The Atomic Density on the Thomas--Fermi Length Scale for the Chandrasekhar Hamiltonian
aa r X i v : . [ m a t h - ph ] O c t THE ATOMIC DENSITY ON THE THOMAS–FERMI LENGTHSCALE FOR THE CHANDRASEKHAR HAMILTONIAN
KONSTANTIN MERZ AND HEINZ SIEDENTOP
Abstract.
We consider a large neutral atom of atomic number Z , modeledby a pseudo-relativistic Hamiltonian of Chandrasekhar. We study its suitablyrescaled one-particle ground state density on the Thomas–Fermi length scale Z − / . Using an observation by Fefferman and Seco [2], we find that the den-sity on this scale converges to the minimizer of the Thomas–Fermi functionalof hydrogen as Z → ∞ when Z/c is fixed to a value not exceeding 2 /π . Thisshows that the electron density on the Thomas–Fermi length scale does notexhibit any relativistic effects. Introduction
The energy of heavy atoms as well as the distribution of its electrons are offundamental interest both in physics and in quantum chemistry. However, as in theclassical Kepler problem, one cannot hope for an exact solution of the Schr¨odingerequation involving more than two particles. For this reason, one needs to devisemodels for many-body quantum systems which are easier to solve but still describethe system accurately.Lieb and Simon [10] showed that the atomic ground state density convergeson the length scale Z − / to the minimizer of the Thomas–Fermi functional ofhydrogen. This results is derived by controlling the atomic energy to leading orderin Z and its derivative with respect to small perturbations.However, it is questionable to describe large Z atoms non-relativistically, sincethe large nuclear charge forces the bulk of the electrons on orbits on the lengthscale Z − / from the nucleus. Thus, electrons close to the nucleus are movingfaster than a substantial fraction of the velocity of light c . This suggests that arelativistic description is necessary.On the other hand, Sørensen [11] showed in the context of the simplest relativisticmodel, namely the Chandrasekhar operator, that energetically this worry is notjustified, at least not to leading order in the energy: the atomic ground state energyof the Chandrasekhar operator is still described by the Thomas–Fermi energy forlarge Z and γ := Z/c fixed to a value not exceeding the critical coupling constant γ c := 2 /π . A similar result for the Brown–Ravenhall operator was proven byCassanas et al [1].Schwinger [13] predicted that relativistic effects occur only in sub-leading order.Frank et al [4] and Solovej et al [14] showed, using completely different approaches,that this is indeed the case. In particular, the authors showed that the coefficientof this order is less than the non-relativistic one which reflects the fact that therelativistic kinetic energy is lower than the non-relativistic one, especially for highmomenta.The question arises whether the density on the Thomas–Fermi length scale Z − / is also unchanged by relativistic effects. This might be conjectured, since the leadingenergy correction is generated by the fast electrons close to the nucleus. Our mainresult is a positive answer to this question: we show that the suitably rescaleddensity of the atomic Chandrasekhar operator converges for large Z and γ fixed to a value not exceeding γ c to the minimizer of the Thomas–Fermi functional ofhydrogen. 2. Definition and main result
Our system consists of a neutral atom, i.e., a nucleus of charge Z located at theorigin with N = Z electrons with q spin states whose motion is described by theChandrasekhar operator. It is given by the Friedrichs extension of the quadraticform associated to(1) C c,Z := N X ν =1 (cid:18)p − c ∆ ν + c − c − Z | x ν | (cid:19) + X ≤ ν<µ ≤ N | x ν − x µ | in the Fermionic Hilbert space V Nν =1 ( L ( R ) : C q ) . (Throughout we use atomicunits, i.e., ~ = e = m = 1.) The constant c denotes the velocity of light whichin these units is the inverse of Sommerfeld’s fine-structure constant α . Here wefocus on N = Z . The form is bounded from below, if and only if γ ≤ γ c (Kato [9,Chapter Five, Equation (5.33)], Herbst [8, Theorem 2.5], Weder [15]). For γ < γ c ,its form domain is H / ( R N : C q N ) ∩ V Nν =1 ( L ( R : C q )) by the KLMN theorem.In fact, Hardekopf and Sucher [7] indicated numerically and gave arguments andRaynal et al [12] showed that the one-particle operator is strictly bigger than − γ = γ c .A general fermionic ground state density matrix can be written as M X µ =1 w µ | ψ µ ih ψ µ | where the ψ µ constitute an orthonormal basis of the ground state eigenspace andthe w µ are non-negative weights such that P Mµ =1 w µ = 1. The corresponding one-particle density ρ is given by ρ Z ( x ) := N M X µ =1 w µ q X σ ,...,σ N =1 Z R N − | ψ µ ( x, σ ; x , σ ; ... ; x N , σ N ) | d x ... d x N . The ground state energy of this system for fixed γ is written as E ( Z ) := inf σ ( C c,Z ).Solovej et al [14] and Frank et al [4] determined the first two terms of the expansionof E ( Z ) for Z → ∞ and γ ≤ γ c fixed to be(2) E ( Z ) = E TF ( Z ) + (cid:16) q − s ( γ ) (cid:17) Z + O ( Z / )where s ( γ ) := γ − tr "(cid:18) p − γ | x | (cid:19) − − (cid:18)p p + 1 − − γ | x | (cid:19) − > n -th eigenvalues of (cid:18) − ∆ − γ | x | (cid:19) ⊗ q and (cid:18) √− ∆ + 1 − − γ | x | (cid:19) ⊗ q and E TF ( Z ) is the infimum of the atomic Thomas–Fermi functional E TF Z on itsnatural domain I , i.e., E TF ( Z ) := inf( E TF Z ( I ))with E TF Z ( ρ ) := Z R (cid:18) γ TF ρ / ( x ) − Z | x | ρ ( x ) (cid:19) d x + D ( ρ, ρ )and I := { ρ ∈ L / ( R ) (cid:12)(cid:12) D ( ρ, ρ ) < ∞ , ρ ≥ } . TOMIC DENSITY OF THE CHANDRASEKHAR HAMILTONIAN 3
Here γ TF := (6 π /q ) / is the Thomas–Fermi constant and D ( ρ, ρ ) is the electro-static selfenergy of the charge density ρ , i.e., D ( ρ, σ ) = 12 Z R Z R ρ ( x ) σ ( y ) | x − y | d x d y. Note that D defines a scalar product and thus induces a norm, the so-calledCoulomb norm k ρ k C := D ( ρ, ρ ) / . The minimizer of E TF Z is denoted by ρ TF Z .It obeys the scaling relation ρ TF Z ( x ) = Z ρ TF1 ( Z / x ) where ρ TF1 is the Thomas–Fermi density of hydrogen, i.e., Z = 1 (Gombas [5]). These scaling relations and theleading order of E ( Z ) show that the Thomas–Fermi theory is energetically correctin leading order even, if relativistic effects are taken into account. Our result onthe convergence of the ground state density shows that it is also a valid model forthe density on this length scale.We write(3) ˆ ρ Z ( x ) := Z − ρ Z ( Z − / x )for the rescaled quantum density on the Thomas–Fermi scale. This allows to for-mulate our main observation: Theorem 1.
Let γ ∈ (0 , γ c ] , then ˆ ρ Z → ρ TF1 in Coulomb norm. In fact, k ˆ ρ Z − ρ TF1 k C = O ( Z − / ) as Z → ∞ . Before proving this claim, we remark that the Schwarz inequality implies alsoweak convergence: suppose σ has finite Coulomb norm, i.e., k σ k C < ∞ . Then D ( σ, ˆ ρ Z − ρ TF1 ) = O ( Z − / ) . (Note that the Hardy–Littlewood–Sobolev inequality ensures that this is the casefor all σ ∈ L / ( R ) but that this is not exhaustive. For example, σ might also bea uniform charge distribution on a sphere.)Finally, setting σ := − (1 / π )∆ U with U vanishing at infinity gives Z U ρ → Z U ρ
TF1 as Z → ∞ for all such U . Proof of Theorem 1.
The basic observation is, that also in this case – as in the non-relativistic case done by Fefferman and Seco [2] – it is useful to keep some positiveterm in the lower bound in the proof of an asymptotic energy formula: tracing thelower bound, the proof of the Scott conjecture by Frank et al does not only give theScott formula (2). If one does not drop the positive term in Onsager’s inequality –unlike as is done there, we get for fixed γ ∈ (0 , /π ] the two bounds(4) E TF ( Z ) + (cid:16) q − s ( γ ) (cid:17) Z + k ρ TF Z − ρ Z k C − const Z / ≤ E ( Z ) ≤ E TF ( Z ) + (cid:16) q − s ( γ ) (cid:17) Z + const Z / . We observe that the left and right side have identical terms up to order Z . Sub-tracting them and rearranging gives(5) k ρ TF Z − ρ Z k C ≤ const Z / . K. MERZ AND H. SIEDENTOP
Since ρ TF Z ( x ) = Z ρ TF1 ( Z / x ) and by definition of ˆ ρ Z in (3), we obtain by changeof variables(6) k ρ TF Z − ρ Z k C = 12 Z d x Z dy ( ρ TF Z ( x ) − ρ Z ( x ))( ρ TF Z ( y ) − ρ Z ( y ) | x − y | = Z / Z d x Z dy ( ρ TF1 ( x ) − ˆ ρ ( x ))( ρ TF1 ( y ) − ˆ ρ ( y ) | x − y | . Combining this with (5), dividing by Z / , and taking the root gives the claimedconvergence. (cid:3) We conclude with two remarks:1. The proof of Solovej et al [14] has the same property as the one used here andyields a generalization for the multi-center case when the distance between nucleiare kept on the Thomas–Fermi scale.2. Also the proof of the Scott conjecture of the two more elaborate models ofatoms, the Brown–Ravenhall operator treated in [3] and the no-pair operator inthe Furry picture treated in [6], have the same property that the missing errorterm in Onsager’s inequality can be added. Repeating the same argument gives theanalogues of Theorem 1. One merely needs to adapt the range of allowed constants γ to (0 , / ( π/ /π )] and (0 ,
1) respectively and change the meaning of ˆ ρ Z to therespective ground state densities. Acknowledgment:
Partial support of the DFG, grant SI 348/15-1, is grate-fully acknowledged.
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