cances minimizer proof

8/12/2019 Cances Minimizer Proof

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Ann. I. H. Poincar AN 26 (2009) 24252455www.elsevier.com/locate/anihpc

Existence of minimizers for KohnSham models in quantumchemistry

Arnaud Anantharaman, Eric Cancs

Universit Paris-Est, CERMICS, Project-team Micmac, INRIA-Ecole des Ponts, 6 & 8 avenue Blaise Pascal,77455 Marne-la-Valle Cedex 2, France

Received 3 October 2008; received in revised form 29 April 2009; accepted 10 June 2009

Available online 21 June 2009

Abstract

This article is concerned with the mathematical analysis of the KohnSham and extended KohnSham models, in the local den-sity approximation (LDA) and generalized gradient approximation (GGA) frameworks. After recalling the mathematical derivationof the KohnSham and extended KohnSham LDA and GGA models from the Schrdinger equation, we prove that the extendedKohnSham LDA model has a solution for neutral and positively charged systems. We then prove a similar result for the spin-unpolarized KohnSham GGA model for two-electron systems, by means of a concentration-compactness argument. 2009 Elsevier Masson SAS. All rights reserved.

Keywords: Electronic structure; Density functional theory; KohnSham; Variational methods; Concentration-compactness

1. Introduction

Density Functional Theory (DFT) is a powerful, widely used method for computing approximations of groundstate electronic energies and densities in chemistry, materials science, biology and nanosciences.

According to DFT [12,17], the electronic ground state energy and density of a given molecular system can beobtained by solving a minimization problem of the form

inf F() + R 3

V, 0, H 1 R 3 , R 3

=N

where N is the number of electrons in the system, V the electrostatic potential generated by the nuclei, and F somefunctional of the electronic density , the functional F being universal, in the sense that it does not depend on themolecular system under consideration. Unfortunately, no tractable expression for F is known, which could be used innumerical simulations.

The groundbreaking contribution which turned DFT into a useful tool to perform calculations, is due to Kohn andSham [13], who introduced the local density approximation (LDA) to DFT. The resulting KohnSham LDA model

* Corresponding author. E-mail address: [email protected] (E. Cancs).

0294-1449/$ see front matter

2009 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.anihpc.2009.06.003


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2426 A. Anantharaman, E. Cancs / Ann. I. H. Poincar AN 26 (2009) 24252455

is still commonly used, in particular in solid state physics. Improvements of this model have then been proposedby many authors, giving rise to KohnSham GGA models [14,23,2,22], GGA being the abbreviation of generalizedgradient approximation. While there is basically a unique KohnSham LDA model, there are several KohnShamGGA models, corresponding to different approximations of the so-called exchange-correlation functional. A givenGGA model will be known to perform well for some classes of molecular system, and poorly for some other classes.

In some cases, the best result will be obtained with LDA. It is to be noticed that each KohnSham model existsin two versions: the standard version, with integer occupation numbers, and the extended version with fractionaloccupation numbers. As explained below, the former one originates from LevyLiebs (pure state) construction of thedensity functional, while the latter is derived from Liebs (mixed state) construction.

There are three main mathematical difculties encountered when studying these models from a theoretical pointof view: the nonlinearity, the nonconvexity, and the possible loss of compactness at innity of the models. To ourknowledge, very few results on KohnSham LDA and GGA models exist in the mathematical literature. In fact, weare only aware of a proof of existence of a minimizer for the standard KohnSham LDA model by Le Bris [15]. In thiscontribution, we prove the existence of a minimizer for the extended KohnSham LDA model, as well as for the two-electron standard and extended KohnSham GGA models, under some conditions on the GGA exchange-correlationfunctional.

Our article is organized as follows. First, we provide a detailed presentation of the various KohnSham models,which, despite their importance in physics and chemistry [26], are not very well known in the mathematical com-munity. The mathematical foundations of DFT are recalled in Section 2.1, and the derivation of the (standard andextended) KohnSham LDA and GGA models is discussed in Section 2.2. We state our main results in Section 3, andpostpone the proofs until Section 4.

We restrict our mathematical analysis to closed-shell, spin-unpolarized models. All our results related to the LDAsetting can be easily extended to open-shell, spin-polarized models (i.e. to the local spin-density approximationLSDA). Likewise, we only deal with all electron descriptions, but valence electron models with usual pseudo-potentialapproximations (norm conserving [31], ultrasoft [32], PAW [3]) can be dealt with in a similar way.

2. Mathematical foundations of DFT and KohnSham models

2.1. Density functional theory

As mentioned previously, DFT aims at calculating electronic ground state energies and densities. Recall that theground state electronic energy of a molecular system composed of M nuclei of charges z1 , . . . , z M (zk N \ {0} inatomic units) and N electrons is the bottom of the spectrum of the electronic hamiltonian

H V N = 12

N

i=1r i

N

i=1V (r i ) +

1 i


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It is easy to see that

|H V N | = |H 0N | + R 3

V and Tr H V N = Tr H 0N + R 3

V.

Besides, it can be checked that

R N = Q N , H N = 1, = = { | D N , =}= 0 H 1 R 3 ,

R 3

=N .

It therefore follows that

I N (V ) = inf F LL() + R 3

V, R N (5)

= inf F L()

+ R3

V,

R N (6)

where LevyLiebs and Liebs density functionals [16,17] are respectively dened by

F LL() = inf |H 0N | , Q N , H N = 1, = , (7)F L() = inf Tr H 0N , D N , = . (8)

Note that the functionals F LL and F L are independent of the nuclear potential V , i.e. they do not depend on themolecular system. They are therefore universal functionals of the density. It is also shown in [17] that F L is theLegendre transform of the function V I N (V ) . More precisely, it holds

F L() = sup I N (V )

R 3

V, V L32 R 3 +L R 3 ,

from which it follows in particular that F L is convex on the convex set R N (and can be extended to a convex functionalon L 1(R 3) L 3(R 3)).Formulae (5) and (6) show that, in principle, it is possible to compute the electronic ground state energy (and thecorresponding ground state density if it exists) by solving a minimization problem on R N . At this stage no approxi-mation has been made. But, as neither F LL nor F L can be easily evaluated for the real system of interest ( N interactingelectrons), approximations are needed to make of the density functional theory a practical tool for computing elec-tronic ground states. Approximations rely on exact, or very accurate, evaluations of the density functional for referencesystems close to the real system:

in ThomasFermi and related models, the reference system is a homogeneous electron gas; in KohnSham models (by far the most commonly used), it is a system of N non-interacting electrons.

2.2. KohnSham models

For a system of N non-interacting electrons, universal density functionals are obtained as explained in the previoussection; it sufces to replace the interacting hamiltonian H 0N of the physical system (formula (4)) with the hamiltonianof the reference system

T N = N

i=112 r i

. (9)

The analogue of the LevyLieb density functional (7) then is the KohnSham type kinetic energy functional

T KS () = inf |T N | , Q N , H N = 1, = , (10)


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A. Anantharaman, E. Cancs / Ann. I. H. Poincar AN 26 (2009) 24252455 2429

while the analogue of the Lieb functional (8) is the Janak kinetic energy functional

T J() = inf Tr(T N ), D N , = .Let be in the above minimization set. The energy Tr (T N ) can be rewritten as a function of the one-electronreduced density operator associated with . Recall that is the self-adjoint operator on L 2(R 3 ) with kernel

(x, x ) =N (R 3 )

N 1 ( x, x2, . . . , xN ;x , x2, . . . , xN ) d x2 . . . d xN .

Indeed, a simple calculation yields Tr (T N ) = Tr(12 r ) , where r is the Laplace operator on L 2(R 3 )-actingon the space coordinate r . Besides, it is known (see e.g. [6]) that

{ | D N , =} = { RD N | =}, (11)where

RD N

=

S L 2 R 3 0 1, Tr( )

=N, Tr(

r ) T J() .


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The density functionals T KS and T J associated with the non-interacting hamiltonian T N are expected to provideacceptable approximations of the kinetic energy of the real (interacting) system. Likewise, the Coulomb energy

J() =12

R 3

R 3

( r )( r )

|r r |d r d r

representing the electrostatic energy of a classical charge distribution of density is a reasonable guess for theelectronic interaction energy in a system of N electrons of density . The errors on both the kinetic energy and theelectrostatic interaction are put together in the exchange-correlation energy dened as the difference

E xc() =F LL() T KS () J(), (15)or

E xc() =F L() T J() J(), (16)depending on the choices for the interacting and non-interacting density functionals. We nally end up with the so-called KohnSham and extended KohnSham models

I KSN (V ) = inf 12N

i=1 R 3 i (x)2 d x + R 3 V +J ( ) +E xc( ), =( 1 , . . . , N ) W N , (17)

and

I EKSN (V ) = inf Tr 12 r

+ R 3

V +J ( ) +E xc( ), RD N . (18)

Up to now, no approximation has been made, in such a way that for the exact exchange-correlation functionals((15) or (16)), I KSN (V ) =I EKSN (V ) =I N (V ) for any molecular system containing N electrons. Unfortunately, there isno tractable expression of E xc() that can be used in numerical simulations. Before proceeding further, and for thesake of simplicity, we will restrict ourselves to closed-shell, spin-unpolarized, systems. This means that we will onlyconsider molecular systems with an even number of electrons N = 2N p , where N p is the number of electron pairs inthe system, and that we will assume that electrons go by pairs. In the KohnSham formalism, this means that theset of admissible states reduces to

=( 1, 1 , . . . , N p , N p ) i H 1 R 3 , R 3

i j =ij

where ( | ) = 1, ( | ) = 0, ( | ) = 0 and ( | ) = 1, yielding the spin-unpolarized (or closed-shell, or restricted)KohnSham model

I RKSN (V )

= inf

N p

i=1 R 3 | i

|2

+ R 3 V

+J ( )

+E xc( ),

=( 1 , . . . , N p ) H

1 R 3 N p ,

R 3

i j =ij , = 2N p

i=1|i |2 , (19)

where the factor 2 in the denition of accounts for the spin. Likewise, the constraints on the one-electron reduceddensity operators originating from the closed-shell approximation read:

r , | , r , | = r , | , r , | and r , | , r , | = r , | , r , | = 0.Introducing ( r , r ) = (r , | , r , | ) and denoting by (r ) = 2 ( r , r ) , we obtain the spin-unpolarized extendedKohnSham model

I REKSN (V ) = inf E (), K N p

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where

E ( ) = Tr( ) + R 3

V +J ( ) +E xc( ),

and

K N p = S L 2 R 3 0 1, Tr( ) =N p , Tr( ) < .Note that any K N p is of the form

=+i=1

n i | i i |with

i H 1 R 3 , R 3

i j =ij , n i [0, 1],+

i=1n i =N p ,

+

i=1n i i 2L 2 < .

In particular,

(r ) = 2+

i=1n i i (r )

2 .

Let us also remark that problem (19) can be recast in terms of density operators as follows

I RKSN (V ) = inf E ( ), P N p (20)where

P N p = S L 2 R 3 2 = , Tr( ) =N p , Tr( ) < is the set of nite energy rank- N p orthogonal projectors (note that K N p is the convex hull of P N p ). The connection

between (19) and (20) is given by the correspondence =N pi=1| i i |, i.e. is the orthogonal projector on thevector space spanned by the i s. Indeed, as | | = ()

12 , it holds

Tr( ) = Tr | | | | =N p

i=1| | i

2L 2 =

N p

i=1 i 2L 2 =

N p

i=1 R 3 | i |2.Let us now address the issue of constructing relevant approximations for E xc() . In their celebrated article [13],

Kohn and Sham proposed to use an approximate exchange-correlation functional of the form

E xc() = R 3

g ( r ) d r (LDA exchange-correlation functional) (21)

where 1g() is the exchange-correlation energy density for a uniform electron gas with density , yielding the so-called local density approximation (LDA). In practical calculations, it is made use of approximations of the function g() (from R + to R ) obtained by interpolating asymptotic formulae for the low and high density regimes (seee.g. [7]) and accurate quantum Monte Carlo evaluations of g() for a small number of values of [5]. Severalinterpolation formulae are available [25,24,33], which provide similar results. In the 80s, rened approximationsof E xc have been constructed, which take into account the inhomogeneity of the electronic density in real molecularsystems. Generalized gradient approximations (GGA) of the exchange-correlation functional are of the form

E xc() = R 3

h ( r ),12 ( r ) 2 dx (GGA exchange-correlation functional) . (22)

Contrarily to the situation encountered for LDA, the function (, )

g(, ) (from R

+ R

+ to R ) does not have

a denitive denition. Several GGA functionals have been proposed and new ones come up periodically.

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Remark 1. We have chosen the form (22) for the GGA exchange-correlation functional because it is well suited forthe study of spin-unpolarized two-electron systems (see Theorem 2 below). In the physics literature, spin-unpolarizedLDA and GGA exchange-correlation functionals are rather written as follows

E xc() =E x() +E c()with

E x() = R 3

( r ) x ( r ) F x s (r ) d r , (23)

E c() = R 3

( r ) c r (r ) +H r (r ), t (r ) d r . (24)

In the above decomposition, E x is the exchange energy, E c is the correlation energy, x and c are respectively theexchange and correlation energy densities of the homogeneous electron gas, r (r ) = ( 43 ( r ))

13 is the Wigner

Seitz radius, s (r ) = 12(3 2)1/ 3 | ( r )|

( r )4/ 3 is the (non-dimensional) reduced density gradient, t (r ) = 14(3 1 )1/ 6 | ( r )|

( r )7/ 6 is

the correlation gradient, F x is the so-called exchange enhancement factor, and H is the gradient contribution to thecorrelation energy. While x has a simple analytical expression, namely

x() = 34

3

13

13

c has to be approximated (as explained above for the function g ). For LDA, F x is everywhere equal to one and H = 0.A popular GGA exchange-correlation energy is the PBE functional [22], for whichF x(s) = 1 +

s 2

1 + 1s2,

H(r,t)

= ln 1

+

t 2 1+A(r)t 2

1 +A(r)t 2

+A(r)2t 4 with A(r)

=

e c(r)/

1 1,

the values of the parameters 0.21951, 0.804, = 2(1 ln 2) and = 3 2 following from theoreticalarguments.3. Main results

Let us rst set up and comment on the conditions on the LDA and GGA exchange-correlation functionals underwhich our results hold true:

the function g in (21) is a C 1 function from R + to R , twice differentiable and such thatg( 0)

= 0, (25)

g 0, (26)

0 < + 2 electrons still holds. The main argument is that, using

[10, Theorem 2.5], the condition (34) still ensures the lower semicontinuity of the energy w.r.t. | |2 for the weak topology of H 1(R 3;R N p ) . Therefore, for all N p N , if a minimizing sequence ( n )n N is compact in L 2(R 3;R N p ) ,then its limit is a minimizer of the problem.

In our proof of compactness in the case N p = 1, we use in a crucial way the properties of the solutions of theEuler equation (39), among which boundedness in L(R 3) and exponential decay at innity. When N p > 1, denotingthe state vector by = (1 , . . . , N p ) and assuming that the energy is differentiable, the EulerLagrange optimalityconditions turn into the following system: i 1, N p ,

12

div i +h

,12 | |

2 k k kk

2k

i +12

h

,12 | |

2 k k kk

2k

i

12

h

,12 | |2 k

k

k

k 2k

2

i + V + |r |1 +h

,12 | |2 i = i i . (40)

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The study of (40) is much more involved than that of (39). We were not able to prove that solutions of (40) stillhave the required regularity properties and behaviour at innity, and thus to extend our proof from the scalar case tothe vector case.

4. Proofs

For clarity, we will use the following notation

E LDAxc () = R 3

g ( r ) d r ,

E GGAxc () = R 3

h ( r ),12 ( r )

2 d r ,

E LDA ( ) = Tr( ) + R 3

V +J ( ) + R 3

g (r ) d r ,

E GGA ( ) = Tr( ) + R 3 V +J ( ) + R 3 h (r ), 12 (r )2 d r .

The notations E xc() and E ( ) will refer indifferently to the LDA or the GGA setting.

4.1. Preliminary results

Most of the results of this section are elementary, but we provide them for the sake of completeness. Let us denoteby S 1 the vector space of trace-class operators on L 2(R 3) (see e.g. [27]) and introduce the vector space

H = S 1 | | | | S 1endowed with the norm H = Tr(| |) +Tr(|| || || ) , and the convex set

K = S L 2 R 3 0 1, Tr( ) < , Tr | | | | < .Lemma 2. For all K , H 1(R 3) and the following inequalities hold true :

Lower bound on the kinetic energy :

12

2L 2 Tr(). (41)

Upper bound on the Coulomb energy :

0 J ( ) C( Tr )32

Tr( )12

. (42) Bounds on the interaction energy between nuclei and electrons :

4Z( Tr )12 Tr( )

12

R 3

V 0. (43)

Bounds on the exchange-correlation energy :

C (Tr ) 12 Tr( )

32 +(Tr ) 1+2 Tr( )

3+2 E xc( ) 0. (44)

Lower bound on the energy :

E ( )12 Tr( )

1

2 4Z( Tr )12

2

8Z 2 Tr C (Tr )2

23 +(Tr )2+

23+ . (45)

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Lower bound on the energy at innity :

E ( ) 12

Tr( ) C (Tr )223 +(Tr )

2+23+ , (46)

for a positive constant C independent of . In particular, the minimizing sequences of (35) and those of (36) arebounded in H .

Proof. (41) is a straightforward consequence of CauchySchwarz inequality; a proof can be found for instance in [4].Using HardyLittlewoodSobolev [18], interpolation, and GagliardoNirenbergSobolev inequalities, we obtain

J ( ) C1 2L

65

C1 32L 1

12L 3 C2

32L 1 L 2 .

Hence (42), using (41) and the relation L 1 = 2 Tr( ) . It follows from CauchySchwarz and Hardy inequalitiesand from the above estimates that

R 3

| R k|2

12L 1 L 2 4(Tr )

12 Tr( )

12 .

Hence (43). Conditions (25)(27) for LDA and (29)(31) for GGA imply that Exc

() 0 and there exists 1 0 and 0 > 0 such that for all 0 1 and all 0 0 ,I

E

( ,

) 2

R 3 |

|2

+ 2 J

2|

|2

c 3(

1)

R 3 |

|2 .

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Therefore I < 0 for positive and small enough. It follows from (37) and (48) that the functions I and I are decreasing, and that for all > 0,

 0 and ( n )n N be a minimizing sequence for (35). Then the sequence ( n )n N cannot vanish,which means (see [20]) that

R > 0 s.t. limn

supx R 3

x+BR n > 0.

The same holds true for the minimizing sequences of (36).

Proof. Let ( n )n N be a minimizing sequence for I . By contradiction, assume that the sequence n vanishes, i.e.

R > 0, limnsupx R 3 x+BR n = 0.

We know from Lemma 2 that n is bounded in H , and thus that n is bounded in H 1(R 3) . According to Lemma I.1in [20], this and the fact that n is vanishing imply that n converge strongly to 0 in L

p (R 3) for 1 < p < 3. Inparticular, it follows from (47) and from the fact that V L r (R 3) +L q (R 3) for some 32 < r,q < +, that

limn

R 3

n V +E xc( n ) = 0.

As

E ( n ) R 3 n V +E xc( n ),we obtain that I 0. This is in contradiction with the previously proved result stating that I < 0. Hence( n )n N cannot vanish. The case of problem (36) is easier since the only non-positive term in the energy functionalis E xc() .

We can now prove that I 0 and R > 0, such that for n large enough, there exists xn R 3 such that

xn+BR

n .

Let us introduce n = x1xn n xn x1 . Clearly n K and E ( n ) E ( n ) z1

R . Thus,

I I z1R

< I .

It remains to prove that the functions I and I are continuous. We will deal here with the former one, thesame arguments applying to the latter one. The proof is based on the following lemma.Lemma 5. Let ( k )k N be a sequence of positive real numbers converging to 1 , and ( k )k N a sequence of non-negative densities such that ( k )k N is bounded in H 1(R 3). Then

limkE xc( kk ) E xc( k ) = 0.

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Proof. In the LDA setting, we deduce from (27) that there exists 1 < p p + < 53 and C R + such that for k largeenough

E LDAxc ( kk ) E LDAxc ( k) C | k 1| R 3

p k +p +k .

In the GGA setting, we obtain from (31) and (33) that there exists 1 < p p + < 53 and C R + such that for k largeenough

E GGAxc ( kk ) E GGAxc ( k) C | k 1| R 3

p k +p +k + | k|2 .

As ( k )k N is bounded in H 1(R 3) , ( k )k N is bounded in L p (R 3) for all 1 p 3 and ( k )k N is bounded in(L 2(R 3)) 3 , hence the result.

We can now complete the proof of Lemma 1.Let > 0, and ( k)k N be a sequence of positive real numbers converging to . Let > 0 and K such that

I E ( ) I +2 .For all k N , k = k1 is in K k so that k N , I k E ( k ) . Besides, it is easy to see that E ( k ) tends to E ( )in virtue of Lemma 5. Thus I k I + for k large enough. Now, for each k N , we choose k K k such thatE ( k ) I k + 1k . For all k N , we set k = 1k k . As k K , it holds I E ( k ) . We then deduce from Lemma 5that limk(E ( k ) E ( k )) = 0, so that for k large enough we get I I k . This proves the continuity of I on R + \ {0}. Lastly, it results from the estimates established in Lemma 2 that lim 0+ I = 0.4.3. Proof of Theorem 1

Let us rst prove the following lemma, which relies on classical arguments.

Lemma 6. Let ( n )n N be a sequence of elements of K , bounded in H , which converges to for the weak- topologyof H . If limnTr( n ) = Tr( ) , then ( n )n N converges to strongly in L p (R 3) for all 1 p < 3 and

E LDA ( ) liminf n

E LDA ( n ) and E LDA,( ) liminf n

E LDA,( n ).

Proof. The fact that ( n )n N converges to for the weak- topology of H means that for any compact operator Kon L 2(R 3) ,

limn

Tr( n K) = Tr(K) and limnTr | | n | | K = Tr | | | | K .For all W Cc (R 3) , the operator (1 + | |)1W (1 + | |)1 is compact (it is even in the Schatten class S p for allp > 32 in virtue of the KatoSeilerSimon inequality [29]), yielding

R 3 n W = 2 Tr( n W ) = 2 Tr 1 +| | n 1 +| | 1 +| | 1W 1 +| | 1n 2 Tr 1 +| | 1 +| | 1 +| |

1W 1 +| |

1

= 2 Tr(W) = R 3

W.

Hence, ( n )n N converges to in D (R3) . As by (41), ( n )n N is bounded in H 1(R 3) , it follows that ( n )n N

converges to weakly in H 1(R 3) , and strongly in L ploc(R 3) for all 2 p < 6. In particular, ( n )n N convergesto weakly in L 2(R 3) . But we also know that

limn

n2L 2 = limn R 3

n = 2 limnTr

( n

)

= 2 Tr( )

= R 3

=

2L 2

.

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Therefore, the convergence of ( n )n N to holds strongly in L 2(R 3) . Using Hlders inequality and the bound-edness of ( n )n N in H 1(R 3) , we obtain that ( n )n N converges strongly to in L p (R 3) for all 2 p < 6,hence that ( n )n N converges to strongly in L

p (R 3) for all 1 p < 3. This readily implies

limn R 3

n V

= R 3 V , lim

nJ ( n )

=J ( ), lim

nE LDAxc ( n )

=E LDAxc ( ).

Lastly, Fatous theorem for non-negative trace-class operators yields

Tr | | | | liminf nTr | | n | | .We thus obtain the desired result.

We will also need the following result.

Lemma 7. Consider > 0 and > 0 such that + N p Z/ 2. If I and I have minimizers, then

I + < I +I .Proof. Let be a minimizer for I . In particular satises the Euler equation

= 1(, F) (H ) +for some Fermi level F R , where

H = 12 +V + |r |

1 +g ( ),and where 0 1, Ran() Ker(H F) . As V + |r |1 + g ( ) is -compact, the essential spectrumof H is [0, +) . Besides, H is bounded from below,

H 12 +V + |r |1 ,and we know from [19, Lemma II.1] that as

M k=1 zk + R 3 = Z + 2 < Z + 2N p 0, the right-hand sideoperator has innitely many negative eigenvalues of nite multiplicities. Therefore, so has H . Eventually, F < 0

and

=n

i=1| i i |+

m

i=n+1n i |i i |

where 0 n i 1 and where

1

2

i

+V i

+

|r

|1 i

+g ( ) i

=i i

1 < 2 3 < 0 denoting the negative eigenvalues of H including multiplicities (by standard arguments theground state eigenvalue of H is non-degenerate). It then follows from elementary elliptic regularity results that allthe i s, hence , are in H 2(R 3) and therefore vanish at innity. Using Lemma 13, all the i decay exponentiallyfast to zero at innity.

Now consider a minimizer for I . satises = 1(, F ) H +

where

H = 12 + |r |

1 +g ( ),and where 0 1, Ran( ) Ker(H F) , and F 0.

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First consider the case F < 0. Then

=n

i=1 i i +

m

i=n +1n i i i ,

all the i s being in C (R3) and decaying exponentially fast at innity. For n N large enough, the operator

n = min 1, + ne ne 1 ( + ne ne)then is in K and Tr ( n ) ( +) , which implies I + I Tr( n ) due to Lemma 1. As both the i s and the i s decayexponentially fast to zero, a simple calculation shows that there exists some > 0 such that for n large enough

E LDA ( n ) =E LDA( ) +E LDA,( ) 2(Z 2)

n +O en =I +I

2(Z 2)n +O e

n .

Since 2 < 2N p Z , we have for n large enough

I + I Tr( n ) E LDA( n ) 0. For 0 < < , +|m+1 m+1| and | | are in K and it is easy to see thatE LDA +|m+1 m+1| =I +2 m+1 +o()

and

E LDA, | | =I +o().Since Tr ( +|m+1 m+1|) = + and Tr( | |) = , we deduce

I + I +2 m+1 +o() and I I +o().Then, according to Lemma 1, we obtain for small enough

I + I + +I I +I +2 m+1 +o() < I +I . We are now in position to prove Theorem 1, and even more generally that problem (35) with (21) has a minimizer

for N p . Let ( n )n N be a minimizing sequence for I with N p . We know from Lemma 2 that ( n )n N isbounded in H and that ( n )n N is bounded in H 1(R 3) . Replacing ( n )n N by a suitable subsequence, we canassume that ( n ) converges to some K for the weak- topology of H and that ( n )n N converges to weakly in H 1(R 3) , strongly in L ploc(R

3) for all 2 p < 6 and almost everywhere.If Tr( ) = , then K and according to Lemma 6,

E LDA ( ) liminf n+

E LDA ( n ) =I yielding that is a minimizer of (35).

The rest of the proof consists in ruling out the eventuality when Tr ( ) < .Let us rst rule out the case Tr ( ) = 0. By contradiction, assume that Tr ( ) = 0, which implies = 0. Then nconverges to 0 strongly in L ploc(R 3) for 1 p < 6, from which we deduce

limn+

R 3

n V = 0.

Consequently,

I limn+E LDA,( n ) = limn+

E LDA( n ) =I which contradicts the rst assertion of Lemma 1.

Let us now set

= Tr( ) and assume that 0 < < . Following e.g. [9], we consider a quadratic partition of

the unity 2 + 2 = 1, where is a smooth, radial function, nonincreasing in the radial direction, such that (0) = 1,


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R 3

g( 1,n + 2,n ) g( 1,n ) g( 2,n )

BR

g( 1,n + 2,n ) g( 1,n ) + BR

g( 2,n ) + B

cR

g( 1,n + 2,n ) g( 2,n ) + B

cR

g( 1,n )

C BR

2,n + 2 2,n + 1,n L 2 BR

2 2,n

12

+C BR

p 2,n +p + 2,n

+C B cR

1,n + 2 1,n + 2,n L 2 B cR

2 1,n

12

+C B cR

p 1,n +p + 1,n

for some constant C independent of R and n . Yet, we know that ( 1,n )n N and ( 2,n )n N are bounded in H 1(R 3) ,that ( 1,n )n N converges to in L

p (R 3) for all 1 p < 3 and that ( 2,n )n N converges to 0 in Lploc(R

3) for all

1 p < 3. Consequently, there exists for all > 0, some N N

such that for all n N ,E LDA ( n ) E LDA( 1,n ) +E LDA,( 2,n ) I +I .

Letting n go to innity, go to zero, and using (37), we obtain that I =I +I and that ( 1,n )n N and ( 2,n )n Nare minimizing sequences for I and I respectively. It also follows from (50) that is a minimizer for I .Let us now analyze more in details the sequence ( 2,n )n N . As it is a minimizing sequence for I , ( 2,n )n Ncannot vanish, so that there exists > 0, R > 0 such that for all n N , yn+BR 2,n for some yn R 3 . Thus, thesequence ( yn 2,n yn )n N converges for the weak- topology of H to some K satisfying Tr ( ) > 0. Let = Tr( ) . Reasoning as above, one can easily check that is a minimizer for I , and that I =I +I +I .On the other hand, Lemma 7 yields I + I

+

+I

, which contradicts Lemma 1. The proof is complete.

4.4. Proof of Theorem 2

For H 1(R 3) , we set (x) = 2|(x) |2 and

E() = R 3

| |2 + R 3

V +J ( ) +E GGAxc ( ).

For all H 1(R 3) such that L 2 = 1, = | | K 1 and E ( ) =E() . Therefore,

I 1 inf E(), H 1 R 3 , R 3 | |2 = 1 .

Conversely, for all K 1 , = 2 satises H 1(R 3) , L 2 = 1 andE GGA ( ) =E GGA | | +Tr( )

12

R 3| |2 E GGA | | =E( ).

Consequently,

I 1 = inf

E(),

H 1 R 3 ,

R 3 |

|2

= 1 (51)


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and (20) has a minimizer for N p = 1, if and only if (51) has a minimizer ( then is a minimizer of (20) for N p = 1).We are therefore led to study the minimization problem (51). In the GGA setting we are interested in, E() can berewritten as

E() =

R 3

| |2 +

R 3

V +J ( ) +

R 3

h , | |2 .

Conditions (29)(33) guarantee that E is Frchet differentiable on H 1(R 3) (see [1] for details) and that for all (,w)H 1(R 3) H 1(R 3) ,

E () w = 212

R 3

1 +h

, | |2 w + R 3

V + |r |1 +h

, | |2 w .

We now embed (51) in the family of problems

J = inf E(), H 1 R 3 , R 3

| |2 = (52)

and introduce the problem at innity

J = inf E (), H 1 R 3 , R 3

| |2 = (53)

where

E () = R 3

| |2 +J ( ) + R 3

h , | |2 .

Note that reasoning as above, one can see that J =I and J =I for all 0 1 (while these equalities do nota priori hold true for > 1).

The rest of this section consists in proving that (52) has a minimizer for all 0 1. Let us start with a simplelemma.

Lemma 8. Let 0 1 and let ( n )n N be a minimizing sequence for J (resp. for J ) which converges to some H 1(R 3) weakly in H 1(R 3). Assume that 2L 2 = . Then is a minimizer for J (resp. for J ).Proof. Let ( n )n N be a minimizing sequence for J which converges to weakly in H 1(R 3) . For almost all x R 3 ,the function z |z|2 + h( (x), |z|2) is convex on R 3 due to (34). Besides the function t t + h( (x),t) isLipschitz on R +, uniformly in x due to (33). It follows that the functional

R 3

| |2 +h , | |2

is convex and continuous on H 1(R 3) . As ( n )n N converges to weakly in H 1(R 3) , we get

R 3

| |2 +h , | |2 liminf n R 3

| n |2 +h , | n |2 .

Besides, we deduce from (31) that

R 3

h n , | n |2 h , | n |2 C n L 2 ,

where the constant C only depends on h and on the H 1 bound of (n

)n

N . As (n

)n

N converges to weakly inL 2(R 3) and as L 2 = n L 2 for all n N , the convergence of ( n )n N to holds strongly in L 2(R 3) . Therefore,

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R 3

| |2 +E GGAxc ( ) = R 3

| |2 +h , | |2

liminf n

R 3| n |2 +h , | n |2 + limn

R 3

h n , | n |2 h , | n |2

= liminf n R 3 | n |2 +E GGAxc ( n ).Finally, as (n )n N is bounded in H 1 and converges strongly to in L2(R 3) , we infer that the convergence holdsstrongly in L p (R 3) for all 2 p < 6, yielding

limn

R 3

n V +J ( n ) = R 3

V +J ( ).

Therefore,

E() liminf nE( n ) =I .As 2L 2 = , is a minimizer for J . Obviously, the same arguments can be applied to a minimizing sequencefor J .

Next, we show that the equivalent of Lemma 7 in the GGA setting holds.

Lemma 9. Consider > 0 and > 0 such that + 1. If J and J have minimizers, thenJ + < J +J .

Proof. Let u and v be minimizers for J and J respectively. Since E() =

E(

|

|)

H 1(R 3) , we can assume

that u and v are non-negative. u satises the Euler equation

12

div 1 +h

u , | u |2 u + V +u |r |1 +h

u , | u |2 u + 1u = 0 (54)and v satises the Euler equation

12

div 1 +h

v , | v|2 v + v |r |1 +h

v , | v|2 v + 2v = 0 (55)where 1 and 2 are two Lagrange multipliers.

Using properties (31) and (33) and classical elliptic regularity arguments [11] (see also the proof of Lemma 13below), we obtain that both u and v are in C 0, (R 3) for some 0 < < 1 and vanish at innity.

Using again (31), this implies that h ( u , | u|

2)u vanishes at innity. Since it is a non-positive function, applying

Lemma 12 (proved in Appendix A) to (54) then yields 1 > 0.Moreover, the function J being decreasing on [0, 1], 2 is non-negative.Let us rst assume 2 > 0. Applying Lemma 13, we then obtain that there exists > 0, f 1 H 1(R 3) , f 2 H 1(R 3) ,

g1 (L 2(R 3)) 3 and g2 (L 2(R 3)) 3 such thatu =e ||f 1 , v =e ||f 2 , u =e ||g1, v =e ||g2 . (56)

In addition, as u 0 and v 0, we also have f 1 0 and f 2 0. Let e be a given unit vector of R 3 . For t > 0, we set

w t (r ) = t u( r ) +v( r t e) where t =( +)12 u +v( t e)

1L 2 .

Obviously, w t

H 1(R 3) and

w t L 2

=

+ , so that

E(w t ) J + . (57)

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Besides, a little calculation (see [1] for details) shows that

E(w t ) =J +J + R 3

V v( t e)2

+D( u , v( t e) ) +O e t ,

the main difculty being to verify that (31), (33), (56) and the boundedness of u and v in L

(R 3) yield

R 3

h w t , | w t |2 h u , | u |2 h v( t e) , v( t e)2

=O e t .

Next, using (56), we get

R 3

Vv( t e) +D( u , v( t e) ) = Zt 1 R 3

u +t 1 R 3

v R 3

u +o t 1

= 2(Z 2)t 1 +o t 1 .Finally, for t large enough and since 2 < 2 Z ,

J + E(w t ) J +J 2(Z 2)t 1 +o t 1 < J +J .Let us now assume that 2 = 0. Using (54) and (55), we easily get that for > 0 small enough,

J (1+) 2 E(u +u) =E(u) 1 +o() =J 1 +o()while

J (12 ) 2E v 2

v =E (v) +o() =J +o().Lemma 1 then yields

J (1+) 2+(12 ) 2 J (1+) 2 +J (12 ) 2 J +J 1 +o(),

and for small enough, it holds (1 +) 2 +(1 2 ) 2 + so thatJ + J (1+) 2+(12 ) 2 J +J 1 +o()< J +J .

In order to prove that the minimizing sequences for J (or at least some of them) are indeed precompact in L 2(R 3)and to apply Lemma 8, we will use the concentration-compactness method due to P.-L. Lions [20], for the simplermethod used in the LDA setting does not seem to work anymore. Consider an Ekeland sequence (n )n N for (52),that is [8] a sequence ( n )n N such that

n N , n H 1 R 3 and R 3

2n =, (58)

limn+E( n ) =J , (59)lim

n+E ( n ) + n n = 0 in H 1 R 3 (60)

for some sequence ( n )n N of real numbers. As in the proof of Lemma 9, we can assume that

n N , n 0 a.e. on R 3 and n 0. (61)

Lastly, up to extracting subsequences, there is no restriction in assuming the following convergences:

n weakly in H 1 R 3 , (62)

n strongly in Lploc

R 3 for all 2 p < 6, (63)

n

a.e. in R 3, (64)

n in R , (65)

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and it follows from (61) that 0 a.e. on R 3 and 0. Note that the Ekeland condition (60) also reads

12

div 1 +h

n , | n |2 n + V +n |r |1 +h

n , | n |2 n + n n =nwith n

n

0 0 in H

1 R 3 . (66)

We can then apply the concentration-compactness method to the sequence ( n )n N and obtain the following lemma.

Lemma 10. Consider (n )n N satisfying (58)(65) . Then, using the terminology introduced in the concentration-compactness lemma in [20] ,

1. if some subsequence (nk )k N of (n )n N satises the compactness condition, then (nk )k N converges to strongly in L p (R 3) for all 2 p < 6;

2. a subsequence of ( n )n N cannot vanish ;3. a subsequence of ( n )n N cannot satisfy the dichotomy condition.

Consequently, ( n )n N converges to strongly in L p (R 3) for all 2 p < 6. It follows that is a minimizer to (52).

Proof of the rst two assertions of Lemma 10. Assume that there exists a sequence (y k )k N in R 3 , such that for all > 0, there exists R > 0 such that

k N , yk+BR

2nk .

Two situations may be encountered: either (y k )k N has a converging subsequence, or lim k|yk| = . In the lattercase, we would have = 0, and thereforelim

k R 3

2nk V = 0.

Hence

I limkE ( nk ) = limk

E( n k ) =I ,which is in contradiction with the rst assertion of Lemma 1. Therefore, (y k )k N has a converging subsequence. It isthen easy to see, using the strong convergence of ( n )n N to in L 2loc(R

3) , that

R 3

2 y+BR

2 ,

where y is the limit of some converging subsequence of (yk )k N . This implies that 2L 2 = , hence that (n )n Nconverges to strongly in L

2(R

3) . As ( n )n N is bounded in H

1(R

3) , this convergence holds strongly in L

p(R

3) for

all 2 p < 6.Assume now that (nk )k N is vanishing. Then we would have = 0, an eventuality that has already been ex-cluded.

Proof of the third assertion of Lemma 10. Let us rst introduce two functions and in C (R 3) such that0 , 1, (x) = 1 if |x | 1, (x) = 0 if |x | 2, (x) = 0 if |x | 1, (x) = 1 if |x | 2, L 2 and L 2. For R > 0, we denote by R () =( R ) and R () =( R ) .Replacing ( n )n N with a subsequence and using the detailed construction of the dichotomy case given in [20], we

can assume that in addition to (58)(65), there exist

]0,

[,

a sequence (y n )n N of points in R 3 ,

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two increasing sequences of positive real numbers (R 1,n )n N and (R 2,n )n N such thatlim

nR1,n = and limn

R 2,n2 R 1,n =

such that the sequences 1,n

= R 1,n (

yn ) n and 2,n

= R 2,n / 2(

yn ) n satisfy

n =1,n on yn +BR 1,n ,n =2,n on R 3 \ (y n +BR 2,n ),lim

n R 3

21,n =, limn R 3

22,n = ,

limn

n ( 1,n +2,n ) L p (R 3) = 0 for all 2 p < 6,lim

nn L p (y n+(B R2,n \B R1,n )) = 0 for all 2 p < 6,

limn

dist(Supp 1,n , Supp 2,n ) = ,liminf n R 3 |

n

|2

| 1,n

|2

| 2,n

|2 0.

Besides, it obviously follows from the construction of the functions 1,n and 2,n that

n N , 1,n 0 and 2,n 0 a.e. on R 3. (67)

A straightforward calculation leads to

E( n ) =E ( 1,n ) + R 3

1,n V +E ( 2,n ) + R 3

2,n V + R 3

| n |2 | 1,n |2 | 2,n |2 + R 3

n V

+D( 1,n , 2,n ) +D( n , 1,n +2,n ) +12

D( n , n )

+ R 3 h n , | n |2 h 1,n , | 1,n |2 h 2,n , | 2,n |2 , (68)

where we have denoted by n =n 1,n 2,n . As|n | 31 yn+(B R2,n \B R1,n ) |n |2,where 1 yn +(B R2,n \B R1,n ) is the characteristic function of yn + (B R 2,n \ B R 1,n ) , the sequence (n )n N goes to zeroin L p (R 3) for all 1 p < 3, yielding

R 3

n V +D( n , 1,n +2,n ) +12

D( n , n ) n 0.

Besides,

D( 1,n , 2,n ) 4dist(Supp 1,n , Supp 2,n )1 1,n 2L 2 2,n2L 2 n 0

and

R 3

h n , | n |2 h 1,n , | 1,n |2 h 2,n , | 2,n |2

yn+(B R2,n \B R1,n )

h n , | n |2 + h 1,n , | 1,n |2 + h 2,n , | 2,n |2

C n p L p(y n+(B R2,n \B R1,n )) + np +L p +(y n+(B R2,n \B R1,n )) n 0

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(recall that 1 < p = 1 + < 53 ). Lastly, as lim ndist(Supp 1,n , Supp 2,n ) = ,min

R 3

1,n V , R 3

2,n V n 0.

It therefore follows from (68) and from the continuity of the functions J and J that at least one of theinequalities below holds trueJ J +J (case 1) or J J +J (case 2). (69)

As the opposite inequalities are always satised, we obtain

J =J +J (case 1) or J =J +J (case 2) (70)and (still up to extraction)

limn

E( 1,n ) =J ,lim

nE ( 2,n ) =J

(case 1) or limn

E ( 1,n ) =J ,lim

nE( 2,n ) =J

(case 2) . (71)

Let us now prove that the sequence ( n )n N , where n =n ( 1,n +2,n ) , goes to zero in H 1(R 3) . For convenience,we rewrite n as n =en n where en = 1 R 1,n (yn ) R 2,n / 2(yn ) and Ekelands condition (66) as

div(a n n ) +V n + n |r |1 n +V n 1+2n +V +n

1+2+n + n n =n (72)where

a n =12

1 +h

n , | n |2 ,V n = 2

nh

n , | n |2 n 1,

V +n =

2+ +n

h

n ,

| n

|2 n > 1.

Due to assumption (32), V n and V +n are bounded in L(R 3) .The sequence (V n +( n |r |1) n +V n

1+2n +V +n 1+2+n + n n )n N is bounded in L 2(R 3) , ( n )n N goesto zero in H 1(R 3) , and the sequence ( n )n N is bounded in H 1(R 3) and goes to zero in L 2(R 3) . We therefore infer

from (72) that

R 3

a n n n n 0.

Besides n =en n +n en with 0 en 1 and en L 0. Thus

R 3 an en | n |

2

n 0.As

0 0 for all k N , we obtain that C k 2

L + k 2+L k 0,

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which obviously contradicts (75). We therefore conclude from this analysis that, if dichotomy occurs, ( n )n N splitsin a nite number, say K , of compact bits having mass k > 0 with Kk=1 k = . We are now going to prove that thiscannot be.

If this was the case, there would exist two sequences (u 1,n )n N and (u 2,n )n N such that

n N , u 1,n H 1

R3

, R 3 |u1,n |

2

=1, u 1 0 a.e. on R3,

limn

E(u 1,n ) =J 1and

n N , u 2,n H 1 R 3 , R 3

|u 2,n |2 =2, u 2 0 a.e. on R 3 ,lim

nE (u 2,n ) =J 2

and converging weakly in H 1(R 3) to u1 and u2 respectively, with u 1 2L 2 =1 and u2 2L 2 =2 (as the weak limit of (

n)

nN in L 2(R 3) is nonzero, one bit stays at nite distance from the nuclei). It then follows from Lemma 8 that u

1and u2 are minimizers for J 1 and J 2 , and from Lemma 9 that J 1+2 < J 1 +J 2 .Applying (70) twice, we also have J = J 1 + J 2 + J 12 , so that we infer J > J 1+2 + J 12 which isa contradiction to Lemma 1.

End of the proof of Lemma 10. As a consequence of the concentration-compactness lemma and of the rst threeassertions of Lemma 10, the sequence (n )n N converges to weakly in H 1(R 3) and strongly in Lp (R 3) for all2 p < 6. In particular,

R 3

2 = limn R 3

2n =.

It follows from Lemma 8 that is a minimizer to (52).

Acknowledgements

The authors are grateful to C. Le Bris, M. Lewin and F. Murat for helpful discussions. We also thank the anonymousreferee for valuable suggestions to improve the rst version of our manuscript. This work was completed while E.C.was visiting the Applied Mathematics Division at Brown University, whose support is gratefully acknowledged.

Appendix A

In this appendix, we state three technical lemmas, which we make use of in the proof of Theorem 2. These lemmasare concerned with second-order elliptic operators of the form

div(A

) . For the sake of generality, we deal with

the case when A is a matrix-valued function, although A is a real-valued function in the two-electron GGA model.For an open subset of R 3 and 0 < < , we denote by M s (,,) the closed convex subset of

L (, R 33) consisting of the symmetric matrix elds A L (, R 33) such that for almost all x , A(x) .

The rst lemma is a H -convergence result which allows to pass to the limit in the Ekeland condition (66). We shallnot give the proof, for it is very similar to the proofs that can be found in the original article by Murat and Tartar [21].Recall that a sequence (A n )n N of elements of M s (,,) is said to H -converge to some A M s ( , , ) , whichis denoted by An H A, if for every the following property holds: f H 1() , the sequence (u n )n N of the elements of H 10 () such that div(A n u n ) =f | in H 1() , satises

u n u weakly in H 10 (),

An u n A u weakly in L 2()

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where u is the solution in H 10 () to div(A u) =f | . It is known [21] that from any bounded sequence (A n )n N inM s (,,) one can extract a subsequence which H -converges to some A M s (, 1 2 , ) .

Lemma 11. Let be an open subset of R 3 , 0 < < , 0 < < , and (A n )n N a sequence of elementsof M s (,,) which H -converges to some A M s ( , , ) . Let (u n )n N , (f n )n N and (g n )n N be sequences of elements of H 1() , H 1() and L 2() respectively, and u H 1() , f H 1() and g L 2() such that

div(A n u n ) =f n +gn in H 1() for all n N ,u n u weakly in H 1(),f n f strongly in H 1(),gn g weakly in L 2().

Then div(A u) =f +g and A n un A u weakly in L 2() .The second lemma is an extension of [19, Lemma II.1] and of a classical result on the ground state of Schrdinger

operators [28]. Recall that

L 2 R 3 +L R 3 = W > 0, (W 2 , W ) L 2 R 3 L R 3 s.t. W L , W =W 2 +W .Lemma 12. Let 0 < < , A M s (,, R 3) , W L 2(R 3) +L (R 3) such that W + = max(0, W ) L 2(R 3) +L 3(R 3) and a positive Radon measure on R 3 such that ( R 3) < Z =

M k=1 zk . Then,

H = div(A ) +V + |r |1 +W denes a self-adjoint operator on L 2(R 3) with domain

D(H) = u H 1 R 3 div(A u) L 2 R 3 . Besides, D(H ) is dense in H 1(R 3) and included in L (R 3) C 0, (R 3) for some > 0 , and any function of D(H )vanishes at innity. In addition,

1. H is bounded from below, ess (H ) [0, ) and H has an innite number of negative eigenvalues ;2. the lowest eigenvalue 1 of H is simple and there exists an eigenvector u1 D(H) of H associated with 1 suchthat u1 > 0 on R 3;

3. if w D(H) is an eigenvector of H such that w 0 on R 3 , then there exists > 0 such that w =u 1 .The third lemma is used to prove that the ground state density of the GGA KohnSham model exhibits exponential

decay at innity (at least for the two-electron model considered in this article).

Lemma 13. Let 0 < < , A M s (,, R 3) , V a function of L65loc(R

3) which vanishes at innity, > 0 and u H 1(R 3) such that

div(A u) +V u +u = 0 in D R 3 .Then there exists > 0 depending on (,, ) such that e |r |u H 1(R 3) .

Proof of Lemma 12. The quadratic form q0 on L 2(R 3) with domain D(q 0) =H 1(R 3) , dened by(u,v) D(q 0) D(q 0), q 0(u,v) =

R 3

A u v,

is symmetric and positive. It is also closed since the norm 2L 2 +q0() is equivalent to the usual H 1 norm.This implies that q0

is the quadratic form of a unique self-adjoint operator H 0

on L2(R 3) , whose domain D(H 0)

is dense in H 1(R 3) . It is easy to check that D(H 0) = {u H 1(R 3) | div(A u) L 2(R 3)} and that u D(H 0) ,

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H 0u = div(A u) . Using classical elliptic regularity results [11], we obtain that there exist two constants 0 < < 1and C R + (depending on and ) such that for all regular bounded domains R 3 , and all v H 1() such thatdiv(A v) L 2() ,v C 0, ()

:= sup

|v

|+ sup

(r , r ) |v( r ) v( r )|

|r r |

C v L 2()

+div(A

v) L 2() .

It follows that on the one hand, D(H 0) L (R 3) C 0, (R 3) , withu D(H 0), u L (R 3) + sup(r , r ) R 3R 3

|v( r ) v( r )||r r |

C u L 2 + H 0u L 2 , (78)

and that on the other hand, any u D(H 0) vanishes at innity.Let us now prove that the multiplication by W =V + |r |1 +W denes a compact perturbation of H 0 . For this

purpose, we consider a sequence (u n )n N of elements of D(H 0) bounded for the norm H 0 =( 2L 2 + H 0 2L 2 )12 .

Up to extracting a subsequence, we can assume without loss of generality that there exists u D(H 0) such that:

u n u in H 1 R 3 and L p R 3 for 2 p 6,u n u in L ploc R 3 with 2 p < 6 and a.e.

Besides, it is then easy to check that the potential W = V + |r |1 +W belongs to L 2 +L (R 3) . Let > 0 and(W 2, W ) L

2(R 3) L(R 3) such that W L and W =W 2 +W . On the one hand, W (u n u) L 22 supn N un H 0 , and on the other hand lim nW 2(u n u) L 2 = 0. The latter result is obtained from Lebesguesdominated convergence theorem, using the fact that it follows from (78) that (u n )n N is bounded in L (R 3) . Conse-quently,

limn

W u n W u L 2 = 0,which proves that W is a H 0-compact operator. We can therefore deduce from Weyls theorem that H = H 0 +W denes a self-adjoint operator on L 2(R 3) with domain D(H)

=D(H 0) , and that ess (H )

= ess (H 0) . As q0 is positive,

(H 0) R + and therefore ess(H ) R +.Let us now prove that H has an innite number of negative eigenvalues which form an increasing sequence con-verging to zero. First, H is bounded below since for all v D(H) such that v L 2 = 1,

v|H |v = R 3

A v v + R 3

W v2 v 2L 2 W 2 L 2 v32L 2

27

2563 W 2 4 .

In order to prove that H has at least N negative eigenvalues, including multiplicities, rst notice that we have

H

+V

+

|r

|1

+W

+ (79)

with W + L 2(R 3) + L 3(R 3) . It is proven in [19, Lemma II.1] that the operator in the right-hand side of (79) hasinnitely many eigenvalues including multiplicities. Therefore by the minimax principle, H also has innitely manynegative eigenvalues, including multiplicities.

The lowest eigenvalue of H , which we denote by 1 , is characterized by

1 = inf R 3

A u u + R 3

W |u|2 , u H 1 R 3 , u L 2 = 1 , (80)

and the minimizers of (80) are exactly the set of the normalized eigenvectors of H associated with 1 . Let u1 bea minimizer (80). As for all u H 1(R 3) , |u | H 1(R 3) and | u | = sgn(u) u a.e. on R 3 , |u1| also is a minimizerto (80). Up to replacing u1 with

|u1

|, there is therefore no restriction in assuming that u1 0 on R 3 . We thus have

u 1 H 1 R 3 C 0 R 3 , u 1 0 and div(A u 1) +gu 1 = 0

30/31


with g =W 1 Lploc(R

3) for some p > 32 (take p = 2). A Harnack-type inequality due to Stampacchia [30] thenimplies that if u1 has a zero in R 3 , then u1 is identically zero. As u 1 L 2 = 1, we therefore have u1 > 0 on R 3 . Usingclassical arguments (see e.g. [28]), it is then not difcult to prove that 1 is simple. The proof of the third assertion of the lemma then is straightforward.

Proof of Lemma 13. Consider R > 0 large enough to ensure that 2 V (r ) + 3 2 a.e. on B cR :=R 3 \ B R . It isstraightforward to see that u is the unique solution in H 1(B cR ) to the elliptic boundary problemdiv(A v) +V v +v = 0 in B cR ,v =u on BR .

Let > 0, u =u exp ( ||R) and w =u u . The function w is in H 1(R 3) and is the unique solution in H 1(B cR ) todiv(A w) +V w +w = div(A u) V u u in B cR ,w = 0 on BR .

(81)

Let us now introduce the weighted Sobolev space W 0 (BcR ) dened by

W 0 B cR = v H 10 B cR e ||v H 1 B cRendowed with the inner product (v,w) W 0 (B cR ) = B cR e |r |(v( r )w( r ) + v( r ) w( r )) d r . Multiplying (81) by e2 ||with D (B cR ) and integrating by parts, we obtain

B cR

Ae |r | w e |r | +2 B cR

Ae |r | w r

|r |e |r | +

B cR

(V +)e |r |we |r |

= B cR

Ae |r | u e |r | 2 B cR

Ae |r | u r

|r |e |r |

B cR

(V +)e |r |ue |r |. (82)

Due to the denitions of W 0 (B

cR ) and u , (82) actually holds for (w,) W

0 (B

cR ) W

0 (B

cR ) , and it is straightforwardto see that (82) is a variational formulation equivalent to (81). It is also easy to check that the right-hand side in (82) is

a continuous form on W 0 (BcR ) , so that we only have to prove the coercivity of the bilinear form in the left-hand side

of (82) to be able to apply LaxMilgram lemma. We have for v W 0 (B

cR )

B cR

Ae |r | v e |r | v +2 B cR

Ae |r | v r

|r |e |r |v +

B cR

(V +)e |r |ve |r |v

e |r | v2L 2(B cR ) 2 e |r | v L 2(B cR ) e |r |v L 2(B cR ) +

2

e |r |v 2L 2(B cR )

( ) e |r | v2L 2(B cR ) +

2 e

|r |v 2L 2(B cR ) .

Thus the bilinear form is clearly coercive if < min( , 2 ) , and there is a unique w solution of (81) in W

0 (B

cR ) for

such a . Now since u =w + u , it is clear that e ||u H 1(B cR ) , and then e ||u H 1(R 3) . References

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cances minimizer proof

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