The Local Density Approximation in Density Functional Theory Robert - PowerPoint PPT Presentation

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 # 1

Density Functional Theory Formulate energy minimization in terms of the particle density only: Z R 3( N − 1) | Ψ ( x, x 2 , . . . , x N ) | 2 dx 2 · · · dx N Ψ ( x 1 , . . . , x N ) 7! ⇢ Ψ ( x ) = N Levy–Lieb formulation of the ground state energy: inf Ψ = inf ⇢ inf Ψ , ⇢ Ψ = ⇢ : ⇢ � Z E V ( N ) = inf � � ⌦ ↵ � Ψ � H N = inf F LL ( ⇢ ) + R 3 V ( x ) ⇢ ( x ) dx Ψ V Ψ ⇢ ∈ R N where � � * + � N � 1 X X � � Ψ , ⇢ Ψ = ⇢ h Ψ | H N r 2 F LL ( ⇢ ) := min 0 | Ψ i = min � x i + Ψ Ψ � � | x i � x k | � � Ψ , ⇢ Ψ = ⇢ � � i =1 1 ≤ i<j ≤ N and Z Z R 3 | rp ⇢ | 2 < 1 } R N = { ⇢ � 0 , R 3 ⇢ = N, [Lieb ’83]. R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 2

Local Density Approximation For slowly varying densities ⇢ , ZZ Z F LL ( ⇢ ) ⇡ 1 ⇢ ( x ) ⇢ ( y ) | x � y | dx dy + R 3 f ( ⇢ ( x )) dx 2 R 3 × R 3 | {z } | {z } local non-local energy per unit volume classical Coulomb energy of uniform electron gas R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 3

Main Result THEOREM (Justification of LDA) There exists a universal constant C > 0 and a universal function f : R + ! R such that � � ZZ Z F LL ( ⇢ ) � 1 ⇢ ( x ) ⇢ ( y ) � � � ˜ | x � y | dx dy � R 3 f ( ⇢ ( x )) dx � � 2 � R 3 × R 3 Z Z Z � � � � dx + C (1 + " ) 2 dx + C 4 p p � ⇢ ( x ) + ⇢ ( x ) 2 � � � � �  " � r ⇢ ( x ) � r ⇢ ( x ) dx � � " 15 " R 3 R 3 R 3 for every " > 0 and every ⇢ 2 L 1 \ L 2 ( R 3 ) such that rp ⇢ 2 L 2 \ L 4 ( R 3 ) . Remarks: • Last term can be replaced by " 1 − 4 p R | r ⇢ ✓ | p with p > 3 , ✓ > 0 and 2  p ✓  1+ p/ 2 . • ˜ F LL grand-canonical version (convex hull), but same result expected for F LL . • For ⇢ ( x ) = � ( x/N 1 / 3 ) we find ZZ Z F LL ( ⇢ ) = N 5 / 3 � ( x ) � ( y ) ˜ R 3 f ( � ( x )) dx + O ( N 11 / 12 ) | x � y | dx dy + N 2 R 3 × R 3 R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 4

Energy of the Uniform Electron Gas For ⇢ 0 > 0 we have ✓ ◆ ZZ F LL ( ⇢ 0 1 ` Ω ⇤ � ) � ⇢ 2 1 1 ` Ω ⇤ � ( x ) 1 ` Ω ⇤ � ( y ) ˜ 0 f ( ⇢ 0 ) = lim dx dy | ` Ω | 2 | x � y | ` →∞ R 3 × R 3 The limit exists and is independent of Ω and � [Hainzl-Lewin-Solovej ’09]. • c TF = 3 5 (3 ⇡ 2 ) 2 / 3 • c D = 3 4 (3 / ⇡ ) 1 / 3 • 1 . 4442  c SCE  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 # 5

Phase Diagram Zong, Lin, Ceperley, PRE (2002) R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 6

Exchange-Correlation Energy In practice one often considers the exchange-correlation energy ZZ F LL ( ⇢ ) � 1 ⇢ ( x ) ⇢ ( y ) E xc ( ⇢ ) = ˜ | x � y | dx dy � T ( ⇢ ) 2 R 3 × R 3 where T ( ⇢ ) is the ( Kohn-Sham ) kinetic energy functional Tr( �r 2 ) � T ( ⇢ ) = min 0 ≤ � ≤ 1 ⇢ � = ⇢ Our result on the LDA applies to E xc ( ⇢ ) as well, since THEOREM (LDA for kinetic energy) [Nam ’18, LLS ’19]. For any " > 0 , Z Z C p R 3 ⇢ ( x ) 5 / 3 dx � ⇢ ( x ) | 2 dx T ( ⇢ ) � c TF (1 � " ) R 3 | r " 13 / 3 Z Z R 3 ⇢ ( x ) 5 / 3 dx + C (1 + " ) p ⇢ ( x ) | 2 dx T ( ⇢ )  c TF (1 + " ) R 3 | r " R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 7

Upper Bound on T ( ρ ) Recall that ✓ ◆ �r 2  5 3 c TF t 2 / 3 P t = 1 has density ⇢ P t = t and kinetic energy density c TF t 5 / 3 . For the upper bound on T ( ⇢ ) , use as a trial density matrix the ‘layer cake’ trial state s ◆ s Z ∞ ✓ ◆ ✓ ✓ ◆ t �r 2  5 t 3 c TF t 2 / 3 t − 1 dt � = ⌘ 1 ⌘ ⇢ ( x ) ⇢ ( x ) 0 R ∞ R ∞ ⌘ ( t ) t − 1 dt  1 and optimize over the choice of ⌘ with ⌘ ( t ) dt = 1 and 0 0 R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 8

Strategy of the Proof (of the Main Theorem) The key is to prove an approximate locality of the indirect energy ZZ F LL ( ⇢ ) � 1 ⇢ ( x ) ⇢ ( y ) F ind ( ⇢ ) = ˜ | x � y | dx dy 2 R 3 × R 3 • For a tiling { Ω ` ,j } of R 3 with boxes of size ` = ` ( " ) , X F ind ( ⇢ ) ⇡ F ind ( ⇢ 1 Ω ` ,j ⇤ � ) j • In each box, estimate di ff erence of F ind ( ⇢ 1 Ω ` ,j ⇤ � ) and F ind (¯ ⇢ 1 Ω ` ,j ⇤ � ) in terms of derivatives of ⇢ • Compare F ind (¯ ⇢ 1 Ω ` ,j ⇤ � ) with f (¯ ⇢ ) | Ω ` ,j | . R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 9

Locality: Lower Bound THEOREM (Graf-Schenker ’94) Let { ∆ n } be a tiling of R 3 of tetrahedra (of size 1 ). Then, for all N � 2 , z j 2 R and x j 2 R 3 , 0 1 N Z | x i � x j | � 1 z i z j z i z j 1 g ` ∆ n ( x i ) 1 g ` ∆ n ( x j ) A dg � C X X X X z 2 @ j ` 3 | x i � x j | ` [0 , ` ] 3 × SO (3) n 1 ≤ i<j ≤ N 1 ≤ i<j ≤ N i =1 Tiling with tetrahedra, averaged over translations and rotations. Local number of particles not fixed ! grand-canonical description ` R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 10

Difficulties in the Upper Bound For a suitable tiling we want to prove that X F ind ( ⇢ ) . F ind ( ⇢ 1 Ω ` ,j ⇤ � ) j Di ffi culties: • 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 • Di ffi cult to do canonically R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 11

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 di ff erence 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 / 3 x ) . • Phase transitions, Wigner crystal, . . . R. Seiringer — Local Density Approximation in Density Functional Theory — Oct. 21, 2019 # 12