Reinhold Schneider, MATHEON TU Berlin. Direct Minimization for effective single particle models (DFT)

Transcription

1 Direct Minimization for effective single particle models (DFT)

2 Partners joint work with: J. Blauert, T. Rohwedder (TU Berlin), A. Neelov (U Basel) joint EU NEST project BigDFT together with Dr. Thierry Deutsch, Co-ordinator, Atomistic Simulation group at CEA-Grenoble, Prof. Stefan Goedecker, Computational Physics Group at the Basel University, Switzerland; Prof. Xavier Gonze, Uni. de Phy.-Ch.et de Phys. des Mat., Univ. Catholique de Louvain; Joint project with H.J. Flad (Matheon - TUB) and W. Hackbusch (MPI Leipzig) DFG Priority Program: Modern and Universal First-Principle-Methods for Many-Electron Systems in Chemistry and Physics

3 Basic model - electronic Schrödinger equation Electronic Schrödinger equation N nonrelativistic electrons + Born Oppenheimer approximation HΨ = EΨ The Hamilton operator H = 1 2 i i i K ν=1 Z ν x i a ν x i x j i,j acts on anti-symmetric wave functions Ψ H 1 (R 3 {± 1 2 })N, Ψ(x 1, s 1,..., x N, s N ), x i = (x i, s i ) R 3 {± 1 2 }.

4 Hartree-Fock- (HF) Approximation Ground state energy E 0 = min{ Hψ, ψ : ψ, ψ = 1} approximation of ψ by a single Slater determinant Ψ SL Φ (x 1, s 1,..., x N, s N ) := 1 N! det(φ i(x j, s j )) Closed Shell Restricted HF (RHF): N := N 2 electron pairs minimization of the functional J HF (Φ) D E Φ J HF (Φ) := HΨ SL Φ, ΨSL Φ = + NX Z j=1 R 3 " φj (y) 2 φ i (x) 2 x y w.r.t. orthogonality constraints NX Z ` φi (x) 2 + 2V core(x) φ i (x) 2 + i=1 # 1 φ i (x)φ i (y) φ j (x)φ j (y) dy dx 2 x y Φ = (φ i ) N i=1 H1 (R 3 ) N and φ i, φ j = δi,j

5 Kohn-Sham model Theorem (Kohn-Hohenberg) The ground state energy E 0 is a functional of the electron density n. 1 2 E 0 E KS = inf{j KS (Φ) : φ i, φ j = δ ij } minimization of the Kohn Sham energy functional J KS (Φ) J KS (Φ) = ( Z 1 2 NX Z φ i 2 + nv core + 1 Z Z 2 i=1 n(x)n(y) x y dx dy E xc(n) φ i H 1 (R 3 ), electron density n(x) := N i=1 φ i(x) 2 E xc (n) exchange-correlation-energy (not known explicitly) e.g. LDA (local density approximation) n E xc (n) : R R )

6 General setting Φ := (φ 1,..., φ N ) (H 1 (R 3 )) N = V N Gelfand triple V := H 1 (R 3 ) L 2 (R 3 ) H 1 (R 3 ) = V Φ T Ψ := ( φ i, ψ j )i,j R N N scalar product Φ, Ψ := tr Φ T Ψ = N i=1 φ i, ψ i R AΦ := (Aφ 1,..., Aφ N ), A : V V Simplified Problem: minimize J SCF A (Φ) := N i=1 Aφ i, φ i = tr Φ T AΦ = Φ, AΦ w.r.t. to orthogonality constraints Φ T Φ = I.

7 Remarks J = J HF, J KS, J SCF are invariant under unitary transformations U U N J (Φ) = J (ΦU), ΦU := ( φ i u ij ) j=1,...,n i=1 gradient J (Φ) = A [Φ] Φ, A [Φ] : V V The density matrix operator D Φ := N φ i, L 2 φ i i=1 projects onto span Φ := span{φ i : 1 i N}, D 2 = D, tr D = N, D T = D

8 Stiefel and Grassmann manifolds Definition Stiefel manifold S V,N := S := {Φ = (φ i ) N i=1 φ i V, Φ T Φ I N N = 0 R N N } Grassmann manifold is a quotient manifold G V,N := G := S V,N /, Φ Φ Φ = ΦU, U U(N) (identify ONB spanning the same subspace span Φ) see [Edelman, Arias, Smith] for V = R n. There is a one-to-one correspondence between [Φ] G D Φ (density matrix operator)

9 Tangent space Proposition (Edelman, Arias, Smith (98); Blauert, Neelov, Rohwedder, S. (08)) tangent space T [Φ] G = {δψ V N (δψ) T Φ = 0 R N N } (I D Φ ) : V N T [Φ] G, is an orthogonal projection onto the tangent space T [Φ] G tangent space T Φ S = T [Φ] G + {ΦA : A T = A} = {Θ V N : Θ T Φ = Φ T Θ }

10 Existence and uniqueness known results Problem Minimize J : G R, J = J HF, J KS, J SCF A Theorem (Lieb-Simon (78), P.L. Lions (89)) There exists a minimizer Φ (H 1 (R 3 )) N for J HF provided that N Z ν. Uniqueness is not know yet? J SCF A has unique minimizer on Grassmann manifold G if there is a gap for the lowest eigenvalues of A: λ N < λ N+1 no uniqueness on Stiefel manifold S.

11 1st order optimality conditions For [Φ] G there hold J (Φ) = A [Φ] Φ (V ) N where A SCF [Φ] and A HF [Φ], AKS [Φ] are the Hamilton Fock operators = A Definition E.g. the Kohn-Sham Hamilton Fock operator is given by ( A KS [Φ] := V core + n 1 ) + v xc (n) = V (n) Nec. cond.: If [Ψ] = argmin {J (Φ) : [Φ] G} V N (V N h ) then A [Ψ] Ψ, δφ = 0 δφ T [Ψ] G V N (V N h ) (I D Ψ )A [Ψ] Ψ, δφ = 0 δφ V N (V N h )

12 Canonical HF - KS equations Lagrangian L(Φ, Λ) := J (Φ) trλ( Φ T Φ I) At a (local) minimum Ψ V, i.e. at stationary points (Ψ, λ) of L there holds Λ = A [Ψ] ˆψ i, ˆψ j. There exists a representation Ψ = ˆΨU of [ ˆΨ] G, s. t. λ i,j = λ i δ ij and λ 1 λ 2... s.t. There hold the canonical (HF or KS ) equations A [Ψ] ψ i = λ i ψ i, λ 1 λ 2... λ N < 0. Self consistent iteration: A [Ψ (n) ] ψ(n+1) i (Cances-LeBris) = λ (n+1) i ψ (n+1) i

13 Self consistent iteration - SCF We define A D n such that A [Ψ] = A DΨ for an appropriate D n P := {D : D = D, trd = N, D 2 D}. Algorithm (Self consistent iteration) 1 Ψn+1 := argmin {JA SCF (Φ) : Φ G} = D n argmin { A D nφ, Φ : Φ G} 2 D n+1 := µd Ψ n + (1 µ)d Ψn+1 P for some µ [0, 1] Roothaan algorithm: A [Ψ ( n)] ψ(n+1) i = λ (n+1) i ψ (n+1) i fails often Optimal damping algorithm (Cances-LeBris 02) converges D n+1 := µd Ψ n + (1 µ)d Ψn+1 P with optimal choice of µ [0, 1]

14 Gradient flow The minimizer Ψ is spanning an invariant subspace V Ψ := span{ψ 1,..., ψ N } of A [Ψ] corresponding to the N lowest eigenvalues. A local minimizer Ψ is a fix point of the gradient flow [Φ](t), Ψ = lim t Φ(t) or d dt Φ A [Φ]Φ(t), θ = 0 θ T [Φ] G V N (V N h ) d dt Φ(t) (I D Φ)A [Φ] Φ, θ = 0 θ V N (V N h )

15 Direct Minimization Algorithm (direct minimization) 1 A := J (Φ (n) ) = A [Φ (n) ] : V N V N 2 Λ (n) := Φ T AΦ = ( Aφ i, φ j )i,j=1,...,n R N N 3 ˆΦ (n+1) := Φ (n) αb 1 (AΦ (n) Φ (n) Λ (n) ), preconditioner B 4 ˆΦ (n+1) Φ (n+1), Φ (n+1) := P ˆΦ (n+1) by projection onto G, e.g. orthogonalization (AΦ (n) Φ (n) Λ (n) ) = [A, D Φ (n)]φ (n) = (I D Φ (n))aφ (n) B, ˆB : V V, B : V N V N, ˆB := (I D)B(I D) + DBD, D = D Ψ

16 Projection onto the Stiefel manifold Φ = Φ (n+1) = ( ˆφ i ) N i=1, and P Φ =: Φ V resp. G s.t. span { ˆφ i : i = 1,..., N} = span {φ i : i = 1,..., N} Projection P : V N S,resp. G Gram Schmidt orthogonalization Löwdin transformation Φ = L 1 Φ where LL T = Φ T Φ Diagonalization of Λ (n+1) = Φ T A [Φ] Φ, yields the first N eigenvalues λ (n) 1... λ (n) N of A [Φ].

17 Comment on direct minimization improvement by subspace acceleration: e.g. line search and DIIS Φ (n+1) = Function(ˆΦ (n+1), Φ (n),..., Φ (0) ) gradient directed convergence with Armijo line search B : V V, φ 2 B = φ, φ B := Bφ, φ φ 2 H 1 e.g.: B C, e.g. multigrid or convolution by FFT also valid if V := V h is a finite dimensional subspace for the simplified problem cf. e.g. Knyazev et al.

18 Convergence results Definition measure of the error between subspaces spanned by Φ, Ψ G (I D Ψ )Φ 2ˆB := N φ i D Ψ φ i 2ˆB i=1 Theorem ( Blauert, Neelov, Rohwedder, S. (08)) If Φ (0) U δ (Ψ), and (J (Ψ) Λ)Φ, Φ γ Φ 2 H 1N Φ T [Ψ] G, then there exists χ < 1 such that for all (I D Ψ )Φ (n+1) ˆB χ (I D Ψ )Φ (n) ˆB

19 Energy error Constrained optimization problem (revisited): u = argmin{j(v) : G(v) = 0} Lagrangian L(x) := L(u, Λ) = J(u) ΛG(u) (x = (u, Λ) X) Theorem (Rannacher et al. ) If L (x)y = 0 y X and L (x n )x n = 0, L(x) L(x n ) = 1 2 L (x n )(x x n ) + O( x x n 3 X ) J(u) J(u n ) = 1 J 2[ (u n )(u u n ) Λ n G (u n )(u u n ) (Λ Λ n )G(u n ) ] Here L(Φ, Λ) = J (Φ) + trλ( Φ T, Φ I)

20 Energy error Theorem ( Blauert, Neelov, Rohwedder, S. (08)) Let J C 2 (V N, R), then J (Φ (n) ) J (Ψ) + R 2 = 2 (Ψ Φ (n) ), (A [Φ (n) ] Φ(n) Φ (n) Λ (n) ) (I D Ψ )Φ (n) 2 V N where R 2 = O( (I D Ψ )Φ (n) 2 ). V N Corresponding results for a priori and a posterriori estimates for the Galerkin solution together with S. Schwinger and W.Hackbusch (08).

21 Pseudo-potentials pseudo-potentials (nuclei + core electrons) K V core (x) = Z j f Rj (x) + P j f Rj (x) = j=1 1 x R j if x R j > r j consider only N - valence electrons (pairs), 1 Φ(x) 1 P VL Φ(x) 1 (I P VL )Φ(x) (I P VL )Φ(x) x x x + non-local pseudo-potentials solutions are smooth orbitals! This allows e.g. relativistic corrections! But modeling error!!!

22 Discretization Let ψ = (ψ λ ) λ I be basis in V h V, dimv h = N <. Φ (n) = (ϕ n 1,..., ϕn N ) be given ϕ(n) i = λ I c(n) i,λ ψ λ = Ψ T c (n) i We build the Galerkin matrices V (n) H T = ( 1 2 ψ λ, ψ λ ), V c = ( V core (x)ψ λ, ψ λ ), (n) = ( V (x)ψ λ, ψ λ ), W (n) = ( ( W (n) + V (n) xc )ψ λ, ψ λ ) H with the Hartree potential V H and the exchange energy terms Hamilton Fock matrix F (n) = F Φ (n) = T + V c + V (n) H + W(n)

23 Basis functions Local Atomic Orbitals (LAO), Gaussian type orbitals (GTO): ψ λ (x) = e a λ R j x 2 R j x 2n Y l,m (x) adapted to atoms; 3,..., 20 basis functions per electron plane waves, trigonometric polynomials, ψ λ = e ik x C for crystals, periodic setting, requires (non-local) pseudo potentials (only valence electrons) 100,... 10, 000 basis functions per atom. local basis functions, finite elements, splines,wavelets, grid methods etc., without pseudo potentials , with pseudo potentials basis functions per atom

24 BigDFT EU NEST Project for developing a linear scaling wavelet code for density functional (DFT) Sim/BigDFT/ Code is incorporated in ABINIT: ( an Open Source plane wave DFT code (molecules and crystals),

25 Figure: BigDFT orth. wavelets representing φ i, interpolets for the density n, convergence rate for E 0 about O(h 14 )! (non-local) pseudo-potentials, semi-adaptivity (2 level of resolution),

26 BigDFT approximate Galerkin approximate V local G 1 G 2 G 3 diag(v(x j k )) GT 1 GT 2 GT 3 (magic filter G = (g[i j])) by Kronecker tensor products. Fast evaluation of nonlocal pseudo potentials Preconditioner C, C ( µi) 1 based on Multilevel preconditioning. Hartree potential V H (x) = R 3 n(y) x y dy for updating the Hamilton Fock operator is a global procedure: Fast Poisson Solver Newton potential based on Kronecker tensor products

27 BigDFT direct minimization, linear scaling (in progress),

28 Numerical results Figure: example: C 19 H 22 N 2 O (N = 55)

29 Numerical results Convergence history for the direct minimization scheme and together with DIIS acceleration 10 1 steepest descent 10 1 history size for DIIS = hgrid = 0.3 hgrid = 0.45 hgrid = hgrid = 0.3 hgrid = 0.45 hgrid = 0.7 absolute error absolute error iter, iter Figure: convergence

30 Numerical results Required resources 5000 memory 8000 runtime DIIS, hist. size = 6 steepest descent 7000 DIIS, hist. size = 6 steepest descent MByte sec hgrid, hgrid Figure: memory and computing time

31 Numerical results first N eigenvalues 0.8 least eigenvalues, hgrid = 0.3 least eigenvalues initial guess HOMO LUMO gap for initial guess final eigenvalues k, initial guess, hgrid = 0.3 HOMO LUMO gap for initial guess,hgrid = 0.3 final eigenvalues, hgrid = 0.3 initial guess, hgrid = 0.7 final eigenvalues, hgrid = 0.7 HOMO LUMO gap for initial guess,hgrid = k Figure: memory and computing time

32 Further comments: Linear scaling Representative of [Φ] G with minimal support Usual complexity scaling: O(N 2 dimv h ) = O(N 3 )! Often there exist orbitals Ψ = (ψ i ) N i=1 with [Ψ] = [Φ], ψ i (x) = e α x R i domain decomposition R 3 = M j=1 Ω j Ω, φ i 0 on Ω. linear scaling w. r. t. the number of particles (for fixed accuracy per atom) BigDFT and Bencteux, Hager, Cances & Lebris... (research in progress)