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SOLUTIONS OF THE DIRAC-FOCK EQUATIONS AND THE

ENERGY OF THE ELECTRON-POSITRON FIELD

MATTHIAS HUBER AND HEINZ SIEDENTOP

Abstract. We consider atoms with closed shells, i.e., the electron number

N is 2, 8, 10,..., and weak electron-electron interaction. Then there exists

a unique solution γ of the Dirac-Fock equations [D(γ)

ditional property that γ is the orthogonal projector onto the first N positive

eigenvalues of the Dirac-Fock operator D(γ)

ergy of the relativistic electron-positron field in Hartree-Fock approximation,

if the splitting of H := L2(R3) ⊗ C4into electron and positron subspace, is

chosen self-consistently, i.e., the projection onto the electron-subspace is given

by the positive spectral projection of D(γ)

g,α. For fixed electron-nucleus coupling

constant g := αZ we give quantitative estimates on the maximal value of the

fine structure constant α for which the existence can be guaranteed.

g,α,γ] = 0 with the ad-

g,α. Moreover, γ minimizes the en-

1. Introduction

Heavy atoms should be described by relativistic quantum electrodynamics. Fol-

lowing this idea, Bach et al. [1] showed that the energy of the relativistic electron-

positron field in Hartree-Fock approximation interacting with the second quantized

Coulomb field of a nucleus is non-negative (if the quantization is chosen with re-

spect to external field) and that the vacuum is a minimizer. Moreover, they showed

that the quantization with respect to the external field is optimal in the sense that

any other quantization yields a lower ground state energy.

Barbaroux et al. [3] addressed the existence of atoms in the above model, i.e.,

they prescribed the charge of the electron-positron field and showed that the corre-

sponding functional has a minimizer which fulfills the no-pair Dirac-Fock equations.

The existence of solutions of the Dirac-Fock equations was shown by Esteban

and S´ er´ e [6] and Paturel [11]. Moreover, Esteban and S´ er´ e [5] considered the non-

relativistic limit of the Dirac-Fock equations. They showed that certain solutions of

the Dirac-Fock equations converge to the energy minimizing solutions of the non-

relativistic Hartree-Fock equations when the speed of light tends to infinity. This

allows them to define the notion of ground state solutions and ground state energy

of the Dirac-Fock equations.

In the spirit of Mittleman [9] the physical energy should be obtained by maxi-

mizing the ground state energy (as defined, e.g., in [3]) over all allowed one-particle

electron subspaces. One might conjecture that a corresponding ground state is

a solution of the Dirac-Fock equations. Moreover, such a solution of the Dirac-

Fock equations should minimize the energy among all solutions of the Dirac-Fock

equations. We call this for brevity the “Mittleman conjecture”.

The validity of Mittleman’s conjecture was already addressed by Barbaroux et al.

[2]. They confirmed it when the atomic shells are closed and the electron-electron

interaction is weak (large velocity of light). In the open shell case it was only proven

by Barbaroux et al. [4] in the case of hydrogen. All other cases are unknown.

Date: December 5, 2005.

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2 M. HUBER AND H. SIEDENTOP

A stronger conjecture – for brevity called in the paper BES conjecture – would

be: the maximin pair ( maximizing Λ and minimizing γ) is a projector onto the

first N eigenfunctions of the self-consistent Dirac-Fock operator and that Λ is the

spectral projector onto the negative spectral subspace of this operator. Barbaroux

et al. [2] showed that this conjecture is incorrect in the open shell case in the

non-relativistic limit. (For N = 1 this result can be extended beyond the limiting

case (Barbaroux et al. [4]).) However, they confirm their conjecture for closed shell

atoms in the non-relativistic limit.

In this paper – following Barbaroux et al. [2] – we also consider the limit of

weak electron-electron interaction. Similarly to Barbaroux et al.[2, Proposition 8]

and Esteban and S´ er´ e [5, Theorem 5] we prove the existence of a unique solution

of the Dirac-Fock equations with the property that the eigenvalues of the solutions

are the lowest eigenvalues of the corresponding self-consistent Dirac-Fock operator

and that the next eigenvalue is strictly bigger. Again similarly to Barbaroux et al.

and Esteban and S´ er´ e [5, Theorem 6] this allows us to prove that this solution is

the minimizer of the Dirac-Fock energy on the set of all solutions of the Dirac-Fock

equations with non-negative eigenvalues. However, we can prove that this solution

minimizes the Dirac-Fock energy even on the set of all charge density matrices (see

the corresponding result of Barbaroux et al. [2, Proposition 8, Equation (15)]) if

the quantization is chosen with respect to this solution. We emphasize that we

do not only admit positrons in the charge density matrices γ; in fact we can drop

the assumption that off-diagonal elements of γ vanish, a requirement inherent in

Barbaroux et al. (Corollary 6). Eventually, we show that the minimizer is uniquely

determined and spherically symmetric in a certain sense. It has eigenfunctions

(orbitals) that respect the Aufbau principle.

The essential novelty of our result is twofold: First, our proof is sufficiently

direct and simple allowing for explicit estimates. This enables us to show not only

existence results (Esteban and S´ er´ e [6] and Paturel [11]) but also to prove important

properties of the solutions. In addition we obtain these properties not only in the

non-relativistic limit (Barbaroux et al. [2]) but we get explicit estimates on the

allowed coupling constants for which these results hold. Second, we can show the

minimization property among all density matrices of the electron-positron field in

the self-consistent quantization.

2. Definition of the Model

The notation and estimates used are mainly those of Barbaroux, Farkas, Helffer,

and Siedentop [3]. For the convenience of the reader we give here nevertheless their

main definitions and results. The technical tools from [3] are listed in an appendix.

For any further details we refer the reader to [3].

The Coulomb-Dirac operator is written as

Dg:= −iα · ∇ + β − g| · |−1.

Physically g = Zα where α is the Sommerfeld fine structure constant and Z is the

atomic number of the considered element. The operator is essentially self-adjoint

on S(R3) ⊗ C4if g ∈ [0,√3/2).

It is convenient to introduce the set G := R3× {1,2,3,4} and the measure

dx := dx ⊗ dµ, where dx is the Lebesgue measure on R3and dµ the counting

measure of the set {1,2,3,4}. We denote the Banach space of trace class operators

on H by S1(H). Furthermore,

F := {γ ∈ S1(H)|γ = γ∗, D0γ ∈ S1(H)}.

Note that Barbaroux et al. [3] use a slightly different definition of the space F.

Moreover, F is a Banach space when equipped with the norm ?γ?F := ?D0γ?1=

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DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD3

?|D0||γ|?1.

0 ≤ g <√3/2 because of Lemma 12.

We write the integral kernel of any given γ ∈ F using its eigenvalues λn and

eigenspinors ξnas

∞

?

The one-particle density associated to γ is

Finally, we note that ?γ?F,g := ?Dgγ?1 is an equivalent norm for

γ(x,y) =

n=1

λnξn(x)ξn(y).

ργ(x) :=

4

?

s=1

∞

?

n=1

λn|ξn(x)|2.

Its electric potential operator is φ(γ):= ργ∗ | · |−1. The exchange operator X(γ)

associated to γ is given by its integral kernel

X(γ)(x,y) := γ(x,y)/|x − y|.

The total interaction operator is defined as

W(γ)= φ(γ)− X(γ).

The Coulomb scalar product is defined as

?

and the exchange scalar product as

?

The total interaction energy is defined as

Q(γ,γ?) := D(ργ,ργ?) − E(γ,γ?)

For α ≥ 0 and γ ∈ F the Dirac-Fock operator is defined as

D(γ)

D(ρ,σ) :=1

2

R3dx

?

R3dyρ(x)σ(y)

|x − y|

E(γ,γ?) :=1

2

G

dx

?

G

dyγ(x,y)γ?(x,y)

|x − y|

.

g,α:= Dg+ αW(γ).

Some useful properties of the operators defined above are listed in Appendix B.

For N ∈ N and δ ∈ F we define

?S(δ)

SN:={γ ∈ F |0 ≤ γ, trγ ≤ N},

and

Eg,α(γ) := trDgγ + αQ(γ,γ)

where Λ(δ)

and Λ(δ)

+ is the projector onto the negative spectral subspace.

Moreover, we will frequently use the abbreviations

cg,α,N:= (bg− 4αN)−1,

where bg:=

?(γ)

j

(j = 1,...) the eigenvalues of Dg and D(γ)

counting multiplicities).

We will be interested in solutions of the Dirac-Fock equations.

∂N:={γ ∈ F |−Λ(δ)

?S(δ):={γ ∈ F |−Λ(δ)

−≤ γ ≤ Λ(δ)

−≤ γ ≤ Λ(δ)

+,trγ = N},

+},

+= χ[0,∞)(D(δ)

−= 1 − Λ(δ)

g,α) is the projector on the positive spectral subspace of D(δ)

g,α

˜ cg,α,N:= (π/4)αNcg,α,N

?1 − g2(?4g2+ 9−4g)/3 (see also Lemma 12). We denote by ?0

jand

g,α respectively (ordered by size and

Definition 1. We denote the set of solutions to the Dirac-Fock equations by DF,

i.e.,

DF := {γ ∈ F|γ = γ2, [D(γ)

g,α,γ] = 0, γΛ(γ)

+ = γ}.

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4M. HUBER AND H. SIEDENTOP

For fixed g and small α we get: for closed shell atoms there exists a solution δ ∈

DF such δ is the projection onto the first N positive eigenvalue of D(δ)

1). We will prove this result using the Banach fixed point theorem yielding even

uniqueness of the solution. We also show that the fixed point (Corollary 3) and the

energy functional Eg,α (Theorem 2) are spherically symmetric in a certain sense,

and that the fixed point minimizes Eg,α on DF (Corollary 2). To this end the

uniqueness of the fixed point is crucial.

Moreover, we show (Theorem 3) that this solution minimizes Eg,αeven on the

set?S(δ)

[3].

Note, that the notion of closed shells – as used in this article – refers to the

Coulomb-Dirac operator, i.e., for N ∈ N we are in the closed shell case, if ?0

?0

N. It does not matter, if the gap is the gap between shells with different principal

quantum numbers. This means that N = 2,8,10,.... For brevity, we denote the set

of all such N by CS.

g,α (Theorem

∂qWe emphasize that we do not need to require Λ(δ)

had to be left open in the context of the no-pair Hartree-Fock theory discussed in

+γΛ(δ)

− = 0, a fact that

N+1>

3. Controlling the Spectrum of Dirac-Fock Operators

The aim of this section is to derive some estimates which control the eigenvalues

of Dirac-Fock operators by the corresponding eigenvalues of Coulomb-Dirac oper-

ators. The main tool of this section is the minimax principle of Griesemer and

Siedentop [8], which is formulated in Appendix A (Theorem 4). We are going to

use the Coulomb-Dirac operator as unperturbed operator and the Dirac-Fock oper-

ator as perturbed operator. First, we check the hypotheses of the minimax theorem

(Theorem 4).

Lemma 1. Let A = D(γ)

χ(0,∞)(Dg), Λ−:= χ(−∞,0)(Dg), and h±:= Λ±h and assume 0 / ∈ σ(D(γ)

the hypotheses of Theorem 4 are fulfilled, if (π/2)α?γ?1≤ bg.

Proof. Let f ∈ Q−. Then

(f,D(γ)

≤ (f,Dgf) +π

g,α with 0 ≤ γ ∈ F, h = L2(R3)4, Q = D(A). Let Λ+:=

g,α). Then

g,αf) = (f,Dgf) + α(f,W(γ)f) ≤ (f,Dgf) + α(f,φ(γ)f)

2α?γ?1(f,|∇|f) ≤ (f,Dgf) +π

2α?γ?1

1

bg(f,|Dg|f) ≤ 0

where we used Lemmata 8 and 12, and (17),. The condition

(f,D(γ)

g,αf) > 0

for all f ∈ Q(A)∩H+is trivially fulfilled since W(γ)≥ 0 (Lemma 8). It remains to

check the boundedness of (|D(γ)

As in [7, Lemma 1],

?∞

i.e., we have to estimate the expression

?∞

Now, ?(Dg− iη)−1? ≤ [(?0

[λ0,∞) → R,fη(λ) :=

g,α| + 1)

1

2P−Λ+. To this end we proceed as follows:

P−Λ+= −α

2π

−∞

(D(γ)

g,α− iη)−1W(γ)(Dg− iη)−1dηΛ+,

−∞

(|D(γ)

g,α| + 1)

1

2(D(γ)

g,α− iη)−1W(γ)(Dg− iη)−1dη.

1)2+ η2]−1/2. Moreover, we look at the function

?

(λ + 1)/(λ2+ η2)

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DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD5

where λ0:= inf σ(|D(γ)

the point max{λ0,−1 +

g,α|) > 0 by assumption. This function has its maximum at

?1 + η2}, i.e.,

fη(λ) =

We conclude that

?∞

is finite, which implies the boundedness of (|D(γ)

Lemma 12 shows that the condition 0 ∈ ρ(D(γ)

bg> 4α?γ?1. One can relax this condition adapting an argument of Barbaroux et

al. ([4]); but since it is not the most restrictive condition, we refrain from doing so.

Lemma 1 enables us to control the eigenvalues of the Dirac-Fock operator by the

eigenvalues of the Coulomb-Dirac operator. Since the minimax principle yields the

eigenvalues ordered by size and counting multiplicities, we do not only get some

information on the localization of the eigenvalues but also on the dimension of

the projector onto a given part of the discrete spectrum. Note that the estimate

depends only on ?γ?1but not on γ itself.

Lemma 2. Let 0 ≤ γ ∈ F and let the hypotheses of Lemma 1 be fulfilled. Then,

for all n ∈ N

?0

n

≤ (1 + (π/2)α?γ?1b−1

Proof. Since 0 ≤ X(γ), 0 ≤ φ(γ)and 0 ≤ W(γ)(Lemma 8), we have using (17) and

Lemma 12

sup

λ

?(λ0+ 1)/(λ2

(−1+√

0+ η2)|η| ≤?(λ0+ 1)2− 1

?

(−1+√

1+η2)+1

1+η2)2+η2

|η| >?(λ0+ 1)2− 1.

−∞

?(|D(γ)

g,α| + 1)1/2(D(γ)

g,α− iη)−1??W(γ)??(Dg− iη)−1?dη

g,α| + 1)

g,α) in Lemma 1 is fulfilled, if

1

2P−Λ+.

??

n≤ ?(γ)

g)?0

n.

Dg≤ Dg+ αW(γ)≤ Dg+ αφ(γ)≤ Dg+π

≤ Dg+π

where (Dg)+ and (Dg)− denote the positive and negative part of the Coulomb-

Dirac operator respectively. We choose now h±:= Λ±h. Then the above operator

inequality yields immediately for all n ∈ N the inequality λn(Dg) ≤ λn(D(γ)

(π/2)α?γ?1b−1

4. Now, by Lemma 1 the hypotheses of Theorem 4 are fulfilled for D(γ)

the hypotheses are trivially fulfilled by the choice of h±:= Λ±h. For the operator

(1 + (π/2)α?γ?1/dg)(Dg)++ (1 − (π/2)α?γ?1/dg)(Dg)− the hypotheses are also

fulfilled, since it has got the same positive and negative spectral subspaces as the

operator Dg. Thus Theorem 4 immediately yields the claimed inequality.

2α?γ?1|∇|

g)(Dg)++ (1 −π

2α?γ?1b−1

g|Dg| ≤ (1 +π

2α?γ?1b−1

2α?γ?1b−1

g)(Dg)−

g,α) ≤ (1+

g)λn(Dg), where the λnare the minimax values defined in Theorem

g,α. For Dg

??

Lemma 3. Let 0 ≤ γ ∈ F with ργspherically symmetric and let the hypotheses of

Lemma 1 be fulfilled. Then, for all n ∈ N

?0

n

≤ ?(g − α?γ?1)0

where ?(g − α?γ?1)0

replaced by g − α?γ?1.

Proof. The first inequality is the same as in Lemma 2. For the second inequality

note that φ(γ)≤ ?γ?1| · |−1by Newton’s inequality. Thus, by Lemma 8

n≤ ?(γ)

n,

ndenote the eigenvalues of the Coulomb-Dirac operator with g

D(γ)

g,α= Dg+ αφ(γ)− αX(γ)≤ Dg+ α?γ?1| · |−1.

Since

(f,(D0− g| · |−1+ α?γ?1| · |−1)f) = (f,Dg−α?γ?1f) ≤ 0