Reduced models and numerical analysis in molecular quantum dynamics I. Variational approximation

Transcription

1 Reduced models and numerical analysis in molecular quantum dynamics I. Variational approximation Christian Lubich Univ. Tübingen Bressanone/Brixen, 14 February 2011

2 Computational challenges and buzzwords multiscale high-dimensional highly oscillatory structure preservation interdisciplinary large-scale... all that and more in computational quantum dynamics

3 Programme Variational approximation: Monday Multiconfiguration methods: Tuesday Semiclassical wavepackets: Thursday

4 Reference for this short course EMS, Zurich, 2008

5 Setting Full quantum dynamics: Schrödinger equation i ψ = Hψ, ψ = ψ(x 1,..., x N, t) Intermediate models: via Dirac Frenkel variational principle Born Oppenheimer, MCTDH, Gaussian wavepackets,... Classical molecular dynamics: Newtonian eqs. of motion Mẍ = V (x), x = (x 1 (t),..., x N (t))

6 Outline The Dirac-Frenkel variational approximation principle Abstract formulation Examples: Born Oppenheimer, Hartree, Gaussians Structural properties Approximation properties Variational splitting integrators Preparation: Strang splitting for TDSE Variational splitting: abstract formulation Variational splitting for Gaussian wavepacket dynamics

7 Outline The Dirac-Frenkel variational approximation principle Abstract formulation Examples: Born Oppenheimer, Hartree, Gaussians Structural properties Approximation properties Variational splitting integrators Preparation: Strang splitting for TDSE Variational splitting: abstract formulation Variational splitting for Gaussian wavepacket dynamics

8 Dirac-Frenkel principle: abstract formulation i ψ = Hψ on H = L 2 (R 3N ) M H approximation manifold T u M tangent space at u M, required to be complex linear approximate wavefunction u = u(, t) M determined from i u Hu, v = 0 for all v T u M Galerkin method on a state-dependent approximation space Dirac 1930, Frenkel 1934 (TDHF)

9 Examples Born Oppenheimer Hartree Gaussian wavepackets

10 Molecular Schrödinger equation N nuclei, L electrons x 1,..., x N y 1,..., y L x n, y l R 3 wavefunction Ψ = Ψ(x 1,..., x N, y 1,..., y L, t) i Ψ = HΨ with Hamiltonian H = T + U: T = T N + T e = N n=1 2 2M n xn L l=1 2 2m y l kinetic energy U = U ee + U en + U NN, U ee = j<l e 2 y j y l,... potential CO 2 : N + L = = 25 (x, y) R 75

11 Adiabatic approximation (Born Oppenheimer) H = T N + H e with H e = T e + U, U = U(x, y) H e = H e (x) on L 2 y parametric dependence on nuclear coord. x electronic eigenvalue problem: H e (x)φ(x, ) = E(x)Φ(x, ) variational approximation on M = {u : u(x, y) = ψ(x)φ(x, y), ψ L 2 x} reduced Schrödinger equation for the nuclei: i ψ = (T N + E +... )ψ, ψ = ψ(x 1,..., x N, t) get rid of the electrons (x 1,..., x N, y 1,..., y L )

12 Hartree approximation nuclear SE i ψ = (T + V )ψ, ψ = ψ(x 1,..., x N, t) T = n T n = n V = V (x 1,..., x N ) E variational approximation on 2 2M n xn M = {u : u(x) = φ 1 (x 1 )... φ N (x N )} nonlinear system of low-dimensional SE for φ n : i φ n = (T n + V n )φ n, φ n = φ n (x n, t) (n = 1,..., N) V n (x n ) = k n φ k V k n φ k x n mean field potential separate the nuclei TDH, TDSCF

13 Gaussian wavepackets nuclear SE iɛ ψ = (T + V )ψ, ψ = ψ(x 1,..., x N, t) T = n ɛ2 2M n xn, V = V (x 1,..., x N ) variational approximation by complex Gaussians: M = { u : u(x) = N n=1 ( i exp an x n q n ɛ( 2 ) )} + p n (x n q n ) + c n parameters q n, p n R 3, a n = α n + iβ n C, c n = γ n + iδ n C note q n = u x n u position, p n = u iɛ n u momentum motivation: Gaussian wavepackets exact for quadratic potentials parametrize the wavefunction Heller 1976

14 GWP equations of motion ODEs for parameters q, p, a = α + iβ, c = γ + iδ: q = p/m ṗ = V ȧ = 2a2 m 1 6 V ċ = p 2 2m V + 3ɛia m + ɛ 8β V with V = u V u = V (x) exp ( 2β ɛ x q 2 2δ ) ɛ dx, etc. note: for ɛ 0 obtain classical dynamics q = p/m, ṗ = V (q) semiclassical dynamics Heller 1976

15 Structural properties of variational approximation i u Hu, v = 0 for all v T u M

16 Orthogonal projection Minimum defect condition: determine approximation t u(t) M such that equivalently: u = ϑ T u M with ϑ 1 Hu = min! i Re u 1 i Hu, v = 0 for all v T um or u = P(u) 1 i Hu : orthogonal projection Frenkel 1934 Dirac

17 Cover illustration: tangent space projection u = P(u) 1 i Hu

18 Symplectic projection Im u 1 i Hu, v = 0 for all v T um symplectic 2-form on L 2 (R 3N ): ω(ξ, η) = 2 Im ξ, η Hamilton function: H(u) = Hu, u Hamiltonian system on the manifold M: Consequences: ω( u, v) = dh(u)v for all v T u M energy conservation: H(u(t)) = Const. symplecticity of the flow: for ξ(t), η(t) tangent vectors propagated by the linearized flow along u(t), ω(ξ(t), η(t)) = Const.

19 Action principle action functional S(u) = t1 t 0 i u(t) Hu(t), u(t) dt made stationary with respect to variations of paths on the manifold M with fixed end-points: t1 0 = δs(u) = 2 Im u(t) 1 i Hu(t), δu(t) dt. t 0 Dirac s quantum-mechanical analogue of Hamilton s principle in classical mechanics.

20 Poisson structure coordinates y on M: u = χ(y) total energy: K(y) = u H u structure matrix: B(y) = ( 2 Im χ (y) χ (y) ) 1 equations of motion in coordinates y: ẏ = B(y) K(y) Poisson system: bracket {F, G} = F T B G is skew-symmetric and satisfies the Jacobi and Leibniz identities Poisson flow ϕ t : {F ϕ t, G ϕ t } = {F, G} ϕ t F, G non-canonical Hamiltonian system

21 Example: Gauss as Poisson Gaussian wavepacket: y = (p, q, α, β, γ, δ) Poisson system ẏ = B(y) K(y) with p B(y) = 1 4β ɛ 0 β N(y) 0 0 4β2 3ɛ 0 β 0 p T β 0 4 ɛ 0 0 β ɛ 0 K(y) = u H u, N(y) = u 2 Both K and N are conserved quantities

22 Approximation properties u(t) ψ(t)?

23 Approximation error A posteriori error bound u(t) ψ(t) Quasi-optimality t 0 dist ( 1 i Hu(τ), T u(τ)m ) dτ t u(t) ψ(t) d(t) + Ce ct d(τ) dτ with d(t) = dist(ψ(t), M) the best-approximation error, under assumptions that include Hartree, Gaussians,... 0 L. 2005

24 Example: Approximation by Gaussians u(t) ψ(t) t 0 dist ( 1 iε Hu(τ), T u(τ)m ) dτ local harmonic approximation for Gaussian u centred at q H = T + V with V = U q + W q note Tu T u M and U q u T u M, therefore dist ( 1 iε Hu, T um ) = dist ( 1 iε W qu, T u M ) = O(ε 1/2 ) and hence u(t) ψ(t) C t ε 1/2 cf. Hagedorn 1980

25 Outline The Dirac-Frenkel variational approximation principle Abstract formulation Examples: Born Oppenheimer, Hartree, Gaussians Structural properties Approximation properties Variational splitting integrators Preparation: Strang splitting for TDSE Variational splitting: abstract formulation Variational splitting for Gaussian wavepacket dynamics

26 Strang splitting for the Schrödinger equation i ψ = Hψ, H = T + V (low-dimensional) TDSE ψ(t + t) = exp( i th)ψ(t) ψ n+1 = exp( i 2 tv ) exp( i tt ) exp( i 2 tv ) ψ n ( ) exp( itv )ψ (x) = e itv (x) ψ(x) ( ) F exp( itt )ψ (ξ) = e it ξ 2 (Fψ)(ξ) space discretization FFT Splitting on sparse grid in higher dimension: Gradinaru 2007

27 Error bounds for the Strang splitting without space discretization, for bounded smooth potential V : ψ n ψ(t n ) C t 2 t n max ψ(t) H 2 cf. Crank-Nicolson scheme ψ n+1 = (1 + i 2 th) 1 (1 i 2 th)ψ n : ψ n ψ(t n ) C t 2 t n max d 3 ψ L dt 3 2 Jahnke & L. 2000

28 Variational splitting: the formal game variational approximation: H = T + V i u Hu, v = 0 for all v T u M time step u n u n+1 via 1. half-step with V : u n+1/2 solution at t/2 of i u V u, v = 0 for all v T u M 2. full step with T : u + n+1/2 solution at t of i u T u, v = 0 for all v T u M 3. half-step with V : u n+1 solution at t/2 of i u V u, v = 0 for all v T u M

29 Variational splitting: examples Gaussian wavepackets MCTDH: tomorrow

30 Variational splitting for Gaussian wavepacket dynamics by magic: Substeps for T and V can be solved exactly and explicitly! Faou & L. 2006

31 Gaussian wavepacket integrator (1) First half-step with V : p n+1/2 = p n t 2 V n α + n = α n t 12 V n γ + n = γ n + t ɛ 16β n V n

32 Gaussian wavepacket integrator (2) Step with T : From p n+1/2, a n + = α n + + iβ n, c n + = γ n + + iδ n get q n+1, an+1 = α n+1 + iβ n+1, cn+1 = γ n+1 + iδ n+1: q n+1 = q n + t m p n+1/2 a n+1 = a + n /( ) t m a+ n c n+1 = c + n iɛ log (1 + 2 t m a+ n )

33 Gaussian wavepacket integrator (3) Second half-step with V : p n+1 = p n+1/2 t 2 V n+1 α n+1 = α n+1 t 12 V n+1 γ n+1 = γ n+1 + t ɛ 16β n+1 V n+1 explicit step, because V n+1 and V n+1 depend only on q n+1, β n+1, δ n+1 determined before

34 Properties of Gaussian wavepacket integrator (1) explicit integrator of order 2 time-reversible integrator Poisson/symplectic integrator: y n+1 = Φ t (y n ) with {F Φ t, G Φ t } = {F, G} Φ t F, G (as a composition of exact Poisson flows) L 2 norm of wavepacket is conserved exactly linear and angular momentum are conserved exactly in systems with translational/rotational symmetry

35 Properties of Gaussian wavepacket integrator (2) long-time near-conservation of energy: K(y) = u H u is conserved up to O( t 2 ) over exponentially long times t e c/ t uniformly in ɛ (use backward error analysis: numerical map is Poisson and therefore the almost-exact flow of a modified Poisson system) classical limit ɛ 0: yields Verlet method p n+1/2 = p n t 2 V (q n) q n+1 = q n + t m p n+1/2 p n+1 = p n+1/2 t 2 V (q n+1) standard integrator of classical MD

36 Outline The Dirac-Frenkel variational approximation principle Abstract formulation Examples: Born Oppenheimer, Hartree, Gaussians Structural properties Approximation properties Variational splitting integrators Preparation: Strang splitting for TDSE Variational splitting: abstract formulation Variational splitting for Gaussian wavepacket dynamics

37

38 2 x 10 3 Error x 10 3 Error x 10 3 Error T x 10 4 Hénon-Heiles, t = 0.1, ε = 10 2, 10 3, 10 4

39 ε = 0.01 Verlet ε = q2 2 3 ε = q1 3D Kepler, t = 0.1, classical limit toward the Verlet integrator

40 Reduced models and numerical analysis in molecular quantum dynamics II. Multiconfiguration methods Christian Lubich Univ. Tübingen Bressanone/Brixen, 15 February 2011

41 Outline MCTDH MCTDHF Variational splitting integrator Modelling error of MCTDH

42 Outline MCTDH MCTDHF Variational splitting integrator Modelling error of MCTDH

43 Recap: Dirac-Frenkel principle, abstract formulation i ψ = Hψ on H = L 2 (R 3N ) M H approximation manifold T u M tangent space at u M, required to be complex linear approximate wavefunction u = u(, t) M determined from i u Hu, v = 0 for all v T u M Galerkin method on a state-dependent approximation space

44 Recap: Time-dependent Hartree method i ψ = (T + V )ψ, ψ = ψ(x 1,..., x N, t) T = n T n = n 2 2M n xn variational approximation on manifold of Hartree products M = {u : u(x) = φ 1 (x 1 )... φ N (x N )} nonlinear system of low-dimensional SE for φ n : i φ n = (T n + V n )φ n, φ n = φ n (x n, t) (n = 1,..., N) V n (x n ) = k n φ k V k n φ k x n mean field potential separation of variables TDH

45 MCTDH Multi-Configuration Time-Dependent Hartree method approximate by linear combination of Hartree products ψ(x, t) r 1 j 1 =1... r N j N =1 a j1...j N (t) φ (1) j 1 (x 1, t)... φ (N) j N (x N, t) mutually orthogonal single-particle functions φ (n) 1,..., φ(n) r n core tensor (a j1...j N ) of full multilinear rank (r 1,..., r N ) H.D. Meyer et al , MCTDH book 2009

46 MCTDH equations of motion from Dirac Frenkel variational principle on MCTDH manifold: coupled system of ODEs and low-dimensional nonlinear PDEs i da J dt = L Φ J H Φ L a L (n) φ(n) iρ t = (I P (n) ) H xn φ (n) with Φ J (x, t) = N n=1 φ(n) j n (x n, t) for multi-indices J = (j 1,..., j N ) Meyer, Manthe & Cederbaum 1990

47 Density matrices ρ (n) (t) nth unfolding of coefficient tensor (a j1...j N ) (for n = 1,..., N): A (n) C rn k n r k set ρ (n) = A (n) A (n) Crn rn hermitian, positive semi-definite ρ (n) positive definite A (n) has full rank (a j1...j N ) has full n-rank

48 MCTDH equations of motion from Dirac Frenkel variational principle on MCTDH manifold: coupled system of ODEs and low-dimensional nonlinear PDEs i da J dt = L Φ J H Φ L a L (n) φ(n) iρ t = (I P (n) ) H xn φ (n) with Φ J (x, t) = N n=1 φ(n) j n (x n, t) for multi-indices J = (j 1,..., j N ) Meyer, Manthe & Cederbaum 1990

49 Existence and regularity assume smooth and bounded potential: V BC 2 initial data in the Sobolev space H 2 There exists a unique strong solution of the MCTDH equations a J C 2 ([0, t ), C), φ (n) j n C 1 ([0, t ), L 2 ) C([0, t ), H 2 ), where either t = or ρ (n) (t) becomes singular as t t. The solution depends on the initial data Lipschitz continuously with respect to the H 2 norm on every compact interval. Moreover, u defined by a J, φ (n) j n indeed solves the Dirac Frenkel variational equation. Koch & L. 2007

50 Outline MCTDH MCTDHF Variational splitting integrator Modelling error of MCTDH

51 Time-dependent Hartree-Fock method fermions: antisymmetric wavefunction ψ(x 1,..., x N, t) consider Coulomb interaction V = j<k 1 x j x k variational approximation on manifold of Slater determinants M = {u : u = φ 1 φ N 1 N! det ( φ j (x k ) ) } nonlinear system of low-dimensional SE for orbitals φ j : i φ j = (T + V j )φ j, V j mean field potential TDHF, Dirac 1930

52 MCTDHF Multi-Configuration Time-Dependent Hartree-Fock method approximate by linear combination of Slater determinants ψ(, t) a j1...j N (t) φ j1 (, t)... φ jn (, t) 1 j 1 <...<j N K mutually orthogonal orbitals φ 1,..., φ K core tensor (a j1...j N ) of full multilinear rank equivalent to MCTDH with antisymmetric core tensor

53 MCTDHF equations of motion from Dirac Frenkel variational principle on MCTDHF manifold: coupled system of ODEs and low-dimensional nonlinear PDEs i da J dt iρ φ t = L Φ J H Φ L a L = (I P) H MF φ with Φ J = N n=1 φ j n for multi-indices J = (j 1 < < j N ) like MCTDH equations, simplified

54 Existence and regularity assume Coulomb interaction initial data in H 2 There exists a unique strong solution of the MCTDHF equations a J C 2 ([0, t ), C), φ j C 1 ([0, t ), L 2 ) C([0, t ), H 2 ), where either t = or ρ (n) (t) becomes singular as t t. Moreover, u defined by a J, φ j indeed solves the Dirac Frenkel variational equation. Bardos, Catto, Mauser & Trabelsi 2010 Koch & L. 2010

55 Outline MCTDH MCTDHF Variational splitting integrator Modelling error of MCTDH

56 Difficulties with standard time integration methods explicit methods: (e.g., explicit Runge-Kutta) step size restriction t c x 2 for stability implicit methods: (e.g., Crank-Nicolson) computationally too expensive, infeasible aim for unconditionally stable explicit method

57 Variational splitting: the formal game variational approximation: H = T + V i u Hu, v = 0 for all v T u M time step u n u n+1 via 1. half-step with V : u n+1/2 solution at t/2 of i u V u, v = 0 for all v T u M 2. full step with T : u + n+1/2 solution at t of i u T u, v = 0 for all v T u M 3. half-step with V : u n+1 solution at t/2 of i u V u, v = 0 for all v T u M

58 Variational splitting for MCTDH and MCTDHF Step with T : since Tu T u M for u M, integration step with T decouples into ȧ J = 0 and i φ(n) j (x n, t) = 2 xn φ (n) j (x n, t) t 2M n low-dimensional free Schrödinger equations FFT Step with V : MCTDH(F) for Hamiltonian V instead of H larger time steps independently of the space discretization explicit, unconditionally stable scheme L. 2004

59 Error of variational splitting for MCTDH(F) second-order error bound u n u(t n ) C t 2 max u(t) H 2 same as for Strang splitting for the linear Schrödinger equation basic tool: Lie-commutator bounds for vector fields corresponding to T and V L (MCTDH with bounded smooth potential) Koch & L (MCTDHF with Coulomb potential)

60 Outline MCTDH MCTDHF Variational splitting integrator Modelling error of MCTDH

61 Convergence of MCTDH to Schrödinger solution??? Naive expectation: Taking more and more terms in the linear combination of Hartree products yields an ever better accuracy Obstructions: approximation properties of basis of Hartree products Φ J (, t)? ill-conditioned coefficient tensor (a J ) ill-conditioned density matrices ρ (n) = A (n) A (n)

62 Quasi-optimality of variational approximation u(t) MCTDH approximation, ψ(t) exact wave function MCTDH error bounded by best-approximation error: t u(t) ψ(t) d(t) + C κ e cκt d(τ) dτ with d(t) = dist(ψ(t), M) and κ curvature of manifold M 0 but κ cond(ρ (n) ) 1/2 as number of configurations L. 2005

63 Objectives error bounds in the case of ill-conditioned density matrices convergence as number of configurations under appropriate assumptions Conte & L. 2010

64 Approximability assumption Exact wavefunction can be written as ψ(t) = v(t) + e(t) where v(t) M has small defect: i v (, t) Hv(, t) t ε Small error e(t) and small defect (ε) for some linear combination of Hartree products

65 Assumptions on the coefficient tensor B (n) (t) nth unfolding of coefficient tensor (b j1...j N (t)) of v(t) M, for n = 1,..., N Bound of the pseudo-inverse (large: small δ!): B (n) (t) 2 = 1 σ rn (B (n) (t)) δ 1 Small time derivatives of the coefficient tensor in components that correspond to small singular values: B (n) (t) Ḃ (n) (t) c, 2

66 Error bound u(t) ψ(t) e(t) + 2tε for t = O(δ/ε) Bound independent of δ (ill-conditioning) for fixed rank, can achieve δ ε by perturbing v(t): t = O(1) Convergence: if we assume δ ε, we can admit arbitrary number of configurations: ε 0

67 Proof of the error bound use estimates for the tangent space projection P(u) to show a quadratic differential inequality d dt u v C δ u v 2 + ε uses bound of the potential V at present no error bound for MCTDHF with Coulomb potential

68 Outline MCTDH MCTDHF Variational splitting integrator Modelling error of MCTDH

69 Computing semiclassical quantum dynamics using Hagedorn wavepackets Christian Lubich, Univ. Tübingen joint work with Erwan Faou and Vasile Gradinaru Bressanone/Brixen, 17 February 2011

70 Outline The Schrödinger equation in the semi-classical regime Hagedorn wavepackets A splitting method for time integration

71 Outline The Schrödinger equation in the semi-classical regime Hagedorn wavepackets A splitting method for time integration

72 Schrödinger equation in semi-classical scaling iε ψ ε2 (x, t) = t 2m xψ(x, t) + V (x)ψ(x, t) for the wavefunction ψ = ψ(x, t), x = (x 1,..., x N ) R N, t 0 initial value problem: ψ specified at time t = 0 SE for the nuclei in a molecule 0 < ε 1

73 Computational challenges solutions are highly oscillatory with wavelengths ε localized with width ε, with velocity 1 wavepacket near classical motion m q = V (q) semi-classical quantum dynamics high dimension: N = 3 n particles no grids! (neither full nor sparse)

74 Rescue? wavefunction is well approximated by complex Gaussian polynomial Hagedorn wavepackets

75 Outline The Schrödinger equation in the semi-classical regime Hagedorn wavepackets A splitting method for time integration

76 Complex Gaussians in Hagedorn s parametrization ϕ 0 [q, p, Q, P](x) = (πε) N/4 (det Q) 1/2 ( i exp 2ε (x q)t PQ 1 (x q) + i ) ε pt (x q), q R N position, p R N momentum Q, P complex N N matrices such that ( ) Re Q Im Q Y = is symplectic: Y T JY = J for J = Re P Im P ( ) 0 I I 0 Consequence: PQ 1 is complex symmetric with positive definite imaginary part Hagedorn 1980

77 Hagedorn wavepackets L 2 -orthonormal set of functions ϕ k (x) = ϕ k [q, p, Q, P](x) for multi-indices k = (k 1,..., k N ), constructed recursively: define the raising operator R = (R j ) = 1 ) (P (x q) + Q ( iε x p) 2ε With j = ( ) the jth unit vector, set ϕ k+ j = 1 kj + 1 R jϕ k. ϕ k are polynomials of degree k k N multiplied with the Gaussian ϕ 0 (N = 1: Hermite functions). Hagedorn 1998

78 Recursive evaluation Q( kj + 1 ϕ k+ j (x) ) N j=1 = 2 ε (x q)ϕ k(x) Q ( kj ϕ k j (x)) N j=1

79 Approximate wavefunction by Hagedorn wavepacket ψ(x, t) e is(t)/ε k K c k (t) ϕ k [q(t), p(t), Q(t), P(t)](x) over multi-index set K in low dimensions, full cube: k j K (j = 1,..., N) in moderate dimensions, hyperbolic cross: (1 + k 1 )... (1 + k N ) K in high dimensions, axes: k j > 0 only for a single component j in each k problem-adapted moving basis functions

80

81 Outline The Schrödinger equation in the semi-classical regime Hagedorn wavepackets A splitting method for time integration

82 Recap: Schrödinger equation with the Hamiltonian iε ψ t = Hψ H = T + V composed of the kinetic energy operator and a smooth potential T = ε2 2m x V = V (x).

83 Bits and pieces H = T + U q(t) + W q(t) We can solve exactly the free Schrödinger equation, with the wavefunction remaining in the Hagedorn wavepacket form with unaltered coefficients c k. For a quadratic potential, we can solve exactly the potential equation with the wavefunction remaining in the Hagedorn wavepacket form with the same coefficients c k. For the non-quadratic remainder, we compute the variational approximation of the potential equation on the linear space spanned by the functions ϕ k with fixed parameters q, p, Q, P, letting the coefficients c k vary.

84 Free Schrödinger equation iε ψ t = ε2 2m ψ A time-dependent Hagedorn wavepacket solves the free Schrödinger equation with modified positions q(t) = q(0) + t m p(0) Q(t) = Q(0) + t m P(0) and unchanged momenta p, P and unchanged coefficients c k. change only position q and Q and phase S

85 Quadratic potential iε ψ t = Uψ For a quadratic potential U(x), a time-dependent Hagedorn wavepacket solves the equation with modified momenta p(t) = p(0) t U(q(0)) P(t) = P(0) t 2 U(q(0))Q(0) and unchanged positions q and Q and unchanged coefficients c k. change only momentum p and P and phase S

86 Galerkin approximation for the remainder iε ψ t = W ψ, W = W (x) fix Gauss parameters q, p, Q, P in ϕ k (x) = ϕ k [q, p, Q, P](x) Galerkin condition: determine u(x, t) = k K c k(t)ϕ k (x) from ϕ k, iε t u Wu = 0 k K

87 Galerkin approximation for the remainder (ctd.) Galerkin condition determines the coefficient vector c = (c k ) as c(t) = exp ( it ) ε F c(0) with the Hermitian matrix F = (f kl ), f kl = W (x) ϕ k (x) ϕ l (x) dx R N The integrals are non-oscillatory, approximated by sparse Gauss Hermite quadrature. F = O(ε 3/2 ) if the quadratic Taylor polynomial of W at q vanishes. Therefore, exp ( it ε F ) c(0) is computed efficiently using just a few Lanczos iterations with F. change only coefficients c k

88 Time-stepping algorithm start from position q 0, momentum p 0, phase S 0, width matrices Q 0, P 0 satisfying the symplecticity condition, and coefficients c 0 k ψ(x, t 0 ) u 0 (x) = e is0 /ε k K c 0 k ϕ k[q 0, p 0, Q 0, P 0 ](x) determine approximation u 1 (x) of the same form after time step t using a splitting algorithm

89 Splitting algorithm 1. Half-step of kinetic part: updates q 1/2, Q 1/2, S 1/2,. 2. Full step of potential part: split the potential V (x) = U 1/2 (x) + W 1/2 (x) into its quadratic Taylor polynomial U 1/2 (x) at q 1/2 and the remainder solve with quadratic potential U 1/2 : updates p 1, P 1, S 1/2,+ Galerkin approximation for the non-quadratic remainder W 1/2 : update coefficients c 1 k 3. Half-step of kinetic part: updates q 1, Q 1, S 1.

90 Properties preserves the symplecticity relation of the matrices Q and P preserves the L 2 norm of the wavepacket time-reversible method for position q and momentum p: Störmer-Verlet method for the corresponding classical Hamiltonian system limit of taking the full basis set ϕ k with all k N N : Strang splitting of the Schrödinger equation robust in the semi-classical limit ε 0: approximation in the potential part becomes exact for ε 0, while the kinetic part is solved exactly for all ε.

91 Comparison with the Fourier method Error vs. time in a 2D example (Henon-Heiles) Hagedorn with 20 basis functions, Fourier with 2 20 basis functions

92 Error behaviour Maximum error vs. number of basis functions at t = 1 and t = 5.

93 Computing time vs. dimension CPU time with a sparse Hagedorn wavepacket at fixed K = 8 in dimensions 2 to 12.

94 Flying carpet Squared absolute values of the approximate wave function evaluated on the flying carpet of quadrature points.

95 References G.A. Hagedorn, Comm. Math. Phys. 71, (1980) Ann. Phys. 269: (1998) C.L., blue book, Eur. Math. Soc., Zurich, 2008 E. Faou, V. Gradinaru, C.L., SIAM J. Sci. Comp. 31, (2009) V. Gradinaru, G.A. Hagedorn, A. Joye, JCP 132, (2010)