Excess Charge for Pseudo-Relativistic Atoms

in Hartree-Fock Theory

Anna Dall’Acqua and Jan Philip Solovej^{1}

Received: January 29, 2010

Communicated by Heinz Siedentop

Abstract. We prove within the Hartree-Fock theory of pseudorelativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge Z and the fine structure constant α as long as Zα is bounded.

2010 Mathematics Subject Classification: 81V45, 35Q40, 81Q10 Keywords and Phrases: Atomic model, Ionization conjecture, Thomas-Fermi theory

1 Introduction

A long standing open problem in the mathematical physics literature is the Ionization conjecture. It can be formulated as follows. Consider atoms with arbitrarily large nuclear charge Z, is it true that the radius (see Definition 1.8) and the maximal negative ionization remain bounded? A positive answer to this question in the non-relativistic Hartree-Fock model has been given by the second author in [23]. One of the aims of the present paper is to extend the result taking into account some relativistic effects. The ionization conjecture for the full Schrödinger theory is still open both in the non-relativistic and relativistic case. See [13], [16], [17], [6], [7] and [22] for some Z-dependent bounds on the maximal negative ionization. The best result is that N(Z) = Z+ O(Z^{a}) with a = 47/56 where N(Z) denotes the maximal number of electrons a nucleus of charge Z binds (see [6], [7] and [22]) .

1 The authors wish to thank Heinz Siedentop for suggesting the problem. Support from the EU IHP network Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems, contract no. HPRN-CT-2002-00277 is gratefully acknowledged. Jan Philip Solovej was supported by a grant from The Danish Council for Independent Research — Natural Sciences.

Documenta Mathematica 15 (2010) 285–345

As a model for an atom with nuclear charge Z and N electrons we consider (in units where ℏ = m = e = 1) the operator

H = ∑_{i=1}^{N} α^{−1}(\sqrt{−Δ_{i} + α^{−2}} − α^{−1} − \frac{Zα}{|x_{i}|}) + ∑_{1≤i<j≤N} \frac{1}{|x_{i} − x_{j}|},  (1)

where α is Sommerfeld’s fine structure constant. The operator H acts on a dense subset of the N body Hilbert space H_{F} := ∧_{i=1}^{N}L^{2}(ℝ^{3}; ℂ^{q}) of antisymmetric wave functions, where q is the number of spin states. The operator H is bounded from below on this subspace if Zα ≤ 2/π (see [9] for N = 1, [5] and [19] for N ≥ 1). In this paper we will consider the sub-critical case Zα < 2/π. Let us notice here that to define the operator H there is an issue. Indeed for Zα < 2/π the nuclear potential is only a small form perturbation of the kinetic energy and hence one needs to work with forms to define the operator H. This has been done in detail in [2]. The quantum ground state energy is the infimum of the spectrum of H considered as an operator acting on H_{F}. In the Hartree-Fock approximation one restricts to wave-functions ψ which are pure wedge products, also called Slater determinants:

ψ(x_{1}, σ_{1}, x_{2}, σ_{2}, …, x_{N}, σ_{N}) = \frac{1}{\sqrt{N!}} det(u_{i}(x_{j}, σ_{j}))_{i,j=1}^{N},  (2)

with {u_{i}}_{i=1}^{N} orthonormal in L^{2}(ℝ^{3}; ℂ^{q}). The u_{i}’s are also called orbitals. Notice that ∥ψ∥_{L^{2}(ℝ^{3N} ,ℂ^{qN})} = 1. The Hartree-Fock ground state energy is

E^{HF}(N, Z, α) := inf{q(ψ, ψ)|ψ ∈ Q(H) and ψ a Slater determinant},

with q the quadratic form defined by H and Q(H) the corresponding form domain. One of the main result of the paper is the following.

Theorem 1.1. Let Z ≥ 1 and α > 0. Let Zα = κ and assume that 0 ≤ κ < 2/π. There is a constant Q > 0 depending only on κ such that if N is such that a Hartree-Fock minimizer exists then N ≤ Z + Q.^{2}

The idea of the proof is the same as in [23]. One shows that the Thomas-Fermi model is a good approximation of the Hartree-Fock model except in the region far away from the nucleus. We first introduce some notation in order to introduce the Hartree-Fock and Thomas-Fermi models.

2 In order to prove this result we need that N < CZ for a positive constant C. We do not include a proof of this fact here for simplicity and since a much stronger result has been proved by Lieb in [13] for αZ < 1/2. The needed extension of this result of Lieb to αZ < 2/π will appear in [3] (see Theorem 1.6 below).

Documenta Mathematica 15 (2010) 285–345

1.1 Notation

Let e be the quadratic form with domain H^{\frac{1}{2}} (ℝ^{3}, ℂ^{q}) such that

e(u, v) = (E(p)^{\frac{1}{2}} u, E(p)^{\frac{1}{2}} v) for all u, v ∈ H^{\frac{1}{2}} (ℝ^{3}, ℂ^{q}),  (3)

where E(p) denotes the operator E(i∇) = \sqrt{−Δ + α^{−2}}. As usual (u, v) denotes the scalar product of u and v in L^{2}(ℝ^{3}, ℂ^{q}). Let V (x) := Zα/|x| and v be the quadratic form with domain H^{\frac{1}{2}} (ℝ^{3}, ℂ^{q}) defined by

v(u, v) = (V ^{\frac{1}{2}} u, V ^{\frac{1}{2}} v) for all u, v ∈ H^{\frac{1}{2}} (ℝ^{3}, ℂ^{q}).  (4)

From [10, 5.33 p.307] we have

∫_{ℝ^{3}} \frac{|f(x)|^{2}}{|x|} dx ≤ \frac{2}{π} ∫_{ℝ^{3}} |p||f̂(p)|^{2} dp for f ∈ H^{\frac{1}{2}} (ℝ^{3}, ℂ)  (5)

with f̂ the Fourier transform of f. Thus since Zα ≤ 2/π and E(p) ≥ |p| it follows that v(u, u) ≤ e(u, u) for all u ∈ H^{\frac{1}{2}} (ℝ^{3}, ℂ^{q}). In the following t denotes the quadratic form associated to the kinetic energy; i.e. for all u, v ∈ H^{\frac{1}{2}} (ℝ^{3}, ℂ^{q})

t(u, v) := α^{−1}e(u, v) − α^{−2}(u, v) = α^{−1}(T (p)^{\frac{1}{2}} u, T (p)^{\frac{1}{2}} v),  (6)

with T (p) := E(p) − α^{−1}. A density matrix γ is a self-adjoint trace class operator that satisfies the operator inequality 0 ≤ γ ≤ Id. A density matrix γ : L^{2}(ℝ^{3}; ℂ^{q}) → L^{2}(ℝ^{3}; ℂ^{q}) has an integral kernel

γ (x, σ, y, τ) = ∑_{j} λ_{j}u_{j}(x, σ)u_{j}(y, τ)^{∗},  (7)

where λ_{j}, u_{j} are the eigenvalues and corresponding eigenfunctions of γ. We choose the u_{j}’s to be orthonormal in L^{2}(ℝ^{3}, ℂ^{q}). Let ρ_{γ} ∈ L^{1}(ℝ^{3}) denote the 1-particle density associated to γ given by

ρ_{γ}(x) = ∑_{σ=1}^{q} ∑_{j} λ_{j}|u_{j}(x, σ)|^{2}.

We define

A := {γ density matrix: Tr[T (p)γ] < +∞} ,  (8)

where for γ ∈ A written as in (7) Tr[T (p)γ] := Tr[E(p)γ] − α^{−1} Tr[γ] and

Tr[E(p)γ] := ∑_{j} λ_{j}e(u_{j}, u_{j}).  (9)

Similarly we use the following notation Tr [V γ] := ∑_{j} λ_{j}v(u_{j}, u_{j}).

Remark 1.2. If γ ∈ A then ρ_{γ} ∈ L^{1}(ℝ^{3}) since γ is trace class and ρ_{γ} ∈ L^{4/3}(ℝ^{3}). The second inclusion follows from Daubechies’ inequality, a generalization of the Lieb-Thirring inequality (see Theorem 2.3).

1.2 Hartree-Fock theory

In Hartree-Fock theory one considers wave functions that are pure wedge products and that satisfy the right statistics: determinantal wave functions as in (2). To define the HF-energy functional it is convenient to use the one to one correspondence between Slater determinants and projections onto finite dimensional subspaces of L^{2}(ℝ^{3}, ℂ^{q}). Indeed if ψ is given by (2) and γ is the projection onto the space spanned by u_{1}, …, u_{N} the energy expectation depends only on γ: (ψ, Hψ) = E^{HF}(γ). Here E^{HF} defines the HF-energy functional

E^{HF}(γ) = α^{−1} Tr[(T (p) − V)γ] + D(γ) − Ex (γ) ,  (10)

where D(γ) is the direct Coulomb energy

D(γ) = \frac{1}{2} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{ρ_{γ}(x)ρ_{γ}(y)}{|x − y|} dxdy,

and Ex(γ) is the exchange Coulomb energy

Ex(γ) = \frac{1}{2} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{Tr_{ℂ^{q}} [|γ(x, y)|^{2}]}{|x − y|} dxdy,

where we think of the integral kernel γ(x, y) as a q × q matrix. Using projections we can define as follows the HF-ground state.

Definition 1.3 (The HF-ground state). Let Z > 0 be a real number and N ≥ 0 be an integer. The HF-ground state energy is

E^{HF}(N, Z, α) := inf {E^{HF}(γ) : γ^{2} = γ, γ ∈ A, Tr[γ] = N} .

If a minimizer exists we say that the atom has a HF ground state described by γ^{HF}.

We may extend the definition of the HF-functional from projections to density matrices in A. We first notice that if γ ∈ A, then all the terms in E^{HF}(γ) are finite. From (5) it follows that

Tr[V γ] = ∑_{j} λ_{j}v(u_{j}, u_{j}) ≤ ∑_{j} λ_{j}e(u_{j}, u_{j}) = Tr[E(p)γ].

On the other hand if γ ∈ A then ρ_{γ} ∈ L^{1}(ℝ^{3}) ∩ L^{\frac{4}{3}} (ℝ^{3}) (see Remark 1.2). By Hölder’s inequality ρ_{γ} ∈ L^{\frac{6}{5}} (ℝ^{3}) and hence D(γ) is bounded by Hardy-Littlewood-Sobolev’s inequality. The boundness of the exchange term follows from 0 ≤ Ex(γ) ≤ D(γ). On the other hand if γ is a density matrix with γ ∉ A then E^{HF}(γ) = ∞. Here we use also that Zα < 2/π. Extending the set where we minimize, we could have lowered the ground state energy and/or changed the minimizer. That this is not the case follows from Lieb’s variational principle.

Theorem 1.4 (Lieb’s variational principle, [12]). For all N non-negative integers it holds that

inf{E^{HF}(γ) : γ ∈ A, γ^{2} = γ, Tr[γ] = N} = inf{E^{HF}(γ) : γ ∈ A, Tr[γ] = N},

and if the infimum over all density matrices is attained so is the infimum over projections.

The following existence theorem for the HF-minimizer in the pseudo-relativistic case has been recently proved in [2].

Theorem 1.5. Let Zα < 2/π and let N ≥ 2 be a positive integer such that N < Z + 1. Then there exists an N-dimensional projection γ^{HF} = γ^{HF}(N, Z, α) minimizing the HF-energy functional E^{HF} given by (10), that is, E^{HF}(N, Z, α) is attained. Moreover, one can write

γ^{HF}(x, σ, y, τ) = ∑_{i=1}^{N} u_{i}(x, σ)u_{i}(y, τ)^{∗},

with u_{i} ∈ L^{2}(ℝ^{3}, ℂ^{q}), i = 1, …, N, orthonormal, such that the HF-orbitals {u_{i}}_{i=1}^{N} satisfy:

1. h_{γ^{HF}}u_{i} = ε_{i}u_{i}, with 0 > ε_{N} ≥ ε_{N−1} ≥ ⋯ ≥ ε_{1} > −α^{−1} and

h_{γ^{HF}} := T (p) − \frac{Zα}{|x|} + ρ^{HF} ∗ |x|^{−1} − K_{γ^{HF}},  (11)

where ρ^{HF} denotes the density of the HF-minimizer and for f ∈ H^{\frac{1}{2}} (ℝ^{3})

(K_{γ^{HF}}f)(x, σ) = ∑_{i=1}^{N} u_{i}(x, σ) ∑_{τ=1}^{q} ∫_{ℝ^{3}} u_{i}(y, τ)^{∗}f(y, τ)|x − y|^{−1}dy.

2. u_{i} ∈ C^{∞}(ℝ^{3} \ {0}, ℂ^{q}) for i = 1, …, N;

3. u_{i} ∈ H^{1}(ℝ^{3} \ B_{R}(0)) for all R > 0 and i = 1, …, N.

In the opposite direction the following result gives an upper bound on the excess charge.

Theorem 1.6. Let αZ < \frac{2}{π}. If N is a positive integer such that N > 2Z + 1 there are no minimizers for the HF-energy functional.

This theorem for Zα < 1/2 was proved by Lieb in [13]. With an improved approximation argument the proof can be extended to Zα < 2/π (see [3]). Notice that both proofs work not only in the Hartree-Fock approximation but for the minimization problem on ∧^{N}L^{2}(ℝ^{3}).

Definition 1.7. Let γ^{HF} be the HF-minimizer. The function

φ^{HF}(x) := \frac{Z}{|x|} − ∫_{ℝ^{3}} \frac{ρ^{HF}(y)}{|x − y|}dy for x ∈ ℝ^{3},

is called the HF-mean field potential and

Φ_{R}^{HF}(x) := \frac{Z}{|x|} − ∫_{|y|<R} \frac{ρ^{HF}(y)}{|x − y|}dy for x ∈ ℝ^{3},

is the HF-screened nuclear potential.

Definition 1.8. We define the HF-radius R_{Z,N}^{HF}(ν) to the ν last electrons by

∫_{|x|≥R_{Z,N}^{HF} (ν)} ρ^{HF}(x) dx = ν.

1.3 A bit of Thomas-Fermi theory

In this subsection we present briefly the Thomas-Fermi theory and especially the result that will be used in the rest of the paper. We refer the interested reader to [11]. Let U be a potential in L^{5/2}(ℝ^{3}) + L^{∞}(ℝ^{3}) with

inf\{∥W ∥_{∞} : U − W ∈ L^{\frac{5}{2}} (ℝ^{3})\} = 0.

Then the TF-energy functional is defined by

E_{U}^{TF}(ρ) = \frac{3}{10}(\frac{6π^{2}}{q})^{\frac{2}{3}} ∫_{ℝ^{3}} ρ(x)^{\frac{5}{3}} dx − ∫_{ℝ^{3}} U(x)ρ(x)dx + \frac{1}{2} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{ρ(x)ρ(y)}{|x − y|} dxdy,

on non-negative functions ρ ∈ L^{5/3}(ℝ^{3}) ∩ L^{1}(ℝ^{3}). As before, q denotes the number of spin states. We recall some properties of the TF-model, see [18].

Theorem 1.9. Let U be as above. For all N′ ≥ 0 there exists a unique nonnegative ρ_{U}^{TF} ∈ L^{5/3}(ℝ^{3}) such that ∫ ρ_{U}^{TF} ≤ N′ and

E_{U}^{TF}(ρ_{V}^{TF}) = inf{E_{U}^{TF}(ρ) : ρ ∈ L^{5/3}(ℝ^{3}), ∫_{ℝ^{3}} ρ(x) dx ≤ N′}.

There exists a unique chemical potential μ_{U}^{TF}(N′), with 0 ≤ μ_{U}^{TF}(N′) ≤ sup U, such that ρ_{U}^{TF} is uniquely characterized by

E_{U}^{TF}(ρ_{U}^{TF}) + μ_{U}^{TF}(N′) ∫_{ℝ^{3}} ρ_{U}^{TF}(x) dx
= inf{E_{U}^{TF}(ρ) + μ_{U}^{TF}(N′) ∫_{ℝ^{3}} ρ(x) dx : 0 ≤ ρ ∈ L^{5/3}(ℝ^{3}) ∩ L^{1}(ℝ^{3})}.

Moreover ρ_{U}^{TF} is the unique solution in L^{5/3}(ℝ^{3}) ∩ L^{1}(ℝ^{3}) to the TF-equation

\frac{1}{2}(\frac{6π^{2}}{q})^{\frac{2}{3}} (ρ_{U}^{TF}(x))^{\frac{2}{3}} = [U(x) − ρ_{U}^{TF} ∗ |x|^{−1} − μ_{U}^{TF}(N′)]_{+} .

If μ_{U}^{TF}(N′) > 0 then ∫ ρ_{U}^{TF} = N′. For all μ > 0 there is a unique minimizer 0 ≤ ρ ∈ L^{5/3}(ℝ^{3}) ∩ L^{1}(ℝ^{3}) to E_{U}^{TF}(ρ) + μ ∫ ρ.

One defines the TF-mean field potential φ_{U}^{TF}, the TF-screened nuclear potential Φ_{U,R}^{TF} and the TF-radius R_{N,Z}^{TF}(ν) to the ν last-electron similarly as in Definitions 1.7 and 1.8 replacing the HF-density with the TF-density.

Theorem 1.10. If U(x) = Z/|x| (the Coulomb potential), then the minimizer of E_{U}^{TF}, under the condition ∫ ρ ≤ N, exists for every N. Moreover, μ_{U}^{TF}(N) = 0 if and only if N ≥ Z.

When U(x) = Z/|x| we denote the minimizer of the TF-functional, under the condition ∫ ρ ≤ Z, simply by ρ^{TF} and ∫ ρ^{TF} = Z. Correspondingly φ^{TF} and Φ_{R}^{TF} denote, respectively, its mean field and screened nuclear potential. With this notation

E^{TF}(ρ^{TF}) = −e_{0}Z^{\frac{7}{3}} ,  (12)

where e_{0} is the total binding energy of a neutral TF-atom of unit nuclear charge. We recall here a result due to Sommerfeld on the asymptotic behavior of the TF-mean field potential, see [23, Th. 4.6].

Theorem 1.11 (Sommerfeld asymptotics). Assume that the potential U is continuous and harmonic for |x| > R and that it satisfies lim_{|x|→∞} U(x) = 0. Consider the corresponding TF-mean field potential φ_{U}^{TF} and assume that μ_{U}^{TF} < lim inf_{r↘R} inf_{|x|=r}φ_{U}^{TF}(x). With ζ = (−7 + \sqrt{73})/2 define

a(R) := lim inf_{r↘R} sup_{|x|=r} [(\frac{φ_{U}^{TF}(x)}{3^{4}2^{−1}q^{−2}π^{2}r^{−4}} )^{− \frac{1}{2}} − 1]r^{ζ}

A(R, μ_{U}^{TF}) := lim inf_{r↘R} sup_{|x|=r} [\frac{φ_{U}^{TF}(x) − μ_{U}^{TF}}{3^{4}2^{−1}q^{−2}π^{2}r^{−4}} − 1]r^{ζ}.

Then we find for all |x| > R

φ_{U}^{TF}(x) ≤ \frac{3^{4}π^{2}}{2q^{2}} (1 + A(R, μ_{U}^{TF})|x|^{−ζ})|x|^{−4} + μ_{U}^{TF} and
φ_{U}^{TF}(x) ≥ max \{\frac{3^{4}π^{2}}{2q^{2}} (1 + a(R)|x|^{−ζ})^{−2}|x|^{−4}, ν(μ_{U}^{TF})|x|^{−1}\},

where

ν(μ_{U}^{TF}) := inf_{|x|≥R} max \{\frac{3^{4}π^{2}}{2q^{2}} (1 + a(R)|x|^{−ζ})^{−2}|x|^{−3}, μ_{U}^{TF}|x|\}.

For easy reference we give here the estimate on the TF-mean field potential corresponding to the Coulomb potential.

Theorem 1.12 (Atomic Sommerfeld estimate, [23, Thm 5.2-5.4]). The atomic TF-mean field potential satisfies the bound

\frac{Z}{|x|} − min \{\frac{Z}{|x|}, \frac{Z^{\frac{4}{3}}}{2β_{0}} \} ≤ φ^{TF}(x) ≤ min \{\frac{3^{4}π^{2}}{2q^{2}} \frac{1}{|x|^{4}} , \frac{Z}{|x|}\},  (13)

with 2β_{0} = π^{\frac{2}{3}} 3^{− \frac{5}{3}} 2^{− \frac{1}{3}} q^{− \frac{2}{3}} , and for |x| ≥ R > 0

φ^{TF}(x) ≥ \frac{3^{4}π^{2}}{2q^{2}} (1 + a(R)|x|^{−ζ})^{−2}|x|^{−4},

where ζ and a(R) are defined in Theorem 1.11.

Corollary 1.13. Let ζ and β_{0} be defined as in Theorem 1.11 and 1.12 respectively. Then the TF-mean field potential satisfies the bound

φ^{TF}(x) ≥ \{ \begin{matrix}\frac{Z}{|x|} − \frac{Z^{\frac{4}{3}}}{2β_{0}} & if |x| ≤ β_{0}Z^{− \frac{1}{3}} \\ \frac{3^{4}π^{2}}{2q^{2}} (1 + aZ^{− \frac{ζ}{3}} |x|^{−ζ})^{−2}|x|^{−4} & if |x| > β_{0}Z^{− \frac{1}{3}} , \end{matrix}

with a = β_{0}^{ζ}(3^{2}π/(qβ0^{\frac{3}{2}} ) − 1).

Corollary 1.14. The TF-screened nuclear potential satisfies

Φ_{|x|}^{TF}(x) ≤ \frac{3^{4}2π^{2}}{q^{2}} |x|^{−4} for all x ∈ ℝ^{3}.

Corollary 1.15. The following estimate holds

∫_{ℝ^{3}}(ρ^{TF}(x))^{\frac{5}{3}} dx ≤ 4\frac{2^{\frac{2}{3}}}{π^{2}} \frac{5}{7}q^{\frac{4}{3}} Z^{\frac{7}{3}} .

Proof. By the TF-equation and since μ^{TF} = 0 we find

∫_{ℝ^{3}}(ρ^{TF}(x))^{\frac{5}{3}} dx = 2^{\frac{5}{2}} (\frac{q}{6π^{2}} )^{\frac{5}{3}} ∫_{ℝ^{3}}(φ^{TF}(x))^{\frac{5}{2}} dx.

The estimate follows from the atomic Sommerfeld upper bound.

1.4 Construction and main results

We present the basic idea for the proof of Theorem 1.1. Let us consider an atomic system with N ≥ 2 fermionic particles and a nucleus of charge Z ≥ 1 with Zα = κ and 0 ≤ κ < 2/π. We assume that N ≥ Z and that N is such that a HF-minimizer exists. That is: there exists a density matrix γ^{HF} ∈ A such that Tr[γ^{HF}] = N and

E^{HF}(γ^{HF}) = inf {E^{HF}(γ) : γ = γ^{∗}, 0 ≤ γ ≤ I, Tr[γ] = N} .

Let ρ^{TF} be the TF-minimizer with potential U(x) = Z/|x| and under the condition ∫ ρ^{TF} = Z. We know that such a minimizer exists and that the corresponding chemical potential is zero (see Theorem 1.10). Denoting by ρ^{HF} the density of the minimizer γ^{HF}, we find for all r > 0

N = ∫_{ℝ^{3}} ρ^{HF}(x)dx
= ∫_{|x|<r} [ρ^{HF}(x) − ρ^{TF}(x)] dx + ∫_{|x|<r} ρ^{TF}(x) dx + ∫_{|x|>r} ρ^{HF}(x) dx.

By the equalities above and since ∫_{|x|<r} ρ^{TF}(x)dx ≤ Z, Theorem 1.1 follows from the following result.

Theorem 1.16. There exist r > 0 and positive constants c_{1} and c_{2} independent of N and Z but possibly depending on κ such that

∫_{|x|<r} [ρ^{HF}(x) − ρ^{TF}(x)] dx ≤ c_{1} and ∫_{|x|>r} ρ^{HF}(x)dx ≤ c_{2}.

The following theorem is the principal ingredient in the proof of the previous one and is the main technical estimate in the paper.

Theorem 1.17. Let Zα = κ, 0 ≤ κ < 2/π. Assume N ≥ Z ≥ 1. Then there exist universal constants α_{0} > 0, 0 < ε < 4 and C_{M} and C_{Φ} depending on κ such that for all α ≤ α_{0}

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C_{Φ}|x|^{−4+ε} + C_{M}.

This main estimate is proven by an iterative procedure. We first prove the estimate for small x (i.e. |x| ≤ β_{0}Z^{− \frac{1}{3}} ), then for intermediate x (i.e. up to a fixed distance independent of Z) and finally for big x. By proving Theorem 1.17 we also get the following interesting results. The proofs of those are given in Section 5.

Theorem 1.18 (Asymptotic formula for the radius). Let Zα = κ, 0 ≤ κ < 2/π. Both lim inf_{Z→∞} R_{Z,Z}^{HF}(ν) and lim sup_{Z→∞} R_{Z,Z}^{HF}(ν) are bounded and behave asymptotically as

3^{\frac{4}{3}} \frac{2^{\frac{1}{2}} π^{\frac{2}{3}}}{q^{\frac{2}{3}}} ν^{− \frac{1}{3}} + o(ν^{− \frac{1}{3}} ) as ν → ∞.

Theorem 1.19 (Bound on the ionization energy of a neutral atom). Let Zα = κ, 0 ≤ κ < 2/π and Z ≥ 1. The ionization energy of a neutral atom E^{HF}(Z − 1, Z) − E^{HF}(Z, Z) is bounded by a universal constant.

Theorem 1.20 (Potential estimate). Let Zα = κ, 0 ≤ κ < 2/π. For all Z ≥ 1 and N with N ≥ Z for which a HF minimizer exists with ∫ ρ^{HF} = N, we have

|φ^{TF}(x) − φ^{HF}(x)| ≤ A_{φ}|x|^{−4+ε_{0}} + A_{1},

with A_{0}, A_{1} and ε_{0} universal constants.

2 Prerequisites

In this section we recall some results that will be used in the rest of the paper. Localization of the kinetic energy. The following is the IMS formula corresponding to the operator T (p).

Theorem 2.1 ([19]). Let χ_{i}, i = 0, …, K, be real valued Lipschitz continuous functions on ℝ^{3} such that ∑_{i=0}^{K} χ_{i}^{2} (x) = 1 for all x ∈ ℝ^{3}. Then for every f ∈ H^{1/2}(ℝ^{3})

t(f, f) = ∑_{i=0}^{K} t(χ_{i}f, χ_{i}f) − α^{−1} ∑_{i=0}^{K}(f, L_{i}f),

where L_{i} is a bounded operator with kernel

L_{i}(x, y) = \frac{α^{−2}}{4π^{2}} \frac{|χ_{i}(x) − χ_{i}(y)|^{2}}{|x − y|^{2}} K_{2}(α^{−1}|x − y|),  (14)

where K_{2} is a modified Bessel function of the second kind.

Remark 2.2. As in [24, App.A, pages 94–98] we use the following integral formula for the modified Bessel function

K_{2}(t) = t ∫_{0}^{∞} e^{−t\sqrt{s^{2}+1}}s^{2} ds , t > 0.

We recall that this function is decreasing and smooth in ℝ^{+}. Moreover,

∫_{0}^{+∞} t^{2}K_{2}(t) dt = \frac{3π}{2} and K_{2} (t) ≤ 16 t^{−2}e^{− \frac{1}{2} t} for t > 0.  (15)

The integral is computed in [21, (A6)] while the estimate follows directly from the integral formula for K_{2} by estimating \sqrt{s^{2} + 1} ≥ \frac{1}{2} + \frac{1}{2}s.

Generalization of the Lieb-Thirring inequality. This result due to Daubechies generalizes the Lieb-Thirring inequality to the pseudo-relativistic case.

Theorem 2.3 (Daubechies’ inequality, [4]). For γ ∈ A

Tr[T (p)γ] ≥ ∫_{ℝ^{3}} G_{α}(ρ_{γ}(x))dx,

where G_{α}(ρ) = \frac{3}{8}α^{−4}Cg(α(ρ/C)^{\frac{1}{3}} ) − α^{−1}ρ with C = .163q, q the number of spin states and g(t) = t(1 + t^{2})^{\frac{1}{2}} (1 + 2t^{2}) − ln(t + (1 + t^{2})^{\frac{1}{2}} ).

Remark 2.4. The function G_{α} defined in the previous theorem is convex and it has the following behavior:

\frac{9}{20} min \{\frac{1}{5}αC^{− \frac{2}{3}} ρ^{\frac{5}{3}} , \frac{1}{2}C^{− \frac{1}{3}} ρ^{\frac{4}{3}} \} ≤ G_{α} (ρ) ≤ \frac{3}{2} min \{\frac{1}{5}αC^{− \frac{2}{3}} ρ^{\frac{5}{3}} , \frac{1}{2}C^{− \frac{1}{3}} ρ^{\frac{4}{3}} \} .

(16)

(The proof of the estimate above is in Appendix A.) Notice that when α ↘ 0 then α^{−1}G_{α}(ρ) tends to a constant times ρ^{5/3}.

Theorem 2.5 (Generalization of the Lieb-Thirring inequality, [4]). Let f^{−1} be the inverse of the function f(t) := \sqrt{t^{2} + α^{−2}} − α^{−1}, t ≥ 0, and define F (s) = ∫_{0}^{s} dt [f^{−1}(t)]^{3}. Then for any density matrix γ it holds

Tr[(T (p) − U)γ] ≥ −Cq ∫_{ℝ^{3}} F (|U(x)|)dx,

with C ≤ 0.163.

Remark 2.6. Since f^{−1}(t) = (t^{2} + 2α^{−1}t)^{1/2} we find for F

F (s) = 2^{\frac{3}{2}} α^{−3/2} ∫_{0}^{s} t^{3/2}(1 + \frac{1}{2}αt)^{3/2} dt for s ≥ 0,  (17)

and since by convexity (1 + \frac{1}{2}αt)^{\frac{3}{2}} ≤ \sqrt{2} + \frac{1}{2}(αt)^{\frac{3}{2}} we have

F (s) ≤ \frac{2^{3}}{5} α^{− \frac{3}{2}} s^{\frac{5}{2}} + \frac{1}{2\sqrt{2}}s^{4} for s ≥ 0.

Hence for any density matrix γ and potential U ∈ L^{\frac{5}{2}} (ℝ^{3}) ∩ L^{4} (ℝ^{3})

Tr[(T (p) − U)γ] ≥ −Cq ∫_{ℝ^{3}} (\frac{2^{3}}{5} α^{− \frac{3}{2}} |U(x)|^{\frac{5}{2}} + \frac{1}{2\sqrt{2}}|U(x)|^{4})dx.  (18)

Coulomb norm estimate. We present here only the definition of Coulomb norm and the result we need. For a more complete presentation we refer to [23, Sec.9].

Definition 2.7. For f, g ∈ L^{\frac{6}{5}} (ℝ^{3}) we define the Coulomb inner product

D(f, g) := \frac{1}{2} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{f(x)\overline{g(y)}}{|x − y|}dxdy,

and the corresponding norm ∥g∥_{C} := D(g, g)^{\frac{1}{2}} .

In the following we write the direct term in the HF-energy functional using the Coulomb scalar product: i.e. D(γ) = D(ρ_{γ}, ρ_{γ}) = D(ρ_{γ}). Similarly, for ρ ∈ L^{1}(ℝ^{3}) ∩ L^{\frac{5}{3}} (ℝ^{3}) the term D(ρ) denotes D(ρ, ρ). The next proposition follows as Corollary 9.3 in [23].

Proposition 2.8. For s > 0, x ∈ ℝ^{3} and f ∈ L^{\frac{6}{5}} (ℝ^{3}) it holds

f ∗ |x|^{−1} ≤ ∫_{|x−y|<s}[f(y)]_{+}(\frac{1}{|x − y|} − \frac{1}{s})dy + \sqrt{2} s^{− \frac{1}{2}} ∥f∥_{C}.

Moreover, for k > 0

∫_{|y|<|x|} \frac{f(y)}{|x − y|}dy ≤ ∫_{A(|x|,k)} \frac{[f(y)]_{+}}{|x − y|} dy + 2^{\frac{3}{2}} k^{−1}|x|^{− \frac{1}{2}} ∥f∥_{C},

where A(|x|, k) denotes the annulus

A(|x|, k) := {y ∈ ℝ^{3} : (1 − 2k)|x| ≤ |y| ≤ |x|} .

2.1 Improved relativistic Lieb-Thirring inequalities

A major difference between the pseudo-relativistic HF-model and the nonrelativistic one studied in [23] is that the boundness of the functional does not yield a bound on the L^{\frac{5}{3}} norm of the HF-density ρ^{HF} in the pseudo-relativistic case. By Theorem 2.3 and Remark 2.4 we see that we can control only the L^{\frac{4}{3}} - norm of ρ^{HF}. Therefore one cannot estimate the term ρ^{HF} ∗ |x|^{−1} in L^{1}-norm simply by Hölder’s inequality with p = 5/2 and q = 5/3. To estimate it we are going to use a combined Daubechies-Lieb-Yau inequality. The following lemma can be found in [24, pages 98–99]^{3}.

Lemma 2.9. For f ∈ S(ℝ^{3}),

∫_{ℝ^{3}} \frac{e^{−μ|x|^{2}}}{|x|} |f(x)|^{2}dx ≤ \frac{π}{2} \frac{1}{\sqrt{2} − 1} (f, T (p)f),

with μ = π^{−1}α^{−2}.

The following is a slight generalization of the Daubechies-Lieb-Yau inequality formulated in Theorem 2.8 in [24].

Theorem 2.10 (Daubechies-Lieb-Yau inequality). Assume that the potential U ∈ L_{loc}^{1}(ℝ^{3}) satisfies

0 ≥ −U(x) ≥ −κ|x|^{−1} for |x| < max{α, R} ,  (19)

for α, R > 0 and 0 ≤ κ ≤ 2/π. Then we have

Tr[T (p) − U]_{−} ≥ −Cκ^{5/2}α^{−3/2}R^{1/2} − Cκ^{4}α^{−1}
−C ∫_{|x|>R} (α^{− \frac{3}{2}} |U(x)|^{\frac{5}{2}} + |U(x)|^{4}) dx.

Proof. If (\sqrt{2} − 1)/π ≤ κ ≤ 2/π then κ^{5/2}α^{−3/2}R^{1/2} + κ^{4}α^{−1} ≥ Cκ^{5/2}α^{−1} and the result follows immediately from Theorem 2.8 in [24] observing that for R > α the two integrals of the potential on {α < |x| < R} are bounded by the constants. If 0 ≤ κ < (\sqrt{2} − 1)/π we write

U(x) = e^{−μ|x|^{2}}U(x)χ_{|x|<R} + (1 − e^{−μ|x|^{2}})U(x)χ_{|x|<R} + U(x)χ_{|x|>R}

with μ = α^{−2}π^{−1}. Using (19) and Lemma 2.9 we find that

T (p) − U(x) ≥ \frac{1}{2}T (p) − κ(1 − e^{−μ|x|^{2}})|x|^{−1}χ_{|x|<R} − U(x)χ_{|x|>R}.

3The result of the lemma and the proof given in [24] are actually due to us, but we communicated the result to the authors of [24], where it is referred to as a a private communication.

Documenta Mathematica 15 (2010) 285–345

Hence from the generalization of the Lieb-Thirring inequality Theorem 2.5 (see (18)) we obtain

Tr[T (p) − U]_{−} ≥ −C ∫_{|x|<R} α^{− \frac{3}{2}} (κ(1 − e^{−μ|x|^{2}})|x|^{−1}))^{\frac{5}{2}} dx
−C ∫_{|x|<R} (κ(1 − e^{−μ|x|^{2}})|x|^{−1})^{4} dx
−C ∫_{|x|>R} (α^{− \frac{3}{2}} |U(x)|^{\frac{5}{2}} + |U(x)|^{4}) dx.

Since the two first integrals above are estimated below by −Cκ^{5/2}α^{−3/2}R^{1/2} − Cκ^{4}α^{−1} we get the result in the theorem.

By Theorem 2.10 we find

κ ∫_{|x−y|<R} \frac{ρ^{HF}(y)}{|x − y|} dy ≤ Tr[T (p)γ^{HF}] + C_{1}κZ^{\frac{3}{2}} R^{\frac{1}{2}} + C_{2}κ^{3}Z,  (20)

with κ ∈ [0, 2/π], κ = Zα and R > 0 parameters to be chosen. This is the inequality that we use to estimate ρ^{HF} ∗ |x|^{−1} (see proof of Lemma 3.2 below).

2.1.1 Bound on the Hartree-Fock energy

As a first application of Theorem 2.10 we can give a lower bound to the HFenergy.

Theorem 2.11 (Bound on the HF-energy). Let N > 0, Z > 0 and such that Zα = κ with 0 ≤ κ ≤ 2/π. Then

E^{HF}(N, Z) ≥ −2C^{\frac{2}{3}} Z^{2}N^{\frac{1}{3}} − Cκ^{2}Z^{2},

with C the constant in Theorem 2.10.

Proof. Let γ be a N-dimensional projection. Since the electron-electron interaction is positive we see that

E^{HF}(γ) ≥ α^{−1} Tr[(T (p) − \frac{Zα}{| · |})γ]
= α^{−1} Tr[(T (p) − \frac{κ}{| · |}χ_{|x|<R})γ] − α^{−1} Tr[\frac{κ}{| · |}(1 − χ_{|x|<R})γ]

with R > 0 a parameter to be chosen. By Theorem 2.10 we find

E^{HF}(γ) ≥ −2C^{\frac{2}{3}} Z^{2}N^{\frac{1}{3}} − Cκ^{2}Z^{2},

using that κ = Zα and by choosing R = C^{− \frac{2}{3}} Z^{−1}N^{\frac{2}{3}} .

3 Near the nucleus

In this section we prove the estimate in Theorem 1.17 in the region near the nucleus (i.e. at distance of Z^{− \frac{1}{3}} ). We again assume that N ≥ Z and that an HF-minimizer γ^{HF} exists for this N and Z. We denote the density of γ^{HF} by ρ^{HF}. We assume throughout that αZ = κ is fixed with 0 ≤ κ < 2/π and Z ≥ 1.

Lemma 3.1. Let Zα = κ be fixed with 0 ≤ κ < 2/π and Z ≥ 1. Let G_{α} be the function defined in Theorem 2.3. Then, there exists α_{0} > 0 such that for all α ≤ α_{0}

α^{−1} ∫_{ℝ^{3}} G_{α}(ρ^{HF}(x))dx ≤ CZ^{7/3}, α^{−1} Tr[T (p)γ^{HF}] ≤ CZ^{7/3}
and ∥ρ^{TF} − ρ^{HF}∥_{C}^{2} ≤ CZ^{2+ \frac{3}{11}} ,

(21)

with C a universal constant depending only on κ.

Proof. Let μ ∈ (0, 1) be such that μ^{−1}κ < 2/π. Notice that here we need κ < 2/π. Splitting the kinetic energy into two parts we find

E^{HF}(γ^{HF}) = (1 − μ)α^{−1} Tr[T (p)γ^{HF}] + D(γ^{HF}) − Ex(γ^{HF})

+μ Tr[(α^{−1}T (p) − \frac{Z}{μ|x|})γ^{HF}] = …,

and introducing ρ ∈ L^{\frac{5}{3}} (ℝ^{3}) ∩ L^{1}(ℝ^{3}), ρ ≥ 0, to be chosen

… = (1 − μ)α^{−1} Tr[T (p)γ^{HF}] + μ∥ρ − ρ^{HF}∥_{C}^{2} + (1 − μ)D(γ^{HF}) (22)

−Ex(γ^{HF}) − μD(ρ) + μ Tr[(α^{−1}T (p) − (\frac{Z}{μ|x|} − ρ ∗ \frac{1}{|x|}))γ^{HF}].

Here ∥ · ∥_{C} denotes the Coulomb norm defined in Definition 2.7 and we used that

∥ρ − ρ^{HF}∥_{C}^{2} = D(ρ) − ∫∫ \frac{ρ^{HF}(x)ρ(y)}{|x − y|} dxdy + D(γ^{HF}).

The estimates in the claim will follow from (22) with different choices of μ and ρ. The main idea is to relate, up to lower order term, the last term on the right hand side of (22) to the TF-energy of a neutral atom of nuclear charge Zμ^{−1}. This has been done in [21]. For completeness and easy reference we repeat the reasoning in Propositions B.1 and B.2 in Appendix B. To prove the first inequality in (21) we choose ρ as the minimizer of the TFenergy functional of a neutral atom with charge μ^{−1}Z. Since the corresponding TF-mean field potential is Z/(μ|x|)−ρ∗1/|x| by Proposition B.2 in Appendix B we find

Tr[(α^{−1}T (p) − (\frac{Z}{μ|x|} − ρ ∗ \frac{1}{|x|}))γ^{HF}] ≥ −C_{1}Z^{\frac{7}{3}} + D(ρ).  (23)

Here we use (12). Since E^{HF}(γ^{HF}) ≤ 0 from (22) and (23) leaving out the positive terms we find

0 ≥ (1 − μ)α^{−1} Tr[T (p)γ^{HF}] − Ex(γ^{HF}) − C_{1}Z^{\frac{7}{3}} .  (24)

From (24) and Theorem 2.3 we get

(1 − μ)α^{−1} ∫_{ℝ^{3}} G_{α}(ρ^{HF}(x)) dx ≤ (1 − μ)α^{−1} Tr[T (p)γ^{HF}] ≤ Ex(γ^{HF}) + C_{1}Z^{\frac{7}{3}} .

(25)

It remains to estimate the exchange term. By the exchange inequality (see [15])

Ex(γ^{HF}) ≤ 1.68 ∫_{ℝ^{3}} (ρ^{HF}(x))^{\frac{4}{3}} dx.

To proceed we separate ℝ^{3} into two regions. Let us define

Σ = \{x ∈ ℝ^{3} : α(C^{−1}ρ^{HF}(x))^{\frac{1}{3}} ≥ \frac{5}{2}\},  (26)

with the same notation as in (16). By Remark 2.4, G_{α}(ρ^{HF}(x)) ≥ C_{2}(ρ^{HF}(x))^{\frac{4}{3}} in Σ and α^{−1}G_{α}(ρ^{HF}(x)) ≥ C_{3}(ρ^{HF}(x))^{\frac{5}{3}} in ℝ^{3} \Σ. Hence by Hölder’s inequality we find

Ex(γ^{HF}) ≤ 1.68 ∫_{Σ}(ρ^{HF}(x))^{\frac{4}{3}} dx
+1.68( ∫_{ℝ^{3}\Σ}(ρ^{HF}(x))^{\frac{5}{3}} dx)^{\frac{1}{2}} ( ∫_{ℝ^{3}\Σ} ρ^{HF}(x) dx)^{\frac{1}{2}}
≤C_{4} ∫_{ℝ^{3}} G_{α}(ρ^{HF}(x)) dx + C_{5}( ∫_{ℝ^{3}} α^{−1}G_{α}(ρ^{HF}(x)) dx)^{\frac{1}{2}} N^{\frac{1}{2}} .(27)

Choosing α_{0} such that 1 − μ > 2C_{4}α for α ≤ α_{0}, from (25) and (27) we find

\frac{1−μ}{2}α^{−1} ∫_{ℝ^{3}} G_{α}(ρ^{HF}(x)) dx ≤ C_{1}Z^{\frac{7}{3}} + C_{5}( ∫_{ℝ^{3}} α^{−1}G_{α}(ρ^{HF}(x)) dx)^{\frac{1}{2}} N^{\frac{1}{2}} .

The first estimate in (21) follows from the estimate above using that x^{2}−bx−c ≤ 0 implies x^{2} ≤ b^{2} + 2c and that N ≤ 2Z + 1 (Theorem 1.6). The second inequality in (21) follows then from (25) and the bound on the exchange term. To prove the third inequality in (21) we estimate from above and from below E^{HF}(γ^{HF}). For the one from below we choose in (22) μ = 1 and ρ = ρ^{TF} the TF-minimizer of a neutral atom with nucleus of charge Z. We find

E^{HF}(γ^{HF}) = ∑_{i=1}^{N}(u_{i}, (α^{−1}T (p) − φ^{TF})u_{i}) + ∥ρ^{HF} − ρ^{TF}∥_{C}^{2} − D(ρ^{TF}) − Ex(γ^{HF}).

(28)

From (28) and the proof of Proposition B.2 (see (B37)), we find

E^{HF}(γ^{HF}) ≥ −\frac{2^{\frac{3}{2}}}{15π^{2}} q ∫ dq(φ^{TF}(q))^{\frac{5}{2}} − CZ^{2+1/5}  (29)
−D(ρ^{TF}) + ∥ρ^{HF} − ρ^{TF}∥_{C}^{2} − Ex(γ^{HF}).

To estimate from above E^{HF}(γ^{HF}) we may proceed exactly as in [23, page 543] using that α^{−1}T (p) ≤ \frac{1}{2}|p|^{2}. For completeness we repeat the main ideas. We consider γ the density matrix that acts identically on each of the spin components as

γ^{j} = \frac{1}{(2π)^{3}} ∫∫_{\frac{1}{2} |p|^{2}≤φ^{TF}(q)} Π_{p,q} dqdp for j = 1, …, q.

Here Π_{p,q} is the projection onto the space spanned by h_{s}^{p,q}(x) := h_{s}(x−q)e^{ip.x} where h_{s} is the ground state (normalized in L^{2}(ℝ^{3})) for the Dirichlet Laplacian on the ball of radius Z^{−s} with s ∈ (1/3, 2/3) to be chosen. One sees that Tr[γ] = Z ≤ N since

ρ_{γ}(x) = \frac{2^{3/2}q}{6π^{2}} (φ^{TF})^{3/2} ∗ h_{s}^{2}(x) = ρ^{TF} ∗ h_{s}^{2}(x),

where we have used the TF-equation. Hence E^{HF}(γ) ≥ E^{HF}(γ^{HF}). Now we estimate from above E^{HF}(γ). Since α^{−1}T (p) ≤ \frac{1}{2}|p|^{2} and Ex(γ) ≥ 0 we find

E^{HF}(γ) ≤ Tr[(−\frac{1}{2}Δ − \frac{Z}{| · |})γ] + D(ρ_{γ}) = …,

and proceeding as in [23, page 543])

⋯ = \frac{q}{(2π)^{3}} ∫∫_{\frac{1}{2} |p|^{2}≤φ^{TF}(q)} \frac{1}{2}|p|^{2} dpdq − \frac{π^{2}}{2} Z^{2s}N − ∫_{ℝ^{3}} \frac{Z}{|x|}ρ_{γ}(x) dx + D(ρ_{γ}).

Computing the integral and summing and subtracting the term ∫ ρ^{TF}φ^{TF} we get

E^{HF}(γ) ≤ \frac{q2^{\frac{1}{2}}}{5π^{2}} ∫_{ℝ^{3}}(φ^{TF}(q))^{\frac{5}{2}} dq − \frac{π^{2}}{2} Z^{2s}N − ∫_{ℝ^{3}} φ^{TF}(x)ρ^{TF}(x) dx
− ∫_{ℝ^{3}} \frac{Z}{|x|}(ρ_{γ}(x) − ρ^{TF}(x))dx − 2D(ρ^{TF}) + D(ρ_{γ}).  (30)

By Newton’s theorem one sees that D(ρ_{γ}) ≤ D(ρ^{TF}) and that

Z ∫_{ℝ^{3}} \frac{ρ^{TF}(x) − ρ_{γ}(x)}{|x|} dx ≤ Z ∫_{|x|≤Z^{−s}} \frac{ρ^{TF}(x)}{|x|} dx ≤ CZ^{\frac{1}{5} (12−s)}.

In the last step we use Hölder’s inequality and Corollary 1.15. From (30) using the TF-equation, that N ≤ 2Z + 1 (Theorem 1.6) and optimizing in s we find

E^{HF}(γ) ≤ −\frac{2^{\frac{3}{2}}}{15π^{2}} q ∫_{ℝ^{3}}(φ^{TF}(q))^{\frac{5}{2}} dq + CZ^{\frac{1}{5} (12− \frac{7}{11} )} − D(ρ^{TF}).  (31)

Hence from (29) and (31) we obtain

∥ρ^{HF} − ρ^{TF}∥_{C}^{2} ≤ CZ^{2+ \frac{3}{11}} + Ex(γ^{HF}).

The last estimate in (21) follows from the estimate above since Ex(γ^{HF}) ≤ CZ^{\frac{5}{3}} using (27) and the estimate just proved on α^{−1} ∫ G_{α}(ρ^{HF}(x)) dx.

Lemma 3.2. Let Zα = κ be fixed with 0 ≤ κ < 2/π and Z ≥ 1. Then, there exists an α_{0} > 0 such that for all α ≤ α_{0}, μ > 0 and x ∈ ℝ^{3} with |x| ≤ βZ^{− \frac{1+μ}{3}} we have

|Φ_{|x|}^{TF}(x) − Φ_{|x|}^{HF}(x)| ≤ Cβ^{\frac{4}{1+μ}} (1 + β^{\frac{9}{22(1+μ)}} |x|^{\frac{2+11μ}{22(1+μ)}} )|x|^{−4+ \frac{4μ}{1+μ}} .

Proof. By the definition of screened nuclear potential we have

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ ∫_{|y|<|x|} \frac{|ρ^{HF}(y) − ρ^{TF}(y)|}{|x − y|}dy = …

and for all k > 0 by Proposition 2.8

… ≤ 2^{\frac{3}{2}} k^{−1}|x|^{− \frac{1}{2}} ∥ρ^{HF} − ρ^{TF}∥_{C} + ∫_{A(|x|,k)} \frac{ρ^{HF}(y) + ρ^{TF}(y)}{|x − y|}dy. (32)

Since ∥ρ^{TF}∥_{L^{\frac{5}{3}} (ℝ^{3})} ≤ CZ^{\frac{7}{5}} (Corollary 1.15) and

∫_{A(|x|,k)} \frac{1}{|x − y|^{\frac{5}{2}}} dy ≤ 8π|x|^{\frac{1}{2}} (2k)^{\frac{1}{2}} .  (33)

(see [23] page 549) one finds

∫_{A(|x|,k)} \frac{ρ^{TF}(y)}{|x − y|}dy ≤ CZ^{\frac{7}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} .  (34)

The term with the HF-density has to be treated differently since we do not have a bound for the L^{\frac{5}{3}} -norm of ρ^{HF}. For a R ∈ ℝ^{+} to be chosen later we consider the splitting

∫_{A(|x|,k)} \frac{ρ^{HF}(y)}{|x − y|}dy = ∫\begin{matrix}A(|x|,k) \\ |x−y|>R \end{matrix} \frac{ρ^{HF}(y)}{|x − y|}dy + ∫\begin{matrix}A(|x|,k) \\ |x−y|<R \end{matrix} \frac{ρ^{HF}(y)}{|x − y|}dy.  (35)

We consider these two terms separately. Let Σ be defined as in (26); i.e. the region where G_{α}(ρ^{HF}) behaves like (ρ^{HF})^{\frac{4}{3}} (Remark 2.4). By Hölder’s inequality we find

∫\begin{matrix}A(|x|,k) \\ |x−y|>R \end{matrix} \frac{ρ^{HF}(y)}{|x − y|}dy ≤ ( ∫\begin{matrix}A(|x|,k) \\ |x−y|>R \end{matrix} \frac{1}{|x − y|^{4}} dy)^{\frac{1}{4}} ( ∫_{y∈Σ} (ρ^{HF}(y))^{\frac{4}{3}} dy)^{\frac{3}{4}}

+( ∫_{A(|x|,k)} \frac{1}{|x − y|^{\frac{5}{2}}} dy)^{\frac{2}{5}} ( ∫_{y∈ℝ^{3}\Σ} (ρ^{HF}(y))^{\frac{5}{3}} dy)^{\frac{3}{5}} .

From the inequality above, Remark 2.4 and estimate (21) we get

∫\begin{matrix}A(|x|,k) \\ |x−y|>R \end{matrix} \frac{ρ^{HF}(y)}{|x − y|}dy ≤ CR^{− \frac{3}{8}} |x|^{\frac{1}{8}} k^{\frac{1}{8}} Z + C|x|^{\frac{1}{5}} k^{\frac{1}{5}} Z^{\frac{7}{5}} .  (36)

On the other hand for the second term on the right hand side of (35) by (20) and Lemma 3.1 we find

∫_{|x−y|<R} \frac{ρ^{HF}(y)}{|x − y|}dy ≤ C(Z^{\frac{4}{3}} + R^{\frac{1}{2}} Z^{\frac{3}{2}} ).  (37)

Hence from (32), Lemma 3.1, (34), (36) and (37), we get

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(\frac{Z^{1+ \frac{3}{22}}}{|x|^{1/2}k} + Z^{\frac{7}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} + R^{− \frac{3}{8}} |x|^{\frac{1}{8}} k^{\frac{1}{8}} Z + R^{\frac{1}{2}} Z^{\frac{3}{2}} + Z^{\frac{4}{3}} ).

(38)

Choosing k such that Z^{\frac{4}{3}} = Z^{\frac{7}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} , i.e. k = |x|^{−1}Z^{− \frac{1}{3}} and R such that R^{− \frac{3}{8}} Z^{1− \frac{1}{24}} = Z^{\frac{4}{3}} , i.e. R = Z^{−1} we find

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(|x|^{\frac{1}{2}} Z^{\frac{4}{3} + \frac{3}{22}} + Z^{\frac{4}{3}} ).

The claim follows using that |x| ≤ βZ^{− \frac{1+μ}{3}} .

Theorem 3.3. Let Zα = κ be fixed with 0 ≤ κ < 2/π and Z ≥ 1. Then there exists an α_{0} > 0 such that for all α ≤ α_{0} and x ∈ ℝ^{3} with |x| ≤ βZ^{− \frac{1}{3}} we have

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ Cβ^{2− \frac{1}{66}} (1 + β^{2} + β^{\frac{5}{2}} + β^{2+ \frac{789}{1936}} |x|^{\frac{179}{1936}} )|x|^{−4+ \frac{1}{66}} . (39)

Moreover if |x| ≤ βZ^{− \frac{1−μ}{3}} for μ < \frac{2}{11} \frac{1}{49}, then

|Φ_{|x|}^{TF}(x) − Φ_{|x|}^{HF}(x)| ≤ Cβ^{2−a(μ)}(1 + β^{2} + β^{\frac{5}{2}} + β^{b(μ)}|x|^{c(μ)})|x|^{−4+a(μ)}, (40)

with a(μ) = \frac{1}{66(1−μ)} − \frac{49μ}{12(1−μ)}, b(μ) = 2 + \frac{3}{176} \frac{24−24μ− \frac{1}{11} + \frac{49}{2} μ}{1−μ} and c(μ) = \frac{1}{11} −

\frac{\frac{3}{11} − \frac{3}{2} 49μ}{22(8−8μ)} strictly positive constants.

Proof. Proceeding as in the proof of Lemma 3.2 up to (36) we get

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(k^{−1}|x|^{− \frac{1}{2}} Z^{1+ \frac{3}{22}} + Z^{\frac{7}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} + R^{− \frac{3}{8}} |x|^{\frac{1}{8}} k^{\frac{1}{8}} Z)
+ ∫_{|x−y|≤R} \frac{ρ^{HF}(y)}{|x − y|}dy,  (41)

for R ∈ ℝ^{+} to be chosen. It remains to estimate the last term on the right hand side of (41). For ‘small’ R which is relevant for small x we already did it in Lemma 3.2, for ‘big’ R which is relevant for big x we use Proposition B.1 in Appendix B.

Take γ ≤ 1/263 to be chosen. If |x| ≤ βZ^{− \frac{1+γ}{3}} then by Lemma 3.2

|Φ_{|x|}^{TF}(x) − Φ_{|x|}^{HF}(x)| ≤ Cβ^{\frac{4}{1+γ}} (1 + β^{\frac{9}{22(1+γ)}} |x|^{\frac{2+11γ}{22(1+γ)}} )|x|^{−4+ \frac{4γ}{1+γ}} .  (42)

If instead |x| > βZ^{− \frac{1+γ}{3}} , let H_{x} be the Hamiltonian defined in (B2) with P = x and ν = Z. Then by the definition of H_{x} and taking the HF-minimizer as a trial wave function we have

inf_{\begin{matrix}ψ∈∧_{i=1}^{N}L^{2}(ℝ^{3}) \\ ∥ψ∥_{2}=1 \end{matrix}}⟨ψ, H_{x}ψ⟩ ≤ E^{HF}(γ^{HF}) − Z ∫_{|x−y|<R} \frac{ρ^{HF}(y)}{|x − y|}dy

= inf_{γ∈A} E^{HF}(γ) − Z ∫_{|x−y|<R} \frac{ρ^{HF}(y)}{|x − y|}dy = ….

Since \frac{1}{2}|p|^{2} ≥ α^{−1}T (p), inf_{γ∈A} E^{HF}(γ) is estimated from above by the HFground state energy of the non-relativistic model (i.e. when the kinetic energy is given by −\frac{1}{2}Δ). Moreover, this last one can be estimated from above by E^{TF}(ρ^{TF}) + CN^{\frac{1}{5}} Z^{2} (see [18] and [11]). Hence we find

⋯ ≤ E^{TF}(ρ^{TF}) + CN^{\frac{1}{5}} Z^{2} − Z ∫_{|x−y|<R} \frac{ρ^{HF}(y)}{|x − y|}dy.

On the other hand since |x| > βZ^{− \frac{1+γ}{3}} choosing for some l > \frac{1+γ}{3} , R < βZ^{−l}/4 from Proposition B.1 it follows that there exists a constant depending only on κ such that for t ∈ ((1 + γ)/3, min{l, 3/5}), and for every ψ ∈ ∧_{i=1}^{N}L^{2}(ℝ^{3}) with ∥ψ∥_{2} = 1 we have

⟨ψ, H_{x}ψ⟩ ≥ E^{TF}(ρ^{TF}) − C(β^{1/2} + β^{−2})Z^{\frac{5}{2} − \frac{t}{2}} ,

Hence combining the two inequalities above we find

∫_{|x−y|≤R} \frac{ρ^{HF}(y)}{|x − y|}dy ≤ C(β^{1/2} + β^{−2})Z^{\frac{1}{2} (3−t)}.  (43)

From (41) and the inequality above we get

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ Ck^{−1}|x|^{− \frac{1}{2}} Z^{1+ \frac{3}{22}} + CZ^{\frac{7}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}}
+CR^{− \frac{3}{8}} |x|^{\frac{1}{8}} k^{\frac{1}{8}} Z + C(β^{1/2} + β^{−2})Z^{\frac{1}{2} (3−t)}.

Choosing k such that Z^{\frac{1}{2} (3−t)} = Z^{\frac{7}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} , i.e k = |x|^{−1}Z^{\frac{1}{2} (1−5t)} and R such that Z^{\frac{1}{2} (3−t)} ∼ R^{− \frac{3}{8}} Z^{1+ \frac{1}{16} (1−5t)}, i.e R = βZ^{− \frac{7}{6} + \frac{1}{2} t}/4 we find

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(|x|^{\frac{1}{2}} Z^{\frac{7}{11} + \frac{5}{2} t} + (β^{1/2} + β^{−2})Z^{\frac{1}{2} (3−t)}). (44)

Notice that R < βZ^{−l}/4 is satisfied choosing l = 4t/3. Then for x such that βZ^{− \frac{1+γ}{3}} ≤ |x| ≤ βZ^{− \frac{1}{3}} we find

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(|x|^{− \frac{31}{22} − \frac{15}{2} t}β^{\frac{21}{11} + \frac{15}{2} t} + (β^{1/2} + β^{−2})β^{\frac{3}{2} (3−t)}|x|^{− \frac{3}{2} (3−t)}).

Optimizing in t gives t = 1/3 + 1/99. For this value of t we get

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(1 + β^{\frac{5}{2}} )β^{2− \frac{1}{66}} |x|^{−4+ \frac{1}{66}} .  (45)

Inequality (39) follows from (42) and (45) choosing γ such that 4γ/(1 + γ) = 1/66, i.e. γ = 1/263. On the other hand from (44) for x such that βZ^{− \frac{1+γ}{3}} ≤ |x| ≤ βZ^{− \frac{1−μ}{3}} we find

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C|x|^{\frac{1}{2} − \frac{3}{1−μ} ( \frac{7}{11} + \frac{5}{2} t)}β^{\frac{3}{1−μ} ( \frac{7}{11} + \frac{5}{2} t)}
+C(β^{1/2} + β^{−2})β^{\frac{3}{2(1−μ)} (3−t)}|x|^{− \frac{3}{2(1−μ)} (3−t)}.

Optimizing in t gives t = 1/3 + 1/99 − \frac{1}{18}μ. For this value of t we get

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(1 + β^{\frac{5}{2}} )β^{2− \frac{1}{66(1−μ)} + \frac{49μ}{12(1−μ)}} |x|^{−4+ \frac{1}{66(1−μ)} − \frac{49μ}{12(1−μ)}} .

Inequality (40) follows from the one above and (42) choosing γ such that 4γ/(1+ γ) = \frac{1}{66(1−μ)} − \frac{49μ}{12(1−μ)}.

4 The exterior part

In this section we complete the proof of Theorem 1.17. We first estimate the exterior integral of the density and study the minimization problem that the exterior part of the minimizer satisfies. Then we prove the main estimate in Theorem 1.17 in an intermediate zone, i.e. far from the nucleus but not further than a fixed distance independent of Z. To study this area we need first to construct a TF-model that gives a good approximation of the HF-density in this intermediate zone. By the estimate on the exterior integral of the density we can then also prove Theorem 1.17 in the region far away from the nucleus.

4.1 The exterior integral of the density

The main result of this section is the following lemma.

Lemma 4.1 (The exterior integral of the density). Assume that for some R, σ, ε′ > 0

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ σ|x|^{−4+ε′} ,  (46)

holds for |x| ≤ R. Then for 0 < r ≤ R

| ∫_{|x|<r}(ρ^{HF}(x) − ρ^{TF}(x)) dx| ≤ σr^{−3+ε′}  (47)

and

∫_{|x|>r} ρ^{HF}(x)dx ≤ C(1 + σr^{ε′})(1 + r^{−3}),  (48)

with C a universal constant.

We proceed similarly as in the proof of Lemma 10.5 in [23]. Since we need to localize we first present some technical lemmas that will take care of the error terms due to the localization. The localization error that will appear in the argument below (see (58)) will be in the form of an operator L similar to the error (14) in the IMS formula. We estimate this error in Lemma 4.3.

Remark 4.2. Let 0 ≤ β_{1} < .. < β_{4} be real numbers with possibly β_{4} = ∞. Let us denote Σ_{r}(β_{i}, β_{j}) = {x ∈ ℝ^{3} : β_{i}r ≤ |x| ≤ β_{j}r}. Then we have

∫∫ \begin{matrix}x ∈ Σ_{r}(β_{1}, β_{2}) \\ y ∈ Σ_{r}(β_{3}, β_{4}) \end{matrix} K_{2}(α^{−1}|x − y|)^{2} dxdy ≤ \frac{4^{6}π^{2}}{3} \frac{β_{2}^{3} − β_{1}^{3}}{β_{3} − β_{2}} α^{4}r^{2}e^{−α^{−1}r(β_{3}−β_{2})} .

The proof of this estimate is given in Appendix A.

Lemma 4.3. Let r > 0 and λ, ν ∈ (0, 1). Let χ_{−} be the characteristic function of B_{r(1−ν)}(0) and χ_{0} be the characteristic function of the sector {x ∈ ℝ^{3} : r(1 − ν) < |x| < r(1 + ν)/(1 − λ)}. Let η be a Lipschitz function such that 0 ≤ η(x) ≤ 1 for all x ∈ ℝ^{3}, η(x) ≡ 0 if |x| ≤ r, η(x) ≡ 1 if |x| ≥ r(1 − λ)^{−1} and ∥∇η∥_{∞} is bounded. Let L denote the operator with integral kernel

L(x, y) = \frac{α^{−2}}{4π^{2}} \frac{(η(x) − η(y))(η(x)|x| − η(y)|y|)}{|x − y|^{2}}K_{2}(α^{−1}|x − y|).  (49)

Then for every function f ∈ L^{2}(ℝ^{3}) we have

α^{−1}|(f, Lf)| ≤ 3 D(η, λ, r) ∥χ_{0}f∥_{2}^{2} + D(η, λ, r)e^{− \frac{1}{2} α^{−1}rν}∥χ_{−}f∥_{2}^{2} + α^{−1}|(f, Qf)|,

with D(η, λ, r) := ∥∇η∥_{∞}(\frac{∥∇η∥_{∞}r}{1−λ} +1) and Q a positive semi-definite operator such that

Tr[Q] ≤ CD(η, λ, r)α^{−1}r^{2}e^{− \frac{1}{2} α^{−1}rν},

with C depending only on λ and ν.

Proof. As a first step we decompose the operator L. We introduce a third cut-off function χ_{+} such that 1 = χ_{−}(x) + χ_{0}(x) + χ_{+}(x) for all x ∈ ℝ^{3}. We decompose the operator L with respect to these characteristic functions as follows:

L = χ_{−}L(χ_{0} + χ_{+}) + (χ_{0} + χ_{+})Lχ_{−} + χ_{0}Lχ_{+} + χ_{+}Lχ_{0} + χ_{0}Lχ_{0}.

We proceed similarly as in [24, Proof of Theorem 2.6 (Localization error)]. For Γ_{1}, Γ_{2} bounded operators from (Γ_{1} − Γ_{2})(Γ_{1} − Γ_{2})^{∗} ≥ 0 it follows

Γ_{1}Γ_{2}^{∗} + Γ_{2}Γ_{1}^{∗} ≤ Γ_{1}Γ_{1}^{∗} + Γ_{2}Γ_{2}^{∗}.  (50)

We are going to use several times this inequality with different choices of Γ_{1} and Γ_{2}.

As a first choice we consider Γ_{1} = \sqrt{ε_{1}}χ_{−} and Γ_{2} = 1/\sqrt{ε_{1}}(χ_{0} + χ_{+})Lχ_{−} with ε_{1} > 0 to be chosen. Using (50) we get

|(f, (χ_{−}L(χ_{0} + χ_{+}) + (χ_{0} + χ_{+})Lχ_{−})f)| ≤ ε_{1}∥χ_{−}f∥_{2}^{2} + \frac{1}{ε_{1}} (f, Q_{1}f), (51)

with Q_{1} = (χ_{0} + χ_{+})Lχ_{−}^{2}L(χ_{0} + χ_{+}). We estimate now the trace of Q_{1}. By the definition of η, χ_{−}, χ_{0} and χ_{+} it follows that

Tr[Q_{1}] = ∫_{|x|≤r(1−ν)} ∫_{|y|≥r} L^{2}(x, y) dxdy ≤ \frac{(16)^{2}}{3π^{2}} \frac{(1−ν)^{3}}{ν} D(η, λ, r)^{2}r^{2}e^{−α^{−1}rν}.

In the last step we use the definition of L, Remark 4.2 and the definition of the constant D(η, λ, r) given in the statement of the lemma. Now we choose Γ_{1} = \sqrt{ε_{2}}χ_{0} and Γ_{2} = 1/\sqrt{ε_{2}}χ_{+}Lχ_{0} with ε_{2} > 0 to be chosen. Proceeding as above we get

|(f, (χ_{+}Lχ_{0} + χ_{0}Lχ_{+})f)| ≤ ε_{2}∥χ_{0}f∥_{2}^{2} + \frac{1}{ε_{2}} (f, Q_{2}f),  (52)

with Q_{2} = χ_{+}Lχ_{0}^{2}Lχ_{+} and such that

Tr[Q_{2}] ≤ \frac{(16)^{2}}{3π^{2}} \frac{1−(1−ν)^{3}(1−λ)^{3}}{ν(1−λ)^{2}} D(η, λ, r)^{2} r^{2}e^{−α^{−1}r \frac{ν}{1−λ}} .

It remains to study the term χ_{0}Lχ_{0}. This one has to be treated differently. By Schwartz’s inequality one gets

|(f, χ_{0}Lχ_{0}f)| ≤ \frac{3α}{2}D(η, λ, r) ∫_{ℝ^{3}} χ_{0}(x)|f(x)|^{2},  (53)

since ∫_{ℝ^{3}} |L(x, y)|dxdy ≤ \frac{3α}{2}D(η, λ, r).

The claim follows from (51), (52) and (53) choosing ε_{1} = D(η, λ, r)αe^{− \frac{1}{2} α^{−1}rν}, ε_{2} = \frac{3α}{2}D(η, λ, r) and with Q := \frac{1}{ε_{1}} Q_{1} + \frac{1}{ε_{2}} Q_{2}.

Definition 4.4 (The localization function). Fix 0 < λ < 1 and let G : ℝ^{3} → ℝ be given by

G(x) := \{ \begin{matrix}0 & if |x| ≤ 1, \\ \frac{π}{2} (|x| − 1)\frac{1}{(1−λ)^{−1}−1} & if 1 ≤ |x| ≤ (1 − λ)^{−1}, \\ \frac{π}{2} & if (1 − λ)^{−1} ≤ |x|. \end{matrix}

Let r > 0 and define the outside localization function θ_{r}(x) := sin(G(\frac{|x|}{r})).

Remark 4.5. From the definition it follows that ∥∇θ_{r}∥_{∞} ≤ \frac{π}{2} \frac{1−λ}{λ}r^{−1}.

Lemma 4.6. For all r > 0 and λ, ν ∈ (0, 1) the density ρ^{HF} of the minimizer satisfies

∫_{|x|>r(1−λ)^{−1}} ρ^{HF}(x)dx ≤ 1 + \frac{2}{λ} + 2 sup_{|x|=r(1−λ)} |x|Φ_{r(1−λ)}^{HF}(x) + R^{\frac{1}{2}}

with

R = 6D(λ)r^{−1} ∫_{r(1−ν)<|x|<r \frac{1+ν}{1−λ}} ρ^{HF}(x) dx + 2D(λ)(r^{−1}N + Crα^{−2})e^{− \frac{1}{2} α^{−1}rν},

with D(λ) := (1 + π/(2λ(1 − λ)))π/(2λ) and C = C(λ, ν).

Proof. Let γ^{HF} be the minimizer. By the variational principle, γ^{HF} is a projection onto the subspace spanned by u_{1}, …, u_{N}. These functions u_{i} satisfy the Euler Lagrange equations h_{γ^{HF}}u_{i} = ε_{i}u_{i}, ε_{i} < 0, for i = 1, …, N, with h_{γ^{HF}} defined in (11). Given η a function in C^{1}(ℝ^{3}) with support away from zero, we find

0 ≥ ∑_{i=1}^{N} ε_{i} ∫_{ℝ^{3}} |u_{i}(x)|^{2}|x|η^{2}(x)dx = ∑_{i=1}^{N} ∫_{ℝ^{3}} u_{i}(x)^{∗}|x|η^{2}(x) h_{γ^{HF}}u_{i}(x)dx.

Since ηT (p)u_{i} ∈ L^{2}(ℝ^{3}) (Theorem 1.5, (3)), using the Euler-Lagrange equations and treating all the terms, except the kinetic energy, as in [23, Formula (63)] we get

0 ≥ α^{−1} ∑_{i=1}^{N}(u_{i}η| · |, ηT (p)u_{i}) − Z ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx

+ ∫_{ℝ^{3}} ∫_{ℝ^{3}} [ρ^{HF}(x)ρ^{HF}(y) − Tr_{ℂ^{q}} |γ^{HF}(x, y)|^{2}] \frac{|y|(1 − η^{2}(x))η^{2}(y)}{|x − y|} dxdy

+\frac{1}{2}( ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx)^{2} − \frac{1}{2} ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx.  (54)

Now we look at the kinetic energy term. For each i ∈ {1, …, N} we may write

Re(u_{i}η| · |, ηT (p)u_{i}) = Re(u_{i}η| · |, T (p)(ηu_{i})) + Re(u_{i}η| · |, [η, T (p)]u_{i}), (55)

where [A, B] denotes the commutator of the operators A and B. The first term on the right hand side of (55) is non-negative by the result of Lieb in [13]. Notice that here we may use that ηu_{i} ∈ H^{1}(ℝ^{3}) (see Theorem 1.5, (3)). Hence, from (54) and (55) we find

0 ≥ α^{−1} ∑_{i=1}^{N} Re(u_{i}η| · |, [η, T (p)]u_{i}) − Z ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx

+ ∫_{ℝ^{3}} ∫_{ℝ^{3}} [ρ^{HF}(x)ρ^{HF}(y) − Tr_{ℂ^{q}} |γ^{HF}(x, y)|^{2}] \frac{|y|(1 − η^{2}(x))η^{2}(y)}{|x − y|} dxdy

+\frac{1}{2}( ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx)^{2} − \frac{1}{2} ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx.  (56)

Documenta Mathematica 15 (2010) 285–345

By a density argument we may choose η = θ_{r} the localization function defined in Definition 4.4. Reasoning as on page 541 of [23], we get

0 ≥ α^{−1} ∑_{i=1}^{N} Re(u_{i}η| · |, [η, T (p)]u_{i}) + \frac{1}{2}( ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx)^{2}

−(\frac{1}{2} + \frac{1}{λ} + sup_{|x|=r(1−λ)} |x|Φ_{r(1−λ)}^{HF}(x)) ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx. (57)

It remains to estimate the first term on the right hand side of (57). With the same arguments used in the proof of the IMS formula, it can be rewritten as

α^{−1} ∑_{i=1}^{N} Re(u_{i}η| · |, [η, T (p)]u_{i}) = −α^{−1} ∑_{i=1}^{N}(u_{i}, Lu_{i}),  (58)

where L is the operator defined in (49). Using Lemma 4.3 and since ∥∇η∥_{∞} = ∥∇θ_{r}∥_{∞} ≤ π/ (2λr) we find, with D(λ) defined as in the statement,

α^{−1}| ∑_{i=1}^{N}(u_{i}, Lu_{i})| ≤ 3D(λ)r^{−1}∥χ_{0}ρ^{HF}∥_{1} + D(λ)r^{−1}e^{− \frac{1}{2} α^{−1}rν}∥χ_{−}ρ^{HF}∥_{1}
+CD(λ)rα^{−2}e^{− \frac{1}{2} α^{−1}rν},  (59)

where χ_{0}, χ_{−} and C are as defined in the statement of Lemma 4.3. Hence combining (57) with (59), using the definition of χ_{0} and that ∥χ_{−}ρ^{HF}∥_{1} ≤ N we have

0 ≥ −3D(λ)r^{−1} ∫_{r(1−ν)<|x|<r \frac{1+ν}{1−λ}} ρ^{HF}(x) dx − D(λ)r^{−1}e^{− \frac{1}{2} α^{−1}rν}N

−CD(λ)rα^{−2}e^{− \frac{1}{2} α^{−1}rν} + \frac{1}{2}( ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx)^{2}

−(\frac{1}{2} + \frac{1}{λ} + sup_{|x|=r(1−λ)} |x|Φ_{r(1−λ)}^{HF}(x)) ∫_{ℝ^{3}} ρ^{HF}(x)η^{2}(x)dx.

The claim follows using that x^{2} − Bx − C ≤ 0 implies x ≤ B + \sqrt{C}.

Proof of Lemma 4.1. We proceed as in [23, page 551]. The first estimate follows directly from the equality

∫_{|x|<r}(ρ^{HF}(x) − ρ^{TF}(x)) dx = \frac{1}{4π} r ∫_{S^{2}}(Φ_{r}^{HF}(rω) − Φ_{r}^{TF}(rω)) dω,

and (46). To prove (48) we use Lemma 4.6. We first notice that for 0 < β < γ and γ such that rγ ≤ R

∫_{rβ<|y|<rγ} ρ^{HF}(y) dy ≤ | ∫_{|y|<rγ}(ρ^{HF}(y) − ρ^{TF}(y)) dy|

+| ∫_{|y|<rβ}(ρ^{HF}(y) − ρ^{TF}(y)) dy| + ∫_{|y|>rβ} ρ^{TF}(y) dy

≤ Cr^{−3}β^{−3}(1 + σr^{ε′}).  (60)

Here we used (47) and that by the TF-equation and (13)

∫_{|y|>rβ} ρ^{TF}(y) dy ≤ \frac{3^{4}2π^{2}}{q^{2}} β^{−3}r^{−3}.

Since ∫_{|x|>r} ρ^{HF} ≤ ∫_{|x|>2r/3} ρ^{HF} to prove the claim we estimate this second integral. By Lemma 4.6 with r replaced by r/2, λ = \frac{1}{4} and ν = \frac{1}{2} we get

∫_{|x|>2r/3} ρ^{HF}(x)dx ≤ 9 + \frac{3}{4}r sup_{|x|=3r/8} Φ_{3r/8}^{HF}(x) + R^{\frac{1}{2}} ,

with R defined as in the statement of Lemma 4.6. By (46) and Corollary 1.14 we find

sup_{|x|=3r/8} Φ_{3r/8}^{HF}(x) ≤ Cσr^{−4+ε′} + sup_{|x|=3r/8} Φ_{3r/8}^{TF}(x) ≤ C(1 + σr^{ε′})r^{−4}.

Moreover, from (60) with β = 1/4 and γ = 1, since N < 2Z + 1 and the boundness of ℝ^{+} ∋ x ↦ x^{p}e^{−x} for all p > 0, we find

R ≤ C(r^{−4}(1 + σr^{ε′}) + r^{−1}).

The claim follows directly.

4.2 Separating the inside from the outside

We consider the exterior part of the minimizer, i.e. the density matrix

γ_{r}^{HF} := θ_{r}γ^{HF}θ_{r},  (61)

with θ_{r} as defined in Definition 4.4. This density matrix almost minimizes a new energy functional where there is no exchange term. Indeed sufficiently far away from the nucleus the electrons are far apart and hence their mutual interaction is small. We define an auxiliary energy functional on A (see (8)) given by

E^{A}(γ) := Tr[(α^{−1}T (p) − Φ_{r}^{HF})γ] + D(ρ_{γ}).  (62)

Theorem 4.7. Let r > 0 and λ, ν ∈ (0, 1). Let χ_{r}^{+} denote the characteristic function of ℝ^{3} \ B_{r}(0). The density matrix γ_{r}^{HF} defined in (61) satisfies

E^{A}(γ_{r}^{HF}) ≤ {E^{A}(γ) : γ ∈ A, supp(ρ_{γ}) ⊂ ℝ^{3} \ B_{r}(0), ∥ρ_{γ}∥_{1} ≤ ∥ρ^{HF}χ_{r}∥_{1}} + R,

where

R = (\frac{π}{2λ} + \frac{C}{λ^{2}} r^{−1})r^{−1} ∫_{r(1−λ)(1−ν)≤|x|} ρ^{HF}(x) dx + c′α^{−2}(1 + αr^{−2})e^{− \frac{1}{2} α^{−1}rd}

+Ex(γ_{r}^{HF}) + C ∫_{r(1−λ)≤|x|≤ \frac{r}{1−λ}} [(Φ_{r(1−λ)}^{HF}(x))^{\frac{5}{2}} + α^{3}(Φ_{r(1−λ)}^{HF}(x))^{4}] dx,

and c′, d are positive constants depending only on ν and λ.

Proof. We proceed as in [23, pages 532-6]. The first step of the proof is a localization. Once again we have to treat carefully the localization error coming from the kinetic energy. This is the main difference with [23]. For completeness we repeat the main ideas of the reasoning. We consider the following partition of unity of ℝ^{3}: 1 = θ_{r}^{2}(x) + θ_{0}^{2}(x) + θ_{−}^{2}(x) with θ_{r} defined as in Definition 4.4 and

θ_{0}(x) := (θ_{r(1−λ)}^{2}(x) − θ_{r}^{2}(x))^{\frac{1}{2}} and θ_{−}(x) := (1 − θ_{r(1−λ)}^{2}(x))^{\frac{1}{2}} .

Associated to this partition of unity we define

γ_{0}^{HF} := θ_{0}γ^{HF}θ_{0} and γ_{−}^{HF} := θ_{−}γ^{HF}θ_{−}.

We prove the claim by showing that for all density matrices γ ∈ A such that supp(ρ_{γ}) ⊂ ℝ^{3} \ B_{r}(0) and ∥ρ_{γ}∥_{1} ≤ ∥ρ^{HF}χ_{r}^{+}∥_{1} it holds that

E^{A}(γ_{r}^{HF}) + E^{HF}(γ_{−}^{HF}) − R ≤ E^{HF}(γ^{HF}) ≤ E^{A}(γ) + E^{HF}(γ_{−}^{HF}). (63)

The proof of the upper bound in (63) is as in [23, page 533]. To prove the lower bound as a first step we localize. By Theorem 2.1 we find

α^{−1} Tr[T (p)γ^{HF}] = α^{−1} Tr[T (p)(γ_{r}^{HF} + γ_{0}^{HF} + γ_{−}^{HF})]

−α^{−1} ∑_{i=1}^{N}(u_{i}, (L_{r} + L_{0} + L_{−})u_{i}),

where L_{r}, L_{0} and L_{−} are defined as the L_{i}’s in (14). We first estimate the error term. The procedure is similar to the one used in the proof of Lemma 4.3. We introduce three cut-off functions: χ_{−} be the characteristic function of B_{r(1−λ)(1−ν)}(0), χ_{r} the characteristic function of ℝ^{3} \ B_{r \frac{1+ν}{1−λ}} (0) and χ_{0} defined by χ_{0}(x) = 1 − χ_{r}(x) − χ_{−}(x) for all x ∈ ℝ^{3}. Notice that χ_{−} and χ_{r} are the characteristic functions of sets where θ_{−}, θ_{0} and θ_{r} are constants. For k ∈ {−, 0, r} we have the following splitting

L_{k} = χ_{−}L_{k}(χ_{0} + χ_{r}) + (χ_{0} + χ_{r})L_{k}χ_{−} + χ_{r}L_{k}χ_{0} + χ_{0}L_{k}χ_{r} + χ_{0}L_{k}χ_{0},

and proceeding as in the proof of Lemma 4.3 with ε_{1,k}, ε_{2,k} to be chosen we find

(f, L_{k}f) ≤ ε_{1,k}∥χ_{−}f∥_{2}^{2} + ε_{1,k}^{−1}(f, Q_{1}f) + ε_{2,k}∥χ_{0}f∥_{2}^{2} + ε_{2,k}^{−1}(f, Q_{2}f)
+\frac{3α}{2}∥∇θ_{k}∥_{∞}^{2}∥χ_{0}f∥_{2}^{2}.

with operators Q_{1} and Q_{2} being positive semi-definite operators with

Tr[Q_{1}] ≤ \frac{(16)^{2}}{3π^{2}} \frac{(1−λ)^{2}(1−ν)^{3}}{ν} ∥∇θ_{k}∥_{∞}^{4}r^{2}e^{−α^{−1}rν(1−λ)}
Tr[Q_{2}] ≤ \frac{(16)^{2}}{3π^{2}} \frac{1}{ν(1−λ)^{2}} ∥∇θ_{k}∥_{∞}^{4}r^{2}e^{−α^{−1}r \frac{ν}{1−λ}} .

Choosing then

ε_{2,k} = \frac{3α}{2}∥∇θ_{k}∥_{∞}^{2} and ε_{1,k} = α∥∇θ_{k}∥_{∞}^{2}e^{− \frac{1}{2} α^{−1}rν(1−λ)},

since (∥∇θ_{r}∥_{∞}^{2} + ∥∇θ_{0}∥_{∞}^{2} + ∥∇θ_{−}∥_{∞}^{2}) ≤ 3π^{2}/(4λ^{2})r^{−2} and ∥ρ^{HF}χ_{−}∥_{1} ≤ N we get

α^{−1} ∑_{i=1}^{N}(u_{i}, (L_{r} + L_{0} + L_{−})u_{i}) ≤ \frac{3π^{2}}{4λ^{2}} r^{−2}∥ρ^{HF}χ_{0}∥_{1} + \frac{3π^{2}}{4λ^{2}} r^{−2}e^{− \frac{1}{2} α^{−1}rν(1−λ)}N
+ cα^{−2}e^{− \frac{1}{2} α^{−1}rν(1−λ)}.

Here c is a constant that depends only on ν and λ. Hence from (64), the inequality above and since N ≤ 2Z + 1 we find

E^{HF}(γ^{HF}) ≥ Tr [(α^{−1}T (p) − \frac{Z}{| · |})(γ_{r}^{HF} + γ_{0}^{HF} + γ_{−}^{HF})] + D(γ^{HF})

−Ex(γ^{HF}) − \frac{3π^{2}}{4λ^{2}} r^{−2}∥ρ^{HF}χ_{0}∥_{1} − c′α^{−2}(1 + αr^{−2})e^{− \frac{1}{2} α^{−1}rd}.

The constants c′, d depend only on λ and ν. Proceeding as in [23] we get

E^{HF}(γ^{HF}) ≥ E^{HF}(γ_{−}^{HF}) + E^{A}(γ_{r}^{HF}) − Ex(γ_{r}^{HF}) − c′α^{−2}(1 + αr^{−2})e^{− \frac{1}{2} α^{−1}rd}
+ Tr [(α^{−1}T (p) − Φ_{r(1−λ)}^{HF}(·))γ_{0}^{HF}]
−(\frac{π}{2λ} + \frac{3π^{2}}{4λ^{2}} r^{−1})r^{−1} ∫_{|x|≥r(1−λ)(1−ν)} ρ^{HF}(x) dx.

The claim follows using Theorem 2.5.

4.3 Comparing with an Outside Thomas Fermi

At this point we introduce an “Outside Thomas Fermi”: a TF-energy functional whose minimizer approximates the HF-density at a certain distance from the nucleus.

Let r > 0 such that

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ σ|x|^{−4+ε′},  (64)

for all |x| ≤ r for some σ > 0 and ε′ > 0. Let V_{r} be the potential defined by

V_{r}(x) = χ_{r}^{+}(x)Φ_{r}^{HF}(x) = \{ \begin{matrix}0 & if |x| < r, \\ Φ_{r}^{HF}(x) & if |x| ≥ r. \end{matrix}  (65)

Here and in the following χ_{r}^{+}(x) := 1 − χ_{r}(x), x ∈ ℝ^{3}, where χ_{r} is the characteristic function of the ball of radius r centered at 0. Notice that V_{r} ∈ L^{\frac{5}{2}} (ℝ^{3}) + L^{∞}(ℝ^{3}) with

inf\{∥W ∥_{∞} : V_{r} − W ∈ L^{\frac{5}{2}} (ℝ^{3})\} = 0.

Let E_{r}^{OTF} be the TF-functional E_{V_{r}}^{TF} corresponding to the potential V_{r} defined in (65). Let ρ_{r}^{OTF} be the unique minimizer of E_{r}^{OTF} under the condition

∫_{ℝ^{3}} ρ(x)dx ≤ ∫_{|y|≥r} ρ^{HF}(y)dy,

(see Theorem 1.9). Then ρ_{r}^{OTF} is solution to the OTF-equation

\frac{1}{2} (\frac{6π^{2}}{q} )^{\frac{2}{3}} (ρ_{r}^{OTF})^{\frac{2}{3}} = [φ_{r}^{OTF} − μ_{r}^{OTF}]_{+} ,  (66)

where

φ_{r}^{OTF}(x) = V_{r}(x) − ∫_{ℝ^{3}} \frac{ρ_{r}^{OTF}(y)}{|x − y|}dy,

is the OTF-mean field potential and μ_{r}^{OTF} is the corresponding chemical potential. From (66) (and μ_{r}^{OTF} ≥ 0) we see that the support of ρ_{r}^{OTF} is contained in ℝ^{3} \ B_{r}(0). In the intermediary zone instead of comparing directly Φ_{|x|}^{HF} and Φ_{|x|}^{TF} we compare first the HF-density with the OTF-density and then the OTF-density with the TF-density. When comparing the TF and OTF there is no difference with the non-relativistic case and for brevity we refer for the proofs to [23]. We start by studying the behavior of the minimizer and mean field potential of the OTF. The proof of the following bounds is in [23, page 557-558] in the case q = 2 and it can be directly generalised to the other values of q.

Lemma 4.8 ([23, Lem.12.1]). For all y ∈ ℝ^{3} we have

φ^{TF}(y) ≤ 3^{4}2^{−1}q^{−2}π^{2}|y|^{−4} and ρ^{TF}(y) ≤ 3^{5}2^{−1}q^{−2}π|y|^{−6}.

Let β_{0} be as defined in Theorem 1.12, then for all |y| ≥ β_{0}Z^{− \frac{1}{3}} we have

φ^{TF}(y) ≥ C|y|^{−4} and ρ^{TF}(y) ≥ C|y|^{−6}.

With r, σ, ε′ such that (64) holds and σr^{ε′} ≤ 1 we have for all |y| ≥ r

ρ_{r}^{OTF}(y) ≤ Cr^{−6} and φ_{r}^{OTF}(y) ≤ |V_{r}(y)| ≤ Cr^{−4}.

Lemma 4.9 ([23, Lem.12.2]). With r, σ, ε′ such that (64) holds for all |x| ≤ r we have

∫_{|y|≥r}(ρ^{TF}(y) − ρ^{HF}(y))dy ≤ σr^{−3+ε′}.

For x ∈ ℝ^{3} with |x| > r we may write

Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x) = A_{1}(r, x) + A_{2}(r, x) + A_{3}(r, x),  (67)

where

A_{1}(r, x) = φ_{r}^{OTF}(x) − φ^{TF}(x),
A_{2}(r, x) = ∫_{|y|>|x|} \frac{ρ_{r}^{OTF}(y) − ρ^{TF}(y)}{|x − y|} dy

and

A_{3}(r, x) = ∫_{r<|y|<|x|} \frac{ρ_{r}^{OTF}(y) − ρ^{HF}(y)}{|x − y|} dy.

4.3.1 Estimate on A_{1} and A_{2}

Lemma 4.10 ([23, Lem.12.4]). Let N ≥ Z. Given ε′, σ > 0 there exists a constant D > 0 such that for all r with β_{0}Z^{− \frac{1}{3}} ≤ r ≤ D for which (64) holds for all |x| ≤ r, then μ_{r}^{OTF} = 0 and

\frac{3^{4}π^{2}}{2q^{2}} |x|^{−4}(1 + ar^{ζ}|x|^{−ζ})^{−2} ≤ φ_{r}^{OTF}(x) ≤ \frac{3^{4}π^{2}}{2q^{2}} |x|^{−4}(1 + Ar^{ζ}|x|^{−ζ}) for |x| > r,

where a, A are universal constants and ζ = (−7 + \sqrt{73})/2.

Lemma 4.11 ([23, Lem.12.5]). Let N ≥ Z. Given ε′, σ > 0 there exists a constant D > 0 depending only on ε′, σ such that for all r with β_{0}Z^{− \frac{1}{3}} ≤ r ≤ D for which (64) holds for |x| ≤ r, then for all |x| ≥ r

|A_{1}(r, x)| ≤ C|x|^{−4−ζ}r^{ζ} and |A_{2}(r, x)| ≤ C|x|^{−4−ζ}r^{ζ},

with ζ = (−7 + \sqrt{73})/2 and C a universal constant.

The proof of the previous lemmas is in [23, p. 558-564].

4.3.2 Estimate on ∥χ_{r}^{+}ρ^{HF} − ρ_{r}^{OTF}∥_{C}

Lemma 4.12. Let G_{α} be the function defined in Theorem 2.3 and ρ_{r}^{HF}(x) be the one-particle density of the density matrix γ_{r}^{HF} defined in (61). Let Zα = κ fixed, 0 ≤ κ < 2/π and Z ≥ 1. Given constants ε′, σ > 0 there exists D < \frac{4}{5} such that for all r with β_{0}Z^{− \frac{1}{3}} ≤ r ≤ D for which (64) holds for |x| ≤ r, it follows that

α^{−1} ∫_{ℝ^{3}} G_{α}(ρ_{r}^{HF}(x)) dx ≤ α^{−1} Tr[T (p)γ_{r}^{HF}]
≤ 2R + Cr^{−7} + Cr^{−4} ∫_{ℝ^{3}} ρ_{r}^{HF}(x) dx,

with C a universal positive constant and R as defined in Theorem 4.7.

Proof. The first inequality follows directly from Theorem 2.3. To prove the second inequality we proceed as in Lemma 3.1. In this case we are interested only in the exterior part of the minimizer. Hence, instead of considering the HF-energy functional we consider the auxiliary functional E^{A}, defined in (62), applied to the “exterior part of the minimizer”γ_{r}^{HF}. Splitting the kinetic energy in two terms we find

E^{A}(γ_{r}^{HF}) ≥ \frac{1}{2}α^{−1} Tr[T (p)γ_{r}^{HF}] + D(ρ_{r}^{HF}) + \frac{1}{2} Tr[(α^{−1}T (p) − 2Φ_{r}^{HF})γ_{r}^{HF}]. (68)

Since Φ_{r}^{HF}(x) is harmonic for |x| > r and going to zero at infinity

Φ_{r}^{HF}(x) ≤ \frac{r}{|x|} sup_{|y|=r} Φ_{r}^{HF}(y) for |x| > r.

Hence, since supp(ρ_{r}^{HF}) ⊂ ℝ^{3} \ B_{r}(0) we find

Tr[(α^{−1}T (p) − 2Φ_{r}^{HF})γ_{r}^{HF}] ≥ Tr[(α^{−1}T (p) − \frac{2r}{| · |} sup_{|y|=r} Φ_{r}^{HF}(y))γ_{r}^{HF}] = ….

Adding and subtracting 2D(ρ, ρ_{r}^{HF}) for ρ ∈ L^{1}(ℝ^{3}) ∩ L^{\frac{5}{3}} (ℝ^{3}), ρ ≥ 0, to be chosen

… = Tr[(α^{−1}T (p) − V_{ρ})γ_{r}^{HF}] − ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{ρ_{r}^{HF}(x)ρ(y)}{|x − y|} dxdy. (69)

where for simplicity of notation here and in the following V_{ρ} is defined as V_{ρ}(x) := \frac{2r}{|x|} sup_{|y|=r} Φ_{r}^{HF}(y) − ρ ∗ \frac{1}{|x|}. From (69), (68) and the definition of the Coulomb norm and scalar product (Definition 2.7) we find

E^{A}(γ_{r}^{HF}) ≥ \frac{1}{2}α^{−1} Tr[T (p)γ_{r}^{HF}] + \frac{1}{2}D(ρ_{r}^{HF}) + \frac{1}{2}∥ρ_{r}^{HF} − ρ∥_{C}^{2}
−\frac{1}{2}D(ρ) + \frac{1}{2} Tr[(α^{−1}T (p) − V_{ρ})γ_{r}^{HF}]  (70)
≥ \frac{1}{2}α^{−1} Tr[T (p)γ_{r}^{HF}] + \frac{1}{2} ∑_{i=1}^{N}(θ_{r}u_{i}, (α^{−1}T (p) − V_{ρ})θ_{r}u_{i}) − \frac{1}{2}D(ρ),

denoting by u_{i} the HF-orbitals. We now choose ρ as the minimizer of the TF-energy functional of a neutral atom with Coulomb potential and nuclear charge 2r sup_{|y|=r} Φ_{r}^{HF}(y). Then V_{ρ} is the corresponding TF-mean field potential and we see that the last two terms on the right hand side of (70) are like the ones in the claim of Proposition B.2. The only difference is due to the presence of the localization function θ_{r}. We now prove that these terms give the TF-energy modulo lower order terms. The method is the same as that of Proposition B.2. We repeat the main steps since in this case the scaling depends on r. Notice that since r > β_{0}Z^{− \frac{1}{3}} the contribution is coming only from the “outer zone”.

Let g ∈ C_{0}^{∞}(ℝ^{3}) be spherically symmetric, normalized in L^{2}(ℝ^{3}) and with support in B_{1}(0). Let us define g_{r}(x) := r^{−3}g(xr^{−2}) and ψ_{r} := g_{r}^{2}. Since V_{ρ} is sub-harmonic on |x| > 0, we see from the support properties of ψ_{r} and θ_{r} that

∑_{i=1}^{N}(θ_{r}u_{i}, (α^{−1}T (p) − V_{ρ})θ_{r}u_{i}) ≥ ∑_{i=1}^{N}(θ_{r}u_{i}, (α^{−1}T (p) − V_{ρ} ∗ ψ_{r})θ_{r}u_{i}) = ….

For p, q ∈ ℝ^{3} we define the coherent states g_{r}^{p,q}(x) := g_{r}(x − q)e^{ip·x}. By the formulas (B16) and (B17) with L_{q} the operator defined in the equation below (B17) we get

… = \frac{1}{(2π)^{3}} α^{−1} ∫_{ℝ^{3}} ∫_{ℝ^{3}} dpdq (T (p) − αV_{ρ}(q)) ∑_{i=1}^{N} ∑_{j=1}^{q} |(θ_{r}u_{i}^{j}, g_{r}^{p,q})|^{2}

− α^{−1} ∑_{i=1}^{N} ∫_{ℝ^{3}} ∫_{ℝ^{3}} dxdq \overline{(θ_{r}u_{i})}(x)(L_{q}θ_{r}u_{i})(x) ,  (71)

where u_{i}^{j} denotes the j-th spin component of the orbital u_{i}. By the choice of the function g_{r} and with the same arguments that led to (B19) in the appendix we find

α^{−1} ∑_{i=1}^{N} ∫_{ℝ^{3}} ∫_{ℝ^{3}} dxdq \overline{(θ_{r}u_{i})}(x)(L_{q}θ_{r}u_{i})(x)

≤ 3 ∑_{i=1}^{N} ∥θ_{r}u_{i}∥_{2}^{2}∥∇g_{r}∥_{∞}^{2}V ol(supp(g_{r})) ≤ Cr^{−4}∥ρ_{r}^{HF}∥_{1}.  (72)

In the first term on the right hand side of (71) the integrand is zero if |q| < \frac{1}{4}r^{2} since in this case supp(θ_{r}) ∩ supp(g_{r}^{q,p}) = ∅ (by the choice D < 4/5). To estimate it further from below we consider only the negative part of the integrand

\frac{1}{(2π)^{3}} α^{−1} ∫_{ℝ^{3}} ∫_{ℝ^{3}} dpdq (T (p) − αV_{ρ}(q)) ∑_{i=1}^{N} ∑_{j=1}^{q} |(θ_{r}u_{i}^{j}, g_{r}^{p,q})|^{2}

≥ \frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}|q|> \frac{1}{4} r^{2} \\ T (p)≤αV_{ρ}(q) \end{matrix} dpdq (T (p) − αV_{ρ}(q)) ,  (73)

where we have used that 0 ≤ ∑_{i=1}^{N} |(θ_{r}u_{i}^{j}, g_{r}^{p,q})|^{2} ≤ 1 (Bessel’s inequality). We split the domain of integration in p as follows

{p ∈ ℝ^{3} : T (p) ≤ αV_{ρ}(q)} = Σ_{1} ∪ Σ_{2}

with Σ_{1}, Σ_{2} disjoint and Σ_{1} = \{p ∈ ℝ^{3} : \frac{1}{2}|p|^{2} ≤ V_{ρ}(q)\}. We treat these two contributions separately. We have

α^{−1} ∫∫\begin{matrix}|q|> \frac{1}{4} r^{2} \\ p∈Σ_{2} \end{matrix} dpdq (T (p) − αV_{ρ}(q)) ≥ − ∫∫\begin{matrix}|q|> \frac{1}{4} r^{2} \\ p∈Σ_{2} \end{matrix} dpdq [V_{ρ}(q)]_{+} = …

and computing the integral, using that (1 + x)^{\frac{3}{2}} ≤ 1 + \frac{3}{2}x + \frac{3}{8}x^{2}

⋯ ≥ −C ∫_{|q|> \frac{1}{4} r^{2}} dq (α^{2}[V_{ρ}(q)]+^{\frac{7}{2}} + α^{4}[V_{ρ}(q)]+^{\frac{9}{2}}) ≥ −Cα^{2}r^{− \frac{23}{2}} − Cα^{4}r^{− \frac{33}{2}} .

(74)

In the last step we used that [V_{ρ}(q)]_{+} ≤ 2\frac{r}{|q|} sup_{|x|=r} Φ_{r}^{HF}(x) and that by the hypothesis and Corollary 1.14

r sup_{|x|=r} Φ_{r}^{HF}(x) ≤ Cr^{−3},  (75)

choosing D such that σr^{ε′} ≤ 1. Since T (p) ≥ \frac{1}{2}α|p|^{2} − \frac{1}{8}α^{3}|p|^{4} we find

α^{−1} ∫∫\begin{matrix}|q|> \frac{1}{4} r^{2} \\ p∈Σ_{1} \end{matrix} dpdq (T (p) − αV_{ρ}(q))

≥ ∫∫\begin{matrix}|q|> \frac{1}{4} r^{2} \\ \frac{1}{2} |p|^{2}≤V_{ρ}(q) \end{matrix} dpdq (\frac{1}{2}|p|^{2} − V_{ρ}(q)) − \frac{1}{8}α^{2} ∫∫\begin{matrix}|q|> \frac{1}{4} r^{2} \\ \frac{1}{2} |p|^{2}≤V_{ρ}(q) \end{matrix} dpdq |p|^{4}. (76)

Computing the last integral we find

α^{2} ∫∫\begin{matrix}|q|> \frac{1}{4} r^{2} \\ \frac{1}{2} |p|^{2}≤V_{ρ}(q) \end{matrix} dpdq |p|^{4} ≤ Cα^{2}r^{−1}(2r sup_{|x|=r} Φ_{r}^{HF}(x))^{\frac{7}{2}} ≤ Cα^{2}r^{− \frac{23}{2}} . (77)

While for the first term on the right hand side of (76), computing the integral with respect to p, we get

∫∫\begin{matrix}|q|> \frac{1}{4} r^{2} \\ \frac{1}{2} |p|^{2}≤V_{ρ}(q) \end{matrix} dpdq (\frac{1}{2}|p|^{2} − V_{ρ}(q)) = −4π \frac{2^{\frac{5}{2}}}{15} ∫_{|q|> \frac{1}{4} r^{2}} dq [V_{ρ}(q)]+^{\frac{5}{2}}.

Hence collecting together (71), (72), (73) (74), (77) and the inequality above we find

Tr[(α^{−1}T (p)−V_{ρ})γ_{r}^{HF}] ≥ −\frac{2^{\frac{3}{2}} q}{15π^{2}} ∫_{ℝ^{3}} dx [V_{ρ}(x)]+^{\frac{5}{2}} −Cr^{−4}∥ρ_{r}^{HF}∥_{1} −Cr^{− \frac{11}{2}} = ….

since β_{0}Z^{− \frac{1}{3}} ≤ r implies β_{0}α^{\frac{1}{3}} ≤ κ^{\frac{1}{3}} r. From the TF-equation that ρ satisfies it follows that

… = \frac{3}{10}(\frac{6π^{2}}{q})^{\frac{2}{3}} ∫_{ℝ^{3}} dx ρ(x)^{\frac{5}{3}} − ∫_{ℝ^{3}} ρ(x)V_{ρ}(x) dx − Cr^{−4}∥ρ_{r}^{HF}∥_{1} − Cr^{− \frac{11}{2}}
= E^{TF}(ρ) + D(ρ) − Cr^{−4}∥ρ_{r}^{HF}∥_{1} − Cr^{− \frac{11}{2}} .

Hence from (70) and the inequality above we get using (12) and (75)

E^{A}(γ_{r}^{HF}) ≥ \frac{1}{2}α^{−1} Tr[T (p)γ^{HF}] − Cr^{−7} − Cr^{−4}∥ρ_{r}^{HF}∥_{1}.

The claim follows since E^{A}(γ_{r}^{HF}) ≤ R by the result of Theorem 4.7 considering as a trial density matrix γ ≡ 0.

Lemma 4.13. Let N′ ∈ ℕ and Zα = κ be fixed, 0 ≤ κ < 2/π and Z ≥ 1. Let e_{j} be the first N′ negative eigenvalues of the operator α^{−1}T (p) − φ_{r}^{OTF} acting on functions with support on {x ∈ ℝ^{3} : |x| ≥ r}. Given constants ε′, σ > 0 there exists D < 4/5 such that for all r with β_{0}Z^{− \frac{1}{3}} ≤ r ≤ D for which (64) holds for |x| ≤ r, for all μ ∈ (0, 1) and s < r we have

∑_{j=1}^{N′} e_{j} ≥ −(\frac{2}{1−μ})^{\frac{3}{2}} \frac{1}{15π^{2}} ∫_{|q|>r}[φ_{r}^{OTF}(q)]+^{\frac{5}{2}} dq − Cr^{−8}sμ^{− \frac{3}{2}} − Cμ^{−3}r^{−5}s
−C(1 − μ)^{− \frac{7}{2}} r^{−5} − C(1 − μ)s^{−2}N′,

with C a positive constant.

Proof. Let f_{j} be the eigenfunctions (normalized in L^{2}(ℝ^{3}, ℂ^{q})) corresponding to the eigenvalues e_{j}, j = 1, .., N′. Let g ∈ C_{0}^{∞}(ℝ^{3}) with support in B_{1}(0) and define g_{s}(x) = s^{− \frac{3}{2}} g(x/s) for a positive parameter s, s < r. We then write for μ ∈ (0, 1)

∑_{j=1}^{N′} e_{j} = ∑_{j=1}^{N′}(f_{j}, (α^{−1}T (p) − φ_{r}^{OTF})f_{j}) = B_{1} + B_{2},

where

B_{1} = ∑_{j=1}^{N′}(f_{j}, ((1 − μ)α^{−1}T (p) − φ_{r}^{OTF} ∗ g_{s}^{2})f_{j}),

B_{2} = ∑_{j=1}^{N′}(f_{j}, (μα^{−1}T (p) − φ_{r}^{OTF} + φ_{r}^{OTF} ∗ g_{s}^{2})f_{j}).

We estimate these two terms separately. Considering for p, q ∈ ℝ^{3} the coherent states g_{s}^{p,q}(x) := e^{ip.x}g_{s}(x − q) using (B16) and (B17), we find

B_{1} = \frac{1}{(2π)^{3}} ∫∫ ((1 − μ)α^{−1}T (p) − φ_{r}^{OTF}(q)) ∑_{j=1}^{N} |(f_{j}, g_{s}^{p,q})|^{2} dqdp

− (1 − μ)α^{−1} ∑_{j=1}^{N′} ∫_{ℝ^{3}} ∫_{ℝ^{3}} dxdq\overline{f_{j}(x)}(L_{q}f_{j})(x) .  (78)

Estimating the error term as done in (B32) and previous inequalities we get

(1 − μ)α^{−1} ∑_{j=1}^{N′} ∫_{ℝ^{3}} ∫_{ℝ^{3}} dxdq\overline{f_{j}(x)}(L_{q}f_{j})(x) ≤ C(1 − μ)s^{−2}N′.

Since we are interested in an estimate from below and φ_{r}^{OTF}(q) ≤ 0 for |q| < r, from (78) we find

B_{1} ≥ \frac{1}{(2π)^{3}} ∫∫_{|q|>r}((1 − μ)α^{−1}T (p) − φ_{r}^{OTF}(q)) ∑_{j=1}^{N} |(f_{j}, g_{s}^{p,q})|^{2} dqdp
−C(1 − μ)s^{−2}N′.  (79)

We estimate now the first term on the right hand side of (79). Considering only the negative part of the integrand and since ∑_{j=1}^{N′} |(f_{j}, g_{s}^{p,q})|^{2} ≤ 1 we get

\frac{1}{(2π)^{3}} ∫∫_{|q|>r}((1 − μ)α^{−1}T (p) − φ_{r}^{OTF}(q)) ∑_{j=1}^{N′} |(f_{j}, g_{s}^{p,q})| dqdp

≥ \frac{1}{(2π)^{3}} ∫∫\begin{matrix}|q|>r, \\ (1−μ)α^{−1}T (p)≤φ_{r}^{OTF}(q) \end{matrix}((1 − μ)α^{−1}T (p) − φ_{r}^{OTF}(q)) dpdq.

Now we split the domain of integration in p as follows

{p ∈ ℝ^{3} : α^{−1}(1 − μ)T (p) ≤ φ_{r}^{OTF}(q)} = Σ_{1} ∪ Σ_{2},

with Σ_{1}, Σ_{2} disjoint and Σ_{1} = {p ∈ ℝ^{3} : (1 − μ)|p|^{2}/2 ≤ φ_{r}^{OTF}(q)}. We treat these two contributions separately. Then

\frac{1}{(2π)^{3}} ∫∫\begin{matrix}|q|>r, \\ p∈Σ_{2} \end{matrix}((1 − μ)α^{−1}T (p) − φ_{r}^{OTF}(q))dpdq

≥ −\frac{1}{(2π)^{3}} ∫∫\begin{matrix}|q|>r, \\ p∈Σ_{2} \end{matrix}[φ_{r}^{OTF}(q)]_{+}dpdq = …

and since in the domain of integration

\frac{2}{1−μ}[φ_{r}^{OTF}(q)]_{+} ≤ |p|^{2} ≤ \frac{2}{1−μ}[φ_{r}^{OTF}(q)]_{+}(1 + \frac{1}{2(1−μ)}α^{2}[φ_{r}^{OTF}(q)]_{+})

we get

… ≥ −\frac{C}{(1−μ)^{\frac{5}{2}}} α^{2} ∫_{|q|>r} dq ([φ_{r}^{OTF}(q)]+^{\frac{7}{2}} + \frac{α^{2}}{8(1−μ)}[φ_{r}^{OTF}(q)]+^{\frac{9}{2}})
≥ −\frac{C}{(1−μ)^{\frac{5}{2}}} α^{2}(r^{−11} + \frac{α^{2}}{1−μ}r^{−15}),  (80)

using Lemma 4.10 in the last step. Since \sqrt{1 + t^{2}} ≥ 1 + (1/2)t^{2} − (1/8)t^{4}, we get

\frac{1}{(2π)^{3}} ∫∫\begin{matrix}|q|>r, \\ p∈Σ_{1} \end{matrix}((1 − μ)α^{−1}T (p) − φ_{r}^{OTF}(q))dpdq

≥ \frac{1}{(2π)^{3}} ∫∫\begin{matrix}|q|>r, \\ p∈Σ_{1} \end{matrix}((1 − μ)\frac{1}{2}|p|^{2} − φ_{r}^{OTF}(q) − \frac{1}{8}(1 − μ)α^{2}|p|^{4})dpdq.

The last term gives by Lemma 4.10

α^{2} ∫∫\begin{matrix}|q|>r \\ p∈Σ_{1} \end{matrix} dpdq |p|^{4} = α^{2} \frac{4π}{7} ∫_{|q|>r} dq (\frac{2}{1−μ})^{\frac{7}{2}} [φ_{r}^{OTF}(q)]+^{\frac{7}{2}} ≤ Cα^{2}(\frac{2}{1−μ})^{\frac{7}{2}} r^{−11}.

(81)

While for the other terms computing the integral with respect to p, we get

\frac{1}{(2π)^{3}} ∫∫\begin{matrix}|q|>r, \\ p∈Σ_{1} \end{matrix}((1 − μ)\frac{1}{2}|p|^{2} − φ_{r}^{OTF}(q))dpdq

= −(\frac{2}{1−μ})^{\frac{3}{2}} \frac{1}{15π^{2}} ∫_{|q|>r} dq [φ_{r}^{OTF}(q)]+^{\frac{5}{2}}.  (82)

For the term B_{2} using Theorem 2.5 and Remark 2.6 we find

B_{2} ≥ −Cq(μ^{− \frac{3}{2}} ∥[φ_{r}^{OTF} − φ_{r}^{OTF} ∗ g_{s}^{2}]_{+}∥\frac{5}{2} \frac{5}{2} + α^{3}μ^{−3}∥[φ_{r}^{OTF} − φ_{r}^{OTF} ∗ g_{s}^{2}]_{+}∥_{4}^{4}).

From the choice of g_{s} it follows that φ_{r}^{OTF} − φ_{r}^{OTF} ∗ g_{s}^{2} ≤ V_{r} − V_{r} ∗ g_{s}^{2} and the term V_{r} − V_{r} ∗ g_{s}^{2} is non-zero only for r − s ≤ |x| ≤ r + s. Hence by Lemma 4.8 and since s < r

∥[φ_{r}^{OTF} −φ_{r}^{OTF} ∗g_{s}^{2}]_{+}∥\frac{5}{2} \frac{5}{2} ≤ ∫_{r−s≤|x|≤r+s}[V_{r}(x)−V_{r} ∗g^{2}(x)]+^{\frac{5}{2}}dx ≤ Cr^{−8}s, (83)

and similarly ∥[φ_{r}^{OTF} − φ_{r}^{OTF} ∗ g_{s}^{2}]_{+}∥_{4}^{4} ≤ Cr^{−14}s. The claim follows from (79), (80), (81), (82) and (83) using that β_{0}α^{\frac{1}{3}} ≤ κ^{\frac{1}{3}} r.

Lemma 4.14. Let G_{α} be the function defined in Theorem 2.3 and ρ_{r}^{HF}(x) the one-particle density of the density matrix γ_{r}^{HF} defined in (61). Let Zα = κ be fixed, 0 ≤ κ < 2/π and Z ≥ 1. There exists α_{0} > 0 such that given ε′, σ > 0 there exists D < 1/4 such that for all α ≤ α_{0} and r with β_{0}Z^{− \frac{1}{3}} ≤ r ≤ D for which (64) holds for |x| ≤ r, we have

∥χ_{r}^{+}ρ^{HF} − ρ_{r}^{OTF}∥_{C} ≤ Cr^{− \frac{7}{2} + \frac{1}{6}}  and
α^{−1} ∫_{ℝ^{3}} G_{α}(χ_{r}^{+}ρ^{HF}(x))dx ≤ Cr^{−7}, α^{−1} Tr[T (p)γ_{r}^{HF}] ≤ Cr^{−7}, (84)

with C a universal positive constant.

Proof. The idea of the proof is the same as that of Lemma 3.1. In this case we are interested only in the exterior part of the minimizer. Hence, instead of considering the HF-energy functional we estimate from above and below the auxiliary one E^{A}, defined in (62), applied on the “exterior part of the minimizer”γ_{r}^{HF}. Step I. Estimate from above on E^{A}(γ_{r}^{HF}). Let us consider γ the density matrix that acts identically on the spin components and on each as

γ^{j} = \frac{1}{(2π)^{3}} ∫∫_{\frac{1}{2} |p|^{2}≤φ_{r}^{OTF}(q)} Π_{p,q} dpdq, where j ∈ {1, …, q} is the spin index, Π_{p,q} is the projection onto the space spanned by h_{s}^{p,q}(x) = h_{s}(x − q)e^{ip.x} where h_{s} is the ground state for the Dirichlet Laplacian on the ball of radius s for 0 < s < r. By the OTF-equation (66) and since μ_{r}^{OTF} = 0 (see Lemma 4.10) we see that ρ_{γ}(x) = ρ_{r}^{OTF} ∗|h_{s}|^{2}(x). Moreover, by Lemma 4.10

Tr[−\frac{1}{2}Δγ] = \frac{3}{10}(\frac{6π^{2}}{q})^{\frac{2}{3}} ∫_{ℝ^{3}}(ρ_{r}^{OTF}(x))^{\frac{5}{3}} dx + Cs^{−2}r^{−3}.  (85)

Since [Φ_{r}^{HF}]_{+} ∈ Lloc^{\frac{5}{2}}(ℝ^{3}), by [23, Lemma 8.5] for λ′ ∈ (0, 1) we may find γ̃ such that supp(ργ̃) ⊂ {x : |x| ≥ r}, ργ̃(x) ≤ ρ_{γ}(x) for x ∈ ℝ^{3} and

Tr[(−\frac{1}{2}Δ − Φ_{r}^{HF})γ̃] ≤ Tr[(−\frac{1}{2}Δ − χ_{r}^{+}Φ_{r}^{HF})γ] + L_{1} ∫_{|x|≤ \frac{r}{1−λ′}} [V_{r}(x)]+^{\frac{5}{2}} dx

+\frac{1}{2}(\frac{π}{2λ′r})^{2} ∫_{|x|≤ \frac{r}{1−λ′}} ρ_{γ}(x) dx.  (86)

Since ∫ ργ̃ ≤ ∫ ρ_{γ} = ∫ ρ_{r}^{OTF} ≤ ∫ χ_{r}^{+}ρ^{HF} we may choose γ̃ as a trial density matrix in Theorem 4.7 and we find for λ, ν to be chosen

E^{A}(γ_{r}^{HF}) ≤ E^{A}(γ̃) + R ≤ Tr[(−\frac{1}{2}Δ − Φ_{r}^{HF})γ̃] + R + D(ργ̃),

since α^{−1}T (p) ≤ \frac{1}{2}|p|^{2}. Notice that R depends on λ and ν. From (86) it follows that

E^{A}(γ_{r}^{HF}) ≤ Tr[(−\frac{1}{2}Δ − χ_{r}^{+}Φ_{r}^{HF})γ] + L_{1} ∫_{|x|≤ \frac{r}{1−λ′}} [V_{r}(x)]+^{\frac{5}{2}} dx

+\frac{1}{2}(\frac{π}{2λ′r})^{2} ∫_{|x|≤ \frac{r}{1−λ′}} ρ_{γ}(x) dx + R + D(ργ̃).  (87)

From the OTF-equation (66) and Lemma 4.10 we get

∫_{|x|≤ \frac{r}{1−λ′}} ρ_{γ}(x) dx ≤ ∫_{|x|≤ \frac{2−λ′}{1−λ′} r} ρ_{r}^{OTF}(x) dx ≤ Cr^{−3}.

While since V_{r}(y) ≤ Cr^{−4} (Lemma 4.8) and is non-zero only for |y| > r

∫_{|x|≤ \frac{r}{1−λ′}} [V_{r}(x)]+^{\frac{5}{2}} dx ≤ Cr^{−7} \frac{λ′}{(1−λ′)^{3}} .

Hence, from (85) and (87) and the inequalities above we find choosing λ′ = r^{\frac{2}{3}}

E^{A}(γ_{r}^{HF}) ≤ \frac{3}{10}(\frac{6π^{2}}{q})^{\frac{2}{3}} ∫_{ℝ^{3}}(ρ_{r}^{OTF}(x))^{\frac{5}{3}} dx − ∫_{ℝ^{3}} V_{r}(x)ρ_{γ}(x) dx + Cs^{−2}r^{−3}

+Cr^{−7+ \frac{2}{3}} + R + D(ργ̃) = ….

Here we used that λ′ ≤ 1/2 which follows by the bound on D. Since ργ̃ ≤ ρ_{γ}, D(ργ̃) ≤ D(ρ_{γ}). Moreover by Newton’s Theorem D(ρ_{γ}) ≤ D(ρ_{r}^{OTF}). Hence we get

… ≤ E^{OTF}(ρ_{r}^{OTF}) + ∫_{ℝ^{3}} V_{r}(x)(ρ_{r}^{OTF}(x) − ρ_{γ}(x)) dx + Cs^{−2}r^{−3}
+Cr^{−7+ \frac{2}{3}} + R.  (88)

We study now the second term on the right hand side of (88). Since ρ_{γ} = ρ^{OTF} ∗ |h_{s}|^{2}, rewriting

∫_{ℝ^{3}} V_{r}(x)(ρ_{r}^{OTF}(x) − ρ_{γ}(x)) dx = ∫_{ℝ^{3}} ρ_{r}^{OTF}(x)(V_{r}(x) − V_{r} ∗ |h_{s}|^{2}(x)) dx.

Since s < r, V_{r} is harmonic on |x| > r and ρ_{r}^{OTF} vanishes for |x| < r one sees that the integrand on the right hand side of the equation above is non-zero only for r < |x| < r + s. Hence by Lemma 4.8

∫_{ℝ^{3}} V_{r}(x)(ρ_{r}^{OTF}(x) − ρ_{γ}(x)) dx ≤ ∫_{r<|x|<r+s} ρ_{r}^{OTF}(x)V_{r}(x) dx ≤ Cr^{−8}s.

Choosing s = r^{\frac{5}{3}} we find from (88) that

E^{A}(γ_{r}^{HF}) ≤ E^{OTF}(ρ_{r}^{OTF}) + Cr^{−7+ \frac{2}{3}} + R.  (89)

It remains to estimate R. From Lemma 4.1, choosing λ, ν ≤ 1/2 and D such that σr^{ε′} ≤ 1 we find

(\frac{π}{2λr} + \frac{C}{λ^{2}r^{2}} ) ∫_{|x|≥r(1−λ)(1−ν)} ρ^{HF}(x) dx ≤ Cr^{−5}λ^{−2}.

By Lemma 4.8, (65) and since λ ≤ 1/2 we get

∫_{r(1−λ)≤|x|≤ \frac{r}{1−λ}} (Φ_{r(1−λ)}^{HF}(x))^{\frac{5}{2}} dx ≤ Cr^{−7}λ,

and similarly

α^{3} ∫_{r(1−λ)≤|x|≤ \frac{r}{1−λ}} (Φ_{r(1−λ)}^{HF}(x))^{4} dx ≤ Cr^{−4}λ,

since r ≥ β_{0}Z^{− \frac{1}{3}} implies αr^{−3} ≤ β_{0}^{−3}κ. Hence from the expression of R and the boundness of t^{p}e^{−t} for t > 0, we find

R ≤ Ex(γ_{r}^{HF}) + Cr^{−5}λ^{−2} + Cr^{−7}λ.  (90)

We estimate now the exchange term. By the exchange inequality ([15] or [23, Th.6.4]) and proceeding as in (27) we find by Lemma 4.1 and Lemma 4.12

Ex(γ_{r}^{HF}) ≤ C ∫_{ℝ^{3}} G_{α}(ρ_{r}^{HF}(x))dx + Cr^{− \frac{3}{2}} (α^{−1} ∫_{ℝ^{3}} G_{α}(ρ_{r}^{HF}(x))dx)^{\frac{1}{2}}
≤ CαR + Cαr^{−7} + Cr^{− \frac{3}{2}} (R + r^{−7})^{\frac{1}{2}} .

Hence choosing α_{0} such that 1 − Cα ≥ 1/2 for all α ≤ α_{0} we get from the inequality above and (90)

\frac{1}{2}R ≤ Cr^{− \frac{3}{2}} (R + r^{−7})^{\frac{1}{2}} + Cr^{−5}λ^{−2} + Cr^{−7}λ ,

that gives

R ≤ C(r^{−5}λ^{−2} + λr^{−7}) .  (91)

The second two inequalities in (84) follow from the estimate above and lemmas 4.1 and 4.12 choosing λ = 1/2 and replacing r with r/2. Step II. Estimate from below on E^{A}(γ_{r}^{HF}). Adding and subtracting D(ρ_{r}^{OTF}) and Tr[ρ_{r}^{OTF} ∗ \frac{1}{|·|}γ_{r}^{HF}] we write

E^{A}(γ_{r}^{HF}) = Tr[(α^{−1}T (p) − φ_{r}^{OTF})γ_{r}^{HF}] + ∥ρ_{r}^{OTF} − ρ_{r}^{HF}∥_{C}^{2} − D(ρ_{r}^{OTF}), (92)

using that V_{r} = Φ_{r}^{HF} on the support of ρ_{r}^{HF}. The first term on the right hand side of (92) is estimated from below by the sum of the first N′ eigenvalues of the operator α^{−1}T (p) − φ_{r}^{OTF} acting on the functions with support on {x : |x| ≥ r}. Here N′ denotes the smallest integer bigger than Tr[γ_{r}^{HF}]. Hence by Lemma 4.13 we find for μ ∈ (0, 1) and s < r

E^{A}(γ_{r}^{HF}) ≥ −(\frac{2}{1−μ})^{\frac{3}{2}} \frac{q}{15π^{2}} ∫_{ℝ^{3}}[φ_{r}^{OTF}(q)]+^{\frac{5}{2}} dq − Cr^{−8}sμ^{− \frac{3}{2}} − Cμ^{−3}r^{−5}s
−C(1 − μ)^{− \frac{7}{2}} r^{−5} − C(1 − μ)s^{−2}( ∫_{ℝ^{3}} ρ_{r}^{HF}(x) dx + 1)
+∥ρ_{r}^{OTF} − ρ_{r}^{HF}∥_{C}^{2} − D(ρ_{r}^{OTF}) = …,

Notice the factor q due to spin. Choosing D such that σr^{ε′} ≤ 1, by lemmas 4.1 and 4.10 we find

∫_{ℝ^{3}} ρ_{r}^{HF}(x) dx ≤ Cr^{−3} and ∫_{ℝ^{3}}[φ_{r}^{OTF}(q)]+^{\frac{5}{2}} dq ≤ Cr^{−7}.

Hence considering μ ≤ 1/2

… ≥ −2^{\frac{3}{2}} \frac{q}{15π^{2}} ∫_{ℝ^{3}}[φ_{r}^{OTF}(q)]+^{\frac{5}{2}} dq − Cr^{−7} − Cr^{−8}sμ^{− \frac{3}{2}} − Cμ^{−3}r^{−5}s
−Cs^{−2}r^{−3} + ∥ρ_{r}^{OTF} − ρ_{r}^{HF}∥_{C}^{2} − D(ρ_{r}^{OTF}) = ….

By the OTF-equation (66) and since ρ_{r}^{OTF} has support where φ_{r}^{OTF} ≥ 0 we find

⋯ = E^{OTF}(ρ_{r}^{OTF}) − Cr^{−7+ \frac{1}{3}} + ∥ρ_{r}^{OTF} − ρ_{r}^{HF}∥_{C}^{2}, choosing μ = \frac{1}{2}r^{− \frac{2}{5}} s^{\frac{2}{5}} and s = r^{\frac{11}{6}} . Hence combining the inequality above with (89) and (91) we find

∥ρ_{r}^{OTF} − ρ_{r}^{HF}∥_{C}^{2} ≤ Cr^{−7+ \frac{1}{3}} + C(r^{−5}λ^{−2} + λr^{−7}).  (93)

We study now ∥χ_{r}^{+}ρ^{HF} − ρ_{r}^{HF}∥_{C}. By Hardy-Littlewood-Sobolev inequality we find

∥χ_{r}^{+}ρ^{HF} − ρ_{r}^{HF}∥_{C} ≤ C∥χ_{r}^{+}ρ^{HF} − ρ_{r}^{HF}∥_{\frac{6}{5}} ≤ C( ∫_{r≤|x|≤ \frac{r}{1−λ}} ρ^{HF}(x)^{\frac{6}{5}} dx)^{\frac{5}{6}} . (94)

To estimate the last term in (94) we are going to use the second estimate in (84) that we have just proved. With Σ defined as in (26) we find by Hölder’s inequality

∫_{r≤|x|≤ \frac{r}{1−λ}} ρ^{HF}(x)^{\frac{6}{5}} dx ≤ ( ∫\begin{matrix}r≤|x|, \\ x∈Σ \end{matrix} ρ^{HF}(x)^{\frac{4}{3}} dx)^{\frac{9}{10}} ( ∫_{r≤|x|≤ \frac{r}{1−λ}} 1 dx)^{\frac{1}{10}}

+( ∫\begin{matrix}r≤|x|, \\ x∈ℝ^{3}\Σ \end{matrix} ρ^{HF}(x)^{\frac{5}{3}} dx)^{\frac{18}{25}} ( ∫_{r≤|x|≤ \frac{r}{1−λ}} 1 dx)^{\frac{7}{25}}

≤ Cr^{− \frac{33}{10}} λ^{\frac{1}{10}} + Cr^{− \frac{21}{5}} λ^{\frac{7}{25}} .

From the estimate above, (93) and (94) it then follows

∥χ_{r}^{+}ρ^{HF} − ρ_{r}^{OTF}∥_{C} ≤ ∥χ_{r}^{+}ρ^{HF} − ρ_{r}^{HF}∥_{C} + ∥ρ_{r}^{HF} − ρ_{r}^{OTF}∥_{C}
≤ Cr^{− \frac{7}{2} + \frac{1}{6}} + C(r^{−5}λ^{−2} + λr^{−7})^{\frac{1}{2}} + C(r^{− \frac{11}{4}} λ^{\frac{1}{12}} + r^{− \frac{7}{2}} λ^{\frac{7}{30}} ),

that gives the claim choosing λ = r^{\frac{5}{7}}

4.3.3 Estimate on A_{3}

Lemma 4.15. Let G_{α} be the function defined in Theorem 2.3. Let Zα = κ fixed, 0 ≤ κ < 2/π and Z ≥ 1. There exists α_{0} > 0 such that given ε′, σ > 0 there exists a constant D < 1/4 depending only on ε′ and σ such that if (64) holds for all |x| ≤ D, then for all α ≤ α_{0}

α^{−1} ∫_{|y|≥|x|} G_{α}(ρ^{HF}(y))dy ≤ C|x|^{−7} for all |x| ≤ D,

with C a universal positive constant.

Proof. If |x| < β_{0}Z^{− \frac{1}{3}} we find by Lemma 3.1

α^{−1} ∫_{|y|>|x|} G_{α}(ρ^{HF}(y))dy ≤ α^{−1} ∫_{ℝ^{3}} G_{α}(ρ^{HF}(y))dy ≤ CZ^{\frac{7}{3}} ≤ C|x|^{−7}.

While if D ≥ |x| ≥ β_{0}Z^{− \frac{1}{3}} the claim follows from the second estimate in (84).

Lemma 4.16. Let Zα = κ fixed, 0 ≤ κ < 2/π, Z ≥ 1 and 0 < μ < \frac{1}{109}. There exists α_{0} such that given ε′, σ > 0 there exists a constant D < 1/4 depending only on ε′ and σ such that for all α ≤ α_{0} and for all r with β_{0}Z^{− \frac{1−μ}{3}} ≤ r ≤ D for which (64) holds for |x| ≤ r, then for all x with |x| ≥ r

|A_{3}(r, x)| ≤ C(\frac{|x|}{r})^{\frac{1}{12}} r^{−4+ \frac{3μ}{1−μ}} ,

with C > 0 a universal constant.

Proof. We proceed similarly as in Theorem 3.3. By the formula for A_{3}, Proposition 2.8 and Lemma 4.14 we get

|A_{3}(r, x)| ≤ ∫_{A(|x|,k)} χ_{r}^{+}(y)\frac{|ρ_{r}^{OTF}(y) − ρ^{HF}(y)|}{|x − y|} dy + Ck^{−1}|x|^{− \frac{1}{2}} r^{− \frac{7}{2} + \frac{1}{6}} . (95)

By Hölder’s inequality, Lemma 4.10, the OTF-equation (66) and (33) we find

∫_{A(|x|,k)} \frac{ρ_{r}^{OTF}(y)}{|x − y|} dy ≤ Cr^{− \frac{21}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} .  (96)

Once again, to estimate ∫_{A(|x|,k)} \frac{χ_{r}^{+}(y)ρ^{HF}(y)}{|x−y|} dy we have to proceed differently

than in [23, Lem.12.7] since ρ^{HF} is not in L^{\frac{5}{3}} (ℝ^{3}). We consider the following splitting

∫_{A(|x|,k)} χ_{r}^{+}(y)\frac{ρ^{HF}(y)}{|x − y|} dy = ∫\begin{matrix}A(|x|,k) \\ |x−y|>R,|y|>r \end{matrix} \frac{ρ^{HF}(y)}{|x − y|} dy + ∫\begin{matrix}|y|>r, \\ |x−y|<R \end{matrix} \frac{ρ^{HF}(y)}{|x − y|} dy,

(97)

for R > 0 to be chosen. By Hölder’s inequality, Theorem 2.3, Remark 2.4, (33) and Lemma 4.14 we get

∫\begin{matrix}A(|x|,k) \\ |x−y|>R,|y|>r \end{matrix} \frac{ρ^{HF}(y)}{|x − y|} dy ≤ Cα^{\frac{3}{4}} R^{− \frac{3}{8}} |x|^{\frac{1}{8}} k^{\frac{1}{8}} r^{− \frac{21}{4}} + Cr^{− \frac{21}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} . (98)

It remains to study the second term on the right hand side of (97). Let ν ∈ ℝ^{+} be such that να ≤ 2/π. We consider the density matrix γ_{r/2}^{HF} defined in (61) with λ = 1/2. From Theorem 2.10 it follows that for x such that |x| ≥ r

Tr[(α^{−1}T (p) − \frac{ν}{| · −x|}χ_{B_{R}(x)}(·))γ_{r/2}^{HF}] ≥ −C(ν^{\frac{5}{2}} R^{\frac{1}{2}} + ν^{4}α^{2}).

Hence we find

ν ∫_{|y−x|<R} χ_{r}^{+}(y)\frac{ρ^{HF}(y)}{|x − y|}dy ≤ ν ∫_{|y−x|<R} \frac{ρ_{r/2}^{HF}(y)}{|x − y|}dy
≤ Tr[α^{−1}T (p)γ_{r/2}^{HF}] + C(ν^{\frac{5}{2}} R^{\frac{1}{2}} + ν^{4}α^{2})

and by Lemma 4.14

∫_{|y−x|<R} χ_{r}^{+}(y)\frac{ρ^{HF}(y)}{|x − y|}dy ≤ Cν^{−1}r^{−7} + C(ν^{\frac{3}{2}} R^{\frac{1}{2}} + ν^{3}α^{2}).  (99)

Hence from (95), (96), (98) and (99) it follows that

|A_{3}(r, x)| ≤ Cν^{−1}r^{−7} + C(ν^{\frac{3}{2}} R^{\frac{1}{2}} + ν^{3}α^{2}) + Cα^{\frac{3}{4}} R^{− \frac{3}{8}} |x|^{\frac{1}{8}} k^{\frac{1}{8}} r^{− \frac{21}{4}}
+Cr^{− \frac{21}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} + Ck^{−1}|x|^{− \frac{1}{2}} r^{− \frac{7}{2} + \frac{1}{6}} .

So choosing ν = 1/2(β_{0}r^{−1})^{\frac{3}{1−μ}} (that gives να < 2/π), k such that r^{− \frac{21}{5}} |x|^{\frac{1}{5}} k^{\frac{1}{5}} = k^{−1}|x|^{− \frac{1}{2}} r^{− \frac{7}{2} + \frac{1}{6}} , i.e. k = |x|^{− \frac{7}{12}} r^{\frac{13}{18}} and R such that α^{\frac{3}{4}} R^{− \frac{3}{8}} |x|^{\frac{1}{8} \frac{5}{12}} r^{− \frac{21}{4} + \frac{1}{8} \frac{13}{18}} = r^{−4− \frac{1}{18}} |x|^{\frac{1}{12}} , i.e. R = α^{2}|x|^{− \frac{1}{12}} r^{− \frac{5}{18}}

|A_{3}(r, x)| ≤ C(r^{−4+ \frac{3μ}{1−μ}} + |x|^{− \frac{1}{24}} r^{− \frac{5}{36} − \frac{9}{2(1−μ)}} α + r^{− \frac{9}{1−μ}} α^{2} + |x|^{\frac{1}{12}} r^{−4− \frac{1}{18}} ).

Finally since r^{−1}α^{\frac{1−μ}{3}} ≤ β_{0}^{−1}κ^{\frac{1−μ}{3}} , the claim follows for |x| ≥ r and μ < 1/(109).

4.4 The intermediate region

Here we prove the main estimate in Theorem 1.17 up to a fixed distance independent of Z.

Lemma 4.17 (Iterative step). Let Zα = κ fixed with 0 ≤ κ < 2/π. Consider μ = \frac{1}{11} \frac{1}{49} and assume N ≥ Z ≥ 1. Then there exists α_{0} > 0 such that for all δ, ε′, σ > 0 with δ < δ_{0}, where δ_{0} is some universal constant, there exists constants ε_{2}, C′_{ϕ} > 0 depending only on δ and a constant D = D(ε′, σ) > 0 depending only on ε′, σ with the following property. For all α ≤ α_{0} and R_{0} < D satisfying that β_{0}Z^{− \frac{1−μ}{3}} ≤ R_{0}^{1+δ} and that (64) holds for all |x| ≤ R_{0}, there exists R′_{0} > R_{0} such that

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C′_{Φ}|x|^{−4+ε_{2}}

for all x with R_{0} < |x| < R′_{0}.

Proof. Let D > 0 depending on σ, ε′ be the smaller of the values of D occurring in Lemma 4.11 and Lemma 4.16. Given δ > 0. We consider R_{0} < D satisfying β_{0}Z^{− \frac{1−μ}{3}} ≤ R_{0}^{1+δ} and such that (64) holds for all |x| ≤ R_{0}. Set R′_{0} = R_{0}^{1−δ} and r = R_{0}^{1+δ}. Then we have β_{0}Z^{− \frac{1}{3}} ≤ β_{0}Z^{− \frac{1−μ}{3}} ≤ r ≤ R_{0} < D we can therefore apply Lemma 4.11 and Lemma 4.16. From (67) we obtain that for all |x| ≥ r and all α ≤ α_{0}

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C|x|^{−4−ζ}r^{ζ} + C(\frac{|x|}{r})^{\frac{1}{12}} r^{−4+ \frac{3μ}{1−μ}} .

Since for R_{0} < |x| < R′_{0} we have

|x|^{\frac{2δ}{1−δ}} ≤ \frac{r}{|x|} ≤ |x|^{δ}

and thus

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C|x|^{−4+δζ} + C|x|^{−4+3 \frac{μ}{1−μ}} |x|^{− \frac{δ}{1−δ} (8+ \frac{1}{6} − \frac{6μ}{1−μ})}.

Hence choosing δ_{0} sufficiently small there are C′_{Φ} and ε_{2} such that the claim holds.

Lemma 4.18. Let Zα = κ fixed with 0 ≤ κ < 2/π. Assume N ≥ Z ≥ 1. Then there exist universal constants α_{0}, ε ∈ (0, 4) and D, C_{Φ} > 0, D < 1/4, such that for all α ≤ α_{0} and x with |x| ≤ D we have

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C_{Φ}|x|^{−4+ε}.

Proof. We fix μ = \frac{1}{11} \frac{1}{49} as in Lemma 4.17. Since μ < \frac{2}{11} \frac{1}{49}, by Theorem 3.3 we know that there exists constants a, b, c > 0 such that for all |x| ≤ βZ^{− \frac{1−μ}{3}}

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(1 + β^{2} + β^{5/2} + β^{b}|x|^{c})β^{2−a}|x|^{−4+a}.  (100)

We first show that we may choose δ small enough such that if we choose R̃^{1+δ} = β_{0}Z^{− \frac{1−μ}{3}} we have for all |x| < R̃ that

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C′′_{Φ}|x|^{−4+ \frac{a}{2}} .  (101)

Let β > 0 be such that (βZ^{− \frac{1−μ}{3}} )^{1+δ} = β_{0}Z^{− \frac{1−μ}{3}} , i.e. β^{1+δ} = β_{0}Z^{δ \frac{1−μ}{3}} . Hence from (100) we find for all |x| ≤ βZ^{− \frac{1−μ}{3}}

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(1 + β^{2} + β^{5/2} + β^{b}|x|^{c})β^{2− \frac{a}{2}} Z^{− \frac{a}{2} \frac{1−μ}{3}} |x|^{−4+ \frac{a}{2}} ,

and by the choice of β (and β_{0} < 1)

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C(1 + Z^{2 \frac{δ}{1+δ} \frac{1−μ}{3}} + Z^{\frac{5}{2} \frac{δ}{1+δ} \frac{1−μ}{3}} + Z^{\frac{δ}{1+δ} \frac{1−μ}{3} (b+c)}Z^{−c \frac{1−μ}{3}} )
Z^{(2− \frac{a}{2} ) \frac{1−μ}{3} \frac{δ}{1+δ}} Z^{− \frac{a}{2} \frac{1−μ}{3}} |x|^{−4+ \frac{a}{2}} .

Hence if δ is small enough we may choose a universal constant C′′_{Φ} such that (101) holds. Let now δ be small enough so that we may apply Lemma 4.17. This give constant ε_{2} and C′_{Φ} (depending only on δ) and for all σ, ε′ > 0 a constant D < 1/4. Now choose σ = max{C′_{Φ}, C′′_{Φ}} and ε′ = min{a/2, ε_{2}}. Now σ, ε′ and D are universal constants. To prove the claim we shall prove that for all |x| ≤ D

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ σ|x|^{−4+ε′}.  (102)

We have to prove that D belongs to the set

M = {0 < R ≤ 1/4 : Inequality (102) holds for all |x| ≤ R}.

We reason by contradiction. If this was not true then D > R_{0} = sup M and in particular R_{0} < 1/4. From (101) and the choice of σ and ε′ it follows that either R̃ > 1/4 or R̃ ∈ M. In the first case then R_{0} = sup M = 1/4 > D that contradicts our hypothesis. On the other hand if R̃ ∈ M, then R_{0}^{1+δ} ≥ R̃^{1+δ} = β_{0}Z^{− \frac{1−μ}{3}} . It then follows from Lemma 4.17 that there exists R′_{0} ∈ M with R′_{0} > R_{0}. This contradicts also our hypothesis.

4.5 The outer zone and proof of Theorem 1.17

The proof of Theorem 1.17 follows directly from Lemma 4.18 and the following result.

Lemma 4.19. Let Zα = κ, 0 ≤ κ < 2/π. Assume N ≥ Z ≥ 1. Let D, ε and C_{Φ} be the constants introduced in Lemma 4.18. Then there exist α_{0} > 0 and a universal constant C_{M} > 0 such that for all α ≤ α_{0} and x with |x| ≥ D we have

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C_{M}.

Proof. Here C_{i}, i = 1, …, 6 denote positive universal constants. We write

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ |Φ_{D}^{HF}(x) − Φ_{D}^{TF}(x)| + ∫_{D<|y|<|x|} \frac{ρ^{TF}(y) + ρ^{HF}(y)}{|x − y|} dy.

(103)

Since Φ_{D}^{HF}(x) − Φ_{D}^{TF}(x) is harmonic for |x| > D and tends to zero at infinity we have by Lemma 4.18

|Φ_{D}^{HF}(x) − Φ_{D}^{TF}(x)| ≤ sup_{|x|=D} |Φ_{D}^{HF}(x) − Φ_{D}^{TF}(x)| ≤ C_{ϕ}D^{−4+ε}.  (104)

For the second term on the right hand side of (103) we write

∫_{D<|y|<|x|} \frac{ρ^{TF}(y) + ρ^{HF}(y)}{|x − y|} dy

≤ ∫\begin{matrix}|x−y|<D/4 \\ |y|>D \end{matrix} \frac{ρ^{TF}(y) + ρ^{HF}(y)}{|x − y|} dy + \frac{4}{D} ∫_{D<|y|}(ρ^{TF}(y) + ρ^{HF}(y)) dy. (105)

By Lemma 4.1, Lemma 4.18, estimate (13) and the TF-equation we find

∫_{D<|y|}(ρ^{TF}(y) + ρ^{HF}(y)) dy ≤ C_{1}(1 + C_{Φ}D^{ε})(1 + D^{−3}) + C_{1}D^{−3}. (106)

It remains to estimate the first term on the right hand side of (105). By Hölder’s inequality, estimate (13) and the TF-equation we get

∫\begin{matrix}|x−y|<D/4 \\ |y|>D \end{matrix} \frac{ρ^{TF}(y)}{|x − y|} dy ≤ C_{2}( ∫_{|y|>D}(ρ^{TF}(y))^{\frac{5}{3}} dy)^{\frac{3}{5}} D^{\frac{1}{5}} ≤ C_{3}D^{−4}. (107)

To estimate the term with the HF-density we use Theorem 2.10. Let γ_{D}^{HF} be the exterior HF-density matrix as defined in (61) with r = D/2 and λ = 1/2. Then by Theorem 2.10 with ν = β_{0}^{3}D^{−3}

α^{−1} Tr[(T (p) − \frac{να}{|x − ·|}χ_{B_{\frac{D}{4}} (x)}(·))γ_{D/2}^{HF}] ≥ −C_{4}(D^{\frac{1}{2}} ν^{\frac{5}{2}} + ν^{4}α^{2}),

and thus

∫_{|x−y|<D/4} \frac{ρ_{D/2}^{HF}(y)}{|x − y|} dy ≤ C_{5}D^{3}α^{−1} Tr[T (p)γ_{D/2}^{HF}] + C_{6}D^{−4},

Here we use that D > 2β_{0}Z^{− \frac{1}{3}} (for α ≤ α_{0}) and D < 1/4. By Lemma 4.14 we conclude

∫_{|x−y|<D/4} χ_{D}^{+}(y)\frac{ρ^{HF}(y)}{|x − y|} dy ≤ ∫_{|x−y|<D/4} \frac{ρ_{D/2}^{HF}(y)}{|x − y|} dy ≤ C_{7}D^{−4}. (108)

The claim follows collecting together formula (103) to formula (108).

5 Proofs of Theorems 1.1, 1.18, 1.19 and 1.20

In this section we always assume the following: Zα = κ with 0 ≤ κ < 2/π and N ≥ Z ≥ 1.

Proof of Theorem 1.1. Assume that a HF-minimizer exists with ∫ ρ^{HF} = N. Let ρ^{TF} be the minimizer of the TF-energy functional of the neutral atom with nuclear charge Z. Then for R > 0 to be chosen

N = ∫_{|x|<R} ρ^{TF}(x) dx+∫_{|x|<R}(ρ^{HF}(x)−ρ^{TF}(x)) dx+∫_{|x|>R} ρ^{HF}(x) dx. (109)

By Theorem 1.17 we know that there exist universal positive constants ε, α_{0}, C_{M} and C_{Φ} such that for all α ≤ α_{0} and x ∈ ℝ^{3}

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ C_{Φ}|x|^{−4+ε} + C_{M}.  (110)

Let Z_{0} be such that Z_{0}α_{0} = κ. Then α ≤ α_{0} corresponds to Z ≥ Z_{0}. Let us choose R such that C_{Φ}R^{−4+ε} = C_{M}. Then from (109), (110) and Lemma 4.1 for all Z ≥ Z_{0} we find

N ≤ ∫_{|x|<R} ρ^{TF}(x) dx + 2C_{Φ}R^{−3+ε} + C(1 + C_{Φ}R^{ε})(R^{−3} + 1) < Z + Q̃.

The claim follows choosing Q = max{Q̃, Z_{0} + 1}.

Proof of Theorem 1.18. Let ρ^{HF} be the density of the HF-minimizer in the neutral case N = Z. We have

| ∫_{|x|>R}(ρ^{HF}(x) − ρ^{TF}(x))dx| = | ∫_{|x|<R}(ρ^{HF}(x) − ρ^{TF}(x))dx|
= |\frac{R}{4π} ∫_{S^{2}} dω(Φ_{R}^{HF}(Rω) − Φ_{R}^{TF}(Rω))|
≤ C_{Φ}R^{−3+ε} + C_{M}R,

where in the last step we have used Theorem 1.17. Notice that for Z sufficiently big α ≤ α_{0} where α_{0} is the constant given in Theorem 1.17. By the TFequation, Theorem 1.12 we then find

3^{4} \frac{2π^{2}}{q^{2}} R^{−3} − C_{Φ}R^{−3+ε} − C_{M}R ≤ ∫_{|x|>R} ρ^{HF}(x)dx
≤ 3^{4} \frac{2π^{2}}{q^{2}} R^{−3} + C_{Φ}R^{−3+ε} + C_{M}R,

from which the claim follows directly by the definition of HF-radius.

Proof of Theorem 1.19. Since E^{HF}(Z − 1, Z) ≥ E^{HF}(Z, Z) the ionization energy is bounded from below by zero. If Z is smaller than a universal constant then we can also bound the ionization energy with a universal constant using Theorem 2.11. It remains to estimate from above the ionization energy when Z is larger than a universal constant. We first construct a density matrix γ such that Tr[γ] ≤ Z − 1. Let θ_{−} := (1 − θ_{r(1−λ)}^{2})^{\frac{1}{2}} for r, λ positive parameters and θ_{r} defined in Definition 4.4. We consider the density matrix γ_{−}^{HF} := θ_{−}γ^{HF}θ_{−} where γ^{HF} is the HF-minimizer in the neutral case. By an opportune choice of r we will then have Tr[γ_{−}^{HF}] ≤ Z − 1. Indeed,

Tr[γ_{−}^{HF}] = ∫_{ℝ^{3}} ρ^{HF}(x) dx − ∫_{ℝ^{3}} θ_{r(1−λ)}^{2}(x)ρ^{HF}(x) dx ≤ Z − ∫_{|x|>r} ρ^{HF}(x) dx.

We now choose λ = \frac{1}{2}. Let R > 0 be such that C_{M} = C_{Φ}R^{−4+ε} where C_{M}, C_{Φ}, ε are the constants in Theorem 1.17. Then R is a universal constant. We consider Z large enough so that β_{0}Z^{− \frac{1}{3}} < R where β_{0} is the constant in Theorem 1.12. This gives that Z has to be larger than some universal constant. For r such that β_{0}Z^{− \frac{1}{3}} < r < R by Theorem 1.17 we find

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ 2C_{Φ}|x|^{−4+ε} for all |x| ≤ r.

Since ∫ ρ^{TF} = ∫ ρ^{HF}, by the choice of r and Lemma 4.1 we get

∫_{|x|>r} ρ^{HF}(x) dx = ∫_{|x|>r} ρ^{TF}(x) dx + ∫_{|x|<r}(ρ^{TF}(x) − ρ^{HF}(x)) dx
≥∫_{|x|>r} ρ^{TF}(x) dx − 2C_{Φ}r^{−3+ε} ≥ Cr^{−3} − 2C_{Φ}r^{−3+ε}. (111)

In the last step we used the TF-equation, Corollary 1.13 and that r > β_{0}Z^{− \frac{1}{3}} . Finally, it follows from (111) by choosing r sufficiently small that ∫_{|x|>r} ρ^{HF} > 1 and hence that Tr[γ_{−}^{HF}] ≤ Z − 1. We may choose r sufficiently small by taking Z large enough. Notice that r can be chosen universally and so Z has to be larger than some universal constant. By the last estimate in the proof of Theorem 4.7 we find

E^{HF}(γ_{−}^{HF}) ≤ E^{HF}(γ^{HF}) − E^{A}(γ_{r}^{HF}) + R,

with R and γ_{r}^{HF} as defined in the statement of Theorem 4.7. Since E^{HF}(γ_{−}^{HF}) ≥ E^{HF}(Z−1, Z) and E^{HF}(γ^{HF}) = E^{HF}(Z, Z) it remains to prove that −E^{A}(γ_{r}^{HF})+ R is bounded from above by some universal constant. Here we use repeteadly that r is a universal constant. By estimate (91) we see that R ≤ Cr^{−7} a universal constant. To estimate from below E^{A}(γ_{r}^{HF}) we first leave out the kinetic energy term and the direct term since these are positive. Moreover, since Φ_{r}^{HF} is harmonic for |x| > r and tends to zero at infinity we see that

Φ_{r}^{HF}(x) ≤ \frac{r}{|x|} sup_{|y|=r} Φ_{r}^{HF}(y) ≤ \frac{r}{|x|} sup_{|y|=r} Φ_{r}^{TF}(y) + \frac{r}{|x|} sup_{|y|=r} |Φ_{r}^{TF}(y) − Φ_{r}^{HF}(y)|,

which is bounded by C′/|x|, C′ a universal constant, by Theorem 1.17 and Corollary 1.14. It then follows that

E^{A}(γ_{r}^{HF}) ≥ − Tr[\frac{C′}{| · |}γ_{r}^{HF}] ≥ −\frac{C′}{r} ∫_{|x|>r} ρ^{HF}(x) dx,

that is bounded from below by a universal constant using Lemma 4.1.

Proof of Theorem 1.20. Let α_{0} be the constant appearing in Theorem 1.17 and Z_{0} be such that α_{0}Z_{0} = κ. The claim follows directly for Z ≤ Z_{0} since both functions are bounded for |x| large, while for |x| small the functions are bounded by a constant times |x|^{−1}. The case Z > Z_{0} corresponds to α < α_{0} and for such values of α we can use the result in Theorem 1.17. We separate the case small x, intermediate x and large x. Once again, comparing with the proof in the non-relativistic case ([23]) we have to do an extra splitting for small x. By the definition of the mean field potential and Proposition 2.8 we find

|φ^{TF}(x) − φ^{HF}(x)| ≤ ∫_{|x−y|<s}(ρ^{TF}(y) + ρ^{HF}(y))(\frac{1}{|x − y|} − \frac{1}{s})

+\frac{\sqrt{2}}{s^{\frac{1}{2}}}∥ρ^{TF} − ρ^{HF}∥_{C}.

Since ρ^{TF} is bounded in L^{\frac{5}{3}} -norm, we find using Hölder’s inequality, Corollary 1.15 and Lemma 3.1 that

|φ^{TF}(x) − φ^{HF}(x)| ≤ ∫_{|x−y|<s} ρ^{HF}(y)(\frac{1}{|x − y|} − \frac{1}{s}) + C(s^{\frac{1}{5}} Z^{\frac{7}{5}} + s^{− \frac{1}{2}} Z^{1+ \frac{3}{22}} ).

(112)

For the integral with the HF-density we need to split the region where the HF-density is bounded in L^{\frac{4}{3}} -norm from the one where it is bounded in L^{\frac{5}{3}} - norm. Proceeding as in the proof of Lemma 3.2 (from (35) to (37) replacing the integrals on A(|x|, k) with integrals on |x − y| < s) using the results of Lemma 3.1 we get with R ∈ (0, s) to be chosen

∫_{|x−y|<s} ρ^{HF}(y)(\frac{1}{|x − y|} − \frac{1}{s}) ≤ C(Z^{\frac{7}{5}} s^{\frac{1}{5}} +R^{− \frac{1}{4}} (αZ^{\frac{7}{3}} )^{\frac{3}{4}} +Z^{\frac{4}{3}} +R^{\frac{1}{2}} Z^{\frac{3}{2}} ). (113)

Recall that Zα = κ is fixed. Choosing s such that Z^{\frac{7}{5}} s^{\frac{1}{5}} = Z^{\frac{4}{3}} (i.e. s = Z^{− \frac{1}{3}} ) and R such that R^{− \frac{1}{4}} Z = R^{\frac{1}{2}} Z^{\frac{3}{2}} (i.e R = Z^{− \frac{2}{3}} ; notice that R < s) we get from (112) and (113)

|φ^{TF}(x) − φ^{HF}(x)| ≤ C(Z^{\frac{4}{3}} + Z^{\frac{7}{6}} ).

The claim follows from this inequality for x ∈ ℝ^{3} such that |x| ≤ β_{0}Z^{− \frac{1+γ}{3}} for γ > 0. We consider γ < \frac{1}{263}. If |x| ≥ β_{0}Z^{− \frac{1+γ}{3}} then proceeding as for very small x and as in the proof of Theorem 3.3 up to inequality (43) we get for t ∈ (\frac{1+γ}{3} , \frac{3}{5}), l > t and R < β_{0}Z^{−l}

|φ^{TF}(x) − φ^{HF}(x)| ≤ C(s^{\frac{1}{5}} Z^{\frac{7}{5}} + s^{− \frac{1}{2}} Z^{1+ \frac{3}{22}} + R^{− \frac{3}{8}} s^{\frac{1}{8}} Z + Z^{\frac{1}{2} (3−t)}).

Here we have also used that Zα is a constant. So choosing s such that s^{\frac{1}{5}} Z^{\frac{7}{5}} = Z^{\frac{1}{2} (3−t)} (i.e. s = Z^{\frac{1}{2} − \frac{5}{2} t}), R such that R^{− \frac{3}{8}} Z^{1+ \frac{1}{16} − \frac{5}{16} t} = Z^{\frac{1}{2} (3−t)} (i.e. R = Z^{− \frac{7}{6} + \frac{1}{2} t}) and optimizing in t (i.e. t = \frac{1}{3} + \frac{4}{3} \frac{1}{77}) we obtain

|φ^{TF}(x) − φ^{HF}(x)| ≤ CZ^{\frac{4}{3} − \frac{2}{3} \frac{1}{77}} .  (114)

Notice that t > \frac{1+γ}{3} , R < s by the choice of t and that R satisfies the condition R < β_{0}Z^{−l}, l > t, for Z sufficiently big. The claim then follows from (114) for x ∈ ℝ^{3} such that |x|^{1+δ} ≤ β_{0}Z^{− \frac{1}{3}} for δ < \frac{1}{153}. We fix δ = \frac{1}{2} \frac{1}{153}. We turn now to study intermediate x. Let D ≤ 1 be such that C_{M} ≤ C_{Φ}D^{−4+ε} with C_{M}, C_{Φ}, ε the constants in Theorem 1.17. Then for all x such that |x| ≤ D

|Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x)| ≤ 2C_{Φ}|x|^{−4+ε}.

Moreover we choose D such that Lemma 4.11 holds. Let x be such that β_{0}Z^{− \frac{1}{3}} ≤ |x|^{1+δ} ≤ D^{\frac{1+δ}{1+μ}} with 0 < μ ≤ δ. We set r = |x|^{1+μ}. Then β_{0}Z^{− \frac{1}{3}} ≤ r ≤ D. We write φ^{TF}(x) − φ^{HF}(x) = φ^{TF}(x) − φ_{r}^{OTF}(x) + φ_{r}^{OTF}(x) − φ^{HF}(x) with φ_{r}^{OTF} the mean field potential of the OTF-problem defined in Subsection 4.3. By the choice of r and D and Lemma 4.11 we get since |x| ≥ r = |x|^{1+μ}

|φ^{TF}(x) − φ_{r}^{OTF}(x)| ≤ C|x|^{−4−ζ}r^{ζ} ,  (115)

for |x| ≥ r with ζ = (7 + \sqrt{73})/2. For the other two terms we see

φ^{HF}(x) − φ_{r}^{OTF}(x) = ∫ \frac{ρ_{r}^{OTF}(y) − χ_{r}^{+}(y)ρ^{HF}(y)}{|x − y|} dy,

Documenta Mathematica 15 (2010) 285–345

and proceeding as for small x with the Coulomb-norm estimate Proposition 2.8, by Lemma 4.14 and inequality (99)

|φ^{HF}(x)−φ_{r}^{OTF}(x)| ≤ C(\frac{s^{\frac{1}{5}}}{r^{\frac{21}{5}}} +\frac{r^{− \frac{7}{2} + \frac{1}{6}}}{s^{\frac{1}{2}}} +R^{− \frac{1}{4}} (αr^{−7})^{\frac{3}{4}} +ν^{−1}r^{−7}+ν^{\frac{3}{2}} R^{\frac{1}{2}} +ν^{3}α^{2}).

Choosing ν = β_{0}^{3}r^{−3 \frac{1+δ}{1+μ}}, so that να ≤ κ < 2/π, s such that s^{\frac{1}{5}} r^{− \frac{21}{5}} = r^{− \frac{7}{2} + \frac{1}{6}} s^{− \frac{1}{2}} (i.e. s = r^{1+ \frac{5}{21}} ), and choosing R such that the two terms where it appears are equal (i.e. R = r^{2+9 \frac{δ−μ}{1+μ}}; notice that R < s) we get

|φ^{HF}(x) − φ_{r}^{OTF}(x)| ≤ C(r^{−4+ \frac{1}{21}} + r^{−4+3 \frac{δ−μ}{1+μ}}),

since αr^{−3 \frac{1+δ}{1+μ}} is bounded and r ≤ 1. Collecting together the inequality above and (115) and using that r = |x|^{1+μ} the claim follows for β_{0}Z^{− \frac{1}{3}} ≤ |x|^{1+δ} ≤ D^{\frac{1+δ}{1+μ}}. We fix μ = δ/2.

It remains to study the case of large x, i.e. |x| ≥ D^{\frac{1+δ}{1+μ}} with D, δ, μ universal constants. For simplicity of notation we fix the universal constant A := D^{\frac{1+δ}{1+μ}}. We first notice that

φ^{HF}(x) − φ^{TF}(x) = Φ_{|x|}^{HF}(x) − Φ_{|x|}^{TF}(x) + ∫_{|y|>|x|} \frac{ρ^{TF}(y) − ρ^{HF}(y)}{|x − y|}dy.

The difference of the first two terms is bounded by a universal constant for |x| ≥ A by the result in Theorem 1.17. To estimate the last integral we split it as follows

∫_{|y|>|x|} \frac{|ρ^{TF}(y) − ρ^{HF}(y)|}{|x − y|}dy ≤ ∫\begin{matrix}|y|>|x| \\ |x−y|<1 \end{matrix} \frac{ρ^{TF}(y)}{|x − y|}dy + ∫\begin{matrix}|y|>|x| \\ |x−y|<1 \end{matrix} \frac{ρ^{HF}(y)}{|x − y|}dy

+ ∫_{|y|>|x|}(ρ^{TF}(y) + ρ^{HF}(y)) dy.

Since |x| ≥ A the third term on the right hand side is bounded by a universal constant by Lemma 4.1 (for ρ^{HF}) and Corollary 1.13 (for ρ^{TF}). We estimate the first term by Hölder’s inequality and Corollary 1.15. We get a bound on the second term proceeding as in (99) (using Theorem 2.10) and choosing ν = \frac{1}{2} and R = 1. We obtain

∫\begin{matrix}|y|>|x| \\ |x−y|<1 \end{matrix} \frac{ρ^{TF}(y) + ρ^{HF}(y)}{|x − y|}dy ≤ C(A^{− \frac{21}{5}} + A^{−7} + α^{2}).

Then there exists a universal contant A′ such that |φ^{HF}(x) − φ^{TF}(x)| ≤ A′ for |x| ≥ A.

Documenta Mathematica 15 (2010) 285–345

A Technical lemmas

Proof of (16) By the definition of the function G_{α} the inequalities in (16) are equivalent to the following ones

\frac{3}{5}t^{4} min\{\frac{2}{5}t, 1\} ≤ g(t) − \frac{8}{3}t^{3} ≤ 2t^{4} min\{\frac{2}{5}t, 1\} for t ≥ 0.  (A1)

As before we use the substitution t = α(ρ/C)^{\frac{1}{3}} . The estimates in (A1) follow directly from the study of the function g separating the cases t < \frac{5}{2} and t ≥ \frac{5}{2}. Proof of Remark 4.2 Using the estimate on K_{2} given in (15) we find

∫∫ \begin{matrix}x ∈ Σ_{r}(β_{1}, β_{2}) \\ y ∈ Σ_{r}(β_{3}, β_{4}) \end{matrix} K_{2}(α^{−1}|x − y|)^{2} dxdy

≤ (16)^{2}α^{4} ∫∫ \begin{matrix}x ∈ Σ_{r}(β_{1}, β_{2}) \\ y ∈ Σ_{r}(β_{3}, β_{4}) \end{matrix} \frac{e^{−α^{−1}|x−y|}}{|x − y|^{4}} dxdy

≤ (16)^{2}α^{4}e^{−α^{−1}r(β_{3}−β_{2})}4π ∫_{r(β_{3}−β_{2})}^{∞} ρ^{−2}dρ ∫_{Σ_{r}(β_{1},β_{2})} dx,

since |x − y| ≥ (β_{3} − β_{2})r. The claim follows computing the two integrals.

A.1 Fourier transform

In the present sub-section we present our notation for the Fourier transform (as in [20]). Given f ∈ L^{2}(ℝ^{3}) we denote its Fourier transform by

f̂(p) = F(f)(p) := \frac{1}{(2π)^{\frac{3}{2}}} ∫_{ℝ^{3}} e^{ip·x}f(x)dx.

Let f, g ∈ L^{2}(ℝ^{3}). The following formulas hold:

1. F(f ∗ g)(p) = (2π)^{\frac{3}{2}} f̂(p)ĝ(p);

2. F(fg)(p) = (2π)^{− \frac{3}{2}} (f̂∗ ĝ)(p);

3. if g(x) = e^{−λ|x|^{2}} then ĝ(p) = (2λ)^{− \frac{3}{2}} e^{−|p|^{2}/(4λ)};

4. |x|^{−α} = π^{\frac{α}{2}} (Γ(\frac{α}{2}))^{−1} ∫_{0}^{+∞} e^{−π|x|^{2}λ}λ^{\frac{α}{2} −1}dλ for 0 < α < n (see [14, page 130]).

Moreover,

F(\frac{f(x)}{|x|})(k) = \frac{1}{2π^{2}} ∫_{ℝ^{3}} \frac{f̂(p)}{|k − p|^{2}} dp.

B Large Z-behavior of the energy

In [21] the author studies the large Z-behavior of the ground state energy for problem (1). In this work we are going to use the same construction in several points (Lemmas 3.1, 4.12, Theorem 3.3, ....) and with, in certain cases, a slightly different Hamiltonian. For convenience we repeat here the main ideas of the proof. We do it as it is needed in the proof of Theorem 3.3 since in this case the proof is more involved. We remark that in our proof we use a localisation less than in [21]. Thanks to Theorem 2.10 and [24, Theorem 2.8] it is sufficient to consider the region near the nuclei and the one far away from the nuclei. There is no need for an intermediate region.

Proposition B.1. Let Zα = κ be fixed with 0 ≤ κ < 2/π and Z ≥ 1. Let us consider P ∈ ℝ^{3}, with |P| ≥ βZ^{− \frac{1+μ}{3}} for β > 0 and μ ∈ (0, 4/5). Let Z ≥ ν > 0 and R > 0 be such that R < βZ^{−l}/4 for some \frac{1+μ}{3} < l. Moreover, let ρ^{TF} denote the minimizer of the TF-energy functional of a neutral atom with nucleus of charge Z. Consider the Hamiltonian

H_{P} := ∑_{i=1}^{N} (α^{−1}T (p_{i}) − \frac{Z}{|x_{i}|} − \frac{ν}{|x_{i} − P|}χ_{B_{R}(P)}(x_{i})) + ∑_{i<j} \frac{1}{|x_{i} − x_{j}|}, (B2)

acting on ∧_{i=1}^{N}L^{2}(ℝ^{3}; ℂ^{q}). Then for all t ∈ (\frac{1+μ}{3}, min\{l, \frac{3}{5}\}) and ψ ∈ ∧_{i=1}^{N}L^{2}(ℝ^{3}), with ∥ψ∥_{2} = 1,

⟨ψ, H_{p}ψ⟩ ≥ E^{TF}(ρ^{TF}) − C(β^{\frac{1}{2}} + β^{−2})Z^{\frac{5}{2} − \frac{1}{2} t},

with C depending only on q and κ.

Proof. Since E^{TF}(ρ^{TF}) = −e_{0}Z^{\frac{7}{3}} (see (12)) to prove the claim it is sufficient to show that the TF-energy gives a lower bound to the quantum energy modulo lower order terms. In the proof we first reduce to a one-particle operator. Then we localize the energy separating the contribution from the regions near the nuclei from the contribution from the region far away from them. Finally we study the contribution of each of these terms. The main contribution to the energy is given by the region far away from the nuclei. This region will give the TF-energy. In the following, s = (3 − t)/4 (t < s < 2/3). In the proof C denotes a generic positive constant depending only on q and κ. Reduction to a one-particle problem. We are going to estimate from below H_{P} by a one-particle operator. This allows us to consider only Slater determinants when minimizing the energy. Let g ∈ C_{0}^{∞}(ℝ^{3}), g ≥ 0 be spherically symmetric with supp(g) ⊂ B_{1}(0) and such that ∥g∥_{2} = 1. Starting from these g we define Φ_{s}(x) := (β/(8Z^{s}))^{−3}g^{2}(8Z^{s}x/β). Then by Newton’s theorem

∑_{i<j} \frac{1}{|x_{i} − x_{j}|} ≥ ∑_{i<j} ∫∫ \frac{Φ_{s}(x_{i} − x)Φ_{s}(x_{j} − y)}{|x − y|} dxdy =

= \frac{1}{2} ∑_{i,j=1}^{N} ∫∫ \frac{Φ_{s}(x_{i} − x)Φ_{s}(x_{j} − y)}{|x − y|} dxdy − \frac{N}{2} ∫∫ \frac{Φ_{s}(x)Φ_{s}(y)}{|x − y|} dxdy = …

and introducing ρ ∈ L^{1}(ℝ^{3}) ∩ L^{\frac{5}{3}} (ℝ^{3}), ρ ≥ 0, to be chosen

… = \frac{1}{2} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{(∑_{i=1}^{N} Φ_{s}(x_{i} − x) − ρ(x))(∑_{j=1}^{N} Φ_{s}(x_{j} − y) − ρ(y))}{|x − y|} dxdy

+ ∑_{i=1}^{N} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{Φ_{s}(x_{i} − x)ρ(y)}{|x − y|} dxdy − D(ρ) − \frac{N}{2} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{Φ_{s}(x)Φ_{s}(y)}{|x − y|} dxdy

≥ ∑_{i=1}^{N} ρ ∗ Φ_{s} ∗ \frac{1}{|x_{i}|} − D(ρ) − C∥g^{2}∥2_{\frac{6}{5}} Nβ^{−1}Z^{s}.  (B3)

In the last inequality we use that the first term on the left hand side of (B3) is non-negative and that

∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{Φ_{s}(x)Φ_{s}(y)}{|x − y|} dxdy = Cβ^{−1}Z^{s} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{g^{2}(x)g^{2}(y)}{|x − y|} dxdy
≤ Cβ^{−1}Z^{s}∥g^{2}∥_{6/5}^{2},

by definition of Φ_{s} and Hardy-Littlewood-Sobolev’s inequality. Hence

H_{P} ≥ ∑_{i=1}^{N} (α^{−1}T (p_{i}) − \frac{Z}{|x_{i}|} − \frac{ν}{|x_{i} − P|}χ_{B_{R}(P)}(x_{i}) + ρ ∗ Φ_{s} ∗ \frac{1}{|x_{i}|})
−D(ρ) − C∥g^{2}∥2_{\frac{6}{5}} Nβ^{−1}Z^{s}.  (B4)

Choice of the localization. The localization will be given by the following functions χ_{1}, χ_{2} ∈ C_{0}^{∞}(ℝ^{3}):

χ_{1}(x) := \{ \begin{matrix}1 & if |x| < \frac{1}{4}βZ^{−t}, \\ 0 & if |x| > \frac{1}{2}βZ^{−t}, \end{matrix} χ_{2}(x) := \{ \begin{matrix}1 & if |x − P| < \frac{1}{4}βZ^{−t}, \\ 0 & if |x − P| > \frac{1}{2}βZ^{−t} \end{matrix} (B5)

and χ_{3} ∈ C^{∞}(ℝ^{3}) such that ∑_{i=1}^{3} χ_{i}^{2}(x) = 1 for all x ∈ ℝ^{3}. Moreover we ask that

∥∇χ_{1}∥_{∞}, ∥∇χ_{2}∥_{∞}, ∥∇χ_{3}∥_{∞} ≤ 2^{5}β^{−1}Z^{t}.  (B6)

Here t is the parameter given in the statement of the proposition. Notice that by the assumptions on R and P the functions defined above give a well defined partition of unity of ℝ^{3}. Moreover, B_{R}(P) is a subset of {x ∈ ℝ^{3} : χ_{2}(x) = 1}.

The localization in the energy expectation. We insert now the localization in the energy expectation. As already observed, since we reduced the operator to a one-particle operator in the energy expectation it is sufficient to consider Slater determinants: i.e. ψ = u_{1} ∧ ⋯ ∧ u_{N} with {u_{i}}_{i=1}^{N} orthonormal functions in L^{2}(ℝ^{3}, ℂ^{q}). We may assume that u_{i} ∈ H^{\frac{1}{2}} (ℝ^{3}, ℂ^{q}) for i = 1, …, N. From (B4) and Theorem 2.1 we find with ψ = u_{1} ∧ ⋯ ∧ u_{N}

⟨ψ, H_{P}ψ⟩ ≥ ∑_{i=1}^{N} ∑_{j=1}^{3}(χ_{j}u_{i}, hχ_{j}u_{i}) − D(ρ) − C∥g^{2}∥2_{\frac{6}{5}} Nβ^{−1}Z^{s}

−α^{−1} ∑_{i=1}^{N} ∑_{j=1}^{3}(u_{i}, L_{j}u_{i}),  (B7)

with

h := α^{−1}T (p) − \frac{Z}{| · |} − \frac{ν χ_{B_{R}(P)}(·)}{| · −P|} + ρ ∗ Φ_{s} ∗ \frac{1}{| · |},

and L_{j} is the operator (defined in Theorem 2.1) that gives the error due to the localization in the kinetic energy. We first estimate this error term. Using the definition of L_{j} we find for all j ∈ {1, 2, 3}, i ∈ {1, …, N}

(u_{i}, L_{j}u_{i}) ≤ \frac{α^{−2}}{4π^{2}} ∥∇χ_{j}∥_{∞}^{2} ∫∫ K_{2}(α^{−1}|x − y|)|u_{i}(y)||u_{i}(x)| dxdy.

We then obtain by using Schwarz’s inequality

α^{−1} ∑_{i=1}^{N} ∑_{j=1}^{3}(u_{i}, L_{j}u_{i}) ≤ \frac{α^{−3}}{4π^{2}} ∑_{j=1}^{3} ∥∇χ_{j}∥_{∞}^{2} ∑_{i=1}^{N} ∫ K_{2}(α^{−1}|z|)dz ≤ CNβ^{−2}Z^{2t},

(B8)

since from (15)

∫_{ℝ^{3}} K_{2}(α^{−1}|z|) dz = α^{3} ∫_{ℝ^{3}} K_{2}(|z|) dz = 4πα^{3} ∫_{0}^{∞} t^{2}K_{2}(t) dt = 6π^{2}α^{3}. (B9)

Collecting together (B7) and (B8) we get

⟨ψ, H_{P}ψ⟩ ≥ ∑_{i=1}^{N} ∑_{j=1}^{3}(χ_{j}u_{i}, hχ_{j}u_{i})−D(ρ)−Cβ^{−2}Z^{1+2t} −Cβ^{−1}Z^{7/4−t/4}. (B10)

Here we used that N ≤ 2Z + 1, the choice of s and that we may choose g such that ∥∇g∥_{2}^{2} ≤ 2π. Near the nuclei. When j = 1 in the summation in the first term on the right hand side of (B10) we find

∑_{i=1}^{N}(χ_{1}u_{i}, hχ_{1}u_{i}) ≥ ∑_{i=1}^{N}(χ_{1}u_{i}, (α^{−1}T (p) − \frac{Z}{| · |})χ_{1}u_{i}),

since χ_{B_{R}(P)}χ_{1} ≡ 0 by the choice of χ_{1}, and the term Φ_{s} ∗ρ∗ \frac{1}{|·|} is non-negative. Then by Theorem 2.10 we find

∑_{i=1}^{N}(χ_{1}u_{i}, hχ_{1}u_{i}) ≥ Tr[α^{−1}T (p) − \frac{Z}{| · |}χ_{|x|< \frac{1}{2} βZ^{−t}}]_{−}
≥ −Cβ^{1/2}Z^{5/2−t/2} − Cκ^{2}Z^{2}.  (B11)

To estimate from below the term corresponding to j = 2 in the sum on the right hand side of (B10) we use [24, Theorem 2.8]. Here we need the result in [24] (instead of Theorem 2.10) because of the presence of the two nuclei. Notice that Theorem 2.10 can be extended to include also different nuclei. We have

∑_{i=1}^{N}(χ_{2}u_{i}, hχ_{2}u_{i}) ≥ ∑_{i=1}^{N}(χ_{2}u_{i}, (α^{−1}T (p) − \frac{Z}{|x|} − \frac{ν}{|x − P|}χ_{B_{R}(P)})χ_{2}u_{i})

≥ Tr[α^{−1}T (p) − \frac{Z}{|x|}χ_{|x−P|< \frac{1}{2} βZ^{−t}} − \frac{ν}{|x − P|}χ_{B_{R}(P)}]_{−},

and by [24, Theorem 2.8] we get

∑_{i=1}^{N}(χ_{2}u_{i}, hχ_{2}u_{i}) ≥ −CZ^{5/2}α^{1/2} − C ∫_{\frac{1}{2} βZ^{−t}>|x−P|>α} (\frac{Z^{5/2}}{|x|^{5/2}} + α^{3} \frac{Z^{4}}{|x|^{4}} ) dx

−C ∫_{R>|x−P|>α} (\frac{ν^{5/2}}{|x − P|^{5/2}} + α^{3} \frac{ν^{4}}{|x − P|^{4}} ) dx

≥ −Cκ^{1/2}Z^{2} − Cβ^{1/2}Z^{5/2−t/2} − Cκ^{2}Z^{2}.  (B12)

Here we used that t < l and Zα = κ. The outer zone. This region gives the main contribution to the energy. The term in (B10) that we still have to study is

∑_{i=1}^{N}(χ_{3}u_{i}, hχ_{3}u_{i}) − D(ρ)  (B13)

We start by estimating the first term in (B13) using coherent states. We consider again the function g ∈ C_{0}^{∞}(ℝ^{3}) introduced at the beginning of the proof and we define the function

g_{s}(x) := (β/(8Z^{s}))^{− \frac{3}{2}} g(8Z^{s}x/β) = Φs^{\frac{1}{2}} (x),  (B14)

with s the same parameter as before. For simplicity of notation we write Ṽ := Z/|x| − ρ ∗ 1/|x|. Then

\frac{Z}{|x|} − ρ ∗ Φ_{s} ∗ \frac{1}{|x|} = Ṽ ∗ Φ_{s} − ZΦ_{s} ∗ \frac{1}{|x|} + \frac{Z}{|x|}.

Since supp(g_{s}) ∩ supp(χ_{3}) = ∅ by Newton’s Theorem we find

∑_{i=1}^{N}(χ_{3}u_{i}, hχ_{3}u_{i}) = ∑_{i=1}^{N}(χ_{3}u_{i}, (α^{−1}T (p) − Ṽ ∗ Φ_{s})χ_{3}u_{i}).  (B15)

We consider the coherent states g_{s}^{p,q} defined for p, q ∈ ℝ^{3} by

g_{s}^{p,q}(x) = g_{s}(x − q)e^{−ip.x}.

The following formulas hold for f ∈ H^{\frac{1}{2}} (ℝ^{3}, ℂ)

(f, f) = \frac{1}{(2π)^{3}} ∫_{ℝ^{3}} dp ∫_{ℝ^{3}} dq (f, g_{s}^{p,q}) (g_{s}^{p,q}, f),
(f, V ∗ g_{s}^{2}f) = \frac{1}{(2π)^{3}} ∫_{ℝ^{3}} dp ∫_{ℝ^{3}} dq V (q) (f, g_{s}^{p,q}) (g_{s}^{p,q}, f) (B16)

and

(f, T (p)f) = \frac{1}{(2π)^{3}} ∫_{ℝ^{3}} dp ∫_{ℝ^{3}} dq T (p) (f, g_{s}^{p,q}) (g_{s}^{p,q}, f)
− ∫_{ℝ^{3}} dx ∫_{ℝ^{3}} dq\overline{f(x)}(L_{q}f)(x),  (B17)

where L_{q} has integral kernel

L_{q}(x, y) = \frac{α^{−2}}{4π^{2}} |g_{s}(x − q) − g_{s}(y − q)|^{2} \frac{K_{2}(α^{−1}|x − y|)}{|x − y|^{2}}.

Using these formulas we can rewrite (B15) as follows

∑_{i=1}^{N}(χ_{3}u_{i}, (α^{−1}T (p) − Ṽ ∗ Φ_{s})χ_{3}u_{i})

= \frac{1}{(2π)^{3}} α^{−1} ∫_{ℝ^{3}} dp ∫_{ℝ^{3}} dq(T (p) − αṼ (q)) ∑_{j=1}^{q} ∑_{i=1}^{N} |(χ_{3}u_{i}^{j}, g_{s}^{p,q})|^{2}

− α^{−1} ∑_{i=1}^{N} ∫_{ℝ^{3}} dx ∫_{ℝ^{3}} dq \overline{χ_{3}u_{i}(x)}(L_{q}χ_{3}u_{i})(x),  (B18)

Here u_{i}^{j} is the j-th spin component of u_{i}. We start by estimating the error term, the last term on the right hand side of (B18). From the definition of L_{q} it follows

L_{q}(x, y) ≤ \frac{α^{−2}}{4π^{2}} ∥∇g_{s}∥_{∞}^{2}K_{2}(α^{−1}|x − y|)(χ_{supp(g_{s})}(x − q) + χ_{supp(g_{s})}(y − q)),

and by the definition of the function g_{s}

∫_{ℝ^{3}} L_{q}(x, y) dq ≤ C∥∇g∥_{∞}^{2}α^{−2}β^{−2}Z^{2s}K_{2}(α^{−1}|x − y|).

By the estimate above, Schwarz’s inequality, (B9) and the choice of s we find

α^{−1} ∑_{i=1}^{N} ∫_{ℝ^{3}} dx ∫_{ℝ^{3}} dq \overline{χ_{3}u_{i}(x)}(L_{q}χ_{3}u_{i})(x) ≤ C∥∇g∥_{∞}^{2}β^{−2}Z^{3/2−t/2}N. (B19)

It remains to study the first term on the right hand side of (B18). In order to get an estimate from below we consider only the negative part of the integrand. Moreover, since if |q| < βZ^{−t}/8 then supp(χ_{3}g_{s}^{p,q}) = ∅ (because Z^{−t} > Z^{−s} since s > t) we find

\frac{1}{(2π)^{3}} α^{−1} ∫_{ℝ^{3}} dp ∫_{ℝ^{3}} dq (T (p) − αṼ (q)) ∑_{j=1}^{q} ∑_{i=1}^{N} |(χ_{3}u_{i}^{j}, g_{s}^{p,q})|^{2}

≥ \frac{q}{(2π)^{3}} α^{−1} ∫_{|q|≥ \frac{1}{8} βZ^{−t}} dq ∫_{T (p)−αṼ (q)≤0} dp (T (p) − αṼ (q)) = …, (B20)

where we also use that ∑_{i=1}^{N} |(χ_{3}u_{i}^{j}, g_{s}^{p,q})|^{2} ≤ 1 (Bessel’s inequality). We split now the integral as a sum of two terms

… = \frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}\frac{1}{2} |p|^{2}−Ṽ (q)≤0 \\ |q|≥ \frac{1}{8} βZ^{−t} \end{matrix} dqdp (T (p) − αṼ (q))

+ \frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}\frac{α}{2} |p|^{2}≥αṼ (q)≥T (p) \\ |q|≥ \frac{1}{8} βZ^{−t} \end{matrix} dqdp (T (p) − αṼ (q)). (B21)

We consider these two terms separately. The second term in (B21) gives a lower order contribution. Indeed

\frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}\frac{α}{2} |p|^{2}≥αṼ (q)≥T (p) \\ |q|≥ \frac{1}{8} βZ^{−t} \end{matrix} dqdp (T (p) − αṼ (q))

≥ −\frac{q}{(2π)^{3}} ∫∫\begin{matrix}(α^{2}[Ṽ (q)]_{+}^{2}+2[Ṽ (q)]_{+})^{\frac{1}{2}} ≥|p|≥(2[Ṽ (q)]_{+})^{\frac{1}{2}} \\ |q|≥ \frac{1}{8} βZ^{−t} \end{matrix} dqdp [Ṽ (q)]_{+} = …,

and computing the p-integral

⋯ = −C ∫_{|q|≥ \frac{1}{8} βZ^{−t}} dq [Ṽ (q)]+^{\frac{5}{2}}((1 + \frac{α^{2}}{2} [Ṽ (q)]_{+})^{\frac{3}{2}} − 1) = ….

Using (1 + x)^{\frac{3}{2}} ≤ 1 + \frac{3}{2}x + \frac{3}{8}x^{2} and that [Ṽ (q)]_{+} ≤ Z/|q| we get computing the integral

\begin{aligned}… & = & −Cα^{2} ∫_{|q|≥ \frac{1}{8} βZ^{−t}} dq [Ṽ (q)]+^{\frac{7}{2}}(1 + \frac{α^{2}}{8} [Ṽ (q)]_{+}) \\ & ≥ & −Cβ^{− \frac{1}{2}} κ^{2}Z^{3/2+t/2} − Cκ^{4}β^{− \frac{3}{2}} Z^{1/2+3t/2}. \end{aligned}  (B22)

Here we use that Zα = κ. Since \sqrt{1 + x} ≥ 1 + x/2 − x^{3}/8 for all x > 0, we have

T (p) ≥ α\frac{1}{2}|p|^{2} − α^{3} \frac{1}{8}|p|^{4},

and, for the first term on the right hand side of (B21), we obtain

\frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}\frac{1}{2} |p|^{2}−Ṽ (q)≤0 \\ |q|≥ \frac{1}{8} βZ^{−t} \end{matrix} dqdp (T (p) − αṼ (q)) ≥

≥ \frac{q}{(2π)^{3}} ∫∫\begin{matrix}\frac{1}{2} |p|^{2}−Ṽ (q)≤0 \\ |q|≥ \frac{1}{8} βZ^{−t} \end{matrix} dqdp (\frac{1}{2}|p|^{2} − \frac{1}{8}α^{2}|p|^{4} − Ṽ (q)) = ….

Computing now the integral with respect to p, we find

⋯ = −\frac{2^{\frac{3}{2}} q}{15π^{2}} ∫_{|q|> \frac{1}{8} βZ^{−t}}[Ṽ (q)]+^{\frac{5}{2}} dq − Cα^{2} ∫_{|q|> \frac{1}{8} βZ^{−t}}[Ṽ (q)]+^{\frac{7}{2}} dq. (B23)

We see that the second term on the right hand side of (B23) gives a lower order contribution since it is of the same order as the one in (B22). Collecting together (B10), (B11), (B12), (B15), (B18), (B19), (B22) and (B23)

⟨ψ, H_{P}ψ⟩ ≥ −C(β^{\frac{1}{2}} + β^{−2})Z^{5/2−t/2} − \frac{2^{\frac{3}{2}} q}{15π^{2}} ∫_{ℝ^{3}}[Ṽ (q)]+^{\frac{5}{2}} dq − D(ρ) . (B24)

Here we used also that N < 2Z + 1, the choice of s and that t ≤ 3/5. Now we choose ρ = ρ^{TF} the minimizer of the TF-energy functional of a neutral atom with Coulomb potential and nuclear charge Z. Hence ρ^{TF} satisfies the TF-equation

\frac{1}{2}(\frac{6π^{2}}{q} )^{\frac{2}{3}} ρ^{TF}(x)^{\frac{2}{3}} = [Ṽ (x)]_{+},

since Ṽ is the TF-mean field potential. Notice that here we use that the chemical potential of a neutral atom is zero. By the choice of ρ from the TF-equation it follows from (B24) that

⟨ψ, H_{P}ψ⟩ ≥ −C(β^{\frac{1}{2}} + β^{−2})Z^{5/2−t/2} + \frac{3}{10}(\frac{6π^{2}}{q} )^{\frac{2}{3}} ∫_{ℝ^{3}} dx ρ^{TF}(x)^{\frac{5}{3}}

−Z ∫_{ℝ^{3}} \frac{ρ^{TF}(x)}{|x|} dx + D(ρ^{TF})

= E^{TF}(ρ^{TF}) − C(β^{\frac{1}{2}} + β^{−2})Z^{5/2−t/2} .

The claim follows.

Proposition B.2. Let ρ^{TF} be the minimizer of the TF-energy functional of a neutral atom with nuclear charge Z. Let Zα = κ be fixed with 0 ≤ κ < 2/π and Z ≥ 1.

Then there is a constant depending only on κ and q such that for all {u_{i}}_{i=1}^{N} ⊂ H^{\frac{1}{2}} (ℝ^{3}; ℂ^{q}) orthonormal in L^{2}(ℝ^{3}) we have

∑_{i=1}^{N}(u_{i}, (α^{−1}T (p) − φ^{TF})u_{i}) − D(ρ^{TF}) ≥ E^{TF}(ρ^{TF}) − CZ^{2+ \frac{1}{5}} ,

with D(·) = D(·, ·) the Coulomb scalar product.

Proof. Since E^{TF}(ρ^{TF}) = −e_{0}Z^{\frac{7}{3}} (see (12)) to prove the claim it is sufficient to show that the TF-energy gives a lower bound to the quantum energy modulo lower order terms. In the proof we localize the energy separating the contribution from the region near the nucleus to the one far away. The region far away from the nuclei will give the TF-energy. In the proof C denotes a generic universal positive constant. Choice of the localization. The localization will be given by the functions χ_{1} ∈ C_{0}^{∞}(ℝ^{3}) and χ_{2} ∈ C^{∞}(ℝ^{3}) such that: 0 ≤ χ_{1}, χ_{2} ≤ 1, χ_{1}^{2} + χ_{2}^{2} = 1 in ℝ^{3},

χ_{1}(x) := \{ \begin{matrix}1 & if |x| < 2Z^{−3/5}, \\ 0 & if |x| > 3Z^{−3/5}. \end{matrix}  (B25)

Moreover we ask that

∥∇χ_{1}∥_{∞}, ∥∇χ_{2}∥_{∞} ≤ 2^{2}Z^{3/5}.  (B26)

The localization in the energy expectation. We insert now the localization in the energy expectation. From Theorem 2.1 we find

∑_{i=1}^{N}(u_{i}, (α^{−1}T (p) − φ^{TF})u_{i}) − D(ρ^{TF})  (B27)

≥ ∑_{i=1}^{N} ∑_{j=1}^{2}(χ_{j}u_{i}, (α^{−1}T (p) − φ^{TF})χ_{j}u_{i}) − D(ρ^{TF}) − α^{−1} ∑_{i=1}^{N} ∑_{j=1}^{2}(u_{i}, L_{j}u_{i}),

with L_{j} is the operator (defined in Theorem 2.1) that gives the error due to the localization in the kinetic energy. We first estimate this error term. Since N ≤ 2Z + 1 we find as in (B8) that

α^{−1} ∑_{i=1}^{N} ∑_{j=1}^{2}(u_{i}, L_{j}u_{i}) ≤ CZ^{6/5}N ≤ CZ^{2+1/5} .  (B28)

Near the nucleus. Since

∑_{i=1}^{N}(χ_{1}u_{i}, (α^{−1}T (p) − φ^{TF})χ_{1}u_{i}) ≥ Tr[α^{−1}T (p) − φ^{TF}χ_{|x|<3Z^{−3/5}}]_{−},

by Theorem 2.10 with R = 3Z^{−3/5} we find

∑_{i=1}^{N}(χ_{1}u_{i}, (α^{−1}T (p) − φ^{TF})χ_{1}u_{i}) ≥ −CZ^{2+1/5} − Cκ^{2}Z^{2}.  (B29)

Here we use that Zα = κ. The outer zone. This region gives the main contribution to the energy. Let g ∈ C_{0}^{∞}(ℝ^{3}), g ≥ 0 be spherically symmetric with supp(g) ⊂ B_{1}(0) and such that ∥g∥_{2} = 1. Starting from these g we define Φ_{Z}(x) := (Z^{−3/5})^{−3}g^{2}(xZ^{3/5}) and

g_{Z}(x) := (Z^{−3/5})^{− \frac{3}{2}} g(xZ^{3/5}) = ΦZ^{\frac{1}{2}}(x).

Since supp(g_{Z}) ∩ supp(χ_{2}) = ∅ by Newton’s Theorem we find

∑_{i=1}^{N}(χ_{2}u_{i}, (α^{−1}T (p) − φ^{TF})χ_{2}u_{i}) = ∑_{i=1}^{N}(χ_{2}u_{i}, (α^{−1}T (p) − φ^{TF} ∗ Φ_{Z})χ_{2}u_{i}).

(B30)

We consider the coherent states g_{Z}^{p,q} defined for p, q ∈ ℝ^{3} by

g_{Z}^{p,q}(x) = g_{Z}(x − q)e^{−ip.x}.

Using formulas (B16) and (B17) we can rewrite (B30) as follows

∑_{i=1}^{N}(χ_{2}u_{i}, (α^{−1}T (p) − φ^{TF} ∗ g_{Z}^{2})χ_{2}u_{i})

= \frac{1}{(2π)^{3}} α^{−1} ∫_{ℝ^{3}} dp ∫_{ℝ^{3}} dq(T (p) − αφ^{TF}(q)) ∑_{j=1}^{q} ∑_{i=1}^{N} |(χ_{2}u_{i}^{j}, g_{Z}^{p,q})|^{2}

− α^{−1} ∑_{i=1}^{N} ∫_{ℝ^{3}} dx ∫_{ℝ^{3}} dq \overline{χ_{2}u_{i}(x)}(L_{q}χ_{2}u_{i})(x),  (B31)

Here u_{i}^{j} is the j-th spin component of u_{i}. We start by estimating the error term, the last term on the right hand side of (B31). We find as in (B19) that

α^{−1} ∑_{i=1}^{N} ∫_{ℝ^{3}} dx ∫_{ℝ^{3}} dq \overline{χ_{2}u_{i}(x)}(L_{q}χ_{2}u_{i})(x) ≤ C∥∇g∥_{∞}^{2}Z^{6/5}N.  (B32)

It remains to study the first term on the right hand side of (B31). In order to get an estimate from below we consider only the negative part of the integrand. Moreover, since if |q| < Z^{−3/5} then supp(χ_{2}g_{Z}^{p,q}) = ∅ we find

\frac{1}{(2π)^{3}} α^{−1} ∫_{ℝ^{3}} dp ∫_{ℝ^{3}} dq (T (p) − αφ^{TF}(q)) ∑_{j=1}^{q} ∑_{i=1}^{N} |(χ_{2}u_{i}^{j}, g_{Z}^{p,q})|^{2}

≥ \frac{q}{(2π)^{3}} α^{−1} ∫_{|q|≥Z^{−3/5}} dq ∫_{T (p)−αφ^{TF}(q)≤0} dp (T (p) − αφ^{TF}(q)) = …, (B33)

where we also use that ∑_{i=1}^{N} |(χ_{3}u_{i}^{j}, g_{Z}^{p,q})|^{2} ≤ 1 (Bessel’s inequality). We split now the integral as a sum of two terms

… = \frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}\frac{1}{2} |p|^{2}−φ^{TF}(q)≤0 \\ |q|≥Z^{−3/5} \end{matrix} dqdp (T (p) − αφ^{TF}(q))

+ \frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}\frac{α}{2} |p|^{2}≥αφ^{TF}(q)≥T (p) \\ |q|≥Z^{−3/5} \end{matrix} dqdp (T (p) − αφ^{TF}(q)). (B34)

We consider these two terms separately. The second term in (B34) gives a lower order contribution. Indeed

\frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}\frac{α}{2} |p|^{2}≥αφ^{TF}(q)≥T (p) \\ |q|≥Z^{−3/5} \end{matrix} dqdp (T (p) − αφ^{TF}(q))

≥ −\frac{q}{(2π)^{3}} ∫∫\begin{matrix}(α^{2}[φ^{TF}]_{+}^{2}+2[φ^{TF}]_{+})^{\frac{1}{2}} ≥|p|≥(2[φ^{TF}(q)]_{+})^{\frac{1}{2}} \\ |q|≥Z^{−3/5} \end{matrix} dqdp [φ^{TF}(q)]_{+} = …,

and computing the integral in p

⋯ = −C ∫_{|q|≥Z^{−3/5}} dq [φ^{TF}(q)]+^{\frac{5}{2}}((1 + \frac{α^{2}}{2} [φ^{TF}(q)]_{+})^{\frac{3}{2}} − 1) = ….

Using (1 + x)^{\frac{3}{2}} ≤ 1 + \frac{3}{2}x + \frac{3}{8}x^{2} and that [φ^{TF}(q)]_{+} ≤ Z/|q| we get computing the integral

\begin{aligned}… & = & −Cα^{2}∫ _{|q|≥Z^{−3/5}}dq [φ^{TF}]+^{\frac{7}{2}}(1 + \frac{α^{2}}{8} [φ^{TF}(q)]_{+}) \\ & ≥ & −Cκ^{2}Z^{2− \frac{1}{5}} − Cκ^{4}Z^{\frac{7}{5}} . \end{aligned}  (B35)

Since \sqrt{1 + x} ≥ 1 + x/2 − x^{3}/8 for all x ≥ 0, we have

T (p) ≥ α\frac{1}{2}|p|^{2} − α^{3} \frac{1}{8}|p|^{4},

and, for the first term on the right hand side of (B34), we obtain

\frac{q}{(2π)^{3}} α^{−1} ∫∫\begin{matrix}\frac{1}{2} |p|^{2}−φ^{TF}(q)≤0 \\ |q|≥Z^{−3/5} \end{matrix} dqdp (T (p) − αφ^{TF}(q)) ≥

≥ \frac{q}{(2π)^{3}} ∫∫\begin{matrix}\frac{1}{2} |p|^{2}−φ^{TF}(q)≤0 \\ |q|≥Z^{−3/5} \end{matrix} dqdp (\frac{1}{2}|p|^{2} − \frac{1}{8}α^{2}|p|^{4} − φ^{TF}(q)) = ….

Computing now the integral with respect to p, we find

⋯ = −\frac{2^{\frac{3}{2}} q}{15π^{2}} ∫_{|q|>Z^{−3/5}}[φ^{TF}(q)]+^{\frac{5}{2}} dq − Cα^{2} ∫_{|q|>Z^{−3/5}}[φ^{TF}(q)]+^{\frac{7}{2}} dq. (B36)

We see that the second term on the right hand side of (B36) gives a lower order contribution since it is of the same order as the one in (B35). Starting from (B27), by (B28), (B29), (B32), (B35) and (B36) we find

∑_{i=1}^{N}(u_{i}, (α^{−1}T (p) − φ^{TF})u_{i}) − D(ρ^{TF})  (B37)

≥ −C(Z^{2+1/5} + Z^{2} + Z^{2−1/5} + Z^{7/5}) − \frac{2^{\frac{3}{2}} q}{15π^{2}} ∫_{ℝ^{3}}[φ^{TF}(q)]+^{\frac{5}{2}} dq − D(ρ^{TF}).

The result follows from the TF-equation.

References

[1] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York 1964.

[2] A. Dall’Acqua, T. Østergaard Sørensen, E. Stockmeyer, Hartree-Fock theory for pseudorelativistic atoms. Ann. Henri Poincaré 9 (2008), 711–742.

[3] A. Dall’Acqua, T. Østergaard Sørensen, E. Stockmeyer, A remark on the maximum negative ionization of pseudo-relativistic atoms. in preparation.

[4] I. Daubechies, An uncertainty principle for Fermions with generalized kinetic energy. Comm. Math. Phys. 90 (1983), 511–520.

[5] I. Daubechies, E.H. Lieb, One-Electron Relativistic Molecules with Coulomb Interaction, Comm. Math. Phys. 90 (1983), 497–510.

[6] C.L. Fefferman, L.A. Seco, An upper bound for the number of electrons in a large ion. Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 10, 3464–3465.

[7] C.L. Fefferman, L.A. Seco, Asymptotic neutrality of large ions. Comm. Math. Phys. 128 (1990), no. 1, 109–130.

[8] A. Gray, G.B. Mathews, A treatise on Bessel Functions and their applications to Physics, 2nd ed. prepared by A. Gray and T.M. Mac-Robert, Dover, New York, 1966.

[9] I.W. Herbst, Spectral theory of the Operator (p^{2} + m^{2})^{1/2} − Ze^{2}/r. Comm. Math. Phys. 53 (1977), 285–294.

[10] T. Kato, Perturbation theory for linear operators, Classic in Mathematics. Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[11] E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. Vol. 53 (1981), no. 4, 603–641.

[12] E.H. Lieb, Variational Principle for Many-Fermion Systems. Phys. rev. Let. Vol. 46 (1981), no. 7, 457–459.

[13] E.H. Lieb, Bound on the maximum negative ionization of atoms and molecules, Phys. Rev. A 29 (1984), 3018–3028.

[14] E.H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics. 14. American Mathematical Society, Providence, RI, 2001.

[15] E.H. Lieb, S. Oxford, An improved lower bound on the indirect Coulomb energy. Internat. J. Quantum Chem. 19 (1981), 427–439.

[16] E.H. Lieb, I.M. Sigal, B. Simon, W. Thirring, Asymptotic neutrality of large-Z ions. Phys. Rev. Lett. 52 (1984), no. 12, 994–996.

[17] E.H. Lieb, I.M. Sigal, B. Simon, W. Thirring, Approximate neutrality of large-Z ions. Comm. Math. Phys. 116 (1988), no. 4, 635–644.

[18] E.H. Lieb, B. Simon, Thomas-Fermi theory of atoms, molecules and solids. Adv. in Math. 23 (1977), 22–116.

[19] E.H. Lieb, H.-T. Yau, The Stability and Instability of Relativistic Matter. Comm. Math. Phys. 118 (1988), 177–213.

[20] M. Reed, B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[21] T. Østergaard Sørensen, The large Z-behavior of pseudorelativistic atoms, J. Math. Phys. 46 (2005), 052307, 24 pp.

[22] L.A. Seco, I.M. Sigal, J.P. Solovej, Bound on the ionization energy of large atoms. Comm. Math. Phys. 131 (1990), no. 2, 307–315.

[23] J.P. Solovej, The ionization conjecture in Hartree-Fock theory. Ann. Math. 158 (2003), 509–576.

[24] J.P. Solovej, W. Spitzer and T. Østergaard Sørensen, The relativistic Scott correction for atoms and molecules. Comm. Pure App. Math. 63 (2010), no.1, 39–118.

Faculty of Mathematics, Otto-von-Guericke Universität Universitätsplatz 2 D-39016 Magdeburg Germany anna.dallacqua@ovgu.de Department of Mathematics University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Denmark solovej@math.ku.dk 346

Documenta Mathematica 15 (2010)