Convergence of Levy–Lieb to Thomas–Fermi density functional

Abstract

We prove that the Levy–Lieb density functional Gamma-converges to the Thomas–Fermi functional in the semiclassical mean-field limit. In particular, this aides an easy alternative proof of the validity of the atomic Thomas–Fermi theory which was first established by Lieb and Simon.

1 Introduction

From first principles of quantum mechanics, the total energy of N identical fermions in ${\mathbb R }^d$ with spin $q\geqslant 1$ can be described by a Hamiltonian $H_N$ in the Hilbert space

$$\begin{aligned} \mathfrak {H}^N_a= L^2_a\left( ({\mathbb R }^{d}\times \{1,2,\ldots ,q\})^N;{\mathbb C }\right) , \end{aligned}$$

which contains wave functions which are anti-symmetric under the permutations of space-spin variables:

$$\begin{aligned} \Psi _N(\ldots ,(x_i,\sigma _i),\ldots ,(x_j,\sigma _j),\ldots )=-\Psi _N(\ldots ,(x_i,\sigma _i),\ldots ,(x_j,\sigma _j),\ldots ). \end{aligned}$$

In particular, the ground state energy of the system is

$$\begin{aligned} E_N^\mathrm{QM}&= \inf \left\{ \langle \Psi _N, H_N \Psi _N \rangle : \Psi _N \in S_N\right\} , \end{aligned}$$

(1)

where $S_N$ is the set of all (normalized) wave functions in the quadratic form domain of $H_N$.

Although the above microscopic theory is very precise, it usually becomes too complicated for practical calculations when N is large. Therefore, it is desirable to develop effective theories which depend on less variables but still capture some collective properties of the system in certain regimes.

1.1 Levy–Lieb and Thomas–Fermi density functionals

In density functional theory, instead of considering a complicated wave function $\Psi _N\in S_N$ one simply looks at its one-body density

$$\begin{aligned} \rho _{\Psi _N}(x)= N \sum _{\sigma _1,\ldots ,\sigma _N\in \{1,\ldots ,q\}} \int _{({\mathbb R }^d)^{N-1}} |\Psi _N((x,\sigma _1),(x_2,\sigma _2),\ldots ,(x_N,\sigma _N))|^2 d x_2 \ldots d x_N, \end{aligned}$$

which satisfies the simple constraints

$$\begin{aligned} \rho _{\Psi _N}(x)\geqslant 0, \quad \int _{{\mathbb R }^d} \rho _{\Psi _N}(x) d x =N. \end{aligned}$$

The idea of describing a quantum state using its one-body density goes back to Thomas [34] and Fermi [9] in 1927. It was conceptually pushed forward by a variational principle of Hohenberg and Kohn [17] in 1964, and since then many variations have been proposed.

In this paper, we are interested in the Levy–Lieb density functional [20, 22]

$$\begin{aligned} \mathcal {L}_N(\rho ) = \inf \left\{ \langle \Psi _N, H_N \Psi _N \rangle : \Psi _N \in S_N , \rho _{\Psi _N}=\rho \right\} . \end{aligned}$$

(2)

This is nicely related to the ground state problem via the identity

$$\begin{aligned} E_N^\mathrm{QM} = \inf \left\{ {\mathcal {L}_N}(\rho ): \rho \geqslant 0, \int _{{\mathbb R }^d} \rho =N \right\} , \end{aligned}$$

(3)

but we will consider (5) in a general context (without limiting to the ground state problem). The complication of the many-body problem is now hidden in the determination of ${\mathcal {L}_N}$, and finding a good approximation is desirable.

In this paper, we will focus on the typical situation when the particles are governed by the non-relativistic kinetic operator, an external potential and a pair-interaction potential, namely the Hamiltonian of the system reads (in appropriate units)

$$\begin{aligned} H_N=\sum _{i=1}^N \Big (- h^2\Delta _{x_i}+V(x_i)\Big )+\lambda \sum _{1\leqslant i<j\leqslant N} w(x_i-x_j). \end{aligned}$$

(4)

Here $h>0$ plays the role of Planck’s constant, and $\lambda >0$ corresponds to the strength of the interaction. The Thomas–Fermi approximation [9, 34] suggests that

$$\begin{aligned} {\mathcal {L}_N}(\rho ) \approx K_\mathrm{cl} h^2 \int _{{\mathbb R }^d} \rho ^{1+2/d} + \int _{{\mathbb R }^d} V\rho + \frac{\lambda }{2}\iint _{{\mathbb R }^d \times {\mathbb R }^d} \rho (x)\rho (y) w(x-y) d x d y \end{aligned}$$

(5)

where

$$\begin{aligned} K_\mathrm{cl} = \frac{d}{d+2} \cdot \frac{(2\pi )^2}{(q|B_{{\mathbb R }^d}(0,1)|)^{2/d}} . \end{aligned}$$

Historically, the Thomas–Fermi approximation was proposed for the atomic Hamiltonian, when V and w are Coulomb potentials in ${\mathbb R }^3$, but we may expect that it holds in a more general context. In this paper, we aim at giving rigorous justifications for (5) in the semiclassical mean-field regime

$$\begin{aligned} N\rightarrow \infty , \quad h \sim N^{-1/d}, \quad \lambda \sim N^{-1}, \end{aligned}$$

which is natural to make all three terms on the right side of (5) comparable.

To formulate our statements precisely, let us denote $\rho =Nf$ and rewrite (5) as

$$\begin{aligned} {\mathcal {E}_N}(f) \approx \mathcal {E}^\mathrm{TF} (f) \end{aligned}$$

(6)

where

$$\begin{aligned} \mathcal {E}_N(f)&= N^{-1}\mathcal {L}_N(Nf ) = \inf \left\{ N^{-1} \langle \Psi _N, H_N \Psi _N \rangle : \Psi _N \in S_N, \rho _{\Psi _N}=Nf \right\} ,\nonumber \\ \mathcal {E}^\mathrm{TF}(f)&= K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d} + \int _{{\mathbb R }^d} Vf + \frac{1}{2}\iint _{{\mathbb R }^d \times {\mathbb R }^d} f(x)f(y) w(x-y) d x d y. \end{aligned}$$

(7)

1.2 An open problem

We expect that (6) holds for a very large class of potentials. However, to make the discussion concrete, let us assume the following conditions in the rest of the paper.

Conditions on potentials The potentials $V,w:{\mathbb R }^d \rightarrow {\mathbb R }$ belong to $L^{p}({\mathbb R }^d)+L^{q}({\mathbb R }^d)$ with $p,q\in [1+d/2,\infty )$. Moreover, w admits the decomposition

$$\begin{aligned} w(x) = \int _0^{\infty } (\chi _r * \chi _r) (x) dr, \end{aligned}$$

(8)

for a family of radial functions $0\leqslant \chi _r \in L^{p}({\mathbb R }^d)+L^{q}({\mathbb R }^d)$ with $p,q\in [2+d,\infty )$

These assumptions hold for the Coulomb potentials in ${\mathbb R }^3$; in particular we have the Fefferman-de la Llave formula [11]

$$\begin{aligned} \frac{1}{|x|} = \frac{1}{\pi }\int _0^{\infty } \frac{1}{r^5} ({\mathbb {1} }_{B_r} * {\mathbb {1} }_{B_r})(x) dr \end{aligned}$$

where ${\mathbb {1} }_{B_r}$ is the characteristic function of the ball B(0, r) in ${\mathbb R }^3$. In fact, (8) holds true for a large class of radial positive functions; see [15] for details.

Recall $\mathcal {E}_N$ and $\mathcal {E}^\mathrm{TF}$ in (7). We expect that the following holds true.

Conjecture 1

(Semiclassical mean-field limit of Levy–Lieb functional) For all $d\geqslant 1$, in the limit $N\rightarrow \infty$, $hN^{1/d}\rightarrow 1$, $\lambda N \rightarrow 1$, we have

$$\begin{aligned} \mathcal {E}_N(f)\rightarrow \mathcal {E}^\mathrm{TF}(f) \end{aligned}$$

(9)

for every function f satisfying $f\geqslant 0$, $\sqrt{f}\in H^1({\mathbb R }^d)$ and $\int _{{\mathbb R }^d} f=1$.

Here are some immediate remarks on Conjecture 1.

(1) By the Hoffmann-Ostenhof inequality [16]

$$\begin{aligned} \left\langle \Psi _N, \sum _{i=1}^N (-\Delta _{x_i}) \Psi _N \right\rangle \geqslant \int _{{\mathbb R }^d} |\nabla \sqrt{\rho _{\Psi _N}}|^2, \end{aligned}$$

(10)

the condition $\sqrt{f}\in H^1({\mathbb R }^d)$ in Conjecture 1 is necessary to ensure that $\mathcal {E}_{N}(f)<\infty$.

(2) In the ideal Fermi gas (i.e. $V=w=0$), the Levy–Lieb functional boils down to the kinetic density functional

$$\begin{aligned} \mathcal {K}_N(f):= \inf \left\{ N^{-1-2/d} \left\langle \Psi _N, \sum _{i=1}^N (-\Delta _{x_i}) \Psi _N \right\rangle : \Psi _N \in S_N,\rho _{\Psi _N}=Nf \right\} . \end{aligned}$$

(11)

Conjecture 1 for the ideal Fermi gas states that, for all $d\geqslant 1$,

$$\begin{aligned} \mathcal {K}_N(f)\rightarrow K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d}. \end{aligned}$$

(12)

In fact, the following stronger, quantitative bounds are expected to hold [22, 28, 32]

$$\begin{aligned} K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d} \leqslant \mathcal {K}_N(f) \leqslant K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d} + N^{-2/d}\int _{{\mathbb R }^d} |\nabla \sqrt{f}|^2, \end{aligned}$$

(13)

for all $d\geqslant 1$ for the upper bound and all $d\geqslant 3$ for the lower bound.

The upper bound in (13) was proposed by March and Young in 1958 [32]. Their proof works for $d=1$, but fails in higher dimensions (see [22] for a discussion, and see [1, 13] for numerical investigations in $d=3$).

The lower bound in (13) was conjectured by Lieb and Thirring in 1975 [28, 29] and they proved the bound with a universal constant (different from $K_\mathrm{cl}$). Note that in $d=1$, the sharp constant in the lower bound in (13) is known to be smaller than $K_\mathrm{cl}$ (it is conjectured to be the optimal constant in a Sobolev–Gagliardo–Nirenberg inequality [29]). Despite several improvements over the constant (see [6] for the best known result), the sharp constant in the lower bound in (13) is still open in all $d\geqslant 1$. On the other hand, recently we proved that [33]

$$\begin{aligned} \mathcal {K}_N(f) \geqslant (K_\mathrm{cl}-\varepsilon ) \int _{{\mathbb R }^d} f^{1+2/d} - C_\varepsilon N^{-2/d}\int _{{\mathbb R }^d} |\nabla \sqrt{f}|^2 \end{aligned}$$

(14)

for all $d\geqslant 1$ and all $\varepsilon >0$. This implies the lower bound in (12). The upper bound in (12) is open for all $d\geqslant 2$.

(3) If we ignore the kinetic part, the Levy–Lieb functional reduces to the classical interaction functional

$$\begin{aligned} \mathcal {I}_N(f):= \inf \left\{ N^{-2}\left\langle \Psi _N, \sum _{1\leqslant i<j\leqslant N} w(x_i-x_j) \Psi _N \right\rangle : \Psi _N \in S_N, \rho _{\Psi _N}=Nf \right\} . \end{aligned}$$

(15)

Conjecture 1 becomes

$$\begin{aligned} \lim _{N\rightarrow \infty } \mathcal {I}_N(f) = \frac{1}{2} \iint _{{\mathbb R }^d\times {\mathbb R }^d} f(x)f(y) w(x-y) d x d y, \end{aligned}$$

(16)

which can be proved rigorously. In fact, when w is the Coulomb potential in ${\mathbb R }^3$, the lower bound in (16) is a direct consequence of the Lieb–Oxford inequality [23]

$$\begin{aligned} \mathcal {I}_N(f) \geqslant \frac{1}{2} \iint _{{\mathbb R }^3\times {\mathbb R }^3} f(x)f(y) |x-y|^{-1} d x d y - 1.68 N^{-2/3} \int _{{\mathbb R }^3} f^{4/3} \end{aligned}$$

(17)

and the upper bound in (16) can be achieved easily by choosing a Slater determinant (a wave function of the form $\Psi _N=u_1 \wedge u_2 \wedge \ldots \wedge u_N$) with the density Nf. The proof of (16) for more general w can be extracted from the proof of our main result below. Other approaches to (16) based on the optimal transportation have recently attracted a lot of attention, see [3,4,5, 8, 10].

If we do not completely ignore the kinetic part, but take $h\rightarrow 0$ and fix N, then the Levy–Lieb functional functional $\mathcal {E}_N(f)$ converges to the interaction functional $\mathcal {I}_N(f)$ in (15). Results of this kind can be found in remarkable recent works [2, 4, 18].

The significance of Conjecture 1, as well as our main result below, is the fact that we take the proper semiclassical limit $h\sim N^{-1/d}$ as $N\rightarrow \infty$, which is crucial to obtain the full Thomas–Fermi functional.

1.3 Main result

While we could not prove the pointwise-type convergence in Conjecture 1, we will provide another justification for the Thomas–Fermi functional from the Levy–Lieb functional in the sense of the Gamma-Convergence.

Recall that $\mathcal {E}_N$ and $\mathcal {E}^\mathrm{TF}$ are defined in (7). We have

Theorem 2

(Gamma convergence from Levy–Lieb to Thomas–Fermi functional) For all $d\geqslant 1$, in the limit $N\rightarrow \infty$, $hN^{1/d}\rightarrow 1$, $\lambda N \rightarrow 1$, the Levy–Lieb functional $\mathcal {E}_N$ Gamma-converges to the Thomas–Fermi functional $\mathcal {E}^\mathrm{TF}$ in

$$\begin{aligned} \mathcal {B}=\left\{ 0\leqslant f\in L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d), \int _{{\mathbb R }^d} f=1 \right\} . \end{aligned}$$

(18)

More precisely, we have

• (i) (Lower bound) For every sequence $f_N \in \mathcal {B}$ such that $f_N \rightharpoonup f$ weakly in $L^{1+2/d}({\mathbb R }^d)$, then

  $$\begin{aligned} \liminf _{N\rightarrow \infty }\mathcal {E}_N(f_N) \geqslant \mathcal {E}^\mathrm{TF}(f). \end{aligned}$$

  (19)

• (ii) (Upper bound) For every $f\in \mathcal {B}$, there exists a sequence of Slater determinants $\Psi _N\in S_N$ such that $f_N=N^{-1}\rho _{\Psi _N}\rightarrow f$ strongly in $L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d)$, and

  $$\begin{aligned} \limsup _{N\rightarrow \infty } \mathcal {E}_N(f_N)\leqslant \mathcal {E}^\mathrm{TF}(f). \end{aligned}$$

  (20)

The notion of Gamma convergence is sufficient for many applications. In particular, we can come back to the ground state problem and immediately obtain

Corollary 3

(Convergence of ground state energy and ground states) For all $d\geqslant 1$, in the limit $N\rightarrow \infty$, $hN^{1/d}\rightarrow 1$, $\lambda N \rightarrow 1$, the ground state energy $E_N^\mathrm{QM}$ of $H_N$ converges to the Thomas–Fermi energy:

$$\begin{aligned} \lim _{N\rightarrow \infty }\frac{E_N^\mathrm{QM}}{N}= E^\mathrm{TF} = \inf \left\{ \mathcal {E}^\mathrm{TF}(f): f\in \mathcal {B} \right\} . \end{aligned}$$

(21)

Moreover, if $\Psi _N$ is a ground state for $E_N^\mathrm{QM}$ and if $f^\mathrm{TF}$ is a Thomas–Fermi minimizer, then $N^{-1}\rho _{\Psi _N}\rightharpoonup f^\mathrm{TF}$ weakly in $L^{1+2/d}({\mathbb R }^d)$.

Corollary 3 covers the seminal result of Lieb and Simon [27] on the validity of Thomas–Fermi in the atomic case (when V, w are Coulomb potentials in ${\mathbb R }^3$).

More general results on the ground state problem have been achieved recently by Fournais, Lewin and Solovej [12] by means of other techniques. Their method is based on a fermionic version of the de Finetti–Hewitt–Savage theorem for classical measures (which should be compared to a weak quantum de Finetti theorem for bosons [19], although the classical version is sufficient for fermions). Consequently, they can treat a very large class of interaction potentials; in particular no form of positivity, e.g. (8), is needed. In fact, negative interaction potentials can be handled by a clever technique of interchanging two-body to one-body potentials. The latter technique goes back to [7, 21, 30, 31] and seems rather specific for the ground state problem.

In contrast, our Theorem 2 applies to a more restrictive class of interaction potentials, but it is not limited to ground states (it can be applied to excited, or high energy states as well).

The rest of the paper is devoted to the proof of our main result. First, we will study the ideal gas separately in Sect. 2. Then the proof of Theorem 2 and Corollary 3 are given in Sect. 3.

In the following proof, we consider spinless particles for simplicity (adding a fixed spin $q\geqslant 1$ requires only straightforward modifications).

2 Kinetic density functional

In this section we prove Theorem 2 in the special case of the ideal Fermi gas, which has its own interest. Recall the kinetic density functional in (11)

$$\begin{aligned} \mathcal {K}_N(f)= \inf \left\{ \frac{1}{N^{1+2/d}} \left\langle \Psi _N, \sum _{i=1}^N (-\Delta _{x_i}) \Psi _N \right\rangle : \Psi _N \in S_N,\rho _{\Psi _N}=Nf \right\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {B}=\left\{ 0\leqslant f\in L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d), \int _{{\mathbb R }^d} f=1 \right\} . \end{aligned}$$

We will prove

Theorem 4

(Gamma convergence of kinetic density functional) For all $d\geqslant 1$, the following convergences hold when $N\rightarrow \infty$.

• (i) (Lower bound) If $f_N \in \mathcal {B}$ and $f_N \rightharpoonup f$ weakly in $L^{1+2/d}({\mathbb R }^d)$, then

  $$\begin{aligned} \liminf _{N\rightarrow \infty }\mathcal {K}_N(f_N) \geqslant K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d}. \end{aligned}$$

  (22)

• (ii) (Upper bound) For every $f\in \mathcal {B}$, there exists a sequence of Slater determinants $\Psi _N\in S_N$ such that $f_N=N^{-1}\rho _{\Psi _N}\rightarrow f$ strongly in $L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d)$, and

  $$\begin{aligned} \limsup _{N\rightarrow \infty } \mathcal {K}_N(f_N) \leqslant \limsup _{N\rightarrow \infty } \frac{1}{N^{1+2/d}} \left\langle \Psi _N, \sum _{i=1}^N -\Delta _{x_i} \Psi _N \right\rangle \leqslant K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d}. \end{aligned}$$

  (23)

Proof

Lower bound The lower bound (22) is a consequence of Weyl’s law for Schrödinger eigenvalues. Let $\Psi _N$ be a N-body wave function with density $\rho _{\Psi _N}=Nf_N$. We can define the one-body density matrix $\gamma _{\Psi _N}$ as a trace class operator on $L^2({\mathbb R }^d)$ with kernel

$$\begin{aligned} \gamma _{\Psi _N}^{(1)}(x,y)= \int _{({\mathbb R }^d)^{N-1}} \Psi _N(x,x_2,\ldots ,x_N) \overline{\Psi _N (y,x_2,\ldots ,x_N)} d x_2\ldots dx_N. \end{aligned}$$

(24)

Then for every function $0\leqslant U\in C_c^\infty ({\mathbb R }^d)$, we can write

$$\begin{aligned} h^2 \left\langle \Psi _N, \sum _{i=1}^N -\Delta _{x_i} \Psi _N \right\rangle = {{\mathrm{\mathrm{Tr}}}}\left[ (-h^2 \Delta - U)\gamma _{\Psi _N}^{(1)}\right] + N \int _{{\mathbb R }^d} U f_N. \end{aligned}$$

(25)

On the other hand, the anti-symmetry of $\Psi _N$ implies Pauli’s exclusion principle [26, Theorem 3.2]

$$\begin{aligned} 0\leqslant \gamma _{\Psi _N}^{(1)} \leqslant 1. \end{aligned}$$

(26)

Consequently, by the min-max principle [25, Theorem 12.1] and Weyl’s law on the sum of negative eigenvalues of Schrödinger operators [25, Theorem 12.12] we can estimate

$$\begin{aligned} {{\mathrm{\mathrm{Tr}}}}\left[ (-N^{-2/d} \Delta - U)\gamma _{\Psi _N}^{(1)}\right]&\geqslant {{\mathrm{\mathrm{Tr}}}}[-N^{-2/d} \Delta - U]_-\nonumber \\&= - N\frac{|B_{{\mathbb R }^d}(0,1)|}{(2\pi )^d(1+d/2)} \left[ \int _{{\mathbb R }^d} U^{1+d/2} + o(1)_{N\rightarrow \infty }\right] . \end{aligned}$$

(27)

From (25) and (27) we deduce that

$$\begin{aligned} \liminf _{N\rightarrow \infty } \mathcal {K}_N(f_N) \geqslant - \frac{|B_{{\mathbb R }^d}(0,1)|}{(2\pi )^d(1+d/2)} \int _{{\mathbb R }^d} U^{1+d/2} + \int _{{\mathbb R }^d} U f. \end{aligned}$$

Optimizing over U leads to the desired lower bound (22).

Upper bound We can follow the coherent state approach in the proof of Weyl’s law [25, Theorem 12.12] to deduce the upper bound (23), but the important conclusion that the density $Nf_N$ comes from a Slater determinant is not easily achieved in this way. In the following, we will provide a direct proof of the upper bound in Theorem 4, using explicit computations of the ground states of the ideal Fermi gas in cubes. This idea goes back to the heuristic argument of Thomas–Fermi [9, 34] and March-Young [32] (the same argument can be used to give a direct proof of the lower bound (22); see [14] for details).

Step 1 (Slater determinants with step-function densities) Recall that that the Dirichlet Laplacian $-\Delta$ on the cube $Q=[0,L]^d$ has eigenvalues $|\pi k/L|^2$, $k\in \mathbb {N}^d$, with eigenfunctions

$$\begin{aligned} u_k(x)= \prod _{i=1}^d \left[ \sqrt{\frac{2}{L}}\sin \bigg (\frac{\pi k^i x^i}{L}\bigg ) \right] , \quad k=(k^i)_{i=1}^d, x=(x^i)_{i=1}^d \in {\mathbb R }^d. \end{aligned}$$

The ground state of the M-body kinetic operator $\sum _{j=1}^M (-\Delta _{x_j})$ is the Slater determinant $\Psi _M^\mathrm{S}$ made of the first M eigenfunctions $\{u_k\}$. It is straightforward to see that when $M\rightarrow \infty$,

$$\begin{aligned} \frac{1}{M}\rho _{\Psi _M^{S}}= \frac{1}{M}\sum _{k\in S_M} |u_k|^2 \rightarrow \frac{{\mathbb {1} }_Q}{|Q|} \end{aligned}$$

(28)

strongly in $L^p(Q)$ for all $1\leqslant p<\infty$, and

$$\begin{aligned} \frac{1}{M^{1+2/d}} \left\langle \Psi _M^{S},\sum _{i=1}^M(-\Delta _{x_i}) \Psi _M^{S}\right\rangle = \frac{1}{M^{1+2/d}} \sum _{k\in S_M} \left| \frac{\pi k}{L}\right| ^2 \rightarrow \frac{K_{cl}}{|Q|^{2/d}}. \end{aligned}$$

(29)

Now let $f\in \mathcal {B}$. Let $\{Q\}$ be a finite family of disjoint cubes, whose construction will be specified in the next step. In the following we only consider cubes Q such that

$$\begin{aligned} \overline{f}^Q := \frac{1}{|Q|}\int _Q f>0,. \end{aligned}$$

We can find an integer number $M_Q\in (N |Q| \overline{f}^Q -1, N|Q| \overline{f}^Q+1]$ such that

$$\begin{aligned} \sum _{Q}M_Q= \sum _{Q} N |Q| \overline{f}^Q = N \int _{{\mathbb R }^d} f=1. \end{aligned}$$

Now for every Q, consider the first $M_Q$ eigenfunctions $\{u_j^Q\}_{j=1}^{M_Q}$ of the Dirichlet Laplacian $-\Delta$ on Q. These functions can be trivially extended to zero outside Q to become a function in $H^1_0(Q_0)$. Since $\{u_j^Q\}_{j=1}^{M_Q}$ are orthogonal for every Q and the subcubes $\{Q\}$ are disjoint, the collection $\bigcup _{Q} \{u_j^Q\}_{j=1}^{M_Q}$ is an orthonormal family of N functions in $H^1_0(Q_0)$. Let $\Psi _N^{S}\in S_N$ be the Slater determinant made of this orthogonal family. Then in the limit $N\rightarrow \infty$, using the fact that $M_Q/N\rightarrow |Q| \overline{f}^Q >0$ and  (28), (29), we get the following

$$\begin{aligned} \frac{1}{N}\rho _{\Psi _N^S} = \sum _Q \sum _{i=1}^{M_Q} \frac{|u_i^Q|^2}{M_Q} \cdot \frac{M_Q}{N}&\rightarrow \sum _Q \frac{{\mathbb {1} }_Q}{|Q|} \cdot |Q|\overline{f}^Q = \sum _Q {\mathbb {1} }_Q \overline{f}^Q \end{aligned}$$

(30)

strongly in $L^p({\mathbb R }^d)$ for all $1\leqslant p<\infty$, and

$$\begin{aligned}&\frac{1}{N^{1+2/d}} \left\langle \Psi _N^{S}, \sum _{i=1}^N - \Delta _{x_i} \Psi _N^{S} \right\rangle = \frac{1}{N^{1+2/d}} \sum _Q \sum _{i=1}^{M_Q}||\nabla u_i^Q||^2 \nonumber \\&\quad = \sum _Q \left[ \frac{1}{M_Q^{1+2/d}} \sum _{i=1}^{M_Q}||\nabla u_i^Q||^2 \right] \left| \frac{M_Q}{N}\right| ^{1+2/d} \rightarrow \sum _Q \frac{K_\mathrm{cl}}{|Q|^{2/d}} \cdot \left| |Q| \overline{f}^Q \right| ^{1+2/d} \nonumber \\&\quad = K_\mathrm{cl} \int _{{\mathbb R }^d} \Big | \sum _Q {\mathbb {1} }_Q \overline{f}^Q \Big |^{1+2/d} \leqslant K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d} . \end{aligned}$$

(31)

Step 2 (Approximating f by step functions and concluding) Since $0\leqslant f\in L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d)$, for every $k \geqslant 1$ we can find a finite family of disjoint cubes $\{Q\}$ such that

$$\begin{aligned} \left\| f - \sum _{Q} {\mathbb {1} }_Q \overline{f}^Q \right\| _{L^1} + \left\| f - \sum _{Q} {\mathbb {1} }_Q \overline{f}^Q \right\| _{L^{1+2/d}} \leqslant k^{-1}. \end{aligned}$$

(32)

Using this collection of cubes, for every $N\geqslant 1$ we can construct a Slater determinant $\Psi _N^\varepsilon \in S_N$ as in Step 2. From the convergence (30), (31), we deduce that there exists $M_k >0$ such that for every $N\geqslant M_k$,

$$\begin{aligned} \frac{1}{N^{1+2/d}} \left\langle \Psi _N^k, \sum _{i=1}^N - \Delta _{x_i} \Psi _N^k \right\rangle \leqslant K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d} + k^{-1} \end{aligned}$$

(33)

and

$$\begin{aligned} \left\| \frac{1}{N}\rho _{\Psi _N^k} - \sum _Q {\mathbb {1} }_Q \overline{f}^Q \right\| _{L^1} + \left\| \frac{1}{N}\rho _{\Psi _N^k} - \sum _Q {\mathbb {1} }_Q \overline{f}^Q \right\| _{L^{1+2/d}} \leqslant k^{-1}. \end{aligned}$$

The latter estimate and (32) imply that for every $N\geqslant M_k$,

$$\begin{aligned} \left\| \frac{1}{N}\rho _{\Psi _N^k} - f \right\| _{L^1} + \left\| \frac{1}{N}\rho _{\Psi _N^k} - f \right\| _{L^{1+2/d}} \leqslant 2 k^{-1}. \end{aligned}$$

(34)

Now we conclude using a standard diagonal argument. By induction we can choose the above sequence $M_k$ such that $M_{k+1}\geqslant M_k+1$. Now for every $N\in \mathbb {N}$, we take $k=k_N$ the smallest number such that $N\geqslant M_k$. Obviously we have $k_N\rightarrow \infty$ as $N\rightarrow \infty$. Moreover, we can choose the Slater determinant $\Psi _N=\Psi _N^{k_N}\in S_N$ as above, and obtain from (33), (34) that

$$\begin{aligned} \frac{1}{N^{1+2/d}} \left\langle \Psi _N, \sum _{i=1}^N - \Delta _{x_i} \Psi _N \right\rangle \leqslant K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d} + k_N^{-1} \rightarrow K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d} \end{aligned}$$

and

$$\begin{aligned} \left\| \frac{1}{N}\rho _{\Psi _N} -f \right\| _{L^1} + \left\| \frac{1}{N}\rho _{\Psi _N} - f \right\| _{L^{1+2/d}} \leqslant 2k_N^{-1} \rightarrow 0 \end{aligned}$$

when $N\rightarrow \infty$. This completes the proof of Theorem 4. $\square$

3 Full density functional

In this section we prove our main result.

Proof of Theorem 2

Lower bound Let $\Psi _N\in S_N$ such that $\rho _{\Psi _N}=Nf_N$ and $f_N\rightharpoonup f$ weakly in $L^{1+2/d}({\mathbb R }^d)$. By Theorem 4, we have

$$\begin{aligned} \liminf _{N\rightarrow \infty } N^{-1-2/d} N^{2/d}h^2 \left\langle \Psi _N, \sum _{i=1}^N -\Delta _{x_i} \Psi _N \right\rangle \geqslant K_\mathrm{cl} \int _{{\mathbb R }^d} f^{1+2/d}. \end{aligned}$$

(35)

Moreover, since $f_N\rightharpoonup f$ weakly in $L^{1+2/d}({\mathbb R }^d)$ and $\Vert f_N\Vert _{L^1}=1$, by interpolation we have $f_N\rightharpoonup f$ weakly in $L^{r}({\mathbb R }^d)$ for all $r \in (1, 1+2/d]$. Under the condition $V\in L^p({\mathbb R }^d)+L^q({\mathbb R }^d)$ with $p,q\in [1+d/2,\infty )$, we deduce that

$$\begin{aligned} \lim _{N\rightarrow \infty } N^{-1} \left\langle \Psi _N, \sum _{i=1}^N V(x_i) \Psi _N \right\rangle = \lim _{N\rightarrow \infty } \int _{{\mathbb R }^d} Vf_N = \int _{{\mathbb R }^d} Vf. \end{aligned}$$

(36)

It remains to consider the interaction terms. We will use an idea of Lieb, Solovej and Yngvason [24], which has been used to give an alternative proof of the Lieb-Oxford inequality. From the Fefferman-de la Llave type decomposition (8), we can write

$$\begin{aligned} w(x-y) = \int _0^{\infty } dr \int _{{\mathbb R }^d} dz \chi _r(x-z) \chi _r(y-z) dz \end{aligned}$$

(37)

and hence

$$\begin{aligned} \left\langle \Psi _N, \sum _{1\leqslant i<j\leqslant N}w(x_i-x_j)\Psi _N \right\rangle = \int _0^{\infty } dr \int _{{\mathbb R }^d} dz \left\langle \Psi _N, \sum _{1\leqslant i<j\leqslant N}\chi _r(x_i-z) \chi _r(x_j-z)\Psi _N \right\rangle . \end{aligned}$$

By the Cauchy–Schwarz inequality we get

$$\begin{aligned}&\left\langle \Psi _N, \sum _{1\leqslant i<j\leqslant N}\chi _r(x_i-z) \chi _r(x_j-z)\Psi _N \right\rangle \\&\quad = \left[ \frac{1}{2} \left\langle \Psi _N, \left( \sum _{i=1}^N \chi _r(x_i-z) \right) ^2 \Psi _N \right\rangle - \left\langle \Psi _N, \sum _{i=1}^N \chi _r^2(x_i-z) \Psi _N \right\rangle \right] _+\\&\quad \geqslant \left[ \frac{1}{2} \left\langle \Psi _N, \sum _{i=1}^N \chi _r(x_i-z) \Psi _N \right\rangle ^2 - \left\langle \Psi _N, \sum _{i=1}^N \chi _r^2(x_i-z) \Psi _N \right\rangle \right] _+\\&\quad = \left[ \frac{N^2}{2}(f_N*\chi _r)^2(z) - N (f_N*\chi _r^2)(z) \right] _+. \end{aligned}$$

For every fixed $r>0$ and $z\in \mathbb {{\mathbb R }^d}$, since $f_N\rightharpoonup f$ weakly in $L^{r}({\mathbb R }^d)$ for all $1<r\leqslant 1+2/d$, and $\chi _r,\chi _r^2 \in L^p({\mathbb R }^d)+L^q({\mathbb R }^d)$ with $p,q\in [1+d/2,\infty )$, we find that

$$\begin{aligned} \lim _{N\rightarrow \infty } (f_N*\chi _r)(z)&= (f*\chi _r)(z), \\ \lim _{N\rightarrow \infty } (f_N*\chi _r^2)(z)&= (f*\chi _r^2)(z), \end{aligned}$$

and hence

$$\begin{aligned} \lim _{N\rightarrow \infty } N^{-2} \lambda N \left[ \frac{N^2}{2}(f_N*\chi _r)^2(z) - N (f_N*\chi _r^2)(z) \right] _+ = \frac{1}{2}(f*\chi _r)^2(z) \end{aligned}$$

for every $z\in {\mathbb R }^d$. Therefore, by Fatou’s lemma,

$$\begin{aligned}&\liminf _{N\rightarrow \infty } N^{-2} \lambda N \left\langle \Psi _N, \sum _{1\leqslant i<j\leqslant N}w(x_i-x_j)\Psi _N \right\rangle \nonumber \\&\quad =\liminf _{N\rightarrow \infty } \int _0^{\infty } dr \int _{{\mathbb R }^d} dz N^{-2} \lambda N \left[ \frac{N^2}{2}(f_N*\chi _r)^2(z) - N (f_N*\chi _r^2)(z) \right] _+ \nonumber \\&\quad \geqslant \int _0^{\infty } dr \int _{{\mathbb R }^d} dz \frac{1}{2}(f*\chi _r)^2(z)= \frac{1}{2} \iint _{{\mathbb R }^d\times {\mathbb R }^d} f(x)f(y) w(x-y) d x dy. \end{aligned}$$

(38)

Here in the last identity we have used (8) again.

Putting (35), (36) and (38) together, we conclude that

$$\begin{aligned} \liminf _{N\rightarrow \infty } N^{-1}\langle \Psi _N, H_N \Psi _N\rangle \geqslant \mathcal {E}^\mathrm{TF}(f). \end{aligned}$$

Since $\Psi _N\in S_N$ can be chosen arbitrarily under the sole condition $\rho _{\Psi _N}=Nf_N$, this leads the desired lower bond

$$\begin{aligned} \liminf _{N\rightarrow \infty } \mathcal {E}_N (f_N) \geqslant \mathcal {E}^\mathrm{TF}(f). \end{aligned}$$

Upper bound Let $0\leqslant f\in L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d)$ with $\int _{{\mathbb R }^d} f=1$. Then by Theorem 4 there exists a sequence of Slater determinants $\Psi _N\in S_N$, such that $f_N:= N^{-1}\rho _{\Psi _N} \rightarrow f$ strongly in $L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d)$ and

$$\begin{aligned} \limsup _{N\rightarrow \infty } N^{-1-2/d} N^{2/d}h^2 \left\langle \Psi _N, \sum _{i=1}^N (-\Delta _{x_i}) \Psi _N \right\rangle \leqslant K_\mathrm{cl}\int _{{\mathbb R }^d} f^{1+2/d}. \end{aligned}$$

(39)

Since $f_N\rightarrow f$ in $L^r({\mathbb R }^d)$ for all $r\in [1,1+2/d]$ and $V\in L^p({\mathbb R }^d)+L^q({\mathbb R }^d)$ with $p,q\in [1+d/2,\infty )$, we have

$$\begin{aligned} \lim _{N\rightarrow \infty } N^{-1} \left\langle \Psi _N, \sum _{i=1}^N V(x_i) \Psi _N \right\rangle = \lim _{N\rightarrow \infty } \int _{{\mathbb R }^d} Vf_N = \int _{{\mathbb R }^d} Vf. \end{aligned}$$

(40)

Finally, for the interaction terms, since $\Psi _N$ is a Slater determinants and w is non-negative, an explicit computation shows that

$$\begin{aligned} N^{-2} \lambda N \left\langle \Psi _N, \sum _{1\leqslant i<j\leqslant N}w(x_i-x_j)\Psi _N \right\rangle \leqslant \frac{1}{2} \lambda N \iint _{{\mathbb R }^d\times {\mathbb R }^d} f_N(x)f_N(y) w(x-y) d x dy. \end{aligned}$$

(41)

Here since $w\geqslant 0$ we can simply ignored the exchange term in the Hartree–Fock functional to get an upper bound (see e.g. [22, Section 5A] for details). The convergence $f_N\rightarrow f$ in $L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d)$ and the assumption $w\in L^p({\mathbb R }^d)+L^q({\mathbb R }^d)$ imply that $f_N*w\rightarrow f*w$ strongly in $L^\infty ({\mathbb R }^d)$, and hence

$$\begin{aligned}&\frac{1}{2} \lambda N \iint _{{\mathbb R }^d\times {\mathbb R }^d} f_N(x)f_N(y) w(x-y) d x dy = \frac{1}{2} \lambda N \int _{{\mathbb R }^d} f_N (f_N*w) \\&\quad \rightarrow \frac{1}{2} \int _{{\mathbb R }^d} f (f*w) = \frac{1}{2} \iint _{{\mathbb R }^d\times {\mathbb R }^d} f(x)f(y) w(x-y) d x dy. \end{aligned}$$

Putting this together with (39), (40) and (41) we obtain the desired upper bound

$$\begin{aligned} \limsup _{N\rightarrow \infty } N^{-1}\langle \Psi _N, H_N \Psi _N\rangle \leqslant \mathcal {E}^\mathrm{TF}(f), \end{aligned}$$

$\square$

Proof of Corollary 3

The upper bound in (21), $N^{-1} E_N^\mathrm{QM} \leqslant E^\mathrm{TF}+o(1)_{N\rightarrow \infty }$, follows immediately from Theorem 2 (upper bound) and optimizing over f in (20).

To see the lower bound in (21), we take arbitrarily a N-body wave function $\Psi _N$ such that

$$\begin{aligned} N^{-1}\langle \Psi _N, H_N \Psi _N\rangle = N^{-1} E_N^\mathrm{QM}+ o(1)_{N\rightarrow \infty } \leqslant E^\mathrm{TF}+ o(1)_{N\rightarrow \infty }. \end{aligned}$$

(42)

Denote $\rho _{\Psi _N}=Nf_N$. Using $w\geqslant 0$, the Lieb-Thirring inequality for the kinetic energy [29], Hölder’s inequality and the assumption $V \in L^{p}({\mathbb R }^d)+L^{q}({\mathbb R }^d)$ with $p,q\in [1+d/2,\infty )$ we can estimate

$$\begin{aligned} N^{-1}\langle \Psi _N, H_N \Psi _N\rangle&\geqslant N^{-1}\langle \Psi _N, \sum _{i=1}^N (-\Delta _{x_i}+V(x_i)) \Psi _N\rangle \nonumber \\&\geqslant K \int _{{\mathbb R }^d} f_N^{1+2/d} - \int _{{\mathbb R }^d} Vf_N \geqslant (K/2) \int _{{\mathbb R }^d} f_N^{1+2/d} - C \end{aligned}$$

where $K,C>0$ are constants independent of $f_N$. Thus from (42) deduce that $f_N$ is bounded in $L^{1+2/d}({\mathbb R }^d)$.

Up to a subsequence, $f_N \rightharpoonup f$ in $L^{1+2/d}({\mathbb R }^d)$, and hence Theorem 2 (lower bound) implies that

$$\begin{aligned} N^{-1}\langle \Psi _N, H_N \Psi _N\rangle \geqslant \mathcal {E}_N(f_N) \geqslant \mathcal {E}^\mathrm{TF}(f) + o(1)_{N\rightarrow \infty }. \end{aligned}$$

(43)

Next, let us show that

$$\begin{aligned} E^\mathrm{TF}= \inf \left\{ \mathcal {E}^\mathrm{TF}(g): 0\leqslant g \in L^1({\mathbb R }^d)\cap L^{1+2/d}({\mathbb R }^d), \int _{{\mathbb R }^d} g \leqslant 1\right\} . \end{aligned}$$

(44)

This follows from a standard argument. If $\int _{{\mathbb R }^d} g\leqslant 1$, we can take a function

$$\begin{aligned} 0\leqslant \varphi \in C_c^\infty ({\mathbb R }^d), \quad \int _{{\mathbb R }^d} \varphi + \int _{{\mathbb R }^d}g =1. \end{aligned}$$

Take a sequence $\{R_k\}\subset {\mathbb R }^d, |R_k|\rightarrow \infty$. By the variational principle

$$\begin{aligned} E^\mathrm{TF}&\leqslant \lim _{k\rightarrow \infty } \mathcal {E}^\mathrm{TF} (g+ \varphi (\cdot +R_k)) \nonumber \\&= \mathcal {E}^\mathrm{TF}(g) + K_\mathrm{cl} \int _{{\mathbb R }^d} |\varphi |^{1+2/d} + \frac{1}{2}\iint _{{\mathbb R }^d} \varphi (x)\varphi (y) w(x-y) dx dy \nonumber \\&\leqslant \mathcal {E}^\mathrm{TF}(g) + K_\mathrm{cl} \int _{{\mathbb R }^d} |\varphi |^{1+2/d} + C( \Vert \varphi \Vert _{L^r}^2+ \Vert \varphi \Vert _{L^{s}}^2). \end{aligned}$$

In the last estimate we have used Young’s inequality [25, Theorem 4.2] and the assumption $w \in L^{p}({\mathbb R }^d)+L^{q}({\mathbb R }^d)$ with $p,q\in [1+d/2,\infty )$. Here the parameters $r,s>1$ are determined by

$$\begin{aligned} \frac{1}{p}+\frac{2}{r}=2 = \frac{1}{q}+\frac{2}{s} \end{aligned}$$

and the constant $C>0$ depends only on w. By scaling $\varphi \mapsto \ell ^{d}\varphi (\ell \cdot )$ with $\ell \rightarrow 0$, we conclude that $E^\mathrm{TF} \leqslant \mathcal {E}^\mathrm{TF}(g)$. Thus (44) holds.

Note that the weak convergence $f_N \rightharpoonup f$ implies that $\int _{{\mathbb R }^d} f \leqslant 1$. Therefore, combining (43) and (44) we arrive at

$$\begin{aligned} N^{-1}\langle \Psi _N, H_N \Psi _N\rangle \geqslant \mathcal {E}_N(f_N) \geqslant \mathcal {E}^\mathrm{TF}(f) + o(1)_{N\rightarrow \infty } \geqslant E^\mathrm{TF}+ o(1)_{N\rightarrow \infty }. \end{aligned}$$

Thanks to (42), we obtain the convergence (21) and that $\mathcal {E}^\mathrm{TF}(f)=E^\mathrm{TF}$.

Finally, note that $\mathcal {E}^\mathrm{TF}(g)$ is strictly convex in g. This can be seen from the strict convexity of the kinetic term $g\mapsto K_\mathrm{cl}\int _{{\mathbb R }^d} g^{1+2/d}$ and the convexity of the interaction term

$$\begin{aligned} g\mapsto \frac{1}{2}\iint _{{\mathbb R }^d\times {\mathbb R }^d} g(x) g(y)w(x-y) d x dy= \frac{1}{2}\int _0^\infty \left[ \int _{{\mathbb R }^d} |(g*\chi _t)(z)|^2 dz \right] dt. \end{aligned}$$

Here we have used again the Fefferman-de la Llave formula (8). Consequently, if $E^\mathrm{TF}$ has a minimizer $f^\mathrm{TF}$, then using $\mathcal {E}^\mathrm{TF}(f)=E^\mathrm{TF}=\mathcal {E}^\mathrm{TF}(f^\mathrm{TF})$, the strict convexity and (44), we conclude that $f=f^\mathrm{TF}$. Thus $f_N = N^{-1}\rho _{\Psi _N} \rightharpoonup f^\mathrm{TF}$ weakly in $L^{1+2/d}({\mathbb R }^d)$, for every wave function $\Psi _N$ satisfying (42) (not necessarily a ground state of $H_N$). $\square$