Density functional theory and optimal transportation with Coulomb cost. Density functional theory and optimal transportation with Coulomb cost. Codina Cotar (joint works with G. Friesecke, C. Klueppelberg, C. Mendl and B. Pass) April 02, 2014
Density functional theory and optimal transportation with Coulomb cost. Outline 1 Informal introduction to Quantum mechanics- Density Functional Theory (DFT) What do physicists do? Our approach 2 Informal Introduction to Optimal Transport 3 Connection to exchangeable processes
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) Informal introduction to Quantum mechanics All materials systems we study essentially consist of electrons and nuclear charge. Mechanical, electronic, magnetic etc. properties are due to electrons and their interaction with other electrons. In order to define electrons and their interaction we use Schrodinger equation. It allows to predict, e.g., binding energies, equilibrium geometries, intermolecular forces Quantum mechanics for electrons reduces to a PDE (the Schroedinger equation)
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) Density Functional Theory (DFT) If Schrodinger equation for the many electrons problem could be solved accurately and efficiently then almost any property of the materials could be determined determined accurately. Unfortunately, there is neither an accurate nor an efficient method to solve these problems. DFT is a simplified version of quantum mechanics (QM), widely used in molecular simulations in chemistry, physics, materials science Introduced by Hohenberg-Kohn-Sham in the 1960s Feasible system size: up to a million atoms 1998 Nobel Prize for ‘founding father’ Walter Kohn
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) Quantum mechanics-Formal definition The solution for this PDE is the wave function Ψ( x 1 , s 1 , . . . , x N , s N ) ∈ L 2 (( R 3 × Z 2 ) N ; C ) N - number of electrons, x i position of electron i , s i spin of electron i | Ψ( x 1 , s 1 , . . . , x N , s N ) | 2 = probability density that the electrons are at positions x i with spins s i . Ψ is an anti-symmetric function, which makes | Ψ | 2 a symmetric ( N -exchangeable) probability measure.
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) A N = { Ψ ∈ L 2 (( R 3 × Z 2 ) N ; C ) | ∇ Ψ ∈ L 2 , Ψ antisymmetric , || Ψ || L 2 = 1 } Key quantum mechanics quantity is the ground state energy E 0 E 0 = Ψ ∈A N E [Ψ] inf where E [Ψ] = T h [Ψ] + V ee [Ψ] + V ne [Ψ]
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) Kinetic energy: T h [Ψ] = h 2 � |∇ Ψ( x 1 , s 1 , . . . , x N , s N ) | 2 dz 1 dz 2 . . . dz N 2 Electron-electron energy: � 1 � | x i − x j || Ψ | 2 dz 1 . . . dz N V ee [Ψ] = 1 ≤ i < j ≤ N Nuclei-electron energy: � v ( x i ) | Ψ( z 1 , z 2 , . . . , z N ) | 2 dz 1 . . . dz N V ne [Ψ] =
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) N-electrons density � ρ Ψ | Ψ( x 1 , s 1 , . . . , x N , s N ) | 2 . N ( x 1 , .., x N ) = s 1 ,.., s N ∈ Z 2 Pair electrons density � N � � ρ Ψ R 3 ( N − 2 ) ρ Ψ 2 ( x 1 , x 2 ) = N ( x 1 , . . . , x N ) dx 3 . . . dx N 2 Single electron density � ρ Ψ ( x 1 ) = N R 3 ( N − 1 ) ρ Ψ N ( x 1 , . . . , x N ) dx 2 . . . dx N . R N := { ρ : R 3 → R | ρ is the density of some Ψ ∈ A N } Full Scrod. eqn. can be reformulated as a hierarchy of eqn: for ρ in terms of of the pair electrons density ρ 2 , for ρ 2 in terms of ρ 3 etc.
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) Variational formulation of density functional theory (Hohenberg/Kohn 1964, M. Levy 1979, E. Lieb 1983) For any external potential v , the exact Schroedinger eqn. satisfies � � � E 0 = inf F h [ ρ ] + N R 3 v ( x ) ρ ( x ) dx ρ ∈R N with � � F h [ ρ ] : = T h [Ψ] + V ee [Ψ] inf Ψ ∈A N , Ψ �→ ρ � 1 � � | x − y | ρ Ψ = T h [Ψ] + 2 ( dx , dy ) , inf Ψ ∈A N , Ψ �→ ρ R 6 F h [ ρ ] is the famous Hohenberg-Kohn functional. Not useful for computations (definitely still contains the big space of Ψ( x 1 , s 1 , . . . , x N , s N ) ). But useful starting point for model reduction in asymptotic limits.
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) What do physicists do? Correlations in DFT Mathematical structure: Minimize an approximate energy functional F [ ρ ] which depends on the electron density ρ ( x ) , a function on R 3 . Catch: exact QM energy requires knowledge of electron-pair density ρ 2 ( x , y ) , a function on R 6 , which entails correlations. Standard way out: start by assuming independence, add semi-empirical corrections to F [ ρ ] accounting for correlations. Often but not always accurate/reliable.
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) What do physicists do? Popular functionals The mean field approximation: � � 1 2 ( dx , dy ) = 1 1 | x − y | ρ Ψ | x − y | ρ ( dx ) ρ ( dy ) =: J [ ρ ] . 2 R 6 R 6 Local Density Approximation approximation: � � 1 2 ( dx , dy ) = J [ ρ ] − 4 3 ( 3 /π ) 1 / 3 R 3 ρ ( x ) 4 / 3 dx . | x − y | ρ Ψ R 6
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) What do physicists do? Quantum mechanics is becoming so unbelievably complex that it is taking longer and longer to train a quantum theorist. It is taking so long, in fact, to train him to the point where he understands the nature of physical problems that he is already too old to solve them. ( Eugene Wigner )
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) Our approach DFT Results-Our approach Let � � T h [Ψ] + V ne [ ρ Ψ ] + V ee [ ρ Ψ E 0 = 2 ] inf Ψ ∈A N and � � T [Ψ] + V ne [ ρ Ψ ] + E N OT [ ρ Ψ ] E OT = , inf 0 Ψ ∈A N where 1 � 1 � E N OT [ ρ Ψ ] = � inf | x i − x j | d γ ( x 1 , x 2 , . . . , x N ) , � N γ 2 1 ≤ i < j ≤ N subject to equal marginals ρ Ψ .
Density functional theory and optimal transportation with Coulomb cost. Informal introduction to Quantum mechanics- Density Functional Theory (DFT) Our approach Theorem (C, Friesecke, Klueppelberg - CPAM 2013) Fix ρ ∈ R N . Let N ≥ 2 . Then h → 0 F h [ ρ ] = E N OT [ ρ ] lim for every ρ ∈ R N , where recall that � � F h [ ρ ] := T h [Ψ] + V ee [Ψ] . inf Ψ ∈A N , Ψ �→ ρ Theorem (C,Friesecke, Klueppelberg - CPAM 2013) For every N and every v E 0 ≥ E OT 0 . We are the first to link electronic structure to optimal transportation. Seidl’99, Seidl/Perdew/Levy’99, Seidl/Gori-Giorgi/Savin’07
Density functional theory and optimal transportation with Coulomb cost. Informal Introduction to Optimal Transport Optimal transportation γ measure in R 2 d , ρ, ρ ′ measures in R d The Cost Function c : R d × R d → R ∪ { + ∞} Prototype problem: transport mass from a given pile ρ into a given hole ρ ′ so as to minimize the transportation cost � R 2 d c ( x , y ) d γ ( x , y ) subject to the constraint � � R d γ ( x , y ) dx = ρ ′ ( y ) . R d γ ( x , y ) dy = ρ ( x ) and γ ( x , y ) = amount of mass transported from x to y c ( x , y ) = cost of transporting one unit of mass from x to y , e.g. | x − y | , | x − y | 2
Density functional theory and optimal transportation with Coulomb cost. Informal Introduction to Optimal Transport Issues Can we find an optimal measure γ which minimizes � R 2 d c ( x , y ) d γ ( x , y )? Under what conditions will the solution γ be unique? Can the optimal measure γ be characterized geometrically? Can we find γ explicitly?
Density functional theory and optimal transportation with Coulomb cost. Informal Introduction to Optimal Transport Some known results Optimal transport goes back to Monge (1781), Kantorovich (1942) and has recently become a very active area of mathematics,e.g. Villani (2009). c ( x , y ) = | x − y | 2 : an optimal measure exist which is unique and it is characterized through the gradient of a convex function (Brenier, Knott and Smith, Cuesta-Albertos, Rüschendorf and Rachev) c ( x , y ) = h ( x − y ) with h strictly convex, or c ( x , y ) = l ( | x − y | ) with l ≥ 0 strictly concave and increasing (Gangbo and McCann-1996)
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