Toward a pruned PSL basis approach for quantum dynamics

(was: Lanczos propagation in a classically moving basis set)

Jason Cooper and Tucker Carrington

Introduction

• The goal: Efficient solution of the time-dependent Schrödinger equation for wave packet dynamics in higher dimensions

• Applications:– Vibrational spectroscopy– Photodissociation– Inelastic scattering

IntroductionWhy the need for new developments?

• Product basis: ~10 basis functions per DOF– CO2: 10,000 vibrational basis functions

– CH4: 1,000,000,000 vibrational basis functions

– CH3CH3: 1017 vibrational basis functions

– : 1015 byte database

• But, this can be mitigated.

• Semiclassical results good, but not great

Three strategiesMoving gaussian basis- Follows the dynamics naturally- Not orthogonal, nonproduct basis- Hard to evaluate

Simple moving basis- Optionally orthogonal product basis- Can evaluate using DVR- Hard to follow dynamics

Phase-space localized (PSL) basis- Optionally orthogonal product basis- Can evaluate using DVR- More flexibility to follow dynamics

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The pruned PSL basisSimultaneous Diagonalization

• Can (nearly) simultaneously diagonalize two or more operators by Jacobi rotations.

• SD(X,P) yields a PSL basis.

• SD has the advantage that we can also choose to diagonalize, e.g., K and X from any basis.

B. Poirier and A. Salam, J. Chem. Phys. 121 (2004) 1690

Example SD(X,P) basis function

“Weylet” basis function

• Gaussian wavepacket in an n-D Seacrest-Johnson type potential:

• For n = 2:

The pruned PSL basisComparison of SD(X,P) and SD(X,K)

: Primitive basis: SD(X,P) @ 108

: SD(X,K) @ 108

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Vv x ( ) = V0 exp −a x 0 − wn x n

i=1

n−1

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+ V1 x n

( )i=1

n−1

∑

x0x1

The pruned PSL basisTime evolution of basis size

32

1

• Basis set grows by 10%-20% over the simulation interval.

• Due to wavefunction spread?

McCormack, D.A. J. Chem. Phys. 124 (2006) 204101.

SD(X,K) basis, =108

And onward…

Implementing culling will require:

• A method for evaluating the integrals without storing large intermediate vectors.

• An efficient but reliable way to determine which basis functions are kept.

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Acknowledgements

Tucker Carrington

Etienne Lanthier

Sergei Manzhos

Jean Christophe Tremblay

Xiao-Gang Wang

Francois Goyer

Funding:

Centre de Recherches Mathématiques

National Science and Engineering Council

• Long propagation times required for high resolution.

• Wavefunction spreads over time to occupy all allowed space.

• Any sufficiently large basis set need not be dynamic.

x2

x1

x2

x1

x2

x1

TIME

Challenging casesSpectroscopy problems

p1

x1

p1

x1

p1

x1

• Short propagation times are sufficient.

• Wavepacket starts and ends in a well-localized state.

• Near the turnaround, the packet spreads in momentum.

TIME

Challenging casesScattering problems