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

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 mono-deuterated one, respectively.