h-P discontinuous Galerkin finite element method for electronic - PowerPoint PPT Presentation

. . . . . . . . . . . . . . h-P discontinuous Galerkin finite element method for electronic structure calculations Carlo Marcati, Yvon Maday Laboratoire Jacques-Louis Lions, UPMC, France Adaptive algorithms for computational PDEs 5-6 January 2016 Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 20

. . . . . . . . . . . . h-P discontinuous finite elements for electronic structure . calculation We combine results from and apply them to the models used in quantum chemistry. Outline of the presentation: 1. Motivation: models for electronic structure calculations 2. Convergence, regularity 3. Asymptotics of the solution and design of an optimal h-P space from a priori estimates. Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 20 • Numerical approximation of elliptic problems in non smooth domains • Approximation of non linear eigenvalue problems

. . . . . . . . . . . . h-P discontinuous finite elements for electronic structure . calculation We combine results from and apply them to the models used in quantum chemistry. Outline of the presentation: 1. Motivation: models for electronic structure calculations 2. Convergence, regularity 3. Asymptotics of the solution and design of an optimal h-P space from a priori estimates. Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 20 • Numerical approximation of elliptic problems in non smooth domains • Approximation of non linear eigenvalue problems

. . . . . . . . . . . . h-P discontinuous finite elements for electronic structure . calculation We combine results from and apply them to the models used in quantum chemistry. Outline of the presentation: 1. Motivation: models for electronic structure calculations 2. Convergence, regularity 3. Asymptotics of the solution and design of an optimal h-P space from a priori estimates. Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 20 • Numerical approximation of elliptic problems in non smooth domains • Approximation of non linear eigenvalue problems

. . . . . . . . . . . . h-P discontinuous finite elements for electronic structure . calculation We combine results from and apply them to the models used in quantum chemistry. Outline of the presentation: 1. Motivation: models for electronic structure calculations 2. Convergence, regularity 3. Asymptotics of the solution and design of an optimal h-P space from a priori estimates. Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 20 • Numerical approximation of elliptic problems in non smooth domains • Approximation of non linear eigenvalue problems

. The Schrödinger equation . . . . . . . . . . Motivation Motivation: the Schrödinger equation . The Schrödinger equation nuclei. It is therefore hard to approach computationally, even for systems of moderately small size. A fjrst approximation (Born-Oppenheimer) consists in considering the nuclei as fjxed particles, thus calculating only electronic wavefunctions. Many methods have been proposed for the approximation of the electronic wavefunctions: among them approximation, …). Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . 3 / 20 . . . . . . . . . . . . . i ℏ ∂ ∂ t Ψ = − ℏ 2 2 m ∇ 2 Ψ + V Ψ is set in a 1 + 3 ( N + M ) dimensional space for a system of N electrons and M • Hartree-Fock (and post Hartree-Fock) methods, • methods based on density functional theory (Kohn-Sham local density

. The Schrödinger equation . . . . . . . . . . Motivation Motivation: the Schrödinger equation . The Schrödinger equation nuclei. It is therefore hard to approach computationally, even for systems of moderately small size. A fjrst approximation (Born-Oppenheimer) consists in considering the nuclei as fjxed particles, thus calculating only electronic wavefunctions. Many methods have been proposed for the approximation of the electronic wavefunctions: among them approximation, …). Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . 3 / 20 . . . . . . . . . . . . . i ℏ ∂ ∂ t Ψ = − ℏ 2 2 m ∇ 2 Ψ + V Ψ is set in a 1 + 3 ( N + M ) dimensional space for a system of N electrons and M • Hartree-Fock (and post Hartree-Fock) methods, • methods based on density functional theory (Kohn-Sham local density

. The Schrödinger equation . . . . . . . . . . Motivation Motivation: the Schrödinger equation . The Schrödinger equation nuclei. It is therefore hard to approach computationally, even for systems of moderately small size. A fjrst approximation (Born-Oppenheimer) consists in considering the nuclei as fjxed particles, thus calculating only electronic wavefunctions. Many methods have been proposed for the approximation of the electronic wavefunctions: among them approximation, …). Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . 3 / 20 . . . . . . . . . . . . . i ℏ ∂ ∂ t Ψ = − ℏ 2 2 m ∇ 2 Ψ + V Ψ is set in a 1 + 3 ( N + M ) dimensional space for a system of N electrons and M • Hartree-Fock (and post Hartree-Fock) methods, • methods based on density functional theory (Kohn-Sham local density

. . . . . . . . . . . . Models in computational quantum chemistry . Motivation: the Hartree-Fock approximation N 2 dxdy 2 dxdy where N Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . Motivation . . . . . . . . . . . . . 4 / 20 . . . . . . . . . . . . . { ∫ } I HF = inf E HF ( φ 1 , . . . , φ N ) , φ i ∈ H 1 ( R 3 ) , R 3 φ i φ j = δ ij ∫ ∫ ∫ ρ Φ ( x ) ρ Φ ( y ) ∑ E HF = R 3 |∇ φ i | 2 + R 3 ρ Φ V + 1 | x − y | R 3 × R 3 i = 1 ∫ | τ Φ ( x , y ) | 2 − 1 | x − y | R 3 × R 3 ∑ τ Φ ( x , y ) = φ i ( x ) φ i ( y ) ρ Φ ( x ) = τ Φ ( x , x ) i = 1

. . . . . . . . . . . . . . Motivation Models in computational quantum chemistry It can be shown that We therefore have the eigenvalue problem [Flad et al., 2008] showed that around the origin the solutions belong to (a subset of) the countably normed spaces Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . 5 / 20 . . . . . . . . . . . . { } ∫ ∫ R 3 | φ | 2 ≤ 1 , φ i = argmin ⟨F φ, φ ⟩ , φ ∈ H 1 ( R 3 ) , R 3 φ i φ j = 0 , ∀ j ̸ = i , φ where F is the self adjoint operator ( ) ∫ τ Φ ( x , y ) F ψ = − 1 2 ∆ ψ + V ψ + ρ Φ ⋆ 1 ψ − | x − y | ψ ( y ) dy . | x | R 3 F φ i = ε i φ i i = 1 , . . . , N { } K ∞ ,γ = u ∈ D ′ : | x | | α |− γ ∂ α u ∈ L 2 , | α | = s , ∀ s ∈ N .

. . . . . . . . . . . . Motivation . Drawbacks of classic methods Classic finite element and spectral approximations The eigenfunctions are thus not regular in the Sobolev spaces convex polygonal domains or fraction elliptic problems. The convergence speed of “classic” fjnite element and spectral methods is bounded by the regularity of the solution in Sobolev spaces. Classic fjnite element and spectral methods Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 20 H k (Ω) = W k , 2 (Ω) , but share features with the solution of e.g. problems in non If u ∈ H s + 1 (Ω) , the following approximation results hold: • for fjnite element methods of degree r ≤ s and element size h : ∥ u − u h ∥ H 1 (Ω) ≲ h r | u | H r + 1 (Ω) ; • for spectral methods of degree p : ∥ u − u δ ∥ H 1 (Ω) ≲ p − s ∥ u ∥ H s + 1 (Ω) ;

. . . . . . . . . . . . The discontinuous h-P finite elements method . Space and mesh The discontinuous h-P finite elements method Finite element space: towards the center (where the singularity lies), while the polynomial degree usually decreases with a slope s . Graded mesh, uniform slope: Carlo Marcati (LJLL) h-P dG FE for electronic structure calculations 5 January 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 20 X δ = { v ∈ L 2 (Ω) : v | S ∈ Q k S ( S ) , ∀ S ∈ T } . The mesh is geometrically refjned by a factor σ At the refjnement step ℓ , the elements in I ℓ will have edges of length σ ℓ , while in the outermost element the polynomial degree will be k 0 + ⌊ s ℓ ⌋