The non-uniform electron gas

THE NON-UNIF ORM ELECTR ON GAS MIH ´ AL Y A. CSIRIK ∗ , 1 AND ANDRE LAEST ADIUS 1 , 2 Abstract. The non-uniform (or inhomogeneous) electron gas has received muc h attention in man y-bo dy quan tum mec hanics and quan tum chemistry in the early days of density functional theory , mainly as a theoretical device to construct gradien t appro ximations via linear resp onse theory . In this article, motiv ated by the recen t works of Lewin, Lieb and Seiringer, we propose a deﬁnition of the quantum (resp. classical) non-uniform electron gas through the use of the grand-canonical Levy–Lieb functional (r esp. the grand-canonical strictly correlated electrons functional), establish these systems as rigorous thermodynamic limits and analyze their basic properties. The non-uniformit y of the gas comes from an arbitrary lattice-perio dic background density . 1. Introduction Before the widespread acceptance of the name “densit y functional theory” early researc hers of the ﬁeld referred to what we call DFT to da y as the theory of the “inhomogeneous electron gas” [ 16 , 26 ]. As far as w e can tell, the (weakly) inho- mogeneous electron gas is regarded a “ﬁctitious system”, or a stepping stone, and not a “real” physical system like the uniform electron gas (UEG). F urthermore, it is alwa ys understo o d in a perturbative sense, i.e. through its linear response [ 11 , 9 ]. These densit y p erturbations are measured relative to the constant density of the homogeneous gas. In [ 9 ] “b oth the case of a lo calized p erturbation as well as some p erio dic structure” are claimed to b e co v ered. The treatmen t of a lo cal- ized p erturbation in the thermodynamic limit remains somewhat unclear. Hence, w e will consider the case of a lattice-p eriodic inhomogeneit y in this w ork. Rather than relying on linear resp onse theory , w e treat the inhomogeneous electron gas as a genuinely nonlinear (i.e. non-p erturbative) system. The weakly inhomogeneous electron gas may b e considered within this nonlinear framew ork as a p erturbation of the homogeneous gas by a slowly v arying inhomogeneity . Henceforth, w e shall adopt the term non-uniform ele ctr on gas (NUEG) in order to b etter align with the existing mathematical physics literature p ertaining to the UEG [ 18 ]. The main reason for this is that we will also employ density functionals to deﬁne the NUEG energy . The alternativ e path would b e to consider the thermody- namic limit of an electronic ground-state problem like in the case of the Jellium [ 24 ]. In the Jellium mo del the external potential is obviously constan t. Ho wev er, it is unclear what external p otential (or whether such p oten tial exists) would generate a prescribed lattice-p eriodic inhomogeneity as a ground-state density . This issue is related to the so-called v -representabilit y problem, which is p o orly understo o d at (*) Corresponding author, email: csirik@gmail.com (1) Dep ar tment of Computer Science, Oslo Metropolit an University, Nor w a y (2) Hylleraas Centre for Quantum Molecular Sciences, Dep ar tment of Chemistr y, University of Oslo, Nor w a y 1 2 NON-UNIFORM ELECTRON GAS the presen t—there is no analytic description of the v -represen tabilit y set except in 1D on a ﬁnite in terv al with Neumann b oundary conditions [ 4 , 5 ]. The approach of [ 18 ] elegan tly av oids these complications by deﬁning the UEG energy through the use of density functionals and demanding the densit y to b e constant. The price to pa y is that w e lose the connection with the ground-state problem and hav e to deal with the constrained optimization problem, whic h ma y not ev en ha v e the usual Sc hr¨ odinger equation as its Euler–Lagrange equation. The said connection is still unresolv ed for the quantum UEG, but is settled for the c lassical case [ 6 , 19 ]. Our deﬁnition of the NUEG in the 3D Coulom b case in volv es a ﬁxed lattice- p eriodic inhomo geneity ζ : R 3 → R + ha ving natural in tegrabilit y and regular- it y properties. This ζ replaces the constan t density ρ 0 in the deﬁnitions of [ 18 ] based on the strictly-correlated electrons (SCE) functional (classical case ) and the grand-canonical Levy–Lieb functional [ 21 ] (quan tum case). In order to obtain a w ell-b eha v ed energy p er v olume, we av erage the energy o v er the translations and rotations of ζ . More precisely , the (3D Coulom b) indirect energy p er volume of the classical NUEG with inhomogeneit y ζ in a b ounded domain Ω ⊂ R 3 is taken to b e e cl Ω ( ζ ) = 1 | Ω | Z SO(3) d R − Z R 3 d a " inf P GC prob. ρ P = 1 Ω ζ ( R ( ·− a )) X n ⩾ 2 X 1 ⩽ j 0. W e will mak e use of the mean v alue notation − Z A u = 1 | A | Z A u for an y A ⊂ R d of ﬁnite measure and u : R d → C . The group of orien tation- preserving rotations of R d is denoted b y SO( d ). The group of all p ermutations of { 1 , . . . , N } is denoted b y S N . 2.2. Lattice-p erio dic functions. Let L ⊂ R d b e a lattice, i.e. is a discrete ad- ditiv e subgroup of R d . A unit cell Λ ⊂ R d of the lattice L is a bounded op en set suc h that its lattice translates (Λ + l ) l ∈L form a tiling of R d , i.e. a pairwise disjoin t family of sets suc h that S l ∈L (Λ + l ) = R d . A function u : R d → C is said to b e L -p eriodic if u ( x + l ) = u ( x ) for all l ∈ L and almost all x ∈ R d . Such functions are almost everywhere determined by their v alues in a unit cell Λ. W e will rep eatedly make use of the following elemen tary fact, the pro of of whic h is provided in Section A for the conv enience of the Reader. Theorem 2.1. F or any L -p erio dic c omplex-value d function u ∈ L 1 loc ( R d ) , the me an value of u over R d exists and e quals its me an value over a unit c el l Λ ⊂ R d of L , i.e. − Z R d u := lim L →∞ − Z C L + a u = − Z Λ u, (1) indep endently and uniformly in a ∈ R d . F or a ﬁxed lattice L , we will adopt the notation L p per ( L ) for the space of L - p eriodic functions such that u ∈ L p loc ( R d ) (equiv alently , u ∈ L p (Λ) for a unit cell Λ of L ). The contin uous inclusions L ∞ per ( L ) ⊂ L p per ( L ) ⊂ L q per ( L ) ⊂ L 1 per ( L ) hold true for all 1 < q ⩽ p < ∞ . Similarly , we in tro duce the Sob olev space H 1 per ( L ). 4 NON-UNIFORM ELECTRON GAS 2.3. Classical density functional theory . F ollowing [ 8 ], we sa y that P = ( P n ) n ⩾ 0 is a gr and-c anonic al pr ob ability if P 0 + X n ⩾ 1 P n ( R dn ) = 1 , where P 0 ∈ [0 , 1] and P n is a ﬁnite p ositiv e symmetric Borel measure on ( R d ) n for all n ⩾ 1. The symmetry of P n means that P n ( B σ (1) × . . . × B σ ( n ) ) = P n ( B 1 × . . . × B n ) for every permutation σ ∈ S n . The density of P is then the marginal ρ P ( B ) = P 1 ( B ) + X n ⩾ 2 n P n ( B , R d ( n − 1) ) . Supp ose that 0 < s < d . The Riesz energy of a grand-canonical probability P is giv en by C s ( P ) = X n ⩾ 2 Z R dn X 1 ⩽ j 0 is a universal constant that dep ends only on d and s . Hence- forth, w e shall work with densities ρ ∈ L 1 ( R d ) ∩ L 1+ s/d ( R d ), whic h automatically ha ve ρ ∈ L 2 d 2 d − s ( R d ) as required ab o v e. Aside from the preceding a priori b ound, the most imp ortant property of the indirect energy is its subadditivity E cl ( ρ 1 + ρ 2 ) ⩽ E cl ( ρ 1 ) + E cl ( ρ 2 ) (3) for every ρ 1 , ρ 2 ∈ L 1 ( R d ) ∩ L 1+ s/d ( R d ). Notice that the densities ρ 1 and ρ 2 ma y o verlap. Another useful property of the indirect energy is its simple scaling law, E cl ( λρ ( λ 1 /d · )) = λ s/d E cl ( ρ ) (4) for every λ > 0. In the quan tum case, neither ( 3 ) nor ( 4 ) holds. NON-UNIFORM ELECTRON GAS 5 2.4. Grand-canonical Levy–Lieb functional. In this section, we introduce no- tations and recall basic facts ab out the quan tum v ersion of the ab o v e functionals. Throughout the paper when discussing the quantum case, w e shall restrict ourselv es to the 3D Coulomb case. Also, we only consider spinless electrons for simplicity of notations. Let D denote the set of fermionic F o c k space states on L 2 ( R 3 ) with ﬁnite ki- netic energy ha ving the additional prop ert y that they comm ute with the n umber op erator. F or a state Γ = (Γ n ) n ⩾ 0 in D we denote its one-particle reduced density matrix by γ Γ ( x, y ) = X n ⩾ 1 n Z R 3( n − 1) Γ n ( x, x 2 , . . . , x n ; y , x 2 , . . . , x n ) d x 2 . . . d x n . F urther, w e denote the densit y of Γ b y ρ Γ ( x ) = γ Γ ( x, x ). This has ρ Γ ∈ L 1 ( R 3 , R + ) and by the Hoﬀmann–Ostenhof inequality ∇ √ ρ Γ ∈ L 2 ( R 3 ). F or an y Γ ∈ D let us introduce the notations T ( Γ ) = T r L 2 ( R 3 ) ( − ∆ γ Γ ) , C ( Γ ) = X n ⩾ 2 X 1 ⩽ j 0. As mentioned ab o ve, E ℏ ( ρ ) is not subadditiv e. A substitute for the subadditivit y prop ert y pioneered b y [ 20 ] is discussed in Section 5.3 b elow. 6 NON-UNIFORM ELECTRON GAS 3. Main resul ts 3.1. Classical non-uniform electron gas. Let 0 < s < d and ﬁx a (classic al) inhomo geneity ζ ∈ L 1+ s/d per ( L ). Setting ρ = 1 Ω ζ ( · − a ) in E cl ( ρ ), the b ounds ( 2 ) imply that the function a 7→ | Ω | − 1 E cl ( 1 Ω ζ ( · − a )) is in L 1 per ( L ), hence its mean v alue ov er R d is well-deﬁned and ﬁnite (see Theorem 2.1 ). F urther, replacing ζ by ζ ( R · ) and av eraging o ver R ∈ SO( d ), we arrive at our deﬁnition 1 of the classic al indir e ct ener gy p er volume e cl Ω ( ζ ) = Z SO( d ) d R − Z R d d a E cl  1 Ω ζ ( R ( · − a ))  | Ω | Clearly , e cl Ω ( ζ ) is isometry-in v ariant with resp ect to b oth Ω and ζ (separately) and v eriﬁes the a priori b ound − c LO ( d, s ) − Z R d ζ 1+ s/d ⩽ e cl Ω ( ζ ) ⩽ 0 . (6) Ph ysically sp eaking, our deﬁnition describes the energy of a “ﬂoating crystal” where the lattice-p eriodic inhomogeneity ζ is allow ed to translate and rotate freely in a “con tainer” Ω, so as to neutralize the energy ﬂuctuations coming from the boundary la yer of lattice cells. Notice that the deﬁnition is lattice-indep enden t in the sense that it is meaningful for every lattice-p erio dic inhomogeneity , and the deﬁnition remains exactly the same. This is because only the mean v alue ov er R d is used, and as long as the in tegrand is lattice-p erio dic with resp ect to some lattice, the mean v alue mak es sense. The isometry-inv ariance of the indirect energy per volume allo ws us to deduce the existence of its thermo dynamic limit using standard metho ds. T o state our result, w e recall a geometric condition for proving thermo dynamic limits. Let Ω ⊂ R d b e a b ounded op en and connected set. Let κ : [0 , t 0 ) → R + b e such that κ ( t ) → 0 as t → 0+. The set Ω is said to hav e κ -r e gular b oundary (in the sense of Fisher [ 10 ]) , if for every 0 ⩽ t < t 0 |{ x ∈ R d : dist( x, ∂ Ω) ⩽ | Ω | 1 /d t }| ⩽ κ ( t ) | Ω | . A sequence of bounded domains { Ω N } ⊂ R d is said hav e a uniformly κ -regular b oundary , if Ω N is has a κ -regular b oundary with the same κ for all N . Theorem 3.1 (Classical non-uniform electron gas) . Fix an inhomo geneity ζ ∈ L 1+ s/d per ( L ) . L et { Ω N } ⊂ R d b e a se quenc e of b ounde d domains having uniformly κ -r e gular b oundary and | Ω N | → ∞ . Then the fol lowing thermo dynamic limit exists lim N →∞ e cl Ω N ( ζ ) = e cl NUEG ( ζ ) (7) and is indep endent of the se quenc e { Ω N } . In the 3D Coulom b case, w e can also deriv e a con v ergence rate estimate for dilated tetrahedra, see Theorem 4.5 . Before pro ceeding, we mak e a few remarks. R emark 1 . (i) Using the scaling relation E cl ( λρ ( λ 1 /d · )) = λ s/d E cl ( ρ ) we hav e that e cl NUEG ( λζ ( λ 1 /d · )) = λ 1+ s/d e cl NUEG ( ζ ) . 1 Our deﬁnition is v aguely motiv ated b y the abstract setting of [ 14 ], where only an a verage translation inv ariance is assumed (see Assumption (A3), ibid.) ab out the energy . NON-UNIFORM ELECTRON GAS 7 The ﬁrst observ ation we make is that it is enough to know e cl NUEG ( ζ ) for ζ having − R ζ = 1 . In fact, w e may write ζ = ρ 0 ζ 0 ( ρ 1 /d 0 · ), where − R ζ 0 = − R ζ 0 ( ρ 1 /d 0 · ) = 1 and ρ 0 = − R ζ are uniquely determined, so that e cl NUEG ( ζ ) = ρ 1+ s/d 0 e cl NUEG ( ζ 0 ) . Roughly sp eaking, e cl NUEG ( ζ ) dep ends on − R ζ in the usual, exp ected manner. Ho w- ev er, it depends on the “shape” of ζ in a non trivial wa y . (ii) Without loss we can assume that the lattice L has a unit cell Λ ⊂ R d with nor- malized volume | Λ | = 1. In fact, we can replace an L -p erio dic ζ such that − R ζ = 1 b y ζ ( | Λ | − 1 /d · ) which is ( | Λ | − 1 /d L )-p eriodic, and also has mean v alue 1. (iii) As exp ected, when ζ ≡ ρ 0 is constant, then ζ 0 ≡ 1 and we recov er the uni- form electron gas energy e cl NUEG ( ρ 0 ) = c UEG ρ 1+ s/d 0 , where the negative constant c UEG = e cl NUEG (1) is the classical UEG energy as deﬁned in [ 18 ]. Lieb and Narn- hofer [ 24 ] ga v e an estimate on the Jellium ground-state energy in 3D Coulom b case, whic h implies c UEG ⩾ − 3 5  9 π 3  1 / 3 ≈ − 1 . 4508. (iv) It is clear that the real function ρ 0 7→ c UEG ρ 1+ s/d 0 is subadditive since it is con- ca ve and v anishes at ρ 0 = 0. More generally , the subadditivity prop ert y of E cl ( ρ ) is inherited by the functional ζ 7→ e cl NUEG ( ζ ), i.e. w e hav e e cl NUEG ( ζ + ζ ′ ) ⩽ e cl NUEG ( ζ ) + e cl NUEG ( ζ ′ ) for all ζ , ζ ′ ∈ L 1+ s/d per ( L ). Next, w e sho w how the NUEG-, and UEG energies are related when the inho- mogeneit y is slo wly v arying. Here, w e restrict ourselves to the 3D Coulomb case. Theorem 3.2 (Local density approximation) . Supp ose that d = 3 and s = 1 . L et p > 3 and 0 < θ < 1 such that θ p ⩾ 4 / 3 . F or every ε > 0 and every inhomo geneity ζ such that ∇ ζ θ ∈ L p loc ( R 3 ) the b ound     e cl NUEG ( ζ ) − c UEG − Z R 3 ζ 4 / 3     ⩽ ε − Z R 3  ζ + ζ 4 / 3  + C p,θ ε b − Z R 3 | ∇ ζ θ | p (8) holds true, wher e b = min { 2 p − 1 , (1 + 3 θ ) p − 4 } and the p ositive c onstant C p,θ only dep ends on p and θ . It may b e b ounde d as C p,θ ⩽ 2 . 71 b  10 θ  b . W e can conclude that for slowly v arying inhomogeneities, the NUEG energy is w ell-approximated by the UEG energy via the LDA. Therefore, in this scenario a p erturbativ e treatment (leading to the “weakly non-uniform electron gas”) migh t b e justiﬁed. This goal is not pursued in the presen t work. Replacing ζ by ζ ( λ · ) in the ab ov e theorem we obtain the conv ergence rate     e cl NUEG ( ζ ( λ · )) − c UEG − Z R 3 ζ 4 / 3     ⩽ C λ 1 / 2  − Z R 3  ζ + ζ 4 / 3   1 − 1 / 2 p  − Z R 3 | ∇ ζ θ | p  1 / 2 p . This form shows that indeed for a nonconstan t inhomogeneit y we get an energy diﬀeren t from the UEG energy , even in the slowly v arying limit. The metho d of pro of of Theorem 3.2 is essentially the same as the one in [ 20 , App endix A]. It is an imp ortan t op en problem to improv e the exp onen t ε and the constan t C p,θ . W e would like to point out that a more general result may b e deriv ed without the regularity assumption by follo wing the ideas of [ 17 ]. 8 NON-UNIFORM ELECTRON GAS 3.2. Quan tum non-uniform electron gas. Next, w e in troduce our deﬁnition of the quantum non-uniform gas and consider some of its basic prop erties. As men tioned ab o ve, w e ignore spin for simplicit y as (collinear) spin is straigh tforw ard to handle, but it would mak e pro ofs muc h more tedious. W e call a function ζ : R 3 → R + a (quantum) inhomo geneity if √ ζ ∈ H 1 per ( L ). Recall that for suc h functions, the mean v alues − R R 3 ζ and − R R 3 | ∇ √ ζ | 2 are well- deﬁned and ﬁnite. By the Sob olev inequalit y , w e ha v e R Λ ζ p < ∞ for all 1 ⩽ p ⩽ 3. In particular, − R R 3 ζ 5 / 3 and − R R 3 ζ 4 / 3 are ﬁnite. Next, w e deﬁne the non-uniform electron gas energy per ﬁnite volume. W e oﬀer t wo deﬁnitions follo wing [ 18 , 20 ] which turn out to b e equiv alen t. 3.2.1. Deﬁnition b ase d sme ar e d indic ators. Recall the N -represen tabilit y criterion from [ 23 ], whic h sa ys that a densit y ρ ∈ L 1 ( R 3 , R + ) with R R 3 ρ = N ∈ N comes from an N -particle w a v efunction with ﬁnite kinetic energy precisely if √ ρ ∈ H 1 ( R 3 ). So in the quan tum case, we need to cut oﬀ the inhomogeneity ζ smoothly near the b oundary of our domain. Let 0 ⩽ η ∈ C ∞ c ( R 3 ) b e radial with supp η ⊂ B 1 and R R 3 η = 1. Deﬁne η δ ( x ) = δ − 3 η ( δ − 1 x ). Then supp η δ ⊂ B δ and R R 3 η δ = 1. W e sa y that η δ is a r e gularization function (or molliﬁer) with smearing parameter δ > 0. While the ﬁnite-volume quan tities are clearly dep endent on the particular choice of the regularization function, this dep endence will disapp ear when passing to the ther- mo dynamic limit. Fix a bounded domain Ω ⊂ R 3 and an inhomogeneit y ζ . Clearly , p ( 1 Ω ∗ η δ ) ζ ∈ H 1 ( R 3 ) so we may consider the function f Ω ,η δ ,ζ : R 3 → R given b y f Ω ,η δ ,ζ ( a ) = E ℏ  ( 1 Ω − a ∗ η δ ) ζ  | Ω | = E ℏ  ( 1 Ω ∗ η δ ) ζ ( · − a )  | Ω | . (9) In the second equality , the translation-inv ariance of the Levy–Lieb functional F ℏ LL ( ρ ) and of the Coulom b self-energy D ( ρ ) w as used. As ζ is L -p eriodic, it is obvious that f Ω ,η δ ,ζ is L -p erio dic. Moreo v er, we ha v e the a priori estimate − R R 3 | f Ω ,η δ ,ζ ( a ) | d a < ∞ (see Corollary 5.7 b elo w). Using Theorem 2.1 , w e can deduce that the mean v alue of f Ω ,η δ ,ζ o ver R 3 exists and is ﬁnite. Next, w e av erage o v er all rotations R ∈ SO(3) in − R R 3 f Ω ,η δ ,ζ ( R · ) to make our energy rotationally inv ariant as well. In summary , we arriv e at e ℏ Ω ,η δ ( ζ ) = Z SO(3) d R − Z R 3 d a E ℏ  ( 1 Ω ∗ η δ ) ζ ( R ( · − a ))  | Ω | (10) whic h by construction is isometry-inv ariant with resp ect to b oth Ω and ζ , sepa- rately . In Corollary 5.8 b elow, we state an a priori b ound on e ℏ Ω ,η δ ( ζ ). Clearly , our deﬁnition collapses to the usual indirect energy p er volume of the uniform electron gas (cf. [ 20 ]) by setting ζ ≡ ρ 0 , i.e. e ℏ Ω ,η δ ( ρ 0 ) = | Ω | − 1 E ℏ  ( 1 Ω ∗ η δ ) ρ 0  . 3.2.2. Deﬁnition b ase d on a tr ansition r e gion. Instead of using a smeared indicator function to ac hieve smo oth cutoﬀ, it is also reasonable to do the follo wing. Let Ω ⊂ R 3 b e a b ounded connected domain. F or an y s > 0, introducing the inner set Ω s − = { x ∈ Ω : dist( x, ∂ Ω) ⩾ s } and the outer set Ω s + = Ω ∪ { x ∈ R 3 : dist( x, ∂ Ω) ⩽ s } , NON-UNIFORM ELECTRON GAS 9 w e can giv e another deﬁnition of the indirect energy p er volume, e e ℏ Ω ,s ( ζ ) = Z SO(3) d R − Z R 3 d a inf √ ρ ∈ H 1 ( R 3 ) ζ ( R ( ·− a )) 1 Ω s − ⩽ ρ ⩽ ζ ( R ( ·− a )) 1 Ω s + E ℏ ( ρ ) | Ω | (11) T aking ρ = ( 1 Ω ∗ η δ ) ζ ( R ( · − a )) for a suﬃciently small δ > 0, we may use an a priori upp er b ound (see Proposition 5.6 b elo w), and for the lo w er bound using the Lieb–Oxford inequality , we obtain that the a -integrand is summable, hence again b y Theorem 2.1 the ab ov e mean v alue exists and ﬁnite. In tuitively speaking, the ab o v e minimization allows the density to “relax” in the transition region Ω s + \ Ω s − . W e will require that the scale s of this transition region go es to inﬁnity in the thermo dynamic limit in such a wa y that the ratio s/ | Ω | 1 / 3 b ecomes negligible, for a regular domain sequence Ω. 2 The a -integrand is again L -p eriodic b ecause ζ is. Clearly , e e ℏ Ω ,s ( ζ ) is isometry-inv arian t in b oth ζ and Ω. The conceptual b eneﬁt of this deﬁnition ov er the previous one is that the “shap e” of the transition is allo wed to b e arbitrary . 3.2.3. Thermo dynamic limit and e quivalenc e of deﬁnitions. Next, we consider the existence and equiv alence of the thermo dynamic limits for b oth deﬁnitions. Theorem 3.3 (Quantum non-uniform electron gas) . Fix an inhomo geneity √ ζ ∈ H 1 per ( L ) . L et { Ω N } ⊂ R 3 b e a se quenc e of b ounde d c onne cte d domains with | Ω N | → ∞ , such that Ω N has uniformly κ -r e gular b oundary with κ ( t ) = C t . (i) L et { δ N } ⊂ R + b e any se quenc e such that δ N / | Ω N | 1 / 3 → 0 and δ N | Ω N | 1 / 3 → ∞ . Then the fol lowing thermo dynamic limit exists lim N →∞ e ℏ Ω N ,η δ N ( ζ ) = e ℏ NUEG ( ζ ) , (12) and is indep endent of the se quenc es { Ω N } , { δ N } and the r e gularization function η . (ii) L et { s N } ⊂ R + b e a se quenc e such that s N → ∞ and s N / | Ω N | 1 / 3 → 0 . Then the fol lowing thermo dynamic limit exists lim N →∞ e e ℏ Ω N ,s N ( ζ ) = e ℏ NUEG ( ζ ) , (13) with the same limiting e ℏ NUEG ( ζ ) as in (i). A gain, the limit is indep endent of the se quenc es { Ω N } and { s N } . Again, we mak e a few remarks. R emark 2 . (i) By dropping the Coulomb interaction, w e obtain a similar result for the kinetic energy per volume τ ℏ Ω ,η δ ( ζ ) and therefore the ph ysically most in teresting quan tity , the exchange-c orr elation ener gy p er volume e ℏ , xc Ω ,η δ ( ζ ) = e ℏ Ω ,η δ ( ζ ) − τ ℏ Ω ,η δ ( ζ ) also admits a thermo dynamic limit. (ii) The follo wing a priori bounds hold true (see Corollary 5.8 ). F or ev ery 0 < ε ⩽ 1 / 15 and every √ ζ ∈ H 1 per ( L ) there holds e ℏ NUEG ( ζ ) ⩽ (1 + ε ) c TF ℏ 2 − Z R 3 ζ 5 / 3 + 38 ℏ 2 15 1 ε − Z R 3 | ∇ p ζ | 2 , 2 Recall that for regular domains diam(Ω) ⩽ C | Ω | 1 / 3 , see [ 10 ]. 10 NON-UNIFORM ELECTRON GAS where c TF = 3 5 (2 π ) 2 (4 π / 3) − 2 / 3 is the Thomas–F ermi constant. F or every 0 < ε ⩽ 3 / 5 and every √ ζ ∈ H 1 per ( L ) we ha v e e ℏ NUEG ( ζ ) ⩾ (1 − ε ) c TF ℏ 2 − Z R 3 ζ 5 / 3 − c LO − Z R 3 ζ 4 / 3 − 20 ℏ 2 27 1 ε − Z R 3 | ∇ p ζ | 2 . (iii) In the uniform case, the ab o v e tw o b ounds become c TF ℏ 2 ρ 5 / 3 0 − c LO ρ 4 / 3 0 ⩽ e ℏ UEG ( ρ 0 ) ⩽ c TF ℏ 2 ρ 5 / 3 0 . 3.2.4. F urther r esults. Next, we consider the quantum analog of Theorem 3.2 . Theorem 3.4 (Local density appro ximation) . L et p > 3 and 0 < θ < 1 such that 2 ⩽ pθ ⩽ 1 + p/ 2 . F or every ε > 0 and inhomo geneity ζ such that for al l ∇ ζ θ ∈ L p loc ( R 3 ) the b ound     e ℏ NUEG ( ζ ) − − Z R 3 e ℏ UEG ( ζ ( x )) d x     ⩽ ε − Z R 3  ζ + ζ 2  + C (1 + ε ) ε − Z R 3 | ∇ p ζ | 2 + C ε 4 p − 1 − Z R 3 | ∇ ζ θ | p holds true for some universal c onstant C > 0 which only dep ends on p , θ and ℏ . Plugging in ζ ( λ · ) and ε = λ α , we ﬁnd the conv ergence rate     e ℏ NUEG ( ζ ( λ · )) − − Z R 3 e ℏ UEG ( ζ ( x )) d x     ⩽ λ 1 / 10 − Z R 3  ζ + ζ 2  + C λ 19 / 20  − Z R 3 | ∇ p ζ | 2 + − Z R 3 | ∇ ζ θ | p  for the slowly v arying limit λ → 0. Next, w e consider the question of obtaining e cl NUEG ( ζ ) as the semiclassical limit of e ℏ NUEG ( ζ ) as ℏ → 0. Note ﬁrst that by scaling relation ( 5 ), w e ha v e for all λ > 0 e ℏ Ω ,η δ ( ζ ) = λ 4 / 3 e λ 1 / 6 ℏ λ 1 / 3 Ω ,η λ 1 / 3 δ ( λ − 1 ζ ( λ − 1 / 3 · )) . This suggests to write ζ = ρ 0 ζ 0 ( ρ 1 / 3 0 · ), where − R ζ 0 = − R ζ 0 ( ρ 1 / 3 0 · ) = 1 and ρ 0 = − R ζ = λ , so that in the thermo dynamic limit e ℏ NUEG ( ζ ) = ρ 4 / 3 0 e ρ 1 / 6 0 ℏ NUEG ( ζ 0 ) = ρ 4 / 3 0 b e NUEG ( ρ 1 / 3 0 ℏ 2 ; ζ 0 ) . Here, we hav e set b e NUEG ( µ ; ζ 0 ) = e √ µ NUEG ( ζ 0 ) for all µ ⩾ 0 and − R ζ 0 = 1. F or ﬁxed ζ 0 , the function µ 7→ b e NUEG ( µ ; ζ 0 ) is concav e and increasing. Prop osition 3.5 (Semiclassical bound) . F or every normalize d inhomo geneity √ ζ 0 ∈ H 1 per ( L ) with − R ζ 0 = 1 , ther e holds lim inf µ → 0 b e NUEG ( µ ; ζ 0 ) ⩾ e cl NUEG ( ζ 0 ) . It remains an op en problem to obtain the opp osite bound. The trial state con- struction based on the regularization strategy of [ 18 ] do es not w ork here as that w ould result in a molliﬁed inhomogeneit y . Finally , we mention that the functional ζ 7→ e ℏ NUEG ( ζ ) inherits the weak*-lo wer semicon tinuit y prop ert y of the grand-canonical Levy–Lieb functional [ 21 ]. See The- orem B.1 near the end of the pap er for the precise statement and its pro of. NON-UNIFORM ELECTRON GAS 11 4. Proofs for the classical case The rest of the pap er is devoted to the pro ofs. W e ﬁrst provide pro ofs for our results concerning the classical NUEG. Throughout this section, we omit the designation “cl” from E cl ( ρ ) and e cl Ω ( ζ ) for clarity . 4.1. Kno wn b ounds. The follo wing estimate on the indirect Riesz energy will b e used rep eatedly in the sequel. Theorem 4.1 (Lieb–Oxford inequalit y) . L et 0 < s < d . F or any symmetric gr and- c anonic al pr ob ability P , the b ound C s ( P ) − D s ( ρ P ) ⩾ − c LO ( d, s ) Z R d ρ 1+ s/d P holds true with some universal c onstant c LO ( d, s ) > 0 . The pro of is a straightforw ard mo diﬁcation of the one in [ 25 ]. In the 3D Coulom b case, the constan t c LO = c LO (3 , 1) ⩽ 1 . 58 is the b est known at presen t [ 22 ]. The pro of of [ 22 ] works all the same in the grand-canonical setting. F or the next result, we recall the concept of lo calization of an N -particle sym- metic probability P on R dN . See [ 8 ] for more details. F or a subset A ⊂ R d , the lo c alization (or restriction) of P onto A is a grand-canonical symmetric probabilit y P | A = (( P | A ) n ) n ⩾ 0 and is given by ( P | A ) 0 = P  ( R d \ A ) N  ( P | A ) n ≡  N n  P  · × ( R d \ A ) N − n  on A n ( n = 1 , . . . , N − 1) ( P | A ) N = P on A N and ( P | A ) n ≡ 0 whenever n ⩾ N + 1. Next, we sp ecify a sp ecial tetrahedral tiling of R 3 that will b e used throughout the pro ofs. Recall that C 1 = ( − 1 2 , 1 2 ) 3 denotes the origin-centered op en unit cub e. Let ∆ j = T j ( ∆ ) ( j = 1 , . . . , 24) b e a tiling of C 1 b y 24 congruent op en tetrahedra, where the isometries T j are giv en by T j ( x ) = R j x − z j for appropriate R j ∈ SO(3) and z j ∈ C 1 . Here, ∆ is a “reference” tetrahedron with barycenter 0 and T j is the isometry that maps ∆ to ∆ j . Giv en a scale ℓ > 0, w e may write R 3 = [ z ∈ Z 3 ( C ℓ + ℓz ) = [ z ∈ Z 3 24 [ j =1 ( ℓ ∆ j + ℓz ) . The corresp onding partition of unit y is giv en by X z ∈ Z 3 24 X j =1 1 ℓ ∆ j ( · − ℓz ) ≡ 1 (a.e.) . (14) W e can no w state the following very imp ortant result that allo ws us to spatially decouple the indirect 3D Coulomb energy on the tetrahedral tiling [ 12 , 13 , 18 ]. 12 NON-UNIFORM ELECTRON GAS Theorem 4.2 (Graf–Sc henker inequalit y) . F or any N -p article symmetric pr ob a- bility density P and any ℓ > 0 the b ound C ( P ) − D ( ρ P ) ⩾ Z SO(3) d R − Z C ℓ d τ X z ∈ Z 3 24 X j =1  C ( P | R ⊤ ( ℓ ∆ j + ℓz + τ ) ) − D ( 1 R ⊤ ( ℓ ∆ j + ℓz + τ ) ρ P )  − c GS ℓ Z R 3 ρ P holds true with the c onstant c GS = π 1+2 √ 2 4 ≈ 3 . 0068 . In p articular, ther e exists an isometry ( R, τ ) ∈ SO(3) × C ℓ such that C ( P ) − D ( ρ P ) ⩾ X z ∈ Z 3 24 X j =1  C ( P | R ⊤ ( ℓ ∆ j + ℓz + τ ) ) − D ( 1 R ⊤ ( ℓ ∆ j + ℓz + τ ) ρ P )  − c GS ℓ Z R 3 ρ P . (15) The constant c GS is obtained by a straigh tforw ard but lengthy calculation, using the explicit formulas of [ 12 , 13 ]. A similar estimate holds true for grand-canonical probabilities as well (with the same constant). The second term on the r.h.s. is a kind of lo calization error analogous to the term arising from the lo calization of the kinetic energy (i.e. the IMS form ula). W e will use the follo wing simple observ ations to replace a contin uous inhomo- geneit y ζ lo cally in a bounded domain Ω ⊂ R d . Let ζ = min Ω ζ , and ζ = max Ω ζ denote the minim um and maxim um of ζ ov er Ω. Then by the subadditivity and nonp ositivit y of E ( ρ ), we ha v e E ( 1 Ω ζ ) ⩽ E ( 1 Ω ζ ) , (16) and E ( 1 Ω ζ ) ⩽ E ( 1 Ω ζ ) . (17) In order to control the ﬂuctuations of the density lo cally , we need the follo wing form of Morrey’s inequalit y . Theorem 4.3 (Morrey’s inequality on tetrahedra) . F or every p > 3 and u ∈ W 1 ,p ( ∆ ) , wher e ∆ ⊂ R 3 is a tetr ahe dr on isometric to a dilation of the tetr ahe dr on ∆ 1 ab ove, the b ound | u ( x ) − u ( y ) | ⩽ c Mo | x − y | 1 − 3 /p ∥ ∇ u ∥ L p , holds true for al l x, y ∈ ∆ , wher e the c onstant c Mo > 0 only dep ends on p . The estimate c Mo ⩽ 2 · 24 1 /p p − 1 p +1 holds. The estimate on the constan t is the one obtained in the proof of [ 1 , Lemma 4.28]. 4.2. Thermo dynamic limit for dyadic cub es. T o pro v e that the limit e C 2 N ( ζ ) exists we ﬁrst establish that the energy is nonincreasing for dyadic cub es. Lemma 4.4. e C 2 N +1 ( ζ ) ⩽ e C 2 N ( ζ ) for every N ⩾ 1 . NON-UNIFORM ELECTRON GAS 13 Pr o of. The big cube C 2 N +1 is essentially comp osed of 2 d smaller cub es of side length 2 N with centers placed at the v ertices of C 2 N . More precisely , C 2 N +1 = [ z ∈{± 1 } d  C 2 N + 2 N − 1 z  . Hence, using the subadditivit y and the isometry inv ariance of E ( ρ ), e C 2 N +1 ( ζ ) = Z SO( d ) d R − Z R d d a E  P z ∈{± 1 } d 1 C 2 N +2 N − 1 z ζ ( R ( · − a ))  | C 2 N +1 | ⩽ 1 2 d X z ∈{± 1 } d Z SO( d ) d R − Z R d d a E  1 C 2 N +2 N − 1 z ζ ( R ( · − a ))  | C 2 N | = 1 2 d X z ∈{± 1 } d Z SO( d ) d R − Z R d d a E  1 C 2 N ζ ( R ( · + 2 N − 1 z − a ))  | C 2 N | = 1 2 d X z ∈{± 1 } d Z SO( d ) d R − Z R d d a E  1 C 2 N ζ ( R ( · − a ))  | C 2 N | = e C 2 N ( ζ ) . In the next to last step we used the translational inv ariance of the mean v alue to absorb 2 N − 1 z in the argument of ζ . □ Hence, the numerical sequence { e C 2 N +1 ( ζ ) } N ⩾ 1 is nonincreasing and b ounded from b elo w according to ( 6 ). Therefore, the limit e cl NUEG ( ζ ) = lim N →∞ e C 2 N ( ζ ) exists and is ﬁnite. 4.3. Thermo dynamic limit for general domains. First, we prov e the upp er b ound via an inner appro ximation by dy adic cub es. Let J n,N 0 = { z ∈ Z d : ( C 2 n + 2 n z ) ⊂ Ω N } and Ω ′ N = [ z ∈ J n,N 0  C 2 n + 2 n z  ⊂ Ω N . By the subadditivity and nonp ositivit y of E ( ρ ), w e hav e E ( 1 Ω N ζ ) | Ω N | ⩽ E ( 1 Ω ′ N ζ ) | Ω N | + E ( 1 Ω N \ Ω ′ N ζ ) | Ω N | ⩽ 1 | Ω N | E X z ∈ J n,N 0 1 C 2 n +2 n z ζ ! ⩽ 1 | Ω N | X z ∈ J n,N 0 E ( 1 C 2 n +2 n z ζ ) . After replacing ζ by ζ ( R ( · − a )) and a v eraging o ver all translations and rotations, w e ﬁnd e Ω N ( ζ ) ⩽ 2 n | J n,N 0 | | Ω N | e C 2 n ( ζ ) = | Ω ′ N | | Ω N | e C 2 n ( ζ ) . 14 NON-UNIFORM ELECTRON GAS T o estimate the prefactor on the r.h.s, we use the uniform κ -regularity of the se- quence Ω N in the manner | Ω ′ N | = | Ω N | − X z ∈ Z 3 ( C 2 n +2 n z ) ∩ ( R d \ Ω N )  = ∅ | ( C 2 n + 2 n z ) ∩ Ω N | ⩾ | Ω N | − |{ x ∈ R d : d ( x, ∂ Ω N ) ⩽ 2 n √ d }| ⩾ | Ω N |  1 − κ (2 n √ d | Ω N | − 1 /d )  . Since κ ( t ) → 0 as t → 0, w e obtain after letting N → ∞ and then n → ∞ , that lim sup N →∞ e Ω N ( ζ ) ⩽ e cl NUEG ( ζ ) , where e cl NUEG ( ζ ) is the limit for dy adic cub es, as ab ov e. It remains to pro v e the lo wer b ound. By an appropriate translation of Ω N w e can assume that it is contained in a dyadic cub e C 2 K of side length | Ω N | 1 /d . W e no w take J n,N c = { z ∈ Z d : ( C 2 n + 2 n z ) ⊂ C 2 K \ Ω N } and Ω ′′ N = [ z ∈ J n,N c  C 2 n + 2 n z  . Therefore, we ma y decomp ose C 2 K = Ω N ∪ Ω ′′ N ∪ R N , where the remainder set is R N = [ z ∈ J n,N c ( C 2 n +2 n z ) ∩ ∂ Ω N  = ∅  C 2 n + 2 n z  ∩  C 2 K \ Ω N  . By Lemma 4.4 , subadditivit y and nonp ositivit y of E ( ρ ), we ha v e | C 2 K | | Ω N | e cl NUEG ( ζ ) ⩽ | C 2 K | | Ω N | e C 2 K ( ζ ) ⩽ e Ω N ( ζ ) + | Ω ′′ N | | Ω N | e C 2 n ( ζ ) Rearranging, we obtain e cl NUEG ( ζ ) + | Ω ′′ N | | Ω N | ( e cl NUEG ( ζ ) − e C 2 n ( ζ )) + | R N | | Ω N | e cl NUEG ( ζ ) ⩽ e Ω N ( ζ ) . Here, using the uniform κ -regularity of { Ω N } , we ha v e | R N | = X z ∈ J n,N c ( C 2 n +2 n z ) ∩ ∂ Ω N  = ∅    C 2 n + 2 n z  ∩  C 2 K \ Ω N    ⩽ |{ x ∈ R d : d ( x, ∂ Ω N ) ⩽ 2 n √ d }| ⩽ | Ω N | κ  2 n √ d | Ω N | − 1 /d  . Clearly , | Ω ′′ N | ⩽ | C 2 K | ⩽ C | Ω N | and so in total we get the estimate e cl NUEG ( ζ ) + C  e cl NUEG ( ζ ) − e C 2 n ( ζ )  + κ  2 n √ d | Ω N | − 1 /d  e cl NUEG ( ζ ) ⩽ e Ω N ( ζ ) . T aking the liminf as N → ∞ and n → ∞ , we arrive at e cl NUEG ( ζ ) ⩽ lim inf N →∞ e Ω N ( ζ ) . In summary , the limits lim N →∞ e Ω N ( ζ ) and lim N →∞ e C 2 N ( ζ ) exist and equal to a common num b er e cl NUEG ( ζ ). This completes the pro of of Theorem 3.1 . NON-UNIFORM ELECTRON GAS 15 4.4. Con v ergence rate for tetrahedra. In the 3D Coulomb case, when the do- main is a dilated (reference) tetrahedron, w e can obtain information ab out the rate of conv ergence e ℓ ∆ ( ζ ) → e NUEG ( ζ ) as ℓ → ∞ . Theorem 4.5. Supp ose that d = 3 and s = 1 . F or ℓ > 0 suﬃciently lar ge, the b ounds e NUEG ( ζ ) ⩽ e ℓ ∆ ( ζ ) ⩽ e NUEG ( ζ ) + c GS ℓ − Z R 3 ζ hold true. Pr o of. Using the subadditivity and nonp ositivity of E ( ρ ), e ℓ ′ ∆ ( ζ ) = − Z R 3 d a Z SO(3) d R E  1 ℓ ′ ∆ ζ ( R ( · − a ))  | ℓ ′ ∆ | ⩽ 1 | ℓ ′ ∆ | X ( z ,j ) ∈ J 0 − Z R 3 d a Z SO(3) d R E  1 ℓ ∆ j + ℓz ζ ( R ( · − a ))  = | J 0 | | ℓ ′ ∆ | − Z R 3 d a Z SO(3) d R E  1 ℓ ∆ ζ ( R ( · − a ))  = | J 0 || ℓ ∆ | | ℓ ′ ∆ | e ℓ ∆ ( ζ ) where J 0 = { ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : ( ℓ ∆ j + ℓz ) ⊂ ℓ ′ ∆ } . Recall that e ℓ ∆ ( ζ ) ⩽ 0, so using the estimate | J 0 || ℓ ∆ | | ℓ ′ ∆ | ⩾ ℓ ′ 3 − C ℓ ′ 2 ℓ ℓ ′ 3 = 1 − C ℓ ℓ ′ , and taking the limit ℓ ′ → ∞ gives the stated low er bound. F or the upp er b ound using the Graf–Sc henker inequalit y (Theorem 4.2 ), e ℓ ′ ∆ ( ζ ) = Z SO(3) d Q − Z R 3 d a E  1 ℓ ′ ∆ ζ ( Q ( · − a ))  | ℓ ′ ∆ | ⩾ 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ X z ∈ Z 3 24 X j =1 E  1 ℓ ′ ∆ 1 ℓ ∆ j ( R · − ℓz − τ ) ζ ( Q ( · − a ))  − c GS ℓ 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z R 3 1 ℓ ′ ∆ ζ ( Q ( · − a )) =: (I) + (I I) . Here, (I I) is simply − C ℓ − R R 3 ζ . T o deal with (I), we deﬁne the index set J := ( ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : supp( 1 ℓ ∆ j ( R · − ℓz − τ )) ∩ supp( 1 ℓ ′ ∆ )  = ∅ , for some R ∈ SO(3) and τ ∈ C ℓ ) , whic h collects all the small tetrahedra R ⊤ ( ℓ ∆ j + ℓz + τ ) that p ossibly intersect with the big tetrahedron ℓ ′ ∆ . F urthermore, we deﬁne J 0 := ( ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : supp( 1 ℓ ∆ j ( R · − ℓz − τ )) ⊂ ℓ ′ ∆ , for all R ∈ SO(3) and τ ∈ C ℓ ) , whic h contains the small tetrahedra that are inside ℓ ′ ∆ . 16 NON-UNIFORM ELECTRON GAS The J 0 -part of (I) reads (Ia) = 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ X ( z ,j ) ∈ J 0 E  1 ℓ ∆ j ( R · − ℓz − τ ) ζ ( Q ( · − a ))  = 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a X ( z ,j ) ∈ J 0 E  1 ℓ ∆ ζ ( Q ( · − a ))  = | J 0 || ℓ ∆ | | ℓ ′ ∆ | e ℓ ∆ ( ζ ) . The J \ J 0 -part of (I) can b e b ounded by the Lieb–Oxford inequality as (Ib) = 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ × X ( z ,j ) ∈ J \ J 0 E  1 ℓ ′ ∆ 1 ℓ ∆ j ( R · − ℓz − τ ) ζ ( Q ( · − a ))  ⩾ − c LO 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ × X ( z ,j ) ∈ J \ J 0 Z R 3 1 ℓ ′ ∆ 1 ℓ ∆ j ( R · − ℓz − τ ) ζ ( Q ( · − a )) 4 / 3 = − c LO 1 | ℓ ′ ∆ | Z SO(3) d R − Z C ℓ d τ X ( z ,j ) ∈ J \ J 0 Z R 3 1 ℓ ′ ∆ 1 ℓ ∆ j ( R · − ℓz − τ ) − Z R 3 ζ 4 / 3 ⩾ − c LO | ℓ ∆ || J \ J 0 | | ℓ ′ ∆ | − Z R 3 ζ 4 / 3 . Clearly , the volume ratios may be b ounded as | J 0 || ℓ ∆ | | ℓ ′ ∆ | ⩽ 1 and | ℓ ∆ || J \ J 0 | | ℓ ′ ∆ | ⩽ C ℓℓ ′ 2 ℓ ′ 3 = C ℓ ℓ ′ . Hence, in the limit ℓ ′ → ∞ we obtain the estimate e NUEG ( ζ ) ⩾ e ℓ ∆ ( ζ ) − c GS ℓ − Z R 3 ζ , whic h ﬁnishes the pro of. □ 4.5. Lo cal density approximation. As mentioned ab o v e, the pro of of Theo- rem 3.2 is v ery similar to that of [ 20 , Theorem 4]. Instead of explaining the diﬀer- ences, we adapt the pro of technique to our case and pro duce explicit constants. If ε is suﬃciently large, the estimate follo ws from the Lieb–Oxford inequality . More precisely , from ( 6 ), 0 ⩾ e cl ℓ ∆ ( ζ ) ⩾ − c LO − Z R 3 ζ 4 / 3 = c UEG − Z R 3 ζ 4 / 3 − ( c LO − | c UEG | ) − Z R 3 ζ 4 / 3 , hence the stated estimate ( 8 ) follows from this, whenever ε ⩾ c LO − | c UEG | . Note that c LO ⩾ | c UEG | according to [ 18 ]. So w e may assume that ε < ε 0 := c LO − | c UEG | . First, we prov e the upper b ound. By subadditivity and nonp ositivit y of E ( ρ ) and ( 16 ), E ( 1 ℓ ′ ∆ ζ ) ⩽ X ( z ,j ) ∈ J 0 E ( 1 ℓ ∆ j + ℓz ζ ) ⩽ X ( z ,j ) ∈ J 0 E  1 ℓ ∆ j + ℓz ζ z ,j  , (18) NON-UNIFORM ELECTRON GAS 17 where J 0 = { ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : ( ℓ ∆ j + ℓz ) ⊂ ℓ ′ ∆ } and ζ z ,j = min ℓ ∆ j + ℓz ζ . A sp ecial case of Theorem 4.5 reads c UEG ρ 4 / 3 0 ⩽ E ( 1 ℓ ∆ ρ 0 ) | ℓ ∆ | ⩽ c UEG ρ 4 / 3 0 + c GS ℓ ρ 0 . (19) Inserting the upp er b ound in our estimate ( 18 ), E ( 1 ℓ ′ ∆ ζ ) ⩽ c UEG X ( z ,j ) ∈ J 0 Z ℓ ∆ j + ℓz ζ 4 / 3 z ,j + c GS ℓ X ( z ,j ) ∈ J 0 Z ℓ ∆ j + ℓz ζ z ,j = c UEG Z ℓ ′ ∆ ζ 4 / 3 + | c UEG | X ( z ,j ) ∈ J 0 Z ℓ ∆ j + ℓz  ζ 4 / 3 − ζ 4 / 3 z ,j  + c GS ℓ Z ℓ ′ ∆ ζ . In the second term, the integral measuring the lo cal ﬂuctuations of the inhomo- geneit y is estimated using Morrey’s and H¨ older’s inequality , Z ℓ ∆ j + ℓz  ζ 4 / 3 − ζ 4 / 3 z ,j  ⩽ a p  4 3 θ  p a c p Mo | ∆ | 1 ε p + p a − 1 Z ℓ ∆ j + ℓz | ∇ ζ θ | p +  1 − a p  ε Z ℓ ∆ j + ℓz ζ ( 4 3 − θa ) p p − a (20) for all 0 < a ⩽ 1 and w e hav e used ε = 1 /ℓ . T o pro ceed, w e require that 1 ⩽  4 3 − θ a  p p − a ⩽ 4 3 , or equiv alen tly that 4 3 ⩽ θp ⩽ 1 + p 3 a , so that w e can b ound point wise as ζ ( 4 3 − θa ) p p − a ⩽ ζ + ζ 4 / 3 . Moreov er, we replace ε →  | c UEG |  1 − a p  + c GS  − 1 ε to absorb the prefactor in the ε term. After replacing ζ b y ζ ( R ( · − a )) and a veraging o ver all rotations and translations, and taking the limit ℓ ′ → ∞ , we arriv e at e NUEG ( ζ ) ⩽ c UEG − Z R 3 ζ 4 / 3 + ε − Z R 3  ζ + ζ 4 / 3  + C p,θ,a ε p + p a − 1 − Z R 3 | ∇ ζ θ | p , where C p,θ,a = a p  4 3 θ  p a 2 p e 2 3 5  9 π 3  1 / 3 + π 1 + 2 √ 2 4 ! p + p a − 1 and w e used the Lieb–Narnhofer b ound | c UEG | ⩽ 3 5  9 π 3  1 / 3 from [ 24 ] and  p − 1 p +1  p ⩽ e − 2 . W e replace C p,θ,a with its rough b ound C p,θ,a ⩽ 15 8  10 θ  p + p a − 1 . (21) The optimal choice for a minimizing the exp onent of ε is a = ( p 3( θp − 1) θ p ⩽ 1 + p 3 1 otherwise whic h gives the stated exp onent b . 18 NON-UNIFORM ELECTRON GAS The proof of the lo wer b ound is less direct. The form ( 15 ) of the Graf–Sc henk er inequalit y implies that E ( 1 ℓ ′ ∆ ζ ) ⩾ X z ∈ Z 3 24 X j =1 E  1 ℓ ∆ j ( · − ℓz ) 1 ℓ ′ ∆ ζ  − c GS ℓ Z R 3 1 ℓ ′ ∆ ζ , (22) where we absorb ed the isometry ( R, τ ) into ζ (recall that e ℓ ′ ∆ ( ζ ) is isometry- in v ariant in ζ ). W e would like to use ( 17 ) to replace ζ b y its maxim um ζ z ,j = max ℓ ∆ j + ℓz ζ , and after applying the lo wer b ound of ( 19 ), pro ceed in a similar wa y as ab o v e. How ev er, this does not w ork b ecause w e cannot lo cally control ζ z ,j b y (p ossibly a constant times) ζ ( x ). T o contin ue, let J = { ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : ( ℓ ∆ j + ℓz ) ∩ ( ℓ ′ ∆ )  = ∅} and J 0 exactly as ab ov e. Case I. (Simple tetrahedra) Deﬁne the set J S 0 of all tetrahedra ( z , j ) ∈ J 0 suc h that c LO Z ℓ ∆ j + ℓz ζ 4 / 3 ⩽ ε Z ℓ ∆ j + ℓz  ζ + ζ 4 / 3  + A ε p + p/a − 1 Z ℓ ∆ j + ℓz | ∇ ζ θ | p , (23) where A > 0 is a univ ersal constant depending only on p , θ and a , whic h we sp ecify later. Recall that we put ε = 1 /ℓ . The terms in ( 22 ) belonging to J S 0 can be b ounded as X ( z ,j ) ∈ J S 0 E  1 ℓ ∆ j ( · − ℓz ) ζ  ⩾ − c LO Z ℓ ∆ j + ℓz ζ 4 / 3 ⩾ ε X ( z ,j ) ∈ J S 0 Z ℓ ∆ j + ℓz  ζ + ζ 4 / 3  + A ε p + p/a − 1 X ( z ,j ) ∈ J S 0 Z ℓ ∆ j + ℓz | ∇ ζ θ | p b y the Lieb–Oxford inequalit y and ( 23 ). Clearly , if ζ z ,j ⩽ ( ε/c LO ) 3 , then ( z , j ) ∈ J S 0 . T etrahedra ℓ ∆ j + ℓz for which ( 23 ) holds are called simple , following the ter- minology of [ 17 ]. Case I I. (Main tetrahedra) The alternative is that c LO Z ℓ ∆ j + ℓz ζ 4 / 3 > ε Z ℓ ∆ j + ℓz  ζ + ζ 4 / 3  + A ε p + p/a − 1 Z ℓ ∆ j + ℓz | ∇ ζ θ | p , and ζ z ,j > ( ε/c LO ) 3 holds for ℓ ∆ j + ℓz , which w e call a main tetr ahe dr on (again following [ 17 ]). The collection of main tetrahedra is denoted by J M 0 . The k ey idea of the proof technique in [ 20 , 17 ] is that in a main tetrahedron the inhomogeneity is slo wly v arying. In fact, we ha v e using Morrey’s inequality (Theorem 4.3 ) that ζ θ z ,j − ζ θ z ,j ⩽ c Mo ℓ 1 − p/ 3 Z ℓ ∆ j + ℓz | ∇ ζ θ | p ! 1 /p ⩽ A − 1 /p c Mo ε 1 a − 1 p ℓ − p 3 c LO Z ℓ ∆ j + ℓz ζ 4 / 3 ! 1 /p ⩽ A − 1 /p c Mo c 1 /p LO ε 1 a − 1 p ζ 4 3 p z ,j NON-UNIFORM ELECTRON GAS 19 Our hypothesis implies that 1 < ( c 3 LO ε − 3 ) θ − 4 3 p ζ θ − 4 3 p z ,j . Inserting this in to the r.h.s. ab o v e gives ζ θ z ,j − ζ θ z ,j ⩽ A − 1 /p c Mo c 3 θ − 3 p LO ε 1 a + 3 p − 3 θ ζ θ z ,j Here, the exp onent of ε is p ositive b y our assumptions. Hence, we need A − 1 /p c Mo c 3 θ − 3 p LO ε 1 a + 3 p − 3 θ < 1 , where the exp onen t of ε is nonnegative. Recall that ε < ε 0 , we can c hoose for instance A = 2 c p Mo c 3 θp − 3 LO ε p ( 1 a + 3 p − 3 θ ) 0 , then ζ z ,j ⩽ 1 1 − A − 1 /p c Mo c 3 θ − 3 p LO ε 1 a + 3 p − 3 θ ζ z ,j ⩽ 1 1 − 2 − 1 /p ζ z ,j . T o summarize, ζ z ,j ⩽ (1 − 2 − 1 /p ) − 1 ζ ( x ) for x ∈ ℓ ∆ j + ℓz for an y ( z , j ) ∈ J M 0 . The main tetrahedra con tribute to the sum in ( 22 ) according to X ( z ,j ) ∈ J M 0 E  1 ℓ ∆ j ( · − ℓz ) 1 ℓ ′ ∆ ζ z ,j  ⩾ c UEG X ( z ,j ) ∈ J M 0 Z ℓ ∆ j + ℓz ζ 4 / 3 z ,j ⩾ c UEG X ( z ,j ) ∈ J M 0 Z ℓ ∆ j + ℓz ζ 4 / 3 − | c UEG | X ( z ,j ) ∈ J M 0 Z ℓ ∆ j + ℓz  ζ 4 / 3 z ,j − ζ 4 / 3  , where w e used ( 17 ) and the lo wer bound of ( 19 ). On the ﬂuctuation term, we apply an estimate lik e ( 20 ) but with the constant C p,θ,a m ultiplied b y (1 − 2 − 1 /p ) − 1 > 1. Using the rough estimate A ⩽ 1 . 90 · 4 b , we ﬁnd that the estimate ( 21 ) is larger, hence our ﬁnal constan t in fron t of the gradient term reads 2 . 71 b  10 θ  b . Dividing ( 22 ) by the volume and a v eraging ζ ov er all rotations and translations, the b oundary terms are b ounded in the usual manner 1 | ℓ ′ ∆ | − Z R 3 d a Z SO(3) d R X ( z ,j ) ∈ J \ J 0 E  1 ℓ ∆ j ( · − ℓz ) 1 ℓ ′ ∆ ζ ( R ( · − a ))  ⩾ − c LO | ℓ ′ ∆ | X ( z ,j ) ∈ J \ J 0 − Z R 3 d a Z SO(3) d R Z R 3 1 ℓ ∆ j ( · − ℓz ) ζ ( R ( · − a )) 4 / 3 = − c LO | J \ J 0 || ℓ ∆ | | ℓ ′ ∆ | − Z R 3 ζ 4 / 3 ⩾ − C ℓ ℓ ′ − Z R 3 ζ 4 / 3 , whic h disapp ears in the limit ℓ ′ → ∞ . W e arrive at e cl NUEG ( ζ ) ⩾ c UEG − Z R 3 ζ 4 / 3 − ε − Z R 3  ζ + ζ 4 / 3  − 2 . 71 b  10 θ  b ε b − Z R 3 | ∇ ζ θ | p . b y taking the limit ℓ ′ → ∞ . This completes the proof of the theorem. 20 NON-UNIFORM ELECTRON GAS 5. Proofs for the quantum case F or simplicity of notation, we drop the sup erscript ℏ in E ℏ and e ℏ when no confusion can arise. 5.1. Kno wn b ounds. W e b egin with a Lieb–Thirring kinetic energy inequality with semiclassical leading term and a gradient correction term. The original in- equalit y is due to Nam [ 28 ] and here we state a sp ecial case of an impro v ed version b y Seiringer and Solov ej [ 29 ]. Theorem 5.1 (Semiclassical Lieb–Thirring inequality with gradient correction) . F or any 0 < ε ⩽ 3 / 5 and any one-p article fermionic density matrix 0 ⩽ γ ⩽ 1 on L 2 ( R 3 ) the b ound T r( − ∆ γ ) ⩾ (1 − ε ) c TF Z R 3 ρ 5 / 3 γ − 10 27 1 ε Z R 3 | ∇ √ ρ γ | 2 holds true. Next, we rec all an estimate that giv es an a priori upper b ound on the kinetic energy functional T ( ρ ). It was ﬁrst conjectured by March and Y oung [ 27 ] in 1958. Bok anowski, Greb ert and Mauser [ 3 ] prov ed a version in the p eriodic setting. The pro of w as ﬁnally achiev ed b y Lewin, Lieb and Seiringer [ 20 ] by a technique in ter- esting in its o wn righ t. It was recen tly generalized by the present authors and E. I. T ellgren to the magnetic case, where the so-called paramagnetic current den- sit y is also prescrib ed [ 7 ]. Theorem 5.2 (Lewin–Lieb–Seiringer) . Supp ose that ρ ∈ L 1 ( R 3 ; R + ) and √ ρ ∈ H 1 ( R 3 ) . Then ther e exists a one-p article fermionic density matrix 0 ⩽ γ ⩽ 1 on L 2 ( R 3 ) such that ρ γ = ρ and for every 0 < ε ⩽ 1 / 15 the b ound T r( − ∆ γ ) ⩽ (1 + ε ) c TF Z R 3 ρ 5 / 3 + 19 15 1 ε Z R 3 | ∇ √ ρ | 2 holds true. The result immediately implies an a priori b ound on the kinetic energy functional T ( ρ ). By representing the one-particle density matrix γ by a quasi-free state Γ (see [ 2 ]), and using the fact that quasi-free states hav e nonpositive indirect Coulomb energy (just lik e Slater determinants), the follo wing very useful a priori estimate on the quantum indirect energy ma y b e obtained. Corollary 5.3. Supp ose that ρ ∈ L 1 ( R 3 ; R + ) and √ ρ ∈ H 1 ( R 3 ) . Then for every 0 < ε ⩽ 1 / 15 the b ound E ( ρ ) ⩽ (1 + ε ) c TF ℏ 2 Z R 3 ρ 5 / 3 + 19 15 ℏ 2 ε Z R 3 | ∇ √ ρ | 2 holds true. W e pro ceed b y recalling a spatial decoupling low er b ound on the indirect quan- tum energy . In order to state this result, w e need to use smeared indicators as sharp lo calization would blo w up the kinetic energy . Let the function η ∈ C ∞ c ( R 3 , R + ) b e radial suc h that supp η ⊂ B 1 and R R 3 η = 1. Deﬁne η δ ( x ) = 1 ( bδ ) 3 η  x bδ  , for some 0 < b < 1. Then supp η δ ⊂ B bδ and R R 3 η δ = 1. NON-UNIFORM ELECTRON GAS 21 Let Ω ⊂ R 3 b e an op en set. It is straightforw ard to see that ( 1 Ω ∗ η δ )( x ) = Z Ω η δ ( · − x ) =      1 supp η δ ( · − x ) ⊂ Ω , ∈ [0 , 1] supp η δ ( · − x ) ∩ ∂ Ω  = ∅ , 0 supp η δ ( · − x ) ⊂ R 3 \ Ω . (24) In particular, since supp η δ ⊂ B bδ 1 Ω ∗ η δ ≡ 1 in Ω \ ( ∂ Ω + B bδ ) . (25) More concretely , let Ω = ℓS , where ℓ > 0 and S ⊂ R 3 is a b ounded star-shaped set with center 0. Then ( 25 ) b ecomes the conv enien t 3 fact that 1 ℓS ∗ η δ ≡ 1 in ( ℓ − δ ) S. W e refer to this fact as 1 ℓS ∗ η δ ≡ 1 wel l inside ℓS . Clearly , the integral of the smeared indicator of Ω is R R 3 1 Ω ∗ η δ = | Ω | . Next, we state an elementary , but very important kinetic energy estimate. Lemma 5.4 (Bound on kinetic energy of a smeared set) . Supp ose that Ω ⊂ R 3 is an op en set with a b oundary having ﬁnite 2-dimensional L eb esgue me asur e: | ∂ Ω | < ∞ . Then the fol lowing b ound holds true Z R 3    ∇ p 1 Ω ∗ η δ    2 ⩽ C η | ∂ Ω | δ , wher e the c onstant C η > 0 dep ends only on the r e gularization function η . In particular, √ 1 Ω ∗ η δ ∈ H 1 ( R 3 ) whenever | ∂ Ω | < ∞ . Imp ortan tly , this means that fermionic man y-b ody localization (see [ 15 , App endix A]) with resp ect to the function √ 1 Ω ∗ η δ is meaningful. W e no w return to the discussion of the low er b ound on the indirect energy . F or an y Γ ∈ D let E ℏ ( Γ ) = ℏ 2 T ( Γ ) + C ( Γ ) − D ( ρ Γ ) denote the full quan tum indirect energy functional. Then it can be shown [ 20 ] using the IMS lo calization form ula for the kinetic energy , and the smeared Graf– Sc henker inequality for the indirect Coulomb energy that for any δ > 0 such that 0 < δ /ℓ < 1 /C , E ℏ ( Γ ) ⩾  1 − C δ ℓ  Z SO(3) d R − Z C ℓ d τ X z ∈ Z 3 24 X j =1 E ℏ  Γ   √ 1 ℓ ∆ j ∗ η δ ( R ·− ℓz − τ )  − C ℓ Z R 3 ((1 + ℏ 2 δ − 1 ) ρ Γ + δ 3 ρ 2 Γ ) (26) for some universal constant C > 0. This immediately implies the following decou- pling low er b ound on the indirect energy E ( ρ ). Theorem 5.5 (Decoupling lo w er bound) . Ther e is a universal c onstant C > 0 such that for any ℓ > 0 and δ > 0 such that 0 < δ /ℓ < 1 /C and ρ ∈ L 1 ( R 3 , R + ) 3 This is why we need the factor 0 < b < 1 in the deﬁnition of η δ . 22 NON-UNIFORM ELECTRON GAS with ∇ √ ρ ∈ L 2 ( R 3 ) the b ound E ( ρ ) ⩾  1 − C δ ℓ  Z SO(3) d R − Z C ℓ d τ X z ∈ Z 3 24 X j =1 E  1 ℓ ∆ j ∗ η δ ( R · − ℓz − τ ) ρ  − C ℓ Z R 3 ((1 + ℏ 2 δ − 1 ) ρ + δ 3 ρ 2 ) holds true. In p articular, E ( ρ ) ⩾  1 − C δ ℓ  X z ∈ Z 3 24 X j =1 E  1 ℓ ∆ j ∗ η δ ( R · − ℓz − τ ) ρ  − C ℓ Z R 3 ((1 + ℏ 2 δ − 1 ) ρ + δ 3 ρ 2 ) (27) for some isometry ( R, τ ) ∈ SO(3) × C ℓ . By absorbing the isometry ( R, τ ) into ρ in ( 27 ), we obtain plainly that the total energy is decoupled in to the sum of the energies of “lumps” 1 ℓ ∆ j ∗ η δ ( · − ℓz ) ρ up to an error. The neighboring “lumps” in general will o v erlap. 5.2. A priori b ounds on the energy per volume. First, we consider the quan- tit y f Ω ,η δ ,ζ ( a ) deﬁned in ( 9 ). Prop osition 5.6. The fol lowing a priori b ounds hold true. (i) F or every 0 < ε ⩽ 1 / 15 and every inhomo geneity ζ , ther e holds f Ω ,η δ ,ζ ( a ) ⩽ (1 + ε ) c TF ℏ 2 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) ζ 5 / 3 ( · − a ) + 38 15 ℏ 2 ε 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) | ∇ p ζ | 2 ( · − a ) + 38 15 ℏ 2 ε 1 | Ω | Z R 3 | ∇ p 1 Ω ∗ η δ | 2 ζ ( · − a ) . (ii) F or every 0 < ε ⩽ 3 / 5 and every inhomo geneity ζ , ther e holds f Ω ,η δ ,ζ ( a ) ⩾ (1 − ε ) c TF ℏ 2 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) ζ 5 / 3 ( · − a ) − c LO 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) ζ 4 / 3 ( · − a ) − 20 27 ℏ 2 ε 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) | ∇ p ζ | 2 ( · − a ) + 20 27 ℏ 2 ε 1 | Ω | Z R 3 | ∇ p 1 Ω ∗ η δ | 2 ζ ( · − a ) . Pr o of. Using Corollary 5.3 we ha v e for 0 < ε ⩽ 1 / 15 that f Ω ,η δ ,ζ ( a ) = 1 | Ω | E (( 1 Ω ∗ η δ ) ζ ( · − a )) ⩽ (1 + ε ) c TF ℏ 2 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) 5 / 3 ζ 5 / 3 ( · − a ) + 1 | Ω | 19 15 ℏ 2 ε Z R 3 | ∇ ( p 1 Ω ∗ η δ p ζ ( · − a )) | 2 ⩽ (1 + ε ) c TF ℏ 2 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) ζ 5 / 3 ( · − a ) + 38 15 ℏ 2 ε 1 | Ω | Z R 3 | ∇ p 1 Ω ∗ η δ | 2 ζ ( · − a ) + 38 15 ℏ 2 ε 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) | ∇ p ζ | 2 ( · − a ) , where in the second step we used 0 ⩽ 1 Ω ∗ η δ ⩽ 1 and elementary inequalit y | ∇ ( p 1 Ω ∗ η δ p ζ ) | 2 ⩽ 2 | ∇ p 1 Ω ∗ η δ | 2 ζ + 2( 1 Ω ∗ η δ ) | ∇ p ζ | 2 . (28) NON-UNIFORM ELECTRON GAS 23 F or (ii), w e hav e by the semiclassical Lieb–Thirring inequality (Theorem 5.1 ) and the Lieb–Oxford b ound that f Ω ,η δ ,ζ ( a ) ⩾ (1 − ε ) c TF ℏ 2 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) 5 / 3 ζ 5 / 3 ( · − a ) − 10 27 ℏ 2 ε 1 | Ω | Z R 3 | ∇ ( p 1 Ω ∗ η δ p ζ ( · − a )) | 2 − c LO 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) 4 / 3 ζ 4 / 3 ( · − a ) ⩾ (1 − ε ) c TF ℏ 2 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) 5 / 3 ζ 5 / 3 ( · − a ) − 20 27 ℏ 2 ε 1 | Ω | Z R 3 | ∇ p 1 Ω ∗ η δ | 2 ζ ( · − a ) − 20 27 ℏ 2 ε 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) | ∇ p ζ | 2 ( · − a ) − c LO 1 | Ω | Z R 3 ( 1 Ω ∗ η δ ) ζ 4 / 3 ( · − a ) whenev er 0 < ε ⩽ 3 / 5. Here, in the second step w e used ( 28 ) again. □ Av eraging f Ω ,η δ ,ζ ( a ) ov er a ∈ R 3 , and using Lemma 5.4 we obtain Corollary 5.7. F or every inhomo geneity √ ζ ∈ H 1 per ( L ) , the b ound − Z R 3 | f Ω ,η δ ,ζ ( a ) | d a ⩽ C − Z R 3 ( ℏ 2 ζ 5 / 3 + ζ 4 / 3 ) + C ℏ 2 − Z R 3 | ∇ p ζ | 2 + C ℏ 2 | ∂ Ω | | Ω | δ − Z R 3 ζ holds true for some universal c onstant C > 0 that only dep ends on η . Recall the deﬁnition ( 10 ) of the quantum indirect energy p er ﬁnite volume that w e now simply denote as e Ω ,η δ ( ζ ). Corollary 5.8. F or every inhomo geneity √ ζ ∈ H 1 per ( L ) , the b ound e Ω ,η δ ( ζ ) ⩽ (1 + ε ) c TF ℏ 2 − Z R 3 ζ 5 / 3 + 38 15 ℏ 2 ε − Z R 3 | ∇ p ζ | 2 + C η ℏ 2 ε | ∂ Ω | | Ω | δ − Z R 3 ζ for any 0 < ε ⩽ 1 / 15 , and the b ound e Ω ,η δ ( ζ ) ⩾ (1 − ε ) c TF ℏ 2 − Z R 3 ζ 5 / 3 − c LO − Z R 3 ζ 4 / 3 − 20 27 ℏ 2 ε − Z R 3 | ∇ p ζ | 2 − C η ℏ 2 ε | ∂ Ω | | Ω | δ − Z R 3 ζ for any 0 < ε ⩽ 3 / 5 hold true. The universal c onstant C η > 0 only dep ends on η . By taking Ω = ℓ ∆ , the prefactor of the last terms are prop ortional to ( ℓδ ) − 1 that disapp ear in the thermo dynamic limit, whic h yields the b ounds of Remark 2 (ii), once the existence of the limit is prov ed. 5.3. An improv ed spatial decoupling upp er b ound. In this section, we pro- p ose an estimate whic h slightly impro ves the spatial decoupling upp er b ound in- tro duced in [ 20 ]. This b ound inv olv es shrinking the smeared set 1 ℓ ∆ ∗ η δ so that its supp ort is strictly contained in ℓ ∆ , while keeping the smearing scale δ ﬁxed. This is b ecause in con trast to low er bound of Theorem 5.5 , it w ould b e m uch more diﬃcult to lo calize the density in an ov erlapping manner for an upp er b ound. Let δ ⩽ ℓ/ 2. Using the Lions–Titchmarsh conv olution theorem, supp( 1 ( ℓ − δ ) ∆ ∗ η δ ) = ( ℓ − δ ) ∆ + (supp η ) bδ , where w e used our hypothesis that η is radial so supp η is a closed ball, in particular con vex. Here, (supp η ) bδ denotes the dilation of the set supp η ⊂ B 1 b y bδ . W e also used the fact that supp( 1 ( ℓ − δ ) ∆ ∗ η δ ) is conv ex, which can b e seen from ( 24 ). 24 NON-UNIFORM ELECTRON GAS So in fact w e hav e supp( 1 ( ℓ − δ ) ∆ ∗ η δ ) ⊂ ℓ ∆ . More generally , it is also clear from the ab o v e that decreasing the parameter δ to cδ strictly increases the supp orts as supp( 1 ( ℓ − δ ) ∆ ∗ η δ ) ⋐ supp( 1 ( ℓ − cδ ) ∆ ∗ η cδ ) (29) where 0 < c < 1 and the symbol A ⋐ B means that A ⊂ K ⊂ B for some compact set K ⊂ R 3 . In fact, the distance from the b oundary of supp( 1 ( ℓ − δ ) ∆ ∗ η δ ) to the b oundary of supp( 1 ( ℓ − cδ ) ∆ ∗ η cδ ) is ℓ − cδ + cbδ − ( ℓ − δ + bδ ) = (1 − c )(1 − b ) δ > 0. Our incomplete partition of unity reads − Z C ℓ d τ " X z ∈ Z 3 24 X j =1 1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) + Υ ℓ,δ ( · − τ ) # ≡ 1 , (30) where we ha v e introduced the “skeleton function” Υ ℓ,δ = 1 − (1 − δ /ℓ ) 3 1 − (1 − δ / (2 ℓ )) 3 1 − X z ∈ Z 3 24 X j =1 1 ℓT j (1 − δ / (2 ℓ )) ∆ ∗ η δ / 2 ( · − ℓz ) ! . Our p oint here is that no w the smeared indicators of the ﬁrst sum in ( 30 ) are no longer m ultiplied b y the normalization factor (1 − δ /ℓ ) − 3 as in [ 20 ], because the sk eleton function absorbs these w eigh ts. Clearly , Υ ℓ,δ is a nonnegative, smo oth ( ℓ Z 3 )-p eriodic function with disjoint supp ort from all the other terms in ( 30 ). In fact, the supp ort of 1 ℓT j (1 − δ / (2 ℓ )) ∆ ∗ η δ / 2 strictly contains the supp ort of 1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ according to ( 29 ). The mean v alue of the skeleton function is easily seen to b e − Z C ℓ Υ ℓ,δ = 1 − (1 − δ /ℓ ) 3 , whic h in turn veriﬁes relation ( 30 ). Note also that this tends to zero in the limit δ /ℓ → 0. According to Lemma 5.4 , the mean kinetic energy of Υ ℓ,δ is b ounded as − Z C ℓ | ∇ p Υ ℓ,δ | 2 ⩽ C η ℓδ , whic h v anishes in the limit ℓδ → ∞ . It turns out when decoupling the energy according to ( 30 ), all the con tributions coming from the sk eleton function Υ ℓ,δ are negligible in the thermo dynamic limit. Theorem 5.9 (Improv ed decoupling upp er b ound) . L et 0 < δ ⩽ ℓ/ 2 and 0 < α < 1 / 2 . Then for any ρ ∈ L 1 ( R 3 , R + ) with ∇ √ ρ ∈ L 2 ( R 3 ) the b ounds E ( ρ ) ⩽ − Z 1+ α 1 − α d t t 4 Z SO(3) d R − Z C tℓ d τ X z ∈ Z 3 24 X j =1 E  1 tℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( R · − tℓz − τ ) ρ  + C ℏ 2 δ ℓ Z R 3 ρ 5 / 3 + C ℏ 2 ℓδ Z R 3 ρ + C ℏ 2 δ ℓ Z R 3 | ∇ √ ρ | 2 + C δ 2 log( α − 1 ) Z R 3 ρ 2 and T ( ρ ) ⩽ − Z C ℓ d τ X z ∈ Z 3 24 X j =1 T  1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ  + C ℏ 2 δ ℓ Z R 3 ρ 5 / 3 + C ℏ 2 ℓδ Z R 3 ρ + C ℏ 2 δ ℓ Z R 3 | ∇ √ ρ | 2 hold true for some universal c onstant C > 0 . NON-UNIFORM ELECTRON GAS 25 Here and henceforth, the notation − Z 1+ α 1 − α f ( t ) d t t 4 =  Z 1+ α 1 − α d t t 4  − 1 Z 1+ α 1 − α f ( t ) d t t 4 is used. Pr o of. Using our incomplete partition of unit y ( 30 ), we ma y write ρ ( x ) = − Z C ℓ " X z ∈ Z 3 24 X j =1 1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( x − ℓz − τ ) + Υ ℓ,δ ( x − τ ) # ρ ( x ) d τ ≡ 1 . (31) F or any τ ∈ C ℓ , z ∈ Z 3 and j = 1 , . . . , 24 let Γ τ ,z ,j b e an optimizer of F LL ( 1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ ). Also, let Γ τ , Υ b e an optimizer of F LL (Υ ℓ,δ ( · − τ ) ρ ). Next, w e take Γ τ = O z ∈ Z 3 24 O j =1 Γ τ ,z ,j ⊗ Γ τ , Υ . All of the states Γ τ ,z ,j and Γ τ , Υ ha ve disjoint supp ort for ﬁxed τ , so we may an- tisymmetrize Γ τ , which w e denote by Γ τ ,a . This state has T (Γ τ ,a ) + C (Γ τ ,a ) = T (Γ τ ) + C (Γ τ ) and ρ Γ τ,a = ρ Γ τ . Finally , using Γ = − R C ℓ Γ τ ,a d τ as a trial state, whic h has ρ Γ = ρ , we obtain F LL ( ρ ) ⩽ − Z C ℓ d τ X z ∈ Z 3 24 X j =1 F LL  1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ  + − Z C ℓ F LL  Υ ℓ,δ ( · − τ ) ρ  d τ + − Z C ℓ d τ X ( z ,j ) , ( z ′ ,j ′ ) ∈ Z 3 ×{ 1 ,..., 24 } ( z ,j )  =( z ′ ,j ′ ) D  1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ, 1 ℓT j ′ (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz ′ − τ ) ρ  + 2 − Z C ℓ d τ X z ∈ Z 3 24 X j =1 D  1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ, Υ ℓ,δ ( · − τ ) ρ  . This implies E ( ρ ) ⩽ − Z C ℓ d τ X z ∈ Z 3 24 X j =1 E  1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ  + − Z C ℓ E  Υ ℓ,δ ( · − τ ) ρ  d τ − D ( ρ ) + − Z C ℓ d τ X z ∈ Z 3 24 X j =1 D  1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ  + − Z C ℓ D  Υ ℓ,δ ( · − τ ) ρ  d τ + − Z C ℓ d τ X ( z ,j )  =( z ′ ,j ′ ) D  1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ, 1 ℓT j ′ (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz ′ − τ ) ρ  + 2 − Z C ℓ d τ X z ∈ Z 3 24 X j =1 D  1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ, Υ ℓ,δ ( · − τ ) ρ  . By inserting ( 31 ) into − D ( ρ ) as − D ( ρ ) = D ( ρ ) − 2 D ( ρ, ρ ) = D ( ρ ) − 2 − Z C ℓ d τ D   ρ, X z ∈ Z 3 24 X j =1 1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ + Υ ℓ,δ ( · − τ ) ρ   , 26 NON-UNIFORM ELECTRON GAS w e see that the v arious direct terms giv e in total − Z C ℓ D   X z ∈ Z 3 24 X j =1 1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz − τ ) ρ + Υ ℓ,δ ( · − τ ) ρ − ρ   d τ =: ( ∗ ) . First, we estimate the indirect energy error coming from the Υ ℓ,δ term. Using Corollary 5.3 and the prop erties of Υ ℓ,δ , we ha v e the b ound − Z C ℓ E  Υ ℓ,δ ( · − τ ) ρ  d τ ⩽ C ℏ 2 Z R 3 ρ 5 / 3 − Z C ℓ Υ ℓ,δ ( · − τ ) 5 / 3 d τ + C ℏ 2 Z R 3 ρ − Z C ℓ | ∇ p Υ ℓ,δ | 2 ( · − τ ) d τ + C ℏ 2 Z R 3 | ∇ √ ρ | 2 − Z C ℓ Υ ℓ,δ ( · − τ ) d τ ⩽ C ℏ 2 δ ℓ Z R 3 ρ 5 / 3 + C ℏ 2 ℓδ Z R 3 ρ + C ℏ 2 δ ℓ Z R 3 | ∇ √ ρ | 2 . Finally , w e estimate the direct term ( ∗ ). W e proceed exactly as in [ 20 ]. Inserting ρ = X z ∈ Z 3 24 X j =1 1 ℓ ∆ j ( · − ℓz − τ ) ρ (a.e.) so that ( ∗ ) = − Z C ℓ D  f ( · − τ ) ρ  d τ , where the ( ℓ Z 3 )-p eriodic function f : R 3 → R is given b y f = X z ∈ Z 3 24 X j =1  1 ℓ ∆ j ( · − ℓz ) − 1 ℓT j (1 − δ /ℓ ) ∆ ∗ η δ ( · − ℓz )  − Υ ℓ,δ . Recall the representation from [ 20 ], − Z C ℓ D  f ( · − τ ) ρ  d τ = 2 π X k ∈ (2 π /ℓ ) Z 3     − Z C ℓ f ( x ) e − ik · x d x     2 Z R 3 | b ρ ( p ) | 2 | p − k | 2 d p, v alid for any ( ℓ Z 3 )-p eriodic function f . W e obtain ( ∗ ) = 2 π X k ∈ 2 π Z 3     Z C 1 f ℓ/δ ( x ) e − ik · x d x     2 Z R 3 | b ρ ( p ) | 2 | p − k /ℓ | 2 d p. where we ha v e set f ε = 24 X j =1  1 ∆ j − 1 T j (1 − ε ) ∆ ∗ η ε  − Υ ε . and Υ ε = 1 − (1 − ε ) 3 1 − (1 − ε/ 2) 3 1 − 24 X j =1 1 T j (1 − ε/ 2) ∆ ∗ η ε/ 2 ! . NON-UNIFORM ELECTRON GAS 27 Here, R C 1 f δ /ℓ = 0 and Z C 1 f ℓ/δ ( x ) e − ik · x d x = − (2 π ) 3 24 X j =1 \ 1 T j (1 − ε ) ∆ ( k ) b η 1 ( εk ) + (2 π ) 3 1 − (1 − ε ) 3 1 − (1 − ε/ 2) 3 24 X j =1 \ 1 T j (1 − ε/ 2) ∆ ( k ) b η 1 ( εk / 2) . F rom this p oin t on, the pro of follows similarly to that of [ 20 , Prop osition 1]. In conclusion, after av eraging o v er rotations and dilations ℓ → tℓ and δ → tδ ov er t ∈ (1 − α, 1 + α ) with weigh t t − 4 , we obtain the b ound − Z 1+ α 1 − α d t t 4 Z SO(3) d R − Z C tℓ d τ D  f ( R ( · − τ )) ρ  ⩽ C δ 2 log( α − 1 ) Z R 3 ρ 2 , whic h concludes the pro of. □ 5.4. Con v ergence rate for tetrahedra. In this section w e study the thermody- namic conv ergence of the indirect energy for dilated tetrahedra. Theorem 5.10 (Con v ergence rate for tetrahedra) . Fix an inhomo geneity ζ and a r e gularization function η . L et e NUEG ( ζ ) and e NUEG ( ζ ) denote the liminf and the limsup of e ℓ ∆ ,η δ ( ζ ) as ℓδ → ∞ and δ /ℓ → 0 . (i) F or ℓ/δ suﬃciently lar ge, ther e holds e ℓ ∆ ,η δ ( ζ ) ⩽ e NUEG ( ζ ) + C ℓ (1 + ℏ 2 δ − 1 ) − Z R 3 ζ + δ 3 − Z R 3 ζ 2 + ℏ 2 δ − Z R 3 ζ 5 / 3 + ℏ 2 δ − Z R 3 | ∇ p ζ | 2 ! (ii) F or ℓ/δ suﬃciently lar ge and 0 < α < 1 / 2 , e NUEG ( ζ ) ⩽ − Z 1+ α 1 − α d t t 4 e tℓ ∆ ,η tδ ( ζ ) + C ℏ 2 δ ℓ − Z R 3 ζ 5 / 3 + C ℏ 2 ℓδ − Z R 3 ζ + C ℏ 2 δ ℓ − Z R 3 | ∇ p ζ | 2 + C δ 2 log( α − 1 ) − Z R 3 ζ 2 (iii) F or ℓ suﬃciently lar ge, e ℓ ∆ ,η δ ( ζ ) ⩾ e NUEG ( ζ ) − C δ ℓ − Z R 3  ζ + ζ 2  − C ℓ 2 / 5 − Z R 3  ζ + ζ 2  − C ℓ 4 / 5 − Z R 3 | ∇ p ζ | 2 . (iv) The thermo dynamic limit lim δ /ℓ → 0 ℓδ →∞ e ℓ ∆ ,η δ ( ζ ) = e NUEG ( ζ ) exists and is indep endent of the r e gularization function η . In al l the ab ove b ounds, the generic c onstant C > 0 dep ends only on the r e gulariza- tion function η and is indep endent of ℓ , δ , ζ and α . Before proving the ab o v e theorem, w e derive a b ound comparing the energy of a big tetrahedron to a small tetrahedron. This result can be though t of as the quan tum analog of Lemma 4.4 . 28 NON-UNIFORM ELECTRON GAS Lemma 5.11 (Monotonicity estimate) . Fix an inhomo geneity ζ and a r e gulariza- tion function η . F or any ℓ ′ ≫ ℓ > 0 , δ ′ , δ > 0 such that δ /ℓ ⩽ 1 /C , the b ound e ℓ ′ ∆ ,η δ ′ ( ζ ) ⩾ 1 − C σ δ ℓ − C σ ℓ + δ + δ ′ ℓ ′ ! e ℓ ∆ ,η δ ( ζ ) − C ℓ + δ + δ ′ ℓ ′ − Z R 3 ζ 4 / 3 − C ℓ (1 + ℏ 2 δ − 1 ) − Z R 3 ζ + δ 3 − Z R 3 ζ 2 ! (32) holds true, wher e σ = ( 0 if e ℓ ∆ ,η δ ( ζ ) ⩽ 0 1 otherwise Pr o of. Let ℓ ≪ ℓ ′ suﬃcien tly large, δ /ℓ ⩽ 1 /C and write e ℓ ′ ∆ ,η δ ′ ( ζ ) = 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a E  ( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a ))  ⩾ 1 | ℓ ′ ∆ |  1 − C δ ℓ  Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ × X z ∈ Z 3 24 X j =1 E  1 ℓ ∆ j ∗ η δ ( R · − ℓz − τ )( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a ))  − C ℓ | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z R 3  (1 + ℏ 2 δ − 1 )( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a )) + δ 3 ( 1 ℓ ′ ∆ ∗ η δ ′ ) 2 ζ ( Q ( · − a )) 2  =: (I) + (I I) Using 0 ⩽ 1 ℓ ′ ∆ ∗ η δ ′ ⩽ 1, and ev aluating the mean v alue with respect to a , the terms in (I I) may be estimated as (I I) ⩾ − C ℓ (1 + ℏ 2 δ − 1 ) − Z R 3 ζ + δ 3 − Z R 3 ζ 2 ! Next, we consider (I). Deﬁne the index set J := ( ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : supp(( 1 ℓ ∆ j ∗ η δ )( R · − ℓz − τ )) ∩ supp( 1 ℓ ′ ∆ ∗ η δ ′ )  = ∅ , for some R ∈ SO(3) and τ ∈ C ℓ ) , whic h collects all the small smeared tetrahedra that possibly intersect with the big smeared tetrahedron supp( 1 ℓ ′ ∆ ∗ η δ ′ ). F urthermore, deﬁne J 0 := ( ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : supp(( 1 ℓ ∆ j ∗ η δ )( R · − ℓz − τ )) ⊂ ( ℓ ′ − δ ′ ) ∆ , for all R ∈ SO(3) and τ ∈ C ℓ ) , whic h contains small smeared tetrahedra whic h are w ell inside ℓ ′ ∆ . According to J 0 and J \ J 0 , we may decompose the sum in (I) to (Ia) and (Ib), resp ectiv ely . T o estimate (Ia), note that for ( z , j ) ∈ J 0 , we simply hav e ( 1 ℓ ∆ j ∗ η δ )( R · − ℓz − τ )( 1 ℓ ′ ∆ ∗ η δ ′ ) = ( 1 ℓ ∆ j ∗ η δ )( R · − ℓz − τ ) , NON-UNIFORM ELECTRON GAS 29 since 1 ℓ ′ ∆ ∗ η δ ′ ≡ 1 on ( ℓ ′ − δ ′ ) ∆ . Therefore, (Ia) = 1 | ℓ ′ ∆ |  1 − C δ ℓ  Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ × X ( z ,j ) ∈ J 0 E  1 ℓ ∆ j ∗ η δ ( R · − ℓz − τ ) ζ ( Q ( · − a ))  Using the deﬁnition ( 10 ), this can b e written as (Ia) = 1 | ℓ ′ ∆ |  1 − C δ ℓ  Z SO(3) d R − Z C ℓ d τ X ( z ,j ) ∈ J 0 | ℓ ∆ | e R ⊤ ℓ ∆ j + ℓz + τ ,η δ ( ζ ) Recalling that Ω 7→ e Ω ,η δ ( ζ ) is isometry-inv ariant, w e ﬁnd (Ia) = M 0 ℓ | ℓ ′ ∆ |  1 − C δ ℓ  e ℓ ∆ ,η δ ( ζ ) where we ha v e set M 0 ℓ = | ℓ ∆ || J 0 | . Next, we estimate the con tributions from tetrahedra close to the b oundary . Using the Lieb–Oxford inequality (Theorem 4.1 ) w e ﬁnd (Ib) ⩾ C | ℓ ′ ∆ |  1 − C δ ℓ  Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ × X ( z ,j ) ∈ J \ J 0 " − c LO Z R 3 1 ℓ ∆ j ∗ η δ ( R · − ℓz − τ ) 4 / 3 ( 1 ℓ ′ ∆ ∗ η δ ′ ) 4 / 3 ζ ( Q ( · − a )) 4 / 3 # This can b e further b ounded as (Ib) ⩾ − M ∂ ℓ | ℓ ′ ∆ |  1 − C δ ℓ  c LO − Z R 3 ζ 4 / 3 , where we ha v e set M ∂ ℓ = | ℓ ∆ || J \ J 0 | . Let us b ound the volume ratios M 0 ℓ / | ℓ ′ ∆ | and M ∂ ℓ / | ℓ ′ ∆ | . If e ℓ ∆ ,η δ ( ζ ) ⩽ 0, then ma y use the trivial bound M 0 ℓ / | ℓ ′ ∆ | ⩽ 1. Recall that the index set J \ J 0 collects the small tetrahedra close to the b oundary of ℓ ′ ∆ and are at a distance O ( ℓ + δ + δ ′ ) from ∂ ( ℓ ′ ∆ ). Since the surface area of ∂ ( ℓ ′ ∆ ) is O ( ℓ ′ 2 ), the tetrahedra close to the b oundary ﬁll a volume of M ∂ ℓ = O ( ℓ ′ 2 ( ℓ + δ + δ ′ )). Therefore, M 0 ℓ | ℓ ′ ∆ | ⩾ | ℓ ′ ∆ | − C ℓ ′ 2 ( ℓ + δ + δ ′ ) | ℓ ′ ∆ | = 1 − C ℓ + δ + δ ′ ℓ ′ (33) and M ∂ ℓ | ℓ ′ ∆ | ⩽ C ℓ + δ + δ ′ ℓ ′ . (34) With these at hand, the stated estimate follows. □ Pr o of of The or em 5.10 . Pro of of (i). W e split the l.h.s. of ( 32 ) according to e ℓ ′ ∆ ,η δ ′ ( ζ ) = (1 − C σ δ /ℓ ) e ℓ ′ ∆ ,η δ ′ ( ζ ) + C σ δ ℓ e ℓ ′ ∆ ,η δ ′ ( ζ ) ⩽ (1 − C σ δ /ℓ ) e ℓ ′ ∆ ,η δ ′ ( ζ ) + C ℏ 2 σ δ ℓ " − Z R 3 ζ 5 / 3 + − Z R 3 | ∇ p ζ | 2 + 1 ℓ ′ δ ′ − Z R 3 ζ # 30 NON-UNIFORM ELECTRON GAS where we used Prop osition 5.6 (i) with ε = 1. Rearranging our b ound, w e ﬁnd 1 − C σ δ ℓ − C σ ℓ + δ + δ ′ ℓ ′ ! e ℓ ∆ ,η δ ( ζ ) ⩽ (1 − C σ δ /ℓ ) e ℓ ′ ∆ ,η δ ′ ( ζ )+ C ℓ (1+ ℏ 2 δ − 1 ) − Z R 3 ζ + δ 3 − Z R 3 ζ 2 + ℏ 2 δ − Z R 3 ζ 5 / 3 + ℏ 2 δ − Z R 3 | ∇ p ζ | 2 ! + C ℓ + δ + δ ′ ℓ ′ − Z R 3 ζ 4 / 3 + C ℏ 2 σ δ ℓ 1 ℓ ′ δ ′ − Z R 3 ζ No w taking the liminf as ℓ ′ → ∞ , and dividing b y (1 − C σ δ /ℓ ), we arriv e at e ℓ ∆ ,η δ ( ζ ) ⩽ e UEG ( ζ ) + C ℓ (1 + ℏ 2 δ − 1 ) − Z R 3 ζ + δ 3 − Z R 3 ζ 2 + ℏ 2 δ − Z R 3 ζ 5 / 3 + ℏ 2 δ − Z R 3 | ∇ p ζ | 2 ! , whic h is what we w an ted to show. Pro of of (ii). The pro of of the low er b ound is similar. Using Theorem 5.9 , we ha ve for 0 < α < 1 2 , e ℓ ′ ∆ ,η δ ′ ( ζ ) ⩽ 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a − Z 1+ α 1 − α d t t 4 Z SO(3) d R − Z C t ( ℓ + δ ) d τ × X z ∈ Z 3 24 X j =1 E  ( 1 t ( ℓ + δ ) T j β ∆ ∗ η tδ )( R · − t ( ℓ + δ ) z − τ )( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a ))  + " C ℏ 2 δ ℓ − Z R 3 ζ 5 / 3 + C ℏ 2 ℓδ − Z R 3 ζ + C ℏ 2 δ ℓ − Z R 3 | ∇ p ζ | 2 + C δ 2 log( α − 1 ) − Z R 3 ζ 2 # = (I) + (I I) (35) where w e used the notation β = ℓ/ ( ℓ + δ ), where 2 3 ⩽ β < 1. Notice that the scales are chosen so that the small tetrahedra are of size tℓ and smeared at scale tδ . Again, we split (I) according to J 0 and J \ J 0 , which we deﬁne analogously to ab o v e. First, w e consider the summation ov er J 0 in (I), whic h we denote by (Ia). Using t ( ℓ + δ ) T j (1 − δ / ( ℓ + δ )) ∆ = t ( ℓ + δ ) β R j ∆ + t ( ℓ + δ ) z j , we ma y write ( 1 t ( ℓ + δ ) T j β ∆ ∗ η tδ )( Rx − t ( ℓ + δ ) z − τ ) = ( 1 tℓ ∆ ∗ η tδ )( R ⊤ j Rx − R ⊤ j ( t ( ℓ + δ ) z + τ + t ( ℓ + δ ) z j )) W e ha ve (Ia) = 1 | ℓ ′ ∆ | − Z 1+ α 1 − α d t t 4 Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C t ( ℓ + δ ) d τ X ( z ,j ) ∈ J 0 × − Z R 3 Z SO(3) E  ( 1 tℓ ∆ ∗ η tδ )( R ⊤ j R · − R ⊤ j ( t ( ℓ + δ ) z + τ + t ( ℓ + δ ) z j )) ζ ( Q ( · − a ))  = 1 | ℓ ′ ∆ | − Z 1+ α 1 − α d t t 4 | J 0 | Z SO(3) d Q − Z R 3 d a E  ( 1 tℓ ∆ ∗ η tδ ) ζ ( Q ( · − a ))  = − Z 1+ α 1 − α d t t 4 M 0 tℓ | ℓ ′ ∆ | e tℓ ∆ ,η tδ ( ζ ) NON-UNIFORM ELECTRON GAS 31 In the second equality , we used the isometry-inv ariance of the energy and Theo- rem 2.1 . Here, the volume ratio may be b ounded similarly as b efore M 0 tℓ | ℓ ′ ∆ | ⩽ 1 + C σ tℓ + tδ + δ ′ ℓ ′ ⩽ 1 + C σ ℓ + δ + δ ′ ℓ ′ . Next, we estimate the contributions near the b oundary using Corollary 5.3 , (Ib) = 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a − Z 1+ α 1 − α d t t 4 Z SO(3) d R − Z C t ( ℓ + δ ) d τ X ( z ,j ) ∈ J \ J 0 × × E  ( 1 t ( ℓ + δ ) T j β ∆ ∗ η tδ )( R · − t ( ℓ + δ ) z − τ )( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a ))  ⩽ ℏ 2 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a − Z 1+ α 1 − α d t t 4 Z SO(3) d R − Z C t ( ℓ + δ ) d τ X ( z ,j ) ∈ J \ J 0 × × " C Z R 3 ( 1 t ( ℓ + δ ) T j β ∆ ∗ η tδ )( R · − t ( ℓ + δ ) z − τ )( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a )) 5 / 3 + C Z R 3    ∇ q ( 1 t ( ℓ + δ ) T j β ∆ ∗ η tδ )( R · − t ( ℓ + δ ) z − τ )( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a ))    2 # Here, the gradient term may b e bounded using the Cauc h y–Sch warz inequality as C Z R 3    ∇ q ( 1 t ( ℓ + δ ) T j β ∆ ∗ η tδ )( R · − t ( ℓ + δ ) z − τ )    2 ( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a )) + C Z R 3    ∇ p 1 ℓ ′ ∆ ∗ η δ ′    2 ( 1 t ( ℓ + δ ) T j β ∆ ∗ η tδ )( R · − t ( ℓ + δ ) z − τ ) ζ ( Q ( · − a )) + C Z R 3   ∇ p ζ ( Q ( · − a ))   2 ( 1 t ( ℓ + δ ) T j β ∆ ∗ η tδ )( R · − t ( ℓ + δ ) z − τ )( 1 ℓ ′ ∆ ∗ η δ ′ ) After computing the a verages with resp ect to a and Q , we obtain by Lemma 5.4 , (Ib) ⩽ C ℏ 2 M ∂ tℓ | ℓ ′ ∆ | " − Z R 3 ζ 5 / 3 + 1 ℓδ − Z R 3 ζ + − Z R 3 | ∇ p ζ | 2 # + C ℏ 2 ℓ ′ δ ′ − Z R 3 ζ . (36) Com bining this with the b ound ( 34 ), we arriv e at an es timate on (Ib) that v anishes in the thermo dynamic limit ℓ ′ δ ′ → ∞ , δ ′ /ℓ ′ → 0. Collecting our estimate s and taking the limsup as ℓ ′ δ ′ → ∞ , δ ′ /ℓ ′ → 0, w e obtain e NUEG ( ζ ) ⩽ − Z 1+ α 1 − α d t t 4 e tℓ ∆ ,η tδ ( ζ ) + C ℏ 2 δ ℓ − Z R 3 ζ 5 / 3 + C ℏ 2 ℓδ − Z R 3 ζ + C ℏ 2 δ ℓ − Z R 3 | ∇ p ζ | 2 + C δ 2 log( α − 1 ) − Z R 3 ζ 2 as stated. 32 NON-UNIFORM ELECTRON GAS Pro of of (iii). In order to ge t rid of the a veraging in (ii), we use ( 32 ) with ℓ → tℓ , δ → tδ and av erage ov er t ∈ (1 / 2 , 3 / 2). W e obtain e ℓ ′ ∆ ,η δ ′ ( ζ ) ⩾ 1 − C σ δ ℓ − C σ ℓ + δ + δ ′ ℓ ′ ! − Z 1+ α 1 − α d t t 4 e tℓ ∆ ,η tδ ( ζ ) − C ℓ + δ + δ ′ ℓ ′ − Z R 3 ζ 4 / 3 − C ℓ (1 + ℏ 2 δ − 1 ) − Z R 3 ζ + δ 3 − Z R 3 ζ 2 ! . (37) Using (ii), we may bound the t -av erage from b elo w by e NUEG ( ζ ) − C ℏ 2 δ ℓ − Z R 3 ζ 5 / 3 − C ℏ 2 ℓδ − Z R 3 ζ − C ℏ 2 δ ℓ − Z R 3 | ∇ p ζ | 2 − C δ 2 − Z R 3 ζ 2 , to get after discarding the p ositive terms and rearranging, e ℓ ′ ∆ ,η δ ′ ( ζ ) ⩾ e NUEG ( ζ ) − C ℓ + δ + δ ′ ℓ ′ − Z R 3 ζ 4 / 3 − C ℓ (1 + ℏ 2 δ − 1 ) − Z R 3 ζ + δ 3 − Z R 3 ζ 2 + ℏ 2 δ − Z R 3 ζ 5 / 3 + ℏ 2 δ − Z R 3 | ∇ p ζ | 2 ! − C δ 2 − Z R 3 ζ 2 ⩾ e NUEG ( ζ ) − C ℓ + δ + δ ′ ℓ ′ − Z R 3  ζ + ζ 2  − C 1 + δ − 1 + δ ℓ − Z R 3 ζ − C  δ + δ 3 ℓ + δ 2  − Z R 3 ζ 2 − C δ ℓ − Z R 3 | ∇ p ζ | 2 where in the last step w e made use of ζ α ⩽ ζ + ζ 2 for 1 ⩽ α ⩽ 2 Here, we choose δ = ℓ − 1 / 3 and then ℓ = ( ℓ ′ ) 3 / 5 to obtain e ℓ ′ ∆ ,η δ ′ ( ζ ) ⩾ e NUEG ( ζ ) − C δ ′ ℓ ′ − Z R 3  ζ + ζ 2  − C ( ℓ ′ ) 2 / 5 − Z R 3  ζ + ζ 2  − C ( ℓ ′ ) 4 / 5 − Z R 3 | ∇ p ζ | 2 for suﬃciently large ℓ ′ . Pro of of (iv). It only remains to explain wh y the limit is indep endent of the regularization function η . Supp ose that e η is another regularization function. One can deriv e estimates similar to (i)–(iii) with ﬁnite-volume quantities inv olving e η instead. This implies that the thermo dynamic limit of e ℓ ∆ , e η δ ( ζ ) is the same as that of e ℓ ∆ ,η δ ( ζ ). □ 5.5. The equiv alence of the t w o deﬁnitions of e NUEG for tetrahedra. In a preliminary step to w ards general domain sequences, we pro v e that the thermo dy- namic limit of e e Ω ,s ( ζ ) for a dilated reference tetrahedron exists, and is equal to the limit in Theorem 5.10 . Prop osition 5.12. Fix an inhomo geneity ζ and a r e gularization function η . Then the limits lim L N →∞ s N /L N → 0 e e L N ∆ ,s N ( ζ ) = lim ℓ →∞ e ℓ ∆ ,η ( ζ ) = e NUEG ( ζ ) exist and ar e indep endent of the se quenc es L N and s N and of η . Notice that in the second limit, we simply to ok the constant sequence δ = 1 of smearing scale, which according to Theorem 5.10 conv erges to e NUEG ( ζ ). NON-UNIFORM ELECTRON GAS 33 Pr o of. Recall deﬁnition ( 11 ), which in our case reads e e L N ∆ ,s N ( ζ ) = Z SO(3) d R − Z R 3 d a inf √ ρ ∈ H 1 ( R 3 ) ζ ( R ( ·− a )) 1 ( L N − s N ) ∆ ⩽ ρ ⩽ ζ ( R ( ·− a )) 1 ( L N + s N ) ∆ E ( ρ ) | L N ∆ | F or suﬃcien tly large N , the p oint wise b ound ζ ( R ( · − a )) 1 ( L N − s N ) ∆ ⩽ ( 1 L N ∆ ∗ η ) ζ ( R ( · − a )) ⩽ ζ ( R ( · − a )) 1 ( L N + s N ) ∆ holds true. In other words, ( 1 L N ∆ ∗ η ) ζ ( R ( · − a )) is a competing density in the ab o v e inﬁmum. W e obtain that for N suﬃcien tly large e e L N ∆ ,s N ( ζ ) ⩽ e L N ∆ ,η ( ζ ) . (38) F or the low er b ound on e e L N ∆ ,s N ( ζ ) let Γ b e a state such that ζ ( R ( · − a )) 1 ( L N − s N ) ∆ ⩽ ρ Γ ⩽ ζ ( R ( · − a )) 1 ( L N + s N ) ∆ . Using the low er b ound ( 26 ) with δ = 1 and dividing by the volume w e get E ( Γ ) | L N ∆ | ⩾  1 − C ℓ  Z SO(3) d Q − Z C ℓ d τ X z ∈ Z 3 24 X j =1 E  Γ   √ 1 ℓ ∆ j ∗ η ( Q ·− ℓz − τ )  | L N ∆ | − C ℓ 1 | L N ∆ | Z R 3 ( ρ Γ + ρ 2 Γ ) . As b efore, we split the ( z , j )-sum to in terior-, and b oundary parts and estimate the b oundary con tributions using the Lieb–Oxford b ound. Using our constraint on ρ Γ for the last term, optimizing ov er Γ and then av eraging o ver R and a w e ﬁnd e e L N ∆ ,s N ( ζ ) ⩾  1 − C ℓ   1 − C ℓ + s n L N  e ℓ ∆ ,η ( ζ ) − C ℓ + s n L N − Z R 3 ζ 4 / 3 − C ℓ | ( L N + s N ) ∆ | | L N ∆ | − Z R 3 ( ζ + ζ 2 ) , where we used estimates analogous to ( 33 ) and ( 34 ). T aking the liminf as N → ∞ and the limsup as ℓ → ∞ , we arriv e at lim inf N →∞ e e L N ∆ ,s N ( ζ ) ⩾ lim sup ℓ →∞ e ℓ ∆ ,η ( ζ ) = e NUEG ( ζ ) . Com bining this with ( 38 ), we conclude that the tw o limits exist and equal. □ 5.6. Thermo dynamic limit for general domains. In this section, we pro v e Theorem 3.3 . The pro of is standard and the technique go es bac k to Fisher [ 10 ]. Pro of of (i). Let { Ω N } ⊂ R 3 and { δ N } ⊂ R + b e sequences such that δ N / | Ω N | 1 / 3 → 0 and δ N | Ω N | 1 / 3 → ∞ and that the Fisher regularit y condition | ∂ Ω N + B r | ⩽ C r | Ω N | 2 / 3 holds true for all r ⩽ | Ω N | 1 / 3 /C . It is straightforw ard to see that the monotonicit y b ound ( 32 ) holds true with the large tetrahedron ℓ ′ ∆ replaced by a general domain Ω N , e Ω N ,η δ N ( ζ ) ⩾ 1 − C σ δ ℓ − C σ ℓ + δ + δ N | Ω N | 1 / 3 ! e ℓ ∆ ,η δ ( ζ ) − C ℓ + δ + δ N | Ω N | 1 / 3 − Z R 3 ζ 4 / 3 − C ℓ (1 + ℏ 2 δ − 1 ) − Z R 3 ζ + δ 3 − Z R 3 ζ 2 ! . 34 NON-UNIFORM ELECTRON GAS T aking ℓ = | Ω N | 1 / 6 and δ ﬁxed, we ﬁnd lim inf N →∞ e Ω N ,η δ N ( ζ ) ⩾ e NUEG ( ζ ) , where the r.h.s. is given b y the limit for tetrahedra (see Theorem 5.10 ). The pro of of the upp er b ound is similar to that of Theorem 5.10 (ii), but instead of the big tetrahedron ℓ ′ ∆ , we use the general domain Ω N . W e obtain e Ω N ,η δ N ( ζ ) ⩽  1 + C σ ℓ + δ + δ N | Ω N | 1 / 3  − Z 1+ α 1 − α d t t 4 e tℓ ∆ ,η tδ ( ζ ) + C ℏ 2 ℓ + δ + δ N | Ω N | 1 / 3 " − Z R 3 ζ 5 / 3 + 1 ℓδ − Z R 3 ζ + − Z R 3 | ∇ p ζ | 2 # + C ℏ 2 δ N | Ω N | 1 / 3 − Z R 3 ζ + C δ 2 log( α − 1 ) − Z R 3 ζ 2 . Cho osing ℓ = | Ω N | 1 / 6 and δ = | Ω N | − 1 / 12 , applying ( 37 ) on the t -a veraged energy and taking the limsup as N → ∞ , w e obtain lim sup N →∞ e Ω N ,η δ N ( ζ ) ⩽ e NUEG ( ζ ) . Pro of of (ii). Similarly to the low er b ound in the proof of Proposition 5.12 abov e, w e can deduce lim inf N →∞ e e Ω N ,s N ( ζ ) ⩾ e NUEG ( ζ ) . Next w e show the upp er b ound. Fix N ∈ N large, a ∈ R 3 and Q ∈ SO(3). Then b y setting ρ N = ( 1 Ω N ∗ η δ ) ζ ( Q ( · − a )) w e hav e ζ ( Q ( · − a )) 1 Ω s N − N ⩽ ρ N ⩽ ζ ( Q ( · − a )) 1 Ω s N + N . Hence, inf √ ρ ∈ H 1 ( R 3 ) ζ ( Q ( ·− a )) 1 Ω s N − N ⩽ ρ ⩽ ζ ( Q ( ·− a )) 1 Ω s N + N E ( ρ ) | Ω N | ⩽ E ( ρ N ) | Ω N | . Clearly , w e hav e ρ N ≡ ζ ( Q ( · − a )) on Ω s N − N . Next, using Theorem 5.9 e e Ω N ,s N ( ζ ) ⩽ − Z R 3 d a Z SO(3) d Q E ( ρ N ) | Ω N | ⩽ 1 | Ω N | − Z R 3 d a Z SO(3) d Q − Z 1+ α 1 − α d t t 4 Z SO(3) d R − Z C t ( ℓ + δ ) d τ × X z ∈ Z 3 24 X j =1 E ( 1 t ( ℓ + δ ) T j β ∆ ∗ η δ ( R · − t ( ℓ + δ ) z − τ ) ρ N ) + 1 | Ω N | − Z R 3 d a Z SO(3) d Q " C ℏ 2 δ ℓ Z R 3 ρ 5 / 3 N + C ℏ 2 ℓδ Z R 3 ρ N + C ℏ 2 δ ℓ Z R 3 | ∇ √ ρ N | 2 + C δ 2 log( α − 1 ) Z R 3 ρ 2 N # =: (I) + (I I) NON-UNIFORM ELECTRON GAS 35 where we used the notation β = ℓ/ ( ℓ + δ ) as ab ov e. It is clear that 1 | Ω N | − Z R 3 d a Z SO(3) d Q Z R 3 ρ α N ⩽ − Z R 3 ζ α and also 1 | Ω N | − Z R 3 d a Z SO(3) d Q Z R 3 | ∇ √ ρ N | 2 ⩽ C | Ω N | 1 / 3 δ − Z R 3 ζ + C − Z R 3 | ∇ p ζ | 2 using Lemma 5.4 . Deﬁne the index set J := ( ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : supp( 1 t ( ℓ + δ ) T j β ∆ ∗ η δ ( R · − t ( ℓ + δ ) z − τ )) ∩ Ω s N + N  = ∅ , for some R ∈ SO(3) and τ ∈ C t ( ℓ + δ ) ) , whic h collects all the small smeared tetrahedra that possibly intersect with the outer domain Ω s + N . F urthermore, deﬁne J 0 := ( ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : supp( 1 t ( ℓ + δ ) T j β ∆ ∗ η δ ( R · − tℓz − τ )) ⊂ Ω s N − N , for all R ∈ SO(3) and τ ∈ C t ( ℓ + δ ) ) , whic h contains small smeared tetrahedra that lie inside the inner domain Ω s − N . The J 0 -part of (I) reads (Ia) = 1 | Ω N | − Z R 3 d a Z SO(3) d Q − Z 1+ α 1 − α d t t 4 Z SO(3) d R − Z C t ( ℓ + δ ) d τ × X ( z ,j ) ∈ J 0 E  1 t ( ℓ + δ ) T j β ∆ ∗ η δ ( R · − t ( ℓ + δ ) z − τ ) ζ ( Q ( · − a ))  , whic h gives similarly as ab ov e, | J 0 || ℓ ∆ | | Ω N | − Z 1+ α 1 − α d t t 4 e tℓ ∆ ,η δ ( ζ ) . The ( J \ J 0 )-part of (I) is b ounded similarly as ( 36 ), and we ﬁnd (Ib) ⩽ C ℏ 2 | J \ J 0 || ∆ | | Ω N | " − Z R 3 ζ 5 / 3 + 1 ℓδ − Z R 3 ζ + − Z R 3 | ∇ p ζ | 2 # + C ℏ 2 | Ω N | 1 / 3 δ − Z R 3 ζ . The set J \ J 0 con tains tetrahedra at a distance O ( ℓ + δ + s N ) of ∂ Ω N . Since | ∂ Ω N | ⩽ C | Ω N | 2 / 3 b y Fisher regularit y , the volume fraction is O (( ℓ + δ + s N ) / | Ω N | 1 / 3 ) → 0 as N → ∞ . Applying ( 37 ) on the t -a veraged energy of (I), we obtain after taking the ﬁrst the limsup as N → ∞ and then δ /ℓ → 0, ℓδ → ∞ , lim sup N →∞ e e Ω N ,s N ( ζ ) ⩽ e NUEG ( ζ ) . This completes the pro of of Theorem 3.3 . 5.7. Lo cal densit y approximation. In this section, w e brieﬂy sketc h the proof of Theorem 3.4 . It is more complicated than the pro of of the classical case, whic h w e detailed in Section 4.5 . Here, we do not attempt to pro duce explicit constants. Analogously to the classical case, our b ound follows from a priori estimates of Remark 2 (ii)–(iii) for large ε . Hence, we ma y assume that ε is suﬃcien tly small. 36 NON-UNIFORM ELECTRON GAS T o show the upper b ound, we set ℓ = ε − 3 / 2 and δ = √ ε . and start with [ 20 , Eq. (75)], which implies E (( 1 ℓ ∆ ∗ η δ ) ζ ) ⩽ Z R 3 ( 1 ℓ ∆ ∗ η δ )( x ) e UEG ( ζ ( x )) d x + C ε Z R 3 ( 1 ℓ ∆ ∗ η δ )  ζ + ζ 2  + C ε 4 p − 1 Z ℓ ∆ + B δ | ∇ ζ θ | p + C ε Z R 3 ( 1 ℓ ∆ ∗ η δ ) | ∇ p ζ | 2 + C Z R 3 ζ    ∇ p 1 ℓ ∆ ∗ η δ    2 Replacing ζ by ζ ( R ( · − a )) and av eraging ov er all translations and rotations, w e ﬁnd after dividing b y the v olume e ℓ ∆ ,η δ ( ζ ) ⩽ − Z R 3 e UEG ( ζ ( x )) d x + C ε − Z R 3  ζ + ζ 2  + C ε 4 p − 1 − Z R 3 | ∇ ζ θ | p + C ε − Z R 3 | ∇ p ζ | 2 , whic h is our stated upp er b ound. The lo w er b ound is more complicated. Replacing ℓ → tℓ , δ → tδ and av eraging in ( 27 ) of Theorem 5.5 we obtain for a large smeared tetrahedron of scale ℓ ′ ≫ ℓ E (( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ) ⩾ (1 − C ε ) − Z 3 / 2 1 / 2 d t t 4 X z ∈ Z 3 24 X j =1 E  1 tℓ ∆ j ∗ η tδ ( · − tℓz )( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ  − C ε Z R 3 ( 1 ℓ ′ ∆ ∗ η δ ′ )( ζ + ε 2 ζ 2 ) As b efore, we split the sum in the ﬁrst term according to J 0 and J \ J 0 , where J 0 and J are deﬁned as in the pro of of Lemma 5.11 . The contribution of the J 0 -part is (1 − C ε ) − Z 3 / 2 1 / 2 d t t 4 X ( z ,j ) ∈ J 0 E  1 tℓ ∆ j ∗ η tδ ( · − tℓz ) ζ  . (39) W e claim that if for any ζ the b ound − Z 3 / 2 1 / 2 d t t 4 " E  1 tℓ ∆ j ∗ η tδ ( · − tℓz ) ζ ) − Z R 3 1 tℓ ∆ j ∗ η tδ ( x − tℓz ) e UEG ( ζ ( x )) d x + C Z R 3    ∇ q 1 tℓ ∆ j ∗ η tδ ( · − tℓz )    2 ζ + C ε Z R 3 1 tℓ ∆ j ∗ η tδ ( · − tℓz )( ζ + ζ 2 ) + C ε Z R 3 1 tℓ ∆ j ∗ η tδ ( · − tℓz ) | ∇ p ζ | 2 # ⩾ − C ε 4 p − 1 Z (2 ℓ ∆ j + ℓz )+ B 2 δ | ∇ ζ θ | p (40) holds true for all ( z , j ) ∈ J 0 with a univ ersal constant C > 0, then the pro of is ﬁnished. In fact, b y replacing ζ by ζ ( Q ( · − a )) in ( 40 ) and av eraging ov er all translations and rotations, and plugging into ( 39 ) w e ﬁnd the stated low er b ound in the limit δ ′ /ℓ ′ → 0, ℓ ′ δ ′ → 0. The b oundary con tributions in the J \ J 0 -part result in an error term − C ε − R R 3 ( ζ + ζ 2 ) using the Lieb–Oxford inequality . Also, the term corresp onding to − C ε in ( 39 ) is b ounded using Corollary 5.3 and merged in to the other terms. T o sho w ( 40 ) we distinguish tw o cases just like in the classical case. NON-UNIFORM ELECTRON GAS 37 Case I. (Simple tetrahedra) If ( z , j ) ∈ J 0 is such that Z R 3 1 tℓ ∆ j ∗ η tδ ( · − tℓz )( ζ 4 / 3 + ζ 5 / 3 ) ⩽ C ε Z R 3 1 tℓ ∆ j ∗ η tδ ( · − tℓz )( ζ + ζ 2 ) + 1 ε 4 p − 1 Z (2 ℓ ∆ j + ℓz )+ B 2 δ | ∇ ζ θ | p , (41) then using the a priori bound      E  1 tℓ ∆ j ∗ η tδ ( · − tℓz ) ζ ) − Z R 3 1 tℓ ∆ j ∗ η tδ ( x − tℓz ) e UEG ( ζ ( x )) d x      ⩽ C Z R 3 1 tℓ ∆ j ∗ η tδ ( · − tℓz )( ζ 4 / 3 + ζ 5 / 3 ) + C Z R 3 1 tℓ ∆ j ∗ η tδ ( · − tℓz ) | ∇ p ζ | 2 + C Z R 3    ∇ q 1 tℓ ∆ j ∗ η tδ ( · − tℓz )    2 ζ , whic h in turn follows from Theorem 5.2 and Theorem 4.1 , implies ( 40 ). Case I I. (Main tetrahedra) If the opp osite of ( 41 ) holds in a tetrahedron ( z , j ) ∈ J 0 , then we pro ceed analogously to Section 4.5 . W e omit the details of the rest of the argument for brevity . 5.8. Semiclassical bound. Here, we prov e the b ound of Prop osition 3.5 . Obvi- ously , w e hav e E ℏ ( ρ ) ⩾ E cl ( ρ ) for all √ ρ ∈ H 1 ( R 3 ). The issue is that when we put ρ = ( 1 ℓ ′ ∆ ∗ η δ ) ζ , the classical NUEG energy will inv olv e a smo oth cutoﬀ in- stead of the hard one used to our deﬁnition. But this smooth cutoﬀ will introduce an error which is negligible in the thermo dynamic limit. Using the Graf–Sc henker inequalit y , e ℏ ℓ ′ ∆ ,η δ ′ ( ζ ) ⩾ Z SO(3) d Q − Z R 3 d a E cl  ( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a ))  | ℓ ′ ∆ | ⩾ 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ X z ∈ Z 3 24 X j =1 E cl  ( 1 ℓ ′ ∆ ∗ η δ ′ ) 1 ℓ ∆ j ( R · − ℓz − τ ) ζ ( Q ( · − a ))  − C ℓ 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z R 3 ( 1 ℓ ′ ∆ ∗ η δ ′ ) ζ ( Q ( · − a )) =: (I) + (I I) . Here, (I I) is simply − C ℓ − R R 3 ζ . T o deal with (I), we deﬁne the index set J := ( ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : supp( 1 ℓ ∆ j ( R · − ℓz − τ )) ∩ supp( 1 ℓ ′ ∆ ∗ η δ ′ )  = ∅ , for some R ∈ SO(3) and τ ∈ C ℓ ) , whic h collects all the small (sharp) tetrahedra R ⊤ ( ℓ ∆ j + ℓz + τ ) that p ossibly in tersect with the big smeared tetrahedron supp( 1 ℓ ′ ∆ ∗ η δ ′ ). F urthermore, deﬁne J 0 := ( ( z , j ) ∈ Z 3 × { 1 , . . . , 24 } : supp( 1 ℓ ∆ j ( R · − ℓz − τ )) ⊂ ( ℓ ′ − δ ′ ) ∆ , for all R ∈ SO(3) and τ ∈ C ℓ ) , 38 NON-UNIFORM ELECTRON GAS whic h contains the small tetrahedra that are well inside ℓ ′ ∆ . Using this, the term (I) can b e split in to tw o parts. The ﬁrst of which is (Ia) = 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ X ( z ,j ) ∈ J 0 E cl  1 ℓ ∆ j ( R · − ℓz − τ ) ζ ( Q ( · − a ))  = | J 0 | | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a E cl  1 ℓ ∆ ζ ( Q ( · − a ))  = | J 0 || ℓ ∆ | | ℓ ′ ∆ | e cl ℓ ∆ ( ζ ) . The second part is (Ib) = 1 | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a Z SO(3) d R − Z C ℓ d τ X ( z ,j ) ∈ J \ J 0 E cl  1 ℓ ∆ j ( R · − ℓz − τ ) ζ ( Q ( · − a ))  ⩾ − | J \ J 0 | | ℓ ′ ∆ | Z SO(3) d Q − Z R 3 d a c LO Z R 3 1 ℓ ∆ ζ ( Q ( · − a )) 4 / 3 ⩾ − | J \ J 0 || ℓ ∆ | | ℓ ′ ∆ | c LO − Z R 3 ζ 4 / 3 , using the Lieb–Oxford inequalit y . The v olume ratios ob ey similar b ounds as in the pro of of Lemma 5.11 , hence by collecting the ab o v e estimates and taking the thermo dynamic limit δ ′ /ℓ ′ → 0, ℓ ′ δ ′ → ∞ and ℓ → ∞ , w e obtain e ℏ NUEG ( ζ ) ⩾ e cl NUEG ( ζ ), and from this lim inf µ → 0 b e NUEG ( µ ; ζ 0 ) ⩾ e cl NUEG ( ζ 0 ) as stated. Appendix A. Pr oof of Theorem 2.1 Let ﬁrst a = 0 and consider the set of lattice v ectors of tiles that lie completely inside C L , L 0 = { l ∈ L ′ : (Λ + l ) ⊂ C L } , where L ′ = { l ∈ L : (Λ + l ) ∩ C L  = ∅} , is the set of lattice v ectors of tiles that in tersect with C L . Then w e hav e by the L -p eriodicity of u , − Z C L u = 1 | C L | X l ∈L 0 Z Λ+ l u + 1 | C L | X l ∈L ′ \L 0 Z (Λ+ l ) ∩ C L u = |L 0 || Λ | | C L | − Z Λ u + 1 | C L | X l ∈L ′ \L 0 Z (Λ+ l ) ∩ C L u Noting that the cells corresp onding to L ′ \ L 0 o ccup y a v olume at most C Λ L d − 1 | Λ | for some constant C Λ > 0 dep ending only on Λ, so |L 0 || Λ | | C L | ⩾ 1 − C Λ | Λ | L . In summary , we obtain the estimate     − Z C L u − − Z Λ u     ⩽  1 − |L 0 || Λ | | C L | + |L ′ \ L 0 || Λ | | C L |  − Z Λ | u | ⩽ C Λ | Λ | L − Z Λ | u | whic h pro ves ( 1 ) for a = 0. F or the general case, w e replace u b y u ( · + a ) and note that − R Λ+ a u = − R Λ u by perio dicity . The proof is ﬁnished. NON-UNIFORM ELECTRON GAS 39 Appendix B. Weak-* lower semicontinuity of e ℏ NUEG This app endix is devoted to the pro of of Theorem B.1. The functional ζ 7→ e ℏ NUEG ( ζ ) is we ak-* lower semic ontinuous in the fol lowing sense: for any se quenc e { p ζ j } ⊂ H 1 per ( L ) and √ ζ ∈ H 1 per ( L ) such that p ζ j ⇀ √ ζ in H 1 loc ( R 3 ) , we have e ℏ NUEG ( ζ ) ⩽ lim inf j →∞ e ℏ NUEG ( ζ j ) . W e set ℏ = 1 and drop it from the notation for clarity . W e ﬁrst recall the follo wing weak-* low er semicon tinuit y property of the grand-canonical Levy–Lieb functional from [ 21 ]. Theorem B.2 (Lewin–Lieb–Seiringer) . F or any se quenc e { √ ρ j } ⊂ H 1 ( R 3 ) such that √ ρ j ⇀ √ ρ in ˙ H 1 ( R 3 ) , ther e holds F LL ( ρ ) ⩽ lim inf j →∞ F LL ( ρ j ) . The crucial p oin t here is that R R 3 ρ j need not conv erge to R R 3 ρ , in particular mass ma y b e lost. Still, the weak conv ergence of the gradien t of the “wa v efunction” √ ρ j is suﬃcient to establish the lo w er semicontin uit y of F LL ( ρ ). As a preparation, w e need a simple lemma. Lemma B.3. Supp ose that { p ζ j } ⊂ H 1 per ( L ) and √ ζ ∈ H 1 per ( L ) such that p ζ j ⇀ √ ζ in H 1 loc ( R 3 ) . Then p ( 1 Ω ∗ η δ ) ζ j ⇀ p ( 1 Ω ∗ η δ ) ζ in ˙ H 1 ( R 3 ) . Pr o of. W e hav e for for every φ ∈ ˙ H 1 ( R 3 ), Z R 3  ∇ q ( 1 Ω ∗ η δ ) ζ j − ∇ p ( 1 Ω ∗ η δ ) ζ  · ∇ φ = Z R 3 ( p ζ j − p ζ ) ∇ p 1 Ω ∗ η δ · ∇ φ + Z R 3 ( 1 Ω ∗ η δ )  ∇ p ζ j − ∇ p ζ  · ∇ φ, where in the ﬁrst term, ∥ ∇ p 1 Ω ∗ η δ · ∇ φ ∥ L 2 ⩽ ∥ ∇ p 1 Ω ∗ η δ ∥ ∞ ∥ ∇ φ ∥ L 2 < ∞ . Since supp( ∇ √ 1 Ω ∗ η δ ) is b ounded, our h yp othesis applies and we conclude that the r.h.s. tends to 0. This ﬁnishes the pro of. □ Next, w e show the w eak-* low er semicon tinuit y of the indirect energy per volume. Lemma B.4. Supp ose that { p ζ j } ⊂ H 1 per ( L ) and √ ζ ∈ H 1 per ( L ) such that p ζ j ⇀ √ ζ in H 1 loc ( R 3 ) . Then e ℓ ∆ ,η δ ( ζ ) ⩽ lim inf j →∞ e ℓ ∆ ,η δ ( ζ j ) . Pr o of. Instead of − R R 3 d a we might as w ell write − R Λ d a for a unit cell Λ of L in the deﬁnition of e ℓ ∆ ,η δ ( ζ ). By Theorem B.2 and the contin uity of D ( ρ ), we ha v e e Ω ,η δ ( ζ ) ⩽ Z SO(3) d R − Z Λ d a lim inf j →∞ E  ( 1 ℓ ∆ ∗ η δ ) ζ j ( R ( · − a ))  | ℓ ∆ | = Z SO(3) d R − Z Λ d a lim inf j →∞ f ℓ ∆ ,η δ ,ζ j ( R · ) ( a ) , (42) 40 NON-UNIFORM ELECTRON GAS where f Ω ,η δ ,ζ ( a ) was deﬁned in ( 9 ). Here, the sequence of functions f ℓ ∆ ,η δ ,ζ j ( R · ) can b e b ounded from b elo w by Lieb–Oxford inequality as f ℓ ∆ ,η δ ,ζ j ( R · ) ( a ) ⩾ − C 1 | ℓ ∆ | Z R 3 ( 1 ℓ ∆ ∗ η δ ) ζ 4 / 3 j ( R ( · − a )) . Consider the set of lattice vectors of the rotated tiles that lie well inside 1 ℓ ∆ ∗ η δ , L 0 = { l ∈ L ′ : R ⊤ (Λ + l ) ⊂ ( ℓ − δ ) ∆ } , where L ′ = { l ∈ L : R ⊤ (Λ + l ) ∩ supp( 1 ℓ ∆ ∗ η δ )  = ∅} . W e hav e 1 | ℓ ∆ | Z R 3 ( 1 ℓ ∆ ∗ η δ ) ζ 4 / 3 j ( R ( · − a )) = 1 | ℓ ∆ | X l ∈L 0 Z R ⊤ (Λ+ l ) ζ 4 / 3 j ( R ( · − a )) + 1 | ℓ ∆ | X l ∈L ′ \L 0 Z R ⊤ ( l +Λ) ( 1 ℓ ∆ ∗ η δ ) ζ 4 / 3 j ( R ( · − a )) ⩽ |L 0 || Λ | | ℓ ∆ | − Z Λ ζ 4 / 3 j + 1 | ℓ ∆ | X l ∈L ′ \L 0 Z R ⊤ ( l +Λ) ζ 4 / 3 j ( R ( · − a )) ⩽ − Z Λ ζ 4 / 3 j + |L ′ \ L 0 || Λ | | ℓ ∆ | − Z Λ ζ 4 / 3 j . Here, |L ′ \ L 0 || Λ | ⩽ C Λ δ ℓ 2 , so we obtain the bound f Ω ,η δ ,ζ j ( R · ) ( a ) ⩾ − C Λ δ ℓ − Z Λ ζ 4 / 3 j . By the w eak con v ergence of p ζ j , we hav e R Λ ζ j ⩽ C and R Λ | ∇ p ζ j | 2 ⩽ C . The Sobolev-, and H¨ older inequalities imply that R Λ ζ 4 / 3 j ⩽ C . In conclusion, f ℓ ∆ ,η δ ,ζ j ( R · ) ( a ) ⩾ − C for a ∈ Λ, so F atou’s lemma may b e used to interc hange the liminf and the in tegrations in ( 42 ), which ﬁnishes the pro of. □ By com bining the ab ov e ﬁnite volume weak-* low er semicon tinuit y prop erty with the con vergence rate estimates from Theorem 5.10 , w e can complete the pro of of Theorem B.1 . Pr o of of The or em B.1 . Using our conv ergence rate estimates Theorem 5.10 and Lemma B.4 , we ﬁnd e NUEG ( ζ ) ⩽ e ℓ ∆ ,η δ ( ζ ) + (error term of Theorem 5.10 (iii)) ℓ,δ ⩽ lim inf j →∞ e ℓ ∆ ,η δ ( ζ j ) + (error term of Theorem 5.10 (iii)) ℓ,δ ⩽ lim inf j →∞ h e NUEG ( ζ j ) + (error term of Theorem 5.10 (i)) ℓ,δ,j i + (error term of Theorem 5.10 (iii)) ℓ,δ − − − − − − − − − → δ /ℓ → 0 , ℓδ →∞ lim inf j →∞ e NUEG ( ζ j ) where in the last step we used the fact that the error terms in the con vergence rate b ound are all uniformly b ounded in j . □ NON-UNIFORM ELECTRON GAS 41 References [1] Rob ert A Adams and John JF F ournier. Sob olev sp ac es . Elsevier, 2003. [2] V olker Bac h, Elliott H Lieb, and Jan Philip Solov ej. Generalized Hartree-Fo ck theory and the Hubbard mo del. Journal of statistic al physics , 76:3–89, 1994. [3] Olivier Bok anowski, Beno ˆ ıt Greb ert, and Norb ert J Mauser. Lo cal density appro ximations for the energy of a p eriodic Coulom b mo del. 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