# Existence of Infinitely Many Distinct Solutions to the Quasirelativistic Hartree-Fock Equations

Authors: M. Enstedt; M. Melgaard
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We establish existence of infinitely many distinct solutions to the semilinear elliptic Hartree-Fock equations forN-electron Coulomb systems with quasirelativistic kinetic energy  −α−2Δxn+α−4−α−2 for the nth electron. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Ztot of K nuclei is greater than N−1 and that Ztot is smaller than a critical charge Zc. The proofs are based on a new application of the Fang-Ghoussoub critical point approach to multiple solutions on a noncompact Riemannian manifold, in combination with density operator techniques.


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## Body

## 1. Introduction

In the present paper we prove existence of infinitely many solutions to the quasirelativistic Hartree-Fock equations(1.1)T̃0ϕn+Venϕn+(ρ*1|x|)ϕn-Kxcϕn+λnϕn=0,n=1,…,N,
associated with N electrons interacting with K static nuclei with charges Z1,…,ZK, whereZk>0. The nonlinear coupled equations arise as the Euler-Lagrange equations of the total energy functional ℰ(·) defined as the quantum energy restricted to (antisymmetric) Slater determinants (see Section 3) constructed from L2-orthonormal functions {ϕn}n=1N belonging to the Sobolev space H1/2(ℝ3). Above T̃0=-α-2Δxn+α-4-α-2 is the quasirelativistic kinetic energy of the nth electron located at xn∈ℝ3 (Δxn being the Laplacian with respect to xn), α is Sommerfeld's fine structure constant, Ven is the attractive interaction between an electron and the nuclei, ρ=∑n=1N|ϕn|2 is the density, and Kxc is the exchange operator defined in (4.1) below. For the nonrelativistic setting, a review on classical results on existence of a ground state and its properties is found in Lions [1]. In the latter paper, Lions studied both minimal and nonminimal (excited states) solutions to the equations by using critical point theory in conjunction with Morse data. Lions' idea is to construct convenient min-max levels which yield the desired solutions through abstract critical point theory. For the nonrelativistic HF model, Lions verifies a Palais-Smale (compactness) condition which, roughly speaking, amounts to “being away from the continuous spectrum” or, equivalently (when the so-called Morse information is taken into account), showing that certain Schrödinger operators with Coulomb type potentials have enough negative eigenvalues.The novelty of the present paper is Theorem7.1, wherein we establish the following results for the quasirelativistic Hartree-Fock equations. (1) A ground state exists provided the total charge Ztot of K nuclei is greater than N-1 and Ztot is smaller than a critical charge Zc to be defined below; (2) under the same assumptions, infinitely many distinct solutions to the quasirelativistic Hartree-Fock equations exist; we refer to the theorem for the full statement. We proceed to sketch the proof of Theorem 7.1, starting with the existence of a ground state. We consider the C2-functional ℰ on a (Hilbert) manifold 𝒞N defined in (3.7). Since ℰ is bounded from below, we may try to find a critical point at the level l=inf𝒞Nℰ by determining whether the infimum is achieved. As we will see, it is easy to find an almost critical sequence at the level l, that is, a sequence {h(j)} in 𝒞N satisfying
(1.2)limj→∞ℰ(h(j))=l,limj→∞ℰ′(h(j))=0.
The hard part is to prove existence of a converging subsequence of {h(j)}. Unfortunately, roughly speaking due to ionization, the energy functional will not satisfy a Palais-Smale condition at level l. To make sure that we can extract a convergent subsequence, we use second-order information of ℰ.In the process of implementing these ideas we have to overcome additional technicalities for the quasirelativistic setting compared to the nonrelativistic, for instance, the Coulomb potential is not relatively compact (in the operator sense) with respect to the quasirelativistic energy operator. In particular, compact Sobolev imbeddings are not available (for a recent survey of such problems, we refer to Bartsch et al. [2]). To overcome this  problem, it is necessary to switch to a density operator formalism, as pioneered by Solovej [3], and use that for an enlarged set of admissible density operators, one can, at least for the certain sequences, establish the inequality (6.20) below. A different proof for existence of a ground state was given by Dall'Acqua et al. [4]. Moreover, regularity of the ground state away from the nucleus and pointwise exponential decay of the orbitals were established therein.In the opposite direction, Lieb [5] has proved that for N≥2Ztot+K there never exists a quasirelativistic Hartree-Fock ground state (see Enstedt and Melgaard [6] for an analogous result). For the nonrelativistic setting, Solovej has improved Lieb's result by proving that there exists a universal constant  Q>0 such that N≥Z+Q  ensures that there are no minimizers [7] and Lewin [8] has applied Lions' approach to the nonrelativistic MCSCF equations. For further references, we refer to the survey by Le Bris and Lions [9, Section 3.1.6].We invoke a direct method developed by Fang and Ghoussoub [10, 11] to address the existence of infinitely many nonminimal solutions. Since we are looking for nonminimal (or unstable) critical points, we consider a collection H of compact subsets of 𝒞N which is stable under a specific class of homotopies and then we show that ℰ has a critical point at the level
(1.3)l=lℰ,H=infM∈Hmaxh∈Mℰ(h).
As we will see, the method by Fang and Ghoussoub gives us an almost critical sequence at the level l, that is, a sequence {h(j)} in 𝒞N satisfying (1.2), with additional Morse information (as mentioned above) which is crucial for proving that the sequence is convergent.Work related to our study of semilinear elliptic equations and critical point theory includes existence of solutions with finite Morse indices established by Dancer [12], de Figueiredo et al. [13], Flores et al. [14], and Tanaka [15], existence of multiple solutions established by Cingolani and Lazzo [16] and Ghoussoub and Yuan [17], “relaxed” Palais-Smale sequences as in Lazer and Solimini [18] and Jeanjean [19], and problems on noncompact Riemannian manifolds found in Fieseler and Tintarev [20, 21], Mazepa [22], and Tanaka [23].


## 2. Preliminaries

Throughout the paper we denote byc and C (with or without indices) various positive constants whose precise value is of no importance. Moreover, we will denote the complex conjugate of z∈ℂ by z¯.Function Spaces
For1≤p≤∞, let Lp(ℝ3) be the space of (equivalence classes of) complex-valued functions ϕ which are measurable and satisfy ∫ℝ3|ϕ(x)|pdx<∞ if p<∞ and ∥ϕ∥L∞(ℝ3)=esssup|ϕ|<∞ if p=∞. The measure dx is the Lebesgue measure. For any p the Lp(ℝ3) space is a Banach space with norm ∥·∥Lp(ℝ3)=(∫ℝ3|·|pdx)1/p. In the case p=2, L2(ℝ3) is a complex and separable Hilbert space with scalar product 〈ϕ,ψ〉L2(ℝ3)=∫ℝ3ϕψ¯dx and corresponding norm ∥ϕ∥L2(ℝ3)=〈ϕ,ϕ〉L2(ℝ3)1/2. Similarly, L2(ℝ3)N, the N-fold Cartesian product of L2(ℝ3), is equipped with the scalar product 〈ϕ,ψ〉=∑n=1N〈ϕn,ψn〉L2(ℝ3). The space of infinitely differentiable complex-valued functions with compact support will be denoted C0∞(ℝ3). The Fourier transform is given by
(2.1)(ℱψ)(ξ)=ψ̂(ξ)=(2π)-3/2∫ℝ3e-ixξψ(x)dx.
Define
(2.2)H1/2(ℝ3)={ϕ∈L2(ℝ3):(1+|ξ|)1/2ϕ̂∈L2(ℝ3)},
which, equipped with the scalar product
(2.3)〈ϕ,ψ〉H1/2(ℝ2)=∫ℝ3(1+|ξ|)ϕ̂(ξ)ψ̂(ξ)¯dξ,
becomes a Hilbert space; evidently, H1(ℝ3)⊂H1/2(ℝ3). We have that C0∞(ℝ3) is dense in H1/2(ℝ3) and the continuous embedding H1/2(ℝ3)↪L3(ℝ3) holds; more precisely, the Sobolev inequality
(2.4)∥ϕ∥L3(ℝ3)2≤csob∥ϕ∥H1/2(ℝ3)2
is valid with csob=2-1/2π-2/3. Moreover, we will use any weakly convergent sequence that in H1/2(ℝ3) has a pointwise convergent subsequence.Operators
LetT be a self-adjoint operator on a Hilbert space ℋ with domain 𝔇(T). The spectrum and resolvent set are denoted by σ(T) and ρ(T), respectively. We use standard terminology for the various parts of the spectrum; see, for example, [24, 25]. The resolvent is R(ζ)=(T-ζ)-1. The spectral family associated to T is denoted by ET(λ), λ∈ℝ. For a lower semibounded self-adjoint operator T, the counting function is defined by
(2.5)Coun(λ;T)=dimRanET((-∞,λ)).
The space of trace operators, respectively, Hilbert-Schmidt operators, on 𝔥=L2(ℝ3) is denoted by 𝔖1(𝔥), respectively, 𝔖2(𝔥).We need the following abstract operator result by Lions [1, Lemma II.2].Lemma 2.1.
LetT be a self-adjoint operator on a Hilbert space ℋ, and let ℋ1, ℋ2 be two subspaces of ℋ such that ℋ=ℋ1⊕ℋ2, dimℋ1=h1<∞ and P2TP2≥0, where P2 is the orthogonal projection onto ℋ2. Then T has at most h1 negative eigenvalues.


## 3. The Quasirelativistic Hartree-Fock Model

Within the Born-Oppenheimer approximation, the quantum energy ofN quasirelativistic electrons interacting with K static nuclei with charges Z=(Z1,…,ZK), Zk>0, is, in Rydberg units, given by
(3.1)ℰQM(·):⋀1NH1/2(ℝ3,ℂ)⊂L2(ℝ3N,ℂ)→ℝ:Ψe↦α-1∑n=1N∫ℝ3N|ℱnΨe(x1,…,xn-1,ξ,xn+1,…,xN)|2dx1⋯dμ(ξ)⋯dxN+∫ℝ3N(-α-2+Ven(xn)+12∑m≠nVee(xm-xn))|Ψe(x1,…,xN)|2dx1⋯dxN,
where x=(x1,…,xN)∈ℝ3N, xn=(xn(1),xn(2),xn(3))∈ℝ3 is the position of the nth electron, α is Sommerfeld's fine structure constant, and the potentials Vee and Ven are given by
(3.2)Vee(x)=1|x|,Ven(y)=∑k=1KVk(y),Vk(y)=-Zkα|y-Rk|
with Rk∈ℝ3 being the position of the kth nucleus. Here it is important that Ztot<Zc:=2/(απ). See Section 3.1 for details. In what follows, we ignore the spin variable but the entire contents can be trivially carried over to the spin-valued setting. Above ℱnΨe is the Fourier transform of Ψe(x1,…,xn-1,·,xn+1,…,xN), in the case when N=1 we will just write ℱΨe, and
(3.3)dμ(ξ):=|ξ|2+α-2.
The interpretation of the quadratic form (3.1) is as follows (see Section 3.1 for its well definedness). The first term corresponds to the quasirelativistic kinetic energy of the electrons, the second term is the one-particle attractive interaction between the electrons and the nuclei, and the third term is the standard two-particle repulsive interaction between the electrons. The wave function Ψe:ℝ3N→ℂ in (3.1) belongs to ℋe=⋀NH1/2(ℝ3), that is, the N-particle Hilbert space consist of antisymmetric functions (expressing the Pauli exclusion principle)
(3.4)Ψe(x1,…,xN)=sign(σ)Ψe(xσ(1),…,xσ(N))a.e.,∀σ∈SN,
where SN is the group of permutations of {1,…,N}, with the signature of a permutation σ being denoted by sign(σ), and H1/2(ℝ3) is the Sobolev space introduced in Section 2. The ground state energy is defined as
(3.5)EQM(N,Z)=inf{ℰQM(Ψe):Ψe∈ℋe,∥Ψe∥L2(ℝ3N)=1}.
To determine EQM(N,Z) directly turns out to be too difficult, even for small N. One of the classical approximation methods for determining EQM(N,Z) is the Hartree-Fock theory, introduced by Hartree and improved by Fock and Slater in the late 1920s (see, e.g., [26]), which consists of restricting attention to simple wedge products Ψe∈𝒮N, where
(3.6)𝒮N={Ψe∈ℋe:∃Φ={ϕn}1≤n≤N∈𝒞Ns.t.Ψe=1N!det(ϕn(xm))}
with
(3.7)𝒞N={Φ={ϕn}1≤n≤N,ϕn∈H1/2(ℝ3),〈ϕm,ϕn〉L2=δmn,1≤m,n≤N}.
This space is clearly a complete metric space and also an (Hilbert) manifold. A function Ψe∈𝒮N is sometimes called a Slater determinant, and the ϕn are called orbitals [26].In fact, ifΨe∈𝒮N then, by simple algebraic calculations, ℰQM(Ψe)=ℰ(Ψe), where the quasirelativistic Hartree-Fock functional (or simply the energy functional) ℰ(·):(H1/2(ℝ3)N→ℝ is given by
(3.8)ℰ(Ψe)=ℰ(ϕ1,…,ϕN)=α-1∑n=1N∫ℝ3|ℱϕn|2dμ(ξ)-∫ℝ3α-1|ϕn|2dx+α-1∫ℝ3Ven(x)ρ(x)dx+12∬ℝ3ρ(x)ρ(x′)-|𝒟(x,x′)|2|x-x′|dxdx′.
Here
(3.9)𝒟(x,x′)=∑n=1Nϕn(x)ϕn(x′)¯
is the density matrix, and
(3.10)ρ𝒟(x)=∑n=1N|ϕn(x)|2
is the density associated to the state Ψe; when there is no risk of confusion we will suppress the dependence of 𝒟.By standard arguments (see, e.g., [27, Lemma 4.1]) we obtain the following result on the regularity of quasirelativistic functional ℰ(·).Lemma 3.1.
The functionalℰ(·) belongs to C2(H1/2(ℝ3)N,ℝ).Analogous to (3.5) we define what followsDefinition 3.2 (quasirelativistic Hartree-Fock ground state).
LetZ=(Z1,…,ZK), Zk>0, k=1,…,K, and let N be a nonnegative integer. The quasirelativistic Hartree-Fock ground state energy is
(3.11)E(N,Z)=inf{ℰ(Ψe):Ψe∈𝒮N},
and if it is attained we say that the molecule has a quasirelativistic Hartree-Fock ground state described by Ψe.


### 3.1. Atomic and Molecular Hamiltonians

Byp we denote the momentum operator -i∇ on L2(ℝ3). The operator T0=p2+α-2 is generated by the closed positive form 𝔱0[ϕ,ϕ]=〈T01/2ϕ,T01/2ϕ〉L2(ℝ3) on the form domain 𝔇(𝔱0)=H1/2(ℝ3). Set S(x)=Zα/|x|, Z>0, Zc=2α-1π-1, and let T̃0=T0-α-1. The following facts are well known for the perturbed one-particle operator H1,1,α=T̃0-S(x) [25, 28]. Small Perturbations
IfZ<(π/2)Zc then S is T̃0-bounded with relative bound equal to two. If, on the other hand, (2α)-1<Z<Zc then S is T̃0-form bounded with relative bound less than one.We prove the above-mentioned form boundedness. It follows from the following inequality (first observed, it seems, by Kato [25, paragraph V-5.4]):
(3.12)〈Sϕ,ϕ〉L2(ℝ3)≤(ZZc)∥ϕ∥H1/2(ℝ3)2,∀ϕ∈H1/2(ℝ3).
Indeed, if, for any ψ,ϕ∈H1/2(ℝ3), we define the sesquilinear forms
(3.13)𝔰[ψ,ϕ]:=〈S1/2ψ,S1/2ϕ〉L2(ℝ3),𝔱0[ψ,ϕ]:=〈T01/2ψ,T01/2ϕ〉L2(ℝ3),𝔱̃0[ψ,ϕ]:=𝔱0[ψ,ϕ]-α-1〈ψ,ϕ〉L2(ℝ3),
then (3.12) shows that 𝔰 is well defined and also, by invoking |-i∇|≤T0, we infer that, for all ϕ∈H1/2(ℝ3),
(3.14)𝔰[ϕ,ϕ]<𝔱0[ϕ,ϕ]providedZ<Zc.
This is the Coulomb uncertainty principle in the quasirelativistic setting. The KLMN theorem (see, e.g., [25, paragraph  VI-1.7]) implies that there exists a unique self-adjoint operator, denoted H1,1,α, generated by the closed sesquilinear form
(3.15)𝔥1,1,α[ψ,ϕ]:=𝔱̃0[ψ,ϕ]-𝔰[ψ,ϕ],ψ,ϕ∈𝔇(𝔥1,1,α)=H1/2(ℝ3),
which is bounded below by -α-1. It is well known [28] that(3.16)σ(H1,1,α)∩[-α-1,0)isdiscreteσ(H1,1,α)∩[0,∞)isabsolutelycontinuous.
In particular,
(3.17)σess(H1,1,α)=[0,∞).
The form construction of the atomic Hamiltonian H1,1,α can be generalized to the molecular case, describing a molecule with N electrons and K nuclei of charges Z=(Z1,…,ZK),Zk>0, located at R1,…,RK, Rk∈ℝ3, if we substitute 𝔰 by
(3.18)𝔳en[ψ,ϕ]=∑k=1K〈Vk1/2,ψ,Vk1/2ϕ〉,ψ,ϕ∈H1/2(ℝ3),
where Vk is defined in (3.2) and by assuming that Ztot<Zc. Under the same hypothesis, we note that the discussion on the forms 𝔰, 𝔱0, and 𝔳en immediately gives us that the form (3.1) (and thus ℰ(·)) is well defined closed and bounded from below.


### 3.2. Density Operator Formalism

We can re-expressℰ(·) and the Hartree-Fock ground state energy via the one-to-one correpondence between elements of 𝒞N and projections onto finite-dimensional subspaces of L2(ℝ3). Indeed, given an element {ϕn}n=1N in 𝒞N we can associate a canonical projection operator, 𝒟=∑n=1N〈·,ϕn〉ϕn with trace equal to N. We may therefore write
(3.19)ℰ(𝒟)=α-1(Tr[T̃0𝒟]-Tr[Ven𝒟])+𝒥(𝒟)-𝒦(𝒟),
where
(3.20)Tr[T̃0𝒟]=∑n=1N𝔱0[ϕn,ϕn]-α-1[ϕn,ϕn],Tr[Ven𝒟]=∑n=1N𝔳en[ϕn,ϕn].
The direct Coulomb energy defined in terms of the Coulomb inner product
(3.21)𝒥(𝒟)=𝒥(ρ𝒟,ρ𝒟)=12∬ℝ3ρ𝒟(x)|x-x′|-1ρ𝒟(x′)dxdx′
and the exchange Coulomb energy defined by
(3.22)𝒦(𝒟):=12∬ℝ3|𝒟(x,x′)|2|x-x′|dxdx′.
Furthermore, it is not hard to verify that given a projection operator with trace N defined on L2(ℝ3) we can also find an element in 𝒞N corresponding to this operator. It is therefore clear that the Hartree-Fock ground state energy can be expressed as
(3.23)E(N,Z)=inf{ℰ(𝒟):𝒟*=𝒟=𝒟2,Tr[𝒟]=N}.
More generally, a density operator𝒟 is a trace class operator on 𝔥=L2(ℝ3), in symbols 𝒟∈𝔖1(𝔥), which satisfies the operator inequality 0≤𝒟≤I. This motivates the following definition:
(3.24)ℳadm={𝒟∈𝔖1(𝔥):0≤𝒟≤I,Tr[𝒟]=N}.
Using standard arguments [3] in combination with (3.12), and the Sobolev inequality (2.4), it is easy to show that ℰ(·) is well defined on the following enlarged set of density operators:
(3.25)ℳenl={𝒟∈ℳadm:Tr[T0𝒟]<+∞}.
For later purpose we also introduce
(3.26)ℳproj={𝒟∈ℳenl:𝒟2≥𝒟}.


## 3.1. Atomic and Molecular Hamiltonians

Byp we denote the momentum operator -i∇ on L2(ℝ3). The operator T0=p2+α-2 is generated by the closed positive form 𝔱0[ϕ,ϕ]=〈T01/2ϕ,T01/2ϕ〉L2(ℝ3) on the form domain 𝔇(𝔱0)=H1/2(ℝ3). Set S(x)=Zα/|x|, Z>0, Zc=2α-1π-1, and let T̃0=T0-α-1. The following facts are well known for the perturbed one-particle operator H1,1,α=T̃0-S(x) [25, 28]. Small Perturbations
IfZ<(π/2)Zc then S is T̃0-bounded with relative bound equal to two. If, on the other hand, (2α)-1<Z<Zc then S is T̃0-form bounded with relative bound less than one.We prove the above-mentioned form boundedness. It follows from the following inequality (first observed, it seems, by Kato [25, paragraph V-5.4]):
(3.12)〈Sϕ,ϕ〉L2(ℝ3)≤(ZZc)∥ϕ∥H1/2(ℝ3)2,∀ϕ∈H1/2(ℝ3).
Indeed, if, for any ψ,ϕ∈H1/2(ℝ3), we define the sesquilinear forms
(3.13)𝔰[ψ,ϕ]:=〈S1/2ψ,S1/2ϕ〉L2(ℝ3),𝔱0[ψ,ϕ]:=〈T01/2ψ,T01/2ϕ〉L2(ℝ3),𝔱̃0[ψ,ϕ]:=𝔱0[ψ,ϕ]-α-1〈ψ,ϕ〉L2(ℝ3),
then (3.12) shows that 𝔰 is well defined and also, by invoking |-i∇|≤T0, we infer that, for all ϕ∈H1/2(ℝ3),
(3.14)𝔰[ϕ,ϕ]<𝔱0[ϕ,ϕ]providedZ<Zc.
This is the Coulomb uncertainty principle in the quasirelativistic setting. The KLMN theorem (see, e.g., [25, paragraph  VI-1.7]) implies that there exists a unique self-adjoint operator, denoted H1,1,α, generated by the closed sesquilinear form
(3.15)𝔥1,1,α[ψ,ϕ]:=𝔱̃0[ψ,ϕ]-𝔰[ψ,ϕ],ψ,ϕ∈𝔇(𝔥1,1,α)=H1/2(ℝ3),
which is bounded below by -α-1. It is well known [28] that(3.16)σ(H1,1,α)∩[-α-1,0)isdiscreteσ(H1,1,α)∩[0,∞)isabsolutelycontinuous.
In particular,
(3.17)σess(H1,1,α)=[0,∞).
The form construction of the atomic Hamiltonian H1,1,α can be generalized to the molecular case, describing a molecule with N electrons and K nuclei of charges Z=(Z1,…,ZK),Zk>0, located at R1,…,RK, Rk∈ℝ3, if we substitute 𝔰 by
(3.18)𝔳en[ψ,ϕ]=∑k=1K〈Vk1/2,ψ,Vk1/2ϕ〉,ψ,ϕ∈H1/2(ℝ3),
where Vk is defined in (3.2) and by assuming that Ztot<Zc. Under the same hypothesis, we note that the discussion on the forms 𝔰, 𝔱0, and 𝔳en immediately gives us that the form (3.1) (and thus ℰ(·)) is well defined closed and bounded from below.


## 3.2. Density Operator Formalism

We can re-expressℰ(·) and the Hartree-Fock ground state energy via the one-to-one correpondence between elements of 𝒞N and projections onto finite-dimensional subspaces of L2(ℝ3). Indeed, given an element {ϕn}n=1N in 𝒞N we can associate a canonical projection operator, 𝒟=∑n=1N〈·,ϕn〉ϕn with trace equal to N. We may therefore write
(3.19)ℰ(𝒟)=α-1(Tr[T̃0𝒟]-Tr[Ven𝒟])+𝒥(𝒟)-𝒦(𝒟),
where
(3.20)Tr[T̃0𝒟]=∑n=1N𝔱0[ϕn,ϕn]-α-1[ϕn,ϕn],Tr[Ven𝒟]=∑n=1N𝔳en[ϕn,ϕn].
The direct Coulomb energy defined in terms of the Coulomb inner product
(3.21)𝒥(𝒟)=𝒥(ρ𝒟,ρ𝒟)=12∬ℝ3ρ𝒟(x)|x-x′|-1ρ𝒟(x′)dxdx′
and the exchange Coulomb energy defined by
(3.22)𝒦(𝒟):=12∬ℝ3|𝒟(x,x′)|2|x-x′|dxdx′.
Furthermore, it is not hard to verify that given a projection operator with trace N defined on L2(ℝ3) we can also find an element in 𝒞N corresponding to this operator. It is therefore clear that the Hartree-Fock ground state energy can be expressed as
(3.23)E(N,Z)=inf{ℰ(𝒟):𝒟*=𝒟=𝒟2,Tr[𝒟]=N}.
More generally, a density operator𝒟 is a trace class operator on 𝔥=L2(ℝ3), in symbols 𝒟∈𝔖1(𝔥), which satisfies the operator inequality 0≤𝒟≤I. This motivates the following definition:
(3.24)ℳadm={𝒟∈𝔖1(𝔥):0≤𝒟≤I,Tr[𝒟]=N}.
Using standard arguments [3] in combination with (3.12), and the Sobolev inequality (2.4), it is easy to show that ℰ(·) is well defined on the following enlarged set of density operators:
(3.25)ℳenl={𝒟∈ℳadm:Tr[T0𝒟]<+∞}.
For later purpose we also introduce
(3.26)ℳproj={𝒟∈ℳenl:𝒟2≥𝒟}.


## 4. The Quasirelativistic Fock Operator

Herein we introduce the quasirelativistic Fock operator.Lemma 4.1.
AssumeZtot<Zc. Let
(4.1)Kxc(x,x′)=𝒟(x,x′)|x-x′|
be the integral kernel of the exchange operator Kxc. Then the unique self-adjoint operator F associated with the differential expression
(4.2)α-1T̃0ϕ+α-1Venϕ+ρ*1|x|ϕ-Kxcϕ
is generated by the sesquilinear form
(4.3)𝔣[ϕ,ψ]=α-1𝔱̃0[ϕ,ψ]+α-1∫ℝ3Ven(x)ϕ(x)ψ¯(x)dx+∬ℝ3ρ(x)ϕ(y)ψ¯(y)|x-y|dxdy-∬ℝ3𝒟(x,y)ϕ(y)ψ¯(x)|x-y|dydx.Proof.
Bear in mind the definitions of𝔱0, 𝔱̃0, and 𝔳en from Section 3.1. Define 𝔳ρ*(1/|x|) as the third form on the right-hand side of (4.3). Then (3.12) yields the estimate
(4.4)𝔳ρ*(1/|x|)[ϕ,ϕ]≤C(Zc,N)𝔱0[ϕ,ϕ].
Under the hypothesis, we already know from Section 3.1 that the quadratic form 𝔱0+𝔳en is nonnegative on H1/2(ℝ3). Evidently, 𝔳ρ*(1/|x|) is a nonnegative form and, consequently, 𝔣̃=𝔱0+𝔳en+𝔳ρ*(1/|x|) is a nonnegative form on H1/2(ℝ3). Closedness of the nonnegative quadratic form 𝔣̃ is equivalent to lower semicontinuity of 𝔣̃ on H1/2(ℝ3). In fact, 𝔣̃ is continuous. Indeed, (3.12), respectively, (4.4) enables us to show continuity of the second, respectively, the third terms, in 𝔣̃. For instance, we consider 𝔳en and assume that ϕj→ϕ in H1/2(ℝ3). Then an application of Hölder's inequality and (3.12) yields
(4.5)|𝔳en[ϕj]-𝔳en[ϕ]|≤∫ℝ3|Ven||ϕj-ϕ||ϕj+ϕ|dx≤c(∫ℝ3|Ven||ϕj-ϕ|2dx)1/2≤C∥ϕj-ϕ∥H1/2(ℝ3).
We conclude that 𝔣̃ is a closed quadratic form on H1/2(ℝ3). The first representation theorem [24, Theorem  VI.2.4] informs us that the nonnegative closed form 𝔣̃ is associated to a unique self-adjoint operator, say F̃. Furthermore, the exchange operator Kxc is a Hilbert-Schmidt operator. Indeed, using, in this particular order, the weak Young inequality, the Hölder inequality and (3.12) we find that Kxc(x,x′)∈L2(ℝ3×ℝ3). It is clear that the form 𝔣̃[·,·]-α-1〈·,·〉L2-〈Kxc·,·〉L2 is closed and, once again applying the first representation theorem, we obtain a unique self-adjoint operator F associated with the form in (4.3).


## 5. Lower Spectral Bound

We will later need the following spectral result.Lemma 5.1.
Assumeϑ<Ztot<Zc, and let ρ∈L1(ℝ3)∩L4/3(ℝ3) such that ∫ℝ3ρdx<ϑ. Define the quasirelativistic Schrödinger operator
(5.1)T=α-1T̃0+α-1Ven+ρ*1|x|.
Then, for any κ≥1 and any 0≤ϑ<Ztot, there exists ϵκ,ϑ>0 such that
(5.2)Count(-ϵn,ϑ;T)≥κ.Proof.
By a minor modification of [28, page 291], which carries over the result (3.17) from the one-nucleus to the many-nuclei cases, we deduce that the essential spectrum of T̃0+Ven equals the semiaxis [0,∞). Next, a standard perturbation argument and (yet) an application of Weyl's essential spectrum theorem prove that σess(T)=[0,∞). Let 𝔱μ denote the quadratic form defined by
(5.3)α-1𝔱̃0[ϕ,ϕ]+∫ℝ3(α-1Ven+ρ*1|x|)|ϕ(x)|2dx.
For any κ≥1 and any 0≤ϑ≤Ztot we construct a κ-dimensional subspace ℋκ,ϑ in H1/2(ℝ3) such that
(5.4)𝔱[ϕ,ϕ]<-ϵκ,ϑ<0
for all L2-normalized ϕ∈ℋκ,ϑ. We note that
(5.5)α-1𝔱̃0[ϕ,ϕ]≤C∫ℝ3|∇ϕ|2dx.
As a consequence, by selecting a κ-dimensional subspace of normalized radially symmetric functions in C0∞(ℝ3), we can construct a subspace ℋκ,ϑ of functions satisfying (5.4), away from ϕ=0, by repeating the arguments in [1, Lemma  II.1] (see also [27, Lemma 6.1]). Then the assertion follows by an application of Glazman's Lemma.Within the nonrelativistic context a similar result was first given by Lions [1, Lemma  II.1].


## 6. Relative Compactness of Palais-Smale Type Sequences

In this section we give the main auxiliary result that will be used in the proof of Theorem7.1. We emphasize that the functional ℰ(·) is not weakly lower semicontinuous on (H1/2(ℝ3))N and, in the proof below, it is thus necessary to switch to a density operator formalism. In particular, we use that for a specific sequence of density operators (see the proof for details), one can establish the inequality (6.20) below (replacing the notion of weak lower semicontinuity which is absent).Proposition 6.1.
Assume thatl∈ℝ, that m∈ℕ, and let N-1<Ztot<Zc. Then any sequence {ϕ(j)}j=1∞⊂𝒞N satisfying a Palais-Smale condition at level l and of order less than m is relatively compact in 𝒞N, that is, any sequence {ϕ(j)}j=1∞ in 𝒞N is relatively compact whenever the sequence satisfies the following conditions: (i)
limj→∞ℰ(ϕ(j))=l;(ii)
limj→∞ℰ′(ϕ(j))=0;(iii)
there exists a sequence of positive reals{δ(j)}j=1∞ with δ(j)→0 such that for each j, ℰ′′(ϕ(j)) has at most m eigenvalues below -δ(j).
Moreover, the components of the limit elementϕ=(ϕ1,…,ϕN) of {ϕ(j)}j=1∞ in 𝒞N satisfy the quasirelativistic Hartree-Fock equations
(6.1)Fϕn+λnϕn=0,n=1,…,N,〈ϕm,ϕn〉L2(ℝ3)=δmn,
where λn≥0 for Ztot>N-1, respectively, λn>0 for Ztot>N, and F is the Fock operator defined in Lemma 4.1.Proof.
First we treat the caseN<Ztot. Henceforth we let {ϕn(j)}n=1N∈𝒞N be the canonical sequence associated with an operator 𝒟(j) in ℳproj defined in Section 3.2. The hypotheses (i) and (ii) give us that
(6.2)sup∥ψ∥H1/2(ℝ3)=1|𝔣j[ϕn(j),ψ]+λn(j)〈ϕn(j),ψ〉|→0,∀n,
where λn(j) is a sequence of reals and 𝔣j is a sequence of quadratic forms associated with {ϕn(j)}, defined as in (4.3).
Let us now extract some subsequences that we will need. Let us start by proving existence of0<λ<λ̃<∞ such that λ<λn(j)<λ̃. To prove existence of a lower bound we note that from hypothesis (iii) we get (in particular) that
(6.3)𝔣j[ψ,ψ]+(λn(j)+δn(j))∥ψ∥L2(ℝ3)2≥0
with δn(j)→0 in the standard Euclidean metric for each fixed n and ψ in a closed subspace of H1/2(ℝ3) with finite codimension N+m. By invoking Lemma 2.1 we deduce that the quasirelativistic Schrödinger operator
(6.4)T(j)=T̃0+Ven+ρ(j)*1|x-y|
has at most N+m eigenvalues strictly less than -(λn(j)+δn(j)). Moreover, since
(6.5)∫ℝ3ρ(j)dx=∑n=1N∫ℝ3|ϕn(j)(x)|2=N<Ztot,
Lemma 5.1 ensures that there exists δ>0 (independent of j) such that T(j) has at least N+m eigenvalues strictly below -δ. As a consequence, we infer that
(6.6)λn(j)+δn(j)≥δ,∀j.
Since δn(j)→0 as j→∞, we conclude that, for j large enough,
(6.7)λn(j)≥λ>0,∀n.
We note that the hypothesis Ztot<Zc and the fact that {𝒟(j)}⊂ℳproj satisfies
(6.8)ℰ(𝒟(j))→l
ensure the existence of a constant C, depending on N, such that
(6.9)Tr[T0𝒟(j)]≤C(1+Tr[𝒟(j)]).
To prove existence of an upper bound we note that
(6.10)0≤∑n=1Nλn(j)≤-ℰ(𝒟(j))≤C̃,
which follows from the Cauchy-Schwarz inequality and (6.8). Now, perhaps after going to a subsequence using the Bolzano-Weierstrass theorem, we may assume that
(6.11)λn(j)→λn>0.
We know from (6.9) that
(6.12)Tr[T0𝒟(j)]
is uniformly bounded in j. Then some straightforward calculations give us that
(6.13)Tr[T01/2𝒟(j)T01/2]
is also uniformly bounded in j. Here the T01/2 is defined using Kato's second representation theorem [24, Theorem  VI.2.4] . Hence we may, using the Banach-Alaoglu theorem, extract a subsequence such that T01/2𝒟(j)T01/2 converges weakly in 𝔖2 to an element 𝒟̃. Fix any ψ∈L2(ℝ3), then
(6.14)〈·ψ,ψ〉:𝔖1→ℝ
is a linear bounded functional. We get that
(6.15)limj〈𝒟(j)ψ,ψ〉=limj〈T01/2𝒟(j)T01/2T0-1/2ψ,T0-1/2ψ〉=〈T0-1/2𝒟̃T0-1/2ψ,ψ〉.
Define 𝒟:=T0-1/2𝒟̃T0-1/2 and let {ψn}n=1∞⊂H1/2(ℝ3) be a basis in L2(ℝ3). Then a direct application of Fatou's lemma (with respect to a counting measure) gives us that
(6.16)Tr[𝒟]≤liminfjTr[𝒟(j)]≤N.
Mutatis mutandis it is clear that
(6.17)Tr[T0𝒟]≤liminfjTr[T0𝒟(j)]≤C.
We note that 𝒟(j) converges weakly to 𝒟 in 𝔖2 and hence that the kernels of the operators 𝒟(j) will converge weakly in L2(ℝ3×ℝ3) to the kernel of 𝒟. In view of (6.17) and the fact that Tr[𝒟(j)]≤C we infer that there exists a subsequence such that
(6.18)ϕn(j)→ϕn∈H1/2(ℝ3)
weakly in H1/2(ℝ3) and, by invoking weak compactness (see Section 2), the convergence holds almost everywhere. Since weak limits are unique, we may assume that the kernel associated with 𝒟 can be written as
(6.19)𝒟(x,x′):=∑n=1Nϕn(x)ϕn(x′)¯.
The inequality
(6.20)liminfjℰ(𝒟(j))≥ℰ(𝒟)
can be derived by arguments similar to the ones in [4, pages  722–724], wherein it is proven for a minimizing sequence (bearing in mind the spectral properties of the one-particle operator H1,1,α in (3.15) which we summarized in Section 3.1).  More specically, using arguments by Barbaroux et al. [26] and Solovej [30] , the inequality (6.20) was proved by Dall'Acqua et al [16] (for specfic sequences) in the quasirelativistic setting and their proof carries over to our sequence. As a consequence, we have that
(6.21)limsupj∑n=1Nλn(j)∥ϕn(j)∥L22=-liminfjℰ(𝒟(j))+12(𝒥(𝒟(j))-𝒦(𝒟(j)))≤-ℰ(𝒟)-12(𝒥(𝒟)-𝒦(𝒟))=∑n=1Nλn∥ϕn∥L22≤liminfj∑n=1Nλn(j)∥ϕn(j)∥L22.
From this we conclude that Tr[𝒟]=N and therefore that ∥ϕn∥L2(ℝ3)=1. Repeating the argument above, we obtain the convergence in H1/2(ℝ3)N. We recall the regularity property of ℰ and that the quasirelativistic Hartree-Fock equations are the Euler-Lagrange equations corresponding to this functional. The last assertion then follows from hypothesis (ii) and the relative compactness that was just proved.
Finally, we consider the caseN-1<Ztot. By going to the limit in (6.3), the resulting inequality
(6.22)𝔣[ψ,ψ]+λn∥ψ∥L2(ℝ3)2≥0
holds on a closed subspace of H1/2(ℝ3) with finite codimension; this requires that ϕn(j)→ϕn weakly in H1/2(ℝ3) (inspection of the argument above justifies this). Hence we infer that T (defined similar to T(j) with ρ(j) replaced by ρ) has at most finitely many eigenvalues less than or equal to -λn≤0. If Tr[𝒟]=N, then we are done. If, on the other hand, Tr[𝒟]<N then we apply Lemma 5.1 and repeat the reasoning above. This completes the proof.The density operator argument in the proof of Proposition6.1 is inspired by Solovej [3].Remark 6.2.
It is worth to mention that from the perspective of Physics, there is no difference between the requirementsZtot>N-1 and Ztot≥N because Ztot is integer valued.


## 7. Existence of a Ground State and Excited States

The main result is the following theorem.Theorem 7.1.
Assume that the total nuclear chargeZtot=∑k=1KZk satisfies Ztot<Zc and let N∈ℕ satisfy N-1<Ztot. Then
(1) every minimizing sequence of the quasirelativistic Hartree-Fock functional ℰ(·) is relatively compact in 𝒞N. In particular, there exists a minimizer φ of ℰ(·) on the admissible set 𝒞N and (up to unitary transformations) the components of φ=(φ1,…,φN) satisfy the quasirelativistic Hartree-Fock equations
(7.1)Fφn+λnφn=0,〈φm,φn〉L2(ℝ3)=δmn,
where F is the quasirelativistic Fock operator defined in Lemma 4.1, and the numbers -λn are the N lowest negative eigenvalues of F,
(2) there exists a sequence {φ(k)}k≥1, with entries φ(k)=(φ1(k),…,φN(k)), of distinct solutions of the quasirelativistic Hartree-Fock equations (7.1) in H1/2(ℝ3)N which satisfy the constraints 〈φm(k),φn(k)〉L2(ℝ3)=δmn for all 1≤m,n≤N and, furthermore, the Lagrange multipliers λn(k) are positive, respectively, nonnegative, when Ztot>N, respectively, Ztot>N-1. Moreover, the following properties are valid as k→∞:
(7.2)λn(k)→0,ℰ(φ(k))→0,φ(k)→0weaklyinH1/2(ℝ3)N,
(3) any solution to (7.1) belongs to ℂ∞(ℝ3∖{R1,…,RN})N and φn decays exponentially sufficiently far away from the locations of the nuclei.Before proving assertion1 of Theorem 7.1, let us give a few explanations. To ensure that a Palais-Smale sequence converges, one needs to somehow “improve” it. Since ℰ is a C2-functional, one may try to obtain an almost critical sequence with some information on the second derivative. This enables us to built an almost critical sequence which satisfies (6.3). Due to lack of compactness, one cannot find critical points of ℰ and therefore one perturbs the functional while, simultaneous, ensuring that the new functional has critical points of the kind, one expects for the original one. The way one will obtain such sequences consist in applying a “perturbed variational principle” by Borwein and Preiss [30].Proof of Theorem7.1 (assertion 1).
First of all we note that using (3.12) and the Cauchy-Schwarz inequality that ℰ is bounded from below uniformly on 𝒞N and we may therefore conclude existence of a minimizing sequence, {Φ̃(j)}j=1∞ to (3.11). To prove relative compactness we will now prove that the hypotheses (ii) and (iii) in Proposition 6.1 are satisfied. An application of the Borwein and Preiss variational principle [30, Theorem 2.6] provides us with a new minimization sequence {Φ(j)}j=1∞, such that
(7.3)∥Φ(j)-Φ̃(j)∥H1/2(ℝ3)N→0.
We will also have that Φ(j) minimizes
(7.4)ℰ(·)+γ(j)∥·-Ψ(j)∥H1/2(ℝ3)N2
for some Ψ(j)∈H1/2(ℝ3)N and γ(j)>0, where γ(j)→0. From this we can conclude that hypothesis (ii) is satisfied. If we then follow the idea to prove a lower bound on the reals in the proof of Proposition 6.1 it is not difficult to show that hypothesis (iii) is satisfied for N. Existence of a minimum follows from Proposition 6.1. To show that λn>0 one argues by contradiction as in [27, page 2139]. The last assertion on the Lagrange multipliers and its relation to the Fock operator has been proven in [4].Proceeding towards the second assertion of Theorem7.1 which addresses the existence of infinitely many nonminimal solutions, one would expect from the previous proof that a more involved perturbed variational principle is needed. For our specific setting, however, it suffices to apply the direct method by Fang and Ghoussoub [10] (see also [11, 31]). Again, due to the lack of weakly lower semicontinuity, it is necessary to switch to density operators in the proof below.Proof of Theorem7.1 (assertions 2 and 3).
We will prove that there exists a critical point at infinitely many distinct levels. We will use abstract critical point theory by Fang and Ghoussoub [10]. Consider the C2-functional ℰ on the C2-Riemannian manifold 𝒞N. We consider ℤ2={0,1} to be the compact (0-dimensional) Lie group, with groups actions (0,ϕ)↦ϕ and (1,ϕ)↦-ϕ (ϕ∈𝒞N). We note that the functional is even, in fact it is invariant under unitary transformation, this can be seen by repeating the proof from the nonrelativistic Hartree-Fock case (see, e.g., [27, Lemma 2.3]. Next we make preparations for the min-max principle: For each k∈ℕ, we consider the following homotopic classes of order k(7.5)Hk={M:M=h(𝕊k-1)forsomeoddcontinuoush:𝕊k-1→𝒞N},
where 𝕊k-1 is the unit sphere in the Euclidean space ℝk. Let
(7.6)lk=infM∈Hkmaxϕ∈Mℰ(ϕ).
We claim that -∞<lk≤lk+1<0 for each k∈ℕ and that limk→∞lk=0, the proof of this fact will be given last in this proof. We may of course, after perhaps going to a subsequence, assume that lk<lk+1 for each k. Now, we will use the abstract results by Fang and Ghoussoub [10] (in particular [31, Theorem 11.1 and Remark 11.13]), to extract a sequence satisfying the assumptions of Palais-Smale condition at level lk and of order less than k, but such a sequence is according to Proposition 6.1 relatively compact in 𝒞N.
Let us now prove the properties of the sequence{φ(k)}k≥1 of distinct solutions. We have already seen that we may assume that -∞<lk<lk+1<0 so we may find a sequence such that
(7.7)-∞<lk-1<lk=ℰ(φ(k))<lk+1<0.
We conclude that
(7.8)ℰ(φ(k))→0.
Now using the Cauchy-Schwarz inequality (recall that 0<λn(k))
(7.9)-∑n=1Nλn(k)≤ℰ(φ(k))<0,
and thus λn(k)→0 in k for each n. We note that the right-hand side of
(7.10)2(ℰ(φ(k))+∑n=1Nλn(k))=-∫ℝ6ρ(k)(x)ρ(k)(x′)-|𝒟(k)(x,x′)|2|x-x′|dxdx′
tends to zero. This together with (7.7) allows us to conclude that we can find (perhaps after going to a subsequence) a weak limit, φ∈H1/2(ℝ3)N, for our sequence. Due to the assumption Ztot<Zc we may find a constant C>0 such that
(7.11)ℰ(φ(k))-∬ℝ3ρ(k)(x)ρ(k)(x′)-|𝒟(k)(x,x′)|2|x-x′|dxdx′≤C𝔱0[φ,φ],
and we may therefore conclude that φ=0. This finishes the part on the properties of the sequence. It remains to prove the claim stated above. The monotonicity of {lk}k=1∞ is a direct consequence of how we have defined Hk and since ℰ is uniformly bounded below on 𝒞N, we immediately get that lk>-∞. An application of Lemma 5.1 ensures that there exists a k-dimensional subspace ℋk of H1/2(ℝ3) such that for all ϕ∈ℋk with ∥ϕ∥L2(ℝ3)=1 (we denote the unit sphere in this subspace by 𝕊̃k-1), one has
(7.12)𝔱̃0[ϕ,ϕ]+𝔳e[ϕ,ϕ]+∬ℝ3ρ(x)|ϕ(x′)|2|x-x′|dxdx′≤-ϵk
for some ϵk>0. It is not hard to find a continuous isomorphism g:ℋk→ℝk such that g(𝕊k-1)=𝕊̃k-1, now denote by e the natural embedding of 𝕊̃k-1 into 𝒞N (due to the monotonicity we may assume k to be sufficiently large) and therefore g∘e will be an odd and continuous mapping from 𝕊k-1 into 𝒞N where ℰ negative and therefore we can conclude that lk<0. Hence we can find Mk∈Hk such that
(7.13)lk≤maxMkℰ<lk2.
To prove that limk→∞lk=0, we use the separability of H1/2(ℝ3) by considering a nested sequence of finite-dimensional subspaces 𝒲k of H1/2(ℝ3) such that dim(𝒲k)=k and ∪k𝒲k is dense in H1/2(ℝ3). Define 𝒱k as the orthogonal complement of 𝒲k-1. Now, assume that Mk∩𝒱k=∅, let πk-1 be the orthogonal projection from H1/2(ℝ3) onto 𝒲k-1. Then (𝒱k≔Ker(πk-1))
(7.14)π(Mk)⊂𝒲k∖{0}≅ℝk∖{0},
and by following Rabinowitz [32] and using the Borsuk-Ulam theorem we will now arrive at a contradiction. Using that zero is an upper bound for the functional we can extract a sequence hk∈Mk∩𝒱k such that hk tends weakly to some element that must be equal to 0. By repeating the arguments in Proposition 6.1 we may find a subsequence (which is of course sufficient in our case) such that 𝒟(k), the density operator corresponding to hk, tends weakly to 0 in 𝔖2. We get by the same type of argument as for (6.20) that
(7.15)0=ℰ(0)≤liminfkℰ(𝒟(k))≤0.
The latter together with (7.13) implies that limk→∞,lk=0.
The regularity and decay properties of our sequence were proved in [4] for an atom and it carries over to our setting mutatis mutandis.



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