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

Introduction Why the need for new developments? Product basis: ~10 basis functions per DOF –CO 2 : 10,000 vibrational basis functions –CH 4 : 1,000,000,000 vibrational basis functions –CH 3 CH 3 : vibrational basis functions – : byte database But, this can be mitigated. Semiclassical results good, but not great

Three strategies Moving 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

The pruned PSL basis Simultaneous 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 basis Comparison of SD(X,P) and SD(X,K)  : Primitive basis  : 10  8  : 10  8 x0x0 x1x1

The pruned PSL basis Time evolution of basis size Basis set grows by 10%-20% over the simulation interval. Due to wavefunction spread? McCormack, D.A. J. Chem. Phys. 124 (2006) SD(X,K) basis,  =10  8

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.

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. x2x2 x1x1 x2x2 x1x1 x2x2 x1x1 TIME Challenging cases Spectroscopy problems

p1p1 x1x1 p1p1 x1x1 p1p1 x1x1 Short propagation times are sufficient. Wavepacket starts and ends in a well-localized state. Near the turnaround, the packet spreads in momentum. TIME Challenging cases Scattering problems