A Statistical Theory of Heavy Atoms: Asymptotic Behavior of the Energy and Stability of Matter
aa r X i v : . [ m a t h - ph ] J a n A STATISTICAL THEORY OF HEAVY ATOMS:ASYMPTOTIC BEHAVIOR OF THE ENERGY AND STABILITYOF MATTER
HEINZ SIEDENTOP
Dedicated to Ari Laptev on the occasion of his septuagesimal birthday.His ideas in analysis have inspired many.
Abstract.
We give the asymptotic behavior of the ground state energy ofEngel’s and Dreizler’s relativistic Thomas-Fermi-Weizs¨acker-Dirac functionalfor heavy atoms for fixed ratio of the atomic number and the velocity of light.Using a variation of the lower bound, we show stability of matter. Introduction
Heavy atoms require a relativistic description because of the extremely fast mov-ing inner electrons. However, a statistical theory of the atom in the the spirit ofThomas [29] and Fermi [12, 13] yields a functional which is unbounded from belowbecause the semi-classical relativistic Fermi energy is too weak to prevent mass fromcollapsing into the nucleus. (See Gombas [17, §
14] and [18, Chapter III, Section16.]. Gombas also suggested that Weizs¨acker’s (non-relativistic) inhomogeneity cor-rection would solve this problem. Tomishia [30] carried this suggestion through.)Because of the same reason the relativistic generalization of the Lieb-Thirring in-equality by Daubechis is not directly applicable to Chandrasekhar operators withCoulomb potentials but requires a separate treatment of the singularity. Frankand Ekholm [10] found a way circumventing this problem treating critical poten-tials of Schr¨odinger operators; later Frank et al. [14] accomplished the same forChandrasekhar operators with Coulomb potentials. It amounts to a Thomas-Fermifunctional with a potential whose critical singularity has been extracted. However,there is a drawback, namely the Thomas-Fermi constant of this functional is smallerthan the classical one, i.e., we cannot expect asymptotically correct results.Here we discuss an alternative relativistic density functional which can handleCoulomb potentials of arbitrary strength: Engel and Dreizler [11] derived a func-tional E TFWD c,Z of the electron density ρ from quantum electrodynamics (in contrastto Gombas’ ad hoc procedure of adding the non-relativistic Weizs¨acker correction).It is – in a certain sense – a generalization of the non-relativistic Thomas-Fermi-Weizs¨acker-Dirac functional to the relativistic setting, a feature that it shares withthe functional investigated by Lieb et [22]. However, it does not suffer from theproblem that it becomes unbounded from below for heavy atoms. We will showhere, that it has – unlike the functional which can be obtained from [14] – the sameasymptotic behavior as the atomic quantum energy. The price to pay is the absenceof a known inequality relating it to the full quantum problem. One could speculatethat it might be an upper bound on the ground state energy. The way we provethe upper bound of the asymptotics might nourishes such thoughts. However, evenin the non-relativistic context this is open despite numerical evidence and claimsto the contrary, e.g., by March and Young [25]: the arguments given contain a gap. Date : January 15, 2021.
In other words, such a claim would have – even in the non-relativistic context – atbest the status of a conjecture.Engel’s and Dreizler’s functional is the relativistic TF functional (see Chan-drasekhar [7] [in the ultrarelativistic limit] and Gombas [17, §
14] for the gen-eral case) with an inhomogeneity and exchange correction different from the non-relativistic terms but with an integrand tending pointwise to their non-relativisticanalogue for large velocity of light c . In Hartree units it reads for atoms of atomicnumber Z and electron density ρ (1) E TFWD c,Z ( ρ ) := T W ( ρ ) + T TF ( ρ ) − X ( ρ ) + V ( ρ ) . The first summand on the right is an inhomogeneity correction of the kinetic en-ergy generalizing the Weizs¨acker correction. Using the Fermi momentum p ( x ) :=(3 π ρ ( x )) / it is(2) T W ( ρ ) := Z R d x λ π ( ∇ p ( x )) c f ( p ( x ) /c ) with f ( t ) := t ( t + 1) − + 2 t ( t + 1) − Arsin ( t ) where Arsin is the inverse functionof the hyperbolic sine. The parameter λ ∈ R + is given by the gradient expansionas 1 / T TF ( ρ ) := Z R d x c π T TF ( p ( x ) c )with T TF ( t ) := t ( t + 1) / + t ( t + 1) / − Arsin ( t ) − t .The third summand is a relativistic generalization of the exchange energy. It is(4) X ( ρ ) := Z R d x c π X ( p ( x ) c )with X ( t ) := 2 t − t ( t + 1) − Arsin ( t )] .Eventually, the last summand is the potential energy, namely the sum of theelectron-nucleus and the electron-electron interaction. It is(5) V ( ρ ) := − Z Z R d xρ ( x ) | x | − + Z R d x Z R d yρ ( x ) ρ ( y ) | x − y | − | {z } =: D [ ρ ] . Using F ( t ) := R t d sf ( s ), the functional E TFWD c,Z is naturally defined on(6) P := { ρ ∈ L ( R ) | ρ ≥ , D [ ρ ] < ∞ , F ◦ p ∈ D ( R ) } and bounded from below [9] for all c and Z . In fact Chen et al. [8] obtaineda Thomas-Fermi type lower bound for fixed ratio κ := Z/c . Unfortunately itsThomas-Fermi constant γ e is less than γ TF := (3 π ) , the correct physical value.For comparison we need the non-relativistic Thomas-Fermi functional(7) E TF Z ( ρ ) := 310 γ TF Z R d xρ ( x ) + V ( ρ )defined on I := { ρ ∈ L ( R ) | ρ ≥ , D [ ρ ] < ∞} . The functional is bounded frombelow (Simon [27]) and its infimum fulfills the scaling relation(8) E TF ( Z ) := inf E TF Z ( I ) = − e TF Z where e TF = − E TF (1) (Gombas [17], Lieb and Simon [23]). There exists a uniqueminimizer of E TF Z which we denote by σ .Our first result is TATISTICAL THEORY OF HEAVY ATOMS 3
Theorem 1.
Assume κ := Z/c ∈ R + fixed. Then, as Z → ∞ , (9) inf E TFWD c,Z ( P ) = − e TF Z + O ( Z ) . Our second result is the stability of the second kind of the functional which weaddress in Section 3.From a mathematical perspective it might come as a surprise that Engel’s andDreizler’s density functional – derived by purely formal methods from a quantumtheory which is still lacking a full mathematical understanding – yields a funda-mental feature like the ground state energy of heavy atoms to leading order quanti-tatively correct, in full agreement with the N -particle descriptions of heavy atomslike the Chandrasekhar Hamiltonian and no-pair Hamiltonians (Sørensen [26], [6],Solovej [28], Frank et al. [15, 16], [19]). It remains to be seen whether this is alsotrue for other quantities like the density or whether the functional can be used asa tool to investigate relativistic many particle systems – like Thomas-Fermi theoryin non-relativistic many body quantum mechanics – or, whether it even can shedlight on a deeper understanding of quantum electrodynamics.2. Bounds on the Energy
Upper Bound on the Energy.
We begin with an innocent lemma.
Lemma 1.
Assume ρ : R → R + such that p := (3 π ρ ) has partial derivativeswith respect to all variables at x ∈ R . Then (10) 38 π |∇ p ( x ) | cf ( p ( x ) /c ) ≤ | ( ∇√ ρ )( x ) | . Thus every nonnegative ρ with ∇√ ρ ∈ L ( R ) fulfills(11) T W ( ρ ) ≤ λ Z R |∇√ ρ | . Proof.
We set ψ := √ ρ and compute38 π |∇ p ( x ) | cf ( p ( x ) /c )= 3 |∇ √ ρ ( x ) | ψ ( x ) p p ( x ) /c ) + 2 ψ ( x )( p ( x ) /c ) Arsin ( p ( x ) /c )1 + ( p ( x ) /c ) ! ≤ |∇ ψ ( x ) | max ( √ t + 2 t Arsin ( t )1 + t | t ∈ R + ) ≤ |∇ ψ ( x ) | . (12) (cid:3) Of course the illuminati are hardly impressed by (11), since dominating rela-tivistic energies by non-relativistic ones is common place for them. Presumably noteven use of the numerically correct value 1.658290113 of the maximum in the proofinstead of the estimate 2 would change that.Now we turn to the upper bound on the left side of (9). It will be practical touse the non-relativistic Thomas-Fermi-Weizs¨acker functional(13) E nrTFW Z ( ρ ) := β Z R |∇√ ρ | + E TF Z ( ρ )where β ∈ R + . It is defined on J := { ρ ∈ L ( R ) | ρ ≥ , √ ρ ∈ D ( R ) , D [ ρ ] < ∞} ,has a unique minimizer ρ W with R R ρ W ≤ Z + C , and R R ρ W = O ( Z ) (Benguria[1], Benguria et al. [2], Benguria and Lieb [3]). Moreover,(14) E nrTFW ( Z ) = E nrTFW Z ( ρ W ) = E TF ( Z ) + D β Z + o ( Z ) H. SIEDENTOP for some β -dependent constant D β ∈ R + (Lieb and Liberman [21] and Lieb [20,Formula (1.6)]).In the following we pick β = 2 and use the minimizer ρ W of the non-relativisticThomas-Fermi-Weizs¨acker functional as a test function.We estimate the exchange term first. Since − X ( t ) ≤ t , we get(15) −X ( ρ W ) ≤ (3 π ) π Z R ρ w ≤ C sZ R ρ W Z R ρ W = O ( Z ) . Thus, since T TF ( t ) ≤ t ,(16) inf E TFWD c,Z ( P ) ≤ E TFWD c,Z ( ρ W ) ≤ E nrTFW Z ( ρ W ) + O ( Z ) = E TF ( Z ) + O ( Z )which concludes the proof of the upper bound.2.2. Lower Bound on the Energy.
We set ¯d ξ := d ξ/h = d ξ/ (2 π ) . (Notethat the rationalized Planck constant ~ equals one in Hartree units and, therefore, h = 2 π .)We introduce the notation ( a ) − := min { , a } and write ϕ σ := Z/ | · | − σ ∗ | · | − for the Thomas-Fermi potential of the minimizer σ . We start again with a littleLemma. Lemma 2.
Assume κ = Z/c fixed. Then, as Z → ∞ , Z | x | > Z d x Z R ¯d ξ ( ξ − ϕ σ ( x )) − − Z | x | > Z d x Z R ¯d ξ ( p c ξ + c − c − ϕ σ ( x )) − = O ( Z ) . Again, it does not come as a surprise to the physicist that relativistic and non-relativistic theory give the same result up to errors, if the innermost electrons, i.e.,in particular the fast moving, are disregarded.
Proof.
Since ξ / ≥ p c ξ + c − c , the left side of the claimed inequality cannotbe negative. Thus, we merely need an upper bound: Z | x | > Z d x Z R ¯d ξ ( ξ − ϕ σ ( x )) − − Z | x | > Z d x Z R ¯d ξ (cid:16)p c ξ + c − c − ϕ σ ( x ) (cid:17) − ≤ Z | x | > Z d x ( c Z ξ < ϕσ ( x ) c ¯d ξ [ ξ − ( p ξ + 1 − − Z √ c ξ + c − c <ϕ σ ( x ) ≤ ξ / ¯d ξ (cid:16)p c ξ + c − c − ϕ σ ( x ) (cid:17)) ≤ c Z | x | > Z d x Z ξ < ϕσ ( x ) c ¯d ξ + Z √ ξ +1 − < ϕσ ( x ) c ≤ ξ ¯d ξ [ ξ − ( p ξ + 1 − ≤ c Z | x | > Z d x Z ξ < ϕσ ( x ) c ¯d ξ + Z √ ξ +1 − < ϕσ ( x ) c ≤ ξ ¯d ξ ! | ξ | ≤ c Z | x | > Z d x Z √ ξ +1 − < ϕσ ( x ) c ¯d ξ | ξ | ≤ c κ Z | x | > d x Z √ ξ +1 − < κ | x | ¯d ξ | ξ | (17)where we used ϕ σ ( x ) ≤ Z/ | x | in the last inequality. Moreover, the resulting lastintegral obviously exists and is independent of Z . Thus the left side of the claimedinequality is bounded from above by a constant depending only on κ times Z quoderat demonstrandum. (cid:3) TATISTICAL THEORY OF HEAVY ATOMS 5
We turn to the lower bound on the left side of (9) and follow initially [8]. In fact,apart from minor modifications, we copy the high density part and focus on the lowdensity part. We pick any ρ ∈ P and address the parts of the energy separately.2.2.1. The Weizs¨acker Energy.
Since F ( t ) ≥ t p Arsin ( t ) / T W ( ρ ) ≥ λc π Z R d x p ( x ) Arsin ( p ( x ) c ) | x | = 3 λc π Z R d x ρ ( x ) Arsin ( p ( x ) c ) | x | | {z } =: H ( ρ ) . The Potential Energy.
Since σ is positive, we have ϕ σ ( x ) ≤ Z/ | x | . Then(19) V ( ρ ) = − Z R d xϕ σ ( x ) ρ ( x ) − D ( σ, ρ ) + D [ ρ ] ≥ − Z R d xϕ σ ( x ) ρ ( x ) − D [ σ ] . Splitting the integrals at s , using (19), and Schwarz’s inequality yields V ( ρ ) ≥ − Z p ( x ) /c ( ρ ) − Z p ( x ) /c ( ρ ) := R p ( x ) /c>s d xρ ( x ) .2.2.3. The Thomas-Fermi Term.
First, we note that(21) R + → R + , t T TF ( t ) /t is monotone increasing from 0 to 2. Thus T TF ( ρ ) = Z p ( x ) /c ( ρ ) . (22)2.2.4. Exchange Energy.
Since X is bounded from above and X ( t ) = O ( t ) at t = 0, we have that for every α ∈ [0 ,
4] there is an η such that X ( t ) ≤ η t α . Wepick α = 3 in which case ξ ≈ .
15. Thus, with η := η / (4 π ) ≈ . X ( ρ ) ≤ cη π N = ηcN. The Total Energy.
Adding everything up yields E TFWD c,Z ( ρ ) ≥ λc π H ( ρ ) + 38 T TF ( s ) s γ TF c T > ( ρ ) − Z Arsin ( s ) H ( ρ ) T > ( ρ ) + Z p ( x ) c
The solution is uniquely determined, because of (21) (and the line below) and
Arsin ( s ) is also monotone increasing from 0 to ∞ . Call the corresponding s s .Obviously, s does not depend on c and Z independently but only on the ratio κ := Z/c and is strictly monotone increasing from 0 to ∞ .We set I s,Z := { x ∈ R | p ( x ) /c < s, | x | < /Z } ,A s,Z := { x ∈ R | p ( x ) /c < s, | x | ≥ /Z } . (26)Then(27) E TFWD c,Z ( ρ ) ≥ I + A − D [ σ ] − ξcN with I := Z I s,Z d x (cid:18) c π T TF ( p ( x ) c ) − ϕ σ ( x ) ρ ( x ) (cid:19) (28) A := Z A s,Z d x (cid:18) c π T TF ( p ( x ) c ) − ϕ σ ( x ) ρ ( x ) (cid:19) (29)We estimate I from below by dropping the TF-term, using ϕ σ ( x ) ≤ Z/ | x | , andobserving that x ∈ I s,Z . We get(30) I ≥ − C κ Zc Z Z − d rr = − C κ Z where C κ is a generic constant depending on κ only. In other words, we can pullthe Coulomb tooth paying a negligible price.Next we estimate A from below by keeping A s,Z fixed and minimizing the inte-grand at each point x ∈ A s,Z by varying the values p ( x ) ∈ R + . We get A ≥ Z A s,Z d x Z R ¯d ξ (cid:16)p c ξ + c − c − ϕ σ ( x ) (cid:17) − ≥ Z | x |≥ /Z d x Z R ¯d ξ (cid:16)p c ξ + c − c − ϕ σ ( x ) (cid:17) − . (31)Although at first glance the first inequality might seem abrupt, it is easily checkedthat the Thomas-Fermi functional (relativistic or non-relativistic, restricted to someregion in space M ) with kinetic energy T ( ξ ) and external potential ϕ is merely themarginal functional (integrating out the momentum variable ξ ) of the phase spacevariational principle(32) E Γ ( γ ) := Z M d x Z R ¯d ξ ( T ( ξ ) − ϕ ( x )) γ ( x, ξ )with γ ( M, R ) ⊂ [0 ,
2] and the choice γ ( x, ξ ) := χ { ( x,ξ ) ∈ M × R || ξ |
Stability of Matter
For several atoms the potential term V in (5) is replaced by(34) V ( ρ ) := − K X k =1 Z R d x Z k ρ ( x ) | x − R k | + D [ ρ ] + X ≤ k 4. We also set H k ( x ) := 2 p Y ( | x − R k | /D K ) /D k . We pick theradii maximal, namely D k as half the distance of the k -th nucleus to its nearestneighbor.We also use Lieb and Yau’s electrostatic inequality [24, Formula (4.5)]: we writeΓ , ..., Γ K for the Voronoi cells of R , ..., R K and set(36) Φ( x ) := K X l =1 Γ l ( x ) K X k =1 ,k = l Z | x − R k | , i.e., the nuclear potential at point x of all nuclei except the one from the cell inwhich x lies. With this notation the electrostatic inequality reads(37) D [ ν ] − Z Z R Φ( x )d ν ( x ) + Z X ≤ k Instead of (18) we get(38) T W ( ρ ) ≥ λc π K X k =1 Z B k d xρ ( x ) Arsin ( p ( x ) c ) (cid:0) | x − R k | − − H k ( x ) (cid:1) + | {z } =: H k ( ρ ) where we use (35) and F ( t ) ≥ t p Arsin ( s ) / H. SIEDENTOP The Potential Energy. Using (37) and p ( a − b ) + ≥ a − b for a, b ∈ R + we get V ( ρ ) ≥ − K X k =1 Z B K d x + Z Γ k \ B k d x ! Zρ ( x ) | x − R k | + Z K X k =1 D k ≥ − K X k =1 Z (Z B k , p ( x ) /c ≥ s d xρ ( x ) "s(cid:18) | x − R k | − H k ( x ) (cid:19) + + H k ( x ) + Z B k , p ( x ) /c ≤ s d x ρ ( x ) | x − R k | + Z Γ k \ B k ρ ( x ) | x − R k | − Z D k ) ≥ − K X k =1 Z p Arsin ( s ) p H R k ( ρ ) T R k ( ρ ) + Z Z B k , p ( x ) /c ≥ s d xH k ( x ) ρ ( x )+ Z B k , p ( x ) /c ≤ s d x Zρ ( x ) | x − R k | + Z Γ k \ B k d x Zρ ( x ) | x − R k | ! + Z K X k =1 D k (39)with T R k ( ρ ) := R B k , p ( x ) /c ≥ s d xρ ( x ) .3.3. The Combined Thomas-Fermi and Exchange Terms. We use T TF ( t ) ≥ t − t / 3, (23), and set δ := (3 π ) . This yields(40) T TF ( ρ ) − X ( ρ ) = Z R d x c π T TF ( p ( x ) c ) − ηcN ≥ δc Z R d xρ ( x ) − C c N, The Total Energy. Adding again all up yields E TFWD c,Z ( ρ )(41) ≥ K X k =1 c " λ π H R k ( ρ ) + δ T R k ( ρ ) − κ p Arsin ( s ) p H R k ( ρ ) T R k ( ρ )(42) + Z B k , p ( x ) c 037 (Betheand Salpeter [5, p. 84]), we get(53) Z max ≈ . Theorem 2. There exists a constant C such that for all ρ ∈ P and all pairwisedifferent R , ..., R K ∈ R and Z = ... = Z K ∈ [0 , Z max ] E TFWD R , Z ( ρ ) ≥ C · ( K + N ) . We conclude with two remarks:1. Theorem 2 holds actually for all Z ∈ [0 , Z max ] K . Our proof obviously gener-alizes to that case, since the potential estimate (37) also generalizes in the obviousway.2. It might be surprising that there is no requirement on a minimal velocity oflight which is independent of the value of Z . This is different from other unrenor-malized relativistic models like the Thomas-Fermi functional with inhomogeneitycorrection ( √ ρ, |∇|√ ρ ) investigated by Lieb et al. [22, Formula (2.7)]. There wewere forced to control the exchange energy by the Thomas-Fermi term. Here, this isno longer necessary: due to the renormalization in Engel’s and Dreizler’s derivationthe exchange energy is bounded from below by a multiple of the particle number. Acknowledgments Special thanks go to Rupert Frank for many inspiring discussions, in particularfor pointing out that in addition to separation in high and low density regimes, alocalization in space near the nucleus could be useful, and for critical reading of asubstantial part of the manuscript.Thanks go also to Hongshuo Chen for critical reading of the manuscript.Partial support by the Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) through Germany’s Excellence Strategy EXC - 2111 - 390814868is gratefully acknowledged. References [1] Rafael Benguria. The von Weizs¨acker and Exchange Corrections in the Thomas-Fermi The-ory . PhD thesis, Princeton, Department of Physics, June 1979.[2] Rafael Benguria, Haim Brezis, and Elliott H. Lieb. 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Phys. , 96:431–458, 1935.[32] Katsumi Yonei and Yasuo Tomishima. On the Weizs¨acker correction to the Thomas-Fermitheory of the atom. Journal of the Physical Society of Japan , 20(6):1051–1057, 1965. Mathematisches Institut, Ludwig-Maximilans Universit¨at M¨unchen, Theresienstraße39, 80333 M¨unchen, Germany, and Munich Center for Quantum Science and Technology(MCQST), Schellingstr. 4, 80799 M¨unchen, Germany Email address ::s d x (cid:16) δρ ( x ) − κH k ( x ) ρ ( x ) (cid:17) (44) + Z Γ k \ B k d x (cid:18) δρ ( x ) − κρ ( x ) | x − R k | (cid:19) (45) + K X k =1 Z D k − C c N. (46)We pick s such that(47) 2 s λ π δ = κ p Arsin ( s )which makes (42) a sum of complete squares. Next(43) ≥ δ ( cs ) inf (Z p ( x ) < d x (cid:18) ρ ( x ) − κρ ( x ) csδ | x | (cid:19) (cid:12)(cid:12)(cid:12) ρ ∈ P ) ≥ csκ δ inf (Z p ( x ) < d x (cid:18) ρ ( x ) − ρ ( x ) | x | (cid:19) (cid:12)(cid:12)(cid:12) ρ ∈ P ) ≥ − C csκ δ (48) TATISTICAL THEORY OF HEAVY ATOMS 9 where we replaced p by csp in the first step and x by κ/ ( csδ ) x in the secondstep. Thus (48) yields, after summation, K times a constant which is irrelevant forstability. Furthermore,(49) (44) ≥ − κ · π · δ D k Z d rr (1 + r / = − πκ δ D k and using R Γ k \ B k d x | x − R k | − ≤ π/D k (Lieb et al. [22, Formula (4.6)]) we get(50) (45) ≥ − κ δ Z Γ k \ B k | x − R k | ≥ − πκ δ D k . Thus, the energy per particle is bounded from below uniformly in ρ , K , and N , if(51) c (cid:18) πκ δ + 3 πκ δ (cid:19) ≤ Z , i.e.,(52) Z ≤ Z max := 3 √ π c . Numerically, using the physical value of the velocity of light c = 137 .