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Using High-Dimensional Smolyak Interpolation to Solve the Schroedinger Equation and Represent Potentials

Banff International Research Station for Mathematical Innovation and Discovery

Smolyak or sparse-grid interpolation enables one to accurately represent smooth multi-dimensional functions, without using a direct product basis. Instead, the basis is a pruned product basis. Pruning the basis makes it possible to also reduce the number of interpolation points. This attenuates what is referred to as the "curse of dimensionality", i.e. the exponential increase in the number of basis functions and points with the number of coordinates. In chemical physics Smolyak interpolation can be used to represent potential energy surfaces in sum-of-products form. The number of terms can be minimized by pruning so as to include only the most important product basis functions. Related collocation methods can be used to solve the vibrational Schroedinger equation to study molecules with 6 atoms. The collocation approach enables us to use general potentials (not necessarily of sum-of-products form) and very flexible basis sets. In the collocation approach there is no need to compute the matrix elements and therefore no quadrature.

Mathematics; Quantum theory; Partial differential equations; Applied computer science

video/mp4

Author affiliation: Queen's University

2016-08-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58703

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/