N-representability and stationarity in time-dependent density functional theory

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by  Morrel H. Cohen

N-Representability and stationarity in time-dependent density functional theory
Morrel H. Cohen1,2 and Adam Wasserman3
1Department of Physics and Astronomy, Rutgers University,
126 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA
2Department of Chemistry, Princeton University,
Washington Rd., Princeton, NJ 08544-1009, USA
5 3Department of Chemistry and Chemical Biology, Rutgers University,
0 610 Taylor Rd., Piscataway, NJ 08854-8087, USA
0 (Dated: February 2, 2008)
2
To construct an N-representable time-dependent density-functional theory, a generalization to
n
the time domain of the Levy-Lieb (LL) constrained search algorithm is required. That the action
a
is only stationary in the Dirac-Frenkel variational principle eliminates the possibility of basing the
J
search on the action itself. Instead, we use the norm of the partial functional derivative of the
0
action in the Hilbert space of the wave functions in place of the energy of the LL search. The
2
electron densities entering the formalism are N-representable, and the resulting universal action

functional has a unique stationary point in the density at that corresponding to the solution of
]
r theSchr¨odingerequation. Theoriginal Runge-Gross(RG)formulation issubsumedwithinthenew
e formalism. Concerns in the literature about the meaning of the functional derivatives and the
h
internal consistency of the RG formulation are allayed by clarifying the nature of the functional
t
o derivativesentering theformalism.
.
at PACSnumbers:
m
- I. INTRODUCTION foundations of the theory continue to take place [17].
d
n In their original work, Runge and Gross [13] employed
o the quantum-mechanical action integral (hereon the RG
Density-functional theory (DFT) now provides the
c action) to derive time-dependent Kohn-Sham equations
conceptual, theoretical, and computational framework
[ through the Dirac-Frenkel variational principle [18]. It
for the study of the ground-state properties of a vast ar-
2 rayofquantum-mechanicalsystemsatalllevelsofaggre- was later argued [19] that the RG action led to para-
v doxes when calculating response functions because these
gation from atomic to macroscopic. The foundations for
2 must be causal, whereas second functional derivatives
the contemporary theory of chemical reactivity emerge
3 of the RG action were thought to be symmetric. This
naturally from DFT as well [1, 2, 3]. The essential ele-
5
“symmetry-causality paradox” was resolved first by Ra-
5 ments of DFT are the Hohenberg-Kohn (HK) theorems
jagopal [20], who introduced an action based on the
0 [4], the Kohn-Sham (KS) equations [5], the Levy-Lieb
time path introduced by Jackiw and Kerman [21], and
4 (LL) constrained search algorithm [6, 7] which together
0 with the Harriman-Zumbach-Masche (HZM) construc- subsequently by van Leeuwen [22], who reformulated
/ TDDFT replacing the RG action by a Keldysh action
t tion [8] introduces N-representable densities into DFT;
a [23]. The RG, Jackiw-Kerman, and Keldysh actions are
accurate approximate functionals [9]; and powerful com-
m definedonly fortime-dependentv-representabledensities
putational algorithms [10].
- (TDVR),and,regardedasfunctionalsonly ofthedensity,
d As defined through the LL algorithm, the density are not stationary at the density of the solution of the
n functional E[n] has a unique global minimum at the
time-dependentSchr¨odingerequation[12,19,22,24,25].
o ground-state density within the space N of all allow-
Suchalackofstationarityisadecidedinconvenience,but
c able [7] electron densities n(r). This variational prin-
: evenwithin TDVR TDDFT stationaritycan be restored
v ciple of DFT stands in one-to-one correspondence with
byrecognizingthatthedensityandexternalpotentialcan
i the Rayleigh-Ritz variational principle for the time-
X be treated as independent functions [26]. Nevertheless,
independentSchr¨odingerequationandprovidesthesame
r for reasons analogous to those applying to the ground-
a generality to the derivation of the KS equations. state theory, it is important to generalize the definition
Following the ground-breaking HK paper, a series of the action functional to hold for time-dependent N-
of steps was taken towards a time-dependent density- representabledensities(TDNR).Moreexplicitly,Mearns
functional theory (TDDFT) [11, 12] which culminated and Kohn [27] have shown that small, time-dependent
ina moregeneralformulationby Runge andGross(RG) additions to the ground-state density need not be v-
[13]. TDDFTisnowbeingroutinelyappliedtothecalcu- representable in first order. A suitable generalization
lationof excitation energiesof atoms and molecules [14], can be effected by constructing a constrained-search al-
as well as various physical properties within the linear gorithm for TDDFT analogous to the LL algorithm for
response regime [15] and beyondit [16] (see ref.[17] for a DFT.
survey of recent applications). Apart from restoring stationarity to the action in
In parallel to this success, discussions regarding the TDDFT, N-representability is important because, as in
2
DFT, accurate solution of the KS equations requires it- Hamiltonian becomes
eration to self-consistency. The most convenient start-
Hˆ[v]=Tˆ+Wˆ +Vˆ[v]=Hˆ +Vˆ[v]. (1)
ing densities may well not be v-representable, nor may
the densities be at intermediate stages of the computa-
In Eq.(1), Tˆ is the electron kinetic-energy operator and
tional algorithms. It is then essential to have an action
Wˆ theelectron-electroninteractionoperator. Theopera-
functionalandKSpotentialsdefinedforN-representable
torVˆ[v] is the energyofinteractionofthe electronswith
densities both as a matter of principle and for practical
a time-dependent external potential v(r,t),
reasons.
InthispaperweformulateanN-representableTDDFT
basedontheDirac-Frenkelvariationalprincipleinwhich Vˆ[v]= dr v(r,t)nˆ(r) . (2)
Z
the RG action functional is stationary with respect to n
at thatunique n derivable fromthe solution ofthe time- In Eq.(2), nˆ(r) is the electron-densityoperator. v(r,t) is
dependent Schr¨odinger equation. We establish a one- comprised of the potential energy of an electron in the
to-one invertible map between all densities in a time- fixed nuclear electrostatic potential plus that in a time-
dependent generalization of N and wave functions by dependentpotentialgeneratedbysourcesexternaltothe
useofthenormofthepartialfunctionalderivative[28]of system. For each time t in the interval (t ,t ) under
0 1
theRGactionintheHilbertspaceofthewavefunctions. consideration,the r-dependence of v(r,t) must meet the
Insertion of that map into the action defines the action conditions imposed by Lieb [7]. In addition, we impose
functional. The Runge-Gross formulation of TDVR is the requirement that
subsumed within this TDNR TDDFT, and the desired
stationarity and generality are achieved. v(r,t)→0 , r↑∞ , ∀ t ∈(t0,t1) (3)
In Section II, we begin by reviewing two topics cen-
toeliminateirrelevantphasefactorsinthewavefunctions
tral to our later developments, the Dirac-Frenkel varia-
(see also ref.[31]). The time dependence of v must meet
tional principle and the action and its total and partial
certain implicit integrability conditions discussed below.
derivatives. Via Section II we introduce our notation for
Such acceptable potentials lie in the space V. The space
wavefunctions,operators,functionalderivatives,Hilbert R3 ×(t ,t ) is the support on which the elements v of
spaces,and more generalfunction spaces. We alsointro- 0 1
V are defined. As indicated by our notation, Vˆ[v] is a
duce the notion of mapping between abstract spaces as
linear functional of v, Eq.(2), and so, consequently, is
central to the formulation of TDDFT, following Dreizler
Hˆ[v], Eq.(1).
and Gross for DFT [29]. In Section III, we recapitulate
The wave-functions Φ(t) of the N-electron system are
theRGformulationofv-representableTDDFTandshow
time-dependent, normalized, antisymmetric functions of
explicitly that its unnecessary limitation to the density
the N space and spin coordinates of the electrons,
generated by that v which enters the Hamiltonian de-
stroys the stationarity of the action functional. We also
||Φ(t)||=(Φ(t),Φ(t))=1 , ∀ t∈(t ,t ) . (4)
0 1
provide an explicit explanation of why there are no in-
consistencies in the functional derivatives entering the They satisfy the time-dependent Schr¨odinger equation
theory and why second functional derivatives of the RG (atomic units are used throughout),
actionwithrespecttothedensityarenotsymmetric. Up
tothispoint,ourpaperhasconcerneditselfwiththeclar- i∂tΦ(t)=Hˆ[v]Φ(t) . (5)
ification of existing work on v-representable TDDFT. In
Once the initial condition
Section IV, we turn to the problemof establishing a sat-
isfactory N-representable TDDFT. We begin by stating
Φ(t )=Φ (6)
0 0
a set of criteria that such a theory must meet. Next, we
review existing proposals ([26],[30]) and show that they is imposed, Φ(t) is unique,
do not meet all of the criteria. Finally, we develop the
t
principal result of this paper, a constrained search algo- Φ(t)=T exp −i dt′Hˆ[v] Φ . (7)
rithm which meets all of the criteria. We close with a L (cid:20) Z (cid:21) 0
t0
brief summary of our results in Section V.
In Eq.(7) T is the time-ordering operator, later to the
L
left. Eq.(7) defines implicitly the conditions which v(t),
II. BACKGROUND AND NOTATION Φ(t), and Φ0 must meet. In addition to those conditions
which were specified by Lieb [7], Φ(t) must be differen-
tiable in time. The set of such functions which are so-
The Dirac-Frenkel variational principle
lutions of Eq.(5) for all v in V form a Hilbert space Φ.
They are supported in Φ on the space τ, which is the
Consider a finite system of electrons and nuclei con-
product of (t ,t ) with the configuration and spin space
0 1
taining N electrons. Ignoring nuclear kinetic energy, S of the N-electrons,
keeping the nuclei fixed, and discarding the internuclear
interaction energy as an irrelevant constant, the system τ =S×(t ,t ) . (8)
0 1
3
All scalarproducts like the norm entering Eq.(4) are de- todefineFr´echetandGˆateauxderivatives[28]-[32],which
fined on S. restrict the functions on which they are defined to a sin-
Eqs.(5) plus (6) implicitly, and (7) explicitly, define a gle Banachspace (in the caseof a Fr´echetderivative),or
mapping M : V → Φ. M is surjective; Φ contains no normedspace (in the case of a Gˆateaux derivative). The
1 1
elementwhichisnotassociatedwithanelementofV [33]. Gˆateauxderivativehasbeenregardedasageneralization
ThatM isinjectiveaswell,i.e. one-to-oneandtherefore ofthe conceptofthe partialderivativeofafunction [28].
1
bijective or invertible, can be seen as follows. Suppose Similarly, the Fr´echet derivative has been regarded as a
there is a v′ and therefore a Vˆ′ which yields the same Φ generalization of a total derivative [28]. In our case, the
as solution of Eq.(5) as does v and Vˆ. Subtracting the properties of the action are such that taking derivatives
two Schr¨odinger equations leads to only with respect to n meets the criteria for a Fr´echet
derivative despite the fact that it is a partial functional
Vˆ′−Vˆ Φ= dr[v′(r,t)−v(r,t)]nˆ(r)Φ(t)=0 , derivative. In the following we shall use the terminology
(cid:16) (cid:17) Z partial functional derivative to refer to derivatives with
(9)
respect to a single function of functionals of more than
′
which implies that v must equal v under the conditions
one function. When, however, we map the potential in
onther-dependenceofvrequiredfortheanalogousproof
the action back to the density or vice versa so that the
for the time-independent problem (cf ref.[7] and p.5 of
actionbecomes a functional only of a single function, we
ref.[29]). Thus M−1 exists and Φ ↔V is one-to-one. Φ
1 shallrefertothefunctionalderivativetakenwithrespect
can then be regarded as a functional of v, Φ[v], or v one
to that single function as a total functional derivative.
of Φ, v[Φ].
Thetotaldifferentialcouldthenberepresentedasalinear
Let us now expand the Hilbert space Φ to Ψ which
combination of partial functional derivatives times the
contains all functions Ψ which meet the conditions im-
corresponding differentials of the respective functions.
posed on Φ including Ψ(t ) = Φ , except that the Ψ
0 0
need not satisfy Eq.(5). The Dirac-Frenkel variational
principle states that Ψ satisfies Eq.(5) if and only if III. V-REPRESENTABLE TDDFT
δΨ, i∂ −Hˆ[v] Ψ =0 , (10a)
t 1. v-representability and stationarity of the RG action
(cid:16) h i (cid:17)
WecanrecasttheargumentsofRG[13]asfollows. Re-
(δΨ(t),Ψ(t))=0 , ∀t . (10b) stricttheargumentΨofA[Ψ,v]inEq.(11)tolieinΦ,the
′
spaceofv-representablewavefunctionsΦ[v ],definingan
action functional,
t1
A[Φ,v]= dt Φ, i∂ −Hˆ[v] Φ (14)
The action and its total and partial functional derivatives Zt0 (cid:16) h t i (cid:17)
on the space Φ×V. Stationarity of A[Ψ,v] implies sta-
We can now define the usual quantum-mechanical ac-
tion functional A[Ψ,v] on the space Ψ×V, tionarityofA[Φ,v],i.e. thatitspartialfunctionalderiva-
tive vanishes,
t1
A[Ψ,v]= dt Ψ, i∂t−Hˆ[v] Ψ . (11) ∂Φ∗A[Φ,v]=0 . (15)
Zt0 (cid:16) h i (cid:17)
since Φ⊂Ψ and the stationary point of A[Ψ,v] is in Φ.
A[Ψ,v] is stationary only at Φ[v] in Ψ with respect
A[Φ,v]canbeestablishedasafunctionalofv aloneby
to variations δΨ, δΨ∗ taken at constant v, given that inserting in A[Φ,v] that Φ[v′] for which v′ =v,
Ψ(t )=Φ ∀Ψ∈Ψ, and requiring as well that [34]
0 0
A[v]=A[Φ[v],v] . (16)
(δΨ(t ),Ψ(t ))=0 , (12)
1 1
The stationarity condition (15) then implies that the to-
a lesssevererestrictionthanthat ofEq.(10b). The func- tal functional derivative of A[v] is −n,
∗
tional gradient of A[Ψ,v] along Ψ (v is fixed),
δv(r,t)A[v]=δv(r,t)A[Φ[v],v]=−n(r,t) , (17)
ΘΨ∗ =∂Ψ∗A[Ψ,v]= i∂t−Hˆ[v] Ψ , (13) a generalizationof the Hellmann-Feynman theorem [35].
h i
The total functional derivative (17) does not vanish, ob-
thus vanishes in Ψ at Φ[v], yielding the time-dependent
viously. It is only the partial functional derivative (15)
Schr¨odingerequation. NotetheuseinEq.(13)of∂Ψ∗ asa which yields stationarity [26].
symbolforapartialfunctionalderivative. Wenowclarify
To go on to the density-functional, A[n,v], requires
thenatureofsuchaderivative. Theactionisafunctional establishing that the map M′ :Φ→N ,
2 v
of two functions defined in two different spaces. Accord-
ingly, it does not fit simple examples of functionals used n(r,t)=(Φ(t),nˆ(r)Φ(t)) , (18)
4
is one-to-one and invertible. In Eq.(18), n(r,t) is the v (r,t), starting from a single determinantal state Φ
s 0s
time-dependent electron density, and the symbol N at t . v is a functional of their electron density, v [n ].
v 0 s s s
stands for the subset of all such v-representable densi- TheHZMconstruction[8]allowsidentificationofn (r,t)
s
ties contained in N, the time-dependent generalization with the density of the interacting system,
ofthe space ofdensities ofDFT [7]. Alln(r,t) inN and
Nv obey the initial condition ns(r,t)≡n(r,t) , ∀r,t∈(t0,t1) . (23)
n(r,t0)=n0(r)=(Φ0,nˆ(r)Φ0) . (19) Thusvs canberegardedasafunctionalofn. Combining
Eq.(18) defines what is meant by the phrase TDVR; a
TDVR density is derivable via Eq.(18) from the solu- δn(r,t)Bs[n]=vs(r,t) (24)
tion Φ of the Schr¨odinger equation (5) for some v in
V. Demonstrating the invertibility of M′ directly, how- with Eq.(22) leads to
2
ever,is nontrivial. The HZMconstruction[8]showsthat
Ψ→N is many to one. vs(r,t)=v(r,t)−δn(r,t)(A−As) . (25)
RG followed an alternative path. Substituting Eq.(7)
into Eq.(18) defines a map M = M M′ : V → N . The usual rearrangementsin A−As in turn lead to
3 1 2 v
They then show by a pretty argument that M is one-
3 v (r,t)=v(r,t)+v (r,t)+v (r,t) , (26)
to-oneandinvertibleforallpotentialsv(t)whichpossess s H XC
a Taylor expansion in time about t converging for all
0
t ∈ (t ,t ). Van Leeuwen [25] has pointed out that it is
0 1
sufficientforaTaylorseriestoexistaboutasetofpoints n(r′,t)
′
ti ∈ (t0,t1) for which the radii of convergence overlap vH(r,t)=Z dr |r−r′| , (27)
to cover (t ,t ). M′−1 : N → Φ can then be con-
structed as0M1−1M .2Substituvtion of M′−1, that is Φ[n]
3 1 2
into A[Φ,v], then yields the desired functional A[n,v].
A[n,v]isstationarywithrespecttovariationofnatfixed vXC(r,t)=−δn(r,t)AXC[n] , (28)
v [26], that is its partial functional derivative vanishes,
t1
∂ A[n,v]=0 . (20)
n A [n]= dt{[(Ψ,i∂ Ψ)−(Ψ ,i∂ Ψ )] (29)
XC t s t s
Z
t0
It is important to recognize that A[n,v] is defined for all −[(T −T )−(W −W )]} . (30)
′ s H
ngeneratedviaEqs.(7)and(18)fromsomev ,whichcan
be variedindependently ofv. Itis only atthe stationary T is the kinetic energy and W the energy of electron-
′
point that v =v. electron interaction of the interacting electrons in state
Φ[n]. T is the kinetic energy of the noninteracting elec-
s
tronsinstateΦ [n]. W istheHartreeapproximationto
s H
2. Time-dependent Kohn-Sham equations W using Φ[n] or equivalently Φ [n].
s
It is at this point that concern about the meaning of
Substitution of both M′−1 and M−1, i.e. Φ[n] and the functional derivative defining v , Eq.(28), arises in
2 3 XC
v[n], into A[Φ,v], then yields a functional A[n] of n only the literature [12, 19, 22, 24, 25]. Since in Section IV we
(for a given initial state [36]), the RG action functional. shall base our development of N-representable TDDFT
One thus has the option of using n or v as the indepen- ontheRGactionandsincetheabove-mentionedconcern
dentvariableinthe functional. VanLeeuwen[25]givesa raises doubts about the validity of doing this, we now
simple and elegant argument for the construction of the summarize the debate and show why the RG action is
TDKSequationsfromA[n]withoutinvokingstationarity perfectly suitable for the developments of Section IV.
in N . Switching now to A[n] from A[v], we carry out a
v
Legendre transformationto
3. The symmetry-causality dilemma
t1
B[n]=A[n]+ dt dr v(r,t;[n])n(r,t) . (21)
Zt0 Z Taking the functional derivative of Eq.(26) with re-
spect to n results in [37]
From Eq.(21), it follows that
χ−1(r,t;r′,t′)=χ−1(r,t;r′,t′)+f(r,t;r′,t′) , (31)
s
δn(r,t)B[n]=v(r,t). (22)
where
The TDKS equations [25] follow from (22).
Consider a system of non-interacting electrons de- ′ ′ δn(r,t)
χ(r,t;r,t)=− , (32)
notedbysubscriptswhichmoveinanexternalpotential δv(r′,t′)
5
that the causality of χ−1 imposes a symmetry-causality
dilemmaviaEq.(31)isthatofGross,Dobson,andPeter-
δn(r,t)
′ ′
χ (r,t;r,t)=− , (33) silka [19]. The flaw in their reasoning is the supposition
s δv (r′,t′)
s that Schwarz’s lemma can be applied to the functionals
of TDDFT. Throughout all of density-functional theory,
δ[v +v ](r,t) δ(t−t′) δv (r,t) theFr´echetdefinition[28]ofthefunctionalderivativewas
f(r,t;r′,t′)= H XC = + XC .
δn(r′,t′) |r−r′| δn(r′,t′) implicitly used. In the present instance, the functional
(34) derivatives of Eqs.(20),(22),(24) and (25) are all Fr´echet
From Eqs.(18) and (7), the well known retarded char- derivatives. Takingasecondderivative simply involvesa
acter of the time dependence of the susceptibilities χ singleiterationoftheFr´echetoperation[28]. Forthefirst
and χ follows; they vanish if t′ > t. Their inverses derivative to exist, both the functional and the function
s
χ−1(r,t;r′,t′)andχ−1(r,t;r′,t′)enteringEq.(31)arere- space must meet smoothness criteria. The first deriva-
s
tarded as well. Yet tiveremainsafunctional,whichforthesecondderivative
to exist, must remain smooth. This condition is implic-
δv (r,t) δ2A [n]
fXC(r,t;r′,t′)= δnX(Cr′,t′) =−δn(r′,t′X)Cδn(r,t) (35) iwtleyparsessuummeeditfohrevreXCa[ns]wienlla.llWoef DcoFnTcluadnedtThaDtDaFllTo,fatnhde
secondfunctionalderivativesencounteredinTDDFTare
is formally a second derivative of A [n] according to
XC perfectly well defined iterations of the Fr´echetderivative
Eq.(28). Van Leeuwen [22, 25] assumes that, as f is
XC operation. These include
a second functional derivative, it must be symmetric in
r,t andr′,t′. Suchsymmetryis inconsistentwiththe re- δ2A[v]
tarded nature of χ−1 and χ−1 in Eq.(31). Van Leeuwen χ(r,t;r′,t′)= (36)
s δv(r′,t′)δv(r,t)
[22, 25] describes this inconsistency as a “paradox” and
develops TDVR TDDFT from the Keldysh action in-
and χ as well as f . All have a retarded dependence
s XC
steadoftheRGactiontoavoidit. Thesecondfunctional on t and t′ and are decidedly not symmetric in r,t and
derivatives remain symmetric on the Keldysh time con- r′,t′. Similarly,χ−1 andχ−1 canbe expressedas second
s
tour but become retarded when mapped into real time.
derivatives, e.g.
Gross,Dobson, andPetersilka[19], onthe otherhand,
sbueprpeotsaerdtehdataEnqd.(n3o1t)shyomldmseatnrdictihnatr,ftXCa(nrd,t;rr′,′,t′t.′)Tmhuesyt χ−1(r,t;r′,t′)=−δδnv((rr′,,tt)′) = δn(r′δ,2tB′)[δnn](r,t) , (37)
then conclude by supposing from Schwarz’s lemma [39]
that (1) fXC(r,t;r′,t′) cannot be a second functional have retarded time dependence (Appendix A), and are
derivative andthat (2) the exactvXC[n]cannottherefore not symmetric in r,t and r′,t′.
be a functional derivative. They conclude further that We therefore agree with the main conlcusion of Amu-
thisinturnisincontradictiontotheprincipleofstation- sia and Shaginyan [40],[41] and Harbola and Banerjee
ary action which leads to vXC as a functional derivative. [38],[42] that there is no conflict between the symmetry
To complicate matters further, Harbola and Baner- and the causality. However, the way out of the dilemma
jee [38] have argued that there is no symmetry-causality is not by finding symmetry in the inverse response func-
dilemma because while χ is causal, χ−1 is symmetric. tions, but by recognizing that second-functional deriva-
Amusia and Shaginyan [41], while not disagreeing with tives need not be symmetric functions of the time vari-
this conclusion, have argued that, in contrast, it is pos- ables. To understand how this asymmetry can come
sible to construct a causal χ−1 as well. Harbola [42] has about in a second functional derivative, consider that
responded that reference [41] itself implies a causality in functionalsaredefinedonthreelevels. First, thereis the
thepotentialasafunctionalofthedensity. vanLeeuwen, spaceonwhichthefunctionsaredefined;second,thereis
however, has argued that χ−1 must be rigorously causal the function space on which the functionals are defined;
in analogy with the properties of discrete lower triangu- and third, there is the definition of the functional. For
lar matrices [25], an argument which does not take into example, A[v] is defined through Eq.(11) and the map
account the fact that χ−1 is not a smooth function of M ,onthe functionspaceV within whichthe potentials
1
t − t′ but contains both a delta function and the sec- v are supportedon R3×(t ,t ). For the total functional
0 1
ond derivative of a delta function at t = t′+. We show derivative δA[v]/δv(r,t) to exist andequal −n(r,t), first
in Appendix A that χ−1(t−t′) is causal, consisting of V must be smooth enough that variations δv(r,t) exist
those singular functions at t = t′+ plus a smooth causal which can be taken continuously to zero. Following Lieb
function of (t−t′), so that the dilemma remains. [7],wehavedefinedV forthistobethecase. Second,the
functionalA[v]mustbesmoothenoughthattheresulting
variationinit,δA[v],exists,islinearinδv(r,t), andgoes
4. A way out of the dilemma continuously to zero with δv(r,t). A[v] meets that crite-
rion. The functional derivative δA[v]/δv(r,t) is then de-
We conclude that in the context of TDDFT at the fined through Fr´echet’s theory of linear functionals [28].
RG level, χ−1 is causal. The most forceful argument Similarly, for δn(r,t)/δv(r′,t′)= δ2A[v]/δv(r′,t′)δv(r,t)
6
to exist, n(r,t) need only meet the smoothness criterion Previous work
as a functional of v, which it does throughthe definition
of Nv. Apart from formulations applicable to special classes
The requirement for the applicability of Schwarz’s ofpotentials[43],therearetwoproposalsfortheformula-
lemma, that the second derivative be invariant with re- tionof NR TDDFT. That ofKohlandDreizler [30] does
specttointerchangeoftheorderofdifferentiation,isthat notmeetcriterion2.) and,asaconsequence,cannotmeet
the first level of support, the space on which the func- criterion4.) as well. Thatof GhoshandDhara [26] does
tion is defined, be unchanged by the first functional dif- not produce N-representablility, only v-representability.
ferentiation. That is not the case here, and Schwarz’s Their Theorem 4 can be restated as defining the map
lemma does not apply. In A[v], v is supported on MGD :N →Ψ,
R3 ×(t ,t ), but in n[v], the first derivative, v is sup-
0 1
ported on R3 × (t0,t) precluding the applicability of Ψ[n]=ARG{STATΨ→nB[Ψ]} . (39)
Schwarz’s lemma (see also the discussion in Appendix
B).Ift′ >tinδ2A[v]/δv(r′,t′)δv(r,t),itmustvanish,de- However, since in Eq.(39) one searches only for a sta-
tionary point of B[Ψ] in Ψ, one is allowed to relax the
stroyingsymmetrywhileremainingawell-definedsecond
subsidiary condition (38) by a Legendre transformation.
functional derivative [28]. In the Keldysh action func-
Eq.(39) then becomes
tional used by van Leeuwen [22, 25], the time-ordered
contour on which the action is defined provides the sup-
t1
port for the time-dependence of the potential v. Func- Ψ[n]=ARG STAT B[Ψ]− dt dr Λ(r,t)n(r,t)
Ψ
tionaldifferentiationoftheKeldyshactiondoesnotmod- (cid:26) (cid:26) Zt0 Z (cid:27)(cid:27)
ify this support, and so the second functional derivative (40)
remainssymmetricinthatsupport. Transformationfrom Now,theLagrangemultiplierΛ(r,t)willexistifandonly
theKeldyshtimecontourbacktorealtimeintroducesthe if n(r,t) is v-representable, in which case Eq.(40) be-
asymmetrywithoutchangingthefactthatasecondfunc- comes
tional derivative was taken. Thus, van Leeuwen has, in
Ψ[n]=ARG{STAT A[Ψ,Λ]} (41)
effect,provedthatsecondfunctionalderivativesneednot Ψ
be symmetric. We conclude that all functional deriva-
with Λ ⊂ V, a potential. Thus, the map defined by
tives in the RG formulation are well defined, both first Eq.(39) is identical to the map M′−1 :N →Φ defined
and second, and that Eq.(31), a relation among second 2 v
by RG. Eqs.(39)-(41) should therefore be rewritten with
functional derivatives, contains no inconsistencies. One
Φ[n] replacing Ψ[n]. What Ghosh and Dhara have ac-
thus has a choice - one can base TDVR TDDFT on the
tually accomplished is to find a simpler and more direct
RG action or on the Keldysh action. How to general-
proofofv-representabilitythanthe originalproofofRG.
izetheKeldyshactionsoastoprovideabasisforTDNR
TDDFTisnotnowclear. Accordingly,wechoosetobase
our development of TDNR TDDFT on the RG action.
Our approach
We note that the stationary point Φ[n] of B[Ψ] in
Eq.(39) is unique in the subspace Φ (Ψ → n) of Ψ
n
IV. N-REPRESENTABLE TDDFT for n⊂N . The partialfunctionalderivative ofA[Ψ,v],
v
its gradient in Φ , vanishes uniquely there,
n
TheHZMconstruction[8]establishesthatateachtime
t, there is an infinite set of wave functions Ψ(t) which ∂Ψ∗A[Ψ,v])v,n = ∂Ψ∗B[Ψ])n = i∂t−Hˆ Ψ
h i
yield any preset n(r,t) in N via the mapping M2 :Ψ→ = 0 ; n⊂Nv,Ψ=Φ[n] . (42)
N,
Thus the magnitude squared of the gradient,
n(r,t)=(Ψ(t),nˆ(r)Ψ(t)) , (38) t1 t1
Z dt(∂Ψ∗A,∂Ψ∗A)v,n =Z dt(∂Ψ∗B,∂Ψ∗B)n (43a)
t0 t0
with n(r,t ) = n (r), Eq.(19). The task in constructing
0 0
an N-representableTDDFT is to select a single member has a unique minimum there as well. As the search can
ofthatsetsothatM becomesone-to-oneandinvertible, berestrictedtonormalizedΨ’swithoutpenalty,itfollows
2
i.e. tofindM−1 :N →Ψ. M−1 shouldmeetthefollow- from eq.(43a) that
2 2
ing four criteria. 1.) It shouldbe universal;2.) it should
irteqsuhiorueldsesaurbchsuinmgeotnhleyminapΨpinagndM2n′−o1toinf vΨ-repanredseVnt;ab3l.e) Zt0t1dt(∂Ψ∗A,∂Ψ∗A)v,n =Zt0t1dt(cid:16)Ψ,[i∂t−Hˆ]2Ψ(cid:17)
TDDFT; and 4.) it should provide a stationarity princi- (43b)
ple. holdsaswellbecauseoftheconsequenthermiticityofi∂ .
t
7
On the other hand, no such minimum can exist in the V. SUMMARY
magnitudeofthegradientforanN-representablen⊂N
whichisnotv-representable. Asimilarsituationexistsin
AnN-representabletime-dependentDFThasbeenes-
time-independent DFT. A minimum exists in the func-
tablished,andatime-dependentanalogofthe Levy-Lieb
tional E[Ψ] = Ψ,HˆΨ for Ψ → n if and only if n is
constrained search algorithm has been proposed. The
(cid:16) (cid:17)
v-representable. If we suppose that a minimum exists central quantity in this search is the norm of the par-
under the constraintoffixedn forn notv-representable, tial functional derivative of the Runge-Gross action in
the constant can be eliminated by a Legendre transfor- the Hilbert space of wavefunctions. The proposed con-
mation. TheLagrangemultiplierthensimplyaddstothe strained searchmeets all of the requirements we pose: it
external potential contradicting the hypothesis that n is isuniversal,requiressearchingonlyinoneHilbertspace,
not v-representable as in the arguments associated with subsumesRunge-Grossv-representability,andprovidesa
Eqs.(39)-(41). For a general N-representable n, there is stationarity principle.
only an infimum in both (Ψ,HˆΨ) and (Ψ,HˆΨ) at the
samepoint. TheLevy-Liebconstrainedsearchalgorithm
makesuseofthisinfimumtodefinethedensityfunctional
for N-representable densities:
APPENDIX A: CAUSALITY OF χ−s1 AND χ−1
E[n]=INFΨ→nE[Ψ] ; (44a)
As statedinSectionIII.3 andIII.4, thereis asubstan-
Ψ[n]=ARG{INFΨ→nE[Ψ]} . (44b) tialspreadofopinioninthe literaturewithregardto the
time dependence ofχ−1 andχ−1,differingastowhether
s
Ananalogousconstrainedsearchalgorithmcanbecon-
it is causal or symmetric. We argue here that it is un-
structedforTDDFT fromthe magnitude ofthe gradient equivocally causal, with local singularities at t = t′+.
[44]:
It is easiest to see this explicitly for the χ of the uni-
s
t1 form electron gas, which has the form χs(|r−r′|,t−t′),
Ψ[n] = ARG(cid:26)INFΨ→nZ dt(∂Ψ∗A,∂Ψ∗A)v,n(cid:27) from space-time uniformity. Accordingly, it is diagonal-
t0 ized by Fourier transforming on space and time yielding
t1 the eigenvalues χ (q,ω − iδ), δ ↓ 0. The wave-vector
= ARG(cid:26)INFΨ→nZ dt(∂Ψ∗B,∂Ψ∗B)n(cid:27)(45a) q is introduced bys the Fourier transform on r−r′, the
t0 ′
frequency ω is introduced by that on t−t, and δ is in-
t1
= ARG INFΨ→n dt Ψ,[i∂t−Hˆ]2Ψ ; troduced by the causality of χs, χs(|r−r′|,t−t′) = 0,
(cid:26) Zt0 (cid:16) (cid:17)(cid:27) t>t′+.
Theexplicitformofχ (q,ω−iδ)isknown[47]. When
s
t1
A[n,v]= dt Ψ[n],[i∂ −Hˆ[v]]Ψ[n] . (45b) continued to the entire complex angularfrequency plane
t
Zt0 (cid:16) (cid:17) intheprocessofinvertingtheFouriertransformontime,
its only singularities are a second-order pole at infinity
We note that the quantity over which the search in
and bounded branch cuts just above the real axis. For
Eq.(43a) is done corresponds to the time-integral of the
q < 2k (k is the Fermi wavenumber), there is one
McLachlan functional for Hˆ [45] widely used in formu- F F ~
boundedbranchcutatz =ω+iδ,with|ω|≤ (2k q+
lations of semiclassical dynamics [46]. The proposed q2); for q > 2k there are two, with ~ (−2k2mq+qF2) ≤
constrained searchalgorithm expressed in Eqs.(45a-45b) F 2m F
|ω| ≤ ~ (2k q+q2). χ (q,ω) has no zeros away from
meets all of the criteria imposed above: 1.) It is univer- 2m F s
the branch cuts.
sal, not involving v. 2.) It requires searching only in Ψ.
3.) It subsumes v-representablen for which the infimum χ−1 is also uniform in space and time and therefore
s
becomes a minimum and yields the condition diagonalized by Fourier transformation. It’s eigenvalues
are simply
2
i∂ −Hˆ Ψ=0 s.t. Ψ→n , (46)
t
(cid:16) (cid:17)
whichyieldsthesameΨasEq.(39)or,ultimatelyEq.(5). χ−s1(q,ω)=1/χs(q,ω) . (A1)
Finally, 4.) it provides a stationarity principle since n
in (45b) can be varied independently of v, and the
The second-order pole in χ (q,ω) at ω = ∞ yields the
corresponding partial functional derivative vanishes via following behavior in χ−1(qs,ω) at ∞,
Eq.(13), s
∂ A[n,v]=0 . (47)
n C (q)
χ−1(q,ω) → A (q)ω2+B (q)+ s +... (A2)
Stationarity in TDDFT plays the role that minimality s s s ω2
|ω|→∞
does in DFT regarding error reduction.
δ →0
8
A (q) = − m where χˆ−1(t−t′)′ is nonlocal and causal in time.
s nq2
2E 5~2q2
B (q) = F 3+
s
5 n (cid:20) 8mE (cid:21)
F
APPENDIX B: SECOND-FUNCTIONAL
16E2q2 ~2q2
C (q) = F 3+35 DERIVATIVE ASYMMETRY
s
175nm (cid:20) 8mE (cid:21)
F
where n is the number of electrons per unit volume and
In section III below Eq.(37), we have pointed to the
m is the electron mass. Thus χ−s1 has the form modification of the support of v(r,t) in n[v] by the first
χ−1(q,t−t′)=−A (q)δ′′((t−t′)+)+B (q)δ((t−t′)+) functional derivative of A[v] with respect to v as the ori-
s s s
+χ−1(q,t−t′)′(A.3) gin of the asymmetry of its second derivative, χ. Al-
s ternatively, one can preserve the support on which the
′′
In Eq.(A3), δ is the secondderivative of the delta func- function v(r,t) is defined, but then the second level of
tion. (χ−1)′ arises from the branch cut(s) and is rigor- definition, the function space on which the functional is
s
ously causal because the locations of the branch cuts in defined, must change. Consider, for example, the map
χs−1(q,z) are identical to those of χs(q,z), being in the M1 : V → Φ, Eq.(7), which, together with Eq.(18) de-
upper-half z-plane. fines the map M : V → N , n = n[v]. A more explicit
3 v
The principal change in passing from χs(q,ω) to expression of that map would be
χ(q,ω) for the uniform electron gas is that the free-
particle excitations are replaced by quasi-particle exci- n=n[Hˆ[v]] , (B1)
tations which have finite lifetime except at q = 0. This
causes the branch cuts to extend to infinity, but causes
no change in the formal structure of χ−1(q,t−t′) which according to Eq.(7), in which v(r,t) is supported on
(t ,t). However, Eq.(7) can be rewritten as
isgivenby (A3)withmodificationofA (q) to A(q), etc. 0
s
For a non-uniform extended system for which the ex-
citationspectrumformscontinua,be the systemordered Φ(t)=T exp −i t1dt′H˜ [v] Φ , (B2)
or disordered, there is no change in formal structure of L (cid:20) Z t (cid:21) 0
χ−1(r,r′;t − t′) and χ−1(r,r′;t − t′). Each contains t0
s
the local contributions δ′′((t − t′)+) and δ((t − t′)+)
where
as well as non-local retarded contributions. For finite
systems which have at least one discrete excitation as- H˜ [v] = Hˆ[v] , t′ ∈(t ,t);
sociated with a transition from the ground state to a t 0
′
bound excited state, there is a change. Each eigenvalue = 0 , t ∈ t,t1) . (B3)
of χ (r,r′;ω) or χ(r,r′;ω) switches from +∞ to −∞ as
s
the pole at z = ω +iδ with ~ω equal to that discrete Thus,bychangingtheoperatorspaceonwhichtheargu-
excitation energy is crossed. This forces the existence mentofthefunctional,nowtheoperatorH˜ [v],isdefined,
t
ofa zerobetweendiscrete excitationenergiesorbetween we have formally restored the support of v to (t ,t ).
0 1
thehighestdiscreteexcitationenergyandthecontinuum However,thatdoesnoteliminatetheasymmetry;ittriv-
threshold. Each such zero gives rise to a pole in the cor- ially shifts the location of its origin, viz
responding eigenvalue of χ−1(r,r′;z) or χ−1(r,r′;z) at
s
the same z. Upon Fourier transformto the time-domain δn(r,t) δn(r,t) δHˆ
χ−1(r,r′;t−t′)andχ−1(r,r′;t−t′)eachcontainsacausal = t =0 , t′ >t . (B4)
s δv(r′,t′) δHˆ δv(r′,t′)
contribution from the pole which oscillates with angu- t
lar frequency corresponding to the excitation energy for
′ ′
t≥t and vanishes for t >t.
In conclusion, the causality of χ(r,r′;t − t′) and
χ (r,r′;t − t′) forces the eigenvalues of χ(r,r′;ω) and
s
χ (r,r′;ω) to have singularities only in the upper-half ACKNOWLEDGMENTS
s
complex-frequency plane. The corresponding eigenval-
ues of χ−1(r,r′;ω) and χ−1(r,r′;ω) can therefore also
s We are grateful to Kieron Burke for critical and stim-
havesingularitiesonlyintheupper-halfplaneapartfrom
ulatingdiscussionsinresponsetowhichAppendix Bwas
the second-order pole at ∞. This arises from the fact
added to the paper. We thank Neepa Maitra for call-
that ω enters χ and χ only in the combination ω−iδ,
s ing relevant references to our attention. We also thank
δ ↓ 0, which does not change when their eigenvalues are
A.K. Rajagopal for bringing his pioneering work on the
inverted to obtain χ−1 and χ−1. Those quantities must
s symmetry-causality paradox to our attention. This re-
therefore always be of the form
search was supported in part by ONR grant N00014-01-
χˆ−1(t−t′)=Aˆδ′′((t−t′)+)+Bˆδ((t−t′)+)+χˆ−1(t−t′)′ , 1-0365,ONR/DARPAgrantN00014-01-1-1061andNSF
(A4) grant no. CHE-9875091.
9
[1] R. Parr and W. Yang, Density Functional Theory [21] R.Jackiw andA.Kerman,Phys.Lett. 71A,158(1979).
of Atoms and Molecules, Oxford University Press, [22] R. van Leeuwen, Phys.Rev.Lett.80, 1280 (1998).
NewYork (1989). [23] L.V. Keldysh, Sov.Phys. JETP 20, 1018 (1965).
[2] P. Geerlings, F. De Proft, and W. Langenaeker, Chem. [24] K.BurkeandE.K.U.Gross,inDensityfunctionals: The-
Rev.103, 1793 (2003). ory and applications, ed. D. Joubert, Springer, Berlin
[3] M.H.CohenandA.Wasserman,IsraelJ.Chem.43,219 (1998).
(2003). [25] R. van Leeuwen, Int.J. Mod. Phys.B15, 1969 (2001).
[4] P. Hohenberg and W. Kohn, Phys. Rev. 136B, 864 [26] S.K. Ghosh and A.K. Dhara, Phys. Rev. A 38, 1149
(1964). (1988).
[5] W.KohnandL.J.Sham,Phys.Rev.140A,1133(1965). [27] D.MearnsandW.Kohn,Phys.Rev.A35,4796(1987).
[6] M. Levy, Proc. Nat. Acad. Sci. USA 76, 6062 (1979); [28] P. Blanchard and E. Bru¨ning, Variational Methods in
Phys.Rev.A 26, 1200 (1982). Mathematical Physics, Springer, Berlin (1982).
[7] E.H.Lieb,inPhysics as Natural Philosophy,eds.A.Shi- [29] R.M.DreizlerandE.K.U.Gross,DensityFunctionalThe-
mony and H. Feshback,(MIT Press, Cambridge), p.111 ory, Springer, Berlin (1990), p.10 ff.
(1982); Int. J. Quant. Chem. 24, 243 (1983); in Density [30] H. Kohl and R.M. Dreizler, Phys. Rev. Lett. 56, 1993
Functional Methods inPhysics, NATO ASISeries B123, (1986).
ed. R.M. Dreizler and J. da Providencia, Plenum, New [31] A.K.Dhara and S.K. Ghosh, Phys. Rev. A. 35, 442
York(1985), p.31. (1987).
[8] J.E. Harriman, Phys. Rev. A 24, 680 (1980); G. Zum- [32] I.LindgrenandS.Salomonson, Phys.Rev.A.67,56501
bach andK.Masche, Phys.Rev.A28,544 (1983); erra- (2003).
tum,Phys. Rev.A 29, 1585 (1984). [33] Ref.[29], p.4 ff.
[9] W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 [34] Van Leeuwen [25] has argued that by imposing station-
(1965); J.P.Perdew, K. Burke, and M. Ernzerhof, Phys. arity on A[Ψ,v] with respect to the two variations δΨ
Rev. Lett. 77, 3865 (1996); J. Tao, J.P. Perdew, V.N. and iδΨ, one can derive Eq.(5) without imposing con-
Staroverov, and G.E. Scuseria, Phys. Rev. Lett. 91, dition (12) and thus eliminate distinguishing t1 from
146401 (2003). t ∀t ∈ (t0,t1 . But normalization then requires that
[10] R. Car and M. Parinello, Phys. Rev. Lett. 55, 2471 (δΨ,Ψ)+(Ψ,δΨ)=0 and (iδΨ,Ψ)+(Ψ,iδΨ)=0 hold
(1985). simultaneously,implyingthat(δΨ,Φ)=0∀t∈(t0,t1),a
[11] SeethereviewbyGross,etal.,ref.[12],forabriefhistor- much more severe restriction than Eq.(12). L¨owdin and
ical survey. Mukherjee [48] had proposed thisconstraint earlier.
[12] E.K.U. Gross, C.A. Ullrich, and U.J. Gossmann, [35] H. Hellmann, Z. Physik 85, 180 (1933): R.P. Feynman,
in Density Functional Theory, NATO ASI Series Phys. Rev.56, 340 (1939).
B337,ed.E.K.U.GrossandR.M.Dreizler,PlenumPress, [36] N.T. Maitra and K. Burke, Phys. Rev. A 63, 042501
NewYork (1995), p.149. (2001); 64, 039901 (E).
[13] E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 [37] E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850
(1984). (1985).
[14] M.E.Casida, in Recent Advances in Density Functional [38] M.K. Harbola and A. Banerjee, Phys. Rev. A 60, 5101
Methods,ed.D.P.Chong,p.155(WorldScientific,Singa- (1999).
pore, 1995); M. Petersilka, U.J. Gossmann, and E.K.U. [39] J. Wloka, Functionalanalysis und Anwendungen, de
Gross, Phys. Rev. Lett. 76, 1212 (1996); H. Appel, Gruyter, Berlin (1971).
E.K.U.Gross,andK.Burke,Phys.Rev.Lett.90,043005 [40] M.Ya. Amusia and V.R. Shaginyan, Phys. Lett. A 250,
(2003). 157 (1998).
[15] A.ZangwillandP.Soven,Phys.Rev.A21,1561(1980). [41] M. Ya. Amusia and V.R. Shaginyan, Phys. Rev. A 63,
[16] J.F.Dobson,M.J.Bunner,andE.K.U.Gross,Phys.Rev. 056501 (2001).
Lett.79, 1905 (1997). [42] M.K. Harbola, Phys.Rev.A 63, 056502 (2001).
[17] N.T. Maitra, K. Burke, H. Appel, E.K.U. Gross, and R. [43] L.J. Bartolotti, Phys. Rev.A 24, 1661 (1981).
vanLeeuwen,inReviewsinModernQuantumChemistry: [44] L¨owdin and Mukherjee [48] have already considered the
A celebration of the contributions of R.G. Parr, ed. K. gradient(13),butconstrainedonlytoconstantvandnot
Sen,(World-Scientific,2001). bothv andnconstantasinEq.(42).Theyuseditsnorm
[18] J. Frenkel, Wave Mechanics, Clarendon, Oxford, 1934; inS ,(ΘΨ∗,ΘΨ∗),asameasureoftheerrorforΨ6=Φ[v].
P.A.M.Dirac,Proc.CambridgePhil.Soc.26,376(1930); Minimizing that error separately for each time leads to
S.I. Sawada, R. Heather, B. Jackson, and H. Metiu, J. the Schr¨odinger equation (5). We add below a constant
Chem. Phys. 83, 3009 (1985); F.Y. Hansen, N.E. Hen- n constraint, integration overtime, and theinfimumop-
riksen,andG.D.Billing,J.Chem.Phys.90,3060(1989). eration to generalize theLL search to thetime domain.
[19] E.K.U.Gross,J.F.Dobson,andM.Petersilka, inDensity [45] A.D. McLachlan, Mol. Phys.8, 39 (1964).
Functional Theory, ed. F. Nalewajski, Springer, Berlin [46] E.J. Heller, J. Chem. Phys. 64, 63 (1976).
(1996). [47] See e.g. G.D. Mahan, Many-Particle Physics, 2nd. ed.,
[20] A.K.Rajagopal, Phys.Rev.A54,3916(1996);in”Elec- Plenum Press, Chapter 5 (1990).
tron Density Functional Theory: Recent Progress and [48] P.-O. L¨owdin and P.K. Mukherjee, Chem. Phys. Lett.
NewDirections”,eds.J.F.Dobson,G.Vignale,andM.P. 14, 1 (1972).
Das, p.217, Plenum Press (1998).