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

05 December 2023 par clavaguera
David Lauvergnat et André Nauts ont récemment publié un article dans ChemPhysChem choisi pour représenter l'image de couverture du numéro

The Cover Feature illustrates the quantum sparse grid approach used to compute the tunneling splitting of malonaldehyde. More precisely, it shows the delocalization of the wave function due to the presence of two equivalent minima on the potential energy surface. Our approach is based on the Smolyak scheme, allowing to significantly reduce the size of the basis set and the number of grid points compared to the usual direct product scheme. Cover design by David Lauvergnat. More information can be found in the Research Article :"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.