When the potential energy surface (PES) does not have a special form (e.g. a sum of products), it is common to use quadrature to compute a vibrational spectrum. Direct-product quadrature grids are most popular. The size of a direct-product grid scales exponentially with the number of atoms and it is not not possible to store values of the PES for molecules with more than 5 atoms. In this talk, I shall present collocation methods we are developing. Collocation has advantages: 1) point selection is less important; 2) no integrals, no quadratures, no weights; 3) easy to use with complicated kinetic energy operators; 4) it can be used with any (the best possible) coordinates and basis functions; 5) in many cases fewer collocation than quadrature points are required; 6) the length of the vectors one must store is reduced. Collocation can be used with the Multi-configuration Time-Dependent Hartree (MCTDH) approach. The collocation-based MCTDH method I shall present can be used with general potential energy surfaces. This is imperative if one wishes to compute very accurate spectra. When the basis is good, the accuracy of collocation solutions to the Schrödinger equation is not sensitive to the choice of the collocation points. The original collocation-MCTDH (C-MCTDH) method [J. Chem. Phys. 148, 044115 (2018)] uses, as is also true in standard MCTDH, a direct product basis. Because we do not rely on having a sum-of-products potential energy surface, we also have a direct product grid. By using generalized hierarchical basis functions, that span the same space as the single particle functions we introduced in the first C-MCTDH paper, and a Smolyak grid, we have developed C-MCTDH approach that makes it possible to prune both the basis and the grid.
Prof. Tucker Carrington has been focusing on developing new methods to study quantum dynamics and spectroscopy. He received Diploma at University of Toronto in 1981, and Ph. D degree at University of California at Berkeley in 1985. After that, he became a Research Fellow and Associate Professor at Department of Chemistry - University de Montreal from 1988 to 1998. He became a full Professor at Department of Chemistry - University de Montreal in 1998. He has been a Canada Research Chair (Tier I) in Computational Quantum Dynamics at Queen's University since 2007. He is a Fellow of American Physical Society and Chemical Institute of Canada. He received many awards, including recent prestigious ones such as, Alexander von Humboldt Research Award (2017), John C. Polanyi Award of the Canadian Society for Chemistry (2014), Gerhard Herzberg award of the Canadian Society for Analytical Sciences and Spectroscopy (2013). He is the Member of the Editorial Board of Molecular Physics and Associate Editor of the Journal of Theoretical and Computational Chemistry.