pair
(x,x′) are expressed with respect to the complete basis of occupationally degenerate natural orbitals, i.e., nat-
of ground state natural orbitals φ(i0)(x). For example, ural orbitals for which fj(0) = fk(0). The diagonal
v(1)(ij) = dxdx′φ∗(0)(x)v(1)(x,x′)φ(0)(x′). We may and degenerate components correspond to null eigen-
ω i ω j functions of χ (ω), so they are not LR VR in the KS
now state thRe stationary principle. s
system. Therefore, the appropriate basic variable for
Theorem 3. — If the frequency dependent response
the linear response KS system is γ(1) = γ(1)(x,x′) =
function of the ensemble is invertible for a given ω, then ω ω
the ensemble-weighted second-order quasienergy ′ijγ(ω1)(ij)φ(i0)(x)φ∗j(0)(x′), where the prime indicates
Pthat the diagonal and degenerate components are ex-
K(2) = w K(2) (40) cluded fromthe sum. In effect, γ(1) describes the orbital
v n n ω
Xn degreesoffreedombutnottheoccupationnumbers. Sim-
ilarly,letv(1)denotetheprojectionofthegivenperturba-
is a functional of γω(1) = γω(1)(ij) and satisfies the sta- tionv(1) toωthenondiagonalandnondegeneratesubspace.
ω
tionary condition δKv(2) = 0 for an arbitrary admissible Also,letγ(1) bethelinearresponsecorrespondingtov(1).
ω,v ω
(1) (1) (1)
variation δγω at γω =γω,v. Theorem 4. — If an interactingsystem satisfies con-
Proof. — Let ξ(1) be the first-order steady state ditions (2a) and (2b), then its first-order steady state
n
8
one-matrix γ(0)(ij)+λγ(1)(ij)cosωt can be reproduced obtained for some v(1)(ij). In order for δK(2) to van-
ω ω v
by a KS system with the potential v (ij,t) = v(0)(ij)+ ish for an arbitrary admissible variation, the expression
s s
λv(1)(ij)f(t/τ)sinωt. The contribution to v(1)(ij) from in brackets in (43) must vanish for all “directions” ex-
s s
cept the diagonal and degenerate directions. Eq. (43)
the electron-electron interaction is given by
is identical to the stationary condition for the ensemble-
δK(2) weightedsecond-orderquasienergyofaKSsystem38with
w(1)(ij)= int , (41) the potential v (ij,t) = v(0)(ij)+λv(1)(ij)f(t/τ)sinωt,
δγ(1)(ji) s s s
ω if
where Ki(n2t)[γ(ω1)] is the interaction part of the second- v(s1)(ij)=v(ω1)(ij)+w(1)(ij). (44)
order quasienergy.
Proof. — The proof is analogous to the proof of the- We remark that the KS perturbation in TD 1MFT
orem2. Theexistenceofvs(ij,t)followsfromconditions must be advanced by a phase of π/2 with respect
(2a) and (2b). According to condition (2a), the ground to the given perturbation because the linear response
stateone-matrixcanbereproducedbyaKSsysteminthe λγ(1)(ij)cosωt of the KS system has a phase delay. As
ω
ensemble state ρ(s0) = nws,n|Φs,nihΦs,n|. The Φs,n the KSsystemcanbe interpretedas adoptingan ensem-
can be taken to be SlatePr determinants of N natural or- ble state, its one-matrix is governed by the equation of
bitals. To show that w(1)(ij) is the functional derivative motion
of a universal interaction functional, we first define
i∂ γˆ(t)= tˆ+vˆ (t),γˆ(t) . (45)
t s
SW(2)[γ(ω1)]= T1 tt+Tdt′ ξ(1) Tˆ+Vˆ(0)−E0(0)−i∂t′ ξ(1) , (cid:2) (cid:3)
S(2)[γ(1)]= 1 Rt+Tdt′(cid:10) w(cid:12)(cid:12) (cid:12)(cid:12) (cid:11) which gives, to first order in λ,
0 ω T t s,n
R Xn i φ(0) φ˙(1)(t) = φ(0) vˆ(1)(t) φ(0) , (46)
= 1 t+Td×t′(cid:10)ξs(1,n)f(cid:12)(cid:12)Tˆ(0)+Vˆ(0)−E0(0)−i∂t′(cid:12)(cid:12)ξs(1,n)(cid:11), for any pa(cid:10)iriof| njatura(cid:11)l or(cid:10)biital(cid:12)(cid:12)s sfor w(cid:12)(cid:12)hijch(cid:11)f(0) = f(0).
T t i i 6 j
R Xi Thus, the steady state linear response of the interacting
× φ(i1) tˆ+vˆ(0)−E0(0)−i∂t′ φ(i1) , system can be computed from the expression
where tˆand vˆ(0) are o(cid:10)ne-b(cid:12)(cid:12)ody operators and ξ((cid:12)(cid:12)1) is(cid:11)the γ(ω1)(ij,t) = fj(0) φ(i0) φ(j1)(t) +fi(0) φ(i1)(t) φ(j0)
s,n
first-order steady state nonsecular wave function that = i(f(cid:10)(0) (cid:12)(cid:12)f(0))v((cid:11)1)(ij)co(cid:10)sωt. (cid:12)(cid:12) ((cid:11)47)
would be obtained if the KS system were to start in the ω j − i s
pure state Φ . The contribution to the second-order
s,n While the linear response KS scheme does not give the
quasienergy from the electron-electron interaction is
diagonalanddegeneratecomponentsofγ(1)(ij),theycan
ω
1 t+T be obtained instead by finding the stationary point of
Ki(n2t)[γ(ω1)] = T Z dt′ ξ(1) Wˆ ξ(1) +SW(2)[γ(ω1)] Kv(2)[γω(1)].39 This is analogous to the situation in static
t (cid:10) (cid:12) (cid:12) (cid:11)
S(2)[γ(1)]. (cid:12) (cid:12) (42) 1MFT, where the occupation numbers, which are not
− 0 ω determineddirectlybytheKSequations,canbeobtained
from the minimization of the energy.28
WehavenotpartitionedK(2) intoHartreeandexchange-
int The linear response KS scheme implies the Dyson-like
correlation terms because the linear response density
equation
(1)
n , which appears in the Hartree term, cannot be ex-
ω
pressedintermsofγ(ω1) alone,foritdependsalsoondiag- χ(ij;kl;ω) = χs(ij,kl;ω)+ χs(ij,mn;ω)
onal and degenerate components of γω(1). In terms of the mXnpq
above functionals, the second-order quasienergy can be Λ(mn,pq;ω)χ(pq,kl;ω), (48)
(2) (2) (1) (1) (2) ×
writtenK =S + v (ij)γ (ji)+K . Hence,
v 0 ij ω ω int
the stationary conditioPn (theorem 3) for the interacting where Λ(ω) = δw(1)/δγ(1). Excitation energies can be
ω
system is calculatedfromthe polesofthe responsefunctionbythe
method proposed in Ref. 11. However, if the potential
δS(2) is local, it may be preferable to use (3) rather than (48)
δK(2) = 0 +v(1)(ij)+w(1)(ij) δγ(1)(ji)
v Xij hδγ(ω1)(ji) ω i ω because the single-particle eigenvalues of the DFT KS
system are often good approximations to the exact low-
= 0 (43)
lying spectrum, while the 1MFT KS system provides no
approximation at all due to its total degeneracy. If the
for an arbitrary admissible variation δγ(ω1)(ij). A varia- potentialisnonlocal,theinverseresponsefunction,which
tion δγ(1)(ij) is admissible if γ(1)(ij)+δγ(1)(ij) can be also contains information about the excitations, can be
ω ω,v ω
9
obtained from the second functional derivative of K(2), There are only two natural orbitals,
v
as seen from the following expression:
cos(θ/2)e−iψ/2
K(2) = 1 dx dx′dx dx′ γ(1)(x′,x ) φa = (cid:18) sin(θ/2)eiψ/2 (cid:19) and
v −2Z 1 1 2 2 ω 1 1
sin(θ/2)e−iψ/2
χ−1(x1,x′1;x2,x′2;ω)γω(1)(x2,x′2) φb = (cid:18) cos(θ/2)eiψ/2 (cid:19). (54)
−
+ dx dx′ v(1)(x ,x′)γ(1)(x′,x ). (49)
Z 1 1 ω 1 1 ω 1 1 As a Hermitian 2 2 matrix, the spatial one-matrix can
×
be expressed as
VI. ILLUSTRATIVE EXAMPLE γ = I +A(sinθcosψσx+sinθsinψσy +cosθσz)
= I +~γ ~σ; ~γ =(γ ,γ ,γ ); (55)
x y z
·
In the previous section, it wasfound that the time de-
where A = (f f )/2 and ~σ is the vector of Pauli ma-
pendentKSschemein1MFTdoesnotdirectlydetermine a− b
trices. It is also convenient to express the one-matrix
the linear response of the occupation numbers. To clar-
response function χ(ω) with respect to the Pauli basis,
ify this aspect of the theory, we use the KS scheme to
e.g., χ =δγ /δv . For the ground state Φ , we obtain
calculate the linear response in a simple example. xy x y 0
We consider a simple versionofthe Hubbard model.30 ω30 (x2 y2)2 0 0
The electrons are confined to a discrete lattice, each site ω2−ω320 −
of which can accommodate up to two electrons. The χ(ω)=8 0 ω2ω−2ω0220x2 iω2−ωω220xy ,
electron-electroninteractionis modeled by anon-site in-  0 −iω2−ωω220xy ω2ω−2ω0220y2 
teraction. As a further simplification, we consider that
there are only two sites and only two electrons.40 The whereωk0 =Ek E0. TheKohn-Shamresponsefunction
−
unperturbed Hamiltonian is
0 0 0
4A
Hˆ(0) = −t˜ c†1σc2σ +c†2σc1σ χs(ω)= ω  0 0 i 
Xσ (cid:0) (cid:1) 0 i 0
 − 
+ U(nˆ nˆ +nˆ nˆ )+Vˆ(0), (50)
1↑ 1↓ 2↑ 2↓ has one null vector corresponding to the null linear re-
where c† and c are the creation and annihilation op- sponse of the occupation numbers to a “diagonal” per-
iσ iσ
erators of an electron at site i with spin σ and nˆi = turbation δv|φaihφa|−δv|φbihφb|.
c† c . The first term of the Hamiltonian represents For Vˆ(0) =0, the KS response function becomes
σ iσ iσ 6
Pthekineticenergybyintroducing“hopping”betweenthe
sites with energy parameter t˜. 4A 0 icosθ −isinθsinψ
For Vˆ(0) = 0, the ground state is Φ0 = χs(ω)= ω −icosθ 0 isinθcosψ .
| i isinθsinψ isinθcosψ 0
(1/√2) yc† c† +xc† c† +xc† c† +yc† c† 0 , where  − 
1↑ 1↓ 1↑ 2↓ 2↑ 1↓ 2↑ 2↓ | i
x = co(cid:0)s(π/4 α0/2) and y = sin(π/4 (cid:1)α0/2) with This has the null vector (sinθcosψ,sinθsinψ,cosθ),
tanα0 =U/4t˜.− − which is just the unit vector with polar angle θ and az-
We are interested in the linear response to a non- imuthal angle ψ. We observe that this vector is parallel
local perturbation. For simplicity we shall consider to the~γ of the ground state, cf. (55), which implies that
only spin independent perturbations, so there will be any perturbation of the KS system, even a nonlocal per-
only spatial nonlocality. Thus, the AR perturbation is turbation, can change only the direction of ~γ and not
λVˆ(1)f(t/τ)cosωt with its magnitude. The magnitude ~γ = A is related to the
| |
Vˆ(1) = v(1)c† c difference of the occupation numbers (the sum is fixed,
ij iσ jσ f + f = 2). As noted in Section V, the occupation
Xijσ a b
numbers of the KS orbitals are not changed, to first or-
= v(1)σˆ , (51) der, by any perturbation to the KS system. This is a
α α
Xα general feature of the KS scheme in 1MFT. Neverthe-
where we have introduced Pauli operators, e.g., σˆ = less, the linear response of the occupation numbers can
x
σ(c†1σc2σ+c†2σc1σ). Thespatialone-matrixofageneral be obtained from the stationary condition δKv(2) =0.
Pstate Ψ is defined as
γ(ij)= Ψ c† c Ψ , (52)
jσ iσ VII. ON THE EXISTENCE OF fxc(ω)
Xσ (cid:10) (cid:12) (cid:12) (cid:11)
(cid:12) (cid:12)
which can be expressed in terms of the natural orbitals In this section, we explain why the invertibility of
as χ(t,t′), which is established by the RG theorem, does
γ(ij)= f φ (i)φ∗(j). (53) not imply the invertibility of χ(ω) for a pure frequency
k k k
Xk component.
10
The RG theorem implies that the inverse response been devoted to calculating χ(r,r′;ω) with the Dyson-
function χ−1(t,t′) is defined on the space (Ψ ,t ), like equation (3). This equation contains the exchange-
N t0 0
which consists of all n(1)(r,t) that can be realized for correlation kernel f (ω). In this paper, we have shown
xc
the given initial state Ψ(t0)=Ψt0 by some perturbation that fxc(r,r′;ω) = δ2Kx(2c)/(δn(ω1)(r)δn(ω1)(r′)), where
that is analytic at t = t0. In order to obtain χ−1(ω) K(2)[n(1)] is a universal functional.
from the Fourier transform of χ−1(t,t′), we must have xc ω
χ−1(t,t′) = χ−1(t t′). This will be the case only if The RG theorem establishes the existence of time de-
−
the system is in the ground state (or a stationary state) pendent KS equations, but it has not been possible to
of the unperturbed Hamiltonian at t = t0. Therefore, derive the exchange-correlationpotential froma station-
therelevantspaceis (Ψgs,t0),whereΨgs istheground ary principle. The quantum mechanical action princi-
N
state. Thetime t0 isarbitrarybutfinite. Thus,itcanbe ple does not provide a suitable stationary principle be-
shownthattheinvertibilityofχ(t t′)impliestheinvert- cause its density-functional formulation contains bound-
−
ibility of χ(ω) on the space ω(Ψgs,t0), which consists ary terms.8 For the special case of harmonic pertur-
N
of all n(1)(r,ω) that are the Fourier transform of some bations, we have found a stationary principle for the
n(1)(r,t) (Ψgs,t0). However, ω(Ψgs,t0)istoosmall quasienergy that can be used to derive the first-order
∈N N
to establishthe invertibility ofχ(ω) fora pure frequency exchange-correlationpotential.
component ω. In other words, it does not contain the
elements n(1)(r)(δ(ω Ω)+δ(ω+Ω))/2, corresponding If the external potential of a time dependent system
−
to n(1)(r,t) = n(1)(r)cosΩt. Such elements are absent is nonlocal, then it is not known whether a KS scheme
because a system in a perfect steady state with density exists in general. Although 1MFT has the scope to
n (r)+λn(1)(r)cosΩt+ (λ2) for all time is generally treat nonlocal potentials, a theorem as general as the
gs
O
neverinaninstantaneous groundstate. Therefore,there RG theorem has not been found in TD 1MFT. By ex-
is no time t at which to specify the initial condition as tending the stationary principle for the quasienergy to
0
required above. Hence, the invertibility of χ(ω) and the TD 1MFT, we have shown that there is a KS scheme
existence of f (ω) are not implied by the RG theorem. for the linear response of the natural orbitals to a har-
xc
monic perturbation. The KS system experiences an adi-
abaticallyramped perturbationof the formv (x,x′,t)=
s
VIII. CONCLUSIONS v(0)(x,x′) + λv(1)(x,x′)sinωt. The part of v(1)(x,x′)
s s s
dueto interactionscanbe calculatedfromthe functional
One of the fundamental questions that can be asked derivative of a universal functional. In contrast to the
about a quantum system is: How does it respond to a DFT KS system, the linear response of the 1MFT KS
harmonic perturbation? The first-order response of the system has a phase delay of π/2, so that the KS poten-
density to a weak perturbation is described by the lin- tial must be advanced by a phase of π/2 with respect to
ear response function χ(r,r′;ω). Considerable effort has the given external potential.
∗ Electronic address: [e-mail] 17 A. Banerjee and M. K. Harbola, Phys. Lett. A 236, 525
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