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Asymptotics of the Ground State Energy forRelativistic Atoms and Molecules

PDE and Analysis Seminar,Hebrew University at Jerusalem

Victor Ivrii

Department of Mathematics, University of Toronto

Victor Ivrii (Math., Toronto) Relativistic Atoms and Molecules December 6, 2017 1 / 43

Table of Contents

Table of Contents

1 Introduction2 Thomas-Fermi theory3 Reduction to one-particle

problemReduction to one-particleproblem. IReduction to one-particleproblem. IIReduction to one-particleproblem. III

4 Semiclassics

Singular zoneRegular zoneTrace term and Scottcorrection term

5 Sharper estimate6 Discussion7 Self generated magnetic field

Set-upNon-relativistic caseRelativistic caseDiscussion


Introduction

Introduction

The purpose of this work was:

1 To improve results of J. P. Solovej, T. Ø. Sørensen, W. L. Spitzer[SSS];

2 and L. Erdos, S. Fournais, J. P. Solovej [EFS2].

The functional analytic arguments in these papers are the best (and waybetter than I could produce on my own), and I use them heavily on certainstages of my analysis.On the other hand, the microlocal semiclassical arguments requireimprovements, which we provide.


Introduction

Introduction




The functional analytic arguments in these papers are the best (and waybetter than I could produce on my own), and I use them heavily on certainstages of my analysis.

On the other hand, the microlocal semiclassical arguments requireimprovements, which we provide.


Introduction

Introduction




The functional analytic arguments in these papers are the best (and waybetter than I could produce on my own), and I use them heavily on certainstages of my analysis.On the other hand, the microlocal semiclassical arguments requireimprovements, which we provide.


Introduction

The framework

Let us consider the following operator (quantum Hamiltonian)

H = HN :=∑

1≤j≤N

HV ,xj +∑

1≤j<k≤N

e2

|xj − xk |(1)

on

H =⋀

1≤n≤N

H, H = L2(R3,Cq) ≃ L2(R3 × {1, . . . , q},C) (2)

with

HV = T − V (x) (3)

describing N same type particles in the external field with the scalarpotential −V and repulsing one another according to the Coulomb law; eis a charge of the electron, T is an operator of the kinetic energy.


Introduction

Here xj ∈ R3, and (x1, . . . , xN) ∈ R3N , potential V (x) is assumed to bereal-valued. Except when specifically mentioned we assume that

V (x) =∑

1≤m≤M

Zme2

|x − ym|(4)

where Zme > 0 and ym are charges and locations of nuclei.

The are two crucial questions: the quantum statistics and what is T?

Quantum statistics

Assume that the particles (electrons) are fermions. This means that theHamiltonian should be considered on the Fock space H defined by (2) ofthe functions antisymmetric with respect to all variables(x1, 𝜍1), . . . , (xN , 𝜍N) where 𝜍j ∈ {1, . . . , q}, q = 2, are spin variables.


Introduction

Here xj ∈ R3, and (x1, . . . , xN) ∈ R3N , potential V (x) is assumed to bereal-valued. Except when specifically mentioned we assume that

V (x) =∑

1≤m≤M

Zme2

|x − ym|(4)

where Zme > 0 and ym are charges and locations of nuclei.The are two crucial questions: the quantum statistics and what is T?

Quantum statistics

Assume that the particles (electrons) are fermions. This means that theHamiltonian should be considered on the Fock space H defined by (2) ofthe functions antisymmetric with respect to all variables(x1, 𝜍1), . . . , (xN , 𝜍N) where 𝜍j ∈ {1, . . . , q}, q = 2, are spin variables.


Introduction

Kinetic energy operator

Non-magnertic, non-relativistic theory

In this the most basic case

T =1

2m(−i~∇)2, (5)

where m is the mass of electron, ~ is Planck constant.

Non-magnertic, relativistic theory

In the the relativistic case

T =(c2(−i~∇)2 +m2c4

) 12 −mc2, (6)

where c is the speed of light.


Introduction

Scaling

Using the scaling by the spatial variables and by the energy we can makem = 1

2 , e = ~ = 1; then Zm do not change, in the relativistic theory𝛽 := e2/~c also does not change.

Therefore, in the non-relativistic case there is only one scaling parameterZ = Z1 + . . .+ ZM ≍ N, while in the relativistic case there is also 𝛽.

Remark

However, if M ≥ 2 there is another parameter

d = min1≤m<m′≤M

|ym − ym′ | : ~2/me2. (7)

and in relativistic settings ~2/me2 · 𝛽 = ~/mc is called Comptonwavelength.


Introduction

Ground state energy

We are looking for the ground state energy, which is

inf Spec(HN) := infΨ∈H,‖Ψ‖=1

(HNΨ,Ψ) (8)

with HN = HN,V (so operator is understood if necessary in the sense offorms).

Remark

While non-relativistic operator is always semi-bounded from below, itsrelativistic cousin is semi-bounded from below if and only if

𝛽Zm ≤ 2

𝜋for m = 1, . . . ,M; (9)

see [Herbst, Lieb-Yau]; we assume that d ≥ Z−1.


Thomas-Fermi theory

Thomas-Fermi theory

If electrons were not interacting between themselves but the field potentialwas −W (x) then they would occupy lowest eigenvalues and ground statewave functions would be (anti-symmetrized)𝜑1(x1, 𝜍1)𝜑2(x2, 𝜍2) . . . 𝜑N(xN , 𝜍N) where 𝜑j and 𝜆j are eigenfunctions andeigenvalues of H = T −W (x).

Then the local electron density would be 𝜌Ψ =∑

1≤j≤N |𝜑j(x)|2 andaccording to the pointwise Weyl law

𝜌Ψ(x) ≈ P ′(W + 𝜈) := q(2𝜋)−3

∫{𝜉:T (𝜉)−W (x)≤𝜈}

d𝜉, (10)

where 𝜈 = 𝜆N .

This density would generate potential −|x |−1 * 𝜌Ψ and we would haveW ≈ V − |x |−1 * 𝜌Ψ.


Thomas-Fermi theory

Thomas-Fermi theory



1≤j≤N |𝜑j(x)|2

andaccording to the pointwise Weyl law

𝜌Ψ(x) ≈ P ′(W + 𝜈) := q(2𝜋)−3

∫{𝜉:T (𝜉)−W (x)≤𝜈}

d𝜉, (10)




Thomas-Fermi theory

Thomas-Fermi theory



1≤j≤N |𝜑j(x)|2 andaccording to the pointwise Weyl law

𝜌Ψ(x) ≈ P ′(W + 𝜈) := q(2𝜋)−3

∫{𝜉:T (𝜉)−W (x)≤𝜈}

d𝜉, (10)




Thomas-Fermi theory

Replacing all approximate equalities by a strict ones we arrive toThomas-Fermi equations:

V −W TF = |x |−1 * 𝜌TF, (11)

𝜌TF = P ′(W + 𝜈), (12)∫𝜌TF dx = min(N,Z ), (13)

where 𝜈 ≤ 0 is called chemical potential and in fact approximates 𝜆N .

In the non-relativistic case

P ′(W + 𝜈) =q

6𝜋2(W + 𝜈)

32+, (14)

and in the relativistic case we have

P ′*(W + 𝜈) =

q

6𝜋2(W + 𝜈)

32+(1 + c−2(W + 𝜈))

32 (15)


Thomas-Fermi theory

Replacing all approximate equalities by a strict ones we arrive toThomas-Fermi equations:

V −W TF = |x |−1 * 𝜌TF, (11)

𝜌TF = P ′(W + 𝜈), (12)∫𝜌TF dx = min(N,Z ), (13)

where 𝜈 ≤ 0 is called chemical potential and in fact approximates 𝜆N .In the non-relativistic case

P ′(W + 𝜈) =q

6𝜋2(W + 𝜈)

32+, (14)

and in the relativistic case we have

P ′*(W + 𝜈) =

q

6𝜋2(W + 𝜈)

32+(1 + c−2(W + 𝜈))

32 (15)


Thomas-Fermi theory

We need also to know its primitive which in non-relativistic case is simple

P(W + 𝜈) =q

15𝜋2(W + 𝜈)

52+; (16)

in the relativistic case P*(W + 𝜈) is an elementary function as well and asadistic Calculus instructor can give it on the test.

However it turns outthat we really do not need any separate relativistic Thomas-Fermi theory.

Remark

1 It looks strange because even∫P ′*(W + 𝜈) dx logarithmically diverges

at ym but in the zones {x : |x − ym| ≤ Z−1} effective semiclassicalparameter is & 1, so Weyl approximation is wrong here anyway, andwe will estimate nicely the contribution of these zones.

2 Meanwhile,∫P*(W + 𝜈) dx diverges at 0 in more malicious way but

we will need to regularize both it and∫P(W + 𝜈) dx by subtracting∫

P*(Vm) dx and∫P(Vm) dx respectively, Vm = Zm|x − ym|−1.


Thomas-Fermi theory

We need also to know its primitive which in non-relativistic case is simple

P(W + 𝜈) =q

15𝜋2(W + 𝜈)

52+; (16)

in the relativistic case P*(W + 𝜈) is an elementary function as well and asadistic Calculus instructor can give it on the test. However it turns outthat we really do not need any separate relativistic Thomas-Fermi theory.

Remark

1 It looks strange because even∫P ′*(W + 𝜈) dx logarithmically diverges

at ym but in the zones {x : |x − ym| ≤ Z−1} effective semiclassicalparameter is & 1, so Weyl approximation is wrong here anyway, andwe will estimate nicely the contribution of these zones.

2 Meanwhile,∫P*(W + 𝜈) dx diverges at 0 in more malicious way but

we will need to regularize both it and∫P(W + 𝜈) dx by subtracting∫

P*(Vm) dx and∫P(Vm) dx respectively, Vm = Zm|x − ym|−1.


Reduction to one-particle problem Reduction to one-particle problem. I

Reduction to one-particle problem. I

The original problem can be reduced to one-particle problem. Actually,there are two different reductions: in the estimate from below and in theestimate from above. We start from the the estimate from below.

In the non-relativistic case one uses Lieb’s electrostatic inequality

(∑

1≤j<k≤N

|xj − xk |−1Ψ,Ψ) ≥ 1

2D(𝜌Ψ, 𝜌Ψ)− C

∫𝜌

43Ψ dx (17)

with

D(f , g) :=

∫|x − y |−1f (x)g(y) dxdy (18)

and

𝜌Ψ(x) = N

∫|Ψ(x , x2, . . . , xN)|2 dx2 · · · dxN (19)

which is a spatial density of the system in the state Ψ.



Reduction to one-particle problem. I

The original problem can be reduced to one-particle problem. Actually,there are two different reductions: in the estimate from below and in theestimate from above. We start from the the estimate from below.In the non-relativistic case one uses Lieb’s electrostatic inequality

(∑

1≤j<k≤N

|xj − xk |−1Ψ,Ψ) ≥ 1

2D(𝜌Ψ, 𝜌Ψ)− C

∫𝜌

43Ψ dx (17)

with

D(f , g) :=

∫|x − y |−1f (x)g(y) dxdy (18)

and

𝜌Ψ(x) = N

∫|Ψ(x , x2, . . . , xN)|2 dx2 · · · dxN (19)

which is a spatial density of the system in the state Ψ.



Heuristically the first term in the right-hand expression of (17) is apotential energy of the electron cloud.

This inequality holds for any Ψ ∈ ⊗1≤j≤NL2(R3,Cq), not necessarilyfermionic, or a ground state.But it has proven that for fermionic ground state Ψ in the non-relativisticcase ∫

𝜌43Ψ dx ≤ CZ

53 . (20)

Then

(HNΨ,Ψ) ≥∑k

(HV ,kΨ,Ψ) +1

2D(𝜌Ψ, 𝜌Ψ)− CZ

53 (21)

and the first two terms in the right-hand expression could be rewritten as∑k

(HW ,kΨ,Ψ)− 1

2D(𝜌, 𝜌) +

1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌) (22)

with arbitrary 𝜌 and W = V − |x |−1 * 𝜌.



Heuristically the first term in the right-hand expression of (17) is apotential energy of the electron cloud.This inequality holds for any Ψ ∈ ⊗1≤j≤NL2(R3,Cq), not necessarilyfermionic, or a ground state.

But it has proven that for fermionic ground state Ψ in the non-relativisticcase ∫

𝜌43Ψ dx ≤ CZ

53 . (20)

Then

(HNΨ,Ψ) ≥∑k

(HV ,kΨ,Ψ) +1


53 (21)


(HW ,kΨ,Ψ)− 1

2D(𝜌, 𝜌) +

1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌) (22)




Heuristically the first term in the right-hand expression of (17) is apotential energy of the electron cloud.This inequality holds for any Ψ ∈ ⊗1≤j≤NL2(R3,Cq), not necessarilyfermionic, or a ground state.But it has proven that for fermionic ground state Ψ in the non-relativisticcase ∫

𝜌43Ψ dx ≤ CZ

53 . (20)

Then

(HNΨ,Ψ) ≥∑k

(HV ,kΨ,Ψ) +1


53 (21)


(HW ,kΨ,Ψ)− 1

2D(𝜌, 𝜌) +

1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌) (22)




Since operators HW ,k act by different variables∑k

(HW ,kΨ,Ψ) ≥∑

1≤j≤N′

𝜆j

≥ Tr((HW − 𝜈)−) + 𝜈N

(23)

where N ′ is either N or the number of negative eigenvalues of HW ,whatever is less (we take 𝜌 such that the negative spectrum of HW isdiscrete).

In both cases we have the second inequality (23) with arbitrary𝜈 ≤ 0. So

EN ≥ Tr((HW+𝜈)−) + 𝜈N − 1

2D(𝜌, 𝜌)− CZ

53 . (24)

Remark

Actually, in the right-hand expression is also a “bonus term”12D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌); we call it so because we do not need it but weestimate it also as a bonus.




(HW ,kΨ,Ψ) ≥∑

1≤j≤N′

𝜆j ≥ Tr((HW − 𝜈)−) + 𝜈N (23)

where N ′ is either N or the number of negative eigenvalues of HW ,whatever is less (we take 𝜌 such that the negative spectrum of HW isdiscrete). In both cases we have the second inequality (23) with arbitrary𝜈 ≤ 0.

So

EN ≥ Tr((HW+𝜈)−) + 𝜈N − 1


53 . (24)

Remark





(HW ,kΨ,Ψ) ≥∑

1≤j≤N′

𝜆j ≥ Tr((HW − 𝜈)−) + 𝜈N (23)

where N ′ is either N or the number of negative eigenvalues of HW ,whatever is less (we take 𝜌 such that the negative spectrum of HW isdiscrete). In both cases we have the second inequality (23) with arbitrary𝜈 ≤ 0. So

EN ≥ Tr((HW+𝜈)−) + 𝜈N − 1


53 . (24)

Remark




Consider the semiclassical approximation to the trace term

Tr((HW+𝜈)−) ≈ −

∫P(W + 𝜈) dx (25)

and we arrive to

EN ' −∫

P(W + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N, (26)

and the right-hand expression should be maximized by the choice of 𝜌,W = V − |x |−1 * 𝜌 and 𝜈 ≤ 0.It turns out that the optimal choice are Thomas-Fermi potential W TF anddensity 𝜌TF and a chemical potential 𝜈 i.e. solutions of (11)–(13)




Tr((HW+𝜈)−) ≈ −

∫P(W + 𝜈) dx (25)

and we arrive to

EN ' −∫

P(W + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N, (26)

and the right-hand expression should be maximized by the choice of 𝜌,W = V − |x |−1 * 𝜌 and 𝜈 ≤ 0.

It turns out that the optimal choice are Thomas-Fermi potential W TF anddensity 𝜌TF and a chemical potential 𝜈 i.e. solutions of (11)–(13)




Tr((HW+𝜈)−) ≈ −

∫P(W + 𝜈) dx (25)

and we arrive to

EN ' −∫

P(W + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N, (26)

and the right-hand expression should be maximized by the choice of 𝜌,W = V − |x |−1 * 𝜌 and 𝜈 ≤ 0.It turns out that the optimal choice are Thomas-Fermi potential W TF anddensity 𝜌TF and a chemical potential 𝜈 i.e. solutions of (11)–(13)

V −W TF = |x |−1 * 𝜌TF, (11)

𝜌TF = P ′(W + 𝜈), (12)∫𝜌TF dx = min(N,Z ). (13)




Tr((HW+𝜈)−) ≈ −

∫P(W + 𝜈) dx (25)

and we arrive to

EN ' −∫

P(W + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N, (26)

and the right-hand expression should be maximized by the choice of 𝜌,W = V − |x |−1 * 𝜌 and 𝜈 ≤ 0.It turns out that the optimal choice are Thomas-Fermi potential W TF anddensity 𝜌TF and a chemical potential 𝜈 i.e. solutions of (11)–(13) and wecall the result Thomas-Fermi energy and denote by ℰTF.

It has a magnitude Z73 and the error here consist of two parts:

semiclassical error in the trace term and ≍ Z53 from electrostatic inequality.

Currently semiclassical error is ≍ Z 2 but we will improve it with Scottcorrection term; still 𝜌TF etc will remain our choice.


Reduction to one-particle problem Reduction to one-particle problem. II

Reduction to one-particle problem. II

Let us take a test function Ψ = Ψ(x1, 𝜍1; . . . ; xN , 𝜍N) antisymmetrizedproduct 𝜑1(x1, 𝜍1) · · ·𝜑N(xN , 𝜍N) where 𝜑j are eigenfunctions of HW

corresponding to negative eigenvalues 𝜆j .

If N−(HW ) < N where N−(HW ) is the number of the negative eigenvalues(essential spectrum occupies [0,∞)) then we increase EN replacing N by alesser value N−(HW ).Then

EN ≤∑

1≤j≤N

𝜆j +1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌)− 1

2D(𝜌, 𝜌)

−1

2

∫|x − y |−1 · |eN(x , y)|2 dxdy

where eN(x , y) = e(x , y , 𝜆N + 0) and 𝜌Ψ(x) = tr eN(x , x), tr means thematrix trace.





corresponding to negative eigenvalues 𝜆j .If N−(HW ) < N where N−(HW ) is the number of the negative eigenvalues(essential spectrum occupies [0,∞)) then we increase EN replacing N by alesser value N−(HW ).

Then

EN ≤∑

1≤j≤N

𝜆j +1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌)− 1

2D(𝜌, 𝜌)

−1

2

∫|x − y |−1 · |eN(x , y)|2 dxdy






corresponding to negative eigenvalues 𝜆j .If N−(HW ) < N where N−(HW ) is the number of the negative eigenvalues(essential spectrum occupies [0,∞)) then we increase EN replacing N by alesser value N−(HW ).Then

EN ≤∑

1≤j≤N

𝜆j +1

2D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌)− 1

2D(𝜌, 𝜌)

−1

2

∫|x − y |−1 · |eN(x , y)|2 dxdy




Note that∑1≤j≤N

𝜆j ≤ Tr(H−W+𝜈) + 𝜈N + |𝜆N − 𝜈| · |N−(HW+𝜆N∓0)− N−(HW+𝜈±0)|

where the last factor estimates the number of eigenvalues in [𝜆N , 𝜈] (but𝜈 = 0 is excluded from this interval) and we consider both cases𝜆N ≤ 𝜈 ≤ 0 and 𝜈 < 𝜆N < 0

and

1

2D(eN(x , x)− 𝜌, eN(x , x)− 𝜌) ≤ D(e(x , x , 𝜈)− 𝜌, e(x , x , 𝜈)− 𝜌)+

D(e(x , x , 𝜈)− eN(x , x), e(x , x , 𝜈)− eN(x , x)).



Note that∑1≤j≤N

𝜆j ≤ Tr(H−W+𝜈) + 𝜈N + |𝜆N − 𝜈| · |N−(HW+𝜆N∓0)− N−(HW+𝜈±0)|

where the last factor estimates the number of eigenvalues in [𝜆N , 𝜈] (but𝜈 = 0 is excluded from this interval) and we consider both cases𝜆N ≤ 𝜈 ≤ 0 and 𝜈 < 𝜆N < 0 and

1

2D(eN(x , x)− 𝜌, eN(x , x)− 𝜌) ≤ D(e(x , x , 𝜈)− 𝜌, e(x , x , 𝜈)− 𝜌)+

D(e(x , x , 𝜈)− eN(x , x), e(x , x , 𝜈)− eN(x , x)).



If we skip temporarily dimmed terms,

replace Tr(H−W+𝜈) by its Weyl

approximation, and e(x , x , 𝜈) by its pointwise Weyl approximationP ′(W (x) + 𝜈), we get

EN / −∫

P(W (x) + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N

+D(P ′(W + 𝜈)− 𝜌,P ′(W + 𝜈)− 𝜌)

and minimizing the right-hand expression with respect to W , 𝜈 (recallingthat W = V − |x |−1 * 𝜌) we again arrive to W = W TF etc.



If we skip temporarily dimmed terms, replace Tr(H−W+𝜈) by its Weyl

approximation,

and e(x , x , 𝜈) by its pointwise Weyl approximationP ′(W (x) + 𝜈), we get

EN / −∫

P(W (x) + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N

+D(P ′(W + 𝜈)− 𝜌,P ′(W + 𝜈)− 𝜌)





approximation, and e(x , x , 𝜈) by its pointwise Weyl approximationP ′(W (x) + 𝜈),

we get

EN / −∫

P(W (x) + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N

+D(P ′(W + 𝜈)− 𝜌,P ′(W + 𝜈)− 𝜌)





approximation, and e(x , x , 𝜈) by its pointwise Weyl approximationP ′(W (x) + 𝜈), we get

EN / −∫

P(W (x) + 𝜈) dx − 1

2D(𝜌, 𝜌) + 𝜈N

+D(P ′(W + 𝜈)− 𝜌,P ′(W + 𝜈)− 𝜌)




So in the non-relativistic case both estimates we got the same answer

Tr(H−W+𝜈) + 𝜈N − 1

2D(𝜌, 𝜌) ≈

−∫

P(W + 𝜈) dx + 𝜈N − 1

2D(𝜌, 𝜌) (27)

with W = W TF etc except

1 there is a term −CZ53 in the estimate from below,

2 in the estimate from above there are some semiclassical errors,

3 and the transition from the trace term on the left to its semiclassicalexpression on the right will be reexamined.


Reduction to one-particle problem Reduction to one-particle problem. III

Reduction to one-particle problem. III

But what about relativistic case?

1 It turns out that under assumption

𝛽Zm ≤ 2

𝜋− 𝜖 for m = 1, . . . ,M (9)*

inequality (20)∫𝜌

43Ψ dx ≤ CZ

53 holds for the ground state;

2 But we dont’t know it is the case under (9);

3 And the semiclassical errors are not purely semiclassical anymorebecause while operator is relativistic, we write Weyl expressions fornon-relativistic one.



To remedy the first issue we will use the correlation inequality [SSS]∑1≤j<k≤N

|xj − xk |−1 ≥N∑j=1

(|x |−1 * 𝜌 * 𝜑𝜀)(xj)−1

2D(𝜌, 𝜌)− CN𝜀−1, (28)

where 𝜌 ≥ 0 is any function, 𝜑 ≥ 0 is spherically symmetric, with∫𝜑 dx = 1, 𝜑𝜀(x) = 𝜀−3𝜑(x/𝜀).

Then in operator W = V − |x |−1 * 𝜌 is replaced byW𝜀 = V − |x |−1 * 𝜌 * 𝜑𝜀, which leads to a relative error min(𝜀2ℓ(x)−2, 1)(ℓm(x) = min |x − ym| is the distance to the nearest nucleus) in pointwiseWeyl expressions, and to the error ≍ 𝜀2Z 3 in

∫P(W + 𝜈) dx .

Minimizing 𝜀2Z 3 + 𝜀−1Z , we get 𝜀 = Z− 23 and and an error ≍ Z

53 .

And we do not have “bonus term” 12D(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌), which is a bad

news for some applications.




|xj − xk |−1 ≥N∑j=1

(|x |−1 * 𝜌 * 𝜑𝜀)(xj)−1

2D(𝜌, 𝜌)− CN𝜀−1, (28)


Then in operator W = V − |x |−1 * 𝜌 is replaced byW𝜀 = V − |x |−1 * 𝜌 * 𝜑𝜀,

which leads to a relative error min(𝜀2ℓ(x)−2, 1)(ℓm(x) = min |x − ym| is the distance to the nearest nucleus) in pointwiseWeyl expressions, and to the error ≍ 𝜀2Z 3 in

∫P(W + 𝜈) dx .


53 .






|xj − xk |−1 ≥N∑j=1

(|x |−1 * 𝜌 * 𝜑𝜀)(xj)−1

2D(𝜌, 𝜌)− CN𝜀−1, (28)


Then in operator W = V − |x |−1 * 𝜌 is replaced byW𝜀 = V − |x |−1 * 𝜌 * 𝜑𝜀, which leads to a relative error min(𝜀2ℓ(x)−2, 1)(ℓm(x) = min |x − ym| is the distance to the nearest nucleus) in pointwiseWeyl expressions,

and to the error ≍ 𝜀2Z 3 in∫P(W + 𝜈) dx .


53 .






|xj − xk |−1 ≥N∑j=1

(|x |−1 * 𝜌 * 𝜑𝜀)(xj)−1

2D(𝜌, 𝜌)− CN𝜀−1, (28)


Then in operator W = V − |x |−1 * 𝜌 is replaced byW𝜀 = V − |x |−1 * 𝜌 * 𝜑𝜀, which leads to a relative error min(𝜀2ℓ(x)−2, 1)(ℓm(x) = min |x − ym| is the distance to the nearest nucleus) in pointwiseWeyl expressions, and to the error ≍ 𝜀2Z 3 in

∫P(W + 𝜈) dx .


53 .




Semiclassics Singular zone

Semiclassics: singular zone

First, we need to do something in the singular zone {x : ℓ(x) ≤ Z−1}where P ′

*(W + 𝜇) ≍ ℓ(x)−3, P*(W + 𝜇) ≍ ℓ(x)−4 and all integrals in thesemiclassical expressions are diverging.

But effective semiclassical parameter is & 1 here anyway, so semiclassics iswrong, and we need to estimate properly e(x , x , 𝜏) here where e(x , y , 𝜏) isthe Schwartz kernel of the spectral projector of HW .





*(W + 𝜇) ≍ ℓ(x)−3, P*(W + 𝜇) ≍ ℓ(x)−4 and all integrals in thesemiclassical expressions are diverging.But effective semiclassical parameter is & 1 here anyway, so semiclassics iswrong,

and we need to estimate properly e(x , x , 𝜏) here where e(x , y , 𝜏) isthe Schwartz kernel of the spectral projector of HW .





*(W + 𝜇) ≍ ℓ(x)−3, P*(W + 𝜇) ≍ ℓ(x)−4 and all integrals in thesemiclassical expressions are diverging.But effective semiclassical parameter is & 1 here anyway, so semiclassics iswrong, and we need to estimate properly e(x , x , 𝜏) here where e(x , y , 𝜏) isthe Schwartz kernel of the spectral projector of HW .



Using Lieb-Yau inequality [Lieb-Yau]

√Δ− 2

𝜋|x |≥ AsΔ

s − Bs (29)

for any s ∈ [0, 1/2) and As ,Bs > 0 (recall that Δ = (−i∇)2 is a positiveLaplacian),

we prove that

e(x , x , 𝜏) ≤ CZ 1−𝛿ℓ(x)𝛿−2 for ℓ(x) ≤ Z−1, 𝜏 ≤ C0Z2 (30)

with arbitrarily small 𝛿 > 0.Then contributions of the singular zone to

∫e(x , x , 𝜏) dx ,

D(e(x , x , 𝜏), e(x , x , 𝜏)) and∫ ∫ 𝜏

e(x , x , 𝜏 ′) d𝜏 ′dx do not exceed C , CZand CZ 2 respectively, exactly as in non-relativistic case.Two former are too small to care and the latter will be dealt properly later.




√Δ− 2

𝜋|x |≥ AsΔ

s − Bs (29)

for any s ∈ [0, 1/2) and As ,Bs > 0 (recall that Δ = (−i∇)2 is a positiveLaplacian), we prove that


with arbitrarily small 𝛿 > 0.

Then contributions of the singular zone to∫e(x , x , 𝜏) dx ,

D(e(x , x , 𝜏), e(x , x , 𝜏)) and∫ ∫ 𝜏





√Δ− 2

𝜋|x |≥ AsΔ

s − Bs (29)




∫e(x , x , 𝜏) dx ,

D(e(x , x , 𝜏), e(x , x , 𝜏)) and∫ ∫ 𝜏

e(x , x , 𝜏 ′) d𝜏 ′dx do not exceed C , CZand CZ 2 respectively, exactly as in non-relativistic case.

Two former are too small to care and the latter will be dealt properly later.




√Δ− 2

𝜋|x |≥ AsΔ

s − Bs (29)




∫e(x , x , 𝜏) dx ,

D(e(x , x , 𝜏), e(x , x , 𝜏)) and∫ ∫ 𝜏



Semiclassics Regular zone

Semiclassics: regular zone

In the regular zone {x : ℓ(x) ≥ Z−1} semiclassical errors are exactly as inthe non-relativistic case and the contributions of the regular zone to∫e(x , x , 𝜏) dx , D(e(x , x , 𝜏), e(x , x , 𝜏)) and

∫ ∫ 𝜏e(x , x , 𝜏 ′) d𝜏 ′dx do not

exceed CZ23 , CZ

53 and CZ 2 respectively.

Two former are exactly what we need and the latter will be dealt properlylater.

There are also errors due to the fact that for relativistic operator we writenon-relativistic Weyl expression, but in the regular zone

P ′*(W + 𝜈)− P ′(W + 𝜈) ≍ 𝛽2W

52 , P*(W + 𝜈)− P(W + 𝜈) ≍ 𝛽2W

72 ,

which leads to the error ≍ 𝛽2Z113 . Z

53 in the semiclassical expression for

the trace term, and to the errors which are too small to care in all otherterms.






exceed CZ23 , CZ




P ′*(W + 𝜈)− P ′(W + 𝜈) ≍ 𝛽2W

52 , P*(W + 𝜈)− P(W + 𝜈) ≍ 𝛽2W

72 ,



the trace term,

and to the errors which are too small to care in all otherterms.






exceed CZ23 , CZ




P ′*(W + 𝜈)− P ′(W + 𝜈) ≍ 𝛽2W

52 , P*(W + 𝜈)− P(W + 𝜈) ≍ 𝛽2W

72 ,



the trace term, and to the errors which are too small to care in all otherterms.


Semiclassics Trace term and Scott correction term

Trace term and Scott correction term

To get an error better than O(Z 2) we need to consider

Tr(H−W+𝜈) +

∫P(W + 𝜈) dx =∫ (∫ 0

−∞tr e(x , x , 𝜏) d𝜏 + P(W + 𝜈)

)dx . (31)

Let d ≥ Z−1 be a minimal distance between nuclei, or Z− 13 , whatever is

smaller, and 1 =∑

0≤m≤1 𝜓m where 𝜓m are supported in d/2-vicinities ofym, and 𝜓0 is supported in {x : ℓ(x) ≥ d/4}. Then using the standardtechnique, using cut–of 𝜓0 in the right-hand expression of (31) we get

O(Z32 d− 1

2 ).





Tr(H−W+𝜈) +

∫P(W + 𝜈) dx =∫ (∫ 0

−∞tr e(x , x , 𝜏) d𝜏 + P(W + 𝜈)

)dx . (31)


smaller, and 1 =∑

0≤m≤1 𝜓m where 𝜓m are supported in d/2-vicinities ofym, and 𝜓0 is supported in {x : ℓ(x) ≥ d/4}.

Then using the standardtechnique, using cut–of 𝜓0 in the right-hand expression of (31) we get

O(Z32 d− 1

2 ).





Tr(H−W+𝜈) +

∫P(W + 𝜈) dx =∫ (∫ 0

−∞tr e(x , x , 𝜏) d𝜏 + P(W + 𝜈)

)dx . (31)


smaller, and 1 =∑

0≤m≤1 𝜓m where 𝜓m are supported in d/2-vicinities ofym, and 𝜓0 is supported in {x : ℓ(x) ≥ d/4}. Then using the standardtechnique, using cut–of 𝜓0 in the right-hand expression of (31) we get

O(Z32 d− 1

2 ).



On the other hand, using the same technique as in the non-relativisticcase, see V. Ivrii, [Ivr1], Chapter 25, one can prove that for m = 1, . . . ,Mthe difference between (31) with 𝜓m cut-off, and the same expression butwith and W + 𝜈 replaced by Vm = Zm|x − ym|−1, both in the definition of

operator and Weyl expression, also is O(Z32 d− 1

2 ).

Further, in the same way in the right-hand expressions of (31) for suchoperators we can drop 𝜓m with the same error, thus arriving to the sum of∫ (∫ 0

−∞tr em(x , x , 𝜏) d𝜏 + P(Vm)

)dx . (32)m

In each such expression we can take ym = 0. Then scaling x ↦→ Zmx weget qZ 2

mS(𝛽Zm) where S(k) is delivered by (32)m with 𝛽 := 𝛽Zm,Zm := 1, and q := 1.



On the other hand, using the same technique as in the non-relativisticcase, see V. Ivrii, [Ivr1], Chapter 25, one can prove that for m = 1, . . . ,Mthe difference between (31) with 𝜓m cut-off, and the same expression butwith and W + 𝜈 replaced by Vm = Zm|x − ym|−1, both in the definition of

operator and Weyl expression, also is O(Z32 d− 1

2 ).Further, in the same way in the right-hand expressions of (31) for suchoperators we can drop 𝜓m with the same error, thus arriving to the sum of∫ (∫ 0

−∞tr em(x , x , 𝜏) d𝜏 + P(Vm)

)dx . (32)m

In each such expression we can take ym = 0. Then scaling x ↦→ Zmx weget qZ 2

mS(𝛽Zm) where S(k) is delivered by (32)m with 𝛽 := 𝛽Zm,Zm := 1, and q := 1.



The rest is no different from non-relativistic case and we arrive to

Theorem 1

Let m = 12 , e = ~ = 1, 𝛽 := e2/~c . Assume that (9) holds and d ≥ Z−1.

Then

1

EN = ℰTF +∑

1≤m≤M

qZ 2mS(𝛽Zm) + O(Z

32 d− 1

2 + Z53 ). (33)

2 In particular, for d ≥ Z− 13

EN = ℰTF +∑

1≤m≤M

qZ 2mS(𝛽Zm) + O(Z

53 ). (34)

3 Further, under condition (9)*

D(𝜌Ψ − 𝜌TF, 𝜌Ψ − 𝜌TF) ≤ CZ53 . (35)


Sharper estimate

Sharper estimate

Can we do better than (34)?

In the non-relativistic case we could,

provided d ≫ Z− 13 (we do not reset it to Z− 1

3 ):

EN = ℰTF +∑

1≤m≤M

qZ 2mS(𝛽Zm) + Dirac + Schwinger+

O(Z53−𝛿 + Z

53+ 𝛿

3 d−𝛿) (36)

with Dirac and Schwinger correction terms of magnitude Z53 each:

Dirac = −9

2(36𝜋)

23 q−

13

∫(𝜌TF)

43 dx , (37)

Schwinger = (36𝜋)23 q−

13

∫(𝜌TF)

43 dx (38)


Sharper estimate

Sharper estimate

Can we do better than (34)? In the non-relativistic case we could,

provided d ≫ Z− 13 (we do not reset it to Z− 1

3 ):

EN = ℰTF +∑

1≤m≤M

qZ 2mS(𝛽Zm) + Dirac + Schwinger+

O(Z53−𝛿 + Z

53+ 𝛿

3 d−𝛿) (36)

with Dirac and Schwinger correction terms of magnitude Z53 each:

Dirac = −9

2(36𝜋)

23 q−

13

∫(𝜌TF)

43 dx , (37)

Schwinger = (36𝜋)23 q−

13

∫(𝜌TF)

43 dx (38)


Sharper estimate

What about relativistic case?

The semiclassical errors are O(Z23−𝛿 + Z

23+ 𝛿

3 d−𝛿) in∫e(x , x , 0) dx

and O(Z53−𝛿 + Z

53+ 𝛿

3 d−𝛿) in all other terms.

Schwinger, coming from the semiclassical expression from the traceterm, and Dirac, coming from −Z

s|x − y |−1|e(x , y , 0)|2 dxdy in the

estimate from above remain.

Appears a new, relativistic correction term, again coming from thetrace term, and equal to∫ (

−P*(W + 𝜈) + P(W + 𝜈) + P*(V )− P(V ))dx (39)

which equal modulo the same error to

RCT :=q

6𝜋2c2

∫ (−(W + 𝜈)

72 + V

72

)dx (40)

of the magnitude 𝛽2Z113 .


Sharper estimate

Theorem 2

In the framework of Theorem 1 let condition (9)* be fulfilled and

d ≥ Z− 13 . Then

EN ≤ ℰTF +∑

1≤m≤M

qZ 2mS(𝛼Zm) + Dirac + Schwinger + RCT+

O(Z53−𝛿 + Z

53+ 𝛿

3 d−𝛿) (34)*

andD(𝜌Ψ − 𝜌TF, 𝜌Ψ − 𝜌TF) = O(Z

53−𝛿 + Z

53+ 𝛿

3 d−𝛿). (35)*

Remark

We do not need (9)* for this estimate from above.In the non-relativisticcase to get the estimate from below we use improved electrostaticinequality, due to V. Bach [Bach] and G. M. Graf-J. P. Solovej [GS]. Wecan prove it in the relativistic case under assumption (9)* but we do nothave similarly improved correlation inequality (at least, I don’t knowthose).


Sharper estimate

Theorem 2


d ≥ Z− 13 . Then

EN ≤ ℰTF +∑

1≤m≤M


O(Z53−𝛿 + Z

53+ 𝛿

3 d−𝛿) (34)*


53−𝛿 + Z

53+ 𝛿

3 d−𝛿). (35)*

Remark

We do not need (9)* for this estimate from above.

In the non-relativisticcase to get the estimate from below we use improved electrostaticinequality, due to V. Bach [Bach] and G. M. Graf-J. P. Solovej [GS]. Wecan prove it in the relativistic case under assumption (9)* but we do nothave similarly improved correlation inequality (at least, I don’t knowthose).


Sharper estimate

Theorem 2


d ≥ Z− 13 . Then

EN ≤ ℰTF +∑

1≤m≤M


O(Z53−𝛿 + Z

53+ 𝛿

3 d−𝛿) (34)*


53−𝛿 + Z

53+ 𝛿

3 d−𝛿). (35)*

Remark

We do not need (9)* for this estimate from above.In the non-relativisticcase to get the estimate from below we use improved electrostaticinequality, due to V. Bach [Bach] and G. M. Graf-J. P. Solovej [GS].

Wecan prove it in the relativistic case under assumption (9)* but we do nothave similarly improved correlation inequality (at least, I don’t knowthose).


Sharper estimate

Theorem 2


d ≥ Z− 13 . Then

EN ≤ ℰTF +∑

1≤m≤M


O(Z53−𝛿 + Z

53+ 𝛿

3 d−𝛿) (34)*


53−𝛿 + Z

53+ 𝛿

3 d−𝛿). (35)*

Remark

We do not need (9)* for this estimate from above.In the non-relativisticcase to get the estimate from below we use improved electrostaticinequality, due to V. Bach [Bach] and G. M. Graf-J. P. Solovej [GS]. Wecan prove it in the relativistic case under assumption (9)* but we do nothave similarly improved correlation inequality (at least, I don’t knowthose).


Discussion

Discussion: corollaries

1 Like in non-relativistic case the same results hold in the free nucleimodel when EN is replaced by

EN = EN +∑

1≤m<m′≤M

ZmZm′e2

|ym − ym′ |, (41)

which is minimized by ym. Then the minimal distance between nuclei

d & Z− 521 (provided Zm ≍ Z for all m = 1, . . . ,M), and under

assumption (9)* and in the non-relativistic case d ≥ Z− 521+𝛿.

2 Like in non-relativistic case all correction terms can be calculated foratoms, and if (Z − N)+ ≪ Z , for neutral atoms.

3 Under assumption (9)* we have estimates (35) and even (35)* forD(𝜌Ψ − 𝜌, 𝜌Ψ − 𝜌), and we can estimate excessive negative charge, orexcessive positive charge in the molecule in the free nuclei model, orestimate the difference between 𝜈 and ionization energy.


Self generated magnetic field Set-up

Self generated magnetic field:set-up

Consider now operator with a magnetic field i. e. with non-relativisticone-particle kinetic energy operator

T =1

2mP2 (42)

and relativistic one-particle kinetic energy operator

T = (c2P2 +m2c4)12 −mc2 (43)

where

P =((i∇− eA) · σ

)(44)

and σ = (σ1,σ2,σ3), σj are Pauli matrices.


Self generated magnetic field Non-relativistic case

Non-relativistic case

In the non-relativistic case self-generated magnetic field A was considered,i.e. to E is added the energy of the magnetic field

E(A) = inf Spec(HN) +1

𝛼

∫|∇ × A|2 dx (45)

with HN = HN,A,V , and the result is minimized by A:

E* := infA∈H1

0

E(A). (46)

In [EFS1] it was proven that for 𝛼Zm < 𝜅 (some unknown constant 𝜅 > 0,may be even ∞) ℰTF, while Scott correction term is q

∑m Z 2

mS(𝛼Zm).



The remainder estimate was perfected in [Ivr1], Chapter 27.

The crucial step was the proof that the minimizer exists (but we do notknow if it is unique) and satisfies after rescaling equation

2

𝜅h2ΔAj(x) = Φj :=

− Re tr[σj

((hD − A)x · σe(x , y , 𝜏) + e(x , y , 𝜏) t(hD − A)y · σ

)]y=x

,

(47)

where e(x , y , 𝜏) is the Schwartz kernel of the spectral projector θ(−H) ofH = HA,V and tr is a matrix trace.Then, using newly developed methods of semiclassical microlocal analysis,certain estimates to A and e(., ., 𝜏) where proven, which led to estimatesof the trace.



The remainder estimate was perfected in [Ivr1], Chapter 27.The crucial step was the proof that the minimizer exists (but we do notknow if it is unique) and satisfies after rescaling equation

2


− Re tr[σj


)]y=x

,

(47)

where e(x , y , 𝜏) is the Schwartz kernel of the spectral projector θ(−H) ofH = HA,V and tr is a matrix trace.

Then, using newly developed methods of semiclassical microlocal analysis,certain estimates to A and e(., ., 𝜏) where proven, which led to estimatesof the trace.



The remainder estimate was perfected in [Ivr1], Chapter 27.The crucial step was the proof that the minimizer exists (but we do notknow if it is unique) and satisfies after rescaling equation

2


− Re tr[σj


)]y=x

,

(47)

where e(x , y , 𝜏) is the Schwartz kernel of the spectral projector θ(−H) ofH = HA,V and tr is a matrix trace.Then, using newly developed methods of semiclassical microlocal analysis,certain estimates to A and e(., ., 𝜏) where proven, which led to estimatesof the trace.


Self generated magnetic field Relativistic case

Relativistic case

Consider the same problem in the relativistic case, assuming (9)* and

𝛼Zm ≤ 𝜅(2

𝜋− 𝛽Zm)

3/2 ∀m = 1, . . . ,M. (48)

Again, we prove that the minimizer exists (but we do not know if it isunique) and satisfies after rescaling equation

2


− Re tr[∫ ∞

0σj((hD − A)x · σ)e−𝜆Se(., ., 0)e−𝜆S d𝜆

]x=y

− Re tr[∫ ∞

0σje

−𝜆Se(., ., 0)e−𝜆S t((hD − A)y · σ) d𝜆]

x=y

(49)

whereS = 𝛽2(T + 𝛽−2) =

((𝛽2(hD − A) · σ)2 + 1

) 12 . (50)


Self generated magnetic field Relativistic case

Then, using the same methods of semiclassical microlocal analysis, weprove the similar estimates to A and e(., ., 𝜏), which lead to the sametrace estimates.


Self generated magnetic field Discussion

Discussion

1 The Scott correction term is now q∑

m Z 2mS(𝛽Zm, 𝛼Zm).

2 Then we apply those results to the original multiparticle proble.

3 There is a significant difference in the atomic case (M = 1) and themolecular case (M ≥ 2); in the latter we need to decouple magneticfield between nuclei.

4 Those results enable us to estimate the maximal number of extraelectrons an atom or molecule can bind and find an estimate of theionization energy.

5 Further, they enable us to consider positively charged systems andfind either an estimate or asymptotics of the ionization energy.

6 Finally, in the free nuclei model we can estimate from below thedistance between nuclei, and the excessive positive charge when atomcan bind into molecule.

7 See results and more references in [Ivr3].



Reference I

V. Bach. Error bound for the Hartree-Fock energy of atoms andmolecules. Commun. Math. Phys. 147:527–548 (1992).

L. Erdos, S. Fournais, J. P. Solovej,Scott correction for large atomsand molecules in a self-generated magnetic field . Commun. Math.Physics, 312(3):847–882 (2012).

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L. Erdos, S. Fournais, J. P. Solovej,Scott correction for large atomsand molecules in a self-generated magnetic field . Commun. Math.Physics, 312(3):847–882 (2012).



Reference II

L. Erdos, S. Fournais, J. P. Solovej, Relativistic Scott correction inself-generated magnetic fields. Journal of Mathematical Physics 53,095202 (2012), 27pp.

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Reference III

G. M. Graf, J. P Solovej. A correlation estimate with applications toquantum systems with Coulomb interactions Rev. Math. Phys.,6(5a):977–997 (1994). Reprinted in The state of matter a volumededicated to E. H. Lieb, Advanced series in mathematical physics, 20,M. Aizenman and H. Araki (Eds.), 142–166, World Scientific (1994).

I. W. Herbst. Spectral Theory of the Operator (p+m2)1/2 − Ze/r ,Commun. Math. Phys. 53(3):285–294 (1977).

V. Ivrii. Microlocal Analysis, Sharp Spectral, Asymptotics andApplications.http://www.math.toronto.edu/ivrii/monsterbook.pdf

Chapter 25. Asymptotics of the ground state energy of heavymolecules;


http://www.math.toronto.edu/ivrii/monsterbook.pdf


Reference IV

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https://arxiv.org/abs/1708.07737


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