Smolyak Scheme for solving the Schrödinger equation: Application to Malonaldehyde in Full Dimensionality

Abstract

In 1963 Smolyak introduced an approach to overcome the exponential scaling with respect to the number of variables of the direct product size [S. A. Smolyak Soviet Mathematics Doklady, 4, 240 (1963)]. The main idea is to replace a single large direct product by a sum of selected small direct products. It was first used in quantum dynamics in 2009 by Avila and Carrington [G. Avila and T. Carrington, J. Chem. Phys., 131, 174103 (2009)]. Since then, several calculations have been published by Avila and Carrington and by other groups. In the present study, and to push the limit to larger and more complex systems, this scheme is combined with the use of an on‐the‐fly calculation of the kinetic energy operator and a Block‐Davidson procedure to obtain eigenstates in our home‐made Fortran codes, ElVibRot and Tnum‐Tana. This was applied to compute the tunneling splitting of malonaldehyde in full dimensionality (21D) using the potential of Mizukami et al. [W. Mizukami, S. Habershon, and D.P. Tew, J. Chem. Phys. 141, 1443‐10 (2014)]. Our tunneling splitting calculations, 21.7±0.3 cm‐1 and 2.9±0.1 cm‐1, show excellent agreement with the experimental values, 21.6 cm‐1 and 2.9 cm‐1 for the normal isotopologue and the mondeuterated one, respectively.

Using a pruned basis and a sparse collocation grid with more points than basis functions to do efficient and accurate MCTDH calculations with general potential energy surfaces

“…77 One way to do calculations with a sparse grid is to sum over the smaller grids that together compose the sparse grid. 78,79 Using a nested grid obviates the sum over smaller grids (of which there may be millions even for a 6-D problem) and makes it possible to sum over all the points on the sparse grid by doing sums sequentially. 63 It is common in chemical physics to sum over points on a direct-product grid by doing sums sequentially.…”