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Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems

The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Hongshuo Chen joint work with Heinz Siedentop

Chongqing University, China

October 22 2019, CIRM

Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems

The Thomas-Fermi-Weizsäcker theory

The functional for molecules in non-relativistic Thomas-Fermi-Weizsäcker theory with atomic nuclei of atomic number Zk situated at Rk reads EnTFW (ρ) = ∫R3 ⎛ ⎝ 3 10 γρ

5 3 (x) + 1

2 ∣∇ √ ρ(x)∣2 −

K

∑

k=1

Zkρ(x) ∣x − Rk∣ ⎞ ⎠ dx + D[ρ] + ∑

0≤k<l≤K

ZkZl ∣Rk − Rl∣ where D[ρ] ∶= ∬R3×R3 ρ(x)ρ(y) ∣x − y∣ dxdy. The functional is naturally defined on all non-negative ρ ∈ L5/3(R3) with finite Weizsäcker energy and finite Coulomb energy D[ρ]. The physically γ = (3π2)

2 3 .

The other constants in front of the Weizsäcker term than 1/2 have been also discussed.

Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems

The Thomas-Fermi-Weizsäcker theory

The relativistic Thomas-Fermi-Dirac-Weizsäcker model is given by Engel and

• Dreizler1. Following them, we write it in terms of the Fermi momentum p given

by p(x) ∶= (3π2ρ(x))1/3, instead of density ρ. The energy is

ETFW (p) ∶=T W (p) + T TF (p) −

K

∑

k=1

αZk 3π2 ∫R3 dx p3(x) ∣x − Rk∣ + α 18π4 D[p3] + ∑

0≤k<l≤K

αZkZl ∣Rk − Rl ∣

where α is the Sommerfeld fine structure constant, which is 1/c in Hartree units and has the physical value of about 1/137. The Thomas-Fermi part of the kinetic energy is

T TF (p) ∶= 1 8π2 ∫R3 tTF (p(x))dx

with tTF (s) ∶= s(s2 + 1)3/2 + s3(s2 + 1)1/2 − arsinh(s) − 8

3 s3. The Weizsäcker part

• f the kinetic energy is

T W (p) ∶= 3λ 8π2 ∫R3 dx∣∇p(x)∣2f W (p(x))2

with f W (t) ∶= √

t √ t2+1 + 2 t2 t2+1 arsinh(t). The constant λ is positive. The space

• f allowed p is

P ∶= {p ∈ L4(R3)∣p ≥ 0, D[p3] < ∞, F ○ p ∈ D1(R3)}

where F is the antiderivative F(t) ∶= ∫

t 0 f W (s)ds.

• 1E. Engel and R. M. Dreizler. “ Field-theoretical approach to a relativistic Thomas-Fermi-Dirac-Weizsäcker

model”. In: Physical Review A 35.9 (May 1987), pp. 3607–3618; E. Engel and R. M. Dreizler. “Solution of the relativistic Thomas-Fermi-Dirac-Weizsäcker model for the case of neutral atoms and positive ions”. In: Physical Review A 38.8 (Oct. 1988), pp. 3909–3917.

Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems

The Thomas-Fermi-Weizsäcker theory

The following table shows the asymptotic behaviors of the integrands of the Thomas-Fermi and Weizsäcker term. The relativistic TFW model s → 0 s → ∞ tTF (s)

4 5 s5 − 1 7 s7 + O(s9)

2s4 − 8

3 s3 + 2s2 + 1 4 − ln s − ln 2 − 1 4s2 + O( 1 s4 )

(f W (s))2 s + 3

2 s3 + O(s5)

2 ln s + 2 ln 2 + 1 − 2 ln s+2 ln 2

s2

+ O( 1

s4 )

Comparing them with the non-relativistic model as following. Rel Model Non-rel Model Thomas-fermi

1 8π2 ∫R3 tTF ((3π2ρ(x))

1 3 ) dx

3(3π2)

2 3

10

∫R3 ρ

5 3 (x)dx

Weizsäcker

(3π2)

2 3 λ

24π2

∫R3 dx∣∇ρ(x)∣2 f W ((3π2ρ(x))

1 3 ) 2

ρ

4 3 (x)

1 8 ∫R3 ∣∇ρ(x)∣2 ρ(x)

dx

Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems

Results in the Thomas-Fermi-Weizsäcker theory

Results in the relativistic Thomas-Fermi-Weizsäcker theory

The relativistic energy ETFW (p) for molecules is bounded from below. The energy ETFW

Z

(p) for atoms has a global minimizer. For p ∈ P, there is a p0, such that ETFW

Z

(p0) = inf

P ETFW Z

(p). The ground state energy E TFW

Z

(N) for ∫ ρ = N fixed is monotone decreasing in N. The particle number of the minimizer is NC ∶=

1 cTF ∫ p3 0(x)dx. It satisfies

Z ≤ NC < 2.56Z. An atom of atomic number Z can not bind more than 2.56Z electrons.

Strategy to prove the excess charge problem We follow the strategy given by Benguria. We multiply the Euler equation by ∣x∣F(p)(x) and integrate,

∫ dx (8∣x∣H(p)(x)p3(x)( √ p2(x) + 1 − 1) − 6λF(p)(x)∣x∣∆F(p)(x) −8αZH(p)(x)p3(x) + 4qα 3π2 ∫ dy ∣x∣ ∣x − y∣ H(p)(x)p3(x)p3(y)) = 0

where H(p) ∶=

F(p) pF ′(p) and cF ∶= 0.612 < H < 1.

Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems

Some unsolved problems

The relativistic Weizsäcker term breaks the convexity. The uniqueness of the minimizer is unknown.

Unremovable case The existence of the minimizer for ∫ ρ = N < NC, especially for N ≤ Z.

We are trying to multiply the Euler equation by some other functions than ∣x∣F(p)(x), to get a bound better than N < 2.56Z. The ideal result in our strategy is N < 2Z.

Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems

Thanks for your attention!

Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory