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Multi-layered wavefunction representations and quadratures:

Multi-layered wavefunction representations and quadratures: the multi-configurational time-dependent Hartree approach Uwe Manthe Theoretische Chemie Universität Bielefeld. High-dimensional quantum dynamics: applications. Malonaldehyde intramolecular proton transfer

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Multi-layered wavefunction representations and quadratures:

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  1. Multi-layered wavefunction representations and quadratures: the multi-configurational time-dependent Hartree approach Uwe Manthe Theoretische Chemie Universität Bielefeld

  2. High-dimensional quantum dynamics: applications Malonaldehyde intramolecular proton transfer  tunneling splitting of the vibrational states quantum dynamics in 21D 1 1

  3. Bimolecular reactions, reactive scattering H+CH4H2+CH3 F(3P)+CH4HF+CH3 Reactivity of different initial vibrational state of CH4 Final states of the products: Translational, rotational, and vibrational energy quantum dynamics in 12D, curvilinear coordinates s

  4. Quantum dynamics real time propagation imaginary time propagation Efficient wavefunction representation: Multi-configurational time-dependent Hartree (MCTDH) approach Variational principle  differential equation for wavefunction parameters (equations of motion)

  5. MCTDH: a multi-layer representation Standard wavepacket representation

  6. MCTDH approach (Meyer, Manthe, Cederbaum, CPL 165, 73 (1990) Manthe, Meyer, Cederbaum, JCP 97, 3199 (1992))

  7. Mode-combination MCTDH approach (Worth, Meyer, Cederbaum, JCP 109, 3518 (1998))

  8. Multi-layer MCTDH approach represent the again as MCTDH wavefunctions recursive representation (Wang, Thoss, JCP 119,1289 (2003), Manthe, JCP 128, 164116 (2008))

  9. Equations of motions: matrix elements of the Hamiltonian multi-dimensional integrals (Nf scaling) Hamiltonians with sum of product structure:  matrix elements can be computed via 1D integrals  recursive calculation of all matrix elements in the multi-layer MCTDH

  10. Hamiltonians with general potentials  potential energy matrix elements multi-layer quadrature based on the single-particle functions correlation discrete variable representation (CDVR)

  11. Correlation discrete variable representation discrete variable representation ( DVR )  quadrature grid corresponding to the (time-independent) basis time-dependent DVR  grid corresponding to the (time-dependent) basis

  12. simple quadrature  fails because of inappropriate grid for separable components (example: separable system) correlation DVR ( CDVR ) (Manthe, JCP 105, 6989 (1996))

  13. Multi-layer CDVR (Manthe, JCP 128, 164116 (2008))

  14. Multi-layer / mode-combination CDVR  multi-dimensional “logical” coordinates  multi-dimensional non-direct product DVRs simulaneous diagonalization of multiple coordinate matrices (2D example) transformation to an optimally localized (DVR) basis (Dawes, Carrington, JCP 121, 726 (2004), van Harrevelt, Manthe, JCP 123, 064106 (2005); layered DVR: Manthe, JCP 130, 054109 (2009))

  15. Simulaneous diagonalization: Jacobi rotation based algorithm Problem: convergence can be extremely slow or incomplete Non-unique solutions Example: 3 quadrature points in a symmetric 2D system

  16. Thanks Till Westermann, Ralph Welsch, Robert Wodraszka, Thorsten Hammer, Gerd Schiffel Wolfgang Eisfeld (Bielefeld) Juliana Palma (Quilmes) Alexandra Viel (Rennes) Fermin Huarte (Barcelona) Gunnar Nyman (Göteborg) Finanical Support: DFG, AvH, Univ. Bielefeld

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