Smolyak scheme for solving Schrödinger equation: application to malonaldehyde in full dimensionality

In 1963, Smolyak introduced an elegant approach to overcome the exponential scaling, in terms of the number of variables (i.e. degrees of freedom), of the size of direct products [S. A. Smolyak Soviet Mathematics Doklady, 4, 240 (1963)]. The main idea is to replace a single and large direct product by a sum of selected small direct products. This approach has been applied in various scientific domains and, in particular, in quantum dynamics, where Avila and Carrington used it for the first time in 2009 [G. Avila and T. Carrington, J. Chem. Phys., 131, 174103 (2009)]. Since then, several calculations on systems with 12 degrees of freedom have been published by Avila and Carrington, by other groups and by ourselves. In the present study and to push the limit to larger and complex systems, the Smolyak scheme is combined with the use of an on-the-fly calculations of the kinetic energy operator [A. Nauts and D. Lauvergnat, Mol. Phys. 116, 3701 (2018)] in our home-made Fortran code, ElVibRot-Tnum-Tana. This procedure is applied to compute the tunneling splitting of malonaldehyde in full dimensionality (21D) using a recent potential of Mizukami et al. [W. Mizukami, S. Habershon, and D.P. Tew, J. Chem. Phys. 141, 144310 (2014)]. Our tunneling splitting calculations, 21.7±0.3 cm-1 and 2.9±0.1 cm-1, show an excellent agreement with experimental values, 21.6 cm-1 and 2.9 cm-1for the normal isotopologue and the mono-deuterated one, respectively.