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The Local Density Approximation in Density Functional Theory Robert Seiringer IST Austria Based on joint work with Mathieu Lewin and Elliott Lieb: Journal de l Ecole polytechnique Math ematiques, Tome 5, 79116 (2018)

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The Local Density Approximation in Density Functional Theory

Robert Seiringer IST Austria Based on joint work with Mathieu Lewin and Elliott Lieb: Journal de l’´ Ecole polytechnique – Math´ ematiques, Tome 5, 79116 (2018) arXiv:1903.04046, Pure Appl. Anal. (in press)

Large Coulomb Systems and Related Matters

CIRM Marseille, Oct. 21–25, 2019

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Density Functional Theory

Formulate energy minimization in terms of the particle density only: Ψ(x1, . . . , xN) 7! ⇢Ψ(x) = N Z

R3(N−1) |Ψ(x, x2, . . . , xN)|2dx2 · · · dxN

Levy–Lieb formulation of the ground state energy: infΨ = inf⇢ infΨ,⇢Ψ=⇢: EV (N) = inf

Ψ

⌦ Ψ

  • HN

V

  • Ψ

↵ = inf

⇢∈RN

⇢ FLL(⇢) + Z

R3 V (x)⇢(x)dx

  • where

FLL(⇢) := min

Ψ,⇢Ψ=⇢hΨ|HN 0 |Ψi =

min

Ψ,⇢Ψ=⇢

* Ψ

  • N

X

i=1

r2

xi +

X

1≤i<j≤N

1 |xi xk|

  • Ψ

+ and RN = {⇢ 0, Z

R3 ⇢ = N,

Z

R3 |rp⇢|2 < 1}

[Lieb ’83].

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Local Density Approximation

For slowly varying densities ⇢, FLL(⇢) ⇡ 1 2 ZZ

R3×R3

⇢(x)⇢(y) |x y| dx dy | {z }

non-local classical Coulomb energy

+ Z

R3 f(⇢(x))dx

| {z }

local energy per unit volume

  • f uniform electron gas
  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Main Result

THEOREM (Justification of LDA) There exists a universal constant C > 0 and a universal function f : R+ ! R such that

  • ˜

FLL(⇢) 1 2 ZZ

R3×R3

⇢(x)⇢(y) |x y| dx dy Z

R3 f(⇢(x))dx

  •  "

Z

R3

  • ⇢(x) + ⇢(x)2

dx + C(1 + ") " Z

R3

  • r

p ⇢(x)

  • 2

dx + C "15 Z

R3

  • r

p ⇢(x)

  • 4

dx for every " > 0 and every ⇢ 2 L1 \ L2(R3) such that rp⇢ 2 L2 \ L4(R3). Remarks:

  • Last term can be replaced by "1−4p R

|r⇢✓|p with p > 3, ✓ > 0 and 2  p✓  1+p/2.

  • ˜

FLL grand-canonical version (convex hull), but same result expected for FLL.

  • For ⇢(x) = (x/N 1/3) we find

˜ FLL(⇢) = N 5/3 2 ZZ

R3×R3

(x)(y) |x y| dx dy + N Z

R3 f((x))dx + O(N 11/12)

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Energy of the Uniform Electron Gas

For ⇢0 > 0 we have f(⇢0) = lim

`→∞

1 |`Ω| ✓ ˜ FLL(⇢01`Ω ⇤ ) ⇢2 2 ZZ

R3×R3

1`Ω ⇤ (x)1`Ω ⇤ (y) |x y| dx dy ◆ The limit exists and is independent of Ω and [Hainzl-Lewin-Solovej ’09].

  • cTF = 3

5(3⇡2)2/3

  • cD = 3

4(3/⇡)1/3

  • 1.4442  cSCE  1.4508 (strongly cor-

related electrons)

  • next order for large ⇢ believed to

by ⇢ ln ⇢ [Macke ’50, Bohm-Pines ’53, GellMann-Brueckner ’57]

  • non-smooth because of phase transi-

tions (solid/fluid, ferro/paramagnetic)

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Phase Diagram

Zong, Lin, Ceperley, PRE (2002)

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Exchange-Correlation Energy

In practice one often considers the exchange-correlation energy Exc(⇢) = ˜ FLL(⇢) 1 2 ZZ

R3×R3

⇢(x)⇢(y) |x y| dx dy T(⇢) where T(⇢) is the (Kohn-Sham) kinetic energy functional T(⇢) = min

0≤≤1 ⇢=⇢

Tr(r2) Our result on the LDA applies to Exc(⇢) as well, since THEOREM (LDA for kinetic energy) [Nam ’18, LLS ’19]. For any " > 0, T(⇢) cTF(1 ") Z

R3 ⇢(x)5/3dx

C "13/3 Z

R3 |r

p ⇢(x)|2dx T(⇢)  cTF(1 + ") Z

R3 ⇢(x)5/3dx + C(1 + ")

" Z

R3 |r

p ⇢(x)|2dx

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Upper Bound on T (ρ)

Recall that Pt = 1 ✓ r2  5 3cTFt2/3 ◆ has density ⇢Pt = t and kinetic energy density cTFt5/3. For the upper bound on T(⇢), use as a trial density matrix the ‘layer cake’ trial state = Z ∞ s ⌘ ✓ t ⇢(x) ◆ 1 ✓ r2  5 3cTFt2/3 ◆ s ⌘ ✓ t ⇢(x) ◆ t−1dt and optimize over the choice of ⌘ with R ∞ ⌘(t)dt = 1 and R ∞ ⌘(t)t−1dt  1

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Strategy of the Proof (of the Main Theorem)

The key is to prove an approximate locality of the indirect energy Find(⇢) = ˜ FLL(⇢) 1 2 ZZ

R3×R3

⇢(x)⇢(y) |x y| dx dy

  • For a tiling {Ω`,j} of R3 with boxes of size ` = `("),

Find(⇢) ⇡ X

j

Find(⇢1Ω`,j ⇤ )

  • In each box, estimate difference of Find(⇢1Ω`,j ⇤ ) and Find(¯

⇢1Ω`,j ⇤ ) in terms of derivatives of ⇢

  • Compare Find(¯

⇢1Ω`,j ⇤ ) with f(¯ ⇢)|Ω`,j|.

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Locality: Lower Bound

THEOREM (Graf-Schenker ’94) Let {∆n} be a tiling of R3 of tetrahedra (of size 1). Then, for all N 2, zj 2 R and xj 2 R3, X

1≤i<j≤N

zizj |xi xj| 1 `3 Z

[0,`]3×SO(3)

X

n

@ X

1≤i<j≤N

zizj1g`∆n(xi)1g`∆n(xj) |xi xj| 1 A dgC `

N

X

i=1

z2

j

`

Tiling with tetrahedra, averaged

  • ver translations and rotations.

Local number of particles not fixed ! grand-canonical description

  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Difficulties in the Upper Bound

For a suitable tiling we want to prove that Find(⇢) . X

j

Find(⇢1Ω`,j ⇤ ) Difficulties:

  • Need a trial state with the exact density ⇢
  • Tensor products work badly for fermions if the supports intersect!

Our solution:

  • Partition of unity with holes, average over translations and dilations
  • Averaging of the direct term gives error ⇠ 2 R

⇢2, where = size of holes

  • Difficult to do canonically
  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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Summary and Open Problems

  • We give a mathematically rigorous justification of the Local Density Approxi-

mation in Density Functional Theory.

  • We provide a quantitative estimate on the difference between the (grand-canonical)

Levy–Lieb energy of a given density and the integral over the Uniform Electron Gas energy of this density. Many open problems remain:

  • Extension to canonical, pure state LL energy functional
  • Next order correction terms, expected to scale as N 1/3 for densities of the form

⇢(x) = (N −1/3x).

  • Phase transitions, Wigner crystal, . . .
  • R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019

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