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