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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.

1

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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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16M. HUBER AND H. SIEDENTOP

Appendix B. Properties of Dirac-Fock Operators

We list some useful inequalities from [1] and [3] and some slight improvements

of these.

Lemma 6. For any γ ∈ F we have

(16)

φ(γ)≤π

φ(γ)≤π

2?|∇|γ?1≤π

2?γ?1|∇| ≤π

2?|D0|γ?1=π

2?γ?1|D0|.

2?γ?F,

(17)

Lemma 7. If γ ∈ F, then

X(γ)≤ φ(|γ|), ?X(γ)? ≤ ?φ(|γ|)?, and ?W(γ)? ≤ ?φ(|γ|)?.

Proof. We prove only the third statement. Let γ = γ+− γ−, i.e., γ+and γ−are

the positive and negative parts of γ respectively. Then

W(γ)= φ(γ+)− φ(γ−)− X(γ+)+ X(γ−)≤ φ(γ+)+ X(γ−)≤ φ(γ+)+ φ(γ−)= φ(|γ|),

where we used Lemmata 8 and 7. In the same way we get W(γ)≥ −φ(γ−)−

X(γ+)≥ −φ(γ−)− φ(γ+)= −φ(|γ|), so |(f,W(γ)f) ≤ (f,φ(|γ|)f) for all f ∈ H. This

immediately implies the claim.

??

Lemma 8. Let 0 ≤ γ ∈ F, then 0 ≤ X(γ)≤ φ(γ); in particular 0 ≤ W(γ).

An immediate consequence of the preceding lemmata is

Lemma 9. If γ = γ∗∈ S1(H) and γ?∈ F, then

|D(ργ,ργ?)| ≤π

E(γ,γ?) ≤D(ρ|γ|,ρ|γ?|).

4?γ?1tr(|∇||γ|),

We also need

Lemma 10 (Bach et al. [1]). For all γ ∈ F we have E(γ,γ) ≤π

Lemma 11. Pick γ ∈ F, g ∈ (−√3/2,√3/2), α ∈ R. W(γ)is relatively compact

with respect to D0. The operator D(γ)

H1(R3)4and

σess(D(γ)

(1) Set Cg:= (?4g2+ 9 − 4g)/3 and, for 0 ≤ g <√3/2,

dg:= (1 + C2

(1 − C2

Then, we for g ∈ [0,√3/2] according to Morozov ([10])

|Dg|2≥ d2

If we assume in addition γ ∈ F and dg− 4|α|?γ?1> 0, then

|D(γ)

(2) Setting bg:=

|Dg|2≥ b2

Assuming in addition bg− 4|α|?γ?1> 0 implies the inequalities

|D(γ)

|D(γ)

Acknowledgement: We thank Jean-Marie Barbaroux for explaining the proof

of Equation (13) in [2]. We acknowledge partial support through the European

Union’s IHP network Analysis & Quantum, HPRN-CT-2002-0277.

4tr(γ|∇|γ).

g,α is self-adjoint with D(D(γ)

g,α) = D(Dg) =

g,α) = σess(Dg) = (−∞,−1] ∪ [1,∞).

Lemma 12.

g−

?

g)2+ 4g2C2

g)/2.

g|D0|2.

g,α|2≥ (dg− 4|α|?γ?1)2|D0|2

?1 − g2(?4g2+ 9 − 4g)/3 and g ∈ (0,√3/2) we have

g|∇|2.

g,α|2

g,α|2

≥

≥

(bg− 4|α|?γ?1)2|∇|2,

(1 − 4|α|?γ?1b−1

g)2|Dg|2.

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DIRAC-FOCK EQUATIONS OF THE ELECTRON-POSITRON FIELD17

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Mathematik, Theresienstraße 39, 80333 M¨ unchen, Germany

E-mail address: mhuber@math.lmu.de and h.s@lmu.de