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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 - PowerPoint PPT Presentation
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Multi-layered wavefunction representations and quadratures:
the multi-configurational time-dependent Hartree approach
Uwe MantheTheoretische ChemieUniversität Bielefeld
High-dimensional quantum dynamics: applications
1
1
Malonaldehyde
intramolecular proton
transfer
tunneling splitting of the
vibrational states
quantum dynamics in 21D
sBimolecular reactions,
reactive scattering
H+CH4H2+CH3
F(3P)+CH4HF+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
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)
MCTDH: a multi-layer representation
Standard wavepacket representation
MCTDH approach
(Meyer, Manthe, Cederbaum, CPL 165, 73 (1990)
Manthe, Meyer, Cederbaum, JCP 97, 3199 (1992))
Mode-combination MCTDH approach
(Worth, Meyer, Cederbaum, JCP 109, 3518 (1998))
Multi-layer MCTDH approach
(Wang, Thoss, JCP 119,1289 (2003),
Manthe, JCP 128, 164116 (2008))
represent the
again as
MCTDH
wavefunctions
recursive
representation
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
Hamiltonians with general potentials
potential energy matrix elements
multi-layer quadrature
based on the single-particle functions
correlation discrete variable representation (CDVR)
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
simple quadrature
fails because of inappropriate grid for separable components
(example: separable system)
correlation DVR ( CDVR )
(Manthe, JCP 105, 6989 (1996))
Multi-layer CDVR
(Manthe, JCP 128, 164116 (2008))
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))
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
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