We prove the existence of a ground state within the Hartree-Fock theory for atoms and molecules, in the presence of self-generated magnetic fields, with and without direct spin coupling. The ground state exists provided that the total charge Z of the K nuclei exceeds N, where N is the number of electrons, and, in the spin-polarized case, provided in addition that the nuclear charge is not too high.

In this paper, we study the Hartree-Fock model for a system of electrons interacting with static nuclei through the Coulomb potential, in the presence of direct coupling. In particular, we prove the existence of a ground state when the system is neutral or positively charged. We consider a system of N electrons and K nuclei of charge (Z1,,ZK), Zk>0 for all k, in the Born-Oppenheimer approximation. The many-body Hamiltonian including electron spin-magnetic field interactions that describe such a system system is

HSM=j=1N12[σj(ijA(xj))]2+v(xj)+1i<jN1|xixj|I2×2,
(1)

where

v(x)=k=1KZk|xRk|

is the potential generated by the nuclei, Rk is the position of the kth nucleus, and I2×2 is the 2×2 identity matrix. The Hamiltonian HSM acts on the Hilbert space of normalized, N particle, antisymmetric electron wave functions defined by

H={Ψj=1NL2(R3,C2):ΨL2(R3N,C2N)=1}.

In other words, the electron wave function Ψ(x1,s1,,xN,sN)j=1NL2(R3,C2) is a function of both spatial and spin variables, xjR3 and sj12,12. As a consequence of the Pauli exclusion principle, Ψ is antisymmetric in the indexes labeling the particles. In addition, Ψ is normalized in the following sense:

ΨL2(R3N,C2N)2=s1,,sNR3R3|Ψ(x1,s1,,xN,sN)|2dx1dxN=1.

The inner product on H is defined as

(Ψ,Φ)L2(R3N,C2N)=s1,,sNR3R3Ψ(x1,s1,,xN,sN)Φ¯(x1,s1,,xN,sN)dx1dxN.

The form domain of HSM, which corresponds to the set of admissible wave functions, is

W={Ψj=1NL2(R3,C2):ΨL2(R3N,C2N)=1,ΨL2(R3N,C2N)<+}.

The first term in (1) is the Pauli kinetic energy operator, and A is the magnetic vector potential. In this paper, we assume that there are no externally applied magnetic fields, and therefore the magnetic potential appearing in (1) is the one self-generated by the orbital motion of the electrons.

The Pauli kinetic energy operator σj acts on the j-th variables of ΨW and contains the Pauli matrices

σj=(σjx,σjy,σjz)=0  11  0,0ii0,1001.

The action of this operator on the electron wave function can be easily understood if we think of the electron wave function in terms of spinor: in other words, Ψ is regarded as a 2N-components vector valued function of the space variables. Each component corresponds to a possible configuration of values for the N spin variables (the electron spin has only two possible values: −1/2 or 1/2). Clearly, if we are dealing with an operator that acts on one variable only (such as the Pauli operator), then, in a similar way, we will treat Ψ as a two components vector valued function, one for each possible value of the spin of the variable involved. In the case of the one-electron wave function, for example, ψL2(R3,C2) is a two component vector defined as

ψ(x)=ψ+(x)ψ(x)=ψ(x,1/2)ψ(x,1/2).

The non-relativistic quantum energy of the system is given by

ESM(Ψ,A)=12j=1NR3N|σj(ijA(xj))Ψ|C22+Ψ,j=1Nv(xj)+1i<jN1|xixj|ΨL2(R3N,C2N)+1α2R3|×A|2.
(2)

Note that B=×A is the magnetic field generated by A, and therefore the last term in (2) is the magnetic field energy. The Hamiltonians (1) and (2) are dimensionless in the Hartree units. In particular, in these units, α=e24π𝜖0c1137.04 is the fine structure constant, where is the Planck’s constant divided by 2π, e is the elementary charge, c is the speed of light, and 𝜖0 is the permittivity of vacuum.

If we include the interactions between the electron’s self-generated magnetic field (that is, the magnetic field due to the electron’s orbital motion) and the intrinsic electron spin into our model, then we must define the ground state energy of the system as

ESM=inf{ESM(Ψ,A):(Ψ,A)DSM},
(3)

where

DSM=(Ψ,A):ΨWAL6(R3);A=0;×AL2(R3).

Note that the minimization in (3) is carried among both the wave functions and the magnetic vector potential. For the vector potential A, we choose the Coulomb gauge, i.e., we assume that A=0.

Problem (3) was studied in a series of papers by Fröhlich, Lieb, Loss, Yau, and Solovej (see Refs. 1–4). In Ref. 1, Fröhlich, Lieb, and Loss proved that, in the case of the one-electron atom (N = 1 and K = 1), there is a critical nuclear charge, Zc, for which the atom is stable (i.e., its ground state energy is finite) for Z<Zc and unstable for Z>Zc (Z being the charge of the atom). When the atom is stable, the authors also proved the existence of a ground state wave function. In Ref. 2, Lieb and Loss considered the problem of stability for the many-electrons atom (N arbitrary, K = 1) and the one-electron molecule (N = 1, K arbitrary), under some restrictions on the nuclear charges as well. Finally, in Ref. 3, Loss and Yau showed that the Pauli operator σ(iA) has zero as an eigenvalue. This means that there exist non-zero solutions to σ(iA)Ψ=0. As a consequence of this, the kinetic energy might not be enough to control the potential and magnetic energies. Specifically, if we rescale Ψ, we can drive the total energy to for certain values of the nuclear charge and obtain instability. This is not possible when direct coupling is not present, as a consequence of the Lieb-Thirring and diamagnetic inequalities (see Refs. 5 and 6, respectively) which provide the bound

ρΨL5/3(R3)5/3(iA)ΨL2(R3N,C2N)2,

where ρΨ is the total electronic density associated with Ψ, that is,

ρΨ(x)=Nτ{12,12}N1R3(N1)|Ψ(x,y,τ)|C22dy.

The problem of stability for the many-electrons many-nuclei system was finally solved by Lieb, Loss, and Solovej in Ref. 4. The authors proved stability under suitable values of the nuclear charge (the range of admissible charges was later extended in Ref. 7). They also prove a Lieb-Thirring type of inequality for the Pauli operator. A generalization of the same inequality is showed in Ref. 8 by Bugliaro, Fefferman, Fröhlich, Graf, and Stubbe. In that case, the stability of matter with magnetic fields follows as a by-product of it.

In this paper, we focus on proving the existence of a ground state in the Hartree-Fock theory with direct coupling and self-generated field. A Hartree-Fock theory starts by considering wave functions of the following form:

Ψ(x1,s1,,xN,sN)=1N!det(ψk(xj,sj))=1N!ψ1(x1,s1)ψ1(xN,sN)ψN(x1,s1)ψN(xN,sN),

where ψ1,,ψN is an orthonormal set of one-electron wave functions, that is, ψjH1(R3,C2) for j=1,,N and (ψj,ψk)L2(R3,C2)=δjk.

We denote by WSlater the subset of W consisting of all finite energy Slater determinants. Standard algebraic calculations show that the quantum energy (2) for a Slater determinant Ψ(x1,s1,,xN,sN)=1N!det(ψk(xj,sj)) is given by

ESMHF(ψ1,,ψN,A)=12j=1NR3|σ(iA)ψj|C22+R3ρv,
(4)
+12R3×R3ρ(x)ρ(y)TrC2(|γ(x,y)|2)|xy|dxdy+1α2R3|×A|2,
(5)

where ρ(x)=j=1N|ψj(x)|C22 is the total electron density associated with the Slater determinant and γ(x,y)=j=1Nψj(x)ψj¯(y)2×2(C) is a 2×2 matrix.

We adopt here the same notations as in Ref. 3: for vector fields (A or B)

AL2=(AA)1/2L2(R3),

where A = (A1, A2, A3) and AA=j|Aj|2. For spinors ψ,

ψL2=(ψψ)1/2L2(R3),

where (ψψ)(x)=|ψ(x)|C22=|ψ+(x)|2+|ψ(x)|2. Finally, for gradients A or ψ,

AL2=AL2(R3,R3)=j,k=13|jAk|21/2L2(R3)=j,k=13R3|jAk|21/2

and, analogously,

ψL2=ψL2(R3,C2)=j=13|jψ+|2+|jψ|21/2L2(R3)=j=13R3|jψ+|2+|jψ|21/2.

In the absence of magnetic fields, the existence of a Hartree-Fock ground state for Coulomb systems was proved by Lieb and Simon (see Ref. 9) and Lions (see Ref. 10), under the condition NZ. A different proof, combining non-linear and geometric techniques, has been developed by Friesecke in Ref. 11 and Lewin in Ref. 12.

In Ref. 13, Enstedt and Melgaard extended this result to the case in which a fixed magnetic field is applied (that is, they minimize energy (4) in the wave functions only). However, the authors do not include the spin-field interactions and make some strong assumptions on the vector potential (e.g., they assumed it to be homogeneous of degree −1). An alternative solution to the same problem is presented by Esteban and Lions in Ref. 14: they consider a larger class of vector potentials (i.e., ALloc3(R3)) but do not include the spin in their model, and the field is not determined self-consistently. The condition NZ ensures that the total charge of the nuclei should be sufficiently positive to prevent the electrons from escaping to infinity. It is a sufficient condition, but whether it is necessary is an open question. Results in this direction were presented by Lieb in Ref. 15, where he proved that for N2Z+K there is no ground state for the spinless magnetic Hartree-Fock for bounded vector potentials. This bound was improved in the absence of magnetic fields by Solovej in Ref. 16, who proved that there exists a universal constant Q>0 such that for all positive integers satisfying NZ+Q, there are no minimizers for the Hartree-Fock functional.

Finally, recall that when no magnetic field is present, a fundamental result by Lieb and Simon (see Ref. 17) implies that the Hartree-Fock theory is asymptotically exact for large nuclear charges and electron numbers, in the sense that in the limit for N and Z going to infinity, the Hartree-Fock theory provides the leading terms in the ground state energy. As it was established by Erdós and Solovej in Ref. 18, the inclusion of a self-generated magnetic field does not change the asymptotics of the leading term: in fact, the effect of the magnetic field appears only in the Scott term.

In the context of Kohn-Sham density-functional theory (see Ref. 19), Gontier proved the existence of a ground state in Ref. 20 in the presence of a fixed external magnetic field, extending earlier work by Anantharaman and Cancès on the Kohn-Sham equations to the magnetic case (see Ref. 21).

In our work, we consider spin-orbit interactions and prove the existence of a ground state for a wider class of vector potentials, namely, AL6(R3) and ×AL2(R3), which are self-generated. We define the Hartree-Fock ground state energy as

ESMHF=infESMHF(ψ1,,ψN,A):(ψ1,,ψN,A)DSMHF,
(6)

where

DSMHF=(ψ1,,ψN,A):ψjH1(R3,C2),j=1,,N;(ψj,ψk)L2(R3,C2)=δj,kAL6(R3);A=0;×AL2(R3).

Any minimizer to (6) is called a Hartree-Fock ground state.

Although the classical formulation of the Hartree-Fock problem involves the wave functions, from a mathematical point of view, it is more convenient to work with density matrices. This approach, originally proposed by Gilbert (see Ref. 22) and subsequently generalized by Valone (see Ref. 23), is employed in Ref. 24 for the study of the thermodynamic limit of the reduced Hartree-Fock functional, in Refs. 20 and 21 in the context of Kohn-Sham density functional theory, and in Refs. 16 and 25, where Solovej proves the ionization conjecture within the Hartree-Fock and reduced Hartree-Fock theory of atoms.

To be more specific, to any wave function ΨW we associate the corresponding N-body spin-density matrix ΓΨ, defined as the self-adjoint operator on j=1NL2(R3,C2), whose kernel is given by

ΓΨ(x1,,xN;y1,,yN)=Ψ(x1,,xN)Ψ¯(y1,,yN).

We define the set of pure-state N-body spin-density matrices as

Mpure={ΓΨ:ΨW}.
(7)

Analogously, the set of Slater-state N-body spin-density matrices is defined as

MSlater={ΓΨ:ΨWSlater}.

The convex hull of Mpure is the set of mixed-state N-body spin-density matrices, defined as

Mmixed=k=1+λkΓΨk:0λk1,k=1+λk=1,ΨkW,
(8)

and, as shown by Coleman in Ref. 26, it coincides with the convex hull of MSlater. For every ΓMmixed, we define the associated 1-body spin-density matrix

γΓ=γΓ++γΓ+γΓ+γΓ,

where for a, b = +/−, we have

γΓab(x,y)=Nτ{12,12}N1R3(N1)Γ(x,a,z,τ;y,b,z,τ)dz.

We denote the set of mixed-state 1-body spin-density matrices by D={γΓ:ΓMmixed}.

In Ref. 26, Coleman proves that

D={γS(L2(R3,C2)):0γ1;Tr(γ)=N;Tr(Δγ)<+},
(9)

where S(L2(R3,C2)) is the space of self-adjoint operators on L2(R3,C2). In the equation above, we write Tr(Δγ) for Tr(||γ||)=Tr((Δ)1/2γ(Δ)1/2). The subset of D given by the 1-body spin-density matrices associated with Slater determinants, instead, is equal to the set of rank-N projections

P={γΓ:ΓMSlater}={γD:γ2=γ}.

Note that, as a consequence of the spectral theorem (see Chap. VII of Ref. 27), every γD can be diagonalized in an orthonormal basis {ϕk}kN of L2(R3,C2). In other words, we have that

γ=k=1+λkϕkϕk,
(10)

with 0λk1, k=1+λk=N, ϕk=(ϕk+;ϕk)TL2(R3;C2), (ϕk,ϕ)L2(R3,C2)=δk, and

Tr(Δγ)=k=1+λkϕkL22.

Clearly, ϕk are eigenfunctions of γ with corresponding eigenvalues λk and the components of γ have the form

γab(x,y)=k=1+λkϕka(x)ϕkb(y)¯,

where a, b = +/−.

Simple algebraic calculations imply that for any Slater determinant Ψ=1N!det(ψk), the following identity follows:

ESMHF(ψ1,,ψN,A)=ESM(Ψ,A)=Tr(HSMΓΨ)=ESMHF(γΓΨ,A),

where we define

ESMHF(γΓΨ,A)=12Tr(σ(iA)γΓΨ(iA)σ)+R3vργΓΨ
(11)
+12R3×R3ργΓΨ(x)ργΓΨ(y)TrC2(|γΓΨ(x,y)|2)|xy|dxdy
(12)
+1α2R3|×A|2,
(13)

and

ργΓΨ=ργΓΨ+(x)+ργΓΨ(x)=k=1+λk|ϕk+(x)|2+k=1+λk|ϕk(x)|2

is the total electronic density associated with γΓΨ, with γΓΨ decomposed according to (10).

Note that the definition of the energy (11) can be extended to mixed-states. In other words, (11) is well defined for any (γ,A) such that

(γ,A)C=(γ,A):γDAL6(R3);A=0;×AL2(R3).
(14)

Moreover, Lieb’s variational principle (see Ref. 28) implies that minimizing over mixed-states or rank-N projections does not change the value of the ground state. A simple proof of this is given by Bach in Ref. 29 and it can be easily adapted to the spin-magnetic case. Hence, it is clear that recasting problem (6) in terms of the 1-body spin-density matrices leads us to the following variational problem:

ESMHF=inf{ESMHF(γ,A):(γ,A)C}.
(15)

The main result of this paper is the following theorem, whose proof is outlined in Sec. II:

Theorem I.1

(Existence of HF Minimizers: Spin-Polarized Case).If each nuclear charge satisfiesZkα20.052andNZ=k=1KZk, then there exists a minimizer(γ,A)Cfor (15).

Remark I.2.

The conditionZkα20.052is precisely the condition for magnetic stability derived in Ref. 4. This meansZk975, which includes all elements in the periodic table.

Using similar ideas to those employed in the proof of Theorem I.1, we can also prove the existence of Hartree-Fock minimizers for spinless Coulomb systems interacting with self-generated magnetic fields, as provided by the following theorem:

Theorem I.3
(Existence of HF Minimizers: Spinless Case).IfNZ=k=1KZk, then there exists a minimizer(γ,A)Cfor the spinless counterpart of Problem (15)
EMHF=inf{EMHF(γ,A):(γ,A)C},
(16)
where
EMHF(γ,A)=12Tr((iA)γ(iA))+R3vργ
(17)
+12R3×R3ργ(x)ργ(y)TrC2(|γ(x,y)|2)|xy|dxdy
(18)
+1α2R3|×A|2.
(19)

Remark I.4.

Note that, in the spinless case, we do not need to impose the condition for magnetic stability on the single nuclear charges and the theorem holds true for any neutral or positively charged system.

The article is organized as follows: in Sec. II, we prove Theorem I.1. Our proof is based on variational techniques applied to the Hartree-Fock energy functional. In particular, it relies on the fact that we can relax the trace constraint on the density matrix γ, solve the relaxed problem using the direct method of calculus of variation, and finally show that the minimizer to the relaxed problem has in fact the right trace (i.e., Tr(γ)=N). This last fact is a consequence of Lemma II.15 that characterizes the eigenvalues of the operator defined by the Euler-Lagrange equations associated with (15). This Lemma will be proved in Sec. III. Theorem I.3 follows directly from the proof of Theorem I.1.

As we have seen, the first step in our proof is to relax the trace constraint. In the following lemma, we prove that the ground state is not affected by this relaxation.

Lemma II.1.
The following equality holds
ESMHF=ESMHFR,
where
ESMHFR=infESMHF(γ,A):(γ,A)CR
(20)
is therelaxed problem,
CR=(γ,A):γDRAL6(R3);A=0;×AL2(R3)
and
DR={γS(L2(R3,C2)):0γ1;Tr(γ)N;Tr(γ)<+}.

Remark II.2.

Given two Hermitian matricesR,SCN×N, we say thatRSif(Rξ,ξ)CN(Sξ,ξ)CNfor allξCN.

Proof.
We begin by noticing that we can replace CR in (20) by
C0R=(γ,A):(γ,A)CR;Ran(γ)isfinite;Ran(γ)C0(R3,C2),
where C0(R3,C2) denotes the space of smooth functions with compact support. This follows from the fact that finite rank operators are dense in DR, C0(R3,C2) is dense in H1(R3,C2), and functional (4) is continuous in the strong topology of DR.
It is clear that ESMHFRESMHF since we are minimizing on a larger class of admissible functions. Thus, we just need to prove that for every (γ,A)C0R,
ESMHFESMHF(γ,A).
Fix (γ,A)C0R arbitrarily. Define μ=NTr(γ)0. Since γ is finite rank and compact support, it can be decomposed as
γ=k=1Nγnkψkψk,
where ψk(x)=ψk+(x);ψk(x)T; ψk+, ψk compact support; and (ψk;ψ)L2(R3,C2)=δk.
Since {ψk+/}k=1NγC0(R3), we can find a compact set ΩR3 such that supp(ψk+/)Ω for all k=1,,Nγ. For any given δ>0, consider ϕk(x)=ϕk+(x);ϕk(x)T where ϕk+/ are smooth functions with compact support such that supp(ϕk+/)Ωc for all k=1,,Nγ; (ϕk,ϕ)L2(R3,C2)=δk and such that if we define
γ̃=k=1Nγñkϕkϕk,
with the coefficients ñk conveniently chosen, then Tr(γ̃)=μ.
Moreover, we require that
L=12Tr(σ(iA)γ̃(iA)σ)+12R3×R3ργ̃(x)ργ̃(y)TrC2|γ̃(x,y)|2|xy|dxdyδ.
Note that if A = 0, this can always be achieved by rescaling ϕk, and for general AL6(R3) this can always be achieved by rescaling and translating ϕk.
Let e0R3 be a unit vector such that supp(ϕk+/(+ne0))Ωc for all k=1,,Nγ and for all nN and define ψkn=12ψk+12ϕk(+ne0) for all k=1,,Nγ. Define
γn=k=1Nγ(nk+ñk)ψknψkn.
Note that
(ψkn,ψjn)L2(R3,C2)=(12ψk+12ϕk(+ne0),12ψj+12ϕj(+ne0))L2(R3,C2)=12(ψk,ψj)L2(R3,C2)+12(ϕk(+ne0),ϕj(+ne0))L2(R3,C2)+12(ψk;ϕj(+ne0))L2(R3,C2)+12(ϕk(+ne0),ψj)L2(R3,C2).
Since ϕk and ϕj have compact support, the only surviving terms are 12(ψk,ψj)L2(R3,C2) and 12(ϕk(+ne0),ϕj(+ne0))L2(R3,C2)=12(ϕk;ϕj)L2(R3,C2). Thus,
(ψkn,ψjn)L2(R3,C2)=12(ψk,ψj)L2(R3,C2)+12(ϕk,ϕj)L2(R3,C2)=δkj
and Tr(γn)=k=1Nγ(nk+ñk)=N. In the same way, since AL6(R3) and ×AL2(R3), we have that
ESMHF(γn,A)L+ESMHF(γ,A).
Thus, we obtain that
ESMHFL+ESMHF(γ,A)δ+ESMHF(γ,A).
Since δ>0 is arbitrary, the conclusion follows.□

To prove the existence of minimizers for the relaxed problem, we follow the direct method of the calculus of variations. As a first step, we prove that any minimizing sequence of (20) is bounded in an appropriate norm, to be specified later. Hence, we will be able to extract a weakly convergent subsequence, and the existence of minimizers will be established once we prove that functional ESMHF is weakly lower semicontinuous. To show the boundedness of any minimizing sequence, we use the following result, which is an extension of a similar result proved in Ref. 4 in the many-body case:

Lemma II.3.
IfZkα20.052fork=1,,K, there exist positive constants K1and K2, depending solely on N andα, such that for every(γ,A)C,
ESMHF(γ,A)K1R3|×A|2K2.
(21)

Proof.
From the proof of Theorem 1, part (A) of Ref. 4, we deduce that there exist positive constants K1 and K2 such that for any (Ψ,A)DSM, the following inequality holds:
ESM(Ψ,A)K1R3|×A|2K2.
Moreover, for any (γ,A)C, we have that γD and the proof of Lieb’s variational principle (see Ref. 28) implies that there exists ΓMmixed such that γ=γΓ and Tr(HSMΓ)ESMHF(γ,A).

On the other hand, by the definition of Mmixed, we know that Γ=k=1+λkΨkΨk, where 0λk1, k=1+λk=1, and ΨkW.

Hence, the following inequalities hold:
ESMHF(γ,A)Tr(HSMΓ)=k=1+λk(Ψk,HSMΨk)L2(R3N,C2N)=k=1+λkESM(Ψk,A)K1R3|×A|2K2,
as claimed.□

We are now able to prove the following lemma, which together with Lemma II.3 establishes the coercivity of functional ESMHF in C:

Lemma II.4.
Assume thatZkα20.052fork=1,,K. Then, there exists a positive constantK3>0such that if(γn,An)Cis a minimizing sequence for (20), then
ESMHF(γn,An)14Tr((iAn)γn(iAn))K3.
(22)

Proof.

Since the ground state energy ESM is always a lower bound for ESMHF, it follows from the magnetic stability result proved in Ref. 4 that ESMHFESM>.

Consider a minimizing sequence (γn,An)nNC for (20), that is, for every nN, (γn,An)C and
limn+ESMHF(γn,An)=ESMHF.
Note that, given Lemma II.1, it is always possible to find a minimizing sequence for (20) that is in C rather than CR. Since (γn,An)C, as a consequence of the spectral theorem (see Chap. VII of Ref. 27), we can write
γn(x,y)=γn++(x,y)γn+(x,y)γn+(x,y)γn(x,y),
where
γn=k=1+λknϕknϕkn,
(23)
and 0λkn1, k=1+λkn=N; ϕkn=(ϕkn+;ϕkn)TL2(R3;C2); (ϕkn;ϕn)L2(R3;C2)=δk.
Each component of γn is given by
γnab(x,y)=k=1+λknϕkna(x)ϕknb(y)¯,
(24)
where a, b = +/−.
Given that (γn,An) is a minimizing sequence and the ground state energy is finite, it follows that ESMHF(γn,An) is a bounded sequence, which means that M1>0 such that |ESMHF(γn,An)|M1 for every n. As a consequence of Lemma II.3, we get that for every n,
M1ESMHF(γn,An)K1R3|×An|2K2.
This implies that {×An}nN is uniformly bounded in L2(R3): M2>0 such that
×AnL2M2  nN.
(25)
Besides, simple algebraic calculations provide that for any (γ,A)C, it holds
Tr(σ(iA)γ(iA)σ)=Tr((iA)γ(iA))R3mγ(×A),
(26)
where mγ=TrC2(σRγ) is the spin angular momentum density, Rγ(x)M2×2(C) is the electronic spin-density matrix associated with γ, and the last term in (26) is the direct coupling. For γ trace class, the 2×2 matrix
Rγ(x)=ργ++(x)ργ+(x)ργ+(x)ργ(x)
is defined through the formula
ργab(x)=k=1+λkϕka(x)ϕkb(x)¯,
with a, b = +/− and where γ is decomposed according to (10). The total electronic density associated with γ is defined as
ργ(x)=ργ++(x)+ργ(x)=k=1+λk|ϕk+(x)|2+k=1+λk|ϕk(x)|2.
As a consequence of (26), we can write
ESMHF(γn,An)=12Tr(σ(iAn)γn(iAn)σ)+R3vργn+12R3×R3ργn(x)ργn(y)TrC2(|γn(x,y)|2)|xy|dxdy+1α2R3|×An|2=12Tr((iAn)γn(iAn))12R3mγn(×An)+R3vργn+12R3×R3ργn(x)ργn(y)TrC2(|γn(x,y)|2)|xy|dxdy+1α2R3|×An|2.
We consider each term in the energy separately. First, note that the Cauchy-Schwarz inequality for series implies that
TrC2(|γ(x,y)|2)ργ(x)ργ(y),
(27)
and therefore
12R3×R3ργn(x)ργn(y)TrC2|γn(x,y)|2|xy|0.
(28)
As it concerns the direct coupling term, the Cauchy-Schwarz inequality provides that
12R3mγn×AnmγnL2×AnL2mγnL2M2,
(29)
where the last inequality follows from (25).
By definition, we have that
mγn=TrC2(σRγn)=ργn++ργn+iργn+iργn+ργn++ργn
(30)
and, consequently, that
mγnL224ργn+L22+2ργn++L22+2ργnL22,
(31)
where (31) follows from the fact that ργn+=ργn+¯ and from the standard inequality
2ργn++ργn(ργn++)2+(ργn)2.
Given (31), it is clear that to control the direct coupling term, we need to control the L2-norm of ργnab. In particular, we have the following bound, which is an extension of Lemma 1 in Ref. 20:

Lemma II.5.
For all1p3,Cp>0constant such that for all(γ,A)Cand a, b = +/−, it holds
ργabLpCpN3p2pTr((iA)γ(iA))3(p1)2p.
(32)

Proof.
First, note that for a = +/− and γ decomposed as γ=k=1+λkϕkϕk, it holds
|ργaa|22k=1+λk|ϕka|2k=1+λk||ϕka||2
(33)
4ργaak=1+λk|(iA)ϕka|2,
(34)
where (33) follows from the discrete Cauchy-Schwarz inequality and (34) follows from the diamagnetic inequality. Integrating on both sides leads to
ργaaL22Tr((iA)γaa(iA)),
which together with the Sobolev inequality implies that
ργaaL3M3Tr((iA)γaa(iA)),
(35)
with M3 positive constant. In the same way, when a, b = +/−, we denote
ρ|γ|ab(x)=k=1+λk|ϕka(x)||ϕkb(x)|.
For this function, analogously to (33), we have
|ρ|γ|ab|k=1+λk|(iA)ϕka|2+|(iA)ϕkb|21/2k=1+λk|ϕka|2+|ϕkb|21/2.
If we integrate on both sides, apply Hölder’s inequality and (35), we obtain
ρ|γ|abL3/2Tr((iA)γ(iA))1/2ργaa+ργbbL31/2M4Tr((iA)γ(iA)),
with M4 positive constant. Furthermore, the Sobolev embedding W1,3/2(R3)L3(R3) implies that
ρ|γ|abL3M5Tr((iA)γ(iA)),
(36)
with M5 positive constant. Note that if a = b, ρ|γ|aa=ργaa and we recover (35). Clearly, ρ|γ|ab is also in L1(R3), given the inequality
ρ|γ|abk=1+λk2(|ϕka|2+|ϕkb|2)ργaa+ργbb=ργ,
which implies
ρ|γ|abL1Tr(γ)=N.
(37)
As a consequence, we have that (36) and (37) and the interpolation inequality provide the following bound, for 1p3,
ρ|γ|abLpCpN3p2pTr((iA)γ(iA))3(p1)2p.
(38)
Finally, since |ργab|ρ|γ|ab, the desired inequality for ργabLp follows.□
It is now clear that29,31 Lemma II.5 and Young’s inequality imply that for every η>0, M6>0 constant such that
12R3mγn×AnηTr((iAn)γn(iAn))+M6.
(39)
At this point, the only term we still need to control is the potential term. Note that the potential can be decomposed as
v(x)=k=1KZk|xRk|1{|xRk|1}(x)k=1KZk|xRk|1{|xRk|>1}(x),
where, given a set T, 1T denotes its indicator function.
Thus vL32+𝜖(R3)+L(R3), for every 𝜖>0. Hölder’s inequality implies that
R3v(x)ργn(x)dxvL32+𝜖+LργnL1L3𝜖.
(40)
If we apply (32) with p = 1 and p=3𝜖, respectively, we obtain
R3v(x)ργn(x)dxM7vL32+𝜖+L(1+(Tr((iAn)γn(iAn)))α1),
(41)
where M7 is a positive constant and 0α1<1. Again, Young’s inequality provides that for every η>0, M8>0 constant such that
R3v(x)ργn(x)dxηTr((iAn)γn(iAn))+M8.
(42)
If we collect the inequalities in (28), (39), and (42), we obtain
ESMHF(γn,An)122ηTr((iAn)γn(iAn))K3,
(43)
where K3 = M6 + M8, and the desired result follows by choosing η<1/8.□
As a consequence of Lemmas II.3 and II.4, we get that, given any minimizing sequence {(γn,An)}nNC, the sequences {×An}nN and {Tr((iAn)γn(iAn))}nN are uniformly bounded in L2(R3) and R, respectively. By the Gagliardo-Nirenberg inequality and the fact that An=0, we also obtain that
AnL6CGN×AnL2CGNM2  nN,
(44)
which means that {An}nN is uniformly bounded in L6(R3).
Note that for each ϕkn in (23), it holds that iϕkn=(iAn)ϕkn+Anϕkn, which implies that
ϕknL2(iAn)ϕknL2+AnϕknL2.
(45)
In addition, for every η>0, there exists a constant G1>0, independent of n, such that
AnϕknL2AnL6ϕknL3AnL6ϕknL21/2ϕknL61/2CGN1/2AnL6ϕknL21/2ηϕknL2+G1,
(46)
where the inequalities above follow, respectively, from Hölder’s inequality, interpolation inequality, the fact that ϕkn have unit L2-norm and the Sobolev inequality, uniform boundedness of {An}nN in L6(R3) and Young’s inequality. From (45) and (46), we deduce that for η<1/2,
12ϕknL2(iAn)ϕknL2+G1,
which implies that
Tr(Δγn)G2Tr((iAn)γn(iAn))+G3,
where G2, G3 are positive constants independent of n.
Finally, recalling that {Tr((iAn)γn(iAn))}nN is uniformly bounded, we conclude that the sequence {γn}nN is also uniformly bounded in the Banach space
B={γS(L2(R3,C2)),γB=Tr(|γ|)+Tr(||γ||)<+}.
As a consequence, there are αL6(R3) such that ×αL2(R3) and γ̃B such that, by passing to a subsequence,
Annα     weaklyinL6(R3),
(47)
×Ann×α   weaklyinL2(R3),
(48)
γnnγ̃      weakly*inB.
(49)
Furthermore, the Rellich-Kondrachov theorem, Lemma II.5, and the Banach-Alaoglu theorem imply that all components of Rγn converge to their respective components of Rγ̃ strongly in Llocp(R3) for 1p<3, weakly in Lp(R3) for 1p3, and pointwise almost everywhere. A proof of that can be found in Ref. 20. In addition, α=0 and Tr(γ̃)N.

In the next lemma, we establish the weak lower semicontinuity of functional ESMHF. This clearly implies that (γ̃,α)CR is a minimizer for problem (20).

Lemma II.6.
FunctionalESMHFis*-weakly lower semicontinous. In other words, given a sequence{(γn,An)}nNCand(γ̃,α)CRsuch thatγnn*γ̃weak-*inB,Annαweakly inL6(R3),×Ann×αweakly inL2(R3), then
lim infn+ESMHF(γn,An)ESMHF(γ̃,α).
(50)

Proof.
Recall that the energy is given by
ESMHF(γn,A)=12Tr(σ(iAn)γn(iAn)σ)+R3vργn+12R3×R3ργn(x)ργn(y)TrC2(|γn(x,y)|2)|xy|dxdy+1α2R3|×An|2.
We consider each term separately. First, we look at the potential term, whose weak-* continuity is proved in the following lemma:

Lemma II.7
(Weak-* continuity of the potential term). Letγn,γ̃,An, andαbe the one defined in (47)–(49), then we have that
limn+R3ργnv=R3ργ̃v.
(51)

Proof.
Note that for every η>0, it is possible to decompose v in the following way: v=v1η+v2η, where v1ηL3/2(R3), v2ηL(R3), and v2ηL<η. In fact, v(x)=k=1KZk|xRk| and, for each k, we can write
1|xRk|=1|xRk|1{|xRk|1/η}v1η,k+1|xRk|1{|xRk|>1/η}v2η,k.
It is easy to see that
R31|xRk|1{|xRk|1/η}3/2dx={|xRk|1/η}1|xRk|3/2dx=2π201/η1r3/2r2dr=4π23η3/2<+,
and that v2η,kL<η. Define vjη=k=1KZkvjη,k for j = 1, 2, then it is clear that vjη for j = 1, 2 has the required properties.
Thus, for each η>0, since v1ηL3/2(R3) (the dual space of L3(R3)) and ργnnργ̃ weakly in L3(R3), we obtain that
limn+R3ργnv1η=R3ργ̃v1η.
(52)
On the other hand,
R3ργnv2ηR3ργ̃v2ηR3|ργnργ̃||v2η|ργnργ̃L1v2ηL2Nη.
(53)
The lemma thus follows.

The next step is proving the weak-* lower semi-continuity of the direct and exchange Coulomb term:

Lemma II.8
(Weak-* lower semi-continuity of the direct and exchange Coulomb term). Letγn, γ̃, An, andαbe the one defined in (47)–(49), then we have that
lim infn+R3×R3ργn(x)ργn(y)TrC2(|γn(x,y)|2)|xy|dxdyR3×R3ργ̃(x)ργ̃(y)TrC2(|γ̃(x,y)|2)|xy|dxdy.
(54)

Proof.
Since {γn}nN is bounded in B, its kernel γn(x,y) is bounded in H1(R3×R3,Mat2×2(C)) and ργn is bounded in H1(R3). Hence, up to a subsequence, we can assume that
ργnnργ̃     inLlocp(R3)for1p<3anda.e.,
(55)
γn(x,y)nγ̃(x,y)    inLlocq(R3)for2q<3anda.e.
(56)
Thus, from (27) and Fatou’s lemma, (54) follows.
Given that ×Ann×α weakly in L2(R3), from weakly lower semicontinuity of norms, it also follows that
lim infn+×AnL22×αL22.
(57)
Finally, note that if we prove that
σ(iAn)γn(iAn)σn*σ(iα)γ̃(iα)σ
(58)
weakly-* in S1, the space of trace-class operators on L2(R3,C2), then
lim infn+12Tr(σ(iAn)γn(iAn)σ)12Tr(σ(iα)γ̃(iα)σ)
(59)
will follow from Fatou’s lemma for series.

The weak-* convergence of the kinetic energy term is treated in the following lemma:

Lemma II.9
(Weak-* convergence of the kinetic energy term). For n going to infinity,
σ(iAn)γn(iAn)σn*σ(iα)γ̃(iα)σ
(60)
weakly-*inS1.

Proof.
First note that it is sufficient to prove
(iAn)γn(iAn)n*(iα)γ̃(iα)
(61)
weakly-* in S1. In fact, the Pauli operator σ does not compromise the weak-* convergence since it has the only effect of mixing the order of the components of (iAn)γn(iAn).
Thus, we shall prove (61) directly. In order to do that, for any nN, we define the auxiliary positive operators
τn=(i+(iAn))γn(i+(iAn))
(62)
=γniγn(iAn)+i(iAn)γn+(iAn)γn(iAn).
(63)
As the following lemma proves, these operators are uniformly bounded in S1:

Lemma II.10

(Boundedness in S1). The operators{τn}nNcan be extended toL2(R3,C2)and are uniformly bounded inS1.

Proof.

Note that since (γn,An)C, then, in particular, γn is a positive trace class operator defined on L2(R3,C2) and γnS1=Tr(γn)=N, which means that {γn}nN is uniformly bounded in S1.

On the other hand, the operators γn(iAn) and (iAn)γn are defined in D(iAn), the domain of (iAn), but can be extended to L2(R3,C2), where they are bounded and adjoint of each others. Moreover, they are in S1 and the sequence {Tr(γn(iAn))=Tr((iAn)γn)}nN is uniformly bounded.

In fact, let ψ be in D(iAn) and let γn have decomposition γn=k=1+λknϕknϕkn. Thus,
γn(iAn)ψ=k=1+λkn(ϕkn,(iAn)ψ)L2(R3,C2)ϕkn.
(64)
We want to show that ϕknD(iAn). In fact, there exists a constant D1 such that for every ψD(iAn), (ϕkn,(iAn)ψ)L2(R3,C2)D1ψL2(R3,C2). This follows from the following inequalities (for clarity reasons, we omit the subscript L2(R3,C2) when the inner product or norm we are using is clear):
λkn(ϕkn,(iAn)ψ)2k=1+λkn(ϕkn,(iAn)ψ)2=((iAn)ψ,γn(iAn)ψ)=(ψ,(iAn)γn(iAn)ψ)(iAn)γn(iAn)ψL22,
(65)
where the first inequality follows from λkn>0 and the last one from the boundedness of (iAn)γn(iAn) in S1. In this paper, given an operator T, we denote its operator norm by T and its adjoint by T*.
It follows from (65) that the functional
D(iA)Cψ(ϕkn,(iAn)ψ)
is linear and bounded and can thus be extended to L2(R3,C2). Hence, for each k, ϕknD(iAn)*=D(iAn), since (iAn) is self-adjoint. Thus, (64) becomes
γn(iAn)ψ=k=1+λknϕkn((iAn)ϕkn,ψ)
and
γn(iAn)ψL22=k=1+(λkn)2|((iAn)ϕkn,ψ)|2k=1+(λkn)2(iAn)ϕknL22ψL22k=1+(λkn)(iAn)ϕknL22ψL22=Tr((iAn)γn(iAn))ψL22D2ψL22,
given that {(iAn)γn(iAn)}nN is uniformly bounded in S1.

As a consequence, γn(iAn) can be extended to a bounded operator defined on the entire space L2(R3,C2).

Also, (iAn)γn=(γn(iAn))*. In fact, for every ψ,𝜃L2(R3,C2),
((iAn)γnψ,𝜃)=k=1+λkn(ϕkn,𝜃)((iAn)ϕkn,ψ)=(ψ,(iAn)γn𝜃).
Finally, note that
(γn1/2(iAn))*(γn1/2(iAn))=(iAn)γn(iAn)S1,
which implies that γn1/2(iAn)S2, where S2 is the space of Hilbert-Schmidt operators on L2(R3,C2). Thus, γn(iAn)=(γn1/2)S2(γn1/2(iAn))S2S1 and clearly also (iAn)γn=(γn(iAn))*S1. Moreover
(Tr((iAn)γn))2=(Tr(γn(iAn)))2=Tr(γn1/2γn1/2(iAn))γnS22γn1/2(iAn)S22=Tr(γn)Tr((iAn)γn(iAn)),
which, as we have seen before, are both uniformly bounded. As a consequence, Tr(γn(iAn)) and Tr((iAn)γn) are both uniformly bounded, as claimed.

In the same way, (iAn)γn(iAn) is a positive operator that can be extended to L2(R3,C2) and is uniformly bounded in S1.

In conclusion, the operators τn are positive and can be extended to L2(R3,C2), and given that
Tr(τn)=Tr(γn)iTr(γn(iAn))+iTr((iAn)γn)+Tr((iAn)γn(iAn))
is a sum of terms which are all uniformly bounded in S1, {τn}nN is uniformly bounded in S1.
Consider the following operators:
Rn=(i+(iAn))1,
(66)
R=(i+(iα))1.
(67)
Standard results on self-adjoint operators (see Chap. X of Ref. 30) imply that Rn is a bounded invertible operator and that
Rndist(i,spec((iAn)))1k̃,
(68)
where k̃ is a constant, dist(,) is the Euclidean distance, and spec(iAn) is the spectrum of (iAn). In other words, {Rn}nN is uniformly bounded in S(L2(R3,C2)). Clearly, the same holds for its adjoint, Rn*=(i+(iAn))1, and for R and its adjoint, R*=(i+(iα))1. By definition, it holds that
γn=RnτnRn*.
Note that Lemma II.10 implies that there exists τ̃S1 such that τnn*τ̃ weakly-* in S1. Also, in (49), we proved that there exists γ̃B such that γnn*γ̃ weakly-* in B.
To prove Lemma II.9, we need to identify the weak-* limit in B of (iAn)γn(iAn). This is equivalent to identifying, for every ϕ,ψC0(R3,C2), smooth functions with compact support, the limit of (ϕ,(iAn)γn(iAn)ψ). Let us write
(ϕ,(iAn)γn(iAn)ψ)=(ϕ,τnψ)(ϕ,γnψ)+i(ϕ,γn(iAn)ψ)i(ϕ,(iAn)γnψ).
(69)
In the following lemma, we identify each limit in the right hand side of (69):

Lemma II.11.
Given anyϕ,ψC0(R3,C2), as n goes to infinity, the following limits are obtained:
(ϕ,τnψ)n(ϕ,R1γ̃(R1)*ψ),
(70)
(ϕ,γnψ)n(ϕ,γ̃ψ),
(71)
(ϕ,γn(iAn)ψ)n(ϕ,(Rτ̃iγ̃)ψ),
(72)
(ϕ,(iAn)γnψ)n(ϕ,(τ̃R*+iγ̃)ψ).
(73)

Proof.

Since τnn*τ̃ weakly-* in S1, then clearly (ϕ,τnψ)n(ϕ,τ̃ψ).

We want to show that
τ̃=R1γ̃(R1)*.
(74)
First, note that
(ϕ,γnψ)=(ϕ,RnτnRn*ψ)
(75)
=(Rn*ϕ,τnRn*ψ)
(76)
=(Rn*ϕR*ϕ,τnRn*ψ)+(R*ϕ,τn(Rn*ψR*ψ))+(R*ϕ,τnR*ψ),
(77)
where (76) follows since Rn is bounded. Let us consider the limit for n going to infinity of (77). Note that
0|(Rn*ϕR*ϕ,τnRn*ψ)|Rn*ϕR*ϕL2τnRn*ψL2D3Rn*ϕR*ϕL2ψL2,
(78)
where D3 is a constant. The inequality above follows from the fact that both {τn}nN and {Rn*}nN are uniformly bounded in S(L2(R3,C2)).

In addition, we have strong convergence in L2(R3,C2) of Rn*nR*, as showed by the following lemma:

Lemma II.12

(Strong convergence of Rn).For n going to infinity,RnnRstrongly inL2(R3,C2).

Proof.
Note that
0RnϕRϕL2=Rn(Rn1R1)RϕL2Rn(Rn1R1)RϕL2
(79)
D4(Anα)RϕL2,
(80)
where D4 is a constant and (80) follows since {Rn}nN is uniformly bounded in S(L2(R3,C2)). Moreover, for every ϕL2(R3,C2), RϕHα1(R3,C2), that is, RϕL2(R3,C) and (iα)RϕL2(R3,Mat3×2(C)). As a consequence of the diamagnetic inequality |Rϕ|H1(R3). Thus, if we prove that for every 𝜃H1(R3,C2), (Anα)𝜃L2n0, then we can conclude that RnϕRϕL2n0.
First, consider any 𝜃C0(R3,C2), then
0(Anα)𝜃L2=supp(𝜃)|Anα|2|𝜃|2
(81)
supp(𝜃)|Anα|41/2supp(𝜃)|𝜃|41/2.
(82)
Since {An}nN is uniformly bounded in L6(R3) and Annα weakly in L6(R3), Rellich-Kondrachov theorem implies that Annα in Lloc4(R3). Hence, (82) implies that for every 𝜃C0(R3,C2), (Anα)𝜃L2n0. By density of C0(R3,C2) in H1(R3,C2), the same result follows for any 𝜃H1(R3,C2), as claimed.

It is clear that the same result of Lemma II.12 holds for Rn* and R*. Hence, Lemma II.12 and (78) imply that (Rn*ϕR*ϕ,τnRn*ψ)n0 as n goes to infinity.

In the same way, it is possible to prove that (R*ϕ,τn(Rn*ψR*ψ))n0 and, since τnn*τ̃ weakly-* in S1, then we also have that (R*ϕ,τnR*ψ)n(R*ϕ,τ̃R*ψ).

Now, recall that γnn*γ̃ weakly-* in B and take the limit in (77) for n going to infinity. We obtain that for every ϕ,ψC0(R3,C2),
(ϕ,γ̃ψ)=(R*ϕ,τ̃R*ψ)=(ϕ,Rτ̃R*ψ).
It follows that γ̃=Rτ̃R* and τ̃=R1γ̃(R1)*, which proves (74) and thus (70).

Moreover, since γnn*γ̃ weakly-* in B, then clearly (71) follows.

In addition, note that the following identities hold
(ϕ,γn(iAn)ψ)=(ϕ,γn(Rn1)*ψ)i(ϕ,γnψ)
(83)
=(ϕ,RnRn1γn(Rn1)*ψ)i(ϕ,γnψ)
(84)
=(ϕ,Rnτnψ)i(ϕ,γnψ)
(85)
=(Rn*ϕ,τnψ)i(ϕ,γnψ)
(86)
n(R*ϕ,τ̃ψ)i(ϕ,γ̃ψ),
(87)
where (86) holds since Rn is bounded and (87) follows from Lemma II.12 and the fact that τnn*τ̃ weakly-* in S1. Thus, (72) follows and, in the same way, it is possible to prove (73).
Now, if we take the limit for n going to infinity on both sides of (69), Lemma II.11 implies that for every ϕ,ψC0(R3,C2),
(ϕ,(iAn)γn(iAn)ψ)n(ϕ,(τ̃γ̃+iRτ̃+γ̃iτ̃S+γ̃)ψ)=(ϕ,(τ̃+iRτ̃iτ̃S+γ̃)ψ),
which means that (iAn)γn(iAn)n*τ̃+iRτ̃iτ̃S+γ̃ weakly-* in S1. If we substitute τ̃=R1γ̃S1, we obtain (iAn)γn(iAn)n*(iα)γ̃(iα) weakly-* in S1, as claimed.

From (51), (54), (57), and (59), we deduce that the energy is weakly-* lower semicontinuous and the proof of Lemma II.6 is thus concluded.

We can combine all the previous results in the following theorem:

Theorem II.13.

IfZkα20.052for allk=1,,KandNZ=k=1KZk, the functionalESMHFhas a minimizer inCR.

Proof.
Given Lemma II.1, (47)–(49), Lemma II.6 and the fact that (γn,An)nN is a minimizing sequence for (20), we have that
ESMHFR=lim infn+ESMHF(γn,An)ESMHF(γ̃,α).
Since (γ̃,α)CR, it follows that
ESMHF(γ̃,α)ESMHFR.
Thus, ESMHF(γ̃,α)=ESMHFR=ESMHF and (γ̃,α) is a minimizer to the relaxed problem.

To complete the proof of Theorem I.1, we need to show that there is a minimizer in C. Clearly, it is sufficient to prove that Tr(γ̃)=N. We first need to make an initial remark:

Remark II.14.
We can characterizeγ̃in the following way:
γ̃argminTr(Hγ̃γ):γDR,
(88)
where
Hγ̃=12|σ(iα)|2+v+ργ̃*1|x|I2×2Kγ̃.
In the above,I2×2is the2×2identity matrix andKγ̃is the exchange operator defined by the2×2matrix valued integral kernelKγ̃(x,y)=γ̃(x,y)|xy|.

Proof.

For every γDR, since (γ̃,α) is a minimizer for ESMHF in CR, we have that ESMHF(γ,α)ESMHF(γ̃,α). Therefore, ddt(ESMHF((1t)γ̃+tγ,α))|t=00, which is equivalent to Tr(Hγ̃γ)Tr(Hγ̃γ̃). In other words, γ̃argmin{Tr(Hγ̃γ):γDR}, as claimed.

In addition, we need the following Lemma whose proof we postpone to the end. This Lemma extends a classical result by Lions (see Ref. 10) to the magnetic Hamiltonian.

Lemma II.15.
Letμbe a bounded non-negative measure onR3s.t.μ(R3)<Z. The Hamiltonian
H1=12|σ(iα)|2+v+μ*1|x|I2×2
has an increasing sequence of negative eigenvalues converging to 0.
Let (γ̃,α) be the minimizer for CR, with decomposition
γ̃=k=1+λkϕkϕk,  0<λk1,
(since they do not contribute to it, we are excluding from the sum all the terms with λk=0). First, note that λk and ϕk have the following property:

Proposition II.16.

Ifλk>0, then(Hγ̃ϕk,ϕk)0.

Proof.

Assume there exists k* such that λk*0 but (Hγ̃ϕk*,ϕk*)>0.

Define γ*=γ̃λk*ϕk*ϕk*. Note that Tr(γ*)=k=1,kk*+λk<Tr(γ̃)N and that Tr(Hγ̃γ*)=Tr(Hγ*γ*)λk(Hγ̃ϕk*,ϕk*)<Tr(Hγ*γ̃), which contradicts (88).

Now, assume by contradiction that Tr(γ̃)<N. We distinguish two cases:

Case 1.

γ̃ is not finite rank.

In this case, we have a stronger result than the one proved in Proposition II.16:

Proposition II.17.

There existsk*such that0<λk*<1and(Hγ̃ϕk*,ϕk*)<0.

Proof.

Define the set J={j:λj=1}. Clearly, J is a finite set for otherwise k=1+λk=Tr(γ̃) would not converge. Let the cardinality of J be n¯1 and its elements be reordered as J={λ1,,λn¯1}. Since γ̃ is not finite rank and J is finite, there is at least one index j* in its decomposition such that λj*<1. Assume, by contradiction, that for every 0<λj*<1, (Hγ̃ϕj*,ϕj*)=0.

By Lemma II.15, Hγ̃ has infinitely many negative eigenvalues. Hence, by the min-max principle, we obtain that
0>μn¯(Hγ̃)=supν1,,νn¯1infψD(Hγ̃),ψL2=1ψ[ν1,,νn¯1](ψ,Hγ̃ψ)
(89)
infψD(Hγ̃),ψL2=1ψ[ϕ1,,ϕn¯1](ψ,Hγ̃ψ),
(90)
where ϕ1,,ϕn¯1 are the eigenfunctions of γ̃ associated with the eigenvalues λ1,,λn¯1 in J. It follows that there exists ψ0D(Hγ̃) such that ψ0L2=1, ψ0[ϕ1,,ϕn¯1], and (ψ0,Hγ̃ψ0)<0.

Define γ*=k=1,kJ+ϕkϕk+δψ0ψ0, where δ is chosen in such a way that 0<δ1 and Tr(γ*)=n¯1+δN (note that it is always possible to choose such a δ since, given that γ̃ is not finite rank and Tr(γ̃)N, then n¯1<N). We thus obtain that Tr(Hγ̃γ*)=Tr(Hγ̃γ̃)+δ(ψ0,Hγ̃ψ0)<Tr(Hγ̃γ̃), which is a contradiction to (88). The proof is thus concluded.

Now, choose 0<δ<1 such that λk*+δ1 and Tr(γ̃)+δN (again it is always possible to find such a δ since λk*<1 and we are assuming Tr(γ̃)<N). Define γ*=γ̃+δϕk*ϕk*. Clearly, Tr(γ*)=Tr(γ̃)+δN and Tr(Hγ̃γ*)=Tr(Hγ̃γ̃)+δ(Hγ̃ϕk*,ϕk*)<Tr(Hγ̃γ̃), which contradicts (88). Therefore, the assumption Tr(γ̃)<N cannot hold if γ̃ is not finite rank.

Case 2.
γ̃ is finite rank. In this case, γ̃ has decomposition
γ̃=k=1Kγ̃λkϕkϕk.
As before, from Lemma II.15 and the min-max principle, we deduce that there exists ψ0D(Hγ̃) such that ψ0L2=1, ψ0[ϕ1,,ϕKγ̃], and (ψ0,Hγ̃ψ0)<0. Again, choose 0<δ<1 such that Tr(γ̃)+δN and define γ*=γ̃+δψ0ψ0. Clearly, Tr(γ*)=Tr(γ̃)+δN and Tr(Hγ̃γ*)=Tr(Hγ̃γ̃)+δ(ψ0,Hγ̃ψ0)<Tr(Hγ̃γ̃), which contradicts (88). Hence, we conclude that the assumption Tr(γ̃)<N cannot hold either when γ̃ is finite rank.

In conclusion, we reached a contradiction in both cases, which means that the assumption Tr(γ̃)<N is never true. We thus proved that Tr(γ̃)=N, which implies that (γ̃,α) is a minimizer for (15). This concludes the proof of Theorem I.1.

Remark II.18.
As it concerns Theorem I.3, we note that Lemma II.1 is still valid in the spinless case. Moreover, Lemma II.5 implies that, given a minimizing sequence(γn,An)Cfor (16), we have the bound
EMHF(γn,An)C112Tr((iAn)γn(iAn))+R3|×An|2C2,
where C1, C2are positive constants. Thus,{γn}nN,{×An}nN, and{An}nNare uniformly bounded inB,L2(R3)andL6(R3), respectively. The rest of the proof then follows as in the spin-polarized case. Note that as a consequence of the diamagnetic and Lieb-Thirring inequalities, the kinetic energy is enough to control the potential energy for any value of Z. Therefore we do not need a bound of the typeZα2Cin this case. We only requiteNZ, which ensures that no electron charge escapes to infinity.

In the absence of magnetic fields, this lemma was proved by Lions in Ref. 10. Here we extend the result to magnetic potentials in the previously specified class. Our proof uses a similar technique to the one of Theorem XIII.6 in Ref. 31.

We first note that σess(H1)[0,+). A proof of this can be found in Appendix A of Ref. 32. Choose a function ψ with the following characteristics: smooth and with compact support (ψ=(ψ+,ψ)TC0(R3,C2)), radially symmetric (ψ(x)=ψ(|x|)), and whose support satisfies supp(ψ)={1<|x|<2}. Rescale the function in the following way: for every λ>0, define ψλ=λ3/2ψ(x/λ). It is clear that supp(ψλ)={λ<|x|<2λ}. Then

(H1ψλ,ψλ)L2(R3,C2)=12R3|ψλ|C22+12R3|α|2|ψλ|C22+R3jψλα12R3mψλ(×α)+R3v|ψλ|C22+R3μ*1|x||ψλ|C22,
(91)

where

jψλ(x)=12iTrC2(ψλ¯ψλψλψλ¯)andmψλ=(ψλ,σψλ)C2,  σ=(σx,σy,σz).

We want to consider (91) term by term to establish its asymptotic order when λ+. In this way, we obtain the following:

  • 12R3|ψλ|C22=(1/λ)212R3|ψ|C22=o(1/λ),

  • 12R3|α|2|ψλ|C2212λsupp(ψλ)|α|601/3R3|ψ|C232/3=o(1/λ), where the last equality follows from the fact that αL6(R3,R3) and supp(ψλ) is contained outside any compact set if λ is big enough.

  • R3jψλαλ3/22supp(ψλ)|α|601/6R3|TrC2(ψ¯ψψψ¯)|6/55/6=o(1/λ),

  • 12R3mΨλ(×α)=12R3mψ(x)(×α)(λx)12mψL2R3|(×α)(λx)|21/2=12ψL42λ3/2×αL2=o(1/λ),

  • R3v|ψλ|C22=supp(ψ)v(λx)|ψ(x)|C22=1λZsupp(ψ)|ψ(x)|C22|x|+o(1/λ),since
    v(λx)=k=1KZk|λxRk|=1λk=1KZkxRkλ=supp(ψ)1λZ|x|+o(1/λ).
  • R3μ*1|x||ψλ|C22=1λR3μλ*1|x||ψ|C22  μλ=λ3μ(λ)=1λR3|ψ|C22*1|x|dμλ=1λR3×R3|ψ(y)|C22max(|x|,|y|)1dμλ(x)dy1λμ(R3)R3|ψ(y)|C22|y|dy.

    Now, since by hypothesis (μ(R3)Z)<0, it is clear that if we choose λ big enough (say λ>Q), we have that
    (H1ψλ,ψλ)L2(R3,C2)<0.
    For any nN, define ψn=ψ2nQ. Note that {ψn}n have disjoint supports, are orthonormal, satisfy (ψn,H1ψn)L2(R3,C2)<0 and (ψn,H1ψm)L2(R3,C2)=0 if nm. For any nN, define Vn=span{ψ1,,ψn}. Note that PnH1Pn|Vn has eigenvalues {(ψm,H1ψm)L2(R3,C2)}m=1n. By the Raleigh-Ritz principle (see Theorem XIII.3 in Ref. 31) we obtain that
    μn(H1)sup1mn{(ψm,H1ψm)L2(R3,C2)}<0.
    Since σess(H1)[0,+), and n was arbitrary, this means that H1 has infinitely many negative eigenvalues.

The work of C.J.G.-C. and S.C. was supported by the National Science Foundation (NSF) via Grant No. DMS 1065942. C.J.G.-C. acknowledges support from NSF Grant No. DMS 0645766. This material is based upon work supported by Bizkaia Talent and European Commission through COFUND Programme, under Award No. AYD-000-284 and with the collaboration of BCAM.

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