Scaling limits of density functional theory: cross-over from mean - PowerPoint PPT Presentation

Scaling limits of density functional theory: cross-over from mean field theory to optimal transport Gero Friesecke TU Munich Conference on Nonlinearity, Transport, Physics, and Patterns Fields Institute, Toronto, Octobe 6, 2014 Organizers: Luigi Ambrosio, Bob Jerrard, Felix Otto, Mary Pugh, Robert Seiringer joint work with H.Chen (TUM), C.Cotar (University College London), C.Kl¨ uppelberg (TUM), B.Pass (Alberta) 1

C.Cotar, G.F., C.Kl¨ uppelberg, CPAM 66, 548-599, 2013 G.F., Ch.Mendl, B.Pass, C.C, C.K., J.Chem.Phys. 139, 164109, 2013 C.C., G.F., B.Pass, arXiv 1307.6540, 2013 H.Chen, G.F., on arXiv soon, 2014 2

Density functional theory Dirac 1929 Chemically specific behaviour of atoms and molecules captured, ”in principle”, by quantum mechanics. Emission/absorption spectra, binding energies, equilibrium geometries, interatomic forces,... Catch Curse of dimension. Schr¨ odinger eq. is for N - electron wavefunction Ψ : ( R 3 × Z 2 ) N to C . H 2 O, N=10, PDE in R 30 , 10 gridpts each direction, 10 30 gridpts. Hohenberg, Kohn, Sham 1964/65 Replace Schr¨ od.eq. by closed eq./var.principle for the one-point (or marginal) density ρ : R 3 → R , � � R 3( N − 1) | Ψ( x 1 , s 1 , .., x N , s N ) | 2 dx 2 · · · dx N . ρ ( x 1 ) = N s 1 ,.., s N ∈ Z 2 – Nobel Prize 1998 for W.Kohn – Routinely used in phys., chem., materials, molecular biology; huge non-math.literature – (Ex.: Momany, Carbohyd. Res. 2005) – Theory: ∃ ’exact’ fctnal; practice: clever semi-empirical fctnals: LDA, B3LYP, PBE,... – accuracy not so high; some failures; fctnals not systematically derivable/improvable This talk Behaviour of ’exact’ functional in scaling limits 3

Example: Original semi-empirical Kohn-Sham functional ◮ N-electron molecule, nuclear charges Z 1 , .., Z M > 0, nuclear positions R 1 , .., R M ∈ R 3 ◮ potential exerted by nuclei on electrons: � M Z α | x − R α | − 1 v ( x ) = − α =1 ◮ Ground state energy: � T KS [ ρ ] + 1 ρ ( x ) ρ ( y ) E KS = min ( | x − y | dx dy 0 2 ρ R 6 � 3 � 1 3 � � 3 R 3 ρ 4 / 3 + − v ρ ) 4 π where � N � � 1 R 3 |∇ ψ | 2 : | ψ i ( x ) | 2 = ρ ( x ) , � ψ i , ψ j � = δ ij , ψ i ∈ H 1 ( R 3 ; C 2 ) } T KS [ ρ ] = min { 2 i =1 i 4

Example: Original semi-empirical Kohn-Sham functional ◮ N-electron molecule, nuclear charges Z 1 , .., Z M > 0, nuclear positions R 1 , .., R M ∈ R 3 ◮ potential exerted by nuclei on electrons: � M Z α | x − R α | − 1 v ( x ) = − α =1 ◮ Ground state energy: � T KS [ ρ ] + 1 ρ ( x ) ρ ( y ) E KS = min ( | x − y | dx dy 0 2 ρ R 6 � 3 � 1 3 � � 3 R 3 ρ 4 / 3 + − v ρ ) 4 π where � N � � 1 R 3 |∇ ψ | 2 : | ψ i ( x ) | 2 = ρ ( x ) , � ψ i , ψ j � = δ ij , ψ i ∈ H 1 ( R 3 ; C 2 ) } T KS [ ρ ] = min { 2 i =1 i Where do all these terms come from ...(???)... 5

’Exact’ DFT according to Levy/Lieb ◮ Start from quantum Hamiltonian of N -electron system: N N � � � ( − 1 1 H e ℓ = 2∆ x i ) + | x i − x j | + v ( x i ) i =1 1 ≤ i < j ≤ N i =1 (typically, v ( x i ) = − � M Z α | x i − R α | potential exerted onto α =1 electrons by atomic nuclei) ◮ Ground state energy: � � E 0 = min Ψ , H e ℓ Ψ L 2 Ψ ∈A N where A N = { Ψ ∈ H 1 (( R 3 × Z 2 ) N ; C ) : Ψ antisymmetric, || Ψ || L 2 = 1 } 6

’Exact’ DFT according to Levy/Lieb, ctd. 7

’Exact’ DFT according to Levy/Lieb, ctd. Hohenberg-Kohn-Theorem (1964) For each fixed N , there exists a universal (i.e., molecule-independent) functional F HK of the single-particle density ρ such that for any external potential v , the exact QM ground state en. satisfies � � � F HK [ ρ ] + E 0 = min R 3 v ( x ) ρ ( x ) dx , ρ ∈R N � ρ = N , √ ρ ∈ H 1 ( R 3 ) } . where R N = { ρ ∈ L 1 ( R 3 ) : ρ ≥ 0 , 8

’Exact’ DFT according to Levy/Lieb, ctd. Hohenberg-Kohn-Theorem (1964) For each fixed N , there exists a universal (i.e., molecule-independent) functional F HK of the single-particle density ρ such that for any external potential v , the exact QM ground state en. satisfies � � � F HK [ ρ ] + E 0 = min R 3 v ( x ) ρ ( x ) dx , ρ ∈R N � ρ = N , √ ρ ∈ H 1 ( R 3 ) } . where R N = { ρ ∈ L 1 ( R 3 ) : ρ ≥ 0 , Proof 1. The non-universal part of the energy only depends on ρ Ψ : � � � � v ( x i ) | Ψ( x 1 , .., x N ) | 2 = � Ψ , v ( x i )Ψ � = R 3 v ( x ) ρ Ψ ( x ) dx . i i 9

’Exact’ DFT according to Levy/Lieb, ctd. Hohenberg-Kohn-Theorem (1964) For each fixed N , there exists a universal (i.e., molecule-independent) functional F HK of the single-particle density ρ such that for any external potential v , the exact QM ground state en. satisfies � � � F HK [ ρ ] + E 0 = min R 3 v ( x ) ρ ( x ) dx , ρ ∈R N � ρ = N , √ ρ ∈ H 1 ( R 3 ) } . where R N = { ρ ∈ L 1 ( R 3 ) : ρ ≥ 0 , Proof 1. The non-universal part of the energy only depends on ρ Ψ : � � � � v ( x i ) | Ψ( x 1 , .., x N ) | 2 = � Ψ , v ( x i )Ψ � = R 3 v ( x ) ρ Ψ ( x ) dx . i i 2. Partition the min over Ψ into a double min, first over Ψ subject to fixed 2 ∆ + � := − � 2 ρ , then over ρ : letting H univ 1 | x i − x j | , e ℓ i < j � � � � Ψ , H univ = inf e ℓ Ψ � + v ( r ) ρ Ψ ( r ) dr E 0 Ψ � � � � Ψ , H univ = inf inf e ℓ Ψ � + v ( r ) ρ ( r ) dr . ρ Ψ �→ ρ � �� � =: F HK [ ρ ] 10

Universal map ρ → ρ 2 from densities to pair densities Corollary of the HK theorem There exists a universal (i.e., molecule-independent) map from single-particle densities ρ ( x 1 ) to pair densities ρ 2 ( x 1 , x 2 ) which gives the exact pair density of any N -electron molecular ground state Ψ( x 1 , s 1 , .., x N , s N ) in terms of its single-particle density. 11

Universal map ρ → ρ 2 from densities to pair densities Corollary of the HK theorem There exists a universal (i.e., molecule-independent) map from single-particle densities ρ ( x 1 ) to pair densities ρ 2 ( x 1 , x 2 ) which gives the exact pair density of any N -electron molecular ground state Ψ( x 1 , s 1 , .., x N , s N ) in terms of its single-particle density. Proof Ψ ∗ := minimizer of � Ψ , H univ Ψ � subject to marginal e ℓ constraint Ψ �→ ρ ρ 2 := pair density of minimizer, i.e. ρ 2 ( x 1 , x 2 ) = � � | Ψ ∗ ( x 1 , s 1 , .., x N , s N ) | 2 dx 3 .. dx N s 1 ,.., s N � (Analogously, ... dx k +1 .. dx N gives universal k -pt. density) 12

Universal map ρ → ρ 2 from densities to pair densities Corollary of the HK theorem There exists a universal (i.e., molecule-independent) map from single-particle densities ρ ( x 1 ) to pair densities ρ 2 ( x 1 , x 2 ) which gives the exact pair density of any N -electron molecular ground state Ψ( x 1 , s 1 , .., x N , s N ) in terms of its single-particle density. Proof Ψ ∗ := minimizer of � Ψ , H univ Ψ � subject to marginal e ℓ constraint Ψ �→ ρ ρ 2 := pair density of minimizer, i.e. ρ 2 ( x 1 , x 2 ) = � � | Ψ ∗ ( x 1 , s 1 , .., x N , s N ) | 2 dx 3 .. dx N s 1 ,.., s N � (Analogously, ... dx k +1 .. dx N gives universal k -pt. density) ρ 2 may be nonunique since GS may be degenerate. Hence map multi-valued. Map highly nontrivial and not comp’ly feasible – still uses high-dim. wavefunctions. Pair density gives exact interaction energy � Ψ ∗ , � � ρ 2 ( x , y ) 1 | x i − x j | Ψ ∗ � = | x − y | dx dy R 6 i < j Comp’ly feasible interaction en. fctnals ≈ approximate the map 13

Thinking about the pair density in an elementary way 14

Thinking about the pair density in an elementary way 15

Thinking about the pair density in an elementary way Non-interacting particles Repulsive interactions 16

What does the map ρ �→ ρ 2 look like? Simulations by Huajie Chen/G.F., to appear Ex.: 1D, N electrons, ρ simple ’lump’, scaling parameter α > 0 ρ ( x ) = α N 2 L (1 + cos( α π 2 L x )) , x ∈ [ − α L , α L ] N=2 N=3 N=4 α = 100 α = 1 α = 0 . 1 17

Density scaling For any given density ρ ∈ L 1 ( R d ), let ρ α ( x ) := α d ρ ( α x ), α > 0 F HK [ ρ ] = α F HK [ ρ ] (simple computation) α 2 ∆ + � Ψ ∈ H 1 , Ψ �→ ρ � Ψ , ( − α 1 F HK [ ρ ] = min | x i − x j | )Ψ � L 2 α i < j For dilute systems ( α << 1), ’semiclassical’ behaviour 18

Scaling limit 1: α → 0 In limit α → 0, exact DFT turns into optimal transport. Theorem (Cotar/GF/Kl¨ uppelberg, CPAM 2013)   �  � Ψ , ( − α 1 F HK [ ρ ]  = min 2 ∆ + | x i − x j | )Ψ � L 2 Ψ ∈ H 1 , Ψ �→ ρ i < j � � 1 → | x i − x j | d γ ( x 1 , .., x N ) =: F OT [ ρ ] min α → 0 γ ∈P N ,γ �→ ρ R 3 N 1 ≤ i < j ≤ N where P N is the set of symmetric probability measures on R 3 N . 19